Estimation of the number of principal components in high-dimensional multivariate extremes

TL;DR

This paper proposes using PCA combined with AIC and BIC criteria to estimate the number of significant principal components in high-dimensional multivariate extremes, addressing challenges in modeling dependence structures.

Contribution

It introduces a novel approach applying PCA and information criteria to determine the number of significant components in high-dimensional extreme value analysis, with theoretical consistency results.

Findings

BIC is weakly consistent for fixed dimensions.

AIC and BIC have derived conditions for consistency in high dimensions.

Method performs well in simulations and real precipitation data.

Abstract

For multivariate regularly random vectors of dimension $d$, the dependence structure of the extremes is modeled by the so-called angular measure. When the dimension $d$ is high, estimating the angular measure is challenging because of its complexity. In this paper, we use Principal Component Analysis (PCA) as a method for dimension reduction and estimate the number of significant principal components of the empirical covariance matrix of the angular measure under the assumption of a spiked covariance structure. Therefore, we develop Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) to estimate the location of the spiked eigenvalue of the covariance matrix, reflecting the number of significant components, and explore these information criteria on consistency. On the one hand, we investigate the case where the dimension $d$ is fixed, and on the other hand, where the dimension $d$ converges to $\infty$ under different high-dimensional scenarios. When the dimension $d$ is fixed, we establish that the AIC is not consistent, whereas the BIC is weakly consistent. In the high-dimensional setting, with techniques from random matrix theory, we derive sufficient conditions for the AIC and the BIC to be consistent. Finally, the performance of the different AIC and BIC versions is compared in a simulation study and applied to high-dimensional precipitation data.