Independent Component Analysis Based On Non-Stability And Four Potential Stability Measures

Independent Component Analysis (ICA), as a represent method, has wide applications in fields like neural network, blind source separation, statistical analysis etc., and it has been highly developed in the past twenty years especially recent ten years. But, ICA method is not suitable to solve problems concerning about heavy-tailed signals, which are always modeled byα-stable distribution and have wide applications in computer science, physics, chemistry, economics, finance, geography and so on.We propose a new ICA method based on non-stability in this paper, which finds the independent components through the maximum of the non-stability of the sample. Also, we propose the maximum non-stability principle to take place of the maximum non-Gaussinity one in the classic ICA method, and point out that the later one is a special case of the former one. For the measure of stability, a brand new measure, called Alpha-negentropy, and three approximation stability measures are also proposed. In the end, we design a numerical algorithm based on gradient to realize our new ICA method.Motivated by the Generalized Central Limit Theorem (GCLT), an important extension of classical CLT, we propose a new approach for ICA by maximizing the non-stability contrast function, which should suit heavy-tailed signal source separation problems. In this paper, we challenged the maximum non-Gaussinity principle and changed it into maximum non-stability principle. Also, we demonstrate that the classical ICA based on maximization of non-Gaussianity is a special case of the new approach of ICA which is based on maximization of non-stability with certain constraints. To be able to quantify non-stability, we propose a new measure of stability namely Alpha-negentropy, which is theoretical heretic and designed to be the contrast function. Three other approximation non-stability measures, FLOM method, logarithmic moment method and extreme value method are also introduced since Alpha-negentropy is computational complex. At the end of this paper, a numerical gradient algorithm for the maximization of the Alpha-negentropy with the objective of source separation of heavy-tailed signals is designed. The first part of experiment compares FLOM, logarithmic moment, extreme value method with the differential entropy. The other part evaluates this new ICA method using four non-stability methods as contrast functions, which show that ICA by maximum of non-stability performs successfully in heavy-tailed source separation problems.