Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 117

alpha=0, kurt=0.7306 alpha=10, kurt=0.93051

alpha=20, kurt=1.1106 alpha=30, kurt=1.1866

alpha=40, kurt=1.1227 alpha=50, kurt=0.94904

alpha=60, kurt=0.74824 alpha=70, kurt=0.61611

alpha=80, kurt=0.61603 alpha=90, kurt=0.74861

Figure 4.5
Kurtosis maximization: histograms. Plotted are the random variable w z for
vectors w =(cos( α )sin( α )) and angle α between 0 and 90 degrees. The whitened
mixtures z are shown in figure 4.4. Note that the projection is maximally
non-Gaussian at the demixing angle 30 degrees; the absolute kurtosis is also
maximal there(see also figure 4.4).

Under the assumption of unit variance, E ( y 2 ) = 1, we get

kurt( y )= E ( y 4 )

− 3 ,

which is a sort of normalized fourth-order moment.
Let us consider a two-dimensional example first. Let
q = A b = q q 2

.

Then

y = b x = q s = q 1 s 1 + q 2 s 2 .

Using linearity of kurtosis if the random variables are independent