##################
### DEVETI CAS ###
#################

# 1.
# broj sati ucenja
x <- c(4, 9, 10, 14, 4, 7, 12, 22, 1, 3, 8, 11, 5, 6, 10, 11, 16, 13, 13, 10)
# rezultati testa
y <- c(390, 580, 650, 730, 410, 530, 600, 790, 350, 400, 590, 640, 450, 520, 690, 690, 770, 700, 730, 640)
plot(x, y)
model1 <- lm(y ~ x)
model1

## Interval poverenja za srednju vrednost Y
predict(model1, data.frame(x = 10), interval = "confidence")  
# verujemo s poverenjem od 95% da je prosecan broj poena na testu onih koji su ucili 10 sati za test izmedju 697.6 i 768.5.

## Traka poverenja
interval <- predict(model1, interval = "confidence", data.frame(x = seq(0, 25, 1)))
plot(x, y, xlab = "x", ylab = "y")
abline(model1) # regresiona prava
lines(seq(0, 25, 1), interval[ ,3], col = "purple")
lines(seq(0, 25, 1), interval[ ,2], col = "purple")
# interval je najuzi za x_0 = mean(x)=9.45, jer je tu procena najpreciznija

## Interval predvidjanja 
predvidjanje <- predict(model1, interval = "prediction", data.frame(x = seq(0, 25, 1)))
lines(seq(0, 25, 1), predvidjanje[ ,3], col = "aquamarine3")
lines(seq(0, 25, 1), predvidjanje[ ,2], col = "aquamarine3")
# interval predvidjanja u svakoj tacki siri nego interval poverenja za srednju vrednost

## Testiranje hipoteze o parametrima regresije
summary(model1)
# p-vrednost testa je 0, pa odbacujemo H0, tj. linearni model je koristan za predvidjanje

## Dijagnostika modela
qqnorm(residuals(model1), ylab = "Residuals")
qqline(residuals(model1))
plot(fitted(model1), residuals(model1), xlab = "Fitted", ylab = "Residuals")


# 2.
obim <- c(30, 87, 129, 148, 168, 174, 177)
starost <- c(177, 484, 664, 1004, 1231, 1372, 1582)
plot(starost, obim)
model2 <- lm(obim ~ starost)
abline(model2)
summary(model2)
qqnorm(residuals(model2), ylab = "Residuals")
qqline(residuals(model2))
plot(fitted(model2), residuals(model2), xlab = "Fitted",ylab="Residuals")

model3 <- lm(log(obim) ~ starost)
summary(model3)
qqnorm(residuals(model3), ylab = "Residuals")
qqline(residuals(model3))
plot(fitted(model3), residuals(model2), xlab = "Fitted", ylab = "Residuals")

# 3. za vezbu

# 4. 
head(stackloss)
attach(stackloss)
vmodel1 <- lm(stack.loss ~ Air.Flow + Water.Temp + Acid.Conc.)
summary(vmodel1)
vmodel2 <- lm(stack.loss ~ Air.Flow + Water.Temp)
summary(vmodel2)
anova(vmodel1, vmodel2) #Acid.Conc.ne utice na stack.loss
# predvidjanje
newdata <- data.frame(Air.Flow = c(72, 67, 55), Water.Temp = c(20, 11, 22))
predict(vmodel2, newdata)

cor(stackloss)

# Dijagnostika modela
qqnorm(residuals(vmodel2), ylab = "Residuals")
qqline(residuals(vmodel2))
plot(fitted(vmodel2), residuals(vmodel2), xlab = "Fitted", ylab = "Residuals")

# 5. za vezbu