<s>
In	O
statistics	O
,	O
principal	B-Algorithm
component	I-Algorithm
regression	I-Algorithm
(	O
PCR	O
)	O
is	O
a	O
regression	O
analysis	O
technique	O
that	O
is	O
based	O
on	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	B-Application
)	O
.	O
</s>
<s>
More	O
specifically	O
,	O
PCR	O
is	O
used	O
for	O
estimating	O
the	O
unknown	O
regression	B-General_Concept
coefficients	I-General_Concept
in	O
a	O
standard	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
In	O
PCR	O
,	O
instead	O
of	O
regressing	O
the	O
dependent	O
variable	O
on	O
the	O
explanatory	O
variables	O
directly	O
,	O
the	O
principal	B-Application
components	I-Application
of	O
the	O
explanatory	O
variables	O
are	O
used	O
as	O
regressors	O
.	O
</s>
<s>
One	O
typically	O
uses	O
only	O
a	O
subset	O
of	O
all	O
the	O
principal	B-Application
components	I-Application
for	O
regression	O
,	O
making	O
PCR	O
a	O
kind	O
of	O
regularized	O
procedure	O
and	O
also	O
a	O
type	O
of	O
shrinkage	O
estimator	O
.	O
</s>
<s>
Often	O
the	O
principal	B-Application
components	I-Application
with	O
higher	O
variances	O
(	O
the	O
ones	O
based	O
on	O
eigenvectors	O
corresponding	O
to	O
the	O
higher	O
eigenvalues	O
of	O
the	O
sample	O
variance-covariance	O
matrix	O
of	O
the	O
explanatory	O
variables	O
)	O
are	O
selected	O
as	O
regressors	O
.	O
</s>
<s>
However	O
,	O
for	O
the	O
purpose	O
of	O
predicting	O
the	O
outcome	O
,	O
the	O
principal	B-Application
components	I-Application
with	O
low	O
variances	O
may	O
also	O
be	O
important	O
,	O
in	O
some	O
cases	O
even	O
more	O
important	O
.	O
</s>
<s>
PCR	O
can	O
aptly	O
deal	O
with	O
such	O
situations	O
by	O
excluding	O
some	O
of	O
the	O
low-variance	O
principal	B-Application
components	I-Application
in	O
the	O
regression	O
step	O
.	O
</s>
<s>
In	O
addition	O
,	O
by	O
usually	O
regressing	O
on	O
only	O
a	O
subset	O
of	O
all	O
the	O
principal	B-Application
components	I-Application
,	O
PCR	O
can	O
result	O
in	O
dimension	B-Algorithm
reduction	I-Algorithm
through	O
substantially	O
lowering	O
the	O
effective	O
number	O
of	O
parameters	O
characterizing	O
the	O
underlying	O
model	O
.	O
</s>
<s>
This	O
can	O
be	O
particularly	O
useful	O
in	O
settings	O
with	O
high-dimensional	B-Algorithm
covariates	I-Algorithm
.	O
</s>
<s>
Also	O
,	O
through	O
appropriate	O
selection	O
of	O
the	O
principal	B-Application
components	I-Application
to	O
be	O
used	O
for	O
regression	O
,	O
PCR	O
can	O
lead	O
to	O
efficient	O
prediction	O
of	O
the	O
outcome	O
based	O
on	O
the	O
assumed	O
model	O
.	O
</s>
<s>
Perform	O
PCA	B-Application
on	O
the	O
observed	O
data	B-Algorithm
matrix	I-Algorithm
for	O
the	O
explanatory	O
variables	O
to	O
obtain	O
the	O
principal	B-Application
components	I-Application
,	O
and	O
then	O
(	O
usually	O
)	O
select	O
a	O
subset	O
,	O
based	O
on	O
some	O
appropriate	O
criteria	O
,	O
of	O
the	O
principal	B-Application
components	I-Application
so	O
obtained	O
for	O
further	O
use	O
.	O
</s>
<s>
Now	O
regress	O
the	O
observed	O
vector	O
of	O
