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Two classical methods employed for measuring nongaussianity in ICA are
kurtosis and neguentropy. The kurtosis (fourth-order cumulant) of a random var-
iable u is defined by k ð u Þ¼ E ½ u 4 3E 2 ½ u 2 : It is zero for Gaussian random
variables and non zero for non Gaussian distributions. Random variables with
negative kurtosis are called sub-gaussian or platykurtic, (e.g., the uniform random
variable); and those with positive kurtosis are called super-gaussian or leptokurtic
(e.g., the Laplacian random variable). Thus, functions such as P i
j k s ðÞj and
P i
j k 2 s ðÞ are appropriate contrasts. The gradient algorithm associated with the
absolute value of the kurtosis is:
h
i
E zw T z
3
DWa sign k w T z
ð 2 : 17 Þ
with the projection of W on the unit sphere every step, i.e., it is normalized: w
k w k .
This algorithm finds one component at a time, working with a whitened version of
the mixed sources, z ¼ Vx by finding a column vector W that maximizes the
module of the kurtosis of s ¼ w T z.
The FastIca algorithm uses estimates of neguentropy based on the maximum
entropy principle, which requires the use of appropriate nonlinearities for the
learning rule of the neural network [ 20 , 22 ]. Separation is performed by the
minimization of the neguentropy of the mixture in order to obtain uncorrelated and
independent sources whose amplitude distributions are as non Gaussian as pos-
sible. The non Gaussianity is measured with the differential entropy j, called
neguentropy [ 17 ], which is defined as the difference between the entropy of a
Gaussian random variable u gauss and the differential entropy of a random variable
u, which are both variables of the same correlation (and covariance) matrix
H ð u Þ ð 2 : 18 Þ
where the differential entropy H is defined by H ð u Þ¼ R f ð u Þ log f ð u Þ du : Since
Gaussian random variables have the largest entropy H among all random variables
having equal variance, maximizing J ð u Þ leads to the separation of independent
source signals.
The use of neguentropy has an advantage that is well justified by statistical
theory. However, entropy estimation is computationally very difficult. Thus,
several methods of approximation have been proposed [ 5 ]. One successful
approximation consists of using a nonquadratic function G, which becomes
J ð u Þ¼ Hu gauss
2
J ð u Þ a EG ð u fg EG ð v fg
½
ð 2 : 19 Þ
For optimization, the following algorithm can be obtained
Dw acE zg w T z
ð 2 : 20 Þ
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