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## SHEET 2 ##
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##################################################
## Exercise 1
##################################################

## load data
d.banknoten <- read.table("http://www.stat.math.ethz.ch/~stahel/courses/multivariate/datasets/banknoten.dat", header=TRUE)

## extract unforged notes (CODE==0), get rid of CODE variable
d.banknoten <- d.banknoten[d.banknoten[,"CODE"]==0,-1]

## 1a
r.mean <- mean(d.banknoten)
r.sd <- sd(d.banknoten)
round(r.mean,3)
round(r.sd,3)

## 1b
r.var <- var(d.banknoten)
r.cor <- cor(d.banknoten)
round(r.var, 2)
round(r.cor, 2)

## 1c
C <- solve(t(chol(r.var)))
d.banknoten.st <- as.data.frame(scale(d.banknoten, scale=FALSE) %*% t(C))
       ## the function "scale" type converts the data frame
       ## "d.banknoten" into a matrix, and "as.data.frame" turns the
       ## matrix back into a data frame

par(mfrow=c(2,1))
plot(RIGHT~LEFT,data=d.banknoten.st,main="transformed data")
plot(RIGHT~LEFT,data=d.banknoten,main="original data")

##################################################
## Exercise 2
##################################################

## 2a
t.d <- iris[iris[,"Species"] == "setosa", 1:4]
r.mean <- mean(t.d)
r.sd <- sd(t.d)
round(r.mean,3)
round(r.sd,3)

## 2b
round(var(t.d), 2)
round(cor(t.d), 2)

## 2c
n <- nrow(t.d)
one <- rep(1,n)
t.dc <- as.matrix(t.d) - one %*% t(r.mean)
covEst <- t(t.dc) %*% t.dc / (n - 1)
round(covEst,2)

##################################################
## Exercise 3
##################################################

## 3a
xbar <- c(-1.5,0.3)
Sigma.x <- rbind(c(1.1,-0.3),c(-0.3,1.3))

a <- c(-1,2)
B <- rbind(c(2,0),c(1,-4))

ybar <- a+B%*%xbar
Sigma.y <- B %*% Sigma.x %*% t(B)

ybar
Sigma.y