ABSTRACT
Abstract Max-stable processes are well-established models for spatial extremes. In this chapter, we address the prediction problem: assuming that a max-stable process is observed at some locations only, how can we use these observations to predict the behavior of the process at other unobserved locations? Mathematically, the prediction problem is related to the conditional distribution of the process given the observations. Recently, Dombry and Eyi-Minko (2013) provided an explicit theoretical formula for the conditional distributions of max-stable processes. The result relies on the spectral representation of the max-stable process as the pointwise maxima over an infinite number of spectral functions belonging to a Poisson point process. The effect of conditioning on the Poisson point process is analyzed, resulting in the notions of hitting scenario and extremal or subextremal functions. Due to the complexity of the structure of the conditional distributions, conditional simulation appears at the same time challenging and important to assess characteristics that are analytically intractable such as the conditional median or quantiles. The issue of conditional simulation was considered by Dombry et al. (2013) who proposed a three-step procedure for conditional sampling. As the conditional simulation of the hitting scenario becomes computationally very demanding even for a moderate number of conditioning points, a Gibbs sampler approach was proposed for this step. The results are illustrated on some simulation studies and we propose several diagnostics to check the performance.
