Information Technology Reference
In-Depth Information
wC
2
w
T
wC
1
w
T
þ k
P
ð
w
Þ
J
RCSP2
ð
w
Þ¼
ð
7
:
5
Þ
with
C
i
¼ð
1
cÞ
C
i
þ c
G
i
ð
7
:
6
Þ
In these equations,
P
(
w
) is the penalty term that encodes the prior knowledge.
This a positive function of the spatial
lter
w
, whose value will increase if
w
does
not satisfy the knowledge encoded. Since the
filters are obtained by maximizing
J
RCSPi
, this means that the numerator (which is positive) must be maximized and the
denominator (which is also positive) must be minimized. Since
P
(
w
) is positive and
part of the denominator, this means that
P
ð
w
Þ
will be minimized as well, hence
enforcing that the spatial
filters
w
satisfy the prior knowledge. Matrix
G
i
is another
way of using prior knowledge, in order to stabilize the estimates of the covariance
matrices
C
i
. If we have any idea about how these covariance matrices should be,
this can be encoded in
G
i
in order to de
ne a new covariance matrix C
i
which is a
mix of the matrix
C
i
estimated on the data and of the prior knowledge
G
i
.Wewill
present below what kind of knowledge can be encoded in
P
(
w
) and
G
i
.
For the penalty term
P
(
w
), a kind of knowledge that can be used is spatial
knowledge. For instance, from a neurophysiological point of view, we know that
neighboring neurons tend to have similar functions, which supports the idea that
neighboring electrodes should measure similar brain signals (if the electrodes are
close enough to each other), notably because of the smearing effect. Thus, neigh-
boring electrodes should have similar contributions in the spatial
filters. In other
words, spatial
filters should be spatially smooth. This can be enforced by using the
following penalty term:
X
Prox
ð
i
;
j
Þð
w
i
w
j
Þ
2
P
ð
w
Þ¼
ð
7
:
7
Þ
i
;
j
ð
w
i
w
j
Þ
2
is the
where
Prox
ð
i
;
j
Þ
measures the proximity of electrodes
i
and
j
, and
weight difference between electrodes
i
and
j
, in the spatial
filter. Thus, if two
electrodes are close to each other and have very different weights, the penalty term
P
(
w
) will be high, which would prevent such solutions to be selected during the
optimization of the CSP (Lotte and Guan
2010b
). Another knowledge that can be
used is that for a given mental task, not all the brain regions are involved and useful.
As such, some electrodes are unlikely to be useful to classify some speci
c mental
tasks. This can be encoded in
P
(
w
) as well:
with Dði;
ð
i
;
j
Þ¼
channel i
''
uselessness''
if i
¼
j
P
ð
w
Þ¼
wDw
T
ð
7
:
8
Þ
0
otherwise
