Positive Data Clustering based on Generalized InvertedDirichlet Mixture Model

Recent advances in processing and networking capabilities of computers have caused an accumulation of immense amounts of multimodal multimedia data (image, text, video). These data are generally presented as high-dimensional vectors of features. The availability of these highdimensional data sets has provided the input to a large variety of statistical learning applications including clustering, classification, feature selection, outlier detection and density estimation. In this context, a finite mixture offers a formal approach to clustering and a powerful tool to tackle the problem of data modeling. A mixture model assumes that the data is generated by a set of parametric probability distributions. The main learning process of a mixture model consists of the following two parts: parameter estimation and model selection (estimation the number of components). In addition, other issues may be considered during the learning process of mixture models such as the: a) feature selection and b) outlier detection. The main objective of this thesis is to work with different kinds of estimation criteria and to incorporate those challenges into a single framework. The first contribution of this thesis is to propose a statistical framework which can tackle the problem of parameter estimation, model selection, feature selection, and outlier rejection in a unified model. We propose to use feature saliency and introduce an expectation-maximization (EM) algorithm for the estimation of the Generalized Inverted Dirichlet (GID) mixture model. By using the Minimum Message Length (MML), we can identify how much each feature contributes to our model as well as determine the number of components. The presence of outliers is an added challenge and is handled by incorporating an auxiliary outlier component, to which we associate a uniform density. Experimental results on synthetic data, as well as real world applications involving visual scenes and object classification, indicates that the proposed approach was promising, even though low-dimensional representation of the data was applied. In addition, it showed the importance of embedding an outlier component to the proposed model. EM learning suffers from significant drawbacks. In order to overcome those drawbacks, a learning approach using a Bayesian framework is proposed as our second contribution. This learning is based on the estimation of the parameters posteriors and by considering the prior knowledge about these parameters. Calculation of the posterior distribution of each parameter in the model is done by using Markov chain Monte Carlo (MCMC) simulation methods - namely, the Gibbs sampling and the Metropolis- Hastings methods. The Bayesian Information Criterion (BIC) was used for model selection. The proposed model was validated on object classification and forgery detection applications. For the first two contributions, we developed a finite GID mixture. However, in the third contribution, we propose an infinite GID mixture model. The proposed model simutaneously tackles the clustering and feature selection problems. The proposed learning model is based on Gibbs sampling. The effectiveness of the proposed method is shown using image categorization application. Our last contribution in this thesis is another fully Bayesian approach for a finite GID mixture learning model using the Reversible Jump Markov Chain Monte Carlo (RJMCMC) technique. The proposed algorithm allows for the simultaneously handling of the model selection and parameter estimation for high dimensional data. The merits of this approach are investigated using synthetic data, and data generated from a challenging namely object detection.