Piecewise-linear modeling of multivariate geometric extremes

TL;DR

This paper introduces a flexible, interpretable, and computationally efficient piecewise-linear gauge function for multivariate extreme value modeling, enhancing the analysis of complex extremal dependence structures.

Contribution

It proposes a novel semiparametric, piecewise-linear gauge function for geometric extremal modeling, improving flexibility and interpretability over existing methods.

Findings

Effective modeling of extremal dependence in air pollution data.

Accurate estimation of high radial quantiles at fixed angular values.

Computational efficiency in optimization tasks.

Abstract

A recent development in extreme value modeling uses the geometry of the dataset to perform inference on the multivariate tail. A key quantity in this inference is the gauge function, whose values define this geometry. Methodology proposed to date for capturing the gauge function either lacks flexibility due to parametric specifications, or relies on complex neural network specifications in dimensions greater than three. We propose a semiparametric gauge function that is piecewise-linear, making it simple to interpret and provides a good approximation for the true underlying gauge function. This linearity also makes optimization tasks computationally inexpensive. The piecewise-linear gauge function can be used to define both a radial and an angular model, allowing for the joint fitting of extremal pseudo-polar coordinates, a key aspect of this geometric framework. We further expand the toolkit for geometric extremal modeling through the estimation of high radial quantiles at given angular values via kernel density estimation. We apply the new methodology to air pollution data, which exhibits a complex extremal dependence structure.