outcomes	O
on	O
the	O
selected	O
principal	B-Application
components	I-Application
as	O
covariates	O
,	O
using	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
regression	I-General_Concept
(	O
linear	B-General_Concept
regression	I-General_Concept
)	O
to	O
get	O
a	O
vector	O
of	O
estimated	O
regression	B-General_Concept
coefficients	I-General_Concept
(	O
with	O
dimension	O
equal	O
to	O
the	O
number	O
of	O
selected	O
principal	B-Application
components	I-Application
)	O
.	O
</s>
<s>
Now	O
transform	B-Algorithm
this	O
vector	O
back	O
to	O
the	O
scale	O
of	O
the	O
actual	O
covariates	O
,	O
using	O
the	O
selected	O
PCA	B-Application
loadings	I-Application
(	O
the	O
eigenvectors	O
corresponding	O
to	O
the	O
selected	O
principal	B-Application
components	I-Application
)	O
to	O
get	O
the	O
final	O
PCR	O
estimator	O
(	O
with	O
dimension	O
equal	O
to	O
the	O
total	O
number	O
of	O
covariates	O
)	O
for	O
estimating	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
characterizing	O
the	O
original	O
model	O
.	O
</s>
<s>
Data	O
representation	O
:	O
Let	O
denote	O
the	O
vector	O
of	O
observed	O
outcomes	O
and	O
denote	O
the	O
corresponding	O
data	B-Algorithm
matrix	I-Algorithm
of	O
observed	O
covariates	O
where	O
,	O
and	O
denote	O
the	O
size	O
of	O
the	O
observed	O
sample	O
and	O
the	O
number	O
of	O
covariates	O
respectively	O
,	O
with	O
.	O
</s>
<s>
Data	O
pre-processing	O
:	O
Assume	O
that	O
and	O
each	O
of	O
the	O
columns	O
of	O
have	O
already	O
been	O
centered	B-Algorithm
so	O
that	O
all	O
of	O
them	O
have	O
zero	O
empirical	O
means	O
.	O
</s>
<s>
This	O
centering	B-Algorithm
step	O
is	O
crucial	O
(	O
at	O
least	O
for	O
the	O
columns	O
of	O
)	O
since	O
PCR	O
involves	O
the	O
use	O
of	O
PCA	B-Application
on	O
and	O
PCA	B-Application
is	I-Application
sensitive	I-Application
to	O
centering	B-Algorithm
of	O
the	O
data	O
.	O
</s>
<s>
One	O
frequently	O
used	O
approach	O
for	O
this	O
is	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
regression	I-General_Concept
which	O
,	O
assuming	O
is	O
full	O
column	O
rank	O
,	O
gives	O
the	O
unbiased	O
estimator	O
:	O
of	O
.	O
</s>
<s>
PCA	B-Application
step	O
:	O
PCR	O
starts	O
by	O
performing	O
a	O
PCA	B-Application
on	O
the	O
centered	B-Algorithm
data	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
this	O
,	O
let	O
denote	O
the	O
singular	O
value	O
decomposition	O
of	O
where	O
,	O
with	O
denoting	O
the	O
non-negative	O
singular	O
values	O
of	O
,	O
while	O
the	O
columns	O
of	O
and	O
are	O
both	O
orthonormal	B-Algorithm
sets	I-Algorithm
of	O
vectors	O
denoting	O
the	O
left	O
and	O
right	O
singular	O
vectors	O
of	O
respectively	O
.	O
</s>
<s>
The	O
principal	B-Application
components	I-Application
:	O
gives	O
a	O
spectral	O
decomposition	O
of	O
where	O
with	O
denoting	O
the	O
non-negative	O
eigenvalues	O
(	O
also	O
known	O
as	O
the	O
principal	B-Application
values	I-Application
)	O
of	O
,	O
while	O
the	O
columns	O
of	O
denote	O
the	O
corresponding	O
orthonormal	B-Algorithm
