Biomedical Engineering Reference
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containing source estimates. Ideally, the global system matrix
G
=
HA
between the
original sources
S
and their estimates
Y
will be a permuted scaled identity matrix,
as it can be proven that the order and the original amplitude of the sources cannot be
recovered [4].
In almost all BSS methods, the matrix
H
is obtained as a product of two statistically
based linear transforms:
H
=
JW
with
-
W
performing data orthogonalization: whitening/sphering,
-
J
performing data rotation : independence maximization via higher-order statistics
(HOS) or joint decorrelation of several time (frequency) intervals
The first step (data decorrelation) can be seen as an initialization for the second step. In
theory any orthogonalization technique can be used to initialize the second step but in
this paper we will focus on two popular decorrelation techniques: whitening (classical
solution) and sphering (assumed to be more biologically plausible [13]).
BSS Initialization: Whitening/Sphering
Whitening
In general EEG signals
X
are correlated so their covariance
will not be
a diagonal matrix and their variances will not be normalized. Data whitening means
projection in the eigenspace and normalisation of variances. The whitening transform
can be computed from the eigen-decomposition of the data covariance matrix
Σ
Σ
=
T
:
ΦΛΦ
Λ
−
2
Φ
T
X
w
=
X
,
(2)
where
are the eigenvalues and the eigenvectors matrices respectively. After
(2), the signals are orthogonal and with unit variances (Figure 2(c)).
Λ
and
Φ
Sphering.
completes whitening by rotating data back to the coordinate system defined
by principal components of the original data [14]. In other words, sphered data are
turned as close as possible to the observed data (Figure 2(d)):
ΦΛ
−
2
Φ
T
X
sph
=
X
.
(3)
(a) Original sources
(b) Mixed data
(c) Whitened data
(d) Sphered data
Fig. 2.
Example of different decorrelation approaches for two signals
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