Scatter Matrices and Independent Component Analysis

  • Hannu Oja University of Tampere, Finland
  • Seija Sirkiä University of Jyväskylä, Finland
  • Jan Eriksson Helsinki University of Technology, Finland

Abstract

In the independent component analysis (ICA) it is assumed that the components of the multivariate independent and identically distributed observations are linear transformations of latent independent components. The problem then is to find the (linear) transformation which transforms the observations back to independent components. In the paper the ICA is discussed and it is shown that, under some mild assumptions, two scatter matrices may
be used together to find the independent components. The scatter matrices must then have the so called independence property. The theory is illustrated by examples.

References

Cardoso, J. F. (1989). Source separation using higher order moments. In Proceedings of IEEE International Conference on Acustics, Speech and Signal Processing (p. 2109-2112). Glasgow.

Cardoso, J. F. (1999). High-order contrasts for independent component analysis. Neural Computation, 11, 157-192.

Cardoso, J. F., and Souloumiac, A. (1993). Blind beamforming for non gaussian signals. IEE Proceedings-F, 140(6), 362-370.

Cichocki, A., and Amari, S. (2002). Adaptive blind signal and image processing: Learning algorithms and applications. J. Wiley.

Comon, P. (1994). Independent component analysis, a new concept? Signal Processing, 36(3), 287-314.

Croux, C., Ollila, E., and Oja, H. (2002). Sign and rank covariance matrices: Statistical properties and application to principal component analysis. In Y. Dodge (Ed.), Statistical data analysis based on l1-norm and related methods (p. 257-269). Basel:
Birkhäuser.

Davies, L. (1987). Asymptotic behavior of S-estimates of multivariate location parameters and dispersion matrices. Annals of Statistics, 15, 1269-1292.

Dümbgen, L. (1998). On tyler’s M-functional of scatter in high dimension. Annals of Institute of Statistical Mathematics, 50, 471-491.

Eriksson, J., and Koivunen, V. (2004). Identifiability, separability and uniqueness of linear ICA models. IEEE Signal Processing Letters, 11, 601–604.

Flury, B., and Riedwyl, H. (1988). Multivariate statistics. a practical approach. London: Chapman and Hall.

Hyvärinen, A., Karhunen, J., and Oja, E. (2001). Independent component analysis. J. Wiley.

Kankainen, A., Taskinen, S., and Oja, H. (2005). Tests of multinormality based on location vectors and scatter matrices. submitted.

Locantore, N., Marron, J. S., Simpson, D. G., Tripoli, N., Zhang, J. T., and Kohen, K. L. (1999). Robust principal components for functional data. Test, 8, 1-73.

Lopuhaä, H. P. (1989). On the relation between S-estimators and M-estimators of multivariate location and scatter. Annals of Statistics, 17, 1662-1683.

Lopuhaä, H. P. (1991). Multivariate τ-estimators of location and scatter. Canadian Journal of Statistics, 19, 310-321.

Marden, J. (1999). Some robust estimates of principal components. Statistics and Probability Letters, 43, 349-359.

Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519-530.

Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. Annals of Statistics, 4, 51-67.

Pearson, K. (1895). Contributions to the mathematical theory of evolution ii. skew variation in homogeneous material. Philosophical Transactions of the Royal Society of London, 186, 343-414.

Pham, D. T., and Garat, P. (1997). Blind separation of mixture of independent sources through quasi-maximum likelihood approach. IEEE Transactions of Signal Processing, 45(7), 1712-1725.

Samarov, A., and Tsybakov, A. (2004). Nonparametric independent component analysis. Bernoulli, 10, 565-582.

Tyler, D. E. (1987). A distribution-free m-estimator of multivariate scatter. Annalc of Statistics, 15, 234-251.

Tyler, D. E. (2002). High breakdown point multivariate estimation. Estadística, 54, 213-247.

Visuri, S., Koivunen, V., and Oja, H. (2000). Sign and rank covariance matrices. Journal of Statistical Planning and Inference, 91, 557-575.
Published
2016-04-03
How to Cite
Oja, H., Sirkiä, S., & Eriksson, J. (2016). Scatter Matrices and Independent Component Analysis. Austrian Journal of Statistics, 35(2&3), 175–189. https://doi.org/https://doi.org/10.17713/ajs.v35i2&3.364
Section
Articles