Biomedical Engineering Reference
In-Depth Information
number of quantitative descriptors. These descriptors, or factors can be conceptual-
ized as basic waveforms produced by hypothetical signal generators which, mixed in
correct proportions, would reproduce the original waveforms in the set. FA was used
to describe the evoked potentials [John and Thatcher, 1977]. FA can be performed in
MATLAB using function
factoran
from the Statistical Toolbox.
3.6.2 Independent components analysis (ICA)
3.6.2.1 Definition
Independent component analysis (ICA) is a statistical signal processing method
that decomposes a multichannel signal into components that are statistically inde-
pendent. In simple words two components
s
1
and
s
2
are independent when informa-
tion of the value of
s
1
does not give any information about the value of
s
2
.TheICA
can be represented by a simple generative model:
x
=
Ds
(3.51)
x
1
x
2
x
n
where the
x
=
{
,
,...,
}
is the measured
n
channel signal,
D
is the mixing
s
1
s
2
s
n
matrix, and
s
is the activity of
n
sources. The main assumption
about
s
is that the
s
i
are statistically independent. To be able to estimate the model
we must also assume that the independent components have
nongaussian
distribution
[Hyvarinen and Oja, 2000].
The model implies the following:
=
{
,
,...,
}
•
The signal is a linear mixture of the activities of the sources
•
The signals due to each source are independent
•
The process of mixing sources and the sources themselves are stationary
•
The energies (variances) of the independent components cannot be determined
unequivocally.
3
The natural choice to solve this ambiguity is to fix the magni-
tude of the independent components so that they have unit variance:
E
s
i
[
]=
1.
•
The order of the components is arbitrary. If we reorder both in the same way:
the components in
s
, and the columns in
D
, we obtain exactly the same signal
x
.
The main computational issue in ICA is the estimation of the mixing matrix
D
.Once
it is known, the independent components can be obtained by:
D
−
1
x
s
=
(3.52)
3
This is because the multiplication of the amplitude of the
i
th
source can be obtained either by multiplica-
tion of
s
i
or by multiplication of the
i
th
column of matrix
D
.
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