Biomedical Engineering Reference
In-Depth Information
8.2.2 Coherence and Partial Coherence of the MVAR Process
Using Eq. (
8.7
), we can derive the relationship
H
H
S
(
f
)
=
H
(
f
)
ʣ
(
f
),
(8.8)
where the superscript
H
indicates the Hermitian transpose.
1
In the equation above,
y
H
S
(
f
)
is the cross spectrum matrix, which is given by
S
(
f
)
=
y
(
f
)
(
f
)
where
·
indicates the ensemble average. Also,
ʣ
is the covariance matrix of the residual,
e
H
which is equal to
. Using these definitions, the coherence between
the time series of the
j
th and the
k
th channels is expressed as
ʣ
=
e
(
f
)
(
f
)
h
j
ʣ
h
k
S
j
,
k
S
j
,
j
S
k
,
k
=
ˆ
j
,
k
(
f
)
=
,
(8.9)
h
j
h
k
[
ʣ
h
j
][
ʣ
h
k
]
where
h
j
is the
j
th column of
H
H
such that
H
H
=[
h
1
,...,
h
q
]
.
The partial coherence between the
j
th and the
k
th channels,
ʺ
j
,
k
, is expressed as
[
8
,
9
]
M
j
,
k
M
j
,
j
M
k
,
k
,
ʺ
j
,
k
(
f
)
=
(8.10)
where
M
j
,
k
is the minor of the matrix
S
; the minor is a determinant value of a
matrix formed by the
j
th row and the
k
th column removed from
S
(
f
)
(
f
)
. The partial
coherence
is a measure of the coherence between the time series of the
j
th
and the
k
th channels where the influence of other channels is removed. In other
words, it expresses the direct interaction between these two channels.
Let us derive a convenient formula for computing the partial coherence. We use
the relationship called Cramer's rule [
10
]:
ʺ
j
,
k
(
f
)
S
−
1
j
+
k
M
k
,
j
k
=
(
−
1
)
(
f
)
,
|
S
(
f
)
|
j
,
S
−
1
component of the matrix
S
−
1
where
[
(
f
)
]
j
,
k
indicates the
(
j
,
k
)
(
f
)
. Considering
the fact that
S
(
f
)
is a positive semidefinite Hermitian matrix, the equation above
can be changed to
k
S
−
1
j
+
M
j
,
k
=
(
−
)
(
)
j
,
k
|
(
)
|
.
1
f
S
f
(8.11)
1
The Hermitian transpose is the matrix transpose with the complex conjugation.
