Biomedical Engineering Reference
In-Depth Information
and using this column vector, coherence is expressed as
3
h
j
ʣ
h
k
S
j
,
k
S
j
,
j
S
k
,
k
=
H
j
ˆ
j
,
k
(
f
)
=
=
ʳ
(
f
)
ʣʳ
k
(
f
).
(8.69)
h
j
h
k
[
ʣ
h
j
][
ʣ
h
k
]
This directed coherence
ʳ
j
,
k
is known to represent the directional influence from
the
k
th to the
j
th channels. It contains the residual covariance matrix
whose off-
diagonal terms may represent the instantaneous interaction between two channels,
according to the arguments in Sect.
8.3.3
. Setting
ʣ
ʣ
equal to
I
,wehave
H
j
,
k
(
f
)
ʳ
j
,
k
=
,
(8.70)
h
j
|
h
j
|
which is exactly equal to the normalized DTF in Eq. (
8.66
). Namely, the DTF is a
measure for directional influences obtained by removing instantaneous components
from the directed coherence. Under the assumption that
ʣ
=
I
, coherence
ˆ
j
,
k
(
f
)
can be decomposed of the sum of the product of DTFs, such that
q
m
q
H
k
,
m
(
1
H
j
,
m
(
f
)
f
)
=
1
μ
m
ₒ
j
μ
m
ₒ
k
.
ˆ
j
,
k
(
f
)
=
=
(8.71)
h
j
h
k
[
h
j
][
h
k
]
m
=
8.5.3 Partial Directed Coherence (PDC)
In Sect.
8.5.2
, DTF is expressed using a factorization of coherence. We apply similar
factorization to partial coherence to obtain the partial directed coherence (PDC)
[
6
,
12
]. The starting point is Eq. (
8.14
), which is re-written as,
ʣ
−
1
a
j
¯
a
k
¯
ʺ
j
,
k
(
f
)
=
.
(8.72)
a
j
ʣ
−
1
a
k
ʣ
−
1
[ ¯
¯
a
j
][ ¯
a
k
]
¯
¯
A
=[
A
1
,
j
,...,
A
q
,
j
]
Here, recall that
a
j
is the
j
th column of the matrix
¯
(
f
)
and
¯
a
j
.
We define
A
j
,
k
(
f
)
ˀ
j
,
k
(
f
)
=
a
k
,
(8.73)
a
k
ʣ
−
1
¯
¯
3
Note that since
h
j
is the
j
th column vector of
H
H
,
h
j
is equal to
h
j
H
j
,
1
,...,
H
j
,
q
]
T
.
=[
