Microarray data clustering represents a basic exploratory tool to find groups of genes exhibiting similar expression patterns or to detect relevant classes of molecular subtypes. Among a wide range of clustering approaches proposed and applied in the gene expression community to analyze microarray data, mixture model-based clustering has received much attention to its sound statistical framework and its flexibility in data modeling. However, clustering algorithms following the model-based framework suffer from two serious drawbacks. The first drawback is that the performance of these algorithms critically depends on the starting values for their iterative clustering procedures. Additionally, they are not capable of working directly with very high dimensional data sets in the sample clustering problem where the dimension of the data is up to hundreds or thousands. The thesis focuses on the two challenges and includes the following contributions: First, the thesis introduces the statistical model of our proposed normalized Expectation Maximization (EM) algorithm followed by its clustering performance analysis on a number of real microarray data sets. The normalized EM is stable even with random initializations for its EM iterative procedure. The stability of the normalized EM is demonstrated through its performance comparison with other related clustering algorithms. Furthermore, the normalized EM is the first mixture model-based clustering approach to be capable of working directly with very high dimensional microarray data sets in the sample clustering problem, where the number of genes is much larger than the number of samples. This advantage of the normalized EM is illustrated through the comparison with the unnormalized EM (The conventional EM algorithm for Gaussian mixture model-based clustering). Besides, for experimental microarray data sets with the availability of class labels of data points, an interesting property of the convergence speed of the normalized EM with respect to the radius of the hypersphere in its corresponding statistical model is uncovered. Second, to support the performance comparison of different clusterings a new internal index is derived using fundamental concepts from information theory. This index allows the comparison of clustering approaches in which the closeness between data points is evaluated by their cosine similarity. The method for deriving this internal index can be utilized to design other new indexes for comparing clustering approaches which employ a common similarity measure.

The goal of this thesis research was to develop software to be used with RNA-Seq data for transcriptome quantification that was capable of handling multireads and quantifying isoforms on a more global level. Current software available for these purposes uses various forms of parameter alteration in order to work with multireads. Many still analyze isoforms per gene or per researcher determined clusters as well. By doing so, the effects of multireads are diminished or possibly wrongly represented. To address this issue, two programs, GWIE and ChromIE, were developed based on a simple iterative EM-like algorithm with no parameter manipulation. These programs are used to produce accurate isoform expression levels.

In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.

Dealing with missing data in spatio-temporal time series constitutes important branch of general missing data problem. Since the statistical properties of time-dependent data characterized by sequentiality of observations then any interruption of consecutiveness in time series will cause severe problems. In order to make reliable analyses in this case missing data must be handled cautiously without disturbing the series statistical properties, mainly as temporal and spatial dependencies. In this study we aimed to compare several imputation methods for the appropriate completion of missing values of the spatio-temporal meteorological time series. For this purpose, several missing imputation methods are assessed on their imputation performances for artificially created missing data in monthly total precipitation and monthly mean temperature series which are obtained from the climate stations of Turkish State Meteorological Service. Artificially created missing data are estimated by using six methods. Single Arithmetic Average (SAA), Normal Ratio (NR) and NR Weighted with Correlations (NRWC) are the three simple methods used in the study. On the other hand, we used two computational intensive methods for missing data imputation which are called Multi Layer Perceptron type Neural Network (MLPNN) and Monte Carlo Markov Chain based on Expectation-Maximization Algorithm (EM-MCMC). In addition to these, we propose a modification in the EM-MCMC method in which results of simple imputation methods are used as auxiliary variables. Beside the using accuracy measure based on squared errors we proposed Correlation Dimension (CD) technique for appropriate evaluation of imputation performances which is also important subject of Nonlinear Dynamic Time Series Analysis.

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.

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