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#MATH 5366 Data Mining Notes#
#Principal Components in R  #
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#Generating Example Data
R=cbind(c(0.924,0.381),c(0.381,-0.925))
D=diag(c(29.296,0.149))
Sigma0.5=t(R)%*%sqrt(D)%*%R
temp=t(Sigma0.5%*%matrix(rnorm(200),nrow=2))
x1=temp[,1]
x2=temp[,2]
x1=x1-mean(x1)
x2=x2-mean(x2)


#Plotting Data
plot(x1,x2)
hist(x1)
hist(x2)



#Principal Components
X=cbind(x1,x2)
S=cov(X)
S

eigen(S)

v1=eigen(S)$vectors[,1]
v2=eigen(S)$vectors[,2]


#Checking Orthonormality
t(v1)%*%v1
t(v2)%*%v2
t(v1)%*%v2


plot(x1,x2,asp=1)
xrange=min(x1)+(0:100)/100*(max(x1)-min(x1))
lines(xrange,v1[2]/v1[1] * xrange,col='red')
lines(xrange,v2[2]/v2[1] * xrange,col='red')


#Rotating Data with Principal Components
R=cbind(v1,v2)
W=X%*%R
cov(W)

plot(W,asp=1)

plot(X,asp=1)
head(X)

plot(W,asp=1)
head(W)