Decompositions of Dependence for High-Dimensional Extremes

TL;DR

This paper introduces new methods for summarizing high-dimensional tail dependence using matrix decompositions within the regular variation framework, enabling better interpretation and construction of tail-dependent vectors.

Contribution

It proposes two novel decompositions that summarize tail dependence via a positive semidefinite matrix and its eigendecomposition, facilitating analysis and simulation of high-dimensional extremes.

Findings

Matrix of pairwise tail dependence is positive semidefinite

Eigenbasis provides interpretable tail dependence structure

Decomposition enables construction of vectors with shared tail dependencies

Abstract

Employing the framework of regular variation, we propose two decompositions which help to summarize and describel high-dimensional tail dependence. Via transformation, we define a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regularly varying random vectors preserve regular variation. Rather than model regular-variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. This matrix is positive semidefinite, and eigendecomposition allows one to interpret tail dependence via the resulting eigenbasis. Additionally this matrix is completely positive, and a resulting decomposition allows one to easily construct regularly varying random vectors which share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall data and financial return data.