Biomedical Engineering Reference
In-Depth Information
We then rewrite Eq. (
8.10
)into
k
S
−
1
)
j
,
k
|
j
+
(
−
)
(
(
)
|
1
f
S
f
ʺ
j
,
k
(
f
)
=
2
j
S
−
1
)
j
,
j
|
2
k
S
−
1
)
k
,
k
|
(
−
1
)
(
f
S
(
f
)
|
(
−
1
)
(
f
S
(
f
)
|
S
−
1
)
j
,
k
(
f
=
S
−
1
)
k
,
k
.
(8.12)
)
j
,
j
S
−
1
(
(
f
f
On the other hand, using Eq. (
8.8
), we have the relationship
H
−
1
S
−
1
H
H
(
f
)
=
(
f
)
ʣ
(
f
)
H
−
1
H
¯
A
H
ʣ
−
1
H
−
1
)
ʣ
−
1
¯
A
=
(
)
(
)
=
(
(
).
f
f
f
f
(8.13)
)
=
¯
a
q
. We can
¯
A
¯
A
The
k
th column vector of
(
f
)
is denoted
¯
a
k
, i.e.,
(
f
a
1
,...,
¯
then obtain
S
−
1
a
j
ʣ
−
1
(
f
)
k
= ¯
¯
a
k
,
j
,
and thus derive
a
j
ʣ
−
1
¯
a
k
¯
ʺ
j
,
k
(
f
)
=
.
(8.14)
a
j
ʣ
−
1
a
k
ʣ
−
1
[ ¯
¯
a
j
][ ¯
a
k
]
¯
The equation above is used for deriving partial directed coherence in Sect.
8.5.3
.
8.3 Time-Domain Granger Causality
8.3.1 Granger Causality for a Bivariate Process
Let us first consider bivariate cases in which only a pair of two time series
y
1
(
t
)
and
are determined by using only
their own past values, i.e., the following relationships hold:
y
2
(
t
)
are considered. In the first case,
y
1
(
t
)
and
y
2
(
t
)
∞
y
1
(
t
)
=
A
1
,
1
(
p
)
y
1
(
t
−
p
)
+
e
1
(
t
),
(8.15)
p
=
1
∞
y
2
(
t
)
=
A
2
,
2
(
p
)
y
2
(
t
−
p
)
+
e
2
(
t
).
(8.16)
p
=
1
