Name: Thomas Opitz

Talk Title: Stochastic Geometry of Gaussian Mixture Processes and Spatial Extreme-Value Analysis

Abstract: Excursion sets of spatial stochastic processes above a high threshold allow characterizing extreme event episodes. The distributions of geometric characteristics of these sets, such as their area, perimeter or Euler characteristic,  provide information about the structure of spatial clusters of extreme values. The estimation of these geometric characteristics has strong potential to provide new insights in climatological, environmental and ecological applications. In this work, we consider the setting of independent replicates of Gaussian processes with a random variable embedded for the mean or the variance of the process, leading to Gaussian location or scale mixture processes, respectively. Members of this class of processes have been used for flexible modeling of spatial data. Moreover, standard models in spatial extreme-value theory for threshold exceedances above a high threshold, known as generalized Pareto processes, arise as limits of such constructions when the threshold tends to infinity. Using classical results from the literature for the stochastic geometry of excursion sets in Gaussian processes, we extend the Gaussian theory to Gaussian mixture processes and their limits. In particular, we derive expectation formulas for geometric summaries. Finally, we illustrate estimation of such properties on simulated and real data. This work is joint with Anne Estrade and Elena di Bernardino.

This talk is an invited talk at EVA 2021.