# Aufgabe 1
#
attach(olives)
names(olives)

fat.ac <- olives[4:11]

PCAm <- as.data.frame(as.matrix(normdf(fat.ac)) %*% eigen(cor(fat.ac))$vectors)

write.table(cbind(Region, PCAm), "Oliven.dat", sep=" ", row.name=F, col.names=F, quote=F)
write.table(c("Region", names(PCAm)), sep=" ", row.name=F, col.names=F, quote=F)

plot(PCAm, col=Region)

plot(as.data.frame(predict(princomp(fat.ac, cor=T))), col=Region)

# Aufgabe 2
#
attach(crabs)
names(crabs)

crabsM <- crabs[4:8]

plot(princomp(crabsM, cor=T))

plot(as.data.frame(predict(princomp(crabsM, cor=T))), col=sp-1+2*(sex-1)+1, main="Cor")

plot(as.data.frame(predict(princomp(crabsM, cor=F))), col=sp-1+2*(sex-1)+1, main="Cov")

# Aufgabe 3
#
# 3 Faktoren: Schnell, Kraft, Technik
#
ZKP <- Zehnkampf[14:23]

ZKF2 <- factanal(ZKP, 2)

round(
cor(ZKP) - 
(ZKF2$loadings[,1:2]) %*% t(ZKF2$loadings[,1:2]) -
diag(1 - apply(ZKF2$loadings[,1:2]**2, 1, sum))
, 3)

ZKF3 <- factanal(ZKP, 3)

round(
cor(ZKP) - 
(ZKF3$loadings[,1:3]) %*% t(ZKF3$loadings[,1:3) -
diag(1 - apply(ZKF3$loadings[,1:3]**2, 1, sum))
, 3)

plot(ZKF2$loadings)
text(ZKF2$loadings+0.035, names(ZKP))

plot(as.data.frame(ZKF3$loadings[]))

# Aufgabe 4
#
sfr <- factanal(stocks, 2)

lbd <- det((sfr$loadings[]) %*% t(sfr$loadings[]) + diag(1-apply(sfr$loadings[]**2, 1, sum)))/
det(cor(stocks))

n<-100
p<-5
k<-2

n*log(lbd)
(n-1-(2*p+4*k+5)/6)*log(lbd)
qchisq(0.95, ((p-k)**2 - p - k)/2)

# Aufgabe 5
#
attach(swiss)
names(swiss)

soc <- swiss[, c(2,4,5)]
phy <- swiss[, c(-2,-4,-5)]

round(cor(swiss), 3)

swissCC <- cancor(soc, phy)

U <- as.matrix(soc)%*%swissCC$xcoef

V <- as.matrix(phy)%*%swissCC$ycoef

UV <- cbind(U,V)

round(cor(UV), 3)

plot(as.data.frame(UV))

plot(swiss)