Cette thèse est consacrée à l'évaluation des familles exponentielles pour les problèmes de la modélisation des bruits et de la segmentation des images couleurs. Dans un premier temps, nous avons développé une nouvelle caractérisation des familles exponentielles naturelles infiniment divisible basée sur la fonction trace de la matrice de variance covariance associée. Au niveau application, cette nouvelle caractérisation a permis de détecter la nature de la loi d'un bruit additif associé à un signal où à une image couleur. Dans un deuxième temps, nous avons proposé un nouveau modèle statistique paramétrique mulltivarié basé sur la loi de Riesz. La loi de ce nouveau modèle est appelée loi de la diagonale modifiée de Riesz. Ensuite, nous avons généralisé ce modèle au cas de mélange fini de lois. Enfin, nous avons introduit un algorithme de segmentation statistique d'image ouleur, à travers l'intégration de la méthode des centres mobiles (K-means) au niveau de l'initialisation pour une meilleure définition des classes de l'image et l'algorithme EM pour l'estimation des différents paramètres de chaque classe qui suit la loi de la diagonale modifiée de la loi de Riesz. / This thesis is dedicated to the evaluation of the exponential families for the problems of the noise modeling and the color images segmentation. First, we developed a new characterization of the infinitely divisible natural exponential families based on the trace function of the associated variance-covariance matrix. At the application level, this new characterization allowed to detect the nature of the law of an additive noise associated with a signal or with a color image. Second, we proposed a new parametric multivariate statistical model based on Riesz's distribution. The law of this new model is called the modified diagonal Riesz distribution. Then we generalized this model in the case of a finished mixture of distibution. Finally we introduced an algorithm of statistical segmentation of color images through the integration of the k-means method at the level of the initialization for a better definition of the image classes and the algorithm EM for the estimation of the different parameters of every class which follows the modified diagonal Riesz distribution.

Nous considérions des modèles à espace d'état où les observations sont multicatégorielles et longitudinales, et l'état est décrit par des modèles du type CHARN. Nous estimons l'état au moyen des récursivités de Kalman généralisées. Celles-ci reposent sur l'application d'une variété de filtres particulaires et de l’algorithme EM. Nos résultats sont appliqués à l'estimation du trait latent en qualité de vie. Ce qui fournit une alternative et une généralisation des méthodes existantes dans la littérature. Ces résultats sont illustrés par des simulations numériques et une application aux données réelles sur la qualité de vie des femmes ayant subi une opération pour cause de cancer du sein / We consider state space models where the observations are multicategorical and longitudinal, and the state is described by CHARN models. We estimate the state by generalized Kalman recursions, which rely on a variety of particle filters and EM algorithm. Our results are applied to estimating the latent trait in quality of life, and this furnishes an alternative and a generalization of existing methods. These results are illustrated by numerical simulations and an application to real data in the quality of life of patients surged for breast cancer

