Séance du 22 avril 2024

Séance organisée par Liliane Bel et Alain Célisse 

Lieu : IHP,  amphi Hermite 

14.00 : Timothée Mathieu (INRIA Lille)

Titre :  Robust Multivariate Mean estimation with M-estimators with applications in classification and regression

Résumé : Mean estimation is a fundamental problem in statistics, as it is a tool on which a lot of the statistical procedures are based in particular in Machine Learning. In the well-controlled case of Gaussian random variables (or sub-gaussian random variables), it is known that the empirical mean perform fairly well. On the other hand, as soon as the distribution becomes either heavy-tailed or corrupted, things get complicated, in particular when there are very few samples. This can be a major difficulty because in practice a lot of datasets contains outliers (typically in life sciences there are outliers in most datasets). In this presentation, I will first present usual methods for robust mean estimation in dimension one, then I will explain how to use M-estimators for mean estimation in a multivariate setting and finally how to use robust univariate mean estimators to answer the problem of empirical risk minimization for classification and regression with some illustrations using real datasets. I will also explain the problems that remain open, the mean estimation problem still has challenges even in dimension 1.

15.00 : Alice Le Brigant (Université Panthéon-Sorbonne)

Titre :  Alpha-connections and the L^p Fisher-Rao metrics

Résumé : Information geometry is a geometric framework to study spaces of probability distributions. It provides geometric tools that can be used to obtain new results in statistical inference, as well as gain insight on existing ones. The main objects of information geometry are the Fisher-Rao metric and the alpha-connections. The former is a Riemannian metric that can be used to compare probability distributions inside a given parametric family. The latter is a family of affine connections that can be used, e.g., to describe the EM algorithm in a completely geometric manner. Both the Fisher-Rao metric and the alpha-connections have non-parametric counterparts, which we discuss in this talk. More specifically, we introduce the L^p Fisher-Rao metrics, which generalize the Fisher-Rao metric, and show that their geodesic equations coincide with that of the alpha-connection, for $p=2/(1-\alpha)$, on the space of smooth densities. This result no longer holds on the space of probability densities. This gives a new variational interpretation of alpha-geodesics as being energy minimizing curves.

16.00 : Margaux Bregere (EDF R&D)

Titre : Réconciliation en ligne de prévisions de consommations électriques

Résumé : La réconciliation de prévisions vise à corriger des prévisions (peu importe leur provenance) de variables liées par des contraintes de sommation. Dans ces travaux, nous souhaitons prévoir simultanément la consommation électrique d'agrégats à différents niveaux (type de clients, régional, et national, par exemple) puis réconcilier ces prévisions afin de respecter les contraintes de sommation (la somme des consommations des régions est égale à la consommation nationale, etc.). Généralement, les prévisions sont améliorées à tous les niveaux. Les consommations électriques étant des séries temporelles nous nous intéresserons ensuite à une approche en ligne de la réconciliation.