  Random fields are useful mathematical tools for representing natural
phenomena with complex dependence structures in space and/or time. In
particular, the Gaussian random field is commonly used due to its attractive
properties and mathematical tractability. However, this assumption seems to be
restrictive when dealing with counting data. To deal with this situation, we
propose a random field with a Poisson marginal distribution by considering a
sequence of independent copies of a random field with an exponential marginal
distribution as 'inter-arrival times' in the counting renewal processes
framework. Our proposal can be viewed as a spatial generalization of the
Poisson process.
  Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal
generates a (non)-stationary random field that is mean square continuous and
with Poisson marginal distributions. For the proposed Poisson spatial random
field, analytic expressions for the covariance function and the bivariate
distribution are provided. In an extensive simulation study, we investigate the
weighted pairwise likelihood as a method for estimating the Poisson random
field parameters.
  Finally, the effectiveness of our methodology is illustrated by an analysis
of reindeer pellet-group survey data, where a zero-inflated version of the
proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson
Gaussian copula models. Supplementary materials for this article, include
technical proofs and R code for reproducing the work, are available as an
online supplement.
