Covariance estimation and study of models of deformations between distributions with the Wasserstein distance

La première partie de cette thèse est consacrée à l'estimation de covariance de processus stochastiques non stationnaires. Le modèle étudié amène à estimer la covariance du processus dans différents espaces vectoriels de matrices. Nous étudions dans le chapitre 3 une méthode de sélection de modèle par minimisation d'un critère pénalisé en utilisant des inégalités de concentration, et le chapitre 4 présente une méthode basée sur l'estimation sans biais du risque. Dans les deux cas des inégalités oracles sont obtenues. La seconde partie de cette thèse concerne l'étude de modèles de déformations entre distributions. On suppose observer une quantité aléatoire epsilon à travers une fonction de déformation. C'est l'importance de la déformation, représentée par un paramètre theta, que l'on cherche à retrouver. Nous présentons plusieurs méthodes d'estimation basées sur la distance de Wasserstein en alignant les lois des observations pour retrouver le paramètre de déformation. Dans le cas où les variables aléatoires sont à valeurs réelles, le chapitre 7 donne des propriétés de consistance pour un M-estimateur et sa distribution asymptotique. On y utilise des techniques de Hadamard différentiabilité pour appliquer une Delta-Méthode fonctionnelle. Le chapitre 8 concerne l'étude d'un estimateur de type Robbins-Monro et présente des propriétés de convergence pour un estimateur à noyau de la densité de la variable epsilon obtenu à l'aide des observations. Le modèle est généralisé à des variables dans des espaces métriques complets dans le chapitre 9, puis, dans l'optique de créer un test d'adéquation, le chapitre 10 donne des résultats sur la distribution asymptotique d'une statistique de test.The first part of this thesis concerns the covariance estimation of non stationary processes. We are estimating the covariance in different vectorial spaces of matrices. In Chapter 3, we give a model selection procedure by minimizing a penalized criterion and using concentration inequalities, and Chapter 4 presents an Unbiased Risk Estimation method. In both cases we give oracle inequalities. The second part deals with the study of models of deformation between distributions. We assume that we observe a random quantity epsilon through a deformation function. The importance of the deformation is represented by a parameter theta that we aim to estimate. We present several methods of estimation based on the Wasserstein distance by aligning the distributions of the observations to recover the deformation parameter. In the case of real random variables, Chapter 7 presents properties of consistency for a M-estimator and its asymptotic distribution. We use Hadamard differentiability techniques to apply a functional Delta method. Chapter 8 concerns a Robbins-Monro estimator for the deformation parameter and presents properties of convergence for a kernel estimator of the density of the variable epsilon obtained with the observations. The model is generalized to random variables in complete metric spaces in Chapter 9. Then, in the aim to build a goodness of fit test, Chapter 10 gives results on the asymptotic distribution of a test statistic

Estimation of deformations between distributions by minimal Wasserstein distance

We consider the issue of estimating a deformation operator acting on measures. For this we consider a parametric warping model on an empirical sample and provide a new matching criterion for cloud points based on a generalization of the registration criterion used in [Gamboa-Loubes-Maza 2007]. We study the asymptotic behaviour of the estimator of the deformation and provide some examples to some particular deformation models

Unbiased risk estimation method for covariance estimation

We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (URE) method, we build an estimator of the risk which allows to select an estimator in a collection of model. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology

A parametric registration model for warped distributions with Wasserstein’s distance.

Producción CientíficaWe consider a parametric deformation model for distributions. More precisely, we assume we observe J samples of random variables which are warped from an unknown distribution template. We tackle in this paper the problem of estimating the individual deformation parameters. For this, we construct a registering criterion based on the Wasserstein distance to quantify the alignment of the distributions. We prove consistency of the empirical estimators.Junta de Castilla y León (programa de apoyo a proyectos de investigación – Ref. VA212U13)Ministerio de Economía, Industria y Competitividad (MTM2011-28657-C02-01)Ministerio de Economía, Industria y Competitividad (TM2011-28657-C02-02

Unbiased risk estimation method for covariance estimation

We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (U.R.E.) method, we build an estimator of the risk which allows to select an estimator in a collection of models. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology

Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions

We propose a study of a distribution registration model for general deformation functions. In this framework, we provide estimators of the deformations as well as a goodness of fit test of the model. For this, we consider a criterion which studies the Fréchet mean (or barycenter) of the warped distributions whose study enables to make inference on the model. In particular we obtain the asymptotic distribution and a bootstrap procedure for the Wasserstein variation

Central Limit Theorem and bootstrap procedure for Wasserstein's variations with application to structural relationships between distributions

arXiv admin note: text overlap with arXiv:1508.06465International audienceWasserstein barycenters and variance-like criterion using Wasserstein distance are used in many problems to analyze the homogeneity of collections of distributions and structural relationships between the observations. We propose the estimation of the quantiles of the empirical process of the Wasserstein's variation using a bootstrap procedure. Then we use these results for statistical inference on a distribution registration model for general deformation functions. The tests are based on the variance of the distributions with respect to their Wasserstein's barycenters for which we prove central limit theorems, including bootstrap versions