Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 123

often turns out that the algorithms can be improved by using a measure
that takes even higher order moments into account. Such a measure can,
for example, be the negentropy, defined in definition 3.19 to be

J ( y ):= H ( y gauss ) − H ( y ) .

As seen in section 3.3, the negentropy can indeed be used to measure
deviation from the Gaussian. The smaller the negentropy, the ”less
Gaussian” the random variable.
Algorithm: ( negentropy minimization ) Minimize w J ( w z )on →

S n−1 after whitening.
We can assume that the random variable y has unit variance, so we

get

J ( y ):= 1

2 (1 + log 2 π )

− H ( y ) .

Hence negentropy minimization equals entropy maximization.
In order to see a connection between the two Gaussianity measures
kurtosis and negentropy, Taylor expansion of the negentropy can be used
to get the approximation from equation (3.1):

J ( y )= 1

12 E ( y 3 ) 2 + 1 48 kurt( y ) 2 + ....

If we assume that the third-order moments of y vanish (for exampl,e
for symmetric sources), we see that kurtosis maximization indeed cor-
responds to a first approximation of the more general negentropy mini-
mization.
Other versions of gradient-ascent and fixed-point algorithms can now
easily be developed by using more general approximations [120] of the
negentropy.

Estimation of more than one component

So far we have estimated only one independent component (i.e. one row
of W ). How can the above algorithm be used to estimate the whole
matrix?
By prewhitening W O ( n ), so the rows of the whitened demixing

mapping W are mutually orthogonal. The way to get the whole matrix
W using the above non-Gaussianity maximization is to iteratively search
components as follows.
Algorithm: ( deflation FastICA algorithm ) Perform fixed-point kurto-

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