#MULTIVARIATE NORMAL DISTRIBUTION AND CONFIDENCE ELLIPSE:
# CREATE DATA USING RANDOM NUMBER GENERATOR:

X1=rnorm(1000,0,2)
X2=rnorm(1000,0,1)
X=cbind(X1,X2)

#WRITE DATASET:
write.table(X,file="c:/DATA/Multivariate/randombivariate.txt")

#==============================================================
#START HERE IF YOU WANT TO USE THE SAME DATA MATRIX:

#READ DATASET FROM FILE:
M=read.table("c:/DATA/Multivariate/randombivariate.txt")

plot(M,asp=1,cex=0.5,pch=19,col='blue')

#CALCULATE MEAN AND STANDARD DEVIATION OF REALIZED DATA:
summary(M)
sd(M)

#CONFIDENCE ELLIPSE:
library(latticeExtra)
attach(M)

xyplot(X2 ~ X1, 
       scales = "free",aspect='iso',
       par.settings = list(plot.symbol = list(col='blue',cex = 0.5, pch=19)),
       panel = function(x, y, ...) {
           panel.xyplot(x, y, ...)
           panel.ellipse(x,y,lwd = 1,level=0.68,col='green', ...)
	   panel.ellipse(x,y,lwd = 1,level=0.90,col='brown', ...)
	   panel.ellipse(x,y,lwd = 2,level=0.95,col='red', ...)
       },
       auto.key = list(x = .1, y = .8, corner = c(0, 0)))

#==============================================================
#APPLYING CORRELATION BETWEEN VARIABLES IN MATRIX M 
#USING LINEAR TRANSFORMATION:
A=matrix(c(1,1.5,1.5,1),nrow=2,ncol=2,byrow=TRUE)
A

Mnew1=t(A%*%as.matrix(t(M)))
cor(Mnew1)

#CONFIDENCE ELLIPSE:
library(latticeExtra)
detach(M)

xyplot(Mnew1[,2] ~ Mnew1[,1], 
       scales = "free",aspect='iso',
       par.settings = list(plot.symbol = list(col='blue',cex = 0.5, pch=19)),
       panel = function(x, y, ...) {
           panel.xyplot(x, y, ...)
           panel.ellipse(x,y,lwd = 1,level=0.68,col='green', ...)
	   panel.ellipse(x,y,lwd = 1,level=0.90,col='brown', ...)
	   panel.ellipse(x,y,lwd = 2,level=0.95,col='red', ...)
       },
       auto.key = list(x = .1, y = .8, corner = c(0, 0)))

#==============================================================
#USING A DIFFERENT LINEAR TRANSFORMATION:
A=matrix(c(1,-0.25,-0.25,1),nrow=2,ncol=2,byrow=TRUE)
A

Mnew2=t(A%*%as.matrix(t(M)))
cor(Mnew2)

#CONFIDENCE ELLIPSE:
library(latticeExtra)

xyplot(Mnew2[,2] ~ Mnew2[,1], 
       scales = "free",aspect='iso',
       par.settings = list(plot.symbol = list(col='blue',cex = 0.5, pch=19)),
       panel = function(x, y, ...) {
           panel.xyplot(x, y, ...)
           panel.ellipse(x,y,lwd = 1,level=0.68,col='green', ...)
	   panel.ellipse(x,y,lwd = 1,level=0.90,col='brown', ...)
	   panel.ellipse(x,y,lwd = 2,level=0.95,col='red', ...)
       },
       auto.key = list(x = .1, y = .8, corner = c(0, 0)))