  The Gaussian random field (GRF) and the Gaussian Markov random field (GMRF)
have been widely used to accommodate spatial dependence under the generalized
linear mixed model framework. These models have limitations rooted in the
symmetry and thin tail of the Gaussian distribution. We introduce a new class
of random fields, termed transformed GRF (TGRF), and a new class of Markov
random fields, termed transformed GMRF (TGMRF). They are constructed by
transforming the margins of GRFs and GMRFs, respectively, to desired marginal
distributions to accommodate asymmetry and heavy tail as needed in practice.
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,
which can be used to model Poisson intensity or Bernoulli rate in a spatial
generalized linear mixed model. The method is naturally implemented in a
Bayesian framework. We illustrate the utility of the methodology in an
ecological application with spatial count data and spatial presence/absence
data of some snail species, where the new models are shown to outperform the
traditional spatial models. The validity of Bayesian inferences and model
selection are assessed through simulation studies for both spatial Poisson
regression and spatial Bernoulli regression.
