Biomedical Engineering Reference
In-Depth Information
I
{
x
·
y
}
and this
is called the instantaneous dependence. The following relationship
holds among the causality measures mentioned above,
I
{
x
,
y
}
=
G
x
ₒ
y
+
G
y
ₒ
x
+
I
{
x
·
y
}
.
That is, the total interdependence between the two time series
x
(
t
)
and
y
(
t
)
is
expressed as the summation of the Granger causality from
x
(
t
)
to
y
(
t
)
, the Granger
causality from
y
, and the instantaneous dependence.
The rationale of this instantaneous dependence can be explained in the following
manner. Using the determinant identity in Eq. (C.94), Eq. (
8.42
) is rewritten as
(
t
)
to
x
(
t
)
log
ʣ
−
x
I
{
x
·
y
}
=
log
|
ʔ
|
,
(8.43)
where
x
xy
y
)
−
1
yx
ʔ
=
ʣ
−
ʣ
(
ʣ
ʣ
.
(8.44)
Let us consider the linear regression in which the residual signal
x
(
t
)
is regressed
by the other residual signal
expresses the covariance matrix
of the residual of this linear regression, according to the arguments in Sect. C.3.3.
Let us suppose a case where
x
y
(
t
)
.InEq.(
8.44
),
ʔ
(
t
)
and
y
(
t
)
contain a common instantaneous
interaction
ʽ
(
t
)
, which does not exist in the past values of
x
(
t
)
and
y
(
t
)
. In such a
case, both the residual signals
x
(
t
)
and
y
(
t
)
contain the common component
ʽ
(
t
)
,
ʽ
(
)
x
(
)
y
(
)
and because of this,
t
is regressed out when
t
is regressed by
t
. Therefore,
|
ʔ
|
log
does not contain the influence of this common instantaneous component,
while log
ʣ
does. Accordingly,
x
I
{
x
·
y
}
represents the influence of the instantaneous
component alone.
8.4 Spectral Granger Causality: Geweke Measures
8.4.1 Basic Relationships in the Frequency Domain
Since brain activities have spectral dependence, we naturally like to perform causality
analysis in the spectral domain. The extension of Granger causality analysis into the
spectral domain has been investigated by Geweke [
3
,
4
]. The arguments in this section
are according to Ding [
11
], and we restrict our arguments to a bivariate process
2
:
x
x
e
x
(
∞
(
t
)
(
t
−
p
)
t
)
=
A
(
p
)
+
,
(8.45)
y
(
t
)
y
(
t
−
p
)
e
y
(
t
)
p
=
1
2
The extension of the arguments here to a general multivariate case remains unknown.
