Advanced source separation methods with applications to spatio-temporal datasets

Latent variable models are useful tools for statistical data analysis in many applications. Examples of popular models include factor analysis, state-space models and independent component analysis. These types of models can be used for solving the source separation problem in which the latent variables should have a meaningful interpretation and represent the actual sources generating data. Source separation methods is the main focus of this work. Bayesian statistical theory provides a principled way to learn latent variable models and therefore to solve the source separation problem. The first part of this work studies variational Bayesian methods and their application to different latent variable models. The properties of variational Bayesian methods are investigated both theoretically and experimentally using linear source separation models. A new nonlinear factor analysis model which restricts the generative mapping to the practically important case of post-nonlinear mixtures is presented. The variational Bayesian approach to learning nonlinear state-space models is studied as well. This method is applied to the practical problem of detecting changes in the dynamics of complex nonlinear processes. The main drawback of Bayesian methods is their high computational burden. This complicates their use for exploratory data analysis in which observed data regularities often suggest what kind of models could be tried. Therefore, the second part of this work proposes several faster source separation algorithms implemented in a common algorithmic framework. The proposed approaches separate the sources by analyzing their spectral contents, decoupling their dynamic models or by optimizing their prominent variance structures. These algorithms are applied to spatio-temporal datasets containing global climate measurements from a long period of time.