Modeling Point Referenced Spatial Count Data: A Poisson Process Approach

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 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 counting 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. for this article, including technical proofs and R code for reproducing the work, are available as an online supplement.

Automated summary

Random fields are useful for representing complex dependence structures in natural phenomena. Gaussian random fields are commonly used due to their attractive properties. However, this assumption is restrictive for counting data. To address this, we propose a random field with a Poisson marginal distribution, generating a (non)-stationary random field that is mean square continuous and has Poisson marginal distributions. We provide analytic expressions for the covariance function and bivariate distribution of the proposed Poisson spatial random field. We investigate the weighted pairwise likelihood as a method for estimating the parameters in extensive simulations. Finally, we compare the proposed model with other models using real data.