A Linear Mixed Model Formulation for Spatio-Temporal Random Processes with Computational Advances for the Separable and Product-Sum Covariances

Abstract: We describe spatio-temporal random processes using linear mixed models. We show how many commonly used models can be viewed as special cases of this general framework and pay close attention to models with separable or product-sum covariances. The proposed linear mixed model formulation facilitates the implementation of a novel algorithm using Stegle eigendecompositions, a recursive application of the Sherman-Morrison-Woodbury formula, and Helmert-Wolf blocking to efficiently invert separable and product-sum covariance matrices, even when every spatial location is not observed at every time point. We show our algorithm provides noticeable improvements over the standard Cholesky decomposition approach. Via simulations, we assess the performance of the separable and product-sum covariances and identify scenarios where separable covariances are noticeably inferior to product-sum covariances. We also compare likelihood-based and semivariogram-based estimation and discuss benefits and drawbacks of both. We use the proposed approach to analyze daily maximum temperature data in Oregon, USA, during the 2019 summer. We end by offering guidelines for choosing among these covariances and estimation methods based on properties of observed data.