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

2025-05-28

AI Summary

Estimating the number of principal components in high-dimensional multivariate extreme value analysis is challenging due to the complex dependence structure encoded in the angular measure. Method: This paper proposes a principled approach for selecting the number of principal components based on the spiked covariance structure inherent in extremal angular measures. Contribution/Results: We innovatively extend the AIC and BIC information criteria—originally developed for classical covariance estimation—to the problem of estimating eigenvalue phase transitions in high-dimensional angular measure covariance matrices. We rigorously establish that, under fixed dimensions, BIC is weakly consistent while AIC is inconsistent; further, we derive sufficient conditions for consistency of both criteria under various high-dimensional asymptotic regimes. Integrating random matrix theory with spiked covariance modeling, we validate the method’s effectiveness and robustness through extensive simulations and real-world precipitation data.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

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.

Problem

• Estimating principal components in high-dimensional multivariate extremes
• Developing AIC and BIC criteria for spiked covariance structure
• Assessing consistency of AIC and BIC in varying dimensions

Innovation

• PCA for high-dimensional angular measure reduction
• AIC and BIC for spiked eigenvalue estimation
• Consistency analysis in fixed and infinite dimensions