ICA and ICAMM Methods - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 38

with the projection of w on the unit sphere every step and where c ¼
EGw T ð f g EG ð v fg and v is a standardized Gaussian random variable. The
normalization is necessary to project w to keep the variance of w T z constant. The
non-linearity g ðÞ is the derivative of the function G used in the approximation. It
can be chosen from g 1 ð s Þ¼ tanh a 1 ðÞ where 1 a 1 2 ; g 2 ð s Þ¼ s exp s 2 = 2 ð Þ; or

g 3 ð s Þ¼ s 3

[ 20 , 22 ].

2.2.4 TDSEP

Temporal decorrelation source separation (TDSEP) is one of the ICA algorithms
that exploit the time structure of the signals. It is based on the simultaneous
diagonalization of several time-delayed correlation matrices. The approach relies
on second-order statistics by assuming distinctive spectral/temporal characteristics
of the sources [ 34 , 43 , 44 ]. These algorithms have been successfully applied in
biosignal processing given the inherent time structure of the signals and their
capability to separate signals whose amplitude distribution is near Gaussian.
The TDSEP algorithm uses the property that the cross-correlation functions
vanish for mutually independent signals. It assumes that the signals s ð t Þ have
temporal structure (''non delta'' autocorrelation function). All time delayed cor-
relation matrices R s ð s Þ should be diagonal. This knowledge is used to calculate the
unknown mixing matrix A by a simultaneous diagonalization of a set of correlated
matrices R s ð x Þ ¼ x ð t Þ x ð t s Þ T for different choices of s ; where s ; is a lag
constant, s ¼ 1 ; 2 ; 3 ; .... The diagonal elements of these matrices are formed by
the values of the autocorrelation functions and the off-diagonal elements are the
respective cross correlations,

2
4

3
5

u x 1 ; x 1 ð s Þ ...u x 1 ; x n ð s Þ

u x 1 ; x 2 ð s Þ ...u x 2 ; x n ð s Þ

R s ð x Þ ¼

ð 2 : 21 Þ

.

.

.

. .

u x n ; x 1 ð s Þ ...u x n ; x n ð s Þ

where u denotes the correlation function. If the signals were independent over
time, all time-delayed correlation matrices should be diagonal because the cross-
correlations of independent signals vanish.
The contrast consists of finding a matrix B (considering whitening) so that in
addition to making the instantaneous covariances of s ð t Þ¼ Bx ð t Þ go to zero, the
lagged covariances are made zero as well:

¼ 0 ;

E s i ð t Þ s j ð t s Þ for all i ; j ; s with i 6¼ j ð 2 : 22 Þ