Abstract
Independent component analysis (ICA) solves the blind source separation problem by evaluating higher-order statistics, e.g. by estimating fourth-order moments. While estimation errors of the kurtosis can be shown to asymptotically decay with sample size according to a square-root law, they are subject to two further effects for finite samples. Firstly, errors in the estimation of kurtosis increase with the deviation from Gaussianity. Secondly, errors in kurtosis-based ICA algorithms increase when approaching the Gaussian case. These considerations allow us to derive a strict lower bound for the sample size to achieve a given separation quality, which we study analytically for a specific family of distributions and a particular algorithm (fastICA). We further provide results from simulations that support the relevance of the analytical results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, Mineola (1972)
Amari, S., Cichocki, A., Yang, H.H.: A new learning algorithm for blind signal separation. Advances in Neural Information Processing Systems 8, 757–763 (1996)
Bai, J., Ng, S.: Tests for skewness, kurtosis, and normality for time series data. J. Business & Economic Statistics 23(1), 49–60 (2005)
Bethge, M.: Factorial coding of natural images: how effective are linear models in removing higher-order dependencies? J. Opt. Soc. Am. 23(6), 1253–1268 (2006)
Comon, P.: Independent component analysis - a new concept? Signal Processing 36, 287–314 (1994)
Dodel, S., Herrmann, J.M., Geisel, T.: Comparison of temporal and spatial ica in fmri data analysis. In: ICA 2000. Proc. Second Int. Workshop on Independent Component Analysis and Blind Signal Separation, vol. 13, pp. 543–547 (2000)
Farvardin, N., Modestino, J.: Optimum quantizer performance for a class of non-gaussian memoryless sources. IEEE Trans. Inf. Theory IT-30, 485–497 (1984)
Hyvärinen, A., Karhunen, J., Oja, E.: Independent component analysis. John Wiley & Sons, Chichester (2001)
Hyvärinen, A., Oja, E.: A fast fixed-point algorithm for independent component analysis. Neural Computation 9, 1483–1492 (1997)
Jutten, C., Hérault, J., Comon, P., Sorouchiary, E.: Blind separation of sources, parts I, II and III. Signal Processing 24, 1–29 (1991)
Koldovský, Z., Tichavský, P., Oja, E.: Efficient variant of algorithm fastICA for independent component analysis attaining the Cramér-Rao lower bound. IEEE Transactions on Neural Networks 17(5), 1265–1277 (2006)
MacCallum, R.C., Browne, M.W., Sugawara, H.M.: Power analysis and determination of sample size for covariance structure models. Psychological Methods 1(2), 130–149 (1996)
Molgedey, L., Schuster, G.: Separation of a mixture of independent signals using time delayed correlations. Physical Review Letters 72(23), 3634–3637 (1994)
Regalia, P.A., Kofidis, E.: Monotonic convergence of fixed-point algorithms for ICA. IEEE Transactions on Neural Networks 14(4), 943–949 (2003)
Theis, F.J.: A new concept for separability problems in blind source separation. Neural Computation 16, 1827–1850 (2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Herrmann, J.M., Theis, F.J. (2007). Statistical Analysis of Sample-Size Effects in ICA. In: Yin, H., Tino, P., Corchado, E., Byrne, W., Yao, X. (eds) Intelligent Data Engineering and Automated Learning - IDEAL 2007. IDEAL 2007. Lecture Notes in Computer Science, vol 4881. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77226-2_43
Download citation
DOI: https://doi.org/10.1007/978-3-540-77226-2_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77225-5
Online ISBN: 978-3-540-77226-2
eBook Packages: Computer ScienceComputer Science (R0)