Mathematics in Independent Component Analysis

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208 Chapter 15. Neurocomputing, 69:1485-1501, 2006 3.2.2 Generalized Eigenvector Decomposition The delayed correlation matrices of the matrix pencil are computed with one matrix Xr obtained by eliminating the first ki columns of X and another matrix, Xl, obtained by eliminating the last ki columns. Then, the delayed correlation matrix Rx(ki) = XrXl T will be an NM × NM matrix. Each of these two matrices can be related with a corresponding matrix in the source signal domain: Rx(ki) = ARs(ki)A T = ASrSl T A T (11) Then the two pairs of matrices (Rx(k1), Rx(k2)) and (Rs(k1), Rs(k2)) represent congruent pencils [32] with the following properties: • Their eigenvalues are the same, i.e., the eigenvalue matrices of both pencils are identical: Dx = Ds. • If the eigenvalues are non-degenerate (distinct values in the diagonal of the matrix Dx = Ds), the corresponding eigenvectors are related by the transformation Es = A T Ex. Assuming that all sources are uncorrelated, the matrices Rs(ki) are block diagonal, having block matrices Rmm(ki) = SriSli T along the diagonal. The eigenvector matrix of the GEVD of the pencil (Rs(k1), Rs(k2)) again forms a block diagonal matrix with block matrices Emm forming M × M eigenvector matrices of the GEVD of the pencils (Rmm(k1), Rmm(k2)). The uncorrelated components can then be estimated from linearly transformed sensor signals via Y = Ex T X = Ex T AS = Es T S (12) hence turn out to be filtered versions of the underlying source signals. As the eigenvector matrix Es is a block diagonal matrix, there are M signals in each column of Y which are a linear combination of one of the source signals and its delayed versions. Then the columns of the matrix Emm represent impulse responses of Finite Impulse Response (FIR) filters. Considering that all the columns of Emm are different, their frequency response might provide different spectral densities of the source signal spectra. Then NM output signals y encompass M filtered versions of each of the N estimated source signals. 3.2.3 Implementation of the GEVD There are several ways to compute the generalized eigenvalue decomposition. We resume a procedure valid if one of the matrices of the pencil is symmetric positive definite. Thus, we consider the pencil (Rx(0), Rx(k2)) and perform the following steps: Step 1: Compute a standard eigenvalue decomposition of Rx(0) = V ΛV T , i.e, 11
Chapter 15. Neurocomputing, 69:1485-1501, 2006 209 compute the eigenvectors vi and eigenvalues λi. As the matrix is symmetric positive definite, the eigenvalues can be arranged in descending order (λ1 > λ2 > · · · > λNM). This procedure corresponds to the usual whitening step in many ICA algorithms. It can be used to estimate the number of sources, but it can also be considered a strategy to reduce noise much like with PCA denoising. Dropping small eigenvalues amounts to a projection from a highdimensional feature space onto a lower dimensional manifold representing the signal+noise subspace. Thereby it is tacitly assumed that small eigenvalues are related with noise components only. Here we consider a variance criterion to choose the most significant eigenvalues, those related with the embedded deterministic signal, according to λ1 + λ2 + ... + λl λ1 + λ2 + ...λNM ≥ T H (13) If we are interested in the eigenvectors corresponding to directions of high variance of the signals, the threshold T H should be chosen such that their maximum energy is preserved. Similar to the whitening phase in many BSS algorithms, the data matrix X can be transformed using 1 − Q = Λ 2 V T (14) to calculate a transformed matrix of delayed correlations C(k2) to be used in the next step. The transformation matrix can be computed using either the l most significant eigenvalues, in which case denoising is achieved, or all eigenvalues and respective eigenvectors. Also note, that Q represents a l×NM matrix if denoising is considered. Step 2: Use the transformed delayed correlation matrix C(k2) = QRx(k2)Q T and its standard eigenvalue decomposition: the eigenvector matrix U and eigenvalue matrix Dx. The eigenvectors of the pencil (Rx(0), Rx(k2)), which are not normalized, form the columns of the eigenvector matrix Ex = QT 1 − U = V Λ 2 U. The ICs of the delayed sensor signals can then be estimated via the transformation given below, yielding l (or NM) signals, one signal per row of Y : Y = Ex T X = U T QX = U T 1 − Λ 2 V T X (15) The first step of this algorithm is therefore equivalent to a PCA in a highdimensional feature space [9,38], where a matrix similar to Q is used to project the data onto the signal manifold. 12
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vi Preface
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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1.2. Uniqueness issues in independe
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S U M M A R Y T his �le contains
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Mathematics in Independent Component Analysis

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