Biomedical Engineering Reference
In-Depth Information
estimate is not very far from the real covariance matrix,
C
=
I
n
+
being a symmetric
error matrix. In this case, using the eigenvalues decomposition and the properties of the
trace of a symmetric matrix, equation (4) can be rewritten as:
d
2
(
Σ
N
,Σ
)=
tr
log
2
(
I
n
+
)
,
(5)
=
tr
U
log
2
(
D
I
n
+
)
U
T
(6)
=
tr
log
2
(
D
I
n
+
)
(7)
Σ
i
=1
log
2
(1 +
ii
)
≈
(8)
Now, using the fact that for small
,
log(1 +
)
and assuming that the errors are
equally distributed over the diagonal
D
I
n
+
, (5) becomes:
d
2
(
Σ
N
,Σ
≈
n
2
)
≈
(9)
proportional with the matrix dimension
n
and, in our case, with the number of EEG
channels. In order to avoid this channel number effect, we propose to modify the dis-
tance by multiplying it by
k/n
, with
k
being a user chosen constant ensuring the desired
level of the estimation quality:
n
tr
log
2
−
2
)=
k
−
2
d
n
2
(
Σ
N
,Σ
Σ
N
Σ Σ
N
(10)
As in [15], we adopt an exponential function of the modified distance
d
n
as the local
likelihood
,
Σ
N
)
d
n
2
(
p
(
Σ
N
)
∝
exp
{−
λ
·
Σ
}
.
(11)
with the parameter
λ
fixed to the constant value
λ
=0
.
5
[15]. This
p
(
Σ
N
)
value
Σ
N
). A
p
(
varies between
0
and
1
,
1
meaning perfect estimation (
Σ
N
)
value of
0
.
95
is considered as a well chosen threshold above which the covariance matrices are
considered to be approximately equal.
Σ
=
3.2
Separability Performance Index
In order to measure the global performance of BSS algorithms (orthogonalization plus
rotation), we use the performance index (PI) [4] defined by
⎛
⎞
n
1
n
n
n
1
2
n
(
n
g
ij
|
max
k
|
|
1
2
n
(
n
g
ij
|
max
k
|
|
⎝
⎠
+
PI
=
g
ik
|
−
1
g
kj
|
−
−
1)
−
1)
i
=1
j
=1
j
=1
i
=1
(12)
where
g
ij
is the
(
i,j
)
-element of the global
n
g
ik
|
is the maximum value among the absolute values of the elements in the
i
th row of
G
and
max
k
|
×
n
system matrix
G
=
HA
,
max
k
|
g
kj
|
is the maximum value among the absolute values of the elements in the
j
th column of
G
. Perfect separation yields a null performance index. In practice a
PI
under
10
−
1
means that the separation result is reliable.
Search WWH ::
Custom Search