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226 9 Multivariate Statistics

they could still can be dependent, i.e., nonlinear correlation remains. The
independent component analysis (ICA) was developed for the purpose of
investigating such data. It separates mixed signals into independent signals,
which are then nonlinearly uncorrelated. Fast ICA algorithms use a crite-
rion which estimates how gaussian distributed the joint distribution of the
independent components is. The less gaussian this distribution is, the more
independent the individual components are.
According to the model, n independent signals x(t) are linearly mixed in
m measurements.

and we are interested in the source signals s and in the mixing matrix A.
i
We can, for example, imagine that we are on a party and a lot of people
talk independently with others. We hear a mixing of these talks and perhaps
cannot distinguish the single talks. Now we could install some microphones
and use these measurements in order to separate the single conversations.
Hence, this dilemma is also called the cocktail party problem. Its correct
term is blind source separation that is given by

T
where W is the separation matrix in order to reverse the mixing and get
the original signals. Let us consider a mixing of three signals s , s and s
1 2 3

and their separation using PCA and ICA. At first we create three periodic
signals
clear
i = (1:0.01:10 * pi)';
[dummy index] = sort(sin(i));
s1(index,1) = i/31; s1 = s1 - mean(s1);
s2 = abs(cos(1.89*i)); s2 = s2 - mean(s2);
s3 = sin(3.43*i);
subplot(3,2,1), plot(s1), ylabel('s_1'), title('Raw signals')
subplot(3,2,3), plot(s2), ylabel('s_2')
subplot(3,2,5), plot(s3), ylabel('s_3')
Now we mix these signals and add some observational noise. We get a three-
column vector x which corresponds to our measurement (Fig. 9.5).

randn('state',1);