# Create and plot simulated category data


rm(list=ls())
X<-rnorm(200, mean=2, sd=2)
y1<-5*X+rnorm(200, mean=2, sd=3)
plot(X,y1,col="red",xlim=c(min(X)-5,max(X)+5),ylim=c(min(y1)-5,max(y1)+5))
df<-t(data.frame(rbind(X,y1)))



# recorders = data.frame("X"=c(0,0,1,1), "Y" = c(0,1,1,0),row.names=c("A", "B","C","D"))
# locs = data.frame("X"=c(.3,.5),"Y"=c(.8,.2))
# intensities = data.frame("sine"=sin(0:99*(pi/10))+1.2, "cosine"= .7*cos(0:99*(pi/15))+.9)
# dists = matrix(nrow=dim(locs)[1], ncol=dim(recorders)[1], dimnames=list(NULL, row.names(recorders)))
# for (i in 1:dim(dists)[2]){
#   dists[,i]=sqrt((locs$X-recorders$X[i])^2 + (locs$Y-recorders$Y[i])^2)}
# set.seed(500)
# recorded.data = data.frame(jitter(as.matrix(intensities)%*%as.matrix(exp(-2*dists)),amount=0))
                                  
                                  
# ==========  The Correlation Matrix ==========#
round(cor(df),2)


# ==========  PCA from Scratch ==========#
# Obtain data in a matrix
Xoriginal=as.matrix(df)
# Center the data so that the mean of each row is 0
rMeans=rowMeans(Xoriginal)
X=Xoriginal-matrix(rep(rMeans, dim(df)[2]), nrow=dim(df)[1])
# Calculate P
A=X %*% t(X)
E=eigen(A,TRUE)
P=t(E$vectors)
# Find the new data and standard deviations of the principal components
newdata = P %*% X
sdev = sqrt(diag((1/(dim(X)[2]-1)* P %*% A %*% t(P))))


intensities = data.frame("sine"=sin(0:99*(pi/10))+1.2, "cosine"= .7*cos(0:99*(pi/15))+.9)

# ==========  Internal R PCA  ==========#
pr=prcomp(df)
pr
plot(pr)
barplot(pr$sdev/pr$sdev[1])
pr2=prcomp(df, tol=.1)
plot.ts(pr2$x)
plot.ts(intensities)
plot.ts(df)
plot.ts(cbind(-1*pr2$x,pr2$y1))