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Independent component analysis P7

BASIC INDEPENDENT COMPONENT ANALYSIS Independent Component Analysis. Aapo Hyv¨ rinen, Juha Karhunen, Erkki Oja a Copyright 2001 John Wiley & Sons, Inc. ISBNs: 0-471-40540-X (Hardback); 0-471-22131-7 (Electronic) 7 What is Independent Component Analysis? In this chapter, the basic concepts of independent component analysis (ICA) are defined. We start by discussing a couple of practical applications. These serve as motivation for the mathematical formulation of ICA, which is given in the form of a statistical estimation problem. Then we consider under what conditions this model can be estimated, and what exactly can be estimated. After these basic definitions, we go on to discuss the. | Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic Part II BASIC INDEPENDENT COMPONENT ANALYSIS Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic 7 What is Independent Component Analysis In this chapter the basic concepts of independent component analysis ICA are defined. We start by discussing a couple of practical applications. These serve as motivation for the mathematical formulation of ICA which is given in the form of a statistical estimation problem. Then we consider under what conditions this model can be estimated and what exactly can be estimated. After these basic definitions we go on to discuss the connection between ICA and well-known methods that are somewhat similar namely principal component analysis PCA decorrelation whitening and sphering. We show that these methods do something that is weaker than ICA they estimate essentially one half of the model. We show that because of this ICA is not possible for gaussian variables since little can be done in addition to decorrelation for gaussian variables. On the positive side we show that whitening is a useful thing to do before performing ICA because it does solve one-half of the problem and it is very easy to do. In this chapter we do not yet consider how the ICA model can actually be estimated. This is the subject of the next chapters and in fact the rest of Part II. 7.1 MOTIVATION Imagine that you are in a room where three people are speaking simultaneously. The number three is completely arbitrary it could be anything larger than one. You also have three microphones which you hold in different locations. The microphones give you three recorded time signals which we could denote by xy t x2 i and x3 i with xy x2 and .r3 the amplitudes and the time index. Each of these