Databases Reference
In-Depth Information
The formulae for the reflectional versions are obtained by replacing
R
xy
with
R
xy
, and
R
−
1
yy
with
R
−∗
yy
. The expressions for the total full-rank coefficients are
1
2
p
1
2
p
2
tr
K
2
tr(
R
−
1
xx
R
xy
R
−
1
yy
R
xy
)
ρ
¯
C
1
=
=
,
(4.67)
2
det
1
/
2
(
I
−
K
2
)
det
1
/
2
(
I
−
R
−
1
xx
R
xy
R
−
1
yy
R
xy
)
ρ
¯
C
2
=
1
−
=
1
−
(4.68)
tr
K
2
(
I
K
2
)
−
1
tr
R
xy
R
−
1
yy
R
xy
)
−
1
yy
R
xy
(
R
xx
−
R
xy
R
−
1
−
C
3
=
tr
(
I
K
2
)
−
1
=
tr
R
xx
(
R
xx
−
yy
R
xy
)
−
1
.
ρ
¯
(4.69)
R
xy
R
−
1
−
These coefficients inherit the invariance properties of the canonical correlations. That
is, the rotational and reflectional coefficients are invariant under nonsingular linear
transformation of
x
and
y
, and the total coefficients are invariant under nonsingular
widely linear transformation.
How should we interpret these coefficients? The rotational version of
ρ
C
1
characterizes
the MMSE when constructing a
linear
estimate of the canonical vector
from
y
.This
estimate is
ˆ
R
ξ
y
R
−
1
F
H
R
−
1
/
2
xx
R
xy
R
−
1
F
H
CR
−
1
/
2
=
=
=
=
=
,
(
y
)
yy
y
yy
y
yy
y
KBy
K
(4.70)
and the resulting MMSE is
ˆ
2
R
ξω
R
−
1
R
H
ξω
K
2
)
2
C
1
)
E
(
y
)
−
=
tr(
R
ξξ
−
)
=
tr(
I
−
=
p
(1
−
ρ
.
(4.71)
ωω
from
x
. In a similar fashion, the reflectional version
ρ
C
1
is related to the MMSE of the
conjugate linear
estimator
Since CCA is symmetric in
x
and
y
, the same MMSE is obtained when estimating
ˆ
=
K
(
y
∗
)
∗
,
(4.72)
and the total version ¯
ρ
C
1
is related to the MMSE of the
widely linear
estimator
ˆ
y
∗
)
K
2
∗
.
(
y
,
=
K
1
+
(4.73)
For jointly Gaussian
x
and
y
, the second coefficient,
ρ
C
2
, determines the mutual
information between
x
and
y
,
log
det(
R
xx
−
R
xy
R
−
1
yy
R
xy
)
log det(
I
−
K
2
)
2
I
(
x
;
y
)
=−
=−
=−
log(1
−
ρ
C
2
)
.
det
R
xx
(4.74)
The rotational and reflectional versions only take rotational and reflectional dependences
into account, respectively, whereas the total version characterizes the total mutual infor-
mation between
x
and
y
.
Finally,
ρ
C
3
has an interesting interpretation in the signal-plus-uncorrelated-noise
case
y
N
. It is easy to show that the
eigenvalues of the signal-to-noise-ratio (SNR) matrix
SN
−
1
=
x
+
n
.Let
R
xx
=
R
xy
=
S
and
R
yy
=
S
+
k
i
/
k
i
)
. Hence,
they are invariant under nonsingular linear transformation of the signal
x
, and the
numerator of
are
{
(1
−
}
2
C
3
in (
4.66
)istr(
SN
−
1
). Correlation coefficient
ρ
ρ
C
3
can thus be interpreted
as a normalized SNR.
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