Modeling Point Referenced Spatial Count Data: A Poisson Process Approach

Abstract

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

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

“…Bolin [29] and subsequently Wallin and Bolin [164] provided SPDE-based constructions for non-Gaussian Matérn fields. General classes of non-Gaussian fields with covariance g(M ν,α ), for g(•) a suitable function that preserves the positive definiteness and smoothness properties of the Matérn model, have been provided for instance by Palacios and Steel [120], Xua and Genton [173], Bevilacqua et al [24], Morales-Navarrete et al [113].…”

“…[29] and subsequently [164] have provided SPDE based constructions for non Gaussian Matérn fields. A general class of non Gaussian fields with kernel g(M ν,α ) with g(•) a suitable function that preserves positive definiteness can be obtained through a transformation of (independent replicates of) a Gaussian field (see for instance [120,173,24,113]).…”