Biomedical Engineering Reference
In-Depth Information
b
T
y
, where
a
and
b
are real-valued
vectors. It then computes the correlation between
a
T
x
and
=
=
direction of
a
and
b
such that
x
y
y
. This correlation depends
on the choices of
a
and
b
. The maximum correlation between
x
and
y
is considered
to represent the correlation between
x
and
y
, and this maximum correlation is called
the canonical correlation.
The correlation between
x
and
x
and
y
,
ˁ
, is expressed as
E
(
x
y
)
ˁ
=
E
E
(
x
2
)
(
y
2
)
a
T
E
xy
T
a
T
ʣ
xy
b
(
)
b
=
a
T
E
b
=
a
T
ʣ
yy
b
.
(C.30)
a
b
T
E
ʣ
xx
a
b
T
xx
T
yy
T
(
)
(
)
xy
T
xx
T
yy
T
where
ʣ
xy
=
E
(
)
,
ʣ
xx
=
E
(
)
, and
ʣ
yy
=
E
(
)
and the notation
E
(
·
)
indicates the expectation. Here,
ʣ
xx
and
ʣ
yy
are symmetric matrices, and the rela-
T
tionship
ʣ
xy
=
ʣ
yx
holds. The canonical correlation
ˁ
c
is obtained by solving the
following maximization problem:
a
T
subject to
a
T
1 and
b
T
ˁ
c
=
ʣ
xy
b
ʣ
xx
a
=
ʣ
yy
b
=
.
max
a
1
(C.31)
,
b
To solve this maximization problem, we first compute whitened vectors
ʱ
and
ʲ
,
such that
1
/
2
ʱ
=
ʣ
,
xx
a
(C.32)
1
/
2
ʲ
=
ʣ
b
.
(C.33)
yy
Using these vectors, the optimization problem in Eq. (
C.31
) is rewritten as
T
ʣ
−
1
/
2
xx
ʣ
xy
ʣ
−
1
/
2
T
T
ˁ
c
=
max
ʱ
,
ʲ
ʱ
ʲ
subject to
ʱ
ʱ
=
1 and
ʲ
ʲ
=
1
.
(C.34)
yy
The constrained maximization problem in Eq. (
C.34
) can be solved using the
Lagrange multiplier method. Denoting the Lagrange multipliers
ʽ
1
and
ʽ
2
,the
Lagrangian is defined as
1
1
ʲ
−
ʽ
1
2
−
ʽ
2
2
T
ʣ
−
1
/
2
xx
ʣ
xy
ʣ
−
1
/
2
T
T
L =
ʱ
ʱ
ʱ
−
ʲ
ʲ
−
.
(C.35)
yy
The optimum
ʱ
and
ʲ
are obtained by maximizing
L
with no constraints. The deriv-
atives of
L
with respect to
ʱ
and
ʲ
are expressed as
