Mathematics in Independent Component Analysis

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278 Chapter 20. Signal Processing 86(3):603-623, 2006 JADE NMF NMF∗ sNMF sNMF∗ SCA sources mean kurtosis 10.80 8.48 8.80 8.85 9.64 7.14 11.4 sparseness 0.286 0.268 0.295 0.270 0.302 0.201 0.252 σ(A✷) 2.91 2.02 2.11 1.99 2.36 0.96 3.24 Table 2 Sparseness measures of the recovered sources using the various methods. In the first row, the mean kurtosis is calculated (the higher, the more ‘spiky’ the signal). The second row gives the sparseness index from equation (3) (the higher, the sparser the signal) and the third row the cost function (5) employed in SCA (the lower, the better the SCA criterion is fulfilled). Optimal values are printed in bold face. AJADE ANMF ANMF∗ AsNMF AsNMF∗ ASCA E1(A + ✷A) 1.39 3.27 2.96 3.41 2.56 3.86 JADE clearly outperforms the other methods — this will be explained in the next paragraph. However we have to add that the signal generation additionally involves a slightly nonlinear filter, so we can only estimate the real mixing matrix A from the sources and the mixtures, which yielded a non-negligible error. Hence this result indicates that JADE could best approximate the linearized system. One key question in this work is how the different models induce sparseness. Clearly, sparseness is a rather ambiguous term, so we calculate three indices (kurtosis, ‘sparseness’ from equation (3) and σ from (5)) for the real as well as the recovered sources using the proposed methods, see Tab. 2. As expected, ICA gives the highest kurtosis among all methods, whereas sparse NMF yields highest values of the sparseness criterion. And SCA has the lowest i.e. best value regarding k-sparseness measured by σ(A). We further see that both the kurtosis and the sparseness criterion seem to be somewhat related with regards to the data set as high values in both indices are achieved by both JADE and sNMF. The k-sparseness criterion, which fixes only the zero- (semi)norm i.e. requires a fixed amount of zeros in the data without additional requirements about the other values does not induce as high sparseness when measured using kurtosis/sparseness and vice versa. Finally by looking at the sparseness indices of the real sources, we can now understand why JADE outperformed the other methods in this toy data setting — indeed kurtosis of the sources is high and the mutual information low, see Fig. 6(b). But in terms of sparseness and especially σ, the sources are not as sparse as expected. Hence the (sparse) NMF and mainly the SCA algorithm could not perform as well as JADE when compared to the original sources as noted above. However, we will see that in the case of real s-EMG signals, this distinction will break; furthermore, sparse NMF turns out to be more robust against noise than JADE, as is shown in the following. 17
Chapter 20. Signal Processing 86(3):603-623, 2006 279 Amari index 25 20 15 10 5 0 JADE NMF NMF∗ sNMF sNMF∗ SCA (a) Amari index SNR 30 25 20 15 10 5 0 JADE NMF NMF∗ sNMF sNMF∗ SCA (b) SNR Fig. 7. Boxplot of the separation performance when identifying 5 sources in 8-dimensional observed artificial s-EMG recordings. (a) shows the mean Amari index of the product of the identified separation and the real mixing matrix, and (b) depicts the SNR of the recovered sources versus the real sources. Mean and variance were taken over 100 runs. 3.1.2 Multiple s-EMG experiments We now show performance of the above algorithms when applied to 100 different realizations of artificial s-EMG data sets. The data consists of 8 channels with 5 underlying source activities; more details about data generation are given in section 2.1.1. The algorithm parameters are the same as above with the exception that automated SCA is performed using Mangasarian-style clustering [38] after PCA dimension reduction to 5. The ICA and sparse BSS algorithms from above are applied to these data sets, and Amari index as well as SNR of the recovered sources versus the original sources is stored. In Fig. 7, the means of these two indices as well as their deviations are shown in a box plot, separately for each algorithm. As in the single s-EMG experiment, these statistics confirm that JADE performs best, both in terms of matrix and in terms of source recovery (which is more or less the same due to the fact that we are dealing with the noiseless case so far). The NMF algorithms identify the mixing matrix with acceptable performance, however (sparse) NMF taking only positive samples (sNMF∗) tends to separate the data slightly better than by sample preprocessing using κ from equation (4). SCA cannot detect the source matrix as well as the other BSS algorithms — again the SCA conditions seem to be somewhat violated — however it performs adequately well recovering the sources, which is due to the fact that some sources are recovered very well resulting in a higher SNR than the NMF algorithms. For practical reasons, it is important to check to what extend the signal-to-interference ratio between the channels is improved after applying BSS algorithms. For each run, we monitor the SIR of the original sources and the recoveries by taking the mean over all channels. The two SIR means are 18
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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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Mathematics in Independent Component Analysis

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