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

• URL: http://arxiv.org/abs/2308.01023v1
• Date: Wed, 2 Aug 2023 09:12:03 GMT
• Title: Regular Variation in Hilbert Spaces and Principal Component Analysis for Functional Extremes
• Authors: Stephan Cl\'emen\c{c}on, Nathan Huet, Anne Sabourin
• Abstract summary: We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L2$-norm $|X|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations.
• Score: 1.6734018640023431
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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