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Preprocessing of the data
Some ICA algorithms require a preliminary sphering or whitening of the
data ,
and even those algorithms that do not necessarily need
sphering, often converge better with sphered data.
(Recall that the data has also been assumed to be centered, i.e., made
zeromean.)
Sphering means that the observed variable
of Eq. (11)
is linearly transformed to a variable

(30) 
such that the covariance matrix of
equals unity:
.
This transformation is always possible. Indeed, it can
be accomplished by classical PCA [36,112,110,114]. In
addition to sphering, PCA may allow us to determine the number of
independent components (if m>n): if noise level is low, the energy of
is essentially concentrated on the subspace spanned by the nfirst principal components, with n the number of independent
components in the model (11). Several methods exist for estimating the
number of signals (here, independent components), see
[146,94,144,11]. Thus this reduction of
dimension partially justifies the assumption m=n that was made in
Section 3, and will be retained here.
After sphering we have from (11) and (30):

(31) 
where
is an orthogonal matrix, because
Recall that we assumed that the independent components s_{i} have unit
variance.
We have thus reduced the problem of finding an arbitrary matrix in model (11)
to the simpler problem of finding
an orthogonal matrix .
Once
is found, Eq. (31) is used
to solve the independent components from the observed
by

(32) 
It is also worthwhile to reflect why sphering alone does not
solve the separation problem.
This is because sphering is only defined up to an additional rotation:
if
is a sphering
matrix, then
is also a sphering matrix if and only if
is an orthogonal matrix.
Therefore, we have to find the correct sphering matrix that
equally separates the independent components.
This is done by first finding any sphering matrix ,
and later determining the appropriate
orthogonal transformation from a suitable nonquadratic criterion.
In the following, we shall thus assume in certain sections that the
data is sphered. For
simplicity, the sphered data will be denoted by ,
and the
transformed mixing matrix by ,
as in the definitions of
Section 3. If an algorithm needs preliminary sphering, this
is mentioned in the corresponding section. If no mention of sphering
is made, none is needed.
Next: JuttenHérault algorithm
Up: Algorithms for ICA
Previous: Introduction
Aapo Hyvarinen
19990423