Transformed Gaussian Markov Random Fields and 1 Spatial Modeling

15 The Gaussian random field (GRF) and the Gaussian Markov random field (GMRF) have 16 been widely used to accommodate spatial dependence under the generalized linear mixed 17 model framework. These models have limitations rooted in the symmetry and thin tail of the 18 Gaussian distribution. We introduce a new class of random fields, termed transformed GRF 19 (TGRF), and a new class of Markov random fields, termed transformed GMRF (TGMRF). 20 They are constructed by transforming the margins of GRFs and GMRFs, respectively, to de21 sired marginal distributions to accommodate asymmetry and heavy tail as needed in practice. 22 The Gaussian copula that characterizes the dependence structure facilitates inferences and ap23 plications in modeling spatial dependence. This construction leads to new models such as 24 gamma or beta Markov fields with Gaussian copulas, which can be used to model Poisson 25 intensity or Bernoulli rate in a spatial generalized linear mixed model. The method is natu26 rally implemented in a Bayesian framework. We illustrate the utility of the methodology in an 27 ecological application with spatial count data and spatial presence/absence data of some snail 28 species, where the new models are shown to outperform the traditional spatial models. The 29 validity of Bayesian inferences and model selection are assessed through simulation studies 30 for both spatial Poisson regression and spatial Bernoulli regression. 31 Some key words: Bayesian inference, beta field, gamma field, Gaussian copula, generalized linear mixed model 32 1 ar X iv :1 20 5. 54 67 v1 [ st at .M E ] 2 4 M ay 2 01 2