Clustering of high values in random fields

TL;DR

This paper extends the theory of extreme values in random fields to non-stationary and anisotropic cases, providing new criteria for extremal index existence, and applies these results to max-stable and Gaussian fields with simulation studies.

Contribution

It develops a generalized asymptotic framework for high values in non-stationary, anisotropic random fields, including criteria for extremal index and applications to max-stable models.

Findings

Maximum behaves as the maximum of an approximately independent sequence

Criteria for the existence and estimation of the spatial extremal index

Simulation and estimation study for 1-dependent Gaussian fields

Abstract

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with $\mathbb{Z}^2$, and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of $\mathbb{R}^2$. We show that, under a suitable coordinatewise long range dependence condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levelsThe general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.