Biomedical Engineering Reference
In-Depth Information
We then consider the second case in which the following relationships hold:
∞
∞
y
1
(
t
)
=
A
1
,
1
(
p
)
y
1
(
t
−
p
)
+
A
1
,
2
(
p
)
y
2
(
t
−
p
)
+
1
(
t
),
(8.17)
p
=
1
p
=
1
∞
∞
y
2
(
t
)
=
A
2
,
1
(
p
)
y
1
(
t
−
p
)
+
A
2
,
2
(
p
)
y
2
(
t
−
p
)
+
2
(
t
).
(8.18)
p
=
1
p
=
1
The difference between the first and the second cases above is that in the second
case, time series
y
1
(
t
)
and
y
2
(
t
)
are determined by the past values of both the time
series of
y
1
(
t
)
and
y
2
(
t
)
, while in the first case, the time series
y
1
(
t
)
is determined
only by the past values of
y
1
(
t
)
and the time series
y
2
(
t
)
is determined only by the
past values of
y
2
(
.
We define the variances of the residual terms appearing above, such that
t
)
1
V
(
e
1
(
t
))
=
Σ
e
,
(8.19)
2
V
(
e
2
(
t
))
=
Σ
e
,
(8.20)
1
V
(
1
(
t
))
=
Σ
,
(8.21)
2
V
(
2
(
t
))
=
Σ
,
(8.22)
where
V
(
·
)
indicates the variance. Then, the Granger causality,
G
2
ₒ
1
, is defined as
log
Σ
1
e
G
2
ₒ
1
=
.
(8.23)
1
Σ
The meaning of
G
2
ₒ
1
is that, if the residual variance is reduced by using both the
past values of
y
1
(
t
)
and
y
2
(
t
)
when estimating the current value of
y
1
(
t
)
, the past of
. In such a case, we can conclude that there
is an information flow from the time series
y
2
(
y
2
(
t
)
affects the current value of
y
1
(
t
)
. The above
G
2
ₒ
1
can quantify the amount of this information flow. Similarly, we can define
G
1
ₒ
2
such that
t
)
to the time series
y
1
(
t
)
log
Σ
e
G
1
ₒ
2
=
.
(8.24)
Σ
2
This
G
1
ₒ
2
expresses the information flow from the time series
y
1
(
t
)
to the time series
y
2
(
t
)
.
8.3.2 Multivariate Granger Causality
The idea of Granger causality is extended to a general multivariate case. Let us define
q
-dimensional time series as
y
(
t
)
and
r
-dimensional time series as
x
(
t
)
. We consider
