Biomedical Engineering Reference
In-Depth Information
and the results are given by
⊡
⊤
2
(Σ
x
−
|
ʳ
|
2
y
)
|
H
yx
(
f
)
|
Σ
⊣
1
⊦
.
f
x
ₒ
y
(
f
)
=−
log
−
(8.61)
2
|
y
(
f
)
|
8.5 Other MVAR-Modeling-Based Measures
8.5.1 Directed Transfer Function (DTF)
The directed transfer function (DTF) [
5
] is derived from a definition of causality that
is somewhat different from the Granger causality. For a bivariate AR process, we
have the relationships
P
P
y
1
(
t
)
=
A
1
,
1
(
p
)
y
1
(
t
−
p
)
+
A
1
,
2
(
p
)
y
2
(
t
−
p
)
+
1
(
t
)
p
=
1
p
=
1
P
P
y
2
(
)
=
A
2
,
1
(
)
y
1
(
−
)
+
A
2
,
2
(
)
y
2
(
−
)
+
2
(
).
t
p
t
p
p
t
p
t
p
=
1
p
=
1
It can be intuitively clear that, for example, the influence of the past values of
y
2
(
t
)
on the current value of
y
1
(
t
)
can be assessed by the values of the AR coefficients
A
1
,
2
(
p
)
, (where
p
=
1
,...,
P
). Namely, the causal influence of
y
2
(
t
)
on
y
1
(
t
)
can
be assessed by using
P
A
1
,
2
(
).
p
(8.62)
p
=
1
The spectral domain quantities that play a role similar to the above quantity are
|
A
1
,
2
(
f
)
|
and
|
H
1
,
2
(
f
)
|
,
(8.63)
¯
A
where the matrix
(
f
)
is defined in Eq. (
8.5
) and the transfer matrix
H
(
f
)
is defined
)
=
¯
A
)
−
1
. The directed transfer function (DTF) makes use of
H
as
H
(
f
(
f
(
f
)
to
express the causal relationship.
We define the (unnormalized) directed transfer function (DTF) using the elements
of the transfer function
H
. Namely, the unnormalized directed transfer function
that represents the causal influence of
y
2
(
(
f
)
t
)
on
y
1
(
t
)
is defined as
H
1
,
2
(
f
).
For a
bivariate case, The relationship,
