# Aufgabe 1
############
# jahr <- 1976:1984
offen <- c(3010, 3041, 3439, 4382, 4379, 2911, 1335, 987, 1217)
arblose <- c(15166, 15594, 15420, 13826, 14764, 19899, 28775, 36743, 38917)

# n <- length(offen)

Sxx <- var(offen)
Sxy <- cov(offen, arblose) # 1/(n-1)* sum((offen - mean(offen)) * (arblose - mean(arblose)))

beta <- Sxy / Sxx
alpha <- mean(arblose) - beta * mean(offen)

alpha + beta * 900

plot(offen, arblose)
abline(alpha, beta)


# Aufgabe 4
############
saege <- read.table("saegewerk.csv", header=TRUE)

y <- saege[[1]]

X <- cbind(1, saege[[2]], saege[[3]], saege[[4]])

solve(t(X)%*%X)

t(X)%*%y

solve(t(X)%*%X)%*%t(X)%*%y

# numerisch stabiler (was in dem Beispiel aber nichts ausmacht):
beta <- solve(t(X)%*%X, t(X)%*%y)

# Alternative:
lm(saege[[1]] ~ saege[[2]] + saege[[3]] + saege[[4]])
beta <- lm(saege[[1]] ~ saege[[2]] + saege[[3]] + saege[[4]])$coefficients

t(beta) %*% c(1, 100, 40, 20)