Anne Sabourin

Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization

The probabilistic framework of extreme value theory is well-known: the dependence structure of large events is characterized by an angular measure on the positive orthant of the unit sphere. The family of these angular measures is non-parametric by nature. Nonetheless, any angular measure may be approached arbitrarily well by a mixture of Dirichlet distributions. The semi-parametric Dirichlet mixture model for angular measures is theoretically valid in arbitrary dimension, but the original parametrization is subject to a moment constraint making Bayesian inference very challenging in dimension greater than three. A new unconstrained parametrization is proposed. This allows for a natural prior specification as well as a simple implementation of a reversible-jump MCMC. Posterior consistency and ergodicity of the Markov chain are verified and the algorithm is tested up to dimension five. In this non identifiable setting, convergence monitoring is performed by integrating the sampled angular densities against Dirichlet test functions.

Sparse representation of multivariate extremes with applications to anomaly detection

Capturing the dependence structure of multivariate extreme events is a major concern in many fields involving the management of risks stemming from multiple sources, e.g., portfolio monitoring, insurance, environmental risk management and anomaly detection. One convenient (nonparametric) characterization of extreme dependence in the framework of multivariate Extreme Value Theory (EVT) is the angular measure, which provides direct information about the probable “directions” of extremes, i.e., the relative contribution of each feature/coordinate of the largest observations. Modeling the angular measure in high-dimensional problems is a major challenge for the multivariate analysis of rare events. The present paper proposes a novel methodology aiming at exhibiting a particular kind of sparsity within the dependence structure of extremes. This is achieved by estimating the amount of mass spread by the angular measure on representative sets of directions corresponding to specific sub-cones of R+d. This dimension reduction technique paves the way towards scaling up existing multivariate EVT methods. Beyond a non-asymptotic study providing a theoretical validity framework for our method, we propose as a direct application a first anomaly detection algorithm based on multivariate EVT. This algorithm builds a sparse normal profile of extreme behaviors, to be confronted with new (possibly abnormal) extreme observations. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.

MARGINAL STANDARDIZATION OF UPPER SEMICONTINUOUS PROCESSES. WITH APPLICATION TO MAX-STABLE PROCESSES

In the field of spatial extremes, stochastic processes with upper semicontinuous (usc) trajectories have been proposed as random shape functions for max-stable models. In the literature dealing with usc processes, max-stability is defined via a sequences of scaling constants, rather than functions, only. It is however not clear whether and how extreme-value theory (EVT) for continuous processes extends to usc processes. In particular, classical multivariate and continuous EVT relies on the probability integral transform and Sklar’s theorem. This theorem justifies working with standard marginal distributions, simplifying the task of constructing and characterizing max-stable processes and their domains of attraction. In the present work, we investigate the possibility to follow these steps for usc processes. Unfortunately, the pointwise probability integral transform is not necessarily ‘permitted’: without additional assumptions, the obtained process may not even have usc trajectories. We give sufficient conditions for marginal standardization to be possible, and we state a partial extension of Sklar’s theorem for usc processes, with a particular focus on max-stable ones.

Max K -Armed Bandit: On the ExtremeHunter Algorithm and Beyond

This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.

Feature Clustering for Extreme Events Analysis, with Application to Extreme Stream-Flow Data

The dependence structure of extreme events of multivariate nature plays a special role for risk management applications, in particular in hydrology (flood risk). In a high dimensional context (\(d>50\)), a natural first step is dimension reduction. Analyzing the tails of a dataset requires specific approaches: earlier works have proposed a definition of sparsity adapted for extremes, together with an algorithm detecting such a pattern under strong sparsity assumptions. Given a dataset that exhibits no clear sparsity pattern we propose a clustering algorithm allowing to group together the features that are ‘dependent at extreme level’, i.e.,that are likely to take extreme values simultaneously. To bypass the computational issues that arise when it comes to dealing with possibly \(O(2^d)\) subsets of features, our algorithm exploits the graphical structure stemming from the definition of the clusters, similarly to the Apriori algorithm, which reduces drastically the number of subsets to be screened. Results on simulated and real data show that our method allows a fast recovery of a meaningful summary of the dependence structure of extremes.