Bayesian spatial modeling and interpolation using copulas

Classical Bayesian spatial interpolation methods are based on the Gaussian assumption and therefore lead to unreliable results when applied to extreme valued data. Specifically, they give wrong estimates of the prediction uncertainty. Copulas have recently attracted much attention in spatial statistics and are used as a flexible alternative to traditional methods for non-Gaussian spatial modeling and interpolation. We adopt this methodology and show how it can be incorporated in a Bayesian framework by assigning priors to all model parameters. In the absence of simple analytical expressions for the joint posterior distribution we propose a Metropolis–Hastings algorithm to obtain posterior samples. The posterior predictive density is approximated by averaging the plug-in predictive densities. Furthermore, we discuss the deficiencies of the existing spatial copula models with regard to modeling extreme events. It is shown that the non-Gaussian χ2-copulaχ2-copula model suffers from the same lack of tail dependence as the Gaussian copula and thus offers no advantage over the latter with respect to modeling extremes. We illustrate the proposed methodology by analyzing a dataset here referred to as the Helicopter dataset, which includes strongly skewed radioactivity measurements in the city of Oranienburg, Germany.