Stéphane Lhaut

Stéphane Lhaut, Holger Rootzén, Johan Segers

Economically responsible mitigation of multivariate extreme risks -- extreme rainfall in a large area, huge variations of many stock prices, widespread breakdowns in transportation systems -- requires estimates of the probabilities that such risks will materialize in the future. This paper develops a new method, Wasserstein--Aitchison Generative Adversarial Networks (WA-GAN), which provides simulated values of future $d$-dimensional multivariate extreme events and which hence can be used to give estimates of such probabilities. The main hypothesis is that, after transforming the observations to the unit-Pareto scale, their distribution is regularly varying in the sense that the distributions of their radial and angular components (with respect to the $L_1$-norm) converge and become asymptotically independent as the radius gets large. The method is a combination of standard extreme value analysis modeling of the tails of the marginal distributions with nonparametric GAN modeling of the angular distribution. For the latter, the angular values are transformed to Aitchison coordinates in a full $(d-1)$-dimensional linear space, and a Wasserstein GAN is trained on these coordinates and used to generate new values. A reverse transformation is then applied to these values and gives simulated values on the original data scale. The method shows good performance compared to other existing methods in the literature, both in terms of capturing the dependence structure of the extremes in the data, as well as in generating accurate new extremes of the data distribution. The comparison is performed on simulated multivariate extremes from a logistic model in dimensions up to 50 and on a 30-dimensional financial data set.

Stéphan Clémençon, Hamid Jalalzai, Stéphane Lhaut et al.

The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and, up to logarithmic factors, scale as the square root of the effective sample size. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.