Abstract
Oja, Sirkiä, and Eriksson (2006) and Ollila, Oja, and Koivunen (2007) showed that, under general assumptions, any two scatter matrices with the so called independent components property can be used to estimate the unmixing matrix for the independent component analysis (ICA). The method is a generalization of Cardoso’s (Cardoso, 1989) FOBI estimate which uses the regular covariance matrix and a scatter matrix based on fourth moments. Different choices of the two scatter matrices are compared in a simulation study. Based on the study, we recommend always the use of two robust scatter matrices. For possible asymmetric independent components, symmetrized versions of the scatter matrix estimates should be used.
Highlights
Let x1, x2, . . . , xn be a random sample from a p-variate distribution, and write X = x1 x2 . . . xn for the p × n data matrix
For any diagonal matrix D and for any permutation matrix P . (A permutation matrix P is obtained from identity matrix Ip by permuting its rows.) If Z has independent components, the components of Z∗ = DP Z are independent
We study the behavior of the new estimates Bwith different choices for S1 and S2
Summary
Zn are independent and identically distributed latent random vectors having independent components and A is a full-rank p × p mixing matrix. This model is called the independent component (IC) model. The problem in the so called independent component analysis (ICA) is to find an unmixing matrix B such that Bxi has independent components. A different solution to the ICA problem, called FOBI, was given by Cardoso (1989): After whitening the data as above (stage 1), an orthogonal matrix U is found as the matrix of eigenvectors of a kurtosis matrix (matrix of fourth moments; this will be discussed later).
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