Biomedical Engineering Reference
In-Depth Information
which is a metric using all the eigenvalues. In the arguments in Sect.
7.5.2
,the
relationship between the coherence and the mutual information is given in Eq. (
7.41
).
This relationship can lead to an alternative definition of canonical magnitude coher-
ence based on the mutual information, such that
d
|
ˈ
|
2
=
1
−
exp
[−
I(
x
,
y
)
]=
1
−
1
(
1
−
ʳ
j
).
(7.68)
j
=
This metric uses all the eigenvalues, and since its value is normalized between 0 and
1, the interpretation is intuitively easy, compared to the original mutual information
I(
x
.
Also, the mutual information
,
y
)
in Eq. (
7.67
) depends on
d
, which is the sizes
of the vectors
x
and
y
. Such property may not be appropriate for a brain interaction
metric, and the metric independent of the sizes of vectors can be defined such that,
I(
x
,
y
)
d
1
d
1
I(
x
,
y
)
=
log
−
ʳ
j
.
(7.69)
1
j
=
1
The alternative definition of canonical magnitude coherence in this case is expre-
ssed as
⊡
⊤
1
/
d
d
|
ˈ
|
2
[−
I(
⊣
⊦
=
1
−
exp
x
,
y
)
]=
1
−
1
(
1
−
ʳ
j
)
.
(7.70)
j
=
The above metric may be more effective than the one in Eq. (
7.68
).
7.6.2 Canonical Imaginary Coherence
We next derive a formula to compute the canonical imaginary coherence. Following
the arguments of Evald et al. [
12
] we define
ʣ
xx
=
(
ʣ
xx
)
+
i
(
ʣ
xx
)
=
ʓ
xx
+
i
ʥ
xx
,
(7.71)
ʣ
yy
=
(
ʣ
yy
)
+
i
(
ʣ
yy
)
=
ʓ
yy
+
i
ʥ
yy
,
(7.72)
ʣ
xy
=
(
ʣ
xy
)
+
i
(
ʣ
xy
)
=
ʓ
xy
+
i
ʥ
xy
.
(7.73)
We use real-valued
a
and
b
and express the imaginary part of the coherence between
a
T
x
and
b
T
y
:
ʣ
xy
b
a
T
a
T
ʥ
xy
b
(ˆ)
=
a
T
ʣ
yy
b
=
a
T
ʣ
yy
b
.
(7.74)
ʣ
xx
a
b
T
ʣ
xx
a
b
T
