Agenda

Speaker:
Anne Sabourin , Université Paris Cité

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Meeting ID: 851 5326 8624                     Passcode: SanMarco2

Abstract:
Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as "polar depth". The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a varying regularly random vector beyond asymptotically large thresholds, once its marginals have been appropriately normalized. Not only does the Polar depth function make it easy to order the extreme values taken by a heavy-tailed random variable X and finds natural applications in anomaly detection, but it is also possible to show, as we prove it under appropriate assumptions in this article, that the polar depth of the largest observations, i.e. observations X such that || X || >t , converges to the polar depth of the limiting distribution when t tends to infinity. Although designed to quantify the depth of multivariate extremes, the polar depth is interesting in its own right, insofar as this notion is more relevant for distributions whose support is included in a half-space than the alternatives proposed in the literature, the half-space depth in particular. Here, we demonstrate its properties and analyze statistical issues related to its estimation from both finite-sample and asymptotic points of view. We present numerical results to empirically demonstrate its relevance, particularly for the statistical analysis of extreme observations and more specifically for the identification of anomalies among them.

Joint work with Stephan Clémençon, Carlos Fernàndes, Pavlo Mozharovskyi.

Bio sketch:
Anne Sabourin is Professor at Université Paris Cité. Her main research interests are in the field of extreme value theory, Unsupervised learning for extremes (dimension reduction, clustering), Statistical learning theory for rare events, Supervised learning algorithms and statistical guarantees with extreme covariates or targets, Functional Data, Applications (environmental and industrial risks, anomaly detection).