# Stochastik IV, Übungsblatt 2

# Aufgabe 4

# Einlesen der Daten und Einbinden in Suchpfad

data.crabs <- 
read.table("D:/Univ Augsburg/Lehre/SS07/VO_MultiVar/Datensaetze/crabs.txt", 
header = TRUE, sep = "\t", quote = "")

attach(data.crabs)

# Histogramme und überlagerte Kerndichteschätzer und Normalverteilungen

par(mfrow = c(3,2)) 

# Frontal Lobe Size (FL)

hist(FL, freq = FALSE, col ="grey", xlab = "FL (frontal lobe size, mm)")
curve(dnorm(x, mean(FL), sd(FL)), add = TRUE, col = "red", lwd = 3)
lines(density(FL), lwd = 3)

# Rear Width (RW)

hist(RW, freq = FALSE, col ="grey", xlab = "RW (rear width, mm)")
curve(dnorm(x, mean(RW), sd(RW)), add = TRUE, col = "red", lwd = 3)
lines(density(RW), lwd = 3)

# Carapace Length (CL)

hist(CL, freq = FALSE, col ="grey", xlab = "CL (carapace length, mm)", 
     ylim = c(0.00, 0.06))
curve(dnorm(x, mean(CL), sd(CL)), add = TRUE, col = "red", lwd = 3)
lines(density(CL), lwd = 3)

# Carapace Width (CW)

hist(CW, freq = FALSE, col ="grey", xlab = "CW (carapace width, mm)",
     ylim = c(0.00, 0.06))
curve(dnorm(x, mean(CW), sd(CW)), add = TRUE, col = "red", lwd = 3)
lines(density(CW), lwd = 3)

# Body Depth (BD)

hist(BD, freq = FALSE, col ="grey", xlab = "BD (body depth, mm)")
curve(dnorm(x, mean(BD), sd(BD)), add = TRUE, col = "red", lwd = 3)
lines(density(BD), lwd = 3)

frame()

legend(c(0,1), c(0.3,1), c("Histogram", "Kernel Density Estimate", 
                         "Normal Distribution"), 
       fill = c("grey", "black", "red"))

windows()

# QQ-Plots

par(mfrow = c(3,2)) 

# Frontal Lobe Size (FL)

plot(qnorm(ppoints(FL), mean(FL), sd(FL)), sort(FL), 
xlab = "Theoretical Quantiles of Normal Distribution",
ylab = "Sample Quantiles of FL", main ="QQ-Plot [FL]")
abline(0,1, col = "red", lwd = 2)
lines(c(quantile(FL, 0.25), quantile(FL, 0.75)),
c(qnorm(0.25, mean(FL), sd(FL)), qnorm(0.75, mean(FL), sd(FL))),
lty = 2, col = "blue", lwd = 2)

# Rear Width (RW)

plot(qnorm(ppoints(RW), mean(RW), sd(RW)), sort(RW), 
xlab = "Theoretical Quantiles of Normal Distribution",
ylab = "Sample Quantiles of RW", main ="QQ-Plot [RW]")
abline(0,1, col = "red", lwd = 2)
lines(c(quantile(RW, 0.25), quantile(RW, 0.75)),
c(qnorm(0.25, mean(RW), sd(RW)), qnorm(0.75, mean(RW), sd(RW))),
lty = 2, col = "blue", lwd = 2)

# Carapace Length (CL)

plot(qnorm(ppoints(CL), mean(CL), sd(CL)), sort(CL), 
xlab = "Theoretical Quantiles of Normal Distribution",
ylab = "Sample Quantiles of CL", main ="QQ-Plot [CL]")
abline(0,1, col = "red", lwd = 2)
lines(c(quantile(CL, 0.25), quantile(CL, 0.75)),
c(qnorm(0.25, mean(CL), sd(CL)), qnorm(0.75, mean(CL), sd(CL))),
lty = 2, col = "blue", lwd = 2)

# Carapace Width (CW)

plot(qnorm(ppoints(CW), mean(CW), sd(CW)), sort(CW), 
xlab = "Theoretical Quantiles of Normal Distribution",
ylab = "Sample Quantiles of CW", main ="QQ-Plot [CW]")
abline(0,1, col = "red", lwd = 2)
lines(c(quantile(CW, 0.25), quantile(CW, 0.75)),
c(qnorm(0.25, mean(CW), sd(CW)), qnorm(0.75, mean(CW), sd(CW))),
lty = 2, col = "blue", lwd = 2)

# Body Depth (BD)

plot(qnorm(ppoints(BD), mean(BD), sd(BD)), sort(BD), 
xlab = "Theoretical Quantiles of Normal Distribution",
ylab = "Sample Quantiles of BD", main ="QQ-Plot [BD]")
abline(0,1, col = "red", lwd = 2)
lines(c(quantile(BD, 0.25), quantile(BD, 0.75)),
c(qnorm(0.25, mean(BD), sd(BD)), qnorm(0.75, mean(BD), sd(BD))),
lty = 2, col = "blue", lwd = 2)

frame()

legend(c(0,1), c(0.3,1), c("Graph of QQ-Plot", "Line y = x", 
                         "Line Segment 1. and 3. quartiles"), 
       fill = c("black", "red", "blue"))

# Varianz-Kovarianz Matrix 

cov(data.crabs[, 4:8])