Bayesian cross-validation of geostatistical models

The problem of validating or criticizing models for georeferenced data is challenging as much as conclusions may be sensitive to the partition of data into training and validation cases. This is an obvious issue related to the basic validation scheme which selects a subset of the data to leave out of estimation and to make predictions with an assumed model. In this setup, only a few out-of-sample locations are usually selected to validate the model. On the other hand, the cross-validation approach, which considers several possible configurations of data divided into training and validation observations, is an appealing alternative, but it could be computationally demanding as the estimation of parameters usually requires computationally intensive methods. The purpose of this work is to use cross-validation techniques to choose between competing models and to assess the goodness of fit of spatial models in different regions of the spatial domain. We consider the sampling design for selecting the training and validation sets by assigning a probability distribution to the possible data partitions. To deal with the computational burden of cross-validation, we estimate discrepancy functions in a computationally efficient manner based on the importance weighting of posterior samples. Furthermore, we propose a stratified cross-validation scheme to take into account spatial heterogeneity, reducing the total variance of estimated predictive discrepancy measures. We also illustrate the advantages of our proposal with simulated examples of homogeneous and inhomogeneous spatial processes and with an application to rainfall dataset in Rio de Janeiro. (C) 2019 Elsevier B.V. All rights reserved.