Mathematics in Independent Component Analysis

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224 Chapter 15. Neurocomputing, 69:1485-1501, 2006 (a) 1D slice of the NOESY spectrum of the protein P11 spectrum reconstructed with the algorithm GEVD-MP (b) Corresponding protein spectrum reconstructed with the algorithm dAMUSE Fig. 10. Comparison of denoising of the P11 protein spectrum parameters of the algorithms are collected in table 2 5 Conclusions We proposed two new denoising techniques and also considered KPCA denoising which are all based on the concept of embedding signals in delayed coordinates. We presented a detailed discussion of their properties and also discussed results obtained applying them to illustrative toy examples. Furthermore we compared all three algorithms by applying them to the real world 27
Chapter 15. Neurocomputing, 69:1485-1501, 2006 225 Table 2 Parameter values for the embedding dimension of the feature space of dAMUSE (MdAMUSE), the number (K) of sampling intervals used per delay in the trajectory matrix, the number Npc of principal components retained after the first step of the GEVD and the half-width (σ) of the Gaussian filter used in the algorithms GEVD-MP and dAMUSE Parameter NIC MdAMUSE Npc Nw(GEV D) P11 256 3 148 49 Parameter Nw(dAMUSE) σ SNRGEV D−MP SNRdAMUSE P11 46 0.3 18, 6 dB 22, 9 dB problem of removing the water artifact from NMR spectra and denoising the resulting reconstructed spectra. Although all three algorithms achieved good results concerning the final SNR, in case of the NMR spectra it turned out that KPCA seems to alter the spectral shapes while LICA and dAMUSE do not. At least with protein NMR spectra it is crucial that denoising algorithms do not alter integrated peak intensities in the spectra as the latter form the bases for the structure elucidation process. In future we have to further investigate the dependence of the proposed algorithms on the situation at hand. Thereby it will be crucial to identify data models for which the each one of the proposed denoising techniques works best and to find good measures of how such models suit the given data. 6 Acknowledgements This research has been supported by the BMBF (project ModKog) and the DFG (GRK 638: Nonlinearity and Nonequilibrium in Condensed Matter). We are grateful to W. Gronwald and H. R. Kalbitzer for providing the NMR spectrum of P11 and helpful discussions. References [1] Adel Belouchrani, Karim Abed-Meraim, Jean-François Cardoso, and Eric Moulines. A blind source separation technique using secon-order statistics. IEEE Transactions on Signal Processing, 45(2):434–444, 1997. [2] Andrzej Cichocki and Shun-Ichi Amari. Adaptive Blind Signal and Image Processing. Wiley, 2002. 28
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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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298 BIBLIOGRAPHY Barber, C., Dobkin
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