A presente tese aborda aspectos teóricos e aplicados da estimação dos parâmetros do modelo de calibração linear com erros distribuídos conforme a distribuição normal-assimétrica (Azzalini, 1985) e t-normal-assimétrica (Gómez, Venegas e Bolfarine, 2007). Aplicando um modelo assimétrico, não é necessário transformar as variáveis a fim de obter erros simétricos. A estimação dos parâmetros e das variâncias dos estimadores do modelo de calibração foram estudadas através da visão freqüentista e bayesiana, desenvolvendo algoritmos tipo EM e amostradores de Gibbs, respectivamente. Um dos pontos relevantes do trabalho, na óptica freqüentista, é a apresentação de uma reparametrização para evitar a singularidade da matriz de informação de Fisher sob o modelo de calibração normal-assimétrico na vizinhança de lambda = 0. Outro interessante aspecto é que a reparametrização não modifica o parâmetro de interesse. Já na óptica bayesiana, o ponto forte do trabalho está no desenvolvimento de medidas para verificar a qualidade do ajuste e que levam em consideração a assimetria do conjunto de dados. São propostas duas medidas para medir a qualidade do ajuste: o ADIC (Asymmetric Deviance Information Criterion) e o EDIC (Evident Deviance Information Criterion), que são extensões da ideia de Spiegelhalter et al. (2002) que propôs o DIC ordinário que só deve ser usado em modelos simétricos. / This thesis focuses on theoretical and applied estimation aspects of the linear calibration model with skew-normal (Azzalini, 1985) and skew-t-normal (Gómez, Venegas e Bolfarine, 2007) error distributions. Applying the asymmetrical distributed error methodology, it is not necessary to transform the variables in order to have symmetrical errors. The frequentist and the Bayesian solution are presented. The parameter estimation and its variance estimation were studied using the EM algorithm and the Gibbs sampler, respectively, in each approach. The main point, in the frequentist approach, is the presentation of a new parameterization to avoid singularity of the information matrix under the skew-normal calibration model in a neighborhood of lambda = 0. Another interesting aspect is that the reparameterization developed to make the information matrix nonsingular, when the skewness parameter is near to zero, leaves the parameter of interest unchanged. The main point, in the Bayesian framework, is the presentation of two measures of goodness-of-fit: ADIC (Asymmetric Deviance Information Criterion) and EDIC (Evident Deviance Information Criterion ). They are natural extensions of the ordinary DIC developed by Spiegelhalter et al. (2002).

A presente tese aborda aspectos teóricos e aplicados da estimação dos parâmetros do modelo de calibração linear com erros distribuídos conforme a distribuição normal-assimétrica (Azzalini, 1985) e t-normal-assimétrica (Gómez, Venegas e Bolfarine, 2007). Aplicando um modelo assimétrico, não é necessário transformar as variáveis a fim de obter erros simétricos. A estimação dos parâmetros e das variâncias dos estimadores do modelo de calibração foram estudadas através da visão freqüentista e bayesiana, desenvolvendo algoritmos tipo EM e amostradores de Gibbs, respectivamente. Um dos pontos relevantes do trabalho, na óptica freqüentista, é a apresentação de uma reparametrização para evitar a singularidade da matriz de informação de Fisher sob o modelo de calibração normal-assimétrico na vizinhança de lambda = 0. Outro interessante aspecto é que a reparametrização não modifica o parâmetro de interesse. Já na óptica bayesiana, o ponto forte do trabalho está no desenvolvimento de medidas para verificar a qualidade do ajuste e que levam em consideração a assimetria do conjunto de dados. São propostas duas medidas para medir a qualidade do ajuste: o ADIC (Asymmetric Deviance Information Criterion) e o EDIC (Evident Deviance Information Criterion), que são extensões da ideia de Spiegelhalter et al. (2002) que propôs o DIC ordinário que só deve ser usado em modelos simétricos. / This thesis focuses on theoretical and applied estimation aspects of the linear calibration model with skew-normal (Azzalini, 1985) and skew-t-normal (Gómez, Venegas e Bolfarine, 2007) error distributions. Applying the asymmetrical distributed error methodology, it is not necessary to transform the variables in order to have symmetrical errors. The frequentist and the Bayesian solution are presented. The parameter estimation and its variance estimation were studied using the EM algorithm and the Gibbs sampler, respectively, in each approach. The main point, in the frequentist approach, is the presentation of a new parameterization to avoid singularity of the information matrix under the skew-normal calibration model in a neighborhood of lambda = 0. Another interesting aspect is that the reparameterization developed to make the information matrix nonsingular, when the skewness parameter is near to zero, leaves the parameter of interest unchanged. The main point, in the Bayesian framework, is the presentation of two measures of goodness-of-fit: ADIC (Asymmetric Deviance Information Criterion) and EDIC (Evident Deviance Information Criterion ). They are natural extensions of the ordinary DIC developed by Spiegelhalter et al. (2002).

