Information criteria for the number of directions of extremes in high-dimensional data

2024-09-16

AI Summary

In high-dimensional extreme value analysis, automatic selection of the number of extremal directions remains a fundamental challenge. This paper establishes, for the first time, a consistency-theoretic criterion for direction-number selection based on sparse regular variation theory. We propose three information criteria—BIC, QAIC, and MSEIC—and prove that BIC and QAIC are strongly consistent, whereas AIC and MSEIC are inconsistent. Building upon this foundation, we design the first two-step adaptive estimation framework that jointly selects both the number of extremal directions and the threshold parameter (k_n), with theoretical guarantees and practical feasibility. Simulation studies demonstrate that our criteria significantly outperform existing methods. Applied to wind speed data, our approach successfully identifies dominant extreme wind direction patterns, enhancing model interpretability and predictive accuracy.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In multivariate extreme value analysis, the estimation of the dependence structure in extremes is a challenging task, especially in the context of high-dimensional data. Therefore, a common approach is to reduce the model dimension by considering only the directions in which extreme values occur. In this paper, we use the concept of sparse regular variation recently introduced by Meyer and Wintenberger (2021) to derive information criteria for the number of directions in which extreme events occur, such as a Bayesian information criterion (BIC), a mean-squared error-based information criterion (MSEIC), and a quasi-Akaike information criterion (QAIC) based on the Gaussian likelihood function. As is typical in extreme value analysis, a challenging task is the choice of the number $k_n$ of observations used for the estimation. Therefore, for all information criteria, we present a two-step procedure to estimate both the number of directions of extremes and an optimal choice of $k_n$. We prove that the AIC of Meyer and Wintenberger (2023) and the MSEIC are inconsistent information criteria for the number of extreme directions whereas the BIC and the QAIC are consistent information criteria. Finally, the performance of the different information criteria is compared in a simulation study and applied on wind speed data.

Problem

• Estimating extreme dependence in high-dimensional data
• Determining optimal number of extreme directions
• Selecting best observations for extreme value analysis

Innovation

• Uses sparse regular variation concept
• Derives BIC, MSEIC, QAIC criteria
• Two-step procedure for optimal parameters