Deep hierarchical generalized transformation models for spatio-temporal data with discrepancy errors

"Discrepancy error covariance"refers to the cross-covariance between the signal and the noise terms in an additive model. Traditionally, the signal and noise are assumed independent in additive models to avoid issues with confounding and nonidentifiable expressions in the marginal likelihood. This assumption is made even in settings where it is known that discrepancy error covariances exists. Recently, a model has been proposed that allows for discrepancy error covariances that avoids issues with confounding. These models introduce a telescoping sum within the additive model's expression such that the latent process of interest is dependent on other terms of the telescoping sum that are included as part of the noise. However, when evaluating the telescoping sum one obtains signal and noise terms that are independent, which avoids such concerns with confounding. The current model that allows for discrepancy error covariances only includes two terms in this telescoping sum, and consequently, a natural extension is to include more terms within the telescoping sum, which leads to a deep architecture to the statistical model. We refer to this model as the "deep hierarchical generalized transformation"(DHGT) model due to a relationship with the recently introduced hierarchical generalized transformation model. We show that the DHGT is extremely efficient to implement, and can allow for exact Bayesian implementation without the use of MCMC (i.e., we can sample directly from its posterior distribution). We illustrate the DHGT using a simulation and an analysis of the 2017 Haypress wildfire downloaded from the Geospatial Multi-Agency Coordination (GeoMAC) database. These illustrations show that discrepancy errors that arise from common model misspecifications in the spatio-temporal setting can be leveraged to improve prediction. (c) 2023 Elsevier B.V. All rights reserved.