set	I-Algorithm
of	O
eigenvectors	O
.	O
</s>
<s>
Then	O
,	O
and	O
respectively	O
denote	O
the	O
principal	B-Application
component	I-Application
and	O
the	O
principal	B-Application
component	I-Application
direction	I-Application
(	O
or	O
PCA	B-Application
loading	I-Application
)	O
corresponding	O
to	O
the	O
largest	O
principal	B-Application
value	I-Application
for	O
each	O
.	O
</s>
<s>
Derived	O
covariates	O
:	O
For	O
any	O
,	O
let	O
denote	O
the	O
matrix	O
with	O
orthonormal	B-Algorithm
columns	O
consisting	O
of	O
the	O
first	O
columns	O
of	O
.	O
</s>
<s>
Let	O
denote	O
the	O
matrix	O
having	O
the	O
first	O
principal	B-Application
components	I-Application
as	O
its	O
columns	O
.	O
</s>
<s>
may	O
be	O
viewed	O
as	O
the	O
data	B-Algorithm
matrix	I-Algorithm
obtained	O
by	O
using	O
the	O
transformed	B-Algorithm
covariates	O
instead	O
of	O
using	O
the	O
original	O
covariates	O
.	O
</s>
<s>
The	O
PCR	O
estimator	O
:	O
Let	O
denote	O
the	O
vector	O
of	O
estimated	O
regression	B-General_Concept
coefficients	I-General_Concept
obtained	O
by	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
regression	I-General_Concept
of	O
the	O
response	O
vector	O
on	O
the	O
data	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
,	O
for	O
any	O
,	O
the	O
final	O
PCR	O
estimator	O
of	O
based	O
on	O
using	O
the	O
first	O
principal	B-Application
components	I-Application
is	O
given	O
by	O
:	O
.	O
</s>
<s>
The	O
fitting	O
process	O
for	O
obtaining	O
the	O
PCR	O
estimator	O
involves	O
regressing	O
the	O
response	O
vector	O
on	O
the	O
derived	O
data	B-Algorithm
matrix	I-Algorithm
which	O
has	O
orthogonal	B-Algorithm
columns	O
for	O
any	O
since	O
the	O
principal	B-Application
components	I-Application
are	O
mutually	B-Algorithm
orthogonal	I-Algorithm
to	O
each	O
other	O
.	O
</s>
<s>
Thus	O
in	O
the	O
regression	O
step	O
,	O
performing	O
a	O
multiple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
jointly	O
on	O
the	O
selected	O
principal	B-Application
components	I-Application
as	O
covariates	O
is	O
equivalent	O
to	O
carrying	O
out	O
independent	O
simple	B-General_Concept
linear	I-General_Concept
regressions	I-General_Concept
(	O
or	O
univariate	O
regressions	O
)	O
separately	O
on	O
each	O
of	O
the	O
selected	O
principal	B-Application
components	I-Application
as	O
a	O
covariate	O
.	O
</s>
<s>
When	O
all	O
the	O
principal	B-Application
components	I-Application
are	O
selected	O
for	O
regression	O
so	O
that	O
,	O
then	O
the	O
PCR	O
estimator	O
is	O
equivalent	O
to	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
estimator	O
.	O
</s>
<s>
This	O
is	O
easily	O
seen	O
from	O
the	O
fact	O
that	O
and	O
also	O
observing	O
that	O
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
where	O
indicates	O
that	O
a	O
square	O
symmetric	O
matrix	O
is	O
non-negative	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
Consequently	O
,	O
any	O
given	O
linear	B-Algorithm
form	I-Algorithm
of	O
the	O
PCR	O
estimator	O
has	O
a	O
lower	O
variance	O
compared	O
