setwd("C:\\Users\\authorized user\\Dropbox\\STAT445\\week10\\tutorials")


data <- as.matrix(read.csv("T4-3.csv"))


############################################################
#assess univariate normality of the data
############################################################
qqnorm(data[ ,2])
qqline(data[,2])


qqnorm(data[,3])
qqline(data[,3])



############################################################


############################################################
# Construct the q-q plot for assesing normality of 
#               multivariate data

############################################################
#if all marginal (univariate) distributions are normal
#that does not imply that their join distribution will
#be multivariate normal
#assess multivariate normal by using squared Mahalanobis
#distance.
#that comes from t(data-mu)*sigma(^-1)(data-mu) which is
#squared Mahalanobis distance (generalized distance) 
# and the distribution of this
#is chi-square with degrees of freedom equal to the number
#of variables in the multivariate data set.
#So if we plot the Mahalanobis distance we should see 
#aplot of chi-square
#and if we compare Mahalanobis distance distribution 
#with chi-square with degrees of freedom equal to the number
#of variables in the multivariate data set we should get 
#data and the line matching

############################################################

###Set the values of n (number of observation units),
###and p (the dimension of the observation vector- number of variables).

n<-dim(data)[1]
p<-dim(data)[2]

###Compute the mean vector and variance-covariance matrix.

xbar <- apply(data,2, "mean")
sx <- cov(data)


###Compute the generalized distances from the sample mean.

diffs <- data - matrix(xbar, nrow=n, ncol=p, byrow=TRUE)

gdist <- diag(diffs%*%solve(sx)%*%t(diffs))

###Sort these. 

s.gdist <-sort(gdist)

###Find the matching list of quantiles.

quant <- qchisq(((1:n)-0.5)/n, p)



###Plot the sorted generalized distances vs. the quantiles.
par(mfrow=c(2,1))
plot(density(diffs),main="Distribution of Generalized distances")

plot(quant, s.gdist,pch=20, main="Q-Q Plot of Generalized distances vs. 
     Quantiles of Chi Squared")
###Add a 45-degree line.
lines(quant,quant)