The main part of this thesis concerns parallelization of recursive Bayesian estimation methods, both linear and nonlinear such. Recursive estimation deals with the problem of extracting information about parameters or states of a dynamical system, given noisy measurements of the system output and plays a central role in signal processing, system identification, and automatic control. Solving the recursive Bayesian estimation problem is known to be computationally expensive, which often makes the methods infeasible in real-time applications and problems of large dimension. As the computational power of the hardware is today increased by adding more processors on a single chip rather than increasing the clock frequency and shrinking the logic circuits, parallelization is one of the most powerful ways of improving the execution time of an algorithm. It has been found in the work of this thesis that several of the optimal filtering methods are suitable for parallel implementation, in certain ranges of problem sizes. For many of the suggested parallelizations, a linear speedup in the number of cores has been achieved providing up to 8 times speedup on a double quad-core computer. As the evolution of the parallel computer architectures is unfolding rapidly, many more processors on the same chip will soon become available. The developed methods do not, of course, scale infinitely, but definitely can exploit and harness some of the computational power of the next generation of parallel platforms, allowing for optimal state estimation in real-time applications. / CoDeR-MP

Recursive estimation is an essential procedure in econometrics which appears in many applications when the underlying dataset or model is modi ed. Data arrive consecutively and thus already estimated models will have to be updated with new available information. Moreover, in many cases, data will have to be deleted from a model in order to remove their effect, either because they are old (obsolete) or because they have been detected to be outliers or extreme values and further investigation is required. The aim of this thesis is to develop numerically stable and computationally efficient methods for the recursive estimation of large-scale linear econometric models. Estimation of multivariate linear models is a computationally costly procedure even for moderate-sized models. In particular, when the model needs to be estimated recursively, its estimation will be even more computationally demanding. Moreover, conventional methods yield often, misleading results. The aim is to derive new methods which effectively utilise previous computations, in order to reduce the high computational cost, and which provide accurate results as well. Novel numerical methods for the recursive estimation of the general linear, the seemingly unrelated regressions, the simultaneous equations, the univariate and multivariate timevarying parameters models are developed. The proposed methods are based on numerically stable strategies which provide accurate and precise results. Moreover, the new methods estimate the unknown parameters of the modi ed model even when the variance covariance matrix is singular.

La protection de l’environnement, en particulier celle des systèmes aquatiques, est une des priorités de nos sociétés. L’utilisation de capteurs biologiques permettant de tester la qualité de l’eau en continue est une voie possible de surveillance intégrée des milieux aquatiques. Cette démarche a été mise en place avec succès sur des mollusques bivalves équipés d’électrodes légères qui respectent leur comportement naturel, on parle alors de valvométrie. Le but de cette thèse est de calculer et traiter automatiquement la vitesse de mouvement des valves de mollusques bivalves installés dans divers milieux aquatiques. Les années d’enregistrements déjà acquises nous permettrons, à partir de nos modèles, de détecter s’il existe des variations de la vitesse de mouvement des valves liées aux variations de température. Plus particulièrement, nous avons étudié les dérivées de différents estimateurs non paramétriques d’une fonction de régression : l’estimateur récursif de Nadaraya-Watson, l’estimateur de Johnston, l’estimateur de Wand-Jones ainsi que l’estimateur de Révész. Nous avons aussi pris en compte la version déterministe de l’estimateur de Nadaraya-Watson. Pour chacun des estimateurs nous avons mené une étude sur les comportement asymptotiques en particulier la convergence presque sûre et la normalité asymptotique. Nous avons illustré numériquement ces propriétés et appliqué ces nouvelles méthodes d’estimations sur des données réelles afin de valider, ou non, les hypothèses environnementales émises par les biologistes. / The protection of the environment, in particular aquatic systems, should be tackled as one of the top priorities of our society. The use of biological sensors to continuously test water quality is a possible way of monitoring aquatic environments. This approach has been successfully implemented on bivalve molluscs equipped with light electrodes that respect their natural behaviour, we then speak of valvometry. The purpose of this thesis is to automatically calculate and process the velocity of movement of bivalve mollusc valves in various aquatic environments. Years of recordings already acquired will allow us, from our models, to detect if there are variations in the speed of movement of the valves related to temperature variations. In particular, we studied the derivatives of different non parametric estimators of the regression function : the recursive Nadaraya-Watson estimator, the Johnston estimator, the Wand-Jones estimator and the Révész estimator. We also considered the deterministic version of the Nadaraya-Watson estimator. For each of the estimators we conducted a study on the asymptotic behaviour especially on the almost sure convergence and th asymptotic normality. We digitally illustrated these properties and applied these new estimation methods to real data to validate, or not, the environmental assumptions made by biologists.

