Information Technology Reference
In-Depth Information
and
N
c
p
k
X
i
2
C
k
ð
x
i
x
k
Þð
x
i
x
k
Þ
T
S
w
¼
ð
7
:
11
Þ
k
¼
1
In these equations,
S
b
is the between-class variance,
S
w
the within-class variance,
N
c
is the number of classes,
x
i
is the
i
ith feature vector,
v
is the average of all vectors
v
,
C
k
is the
k
th class, and
p
k
the probability of class
k
.
This criterion is widely used in machine learning in general (Duda et al.
2001
)
and can be used to
filters such that the resulting features maximize this
criterion and thus the discriminability between the classes. This is what the Fisher
spatial
find spatial
filtered EEG time
course (i.e., the feature vector) is maximally different between classes, according to
the Fisher criterion. This is achieved by replacing
x
i
(the feature vector) by
wX
i
(i.e.,
the spatially
filter does. It
finds the spatial
filters such that the spatially
filtered signal) in Eqs.
7.10
and
7.11
. This gives an objective function
J
ð
w
Þ¼
wS
b
w
T
of the form
wS
w
w
T
, which, like the CSP algorithm, can be solved by GEVD.
This has been showed to be very ef
cient in practice (Hoffmann et al.
2006
).
Another option, that has also proved very ef
cient in practice, is the xDAWN
spatial
filter (Rivet et al.
2009
). This spatial
filter, also dedicated to ERP classifi-
-
cation, uses a different criterion from that of the Fisher spatial
filter. xDAWN aims
at maximizing the signal-to-signal plus noise ratio. Informally, this means that
xDAWN aims at enhancing the ERP response, at making the ERP more visible in
the middle of the noise. Formally, xDAWN
finds spatial
filters that maximize the
following objective function:
J
xDAWN
¼
wADD
T
A
T
w
T
wXX
T
w
T
ð
7
:
12
Þ
where
A
is the time course of the ERP response to detect for each channel (esti-
mated from data, usually using a least square estimate) and
D
is a matrix containing
the positions of target stimuli that should evoke the ERP. In this equation, the
numerator represents the signal, i.e., the relevant information we want to enhance.
Indeed,
wADD
T
A
T
w
T
is the power of the time course of the ERP responses after
wXX
T
w
T
spatial
filtering. On the contrary, in the denominator,
is the variance of all
EEG signals after spatial
filtering. Thus, it contains both the signal (the ERP) plus
the noise. Therefore, maximizing
J
xDAWN
actually maximizes the signal, i.e., it
enhances the ERP response, and simultaneously minimizes the signal plus the
noise, i.e., it makes the noise as small as possible (Rivet et al.
2009
). This has
indeed been shown to lead to much better ERP classi
cation performance.
filters have proven to be useful for ERP-based BCI (in par-
ticular for P300-based BCI), especially when little training data are available. From a
theoretical point of view, this was to be expected. Actually, contrary to CSP and band
power which extract nonlinear features (the power of the signal is a quadratic
In practice, spatial
