  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. 
Modelling Point Referenced Spatial Count Data: A Poisson Process   Approach