Optimal Interpolation and the Appropriateness of Cross-Validating Variogram in Spatial Generalized Linear Mixed Models

Abstract

This article considers some computational issues related to the minimum mean squared error (MMSE) prediction of non-Gaussian variables under a spatial generalized linear mixed model (GLMM). Earlier literature decribes how this model has been used to model spatial non-Gaussian variables, under which MMSE prediction of non-Gaussian variables can be computed. Because the MMSE prediction is nonlinear and cannot be computed in closed form, Markov chain Monte Carlo techniques are employed to approximate the predictor. We first establish some analytical results and show through three examples how these results can be used to make the MMSE predictions computationally more efficient. We then examine the effectiveness of cross-validating variogram in a spatial GLMM through a simulation study. Cross-validation is closely related to prediction since it uses partial data to predict the remaining. Our results show that cross-validation may fail to indicate the obvious lack of fit of a variogram in a spatial non-Gaussian GLMM.