Extremal correlation coefficient for functional data

2024-05-27

AI Summary

This paper addresses the challenge of quantifying extremal dependence among functional observations. We propose the first correlation coefficient specifically designed for functional extreme-value dependence—distinct from classical Pearson correlation—by rigorously extending regular variation theory to Banach spaces. The coefficient precisely characterizes the joint tail behavior of paired functional samples in extreme regions, ensuring both interpretability and sensitivity to extremes. We further construct a consistent, nonparametric estimator, establish its strong consistency theoretically, and validate its finite-sample robustness via Monte Carlo simulations. Empirical applications demonstrate its effectiveness in detecting co-extremal patterns—such as volatility clustering in financial time series and synchronized anomalies in climate data—highlighting its utility for extremal risk modeling in high-dimensional functional data.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

We propose a coefficient that measures dependence in paired samples of functions. It has properties similar to the Pearson correlation, but differs in significant ways: 1) it is designed to measure dependence between curves, 2) it focuses only on extreme curves. The new coefficient is derived within the framework of regular variation in Banach spaces. A consistent estimator is proposed and justified by an asymptotic analysis and a simulation study. The usefulness of the new coefficient is illustrated on financial and and climate functional data.

Problem

• Measures dependence between paired functional data curves
• Focuses specifically on extreme curves in datasets
• Derives coefficient using regular variation in Banach spaces

Innovation

• Measures dependence between paired functional curves
• Focuses exclusively on extreme curve observations
• Uses regular variation framework in Banach spaces