CHAPTER 2. DIMENSIONALITY REDUCTION 15
observed mixtures x without knowing the mixing parameters a and having very little knowl-
edge about the nature of the original signals. Independent component analysis (ICA) is one
technique to approach this problem.
Note that ICA is formulated as a linear model – the original source signals are combined
in a linear fashion to obtain the observed signals. We can rewrite the model such that the
observed signals are the “inputs” and we use weights w
11
, w
12
, w
21
, and w
22
to create new
outputs y
1
(t) and y
2
(t):
y
1
(t) = w
11
x
1
(t) + w
12
x
2
(t)
y
2
(t) = w
21
x
1
(t) + w
22
x
2
(t)
(2.2)
We can also write the previous two equations in matrix form:
Y = W X
X = A S
(2.3)
Through linear algebra
1
, it can be shown that if we set the weight matrix W to the inverse
of the original weight matrix A, then Y=S. This means that there exists a set of weights W
that will transform the observed mixtures X to the source signals S. The question now is
how to find the appropriate set of weights W. In the examples below, we will first show some
demonstrations of ICA with a function that will automatically produce the independent
components. We will then give some example code that will illustrate the method behind
ICA.
2.1.3 Matlab examples
There are many different toolboxes for ICA. We will use the fastica toolbox because it
is simple and fast. It is available at http://research.ics.aalto.fi/ica/fastica/code/
dlcode.shtml. Note that when you need to apply ICA to specialized tasks such as EEG
artifact removal or resting-state analysis, there are more suitable Matlab packages for these
tasks.
To get started, let’s give a few demonstrations of ICA in the context of blind source
separation. Listing 2.1 shows Matlab code that produces the visual output show in Figure
2.5. In this example, there are two input signals corresponding to sine waves of a different
wave lengths. These two signals are weighted to produce two mixtures x
1
and x
2
. These are
provided as inputs to the function fastica which produces the independent components y
1
and y
2
. Note how the reconstructed signals are not quite the same as the origal signals. The
first independent component (y
1
) is similar to the second source signal (s
2
) and the second
independent component (y
2
) is similar to the first source signal (s
1
), but in both cases, the
amplitudes are different. This highlights two properties of ICA: the original ordering and
variances of the source signals cannot be recovered.
1
Note that if W = A
−1
, we can write A
−1
X = A
−1
AS = S
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