Non-Gaussian Statistical Modelsand Their Applications

University dissertation from Stockholm : KTH Royal Institute of Technology

Abstract: Statistical modeling plays an important role in various research areas. It provides away to connect the data with the statistics. Based on the statistical properties of theobserved data, an appropriate model can be chosen that leads to a promising practicalperformance. The Gaussian distribution is the most popular and dominant probabilitydistribution used in statistics, since it has an analytically tractable Probability DensityFunction (PDF) and analysis based on it can be derived in an explicit form. However,various data in real applications have bounded support or semi-bounded support. As the support of the Gaussian distribution is unbounded, such type of data is obviously notGaussian distributed. Thus we can apply some non-Gaussian distributions, e.g., the betadistribution, the Dirichlet distribution, to model the distribution of this type of data.The choice of a suitable distribution is favorable for modeling efficiency. Furthermore,the practical performance based on the statistical model can also be improved by a bettermodeling.An essential part in statistical modeling is to estimate the values of the parametersin the distribution or to estimate the distribution of the parameters, if we consider themas random variables. Unlike the Gaussian distribution or the corresponding GaussianMixture Model (GMM), a non-Gaussian distribution or a mixture of non-Gaussian dis-tributions does not have an analytically tractable solution, in general. In this dissertation,we study several estimation methods for the non-Gaussian distributions. For the Maxi-mum Likelihood (ML) estimation, a numerical method is utilized to search for the optimalsolution in the estimation of Dirichlet Mixture Model (DMM). For the Bayesian analysis,we utilize some approximations to derive an analytically tractable solution to approxi-mate the distribution of the parameters. The Variational Inference (VI) framework basedmethod has been shown to be efficient for approximating the parameter distribution byseveral researchers. Under this framework, we adapt the conventional Factorized Approx-imation (FA) method to the Extended Factorized Approximation (EFA) method and useit to approximate the parameter distribution in the beta distribution. Also, the LocalVariational Inference (LVI) method is applied to approximate the predictive distributionof the beta distribution. Finally, by assigning a beta distribution to each element in thematrix, we proposed a variational Bayesian Nonnegative Matrix Factorization (NMF) forbounded support data.The performances of the proposed non-Gaussian model based methods are evaluatedby several experiments. The beta distribution and the Dirichlet distribution are appliedto model the Line Spectral Frequency (LSF) representation of the Linear Prediction (LP)model for statistical model based speech coding. For some image processing applications,the beta distribution is also applied. The proposed beta distribution based variationalBayesian NMF is applied for image restoration and collaborative filtering. Comparedto some conventional statistical model based methods, the non-Gaussian model basedmethods show a promising improvement.