B1276
Title: Regular variation in Hilbert spaces and principal component analysis for functional extremes
Authors: Stephan Clemencon - Telecom ParisTech (France)
Nathan Huet - LTCI - Telecom Paris - Institut polytechnique de Paris (France)
Anne Sabourin - MAP5, UMR 8145, Universite Paris-Cite (France) [presenting]
Abstract: Motivated by the increasing availability of data of a functional nature, a general probabilistic and statistical framework is developed for extremes of regularly varying random elements $X$ in $L^2[0,1]$. Peaks-Over-Threshold framework is utilized, where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. The goal is to propose a dimension reduction framework resulting in finite-dimensional projections for such extreme observations. First, the notion of Regular Variation is investigated for random quantities valued in a general separable Hilbert space, for which a novel concrete characterization is proposed, involving solely stochastic convergence of real-valued random variables. Second, the notion of functional Principal Component Analysis (PCA) is proposed, accounting for the principal `directions' of functional extremes. The statistical properties of the empirical covariance operator of the angular component of extreme functions are investigated by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data further illustrate the applications.