摘要 本文主要考慮群集樣本(clustered samples)的存活分析，而每一群集中又分為兩種組別(groups)。假定同群集同組別內的個體共享相同但不可觀測的隨機脆弱性(frailty)，因此面臨的是雙變量脆弱性變數的多變量存活資料。首先，驗證雙變量脆弱性對雙變量對數存活時間及雙變量存活時間之相關係數所造成的影響。接著，假定雙變量脆弱性服從雙變量對數常態分配，條件存活時間模式為韋伯迴歸模式，我們利用EM法則，推導出雙變量脆弱性之多變量存活模式中母數的估計方法。 關鍵詞：雙變量脆弱性，Weibull迴歸模式，對數常態分配，EM法則 / Abstract Consider survival analysis for clustered samples, where each cluster contains two groups. Assume that individuals within the same cluster and the same group share a common but unobservable random frailty. Hence, the focus of this work is on bivariate frailty model in analysis of multivariate survival data. First, we derive expressions for the correlation between the two survival times to show how the bivariate frailty affects these correlation coefficients. Then, the bivariate log-normal distribution is used to model the bivariate frailty. We modified EM algorithm to estimate the parameters for the Weibull regression model with bivariate log-normal frailty. Key words：bivariate frailty, Weibull regression model, log-normal distribution, EM algorithm.

Nous considérons le problème de la séparation de mélanges audio réverbérants déterminés et sous-déterminés, c'est-à-dire l'extraction du signal de chaque source dans un mélange multicanal. Nous proposons un cadre général de modélisation gaussienne où la contribution de chaque source aux canaux du mélange dans le domaine temps-fréquence est modélisée par un vecteur aléatoire gaussien de moyenne nulle dont la covariance encode à la fois les caractéristiques spatiales et spectrales de la source. A n de mieux modéliser la réverbération, nous nous aff ranchissons de l'hypothèse classique de bande étroite menant à une covariance spatiale de rang 1 et nous calculons la borne théorique de performance atteignable avec une covariance spatiale de rang plein. Les ré- sultats expérimentaux indiquent une augmentation du rapport Signal-à-Distorsion (SDR) de 6 dB dans un environnement faiblement à très réverbérant, ce qui valide cette généralisation. Nous considérons aussi l'utilisation de représentations temps-fréquence quadratiques et de l'échelle fréquentielle auditive ERB (equivalent rectangular bandwidth) pour accroître la quantité d'information exploitable et décroître le recouvrement entre les sources dans la représentation temps-fréquence. Après cette validation théorique du cadre proposé, nous nous focalisons sur l'estimation des paramètres du modèle à partir d'un signal de mélange donné dans un scénario pratique de séparation aveugle de sources. Nous proposons une famille d'algorithmes Expectation-Maximization (EM) pour estimer les paramètres au sens du maximum de vraisemblance (ML) ou du maximum a posteriori (MAP). Nous proposons une famille d'a priori de position spatiale inspirée par la théorie de l'acoustique des salles ainsi qu'un a priori de continuité spatiale. Nous étudions aussi l'utilisation de deux a priori spectraux précédemment utilisés dans un contexte monocanal ou multicanal de rang 1: un a priori de continuité spatiale et un modèle de factorisation matricielle positive (NMF). Les résultats de séparation de sources obtenus par l'approche proposée sont comparés à plusieurs algorithmes de base et de l'état de l'art sur des mélanges simulés et sur des enregistrements réels dans des scénarios variés.

