Bayesian Non-Parametric Inference for Multivariate Peaks-over-Threshold Models

TL;DR

This paper introduces a new Bayesian method for analyzing extreme multivariate data by modeling the angular component of a Pareto distribution using a flexible mixture of projected gamma distributions.

Contribution

A novel Bayesian non-parametric approach for multivariate peaks-over-threshold modeling using a Dirichlet process mixture of projected gamma distributions.

Findings

The proposed method effectively models the angular component of multivariate extremes.

The energy score-based kernel metric provides a reliable way to compare different modeling choices.

The method reveals heterogeneous geographical dependence in extreme IVT data along California's coast.

Abstract

We consider a constructive definition of the multivariate Pareto that factorizes the random vector into a radial component and an independent angular component. The former follows a univariate Pareto distribution, and the latter is defined on the surface of the positive orthant of the infinity norm unit hypercube. We propose a method for inferring the distribution of the angular component by identifying its support as the limit of the positive orthant of the unit p-norm spheres and introduce a projected gamma family of distributions defined through the normalization of a vector of independent random gammas to the space. This serves to construct a flexible family of distributions obtained as a Dirichlet process mixture of projected gammas. For model assessment, we discuss scoring methods appropriate to distributions on the unit hypercube. In particular, working with the energy score criterion, we develop a kernel metric that produces a proper scoring rule and presents a simulation study to compare different modeling choices using the proposed metric. Using our approach, we describe the dependence structure of extreme values in the integrated vapor transport (IVT), data describing the flow of atmospheric moisture along the coast of California. We find clear but heterogeneous geographical dependence.

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