Geometric modelling of spatial extremes

TL;DR

This paper introduces a geometric approach to model spatial extremes, focusing on extremal dependence and extrapolation, and compares its effectiveness with existing methods using geomagnetic data.

Contribution

It adapts the geometric approach for spatial extremes, proposing new models for gauge functions and angular distributions, and demonstrates its advantages over classical models.

Findings

The geometric approach provides unbiased inference with higher uncertainty.

It effectively models extremal dependence in spatial data.

Application to geomagnetic data illustrates practical utility.

Abstract

Recent developments in extreme value statistics have established the so-called geometric approach as a powerful modelling tool for multivariate extremes. We tailor these methods to the case of spatial modelling and examine their efficacy at inferring extremal dependence and performing extrapolation. The geometric approach is based around a limit set described by a gauge function, which is a key target for inference. We consider a variety of spatially-parameterised gauge functions and perform inference on them by building on the framework of Wadsworth and Campbell (2024), where extreme radii are modelled via a truncated gamma distribution. We also consider spatial modelling of the angular distribution, for which we propose two candidate models. Estimation of extreme event probabilities is possible by combining draws from the radial and angular models respectively. We compare our method with two other established frameworks for spatial extreme value analysis and show that our approach generally allows for unbiased, albeit more uncertain, inference compared to the more classical models. We illustrate the methodology on a space weather dataset of daily geomagnetic field fluctuations.