Biomedical Engineering Reference
In-Depth Information
ˀ
k
(
)
and a column vector
f
as
T
ˀ
k
(
f
)
=[
ˀ
1
,
k
,..., ˀ
q
,
k
]
.
(8.74)
We can then express the partial coherence as
H
j
)
ʣ
−
1
ʺ
j
,
k
(
f
)
=
ˀ
(
f
ˀ
k
(
f
).
(8.75)
The equation above indicates that the partial coherence
is factorized, and
the form of the factorization is very similar to the factorization of the coherence in
Eq. (
8.69
).
The partial directed coherence (PDC) is defined using
ʺ
j
,
k
(
f
)
ˀ
j
,
k
in Eq. (
8.73
) with
replacing
with
I
for removing the instantaneous interaction. That is, the PDC
from the
k
th to the
j
th channels,
ʣ
ˀ
k
ₒ
j
, is defined as
A
j
,
k
(
f
)
ˀ
k
ₒ
j
=
a
k
.
(8.76)
a
k
¯
¯
ʣ
−
1
Assuming
=
I
, the partial coherence
ʺ
j
,
k
(
f
)
is factorized using PDC, such
that
m
=
1
A
m
,
j
(
)
A
m
,
k
(
q
f
f
)
1
ˀ
j
ₒ
m
ˀ
k
ₒ
m
.
ʺ
j
,
k
(
)
=
=
f
(8.77)
a
j
a
k
[ ¯
a
j
][ ¯
¯
a
k
]
¯
m
=
As discussed in the previous sections, the non-diagonal elements of the MVAR
coefficient matrix
¯
A
should contain information on causal interaction. The PDC
directly makes use of this information. The DTF also uses this information but it
does so in an indirect manner through the transfer matrix
H
(
f
)
(
f
)
, which is the inverse
¯
A
of
. Although PDC is derived in a somewhat heuristic manner, it represents
causality through
(
f
)
A
j
,
k
(
. The major difference between PDC and DTF is that PDC
only represents the direct causal influence and is not affected by the indirect influence.
f
)
8.6 Transfer Entropy
8.6.1 Definition
Granger causality and its related measures such as DTF and PDC relies on the MVAR
modeling of voxel time series. In that sense, these measures are model-based. In
this section, we introduce transfer entropy [
7
,
13
], which does not use the MVAR
modeling of voxel time series. Explanations on fundamentals of entropy and mutual
information are found in Sect. C.3.2 in the Appendix.
