Modelling 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 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.

Nearest neighbors weighted composite likelihood based on pairs for (non-)Gaussian massive spatial data with an application to Tukey-hh random fields estimation

“…, g q with g i : IR → IR are suitable functions. The class (1) includes several examples of non-Gaussian RFs proposed in the literature such us Bernoulli RFs [25], skew-Gaussian RFs, [26], Tukey g − h RFs [27], Student-t RFs [28], Weibull RFs [29] or Poisson RFs [30] to mention just a few. In addition, the so-called class of trans-Gaussian RFs (see for instance [31] and [32]) or the general class of non-Gaussian-RFs based on Gaussian Copula [33][34][35] and chi-square Copula [36] belong to the class (1).…”

“…In this case the transformation is not continuous and the likelihood evaluation requires computation of 2 n−1 , n-dimensional normal integrals. In other cases the likelihood is completely unknown as, for instance, in the t RFs proposed in [28] or the Poisson RF in [30] To address the abovementioned computational problem we consider the method of composite likelihood (CL) [12,41]. CL is a general class of objective functions based on the likelihood of marginal or conditional events that has been successfully applied in the recent years when estimating (non-)Gaussian RFs.…”

Nearest neighbours weighted composite likelihood based on pairs for (non-)Gaussian massive spatial data with an application to Tukey-hh random fields estimation

“…, g q with g i : IR → IR are suitable functions. The class (1) includes several examples of non-Gaussian RFs proposed in the literature such us Bernoulli RFs [25], skew-Gaussian RFs, [26], Tukey g − h RFs [27], Student-t RFs [28], Weibull RFs [29] or Poisson RFs [30] to mention just a few. In addition, the so-called class of trans-Gaussian RFs (see for instance [31] and [32]) or the general class of non-Gaussian-RFs based on Gaussian Copula [33][34][35] and chi-square Copula [36] belong to the class (1).…”

“…In this case the transformation is not continuous and the likelihood evaluation requires computation of 2 n−1 , n-dimensional normal integrals. In other cases the likelihood is completely unknown as, for instance, in the t RFs proposed in [28] or the Poisson RF in [30] To address the abovementioned computational problem we consider the method of composite likelihood (CL) [12,41]. CL is a general class of objective functions based on the likelihood of marginal or conditional events that has been successfully applied in the recent years when estimating (non-)Gaussian RFs.…”