Many real-world systems experience deterioration with usage and age, which often leads to low product quality, high production cost, and low system availability. Most previous maintenance and reliability models in the literature do not incorporate condition monitoring information for decision making, which often results in poor failure prediction for partially observable deteriorating systems. For that reason, the development of fault prediction and control scheme using condition-based maintenance techniques has received considerable attention in recent years. This research presents a new framework for predicting failures of a partially observable deteriorating system using Bayesian control techniques. A time series model is fitted to a vector observation process representing partial information about the system state. Residuals are then calculated using the fitted model, which are indicative of system deterioration. The deterioration process is modeled as a 3-state continuous-time homogeneous Markov process. States 0 and 1 are not observable, representing healthy (good) and unhealthy (warning) system operational conditions, respectively. Only the failure state 2 is assumed to be observable. Preventive maintenance can be carried out at any sampling epoch, and corrective maintenance is carried out upon system failure. The form of the optimal control policy that maximizes the long-run expected average availability per unit time has been investigated. It has been proved that a control limit policy is optimal for decision making. The model parameters have been estimated using the Expectation Maximization (EM) algorithm. The optimal Bayesian fault prediction and control scheme, considering long-run average availability maximization along with a practical statistical constraint, has been proposed and compared with the age-based replacement policy. The optimal control limit and sampling interval are calculated in the semi-Markov decision process (SMDP) framework. Another Bayesian fault prediction and control scheme has been developed based on the average run length (ARL) criterion. Comparisons with traditional control charts are provided. Formulae for the mean residual life and the distribution function of system residual life have been derived in explicit forms as functions of a posterior probability statistic. The advantage of the Bayesian model over the well-known 2-parameter Weibull model in system residual life prediction is shown. The methodologies are illustrated using simulated data, real data obtained from the spectrometric analysis of oil samples collected from transmission units of heavy hauler trucks in the mining industry, and vibration data from a planetary gearbox machinery application.

Many real-world systems experience deterioration with usage and age, which often leads to low product quality, high production cost, and low system availability. Most previous maintenance and reliability models in the literature do not incorporate condition monitoring information for decision making, which often results in poor failure prediction for partially observable deteriorating systems. For that reason, the development of fault prediction and control scheme using condition-based maintenance techniques has received considerable attention in recent years. This research presents a new framework for predicting failures of a partially observable deteriorating system using Bayesian control techniques. A time series model is fitted to a vector observation process representing partial information about the system state. Residuals are then calculated using the fitted model, which are indicative of system deterioration. The deterioration process is modeled as a 3-state continuous-time homogeneous Markov process. States 0 and 1 are not observable, representing healthy (good) and unhealthy (warning) system operational conditions, respectively. Only the failure state 2 is assumed to be observable. Preventive maintenance can be carried out at any sampling epoch, and corrective maintenance is carried out upon system failure. The form of the optimal control policy that maximizes the long-run expected average availability per unit time has been investigated. It has been proved that a control limit policy is optimal for decision making. The model parameters have been estimated using the Expectation Maximization (EM) algorithm. The optimal Bayesian fault prediction and control scheme, considering long-run average availability maximization along with a practical statistical constraint, has been proposed and compared with the age-based replacement policy. The optimal control limit and sampling interval are calculated in the semi-Markov decision process (SMDP) framework. Another Bayesian fault prediction and control scheme has been developed based on the average run length (ARL) criterion. Comparisons with traditional control charts are provided. Formulae for the mean residual life and the distribution function of system residual life have been derived in explicit forms as functions of a posterior probability statistic. The advantage of the Bayesian model over the well-known 2-parameter Weibull model in system residual life prediction is shown. The methodologies are illustrated using simulated data, real data obtained from the spectrometric analysis of oil samples collected from transmission units of heavy hauler trucks in the mining industry, and vibration data from a planetary gearbox machinery application.

