#	LA rent example

la.rent <- read.table("D:/Courses/regression/data sets/LARENT.txt")

rent.price <- la.rent[,1]
area.bath  <- la.rent[,6]
dist.beach <- la.rent[,8]
dist.ucla  <- la.rent[,9]
 

############################################


lm.12 <- lm(rent.price ~ area.bath + dist.beach)
summary(lm.12)

y <- rent.price
x.mat <- cbind(intercept=rep(1,26),area.bath,dist.beach)

solve( t(x.mat) %*% x.mat ) %*% t(x.mat) %*% y

MSE	<- sum(lm.12$res^2) / (26 - 3)
sqrt(diag(MSE*solve( t(x.mat) %*% x.mat )))


############################################

predict(lm.12,interval = "confidence")[1:4,]

(lambda.3 	<- model.matrix(lm.12)[3,])
(yhat.3	<- t(lambda.3) %*% lm.1$coef)
(se.3		<- sqrt( MSE * t(lambda.3) %*% solve( t(x.mat) %*% x.mat) %*% lambda.3))
yhat.3 - se.3 * qt(0.975,23)
yhat.3 + se.3 * qt(0.975,23)



############################################


summary(lm.12)

(SST	<- sum((y - mean(y))^2))
(SSE	<- sum((y - lm.12$fit)^2))
(SSR	<- sum((lm.12$fit - mean(y))^2))

(SSE + SSR)
(F	<- (SSR / 2)/(SSE / 23)) 
1-pf(F,2,23)


#	R-squared

1 - SSE/SST

#	adjusted R-squared

1 - (SSE/(26-3))/(SST/(26-1))



############################################

lm.1 <- lm(rent.price ~ area.bath)

summary(lm.1)

summary(lm.1)$fstatistic
sqrt(summary(lm.1)$fstatistic[1])

2*(1-pt(1.2,13))
1-pf(1.2^2,1,13)



############################################

lm.123 <- lm(rent.price ~ area.bath + dist.beach + dist.ucla)
summary(lm.123)

(SSE.1	<- sum((y - lm.1$fit)^2))
(SSE.123	<- sum((y - lm.123$fit)^2))

(F = ((SSE.1 - SSE.123)/(24-22))/(SSE.123/22))
1 - pf(F,2,22)

anova(lm.1,lm.123)


############################################

lm.123 <- lm(rent.price ~ area.bath + dist.beach + dist.ucla)
lm.12  <- lm(rent.price ~ area.bath + dist.beach)
lm.1   <- lm(rent.price ~ area.bath )
lm.int <- lm(rent.price ~ 1)

summary(lm.int)
mean(y)
sd(y)

(SSE.int	<- sum((y - lm.int$fit)^2))
var(y)*25

SSE.1		<- sum((y - lm.1$fit)^2)
SSE.12	<- sum((y - lm.12$fit)^2)
SSE.123	<- sum((y - lm.123$fit)^2)

SSE.int - SSE.1
SSE.1   - SSE.12
SSE.12  - SSE.123
SSE.123

anova(lm.123)

(SSE.int - SSE.1)  / (SSE.123/22)
(SSE.1   - SSE.12) / (SSE.123/22)
(SSE.12  - SSE.123)/(SSE.123/22)



############################################


vec.1	<- rep(1,26)
lm.12.noint  <- lm(rent.price ~ vec.1 + area.bath + dist.beach + 0)

summary(lm.12)
summary(lm.12.noint)

anova(lm.12)
anova(lm.12.noint)

sum((mean(y)*vec.1)^2)

(SST.noint	<- sum(y^2))
(SSR.noint	<- sum(lm.12.noint$fit^2))
(SSE.noint	<- sum(lm.12.noint$res^2))

SSR.noint + SSE.noint

SSR.noint/SST.noint

1 - (SSE.noint/(26-3))/(SST.noint/26)

(SSR.noint/3)/(SSE.noint/(26-3))




############################################


















(SSE.123	<- sum((y - lm.123$fit)^2))

(F = ((SSE.1 - SSE.123)/(24-22))/(SSE.123/22))

anova(lm.1,lm.123)
1 - pf(F,2,22)






















#	Geometric interpretation of R^2


lm.1 <- lm(rent.price ~ area.bath + dist.beach)

summary(lm.1)

X.1	<- 0.5*area.bath - 3*dist.beach
X.2	<- 1.5*area.bath - 14.5*dist.beach

lm.1a <- lm(rent.price ~ X.1 + X.2)

summary(lm.1a)

lm.1$fit[1:4]
lm.1a$fit[1:4]

(SSE.1   <- sum(lm.1$res^2))
(SSE.1a  <- sum(lm.1a$res^2))

(SST	<- sum((rent.price - mean(rent.price))^2))
SST  - SSE.1
(SSR	<- sum((lm.1$fit - mean(rent.price))^2))

SSR / SST


#	Regression without intercept


lm.3 <- lm(rent.price ~ area.bath + dist.beach + 0)
summary(lm.3)

x.mat <- cbind(area.bath,dist.beach)
solve( t(x.mat) %*% x.mat ) %*% t(x.mat) %*% y

vec.1	<- rep(1,26)
lm.1b <- lm(rent.price ~ vec.1 + area.bath + dist.beach + 0)
summary(lm.1b)

(SSE.1b   	<- sum(lm.1b$res^2))
(SST.1b	<- sum(rent.price^2))

SST.1b - SSE.1b
(SSR.1b	<- sum(lm.1b$fit^2))

SSR.1b / SST.1b

anova(lm.1)
anova(lm.1b)
sum(rep(mean(rent.price)^2,26))
sum(anova(lm.1b)[,2])







lm.123 <- lm(rent.price ~ area.bath + dist.beach + dist.ucla)
summary(lm.123)