The Poisson-Gaussian Mixture Process: A Flexible and Robust Approach for Non-Gaussian Geostatistical Modeling

Abstract: This paper introduces a novel family of geostatistical models designed to capture complex features beyond the reach of traditional Gaussian processes. The proposed family, termed the Poisson-Gaussian Mixture Process (POGAMP), is hierarchically specified, combining the infinite-dimensional dynamics of Gaussian processes with any multivariate continuous distribution. This combination is stochastically defined by a latent Poisson process, allowing the POGAMP to define valid processes with finite-dimensional distributions that can approximate any continuous distribution. Unlike other non-Gaussian geostatistical models that may fail to ensure validity of the processes by assigning arbitrary finite-dimensional distributions, the POGAMP preserves essential probabilistic properties crucial for both modeling and inference. We establish formal results regarding the existence and properties of the POGAMP, highlighting its robustness and flexibility in capturing complex spatial patterns. To support practical applications, a carefully designed MCMC algorithm is developed for Bayesian inference when the POGAMP is discretely observed over some spatial domain.