Modeling Spatial Data with Cauchy Convolution Processes

Abstract

We study the class of models for spatial data obtained from Cauchy convolution processes based on different types of kernel functions. We show that the resulting spatial processes have some appealing tail dependence properties, such as tail dependence at short distances and independence at long distances with suitable kernel functions. We derive the extreme-value limits of these processes, study their smoothness properties, and consider some interesting special cases, including Marshall-Olkin and H\"usler-Reiss processes. We further consider mixtures between such Cauchy processes and Gaussian processes, in order to have a separate control over the bulk and the tail dependence behaviors. Our proposed approach for estimating model parameters relies on matching model-based and empirical summary statistics, while the corresponding extreme-value limit models may be fitted using a pairwise likelihood approach. We show with a simulation study that our proposed inference approach yields accurate estimates. Moreover, the proposed class of models allows for a wide range of flexible dependence structures, and we demonstrate our new methodology by application to a temperature dataset. Our results indicate that our proposed model provides a very good fit to the data, and that it captures both the bulk and the tail dependence structures accurately.