Biomedical Engineering Reference
In-Depth Information
−|
ˆ
|
−
(ˆ)
−
(ˆ)
2
2
2
I(˃
T
,˃
S
)
=−
(
)
=−
(
)
log
1
log
1
log
1
2
(ˆ)
2
−
(ˆ)
=−
(
−
1
−
(ˆ)
2
1
log
1
(ˆ)
2
−
(ˆ)
2
=−
log
(
1
)
−
−
.
(7.43)
−
(ˆ)
1
2
On the right-hand side, the first term,
−
(ˆ)
2
I
R
(˃
T
,˃
S
)
=−
log
(
1
)
represents a component of the mutual information corresponding to the real part
of coherence. This
can be interpreted as the instantaneous component,
which corresponds to the zero-lag correlation between the target and the seed time
courses, as discussed in Sect.
7.3
.
The second term,
I
R
(˃
T
,˃
S
)
log
(ˆ)
2
I
I
(˃
T
,˃
S
)
=−
(
1
−
−
(ˆ)
2
1
represents a component of the mutual information corresponding to the imaginary
part of coherence. It is interpreted as the non-instantaneous component, which cor-
responds to the nonzero-lag correlation between
u
T
(
t
)
and
u
S
(
t
)
. It is easy to see
log
(ˆ)
2
−
ʾ
2
I
I
(˃
T
,˃
S
)
=−
(
1
−
=−
log
(
1
),
(7.44)
−
(ˆ)
1
2
where
ʾ
is the corrected imaginary coherence. The equation above indicates that the
corrected imaginary coherence can be interpreted as a coherence-domain expression
of the non-instantaneous component of the mutual information. Arguments similar
to those in this section are found in [
9
].
7.5.3 Residual Coherence
We next show that the corrected imaginary coherence can be derived using regression
analysis. Let us regress the target spectrum
˃
T
using the seed spectrum
˃
S
, such that
˃
T
=
ʱ˃
S
+
v,
(7.45)
where
ʱ
is a real-valued constant, and
v
is a residual signal of this regression. The
value for
ʱ
is determined using the least-squares fit:
