On Improved Methods for Blind Separation of Signals with Arbitrary Probability Distributions and Robust Independent Component Analysis

摘要

In this thesis, we concentrate on two important issues related to blind source separation (BSS) using independent component analysis (ICA). First, we note that the separation performance of various ICA estimation methods depends on the probability distributions of source signals. It is therefore important to have algorithms that work well for signals with a wide range of pdfs. Second, it is highly preferable in practical applications to employ a method that is robust in the presence of outliers. In order to address the first problem, we propose two different approaches. The first approach is based on employing certain parametric nonlinear activation functions in the relative gradient algorithm, which is one of the widely used ICA estimation method. The parameters of these activation functions are adapted online so as to ensure that the desired solution remains a locally stable equilibrium point of the algorithm. Numerous simulation results are presented to demonstrate that the proposed methods give good separation performance for signals with many different probability distributions. A central principle for performing BSS by ICA is maximization of non-Gaussianity. Non-Gaussianity is generally (quantitatively) measured by employing a non-quadratic function. However, as we mentioned above, the separation performance depends on the probability distributions of sources and consequently a single nonlinear function may not give adequate results for all signals. Accordingly, we propose to employ an empirical characteristic function based non-Gaussianity measure that directly computes the distance of an arbitrary distribution from the Gaussian in the characteristic function domain. The proposed non-Gaussianity measure is a weighted distance between the characteristic function of a random variable and a Gaussian characteristic function at some appropriately chosen sample points. We derive a fixed-point algorithm to optimize this non-Gaussianity measure and also suggest a procedure to choose adequate sample points online so as to improve the separation performance. Computer experiments with many different types of sources show that the characteristic function based non-Gaussianity measure has the potential to give adequate performance for a wide variety of distributions. Finally, in order to address the problem of robustness, we propose to employ an extension to the natural gradient algorithm in which an exponentially decaying function is introduced to discount the effect of outliers. By appropriately choosing the spread of the exponential function, the separation performance is greatly improved in the presence of outliers; the performance remains almost the same when there are no outliers in the data.