L₁-Module Based Independent Components Analysis

Since Herault & Jutten(1983) founded the theory of Independent Components Analysis,it has been used widely in the field of finance, iatrology,communication and so on.Comon(1994) applied ICA to solve the problem of Blind Source Separation.Hyvarinnen(1999) used it in image denoising.Andrew & Andreas(1997) considered 28 most weighted stocks in Japan market and dealed with the 3-year-return of them.They obtained 24 independent components,by which they gave a new explanation of the stock market's structure.There are many improvements in the ICA theory itself.Giannakis (1987) created an algorithm based on high order accumulant statistics.Lacoume & Ruiz(1989) founded the mathematical frame of ICA.Gaete & Lacoume(1989) tried Maximum Likelihood Method in estimating the model's parameter.Cardoso(1989) considered the algebraic properties of random variable's high order statistics.In Hyvarinen(2001),most of ICA's theoretical results and various methods were collected.In the same book,a widely used algorithm named Fast-ICA,was produced by Hyvarinen.However,most methods in the book are based on high order statistics,which probably make the algorithm unstable.In Schweizer & Wolff(1981),"Copula" was taken as keyword the first time in history,which was translated in chinese as "关联函数" and was used to measure the dependence of random variables.In Schweizer&Wolff(1981),Schweizer&Wolff'sσ,a statistic based on L₁-module,was produced in order to measure the independence of random variables.The contribution of the thesis is creating a new kind of ICA algorithm named S-ICA,based on Schweizer&Wolff'sσ,which makes the algorithm more stable than the classical algorithms based on high order statistics.Detailed introduction of relevant theories and simulation are also provided in the thesis.There are five chapters in the thesis and are arranged as below:·Chapter 1:Introduction to ICAICA's history,background and so on are arranged.Chapter 2 Independent Components AnalysisICA's theoretical frame and most popular algorithms are arranged.·Chapter 3:ICA base on L₁-ModuleThe independence measureσis introduced here and S-ICA algorithm is provided.Chapter 4:SimulationWe compare the S-ICA algorithm and Fast-ICA algorithm by simulation in this chapter.·Chapter 5:Summary and future workSummarize the thesis and claim future work.