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reduced-rank estimation and transform coding if we want to minimize mean-squared
error.
An interesting question in this context is how much of the overall correlation is cap-
tured by
r
coefficients
r
i
=
1
, which can be rotational, reflectional, or total coefficients
defined either through CCA, MLR, or PLS. One could, of course, compute the frac-
tion
{
k
i
}
2
(
r
)
2
(
p
) for all 1
ρ
/ρ
≤
r
<
p
. The following definition, however, provides a more
convenient approach.
Definition 4.4.
The
rotational correlation spread
is defined as
k
i
}
1
p
var(
{
)
2
σ
=
µ
2
(
{
k
i
}
)
0
2
3
5
p
k
i
p
1
p
i
=
1
=
2
−
,
(4.88)
p
p
−
1
k
i
i
=
1
k
i
}
k
i
}
2
(
k
i
}
where
var(
{
)
denotes the variance of the correlations
{
and
µ
{
)
denotes their
squared mean.
The correlation spread provides a
single, normalized measure
of how concentrated
the overall correlation is. If there is only one nonzero coefficient
k
1
, then
2
σ
=
1. If
2
all coefficients are equal,
k
1
=
0. In essence, the correlation
spread gives an indication of how
compressible
the cross-correlation between
x
and
y
is.
The definition (
4.88
) is inspired by the definition of the
degree of polarization
of a
random vector
x
. The degree of polarization measures the spread among the eigenvalues
of
R
xx
. A random vector
x
is said to be completely polarized if all of its energy is
concentrated in one direction, i.e., if there is only one nonzero eigenvalue. On the other
hand,
x
is unpolarized if its energy is equally distributed among all dimensions, i.e., if all
eigenvalues are equal. The correlation spread
k
2
=···=
k
p
, then
σ
=
2
generalizes this idea to the correlation
σ
between two random vectors
x
and
y
.
2
2
Example 4.6.
Continuing Example
4.5
for
R
xx
=
I
, we find
σ
C
=
σ
M
=
0
.
167 for both
1
1
1
1
C
M
CCA and MLR. For
R
xx
=
Diag
(4
,
4
,
4
,
4
,
4
), we find
σ
=
0
.
178 for CCA and
σ
=
2
-value close to 1 indicates that the correlation is highly concentrated.
Indeed, as we have found in Example
4.5
, most of the MLR correlation is concentrated
in a one-dimensional subspace.
0
.
797 for MLR. A
σ
The reflectional correlation spread is defined as a straightforward extension by replac-
ing
k
i
with
k
i
in Definition
4.4
, but the total correlation spread is defined in a slightly
different manner.
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