Abstract
Independent Component Analysis (ICA) designed for complete bases is used in a variety of applications with great success, despite the often questionable assumption of having N sensors and M sources with N≥M. In this article, we assume a source model with more sources than sensors (M>N), only L<N of which are assumed to have a non-Gaussian distribution. We argue that this is a realistic source model for a variety of applications, and prove that for ICA algorithms designed for complete bases (i.e., algorithms assuming N=M) based on mutual information the mixture coefficients of the L non-Gaussian sources can be reconstructed in spite of the overcomplete mixture model. Further, it is shown that the reconstructed temporal activity of non-Gaussian sources is arbitrarily mixed with Gaussian sources. To obtain estimates of the temporal activity of the non-Gaussian sources, we use the correctly reconstructed mixture coefficients in conjunction with linearly constrained minimum variance spatial filtering. This results in estimates of the non-Gaussian sources minimizing the variance of the interference of other sources. The approach is applied to the denoising of Event Related Fields recorded by MEG, and it is shown that it performs superiorly to ordinary ICA.
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References
R. Boscolo, H. Pan, and V. P. Roychowdhury, “Beyond Comon’s Identifiability Theorem for Independent Component Analysis,” Artificial Neural Networks—ICANN 2002 Lecture Notes in Computer Science, vol. 2415, 2002, pp. 1119–1124.
A. W. Bowman and A. Azzalini, Applied Smoothing Techniques for Data Analysis, Oxford University Press, 1997.
J. F. Cardoso, “Blind Signal Separation: Statistical Principles,” Proc. IEEE, vol. 86, 1998, pp. 2009–2025.
J. F. Cardoso and A. Souloumiac, “Blind Beamforming for Non–Gaussian Signals,” IEE Proc. F, vol. 140, 1993, pp. 362–370.
P. Comon,“ Independent Component Analysis, A New Concept?,” Signal Process., vol. 36, 1994, pp. 287–314.
A. Delorme and S. Makeig, “EEGLAB: An Open Source Toolbox for Analysis of Single-trial EEG Dynamics,” J. Neurosci. Methods, vol. 134, 2004, pp. 9–21.
A. Edelman, T. A. Arias, and S. T. Smith, “The Geometry of Algorithms with Orthogonality Constraints,” SIAM J. Matrix Anal. Appl., vol. 20, no. 2, 1998, pp. 303–353.
J. Eriksson and V. Koivunen, “Identifiability and Separability of Linear ICA Models Revisited,” in 4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA 2003), 2003, pp. 23–27.
J. Gross and A. A. Ioannides, “Linear Transformations of Data Space in MEG,” Phys. Med. Biol., vol. 44, 1999, pp. 2081–2097.
M. Grosse-Wentrup and M. Buss, “Subspace Identification through Blind Source Separation,” IEEE Signal Process. Lett., vol. 13, no. 2, 2006, pp. 100–103.
C. J. James and O. J. Gibson, “Temporally Constrained ICA: An Application to Artifact Rejection in Electromagnetic Brain Signal Analysis,” IEEE Trans. Biomed. Eng., vol. 50, 2003, pp. 1108–1116.
T. W. Lee, M. Girolami, and T. J. Sejnowski, “Independent Component Analysis Using an Extended Infomax Algorithm for Mixed Subgaussian and Supergaussian Sources,” Neural Comput., vol. 11, 1999, pp. 417–441.
T. W. Lee, M. S. Lewicki, M. Girolami, and T. J. Sejnowski, “Blind Source Separation of More Sources Than Mixtures Using Overcomplete Representations,” IEEE Signal Process. Lett., vol. 6, 1999, pp. 87–90.
S. Makeig, M. Westerfield, T. P. Jung, S. Enghoff, J. Townsend, E. Courchesne, and T. J. Sejnowski, “Dynamic Brain Sources of Visual Evoked Responses,” Science, vol. 295, 2002, pp. 690–694.
M. J. McKeown, S. Makeig, G. G. Brown, T. P. Jung, S. S. Kindermann, A. J. Bell, and T. J. Sejnowski, “Analysis of fMRI Data by Blind Separation into Independent Spatial Components,” Hum. Brain Mapp., vol. 6, 1998, pp. 160–188.
S. S. Nagarajan, H. T. Attias, and K. E. Hild II, and K. Sekihara, “A Graphical Model for Estimating Stimulus-evoked Brain Responses from Magnetoencephalography Data with Large Background Brain Activity,” NeuroImage, vol. 30, 2006, pp. 400–416.
L. C. Parra and C. V. Alvino, “Geometric Source Separation: Merging Convolutive Source Separation with Geometric Beamforming,” IEEE Trans. Speech Audio Process., vol. 10, 2002, pp. 352–362.
H. Saruwatari, T. Kawamura, T. Nishikawa, A. Lee, and K. Shikano, “Blind Source Separation Based on a Fast-convergence Algorithm Combining ICA and Beamforming,” IEEE Trans. Audio, Speech and Language Processing, vol. 14, 2006, pp. 666—678.
B. D. van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of Brain Electrical Activity via Linearly Constrained Minimum Variance Spatial Filtering,” IEEE Trans. Biomed. Eng., vol. 44, 1997, pp. 867—880.
L. Zhukov, D. Weinstein, and C. Johnson, “Independent Component Analysis for EEG Source Localization—An Algorithm that Reduces the Complexity of Localizing Multiple Neural Sources,” IEEE Eng. Med. Biol. Mag., vol. 19, 2000, pp. 87–96.
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Grosse-Wentrup, M., Buss, M. Overcomplete Independent Component Analysis via Linearly Constrained Minimum Variance Spatial Filtering. J VLSI Sign Process Syst Sign Im 48, 161–171 (2007). https://doi.org/10.1007/s11265-006-0028-3
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DOI: https://doi.org/10.1007/s11265-006-0028-3