to	O
that	O
of	O
the	O
same	O
linear	B-Algorithm
form	I-Algorithm
of	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
estimator	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
columns	O
of	O
the	O
data	B-Algorithm
matrix	I-Algorithm
that	O
correspond	O
to	O
the	O
observations	O
for	O
these	O
covariates	O
tend	O
to	O
become	O
linearly	O
dependent	O
and	O
therefore	O
,	O
tends	O
to	O
become	O
rank	O
deficient	O
losing	O
its	O
full	O
column	O
rank	O
structure	O
.	O
</s>
<s>
This	O
issue	O
can	O
be	O
effectively	O
addressed	O
through	O
using	O
a	O
PCR	O
estimator	O
obtained	O
by	O
excluding	O
the	O
principal	B-Application
components	I-Application
corresponding	O
to	O
these	O
small	O
eigenvalues	O
.	O
</s>
<s>
PCR	O
may	O
also	O
be	O
used	O
for	O
performing	O
dimension	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
To	O
see	O
this	O
,	O
let	O
denote	O
any	O
matrix	O
having	O
orthonormal	B-Algorithm
columns	O
,	O
for	O
any	O
Suppose	O
now	O
that	O
we	O
want	O
to	O
approximate	O
each	O
of	O
the	O
covariate	O
observations	O
through	O
the	O
rank	O
linear	B-Architecture
transformation	I-Architecture
for	O
some	O
.	O
</s>
<s>
is	O
minimized	O
at	O
the	O
matrix	O
with	O
the	O
first	O
principal	B-Application
component	I-Application
directions	O
as	O
columns	O
,	O
and	O
the	O
corresponding	O
dimensional	O
derived	O
covariates	O
.	O
</s>
<s>
Thus	O
the	O
dimensional	O
principal	B-Application
components	I-Application
provide	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
of	O
rank	O
to	O
the	O
observed	O
data	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Thus	O
any	O
potential	O
dimension	B-Algorithm
reduction	I-Algorithm
may	O
be	O
achieved	O
by	O
choosing	O
,	O
the	O
number	O
of	O
principal	B-Application
components	I-Application
to	O
be	O
used	O
,	O
through	O
appropriate	O
thresholding	O
on	O
the	O
cumulative	O
sum	O
of	O
the	O
eigenvalues	O
of	O
.	O
</s>
<s>
Since	O
the	O
smaller	O
eigenvalues	O
do	O
not	O
contribute	O
significantly	O
to	O
the	O
cumulative	O
sum	O
,	O
the	O
corresponding	O
principal	B-Application
components	I-Application
may	O
be	O
continued	O
to	O
be	O
dropped	O
as	O
long	O
as	O
the	O
desired	O
threshold	O
limit	O
is	O
not	O
exceeded	O
.	O
</s>
<s>
The	O
same	O
criteria	O
may	O
also	O
be	O
used	O
for	O
addressing	O
the	O
multicollinearity	O
issue	O
whereby	O
the	O
principal	B-Application
components	I-Application
corresponding	O
to	O
the	O
smaller	O
eigenvalues	O
may	O
be	O
ignored	O
as	O
long	O
as	O
the	O
threshold	O
limit	O
is	O
maintained	O
.	O
</s>
<s>
Since	O
the	O
PCR	O
estimator	O
typically	O
uses	O
only	O
a	O
subset	O
of	O
all	O
the	O
principal	B-Application
components	I-Application
for	O
regression	O
,	O
it	O
can	O
be	O
viewed	O
as	O
some	O
sort	O
of	O
a	O
regularized	O
procedure	O
.	O
</s>
<s>
More	O
specifically	O
,	O
for	O
any	O
,	O
the	O
PCR	O
estimator	O
denotes	O
the	O
regularized	O
solution	O
to	O
the	O
following	O
constrained	B-Application
minimization	I-Application
