Noninvasive Cardiac Signal Analysis Using Data Decomposition Techniques - Modeling in Computational Biology and Biomedicine - page 100

where C mnpq and R ij = R x ] ij denote the fourth-order cumulant and the
covariance of the observation x , respectively. These quantities can easily be
estimated from sufficiently long observation samples. Then it can be shown [ 9 ,
Chap. 3] that

|Ψ KM ( w ) |

is indeed a multiple-input single-output contrast as defined

in Sect. 3.3.3.2 .
In the absence of noise, i.e., if x = Hs holds exactly, where s i are statistically
independent, the fact that the absolute kurtosis can serve as a contrast criterion
accepts an intuitive interpretation based on the Central Limit Theorem. This well-
known theorem in Statistics states that the Gaussianity of a linear mixture increases
as the number of independent variables contributing to the mixture increases. Hence,
looking for the matrix W maximizing the independence among the components of
s = Wx is equivalent to maximizing the non-Gaussianity of every component s i .It
follows that maximizing the absolute kurtosis can be seen, at order 4, as a sensible
criterion to act in the opposite direction of the Central Limit Theorem: reduce the
mixing by decreasing Gaussianity.
When the entries of a random vector z are statistically independent, all its
cumulants γ ijk are null except for i = j = k = , so that the fourth-order cumulant
array will show a diagonal structure. As we have just seen, non-Gaussianity
and independence are closely related, and thus maximizing non-Gaussianity as
described above will implicitly diagonalize the cumulant array of the observations.
Hence, in the same way as PCA finds uncorrelated components by diagonalizing
the observation covariance matrix (Sect. 3.2.2.2 ), ICA looks for independent com-
ponents by diagonalizing the observation cumulant array at orders higher than two.
It is important to stress that if there exist more than one Gaussian source, then
the Gaussian sources cannot be recovered. In fact, as pointed out in Sect. 3.3.3.1 ,
the kurtosis of Gaussian variables is null, and hence cannot be used as an objective
function to estimate this kind of sources. On the other hand, all non-Gaussian
sources can be extracted regardless of the number of Gaussian sources present in
the mixture, as long as the full rank observation assumption holds (Sect. 3.3.1 ).
Before closing this section, note that criterion Ψ KM ( w ) is insensitive to scale,
which makes sense because the actual amplitude of the original sources cannot
be recovered by resorting solely to their statistical independence; this is the scale
ambiguity noted at the end of Sect. 3.3.1 . In order to avoid divergence or instabilities
of vector w during the optimization process, it is hence also desirable to fix its norm,
e.g., to impose

w =1 .

3.3.3.4 Extraction of One Source

An efficient algorithm, the so-called RobustICA [ 26 ], can be employed to maximize
the kurtosis contrast. The key to this algorithm lies in the fact that the contrast
criterion ( 3.24 )-( 3.25 ) is a rational function in vector w . As a result, once a search
direction g has been fixed (for instance the gradient), then the global optimum of
the contrast along the search direction can easily be computed. To see this, it is