Transformed Gaussian Markov random fields and spatial modeling of species abundance

Gaussian random field and Gaussian Markov random field have been widely used to accommodate spatial dependence under the generalized linear mixed models framework. To model spatial count and spatial binary data, we present a class of transformed Gaussian Markov random fields, constructed by transforming the margins of a Gaussian Markov random field to desired marginal distributions that accommodate asymmetry and heavy tail, as needed in many empirical circumstances. The Gaussian copula that characterizes the dependence structure facilitates inferences and applications in modeling spatial dependence. This construction leads to new models such as gamma or beta Markov fields with Gaussian copulas, that are used to model Poisson intensities or Bernoulli rates in hierarchical spatial analyses. The method is naturally implemented in a Bayesian framework. To illustrate our methodology, abundances of variety of gastropod species were collected as counts or presence versus absence from a network of spatial locations in the Luquillo Mountains of Puerto Rico. Gastropods are of considerable ecological importance in terrestrial ecosystems because of their species richness, abundances, and critical roles in ecosystem processes such as decomposition and nutrient cycling. The new models outperform the traditional models based on Bayesian model comparison with conditional predictive ordinate. The validity of Bayesian inferences and model selection were assessed through simulation studies for both spatial Poisson regression and spatial Bernoulli regression.