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In-Depth Information
variance of a signal band-pass
filtered in band
b
is actually the band power of this
signal in band
b
, this means that CSP
filters that lead to optimally
discriminant band-power features since their values would be maximally different
between classes. As such, CSP is particularly useful for BCI based on oscillatory
activity since their most useful features are band-power features. As an example, for
BCI based on motor imagery, EEG signals are typically
finds spatial
filtered in the 8
30 Hz
-
band before being spatially
filtered with CSP (Ramoser et al.
2000
). Indeed, this
band contains both the
rhythms.
Formally, CSP uses the spatial
μ
and
β
filters
w
which extremize the following function:
J
CSP
ð
w
Þ¼
wX
1
X
1
w
T
wX
2
X
2
w
T
¼
wC
1
w
T
ð
7
:
2
Þ
wC
2
w
T
where
T
denotes transpose,
X
i
is the training band-pass
filtered signal matrix for
class
i
(with the samples as columns and the channels as rows), and
C
i
the spatial
covariance matrix from class
i
. In practice, the covariance matrix
C
i
is de
ned as the
average covariance matrix of each trial from class
i
(Blankertz et al.
2008b
). In this
equation,
wX
i
is the spatially
filtered EEG signal from class
i
, and
wX
i
X
i
w
T
is thus
the variance of the spatially
filtered signal, i.e., the band power of the spatially
filtered signal. Therefore, extremizing
J
CSP
ð
w
Þ
, i.e., maximizing and minimizing it,
indeed leads to spatially
filtered signals whose band power is maximally different
J
CSP
ð
w
Þ
between classes.
happens to be a Rayleigh quotient. Therefore, extremizing
it can be solved by generalized eigenvalue decomposition (GEVD). The spatial
are thus the eigenvectors corre-
sponding to the largest and lowest eigenvalues, respectively, of the GEVD of
matrices
C
1
and
C
2
. Typically, six
filters
w
that maximize or minimize
J
CSP
ð
w
Þ
filters (i.e., three pairs), corresponding to the
three largest and three lowest eigenvalues are used. Once these
filters obtained, a
CSP feature
f
is de
ned as follows:
f
¼
log
ð
wXX
T
w
T
Þ¼
log
ð
wCw
T
Þ¼
log
ð
var
ð
wX
ÞÞ
ð
7
:
3
Þ
i.e., the features used are simply the band power of the spatially
filtered signals.
CSP requires more channels than
fixed spatial
filters such as Bipolar or Laplacian,
however in practice, it usually leads to signi
cation perfor-
mances (Ramoser et al.
2000
). The use of CSP is illustrated in Fig.
7.5
. In this
cantly higher classi
filtered with CSP clearly show difference in variance
(i.e., in band power) between the two classes, hence ensuring high classi
figure, the signals spatially
cation
performances.
The CSP algorithm has numerous advantages: First, it leads to high classi
cation
performances. CSP is also versatile, since it works for any ERD/ERS BCI. Finally,
it is computationally ef
cient and simple to implement. Altogether this makes CSP
one of the most popular and ef
cient approach for BCI based on oscillatory activity
(Blankertz et al.
2008b
).
