# SAS implementation of Kernel PCA

February 6, 2010
By

 This post was kindly contributed by SAS Programming for Data Mining Applications - go there to comment and to read the full post.

Kernel method is a very useful technique in data mining that is applicable to any algorithms relying on inner product [1]. The key is applying appropriate kernel function to the inner product of original data space.

I show here SAS/STAT+BASE example code for Kernel PCA implementation. The example used is from Wikipedia @ here. Extention to other algorithms that suitable for Kernel method, such as CCA, ICA, LDA, etc, is straightforward.

Reference:
[1]. Zhu, Mu, “Kernels and Ensembles: Perspectives on Statistical Learning“, The American Statistician. May 1, 2008, 62(2): 97-109

Figure: Result

SAS demo code:

``````
/* Demonstrate kernel PCA. Kernel method can be applied to any algorithm that
relies on inner product */
data original;
do ID=-314 to 314;
x1=sin(ID/100)*1+rannor(8976565)*0.1;
x2=cos(ID/100)*1+rannor(92654782)*0.1;
Class=1; output;
x1=sin(ID/100)*2+rannor(8976565)*0.1;
x2=cos(ID/100)*2+rannor(92654782)*0.1;
Class=2; output;
x1=sin(ID/100)*3+rannor(8976565)*0.1;
x2=cos(ID/100)*3+rannor(92654782)*0.1;
Class=3; output;
end;
run;

/*------------ Linear PCA -----------*/
proc princomp data=original  standard noint cov
outstat=lin_stat(where=(_TYPE_ = 'USCORE'))
noprint;
var x1 x2;
run;

data lin_stat_score;
set lin_stat;
_TYPE_= 'PARMS';
run;

proc score data=original   score=lin_stat_score  type=parms   out=lin_pca;
var x1 x2;
run;

/*--------- Kernel PCA 1.0 ------------*/
/* Inner product based kernel PCA */
proc transpose data=original(drop=ID  Class) out=original_t; run;

proc corr data=original_t
outp=inner(where=(_TYPE_='SSCP' & substr(_NAME_, 1, 3)='COL')
drop=intercept)
sscp noprint;
var col:;
run;

/* apply kernel functon to inner product matrix. Here we use (X'Y+1)^2, the
one used in wikipedia.org example. */
data inner;
set inner;
array _C{*} _numeric_;
_TYPE_='COV';
do j=1 to dim(_C); _C[j]=(_C[j]+1)**2; end;
drop j;
run;

proc princomp data=inner    noint  cov  standard
outstat=k_stat(where=(_TYPE_ in ('EIGENVAL', 'USCORE')) )
noprint;
var col:;
run;

data S(drop=_NAME_)  score;
set k_stat;
if _TYPE_='EIGENVAL' then output S; else output score;
drop _TYPE_;
run;

data _null_;
set S;
array _S{*} _numeric_;
do j=1 to dim(_S);
if _S[j]<0.5*constant('MACEPS') then do;
call symput('maxsingval', j-1); stop;
end;
end;
run;
%put &maxsingval;

data score;
set score; if _n_>&maxsingval then delete; retain _TYPE_ 'PARMS';
run;

proc score data=inner   score=score  type=parms  out=k_U(keep=Prin:);
var col:;
run;

data k_U;
set original(keep=ID Class  x1 x2)  ;
set k_U;
run;

/*
proc gplot data=k_U;
plot Prin1*Prin2=Class;
run;quit;
*/

/*--------- Kernel PCA 2.0------------*/
/* Frobenius norm based kernel PCA */
proc transpose data=original(drop=ID  Class) out=original_t; run;

proc corr data=original_t
outp=inner2(where=(_TYPE_='SSCP' & substr(_NAME_, 1, 3)='COL')
drop=intercept)
sscp noprint;
var col:;
run;

proc means data=original_t noprint ;
var col:;
output out=_uss2(drop=_TYPE_    _FREQ_)    uss= /autoname;
run;

proc transpose data=_uss2  out=_uss2t; run;

proc sql noprint;
select nobs into :NCOLS from sashelp.vtable
where libname='WORK'
and memtype='DATA'
and memname='_USS2T'
;
quit;
%let ncols=%sysfunc(compress(&NCOLS));
%put &ncols;

/* apply kernel functon to inner product matrix. Here we use Gaussian kernel function. */
%let sigma=1;
data inner2;
array _X{&ncols} COL1-COL&ncols;
array _USS0{&ncols}  _temporary_;
if _n_=1 then do;
do j=1 to &ncols;
set  _uss2t(keep=COL1)  point=j;
_USS0[j]=COL1;
end;
end;
set inner2;
_TYPE_='COV';
do j=1 to &ncols;
_X[j]=_USS0[_n_] + _USS0[j] - 2*_X[j];
_X[j]=exp(-0.5*_X[j]/σ);
end;
keep _TYPE_  _NAME_ COL1-COL&ncols;
run;

proc princomp data=inner2    noint  cov  standard
outstat=k_stat2(where=(_TYPE_ in ('EIGENVAL', 'USCORE')) )
noprint;
var col:;
run;

data S2(drop=_NAME_)  score2;
set k_stat2;
if _TYPE_='EIGENVAL' then output S2; else output score2;
drop _TYPE_;
run;

data _null_;
set S2;
array _S{*} _numeric_;
do j=1 to dim(_S);
if _S[j]<0.5*constant('MACEPS') then do;
call symput('maxsingval', j-1); stop;
end;
end;
run;
%put &maxsingval;

data score2;
set score2; if _n_>&maxsingval then delete; retain _TYPE_ 'PARMS';
run;

proc score data=inner2   score=score2  type=parms  out=k_U2(keep=Prin:);
var col:;
run;

data k_U2;
set original(keep=ID Class  x1 x2)  ;
set k_U2;
run;

/*@ Draw figure using R @*/

%macro RScript(Rscript);
data _null_;
file "&Rscript";
infile cards ;
input;
put _infile_;
%mend;

%macro CallR(Rscript, Rlog);
systask command "D:\Progra~1\Analyt~1\R\bin\R.exe CMD BATCH --vanilla --quiet
%mend;

options nosource;
proc export data=original   outfile="c:\o.csv"  dbms=csv; run;
proc export data=lin_PCA    outfile="c:\a.csv"  dbms=csv; run;
proc export data=k_U       outfile="c:\b.csv"  dbms=csv; run;
proc export data=k_U2     outfile="c:\c.csv"   dbms=csv; run;
options source;

%RScript(c:\rscript.r)
cards;
png(file='c:/figure.png')
par(mfcol=c(2,2), mar=c(4,4,3,2))
plot(x1~x2, data=odsn, col=Class, main='Original')
plot(Prin1~Prin2, data=bdsn, col=Class, main='Polynomial Kernel')
plot(Prin1~Prin2, data=cdsn, col=Class, main='Gauss Kernal')
dev.off()
;
run;

%CallR(c:\rscript.r, c:\rlog1.txt);
``````

Chapter 1 & 2 of
Principal Manifolds for Data Visualization and Dimension Reduction by A. Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.) Lecture Notes in Computational Science and Engineering, Springer 2007

 This post was kindly contributed by SAS Programming for Data Mining Applications - go there to comment and to read the full post.

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