| Independent component analysis (ICA) is a recent and powerful technique for recovering latent independent sources given only their mixtures. The basic ICA model assumes that sources are linearly mixed and mutually independent.; In this thesis, first we consider the problem of source separation of post-nonlinear (PNL) mixtures, which is an extension of ICA to the nonlinear mixing case. With a large number of parameters, existing methods are computation-demanding and may be prone to local optima. Based on the fact that linear mixtures of independent variables tend to be Gaussian, we develop a simple and efficient method for this problem, namely extended Gaussianization. With Gaussianization as preprocessing, this method approximates each linear mixture of independent sources by the Cornish-Fisher expansion with only two parameters. Inspired by the relationship between the PNL mixing model and the Wiener system, extended Gaussianization is also proposed for blind inversion of Wiener systems.; Next, we study the subband decomposition ICA (SDICA) model, which extends the basic ICA model to allow dependence between sources by assuming that only some narrow-band source sub-components are independent. In SDICA, it is difficult to determine the subbands of source independent sub-components. We discuss the feasibility of performing SDICA in an adaptive manner. An adaptive method, called band selective ICA, is then proposed for this task. We also investigate the relationship between overcomplete ICA and SDICA and show that band selective ICA can solve the overcomplete ICA problems with sources having specific frequency localizations. Experimental results on separating images of human faces as well as artificial data are presented to verify the powerfulness of band selective ICA.; In practice, the application and extension of the ICA model depend on the problem and the data to be investigated. We finally focus on GARCH models in finance, and show that estimation of univariate or multivariate GARCH models is actually a nonlinear ICA problem; maximizing the likelihood is equivalent to minimizing the statistical dependence in standardized residuals. ICA can then be used for factor extraction in multivariate factor GARCH models. We also develop some extensions of ICA for this task. These techniques for extracting factors from multivariate return series are compared both theoretically and experimentally. We find that the one based on conditional decorrelation between factors behaves best. |