Yes / This correspondence presents an investigation into the comparative performance of an active vibration control (AVC) system using a number of intelligent learning algorithms. Recursive least square (RLS), evolutionary genetic algorithms (GAs), general regression neural network (GRNN), and adaptive neuro-fuzzy inference system (ANFIS) algorithms are proposed to develop the mechanisms of an AVC system. The controller is designed on the basis of optimal vibration suppression using a plant model. A simulation platform of a flexible beam system in transverse vibration using a finite difference method is considered to demonstrate the capabilities of the AVC system using RLS, GAs, GRNN, and ANFIS. The simulation model of the AVC system is implemented, tested, and its performance is assessed for the system identification models using the proposed algorithms. Finally, a comparative performance of the algorithms in implementing the model of the AVC system is presented and discussed through a set of experiments.

In this thesis we investigate the convergence rate of gamma kernel estimators in recursive density estimation. Unlike the traditional symmetric and fixed function, the gamma kernel is a kernel function with bounded support and varying shapes. Gamma kernels have been used to address the boundary bias problem which occurs when a symmetric kernel is used to estimate a density which has support on [0, ?). The recursive density estimation is useful when an 'additional data' (on-line) comes from the population density which we want to estimate. We utilize the ideas and results from the adaptive kernel estimation to show that the L_2 convergence rate of the recursive kernel density estimators which use gamma kernels is n^(-4/5).

Title: Selected problems of financial time series modelling Author: Radek Hendrych Department: Department of Probability and Mathematical Statistics (DPMS) Supervisor: Prof. RNDr. Tomáš Cipra, DrSc., DPMS Abstract: The present dissertation thesis deals with selected problems of financial time series analysis. In particular, it focuses on two fundamental aspects of condi- tional heteroscedasticity modelling. The first part of the thesis introduces and discusses self-weighted recursive estimation algorithms for several classic univariate conditional heteroscedasticity models, namely for the ARCH, GARCH, RiskMetrics EWMA, and GJR-GARCH processes. Their numerical capabilities are demonstrated by Monte Carlo experiments and real data examples. The second part of the thesis proposes a novel approach to conditional covariance (correlation) modelling. The suggested modelling technique has been inspired by the essential idea of the multivariate orthogonal GARCH method. It is based on a suitable type of linear time-varying orthogonal transformation, which enables to employ the constant conditional correlation scheme. The correspond- ing model is implemented by using a nonlinear discrete-time state space representation. The proposed approach is compared with other commonly applied models. It demon- strates its...