Mixture model-based clustering has become an increasingly popular data analysis technique since its introduction almost fifty years ago. Families of mixture models are said to arise when the component parameters, usually the component covariance matrices, are decomposed and a number of constraints are imposed. Within the family setting, it is necessary to choose the member of the family --- i.e., the appropriate covariance structure --- in addition to the number of mixture components. To date, the Bayesian information criterion (BIC) has proved most effective for this model selection process, and the expectation-maximization (EM) algorithm has been predominantly used for parameter estimation. We deviate from the EM-BIC rubric, using variational Bayes approximations for parameter estimation and the deviance information criterion (DIC) for model selection. The variational Bayes approach alleviates some of the computational complexities associated with the EM algorithm. We use this approach on the most famous family of Gaussian mixture models known as Gaussian parsimonious clustering models (GPCM). These models have an eigen-decomposed covariance structure. Cluster-weighted modelling (CWM) is another flexible statistical framework for modelling local relationships in heterogeneous populations on the basis of weighted combinations of local models. In particular, we extend cluster-weighted models to include an underlying latent factor structure of the independent variable, resulting in a novel family of models known as parsimonious cluster-weighted factor analyzers. The EM-BIC rubric is utilized for parameter estimation and model selection. Some work on a mixture of multivariate t-distributions is also presented, with a linear model for the mean and a modified Cholesky-decomposed covariance structure leading to a novel family of mixture models. In addition to model-based clustering, these models are also used for model-based classification, i.e., semi-supervised clustering. Parameters are estimated using the EM algorithm and another approach to model selection other than the BIC is also considered. / NSERC PGS-D

Les modèles à sur-représentation de zéros discrets et continus ont une large gamme d'applications et leurs propriétés sont bien connues. Bien qu'il existe des travaux portant sur les modèles discrets à sous-représentation de zéro et modifiés à zéro, la formulation usuelle des modèles continus à sur-représentation -- un mélange entre une densité continue et une masse de Dirac -- empêche de les généraliser afin de couvrir le cas de la sous-représentation de zéros. Une formulation alternative des modèles continus à sur-représentation de zéros, pouvant aisément être généralisée au cas de la sous-représentation, est présentée ici. L'estimation est d'abord abordée sous le paradigme classique, et plusieurs méthodes d'obtention des estimateurs du maximum de vraisemblance sont proposées. Le problème de l'estimation ponctuelle est également considéré du point de vue bayésien. Des tests d'hypothèses classiques et bayésiens visant à déterminer si des données sont à sur- ou sous-représentation de zéros sont présentées. Les méthodes d'estimation et de tests sont aussi évaluées au moyen d'études de simulation et appliquées à des données de précipitation agrégées. Les diverses méthodes s'accordent sur la sous-représentation de zéros des données, démontrant la pertinence du modèle proposé. Nous considérons ensuite la classification d'échantillons de données à sous-représentation de zéros. De telles données étant fortement non normales, il est possible de croire que les méthodes courantes de détermination du nombre de grappes s'avèrent peu performantes. Nous affirmons que la classification bayésienne, basée sur la distribution marginale des observations, tiendrait compte des particularités du modèle, ce qui se traduirait par une meilleure performance. Plusieurs méthodes de classification sont comparées au moyen d'une étude de simulation, et la méthode proposée est appliquée à des données de précipitation agrégées provenant de 28 stations de mesure en Colombie-Britannique. / Zero-inflated models, both discrete and continuous, have a large variety of applications and fairly well-known properties. Some work has been done on zero-deflated and zero-modified discrete models. The usual formulation of continuous zero-inflated models -- a mixture between a continuous density and a Dirac mass at zero -- precludes their extension to cover the zero-deflated case. We introduce an alternative formulation of zero-inflated continuous models, along with a natural extension to the zero-deflated case. Parameter estimation is first studied within the classical frequentist framework. Several methods for obtaining the maximum likelihood estimators are proposed. The problem of point estimation is considered from a Bayesian point of view. Hypothesis testing, aiming at determining whether data are zero-inflated, zero-deflated or not zero-modified, is also considered under both the classical and Bayesian paradigms. The proposed estimation and testing methods are assessed through simulation studies and applied to aggregated rainfall data. The data is shown to be zero-deflated, demonstrating the relevance of the proposed model. We next consider the clustering of samples of zero-deflated data. Such data present strong non-normality. Therefore, the usual methods for determining the number of clusters are expected to perform poorly. We argue that Bayesian clustering based on the marginal distribution of the observations would take into account the particularities of the model and exhibit better performance. Several clustering methods are compared using a simulation study. The proposed method is applied to aggregated rainfall data sampled from 28 measuring stations in British Columbia.