Multivariate Max-Stable Spatial Processes

Abstract

Analysis of spatial extremes is currently based on univariate processes. Max-stable processes allow the spatial dependence of extremes to be modelled and explicitly quantified, they are therefore widely adopted in applications. For a better understanding of extreme events of real processes, such as environmental phenomena, it may be useful to study several spatial variables simultaneously. To this end, we extend some theoretical results and applications of max-stable processes to the multivariate setting to analyze extreme events of several variables observed across space. In particular, we study the maxima of independent replicates of multivariate processes, both in the Gaussian and Student-t cases. Then, we define a Poisson process construction in the multivariate setting and introduce multivariate versions of the Smith Gaussian extremevalue, the Schlather extremal-Gaussian and extremal-t, and the BrownResnick models. Inferential aspects of those models based on composite likelihoods are developed. We present results of various Monte Carlo simulations and of an application to a dataset of summer daily temperature maxima and minima in Oklahoma, U.S.A., highlighting the utility of working with multivariate models in contrast to the univariate case. Based on joint work with Simone Padoan and Huiyan Sang.