problem	O
:	O
</s>
<s>
Thus	O
,	O
when	O
only	O
a	O
proper	O
subset	O
of	O
all	O
the	O
principal	B-Application
components	I-Application
are	O
selected	O
for	O
regression	O
,	O
the	O
PCR	O
estimator	O
so	O
obtained	O
is	O
based	O
on	O
a	O
hard	O
form	O
of	O
regularization	O
that	O
constrains	O
the	O
resulting	O
solution	O
to	O
the	O
column	O
space	O
of	O
the	O
selected	O
principal	B-Application
component	I-Application
directions	O
,	O
and	O
consequently	O
restricts	O
it	O
to	O
be	O
orthogonal	B-Algorithm
to	O
the	O
excluded	O
directions	O
.	O
</s>
<s>
Given	O
the	O
constrained	B-Application
minimization	I-Application
problem	O
as	O
defined	O
above	O
,	O
consider	O
the	O
following	O
generalized	O
version	O
of	O
it	O
:	O
</s>
<s>
Quite	O
clearly	O
,	O
the	O
resulting	O
optimal	O
estimator	O
is	O
then	O
simply	O
given	O
by	O
the	O
PCR	O
estimator	O
based	O
on	O
the	O
first	O
principal	B-Application
components	I-Application
.	O
</s>
<s>
where	O
,	O
MSE	O
denotes	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Thus	O
in	O
that	O
case	O
,	O
the	O
corresponding	O
would	O
be	O
a	O
more	O
efficient	O
estimator	O
of	O
compared	O
to	O
,	O
based	O
on	O
using	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
as	O
the	O
performance	O
criteria	O
.	O
</s>
<s>
In	O
addition	O
,	O
any	O
given	O
linear	B-Algorithm
form	I-Algorithm
of	O
the	O
corresponding	O
would	O
also	O
have	O
a	O
lower	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
compared	O
to	O
that	O
of	O
the	O
same	O
linear	B-Algorithm
form	I-Algorithm
of	O
.	O
</s>
<s>
it	O
is	O
still	O
possible	O
that	O
,	O
especially	O
if	O
is	O
such	O
that	O
the	O
excluded	O
principal	B-Application
components	I-Application
correspond	O
to	O
the	O
smaller	O
eigenvalues	O
,	O
thereby	O
resulting	O
in	O
lower	O
bias	O
.	O
</s>
<s>
In	O
order	O
to	O
ensure	O
efficient	O
estimation	O
and	O
prediction	O
performance	O
of	O
PCR	O
as	O
an	O
estimator	O
of	O
,	O
Park	O
(	O
1981	O
)	O
proposes	O
the	O
following	O
guideline	O
for	O
selecting	O
the	O
principal	B-Application
components	I-Application
to	O
be	O
used	O
for	O
regression	O
:	O
Drop	O
the	O
principal	B-Application
component	I-Application
if	O
and	O
only	O
if	O
Practical	O
implementation	O
of	O
this	O
guideline	O
of	O
course	O
requires	O
estimates	O
for	O
the	O
unknown	O
model	O
parameters	O
and	O
.	O
</s>
<s>
Unlike	O
the	O
criteria	O
based	O
on	O
the	O
cumulative	O
sum	O
of	O
the	O
eigenvalues	O
of	O
,	O
which	O
is	O
probably	O
more	O
suited	O
for	O
addressing	O
the	O
multicollinearity	O
problem	O
and	O
for	O
performing	O
dimension	B-Algorithm
reduction	I-Algorithm
,	O
the	O
above	O
criteria	O
actually	O
attempts	O
to	O
improve	O
the	O
prediction	O
and	O
estimation	O
efficiency	O
of	O
the	O
PCR	O
estimator	O
by	O
involving	O
both	O
the	O
outcome	O
as	O
well	O
as	O
the	O
covariates	O
in	O
the	O
process	O
of	O
selecting	O
the	O
principal	B-Application
components	I-Application
to	O
be	O
used	O
in	O
the	O
regression	O
step	O
