Multiple channels (multivariate) signals - Practical Biomedical Signal Analysis Using MATLAB - page 98

3.6.2.2 Estimation

Finding the independent components can be considered in the light of the central
limit theorem as finding components of least gaussian distributions. To comprehend
this approach let us follow the heuristic given by [Hyvarinen and Oja, 2000]. For
simplicity, let us assume that the sought independent components have identical dis-
tributions. Let us define y

w T x . Please note, that if w T is one of the columns of
the matrix D − 1 ,then y is one of the sought components. With the change of vari-
ables z

=

z T s . This exposes the fact that y
is a linear combination of the components s i with the weights given by z i .Itstems
from central limit theorem that the sum of independent random variables has more
gaussian character than each of the variables alone. The linear combination becomes
least gaussian when z has only one nonzero element. In that case y is proportional
to s i . Therefore the problem of estimation of the ICA model can be formulated as a
problem of finding w which maximizes the nongaussianity of y

=

D T w we can write y

=

w T x

=

w T Ds

=

w T x . Maximizing
nongaussianity of y gives one independent component, corresponding to one of the
2 n maxima 4 in the optimization landscape. In order to find all independent compo-
nents we need to find all the maxima. Since the components are uncorrelated the
search for consecutive maxima can be constrained to the subspace orthogonal to the
one already analyzed.

=

3.6.2.3 Computation

The intuitive heuristics of maximum nongaussianity can be used to derive differ-
ent functions whose optimization enables the estimation of the ICA model; such a
function may be, e.g., kurtosis. Alternatively, one may use more classical notions
like maximum likelihood estimation or minimization of mutual information to esti-
mate ICA. All these approaches are approximatively equivalent [Hyvarinen and Oja,
2000].
There are several tools for performing ICA in MATLAB. The EEGLAB tool-
box [Makeig, 2000] offers a convenient way to try different algorithms: runica
[Makeig et al., 1996], jader [Cardoso, 1999] and fastica [Hyvarinen, 1999]. The
runica and jader algorithms are a part of the default EEGLAB distribution. To use
the fastica algorithm, one must install the fastica toolbox ( http://www.cis.hut.
fi/projects/ica/fastica/ ) and include it in the MATLAB path. In general, the
physiological significance of any differences in the results of different algorithms (or
of different parameter choices in the various algorithms) have not been thoroughly
tested for physiological signals. Applied to simulated, relatively low dimensional
data sets for which all the assumptions of ICA are exactly fulfilled, all three algo-
rithms return near-equivalent components.
Very important note: As a general rule, finding N stable components (from N -
channel data) typically requires more than kN 2 data sample points (at each channel),
where N 2

is the number of elements in the unmixing matrix that ICA is trying to

4 Corresponding to s i

s i . and −