Robust Spatio-Temporal Distributional Regression

Abstract: Motivated by investigating spatio-temporal patterns of the distribution of continuous variables, we consider describing the conditional distribution function of the response variable incorporating spatio-temporal components given predictors. In many applications, continuous variables are observed only as threshold-categorized data due to measurement constraints. For instance, ecological measurements often categorize sizes into intervals rather than recording exact values due to practical limitations. To recover the conditional distribution function of the underlying continuous variables, we consider a distribution regression employing models for binomial data obtained at each threshold value. However, depending on spatio-temporal conditions and predictors, the distribution function may frequently exhibit boundary values (zero or one), which can occur either structurally or randomly. This makes standard binomial models inadequate, requiring more flexible modeling approaches. To address this issue, we propose a boundary-inflated binomial model incorporating spatio-temporal components. The model is a three-component mixture of the binomial model and two Dirac measures at zero and one. We develop a computationally efficient Bayesian inference algorithm using P\'olya-Gamma data augmentation and dynamic Gaussian predictive processes. Extensive simulation experiments demonstrate that our procedure significantly outperforms distribution regression methods based on standard binomial models across various scenarios.