Biomedical Engineering Reference
In-Depth Information
The alternative canonical imaginary coherence, based on
I
s
(
,
)
x
y
, is obtained as
⊡
⊤
1
/
d
d
ˈ
2
I
⊣
⊦
=
1
−
exp
[−
I
s
(
x
,
y
)
]=
1
−
1
(
1
−
ʶ
j
)
.
(7.88)
j
=
When we can assume
ʶ
j
1 (where
j
=
1
,...,
d
), we have
d
1
d
ˈ
2
I
≈
1
ʶ
j
.
(7.89)
j
=
The connectivity metric above is called the global interaction measure (GIM) [
12
].
7.6.3 Canonical Residual Coherence
We can compute the canonical residual coherence, which is a multivariate version
of the residual coherence described in Sect.
7.5.3
. Let us assume that
y
is the target
spectra,
x
is the seed spectra, and consider the regression
y
=
Ax
+
v
.
(7.90)
The real-valued regression coefficient matrix
A
is obtained by the least-squares fit.
The optimum
A
is obtained as
xx
A
ʣ
yx
ʣ
−
1
=
.
(7.91)
Thus, the residual signal
v
is expressed as
xx
x
−
Ax
ʣ
yx
ʣ
−
1
v
=
y
=
y
−
.
(7.92)
H
x
H
Let us define
ʣ
vv
and
ʣ
v
x
such that
ʣ
vv
=
vv
and
ʣ
v
x
=
v
.Theyare
obtained as
ʣ
xy
−
ʣ
yx
A
T
ʣ
xx
A
T
−
Ax
−
Ax
=
ʣ
yy
−
A
+
A
H
ʣ
vv
=
(
y
)(
y
)
,
(7.93)
and
−
Ax
=
ʣ
yx
−
A
x
H
ʣ
v
x
=
(
)
ʣ
xx
.
y
(7.94)
We just follow the arguments in Sect.
7.6.1
, and derive an expression for the
canonical magnitude coherence between
v
and
x
. Using complex-valued
a
and
b
,
b
H
x
, the magnitude coherence between
a
H
2
,
we define
v
=
v
and
x
=
v
and
x
,
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ˆ
R
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