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localization performance (Greenblatt et al.
2005
). On the other hand, when we
estimate current density by EEG inverse solutions, we estimate current
flowing in
the three orthogonal directions (hence, the
filter is given by three vectors, not one as
here), resulting in a considerable loss of spatial resolution. Linearity allows
switching back from the source space into the sensor space. Substituting (
8.2
) into
(
8.1
) and dropping the noise term in the latter yield
BSS
filtering
v
0
ð
t
Þ
¼
ARsð
t
Þ
¼
ARBvð
t
Þ
;
where
is a diagonal matrix with
mth
diagonal element equal to 1 if the
mth
component is to be retained and equal to 0 if it is to be removed. BSS
R
filtering is
common practice to remove artifacts from the EEG data.
8.5
An Approach for Solving the BSS Problem Based
on Second-Order Statistics Only
It has been known for a long time that in general, the BSS problem cannot be solved
for sources that are Gaussian, independent, and identically distributed (iid) (Dar-
mois
1953
). EEG data are clearly non-iid; thus, we may proceed assuming that
source components are all pair-wise uncorrelated and that either (a) within each
source component, the successive samples are temporally correlated
1
(Molgedey
and Schuster
1994
; Belouchrani et al.
1997
) or (b) samples in successive time
intervals do not have the same statistical distribution, i.e., they are non-stationary
(Matsuoka et al.
1995
; Souloumiac
1995
; Pham and Cardoso
2001
). Provided that
source components have non
-
proportional spectra or the time courses of their
variance (energy) vary differently
, one can show that SOS are suf
cient for solving
the source separation problem (Yeredor
2010
). Since SOS are suf
cient, the method
is able to separate also Gaussian sources, contrary to another well-known BSS
approach named
independent component analysis
(ICA: Comon and Jutten
2010
).
If these assumptions are ful
ed uniquely;
thus, source can be recovered
regardless of the true mixing process
(uniform
performance property: see, e.g., Cardoso
1998
) and regardless of the distribution of
sources, which is a remarkable theoretical advantage. The fundamental question is
therefore whether or not the above assumptions
lled, the separating matrix can be identi
t EEG, ERS/ERD, and ERP data.
Sources are uncorrelated: This assumption may be conceived as a working
assumption. In practice, the BSS output is never exactly uncorrelated, but just as
uncorrelated as possible. What we try to estimate is the coherent signal of large
cortical patches, enough separated in space one from the other. BSS may be
conceived as a spatial
filter minimizing the correlation of the observed mixtures
and recovering the signal emitted from the most energetic and uncorrelated
1
Such processes are called
colored
, in opposition to iid processes, which are called
white
.
