## R-code for assignment 3

#####################################
## Part 1
#####################################

## 2.24
Sigma <- diag(c(4,9,1),3)

# a.)
solve(Sigma)

# b.)
eigen(Sigma)

# c.) 
eigen(solve(Sigma))

## 2.25
Sigma <- matrix(c(25,-2,4,-2,4,1,4,1,9),3,3)

# a.)
V.sqrt <- sqrt(diag(diag(Sigma),3))
Rho <- solve(V.sqrt)%*%Sigma%*%solve(V.sqrt)

# b.)
Sigma.test <- V.sqrt%*%Rho%*%V.sqrt


## 3.18
x.bar <- matrix(c(0.766,0.508,0.438,0.161), 4,1)
S <- matrix(c(0.856, 0.635, 0.173, 0.096, 0.635, 0.568, 0.127, 0.067, 0.173, 0.128, 0.171, 0.039, 0.096, 0.067, 0.039, 0.043), 4,4)

# a.) 
total <- matrix(c(1,1,1,1),4,1) 

# sample mean and variance of total
x.bar.total <- t(total)%*%x.bar
s.sq.total <-  t(total)%*%S%*%total

x.bar.total
s.sq.total


# b.)
excess <- matrix(c(1,-1,0,0),4,1) 

# sample mean of t
x.bar.excess <- t(excess)%*%x.bar
s.sq.excess <-  t(excess)%*%S%*%excess

x.bar.excess
s.sq.excess

# covariance between total and excess
cov.total.excess <- t(total)%*%S%*%excess
cov.total.excess

#########################################
## Part 2
#########################################

glu <- read.table("glucose.txt")

summary(glu)


## boxplot
boxplot(glu, col="yellow", main="Glucose Measurements", cex.main=1.5)

## density plots
par(mfrow=c(3,2))
for(i in 1:6){
plot(density(glu[,i]), lwd=2, main=paste("Glucose", names(glu)[i]))
}

## pairs plot
par(mfrow=c(1,1))
pairs(glu, pch=16)

## spaghetti plot
par(mfrow=c(1,1))
time <- 1:3

plot(time, glu[1,1:3], lwd=2, ylim=c(48,170),type="l", ylab="glucose", xlab="time", cex.lab=1.5, main="Glucose Levels over Time")
lines(time, glu[1,4:6], lwd=2, lty=2)

for(i in 2:50){
lines(time, glu[i,1:3], lwd=2,type="l", col=i)
lines(time, glu[i,4:6], lwd=2, lty=2, col=i)
}

x.bar <- apply(glu, 2, mean)
lines(time, x.bar[1:3], lwd=6, type="l")
lines(time, x.bar[4:6], lwd=6, lty=2)

## plot the difference over time for each individual

diff <- matrix(0, 50,3)

for(i in 1:50){
diff[i,] <- as.numeric(glu[i,1:3] - glu[i,4:6])
}

plot(time, diff[1,], lwd=2, ylim=c(-104,15),type="l", ylab="glucose", xlab="time", cex.lab=1.5, main="Difference in Glucose Levels over Time Pre and Post")

for(i in 2:50){
lines(time, diff[i,], lwd=2,type="l", col=i)
}




##########################
sample.mean <- apply(glu, 2, mean)
S <- cov(glu)
R <- cor(glu)

y.bar <- sample.mean[1:3]
x.bar <- sample.mean[4:6]

sample.mean
y.bar
x.bar

S.yy <- S[1:3, 1:3]
S.yx <- S[4:6, 1:3]
S.xy <- S[1:3, 4:6]
S.xx <- S[4:6, 4:6] 

S
S.yy
S.yx
S.xy
S.xx

R.yy <- R[1:3, 1:3]
R.yx <- R[4:6, 1:3]
R.xy <- R[1:3, 4:6]
R.xx <- R[4:6, 4:6] 

R
R.yy
R.yx
R.xy
R.xx