Cette thèse est consacrée à l'étude de certains modèles de déformation semi-paramétriques. Notre objectif est de proposer des méthodes récursives, issues d'algorithmes stochastiques, pour estimer les paramètres de ces modèles. Dans la première partie, on présente les outils théoriques existants qui nous seront utiles dans la deuxième partie. Dans un premier temps, on présente un panorama général sur les méthodes d'approximation stochastique, en se focalisant en particulier sur les algorithmes de Robbins-Monro et de Kiefer-Wolfowitz. Dans un second temps, on présente les méthodes à noyaux pour l'estimation de fonction de densité ou de régression. On s'intéresse plus particulièrement aux deux estimateurs à noyaux les plus courants qui sont l'estimateur de Parzen-Rosenblatt et l'estimateur de Nadaraya-Watson, en présentant les versions récursives de ces deux estimateurs.Dans la seconde partie, on présente tout d'abord une procédure d'estimation récursive semi-paramétrique du paramètre de translation et de la fonction de régression pour le modèle de translation dans la situation où la fonction de lien est périodique. On généralise ensuite ces techniques au modèle vectoriel de déformation à forme commune en estimant les paramètres de moyenne, de translation et d'échelle, ainsi que la fonction de régression. On s'intéresse finalement au modèle de déformation paramétrique de variables aléatoires dans le cadre où la déformation est connue à un paramètre réel près. Pour ces trois modèles, on établit la convergence presque sûre ainsi que la normalité asymptotique des estimateurs paramétriques et non paramétriques proposés. Enfin, on illustre numériquement le comportement de nos estimateurs sur des données simulées et des données réelles. / This thesis is devoted to the study of some semi-parametric deformation models.Our aim is to provide recursive methods, related to stochastic algorithms, in order to estimate the different parameters of the models. In the first part, we present the theoretical tools which we will use in the next part. On the one hand, we focus on stochastic approximation methods, in particular the Robbins-Monro algorithm and the Kiefer-Wolfowitz algorithm. On the other hand, we introduce kernel estimators in order to estimate a probability density function and a regression function. More particularly, we present the two most famous kernel estimators which are the one of Parzen-Rosenblatt and the one of Nadaraya-Watson. We also present their recursive version.In the second part, we present the results we obtained in this thesis.Firstly, we provide a recursive estimation method of the shift parameter and the regression function for the translation model in which the regression function is periodic. Secondly, we extend this estimation procedure to the shape invariant model, providing estimation of the height parameter, the translation parameter and the scale parameter, as well as the common shape function.Thirdly, we are interested in the parametric deformation model of random variables where the deformation is known and depending on an unknown parameter.For these three models, we establish the almost sure convergence and the asymptotic normality of each estimator. Finally, we numerically illustrate the asymptotic behaviour of our estimators on simulated data and on real data.

Le développement et l'utilisation de modèles intégrés transport-urbanisme sont devenus une norme pour représenter les interactions entre l'usage des sols et le transport de biens et d'individus sur un territoire. Ces modèles sont souvent utilisés comme outils d'aide à la décision pour des politiques de planification urbaine.Les modèles transport-urbanisme, et plus généralement les modèles mathématiques, sont pour la majorité conçus à partir de codes numériques complexes. Ces codes impliquent très souvent des paramètres dont l'incertitude est peu connue et peut potentiellement avoir un impact important sur les variables de sortie du modèle.Les méthodes d'analyse de sensibilité globales sont des outils performants permettant d'étudier l'influence des paramètres d'un modèle sur ses sorties. En particulier, les méthodes basées sur le calcul des indices de sensibilité de Sobol' fournissent la possibilité de quantifier l'influence de chaque paramètre mais également d'identifier l'existence d'interactions entre ces paramètres.Dans cette thèse, nous privilégions la méthode dite à base de plans d'expériences répliqués encore appelée méthode répliquée. Cette méthode a l'avantage de ne requérir qu'un nombre relativement faible d'évaluations du modèle pour calculer les indices de Sobol' d'ordre un et deux.Cette thèse se focalise sur des extensions de la méthode répliquée pour faire face à des contraintes issues de notre application sur le modèle transport-urbanisme Tranus, comme la présence de corrélation entre paramètres et la prise en compte de sorties multivariées.Nos travaux proposent également une approche récursive pour l'estimation séquentielle des indices de Sobol'. L'approche récursive repose à la fois sur la construction itérative d'hypercubes latins et de tableaux orthogonaux stratifiés et sur la définition d'un nouveau critère d'arrêt. Cette approche offre une meilleure précision sur l'estimation des indices tout en permettant de recycler des premiers jeux d'évaluations du modèle. Nous proposons aussi de combiner une telle approche avec un échantillonnage quasi-Monte Carlo.Nous présentons également une application de nos contributions pour le calage du modèle de transport-urbanisme Tranus. / Land Use and Transportation Integrated (LUTI) models have become a norm for representing the interactions between land use and the transportation of goods and people in a territory. These models are mainly used to evaluate alternative planning scenarios, simulating their impact on land cover and travel demand.LUTI models and other mathematical models used in various fields are most of the time based on complex computer codes. These codes often involve poorly-known inputs whose uncertainty can have significant effects on the model outputs.Global sensitivity analysis methods are useful tools to study the influence of the model inputs on its outputs. Among the large number of available approaches, the variance based method introduced by Sobol' allows to calculate sensitivity indices called Sobol' indices. These indices quantify the influence of each model input on the outputs and can detect existing interactions between inputs.In this framework, we favor a particular method based on replicated designs of experiments called replication method. This method appears to be the most suitable for our application and is advantageous as it requires a relatively small number of model evaluations to estimate first-order or second-order Sobol' indices.This thesis focuses on extensions of the replication method to face constraints arising in our application on the LUTI model Tranus, such as the presence of dependency among the model inputs, as far as multivariate outputs.Aside from that, we propose a recursive approach to sequentially estimate Sobol' indices. The recursive approach is based on the iterative construction of stratified designs, latin hypercubes and orthogonal arrays, and on the definition of a new stopping criterion. With this approach, more accurate Sobol' estimates are obtained while recycling previous sets of model evaluations. We also propose to combine such an approach with quasi-Monte Carlo sampling.An application of our contributions on the LUTI model Tranus is presented.

