Mathematics in Independent Component Analysis

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204 Chapter 15. Neurocomputing, 69:1485-1501, 2006 3 Denoising Algorithms 3.1 Local ICA denoising The LICA algorithm we present is based on a local projective denoising technique using an MDL criterion for parameter selection. The idea is to achieve denoising by locally projecting the embedded noisy signal into a lower dimensional subspace which contains the characteristics of the noise free signal. Finally the signal has to be reconstructed using the various candidates generated by the embedding. Consider the situation, where we have a signal x 0 i [l] at discrete time steps l = 0, . . . , L − 1 but only its noise corrupted version xi[l] is measured xi[l] = x 0 i [l] + ei[l] (5) where e[l] are samples of a random variable with Gaussian distribution, i.e. xi equals x 0 i up to additive stationary white noise. 3.1.1 Embedding and clustering First the noisy signal xi[l] is transformed into a high-dimensional signal xi[l] in the M-dimensional space of delayed coordinates according to xi[l] := � xi[l], . . . , xi[l − M + 1 mod L] � T (6) which corresponds to a column of the trajectory matrix in equation 1. To simplify implementation, we want to ensure that the delayed signal, like the original signal, (trajectory matrix) is given at L time steps instead of L − M + 1. This can be achieved by using the samples in round robin manner, i.e. by closing the end and the begin of each delayed signal and cutting out exactly L components in accord with the delay. If the signal contains a trend or its statistical nature is significantly different at the end compared to the beginning, then this leads to compatibility problems of the beginning and end of the signal. We can easily resolve this misfit by replacing the signal with a version where we add the signal in reverse order, hence avoiding any sudden change in signal amplitude which would be smoothed out by the algorithm. The problem can now be localized by selecting K clusters in the feature space of delayed coordinates of the signal {xi[l] | l = 0, . . . , L − 1}. Clustering can be achieved by a k-means cluster algorithm as explained in section 2.2. But k-means clustering is only appropriate if the variance or the kurtosis of a signal do not depend on the inherent signal structure. For other noise 7
Chapter 15. Neurocomputing, 69:1485-1501, 2006 205 selection schemes like choosing the noise components based on the variance of the autocorrelation, it is usually better to find an appropriate partitioning of the set of time steps {0, . . . , L − 1} into K successive segments, since this preserves the inherent time structure of the signals. For other noise selection methods like choosing the noise components based on the variance of the autocorrelation it is usually better to find an appropriate partition of the set of time steps {0, . . . , L − 1} into K successive segments, since this preserves the inherent time structure of the signal. Note that the clustering does not change the data but only changes its time sequence, i.e. permutes and regroups the columns of the trajectory matrix and separates it into K sub-matrices. 3.1.2 Decomposition and denoising After centering, i.e. removing the mean in each cluster, we can analyze the M-dimensional signals in these K clusters using PCA or ICA. The PCA case (Local PCA (LPCA)) is studied in [39] so in the following we will propose an ICA based denoising. Using ICA, we extract M ICs from each delayed signal. Like in all projection based denoising algorithms, noise reduction is achieved by projecting the signal into a lower dimensional subspace. We used two different criteria to estimate the number p of signal+noise components, i.e. the dimension of the signal subspace onto which we project after applying ICA. • One criterion is a consistent MDL estimator of pMDL for the data model in equation 5 ( [39]) pMDL = argmin MDL(M, L, p, (λj), γ) (7) p=0,...,M−1 ⎧ ⎪⎨ argmin p=0,...,M−1 ⎪⎩ −�(M − p)L � ⎛ ⎜ ln ⎝ ΠMj=p+1λ 1 ⎞ M−p j ⎟ 1 �Mj=p+1 ⎠ λj M−p � + pM − p2 � �1 � p + + 1 + ln γ 2 2 2 ⎛ p2 p pM − + + 1 2 2 − ⎝ p 1 2 ln 2 L + ⎞⎫ p� M−1 � ⎬ ln λj − ln λj ⎠ ⎭ j=1 j=1 where λj denote the variances of the signal components in feature space, i.e. after applying the de-mixing matrix which we estimate with the ICA algorithm. To retain the relative strength of the components in the mixture, we normalize the rows of the de-mixing matrix to unit norm. The variances 8
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Statistical machine learning of bio
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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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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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1.3. Dependent component analysis 1
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Table 1.1: BSS algorithms based on
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1.4. Sparseness 29 R 3 R 3 A BSRA R
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1.4. Sparseness 31 Sparse projectio
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298 BIBLIOGRAPHY Barber, C., Dobkin
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Mathematics in Independent Component Analysis

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