Biomedical Engineering Reference
In-Depth Information
(
)
(
)
a case in which
x
t
and
y
t
obey the MVAR process such that
P
x
(
t
)
=
A
x
(
p
)
x
(
t
−
p
)
+
e
x
(
t
),
(8.25)
p
=
1
and
P
y
(
t
)
=
A
y
(
p
)
y
(
t
−
p
)
+
e
y
(
t
).
(8.26)
p
=
1
We define covariance matrices of the residuals in this case, such that
x
e
e
x
(
ʣ
=
e
x
(
t
)
t
)
,
(8.27)
y
e
e
y
(
ʣ
=
e
y
(
t
)
t
)
,
(8.28)
where
indicates the ensemble average.
We next assume that
x
·
(
)
(
)
t
and
y
t
obey the following MVAR process
P
P
x
(
t
)
=
A
x
(
p
)
x
(
t
−
p
)
+
B
y
(
p
)
y
(
t
−
p
)
+
x
(
t
),
(8.29)
p
=
1
p
=
1
and
P
P
y
(
t
)
=
A
y
(
p
)
y
(
t
−
p
)
+
B
x
(
p
)
x
(
t
−
p
)
+
y
(
t
).
(8.30)
p
=
1
p
=
1
We can define covariance matrices of the residuals, such that
x
T
ʣ
=
x
(
t
)
x
(
t
)
,
(8.31)
y
T
ʣ
=
y
(
t
)
y
(
t
)
.
(8.32)
Using the same idea for Eqs. (
8.23
) and (
8.24
), the multivariate Granger causality
G
x
ₒ
y
and
G
y
ₒ
x
are given by
y
e
log
|
ʣ
|
G
x
ₒ
y
=
|
,
(8.33)
y
|
ʣ
e
log
ʣ
G
y
ₒ
x
=
|
,
(8.34)
x
|
ʣ
where
|·|
indicates the matrix determinant.
