Spatial and Space‐Time Threshold Exceedances

Abstract

Abstract Threshold exceedance sets, defined by the coordinates where a certain critical level is exceeded, are of interest in many application fields in relation to risk analysis including deophysics, environmental sciences, engineering, astrophysics, finance, and insurance, among many others. In particular, many risk indicators can be formulated based on certain structural properties of threshold exceedance sets. This article is structured in two parts. The first part reviews the two main approaches to describe the stochastic behavior of temporal threshold excesses, namely, the generalized Pareto distribution and nonhomogeneous Poisson processes, and collects references to some recent developments in the spatial and spatiotemporal contexts. The second part refers to some aspects of interest in relation to the structure of spatial threshold exceedance sets: on the one hand, geometrical measures such as the hypervolume and the Euler characteristic, which are related with certain probabilities of extremal occurrences; on the other hand, two different types of point processes, respectively based on centroids of connected components and on boundary A ‐exit points, which can be used depending on the specific structural properties of interest under study. The article concludes with a discussion and an illustration in the spatiotemporal context.