A finite-dimensional construction of a max-stable process for spatial extremes

From heat waves to hurricanes, often the environmental processes that are the most critical to understand probabilistically are extreme events. Such extremal processes manifestly exhibit spatial dependence. Max-stable processes are a class of asymptotically-justified models that are capable of representing spatial dependence among extreme values. While these models satisfy modeling requirements, they are limited in their utility because their corresponding joint likelihoods are unknown for more than a trivial number of spatial locations, preventing, in particular, Bayesian analyses. In this paper we propose an approximation to the Gaussian extreme value process (GEVP) that, critically, is amenable to standard MCMC and inclusion in hierarchical models. We show that this model is max-stable and approximates the GEVP arbitrarily well. The proposed model also leads to a non-stationary extension, which we use to analyze the yearly maximum temperature in the southeast US for years 1983–2007.