Bayesian Geostatistical Modeling for Discrete-Valued Processes

Abstract: We introduce a flexible and scalable class of Bayesian geostatistical models for discrete data, based on the class of nearest neighbor mixture transition distribution processes (NNMP), referred to as discrete NNMP. The proposed class characterizes spatial variability by a weighted combination of first-order conditional probability mass functions (pmfs) for each one of a given number of neighbors. The approach supports flexible modeling for multivariate dependence through specification of general bivariate discrete distributions that define the conditional pmfs. Moreover, the discrete NNMP allows for construction of models given a pre-specified family of marginal distributions that can vary in space, facilitating covariate inclusion. In particular, we develop a modeling and inferential framework for copula-based NNMPs that can attain flexible dependence structures, motivating the use of bivariate copula families for spatial processes. Compared to the traditional class of spatial generalized linear mixed models, where spatial dependence is introduced through a transformation of response means, our process-based modeling approach provides both computational and inferential advantages. We illustrate the benefits with synthetic data examples and an analysis of North American Breeding Bird Survey data.