.	O
</s>
<s>
Alternative	O
approaches	O
with	O
similar	O
goals	O
include	O
selection	O
of	O
the	O
principal	B-Application
components	I-Application
based	O
on	O
cross-validation	B-Application
or	O
the	O
Mallow	O
's	O
Cp	O
criteria	O
.	O
</s>
<s>
Often	O
,	O
the	O
principal	B-Application
components	I-Application
are	O
also	O
selected	O
based	O
on	O
their	O
degree	O
of	O
association	O
with	O
the	O
outcome	O
.	O
</s>
<s>
In	O
general	O
,	O
PCR	O
is	O
essentially	O
a	O
shrinkage	O
estimator	O
that	O
usually	O
retains	O
the	O
high	O
variance	O
principal	B-Application
components	I-Application
(	O
corresponding	O
to	O
the	O
higher	O
eigenvalues	O
of	O
)	O
as	O
covariates	O
in	O
the	O
model	O
and	O
discards	O
the	O
remaining	O
low	O
variance	O
components	O
(	O
corresponding	O
to	O
the	O
lower	O
eigenvalues	O
of	O
)	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
principal	B-Application
components	I-Application
are	O
obtained	O
from	O
the	O
eigen-decomposition	O
of	O
that	O
involves	O
the	O
observations	O
for	O
the	O
explanatory	O
variables	O
only	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
resulting	O
PCR	O
estimator	O
obtained	O
from	O
using	O
these	O
principal	B-Application
components	I-Application
as	O
covariates	O
need	O
not	O
necessarily	O
have	O
satisfactory	O
predictive	O
performance	O
for	O
the	O
outcome	O
.	O
</s>
<s>
A	O
somewhat	O
similar	O
estimator	O
that	O
tries	O
to	O
address	O
this	O
issue	O
through	O
its	O
very	O
construction	O
is	O
the	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
PLS	O
)	O
estimator	O
.	O
</s>
<s>
The	O
method	O
starts	O
by	O
performing	O
a	O
set	O
of	O
simple	B-General_Concept
linear	I-General_Concept
regressions	I-General_Concept
(	O
or	O
univariate	O
regressions	O
)	O
wherein	O
the	O
outcome	O
vector	O
is	O
regressed	O
separately	O
on	O
each	O
of	O
the	O
covariates	O
taken	O
one	O
at	O
a	O
time	O
.	O
</s>
<s>
Then	O
,	O
for	O
some	O
,	O
the	O
first	O
covariates	O
that	O
turn	O
out	O
to	O
be	O
the	O
most	O
correlated	O
with	O
the	O
outcome	O
(	O
based	O
on	O
the	O
degree	O
of	O
significance	O
of	O
the	O
corresponding	O
estimated	O
regression	B-General_Concept
coefficients	I-General_Concept
)	O
are	O
selected	O
for	O
further	O
use	O
.	O
</s>
<s>
A	O
conventional	O
PCR	O
,	O
as	O
described	O
earlier	O
,	O
is	O
then	O
performed	O
,	O
but	O
now	O
it	O
is	O
based	O
on	O
only	O
the	O
data	B-Algorithm
matrix	I-Algorithm
corresponding	O
to	O
the	O
observations	O
for	O
the	O
selected	O
covariates	O
.	O
</s>
<s>
The	O
number	O
of	O
covariates	O
used	O
:	O
and	O
the	O
subsequent	O
number	O
of	O
principal	B-Application
components	I-Application
used	O
:	O
are	O
usually	O
selected	O
by	O
cross-validation	B-Application
.	O
</s>
<s>
The	O
classical	O
PCR	O
method	O
as	O
described	O
above	O
is	O
based	O
on	O
classical	B-Application
PCA	I-Application
and	O
considers	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
for	O
predicting	O
the	O
outcome	O
based	O
on	O
the	O
covariates	O
