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and β
′
X
()
subject to the unit variance constraints
α
′
X
()
β
′
X
()
, and
V(
)=V(
)=
the kth pair
α
k
, β
k
, which is uncorrelated with the first k
pairs, maximizes the
(
)
−
correlation between α
′
k
X
()
,andisagainsubjecttotheunitvariance con-
straints. he sequence of correlations between α
′
k
X
()
and β
′
i
X
()
describes only the
linear relations between X
()
and X
()
.herearecases wherelinear correlations may
not be adequate for describing the “associations” between X
()
and X
()
.Anatural
alternative, therefore, is to look for nonlinear relations. Kernel methods can provide
aconvenientapproachtononlineargeneralization.Let κ
i
X
()
and β
′
,
and κ
,
betwopos-
(ċ
ċ)
(ċ
ċ)
itive definite kernels defined on
and
, respectively. Let X denote the
X
X
X
X
data matrix given by
x
x
n
X
=
.
n
p
()
j
()
j
Each data point (as a row vector) x
j
=(
x
, x
)
in the data matrix is transformed
into a kernel representation:
()
j
()
j
x
j
γ
j
=(
γ
,γ
)
,
(
.
)
where
(
i
)
j
(
i
)
j
(
i
)
(
i
)
j
, x
(
i
)
n
γ
=(
κ
i
(
x
, x
)
,...,κ
i
(
x
))
, j
=
,...,n,andi
=
,
.
Or, in matrix notation, the kernel data matrix is given by
γ
()
γ
()
K
=[
K
K
]=
,
(
.
)
()
n
()
n
γ
γ
n
n
(
i
)
j
(
i
)
j
′
(
i
)
j
n
n
whereK
i
=[
κ
i
(
x
, x
)]
j, j
′
=
, i
=
,
,arethefullkernelmatricesfordata
x
j
=
.
()
j
()
j
R
n
can be regarded as an alternative
way of recording data measurements with high inputs.
he KCCA procedure consists of two major steps:
(a) Transform the data points into a kernel representation, as in (
.
) or (
.
), in
matrix notation.
(b) Apply the classical CCA procedure to the kernel data matrix K. Note that some
kind of regularization is needed here to solve the associated spectral problem of
extracting leadingcanonical variates andcorrelation coe
cients. Hereweusethe
reducedkernelconceptstatedintheRKHSsectionandinExample
.Onlypartial
columns are computed to form reduced kernel matrices, denoted by K
and K
.
he classical CCA procedure is applied to the reduced kernel matrix
he representation of x
j
by γ
j
=(
γ
,γ
)
K
K
[
]
.
As KCCA is simply the application of classical CCA to kernel data, existing code
from standard statistical packages can be utilized. In the example below we use the
MATLAB m-file “canoncorr” to implement classical CCA on kernel data.
