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Basically, the value of D ( i , i ) is the penalty for the i ith channel. The higher this
penalty, the less likely this channel will have a high contribution in the CSP
filters.
The value of this penalty can be de
ned according to neurophysiological prior
knowledge for instance, large penalties being given to channels unlikely to be
useful and small or no penalty being given to channels that are likely to genuinely
contribute to the
ne the extent of
the penalty from the literature. Another alternative is the use data previously
recorded from other subjects. Indeed, the optimized CSP
filter. However, it may be dif
cult to precisely de
filters already obtained
from previous subject give information about which channels have large contri-
butions on average. The inverse of the average contribution of each channel can be
used as the penalty, hence penalizing channels with small average contribution
(Lotte and Guan 2011 ). Penalty terms are therefore also a nice way to perform
subject-to-subject transfer and re-use information from other subjects. These two
penalties are examples that have proven useful in practice. This usefulness is
notably illustrated in Fig. 7.6 , in which spatial filters obtained with the basic CSP
are rather noisy, with strong contributions from channels not expected from a
neurophysiological point of view. On the contrary, the spatial
filters obtained using
the two RCSP penalties described previously are much cleaner, spatially smoother
and with strong contributions localized in neurophysiologically relevant areas. This
in turns led to higher classi
cation performances, with CSP obtaining 73.1 %
classifi-
cation accuracy versus 78.7 % and 77.6 % for the regularized versions (Lotte
and Guan 2011 ). It should be mentioned, however, that strong contributions from
non-neurophysiologically relevant brain areas in a CSP spatial
filter may be present
to perform noise cancelation, and as such does not mean the spatial
filter is bad per
se (Haufe et al. 2014 ). It should also be mentioned that other interesting penalty
terms have been proposed, in order to deal with known noise sources (Blankertz
et al. 2008a ), non-stationarities (Samek et al. 2012 ) or to perform simultaneous
channel selection (Farquhar et al. 2006 ; Arvaneh et al. 2011 ).
Matrix G i in Eq. 7.6 is another way to add prior knowledge. This matrix can
notably be de
ned as the average covariance matrix obtained from other subjects
who performed the same task. As such it enables to de
ne a good and stable
estimate of the covariance matrices, even if few training EEG data are available for
the target subject. This has been shown to enable us to calibrate BCI system with
2
-
3 times less training data than with the basic CSP, while maintaining classifi-
-
cation performances (Lotte and Guan 2010a ).
Regularizing CSP using a priori knowledge is thus a nice way to deal with some
limitations of CSP such as its sensitivity to overfitting and its non-robustness to
noise. However, these regularized algorithms cannot address the limitation that CSP
only optimizes the use of the spatial information, but not that of the spectral one. In
general, independently of the use of CSP, there are several ways to optimize the use
of the spectral information. Typically, this consists in identifying, in one way or
another, the relevant frequency bands for the current subject and mental tasks
 
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