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

Un algorithme de décomposition des unités motrices constituant un signal électromyographiques intramusculaires (iEMG) a été proposé au laboratoire LS2N. Il s'agit d'un filtrage bayésien estimant l'état d'un modèle de Markov caché. Cet algorithme demande beaucoup de temps d'execution, même pour un signal ne contenant que 4 unités motrices. Dans notre travail, nous avons d'abord validé cet algorithme dans une structure série. Nous avons proposé quelques modifications pour le modèle de recrutement des unités motrices et implémenté deux techniques de pré-traitement pour améliorer la performance de l'algorithme. Le banc de filtres de Kalman a été remplacé par un banc de filtre LMS. Le filtre global consiste en l'examen de divers scénarios arborescents d'activation des unités motrices: on a introduit deux techniques heuristiques pour élaguer les divers scénarios. On a réalisé l'implémentation GPU de cet algorithme à structure parallèle intrinsèque. On a réussi la décomposition de 10 signaux expérimentaux enregistrés sur deux muscules, respectivement avec électrode aiguille et électrode filaire. Le nombre d'unités motrices est de 2 à 8. Le pourcentage de superposition des potentiels d'unité motrice, qui représente la complexité de signal, varie de 6.56 % à 28.84 %. La précision de décomposition de tous les signaux sont plus que 90 %, sauf deux signaux en 30 % MVC , sauf pour deux signaux qui sont à 30 % MVC et dont la précision de décomposition est supérieure à 85%. Nous sommes les premiers à réaliser la décomposition en temps réel pour un signal constitué de 10 unités motrices. / :A sequential decomposition algorithm based on a Hidden Markov Model of the EMG, that used Bayesian filtering to estimate the unknown parameters of discharge series of motor units was previously proposed in the laboratory LS2N. This algorithm has successfully decomposed the experimental iEMG signal with four motor units. However, the proposed algorithm demands a high time consuming. In this work, we firstly validated the proposed algorithm in a serial structure. We proposed some modifications for the activation process of the recruitment model in Hidden Markov Model and implemented two signal pre-processing techniques to improve the performance of the algorithm. Then, we realized a GPU-oriented implementation of this algorithm, as well as the modifications applied to the original model in order to achieve a real-time performance. We have achieved the decomposition of 10 experimental iEMG signals acquired from two different muscles, respectively by fine wire electrodes and needle electrodes. The number of motor units ranges from 2 to 8. The percentage of superposition, representing the complexity of iEMG signal, ranges from 6.56 % to 28.84 %. The accuracies of almost all experimental iEMG signals are more than90 %, except two signals at 30 % MVC (more than 85 %). Moreover, we realized the realtime decomposition for all these experimental signals by the parallel implementation. We are the first one that realizes the real time full decomposition of single channel iEMG signal with number of MUs up to 10, where full decomposition means resolving the superposition problem. For the signals with more than 10 MUs, we can also decompose them quickly, but not reaching the real time level.