Séminaire

Oratrice : Mathilde André (ENS Paris, Université de Vienne et Lmb) TBA

Distributional regression aims at estimating the conditional distribution of a target variable given explanatory co-variates. It is a crucial tool for forecasting of the target variable together with uncertainty quantification. A popular method widely used in practice consists in fitting a parametric model via empirical risk minimization where the risk is measured by the Continuous Rank Probability Score (CRPS). In a regression framework with independent and identically distributed (\iid) observations, we provide concentration results for the estimation error and upper bound for its expectation. Furthermore, we consider model selection which is often performed in practice via minimization of the test error on a validation sample. We also provide concentration bound for the regret in model selection. Our results may be applied to various models such as EMOS, distributional regression networks or distributional random forests. Travail en collaboration avec Ahmed Zaoui et Davit Varron.

Orateur : Sam Allen (ETH Zürich) Tail calibration of probabilistic forecasts Abstract : Probabilistic forecasts comprehensively describe the uncertainty in the unknown outcome, making them essential for decision making and risk management. While several methods have been introduced to evaluate probabilistic forecasts, existing techniques are ill-suited to the evaluation of tail properties of such forecasts. However, these tail properties are often of particular interest to forecast users due to the severe impacts caused by extreme outcomes. In this work, we reinforce previous results related to the deficiencies of proper scoring rules when evaluating forecast tails, and instead introduce several notions of tail calibration for probabilistic forecasts, allowing forecasters to assess the reliability of their predictions for extreme events. We study the relationships between these different notions, and propose diagnostic tools to assess tail calibration in practice. The benefit provided by these diagnostic tools is demonstrated in an application to European weather forecasts.

Nous présentons les propriétés asymptotiques de l'estimateur des moments pour les modèles autorégressifs (AR) intégrant des changements de régime markoviens où les erreurs sont non corrélées, mais pas nécessairement indépendantes, avec l'hypothèse que les régimes ne sont pas directement observables. L'assouplissement des hypothèses concernant la non-indépendance des erreurs et la non-observabilité directe des régimes élargit significativement l'applicabilité de cette classe de modèles AR à changements de régimes. Nous donnons des conditions nécessaires pour prouver la consistance et la normalité asymptotique de l'estimateur des moments dans un cas particulier du modèle étudié. Une attention particulière est portée à l'estimation de la matrice de covariance asymptotique.

Orateur : Landy Rabehasaina (LmB, Besançon) Nous considérons un processus observé de Galton Watson {Y_n, n ∈ ℤ} avec immigration modélisé par un processus corrélé {ε_n, n ∈ ℤ}. Nous présentons des résultats d'estimation du taux de reproduction et l'espérance de l'immigration dans deux situations. La première est lorsque Cov(ε_0,ε_k)=0 pour k supérieur à un certain k_0: nous fournissons un estimateur et prouvons un résultat de normalité asymptotique. Dans un deuxième temps, nous considérons le cas où {ε_n, n ∈ ℤ} a une structure de corrélation générale. Sous des hypothèses de mélange, nous déterminons un estimateur pour le taux de reproduction et nous montrons sa convergence en moyenne quadratique avec vitesse explicite. Lorsque le coefficient de mélange décroit suffisamment vite, un développement d'ordre 2 pour cet estimateur est établi. Travail en collaboration avec Y.Boubacar Maïnassara (UPHF).

Orateur : Maxime Egea (Université de Passau) In this talk, I will present a few results regarding Multilevel methods and their applications. I will start with an introduction that highlights the statistical motivations and the difficulties associated with high dimensions. First, I will provide an overview of the existing tools and results to address these issues. Special attention will be given to describing Multilevel Monte Carlo methods, including their construction and computational cost. Next, I will introduce new multilevel methods based on pathwise averages in a general framework. The complexity of this algorithm will be computed more precisely for Langevin diffusions that satisfy uniform convexity assumptions. Furthermore, I will explore ways to relax the uniform convexity assumption to meet specific statistical objectives. In this challenging context, I will present a penalization approach and a parametric approach.

Orateur : Clément Dombry (LmB, Besançon) Distributional regression aims at estimating the conditional distribution of a target variable given explanatory co-variates. It is a crucial tool for forecasting of the target variable together with uncertainty quantification. A popular method widely used in practice consists in fitting a parametric model via empirical risk minimization where the risk is measured by the Continuous Rank Probability Score (CRPS). In a regression framework with independent and identically distributed (\iid) observations, we provide concentration results for the estimation error and upper bound for its expectation. Furthermore, we consider model selection which is often performed in practice via minimization of the test error on a validation sample. We also provide concentration bound for the regret in model selection. Our results may be applied to various models such as EMOS, distributional regression networks or distributional random forests. Travail en collaboration avec Ahmed Zaoui et Davit Varron.

Orateur : Félix Cheysson (Univ. Gustave Eiffel) Hawkes processes are a family of point processes for which the occurrence of any event increases the probability of further events occurring. Although the linear Hawkes process, for which a representation in the form of a superposition of branching processes exists, is particularly well studied, difficulties remain in estimating the parameters of the process from imperfect data (noisy, missing or aggregated data), since the usual estimation methods based on maximum likelihood or least squares do not necessarily offer theoretical guarantees or are numerically too costly. In this work, we propose a spectral approach well-adapted to this context, for which we prove consistency and asymptotic normality. In order to derive these properties, we show that Hawkes processes can be studied through the scope of mixing, opening the use of central limit theorems that already exist in the literature. I will then present two applications of this approach: to aggregated data (joint work with Gabriel Lang); and to noisy data (joint work with Anna Bonnet, Miguel Martinez and Maxime Sangnier).