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Independent component analysis P8

ICA by Maximization of Nongaussianity In this chapter, we introduce a simple and intuitive principle for estimating the model of independent component analysis (ICA). This is based on maximization of nongaussianity. Nongaussianity is actually of paramount importance in ICA estimation. Without nongaussianity the estimation is not possible at all, as shown in Section 7.5. Therefore, it is not surprising that nongaussianity could be used as a leading principle in ICA estimation. This is at the same time probably the main reason for the rather late resurgence of ICA research: In most of classic statistical theory, random variables are assumed to have. | 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 8 ICA by Maximization of Nongaussianity In this chapter we introduce a simple and intuitive principle for estimating the model of independent component analysis ICA . This is based on maximization of nongaussianity. Nongaussianity is actually of paramount importance in ICA estimation. Without nongaussianity the estimation is not possible at all as shown in Section 7.5. Therefore it is not surprising that nongaussianity could be used as a leading principle in ICA estimation. This is at the same time probably the main reason for the rather late resurgence of ICA research In most of classic statistical theory random variables are assumed to have gaussian distributions thus precluding methods related to ICA. A completely different approach may then be possible though using the time structure of the signals see Chapter 18. We start by intuitively motivating the maximization of nongaussianity by the central limit theorem. As a first practical measure of nongaussianity we introduce the fourth-order cumulant or kurtosis. Using kurtosis we derive practical algorithms by gradient and fixed-point methods. Next to solve some problems associated with kurtosis we introduce the information-theoretic quantity called negentropy as an alternative measure of nongaussianity and derive the corresponding algorithms for this measure. Finally we discuss the connection between these methods and the technique called projection pursuit. 165 166 ICA BYMAXIMIZATION OF NONGAUSSIANITY 8.1 NONGAUSSIAN IS INDEPENDENT The central limit theorem is a classic result in probability theory that was presented in Section 2.5.2. It says that the distribution of a sum of independent random variables tends toward a gaussian distribution under certain conditions. Loosely speaking a sum of two independent random variables usually has a .