.	O
</s>
<s>
However	O
,	O
it	O
can	O
be	O
easily	O
generalized	O
to	O
a	O
kernel	B-Algorithm
machine	I-Algorithm
setting	O
whereby	O
the	O
regression	O
function	O
need	O
not	O
necessarily	O
be	O
linear	O
in	O
the	O
covariates	O
,	O
but	O
instead	O
it	O
can	O
belong	O
to	O
the	O
Reproducing	O
Kernel	O
Hilbert	O
Space	O
associated	O
with	O
any	O
arbitrary	O
(	O
possibly	O
non-linear	O
)	O
,	O
symmetric	O
positive-definite	O
kernel	O
.	O
</s>
<s>
The	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
turns	O
out	O
to	O
be	O
a	O
special	O
case	O
of	O
this	O
setting	O
when	O
the	O
kernel	O
function	O
is	O
chosen	O
to	O
be	O
the	O
linear	O
kernel	O
.	O
</s>
<s>
In	O
general	O
,	O
under	O
the	O
kernel	B-Algorithm
machine	I-Algorithm
setting	O
,	O
the	O
vector	O
of	O
covariates	O
is	O
first	O
mapped	B-Algorithm
into	O
a	O
high-dimensional	O
(	O
potentially	O
infinite-dimensional	O
)	O
feature	O
space	O
characterized	O
by	O
the	O
kernel	O
function	O
chosen	O
.	O
</s>
<s>
The	O
mapping	B-Algorithm
so	O
obtained	O
is	O
known	O
as	O
the	O
feature	B-Algorithm
map	I-Algorithm
and	O
each	O
of	O
its	O
coordinates	O
,	O
also	O
known	O
as	O
the	O
feature	B-Algorithm
elements	I-Algorithm
,	O
corresponds	O
to	O
one	O
feature	O
(	O
may	O
be	O
linear	O
or	O
non-linear	O
)	O
of	O
the	O
covariates	O
.	O
</s>
<s>
The	O
regression	O
function	O
is	O
then	O
assumed	O
to	O
be	O
a	O
linear	O
combination	O
of	O
these	O
feature	B-Algorithm
elements	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
underlying	O
regression	O
model	O
in	O
the	O
kernel	B-Algorithm
machine	I-Algorithm
setting	O
is	O
essentially	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
with	O
the	O
understanding	O
that	O
instead	O
of	O
the	O
original	O
set	O
of	O
covariates	O
,	O
the	O
predictors	O
are	O
now	O
given	O
by	O
the	O
vector	O
(	O
potentially	O
infinite-dimensional	O
)	O
of	O
feature	B-Algorithm
elements	I-Algorithm
obtained	O
by	O
transforming	B-General_Concept
the	O
actual	O
covariates	O
using	O
the	O
feature	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
kernel	O
trick	O
actually	O
enables	O
us	O
to	O
operate	O
in	O
the	O
feature	O
space	O
without	O
ever	O
explicitly	O
computing	O
the	O
feature	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
pairwise	O
inner	O
products	O
so	O
obtained	O
may	O
therefore	O
be	O
represented	O
in	O
the	O
form	O
of	O
a	O
symmetric	O
non-negative	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
also	O
known	O
as	O
the	O
kernel	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
PCR	O
in	O
the	O
kernel	B-Algorithm
machine	I-Algorithm
setting	O
can	O
now	O
be	O
implemented	O
by	O
first	O
appropriately	B-Algorithm
centering	I-Algorithm
this	O
kernel	B-Algorithm
matrix	I-Algorithm
(	O
K	O
,	O
say	O
)	O
with	O
respect	O
to	O
the	O
feature	O
space	O
and	O
then	O
performing	O
a	O
kernel	B-Algorithm
PCA	I-Algorithm
on	O
the	O
centered	B-Algorithm
kernel	I-Algorithm
matrix	I-Algorithm
(	O
K	O
 '	O
,	O
say	O
)	O
