Spatial models with explanatory variables in the dependence structure

Geostatistical models have traditionally been stationary. However, physical knowledge about underlying spatial processes often requires models with non-stationary dependence structures. Thus, there has been an interest in the literature to provide flexible models and computationally efficient methods for non-stationary phenomena. In this work, we demonstrate that the stochastic partial differential equation (SPDE) approach to spatial modelling provides a flexible class of non-stationary models where explanatory variables can be easily included in the dependence structure. In addition, the SPDE approach enables computationally efficient Bayesian inference with integrated nested Laplace approximations (INLA) available through the R-package r-inla. We illustrate the suggested modelling framework with a case study of annual precipitation in southern Norway, and compare a non-stationary model with dependence structure governed by elevation to a stationary model. Further, we use a simulation study to explore the annual precipitation models. We investigate identifiability of model parameters and whether the deviance information criterion (DIC) is able to distinguish datasets from the non-stationary and stationary models.