Regular Variation in Hilbert Spaces and Principal Component Analysis for Functional Extremes

TL;DR

This paper develops a framework for analyzing extreme functional data in Hilbert spaces, introducing a novel notion of regular variation and a PCA method tailored for extremes, with theoretical and empirical validation.

Contribution

It introduces a new characterization of regular variation in Hilbert spaces and proposes a functional PCA method specifically for extreme observations.

Findings

Bounded the estimation error of the empirical covariance operator.

Validated the approach with simulated and real data.

Provided a new probabilistic framework for functional extremes.

Abstract

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.