whereby	O
an	O
eigendecomposition	O
of	O
K	O
 '	O
is	O
obtained	O
.	O
</s>
<s>
Kernel	O
PCR	O
then	O
proceeds	O
by	O
(	O
usually	O
)	O
selecting	O
a	O
subset	O
of	O
all	O
the	O
eigenvectors	O
so	O
obtained	O
and	O
then	O
performing	O
a	O
standard	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
of	O
the	O
outcome	O
vector	O
on	O
these	O
selected	O
eigenvectors	O
.	O
</s>
<s>
The	O
eigenvectors	O
to	O
be	O
used	O
for	O
regression	O
are	O
usually	O
selected	O
using	O
cross-validation	B-Application
.	O
</s>
<s>
The	O
estimated	O
regression	B-General_Concept
coefficients	I-General_Concept
(	O
having	O
the	O
same	O
dimension	O
as	O
the	O
number	O
of	O
selected	O
eigenvectors	O
)	O
along	O
with	O
the	O
corresponding	O
selected	O
eigenvectors	O
are	O
then	O
used	O
for	O
predicting	O
the	O
outcome	O
for	O
a	O
future	O
observation	O
.	O
</s>
<s>
Clearly	O
,	O
kernel	O
PCR	O
has	O
a	O
discrete	O
shrinkage	O
effect	O
on	O
the	O
eigenvectors	O
of	O
K	O
 '	O
,	O
quite	O
similar	O
to	O
the	O
discrete	O
shrinkage	O
effect	O
of	O
classical	O
PCR	O
on	O
the	O
principal	B-Application
components	I-Application
,	O
as	O
discussed	O
earlier	O
.	O
</s>
<s>
However	O
,	O
the	O
feature	B-Algorithm
map	I-Algorithm
associated	O
with	O
the	O
chosen	O
kernel	O
could	O
potentially	O
be	O
infinite-dimensional	O
,	O
and	O
hence	O
the	O
corresponding	O
principal	B-Application
components	I-Application
and	O
principal	B-Application
component	I-Application
directions	O
could	O
be	O
infinite-dimensional	O
as	O
well	O
.	O
</s>
<s>
Therefore	O
,	O
these	O
quantities	O
are	O
often	O
practically	O
intractable	O
under	O
the	O
kernel	B-Algorithm
machine	I-Algorithm
setting	O
.	O
</s>
<s>
Kernel	O
PCR	O
essentially	O
works	O
around	O
this	O
problem	O
by	O
considering	O
an	O
equivalent	O
dual	O
formulation	O
based	O
on	O
using	O
the	O
spectral	O
decomposition	O
of	O
the	O
associated	O
kernel	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Under	O
the	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
(	O
which	O
corresponds	O
to	O
choosing	O
the	O
kernel	O
function	O
as	O
the	O
linear	O
kernel	O
)	O
,	O
this	O
amounts	O
to	O
considering	O
a	O
spectral	O
decomposition	O
of	O
the	O
corresponding	O
kernel	B-Algorithm
matrix	I-Algorithm
and	O
then	O
regressing	O
the	O
outcome	O
vector	O
on	O
a	O
selected	O
subset	O
of	O
the	O
eigenvectors	O
of	O
so	O
obtained	O
.	O
</s>
<s>
It	O
can	O
be	O
easily	O
shown	O
that	O
this	O
is	O
the	O
same	O
as	O
regressing	O
the	O
outcome	O
vector	O
on	O
the	O
corresponding	O
principal	B-Application
components	I-Application
(	O
which	O
are	O
finite-dimensional	O
in	O
this	O
case	O
)	O
,	O
as	O
defined	O
in	O
the	O
context	O
of	O
the	O
classical	O
PCR	O
.	O
</s>
<s>
However	O
,	O
for	O
arbitrary	O
(	O
and	O
possibly	O
non-linear	O
)	O
kernels	O
,	O
this	O
primal	O
formulation	O
may	O
become	O
intractable	O
owing	O
to	O
the	O
infinite	O
dimensionality	O
of	O
the	O
associated	O
feature	B-Algorithm
map	I-Algorithm
.	O
</s>
