Independent Component Analysis is a technique that recovers a set of independent signals from a set of measured signals. It is assumed that each measured signal is a linear combination of each of the independent signals, and that there are an equal number of measured signals and independent signals. The following diagrams show an example of two independent signals and two measured signals that are linear combinations of the independent signals.

These graphs show the measured signals X1 and X2 on the top, and the independent signals S1 and S2 on the bottom. Graphs generated with the MATLAB commands:

x = 0:pi/100:10*pi;
s1 = sin(0.2*x);
s2 = sin(2*x);
x1 = 0.1*s1+0.9*s2;
x2 = 0.9*s1+0.1*s2;

We will refer to each of the original independent signals as Si and to each of the linearly combined mixed signals a Xi. X is a column vector of n measured signals.

Each measured signal can be expressed as a linear combination of the original independent signals:

Xi = a1S1+ a2S2 + ... + anSn

We can express the entire system of n measured signals as:

X = AS

where each row of X is a set of readings for each signal Xi; each row of S is an original signal Si; and A is an nn mixing matrix that generates X from S.

The goal of ICA is, given X, find S and A.

At first glance, this problem seems severely under constrained. However, ICA is looking for specific features in S that allows

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