Mathematics in Independent Component Analysis

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286 Chapter 20. Signal Processing 86(3):603-623, 2006 small errors in the independence assumption — and how they can deal with additional noise. In the first example of artificially created mixtures, we are able to demonstrate that a decomposition analysis using the three models was possible, and give comparisons over larger data sets. Although the recovered sources are all rather alike, Fig. 5 and Fig. 7, we found that ICA outperformed the other methods concerning distance to the real solution, which was mainly due to the fact that the artificial sources — but not in the case of real signals, see below — best fitted to the ICA model, Tab. 2. However, when considering additional noise with increasing power, sparse NMF turned out to be more robust than the ICA model, Fig. 8. We then applied the BSS methods to real s-EMG data sets, and the three different models yielded surprisingly similar results, although in theory these models do not fully overlap. We speculate that this similarity is due to the fact that — as most probably in all applications — the models do not fully hold. This allows the various algorithms to only approximate the model solutions, and hence to arrive at similar solutions, but from different directions. Furthermore, this indicates that the three models look for different properties of the sources, and that these properties are fulfilled to varying extent, see Tab. 3 for numerical details. Comparisons over s-EMG data sets from multiple subjects again confirmed similar equality of performance, where again sparse NMF slightly outperformed the other algorithms in the mean (in nice correspondence with the noise result from above). Note that the aim of the present work is not the full recovery of a target MUAPT (source signal) in its original form. In fact, as shown previously [18], it would be sufficient to increase the amplitude a target MUAPT so that in average it is above the noise and above the other MUAPTs level. Then, we are able to cut the interfering signals with a modified dead zone filter and thus isolating the target MUAPT. And indeed all of the employed sparse BSS methods fulfill this requirement, which is confirmed by the decrease in the number of zero-crossings of the separated signals, see Tab. 4. 5 Conclusion We have compared the effectiveness of various sparse BSS methods for signal decomposition; namely, ICA, NMF, sparse NMF and SCA, when applied to s-EMG signals. Surface EMG signals represent an attractive testing signal as they approximately fulfill all the requirements of these methods and are of major importance in medical diagnosis and basic neurophysiological research. 25
Chapter 20. Signal Processing 86(3):603-623, 2006 287 All methods, in spite of being based on very different approaches, gave similar results in the case of real data and decomposed them sufficiently well. We therefore suggest using sparse BSS as an important preprocessing tool before applying the common template matching technique. In terms of algorithm comparisons, ICA performed better than the other algorithms in the noiseless case, however sparse NMF∗ outperformed the other methods when noise was added and slightly so in the case of multiple real s-EMG recordings. In later work concerning methods of s-EMG decomposition, we therefore want to focus on properties and possible improvements of sparse NMF regarding parameter choice (which level of sparseness to choose) and signal preprocessing (in order to deal with positive signals). Preprocessing to improve sparseness is currently studied. In order to be better able to compare methods, we also plan to apply the methods to artificial sources generated using other available s-EMG generators [47]. Finally extensions to convolutive mixing situations will have to be analyzed. Acknowledgements We would like to thank Dr. S. Rainieri for helpful discussion and K. Maekawa for the first version of the s-EMG generator. This work was partly supported by the Ministry of Education, Culture, Sports, Science, and Technology of Japan (Grant-in-Aid for Scientific Research). G.A.G. is supported by a grant from the same Ministry. F.T. gratefully acknowledges partial financial support by the DFG (GRK 638) and the BMBF (project ‘ModKog’). References [1] A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis, John Wiley & Sons, 2001. [2] A. Cichocki, S. Amari, Adaptive blind signal and image processing, John Wiley & Sons, 2002. [3] P. Georgiev, F. Theis, A. Cichocki, Blind source separation and sparse component analysis of overcomplete mixtures, in: Proc. ICASSP 2004, Vol. 5, Montreal, Canada, 2004, pp. 493–496. [4] P. Georgiev, F. Theis, A. Cichocki, Sparse component analysis and blind source separation of underdetermined mixtures, IEEE Trans. Neural Networks in press. [5] D. Lee, H. Seung, Learning the parts of objects by non-negative matrix factorization, Nature 40 (1999) 788–791. 26
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