/* Filename: 	PCAExample.sas                                                              */
/* Purpose:		IML code that demonstrates PCA                                               */
/* Last Update:	FDN 4.23.02								    								*/


proc iml;
CALL GSTART ;

x= {2 1, 3 4, 5 0, 7 6, 9 2};   *create the data matrix;
print x;

call gstart;
run GXYPLOT(X[,1],X[,2]);	     *plot the data ;
call gclose;

xbar= repeat(x[+,]/nrow(x),nrow(x),1);  *get the means of the cols (variables);

print xbar;

y= x-xbar; 						  *Center the data;
print y;

call gstart;
run GXYPLOT(Y[,1],Y[,2]);	     *plot the centered data ;
call gclose;

sscp= y`*y;				 		 *get the syum of s1qured deviations and sum of cross products;

cov= sscp/(nrow(x)-1);			 *divide by degrees of freedom for the covariance matrix;

print cov;						 *note the variances on the diagonal and the covariances on the off diagnoal;

call eigen(l,u,cov);			 *compute the eigenvalues (l) and vectors (u);

print l;						 *Eigenvalues -- the variance of the scores of the the ojects on the PC's;

print u;						 *the eigenvectors -- the weights fo the linear combinations;

f= y*u;							  *compute the PC's -- these are the linear combinations ;
								  *note the use of the centered data;

								  * f[1,1] = y[1,1]u[1,1] + y[1,2]u[1,2], etc.; 
print f;	 					  *print the PC's;

call gstart;
run GXYPLOT(f[,1],f[,2]);	     *plot the PC's -- check out the rotation! ;
call gclose;

angles= (180/arcos(-1))*(arcos(u)); *convert the cosines to radians to decimal degrees please!
print angles;

s =(f`*f)/(nrow(f)-1);		      *now compute the variances and Covariances of the PC's; 
								 

print s;						  *Note the values -- some look familiar -- why the 0's?;





















print