Biomedical Engineering Reference
In-Depth Information
8.3.3 Total Interdependence
x
T
y
T
T
,
Let us define the
(
q
+
r
)
-dimensional augmented time series
z
(
t
)
=[
(
t
),
(
t
)
]
which obeys
P
z
(
t
)
=
A
z
(
p
)
z
(
t
−
p
)
+
z
(
t
).
(8.35)
p
=
1
z
The covariance matrix of the residual vector
z
(
t
)
is defined as
ʣ
. When the time
T
,
series
x
(
t
)
and
y
(
t
)
are independent,
z
(
t
)
is expressed as
z
(
t
)
=[
e
x
(
t
),
e
y
(
t
)
]
z
and
ʣ
is expressed as
ʣ
e
0
z
z
ʣ
=
z
(
t
)
(
t
)
=
.
(8.36)
e
0
ʣ
When
x
(
t
)
and
y
(
t
)
are not independent, it is expressed as
ʣ
xy
x
ʣ
z
ʣ
=
,
(8.37)
yx
y
ʣ
ʣ
where
xy
T
ʣ
=
x
(
t
)
y
(
t
)
,
(8.38)
yx
T
xy
T
ʣ
=
y
(
t
)
x
(
t
)
=
(
ʣ
)
.
(8.39)
The total interdependence between
x
and
y
,
I
{
x
,
y
}
, is defined such that
e
ʣ
0
log
ʣ
e
ʣ
e
y
e
y
x
0
ʣ
ʣ
ʣ
I
{
x
,
y
}
=
log
=
.
(8.40)
z
z
This total interdependence expresses the deviation of the value of
ʣ
from its value
z
obtained when
x
(
t
)
and
y
(
t
)
are independent. Using Eqs. (
8.33
) and (
8.34
), it is easy
to show the relationship
log
ʣ
ʣ
y
x
ʣ
I
{
x
,
y
}
−
G
x
ₒ
y
−
G
y
ₒ
x
=
.
(8.41)
z
According to Geweke [
3
], the right-hand side of the equation above is defined as
I
{
x
·
y
}
:
log
ʣ
ʣ
y
x
ʣ
I
{
x
·
y
}
=
,
(8.42)
z
