Biomedical Engineering Reference
In-Depth Information
(a)
(b)
1
0.8
0.6
0.4
0.2
0
0
π
/2
π
3
π
/2
2
π
0
π
/2
π
3
π
/2
2
π
(c)
(d)
1
0.8
0.6
0.4
0.2
0
π
/2
π
3
π
/2
2
π
π
/2
π
3
π
/2
2
π
0
0
Fig. 7.1
Plots of the imaginary coherence and corrected imaginary coherence with respect to the
phase for four values of magnitude coherence.
a
|
ˆ
|=
.
The
solid line
shows the corrected imaginary coherence and the
broken line
shows the imaginary
coherence
.
.
b
|
ˆ
|=
.
.
c
|
ˆ
|=
.
.
d
|
ˆ
|=
.
0
99
0
9
0
8
0
6
7.6 Canonical Coherence
7.6.1 Canonical Magnitude Coherence
So far, we have discussed methods to compute voxel-based coherence. However,
when the number of voxels is large and our target is to analyze all-voxel-to-all-
voxel connections, the interpretation and visualization of analysis results may not
be easy because the number of voxel connections becomes huge. One way to reduce
this difficulty is to compute coherence between regions determined based on the
neurophysiology and/or neuroanatomy of a brain. However, in this case, since a brain
region contains multiple voxels, we must compute the coherence between groups of
multiple time courses. We here extend the theory of canonical correlation described
in Sect. C.3 in the Appendix to compute the canonical coherence, which should be
effective for region-based connectivity analysis. The explanation here follows those
in [
10
,
11
].
In this section, random variables
x
and
y
are complex-valued column vectors in
the frequency domain, i.e.,
