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Estimations pour les modèles de Markov cachés et approximations particulaires. Application à la cartographie et à la localisation simultanées.

Abstract : This thesis is dedicated to inference problems in hidden Markov models. The first part is devoted to an online maximum likelihood estimation procedure which does not store the observations. The aim is to produce parameter estimates sequentially as the observations are received. We propose a new Expectation Maximization based method called the Block Online Expectation Maximization (BOEM) algorithm. This algorithm solves the online estimation problem for hidden Markov models with general state-spaces and observation spaces. In many situations, the BOEM algorithm requires the introduction of Sequential Monte Carlo methods (also known as particle methods) to approximate expectations under the fixed interval smoothing distributions. The convergence of the algorithm is shown under the assumption that the Lp mean error due to the Monte Carlo approximation of these expectations can be controlled explicitly as a function of the number of observations and of the number of particles. Therefore, a second part of this thesis establishes such controls for several Sequential Monte Carlo methods (the Forward Filtering Backward Smoothing algorithm and the Forward Filtering Backward Simulation algorithm). This BOEM algorithm combined with Monte Carlo approximations is then used to solve two Simultaneous Localization and Mapping Problems. These problems arise when a mobile (a robot or a human being equipped with sensors) evolves in an unknown environment and seeks to localize itself and to build a map of this environment. In the first problem, the map is made of landmarks and the observations are given by the distance and angular position of each landmark in the neighborhood of the mobile. In the second example, the mobile evolves in a sensor networks and receives WiFi signals from several access points. Finally, the last part of this thesis is dedicated to a nonparametric estimation problem in hidden Markov models. It is assumed that the Markov chain (Xk) is a random walk lying in a compact set with incremental distribution known up to a scaling factor a. At each time step k, the observations Yk is a noisy observation of f(Xk) where f is an unknown function. We establish the identifiability of the statistical model and we propose estimators of f and a based on the pairwise likelihood of the observations. We also show the consistency of these estimators.
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Submitted on : Sunday, January 13, 2013 - 10:45:18 PM
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Sylvain Le Corff. Estimations pour les modèles de Markov cachés et approximations particulaires. Application à la cartographie et à la localisation simultanées.. Statistiques [math.ST]. Telecom ParisTech, 2012. Français. ⟨tel-00773405⟩

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