
#Regression1

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

#############
## Jalase1 ##
#############

x<- c(20,22,24,26,28,30,32,34,36,38,40,42)
y<- c(8.4,9.5,11.8,10.4,13.3,14.8,13.2,
14.7,16.4,16.5,18.9,18.5)
plot(x,y)
cor(x,y)
cor.test(x,y)
M<-lm(y~x)
M
plot(x,y)
abline(M,col="red",lty=2)
abline(a=0,b=0.48,col=4,lty=3)
yhat<-fitted(M)
yhat
plot(x,y)
abline(M,col="red",lty=2)
points(x,yhat,col=4,pch=20,cex=2)

R<-y-yhat
R
mean(R)
round(mean(R),15)

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

#############
## Jalase2 ##
#############

x<- c(15.5,23.75,8,17,5.5,19,24,2.5,7.5,11,13,3.75,25,9.75,22,18,6,12.5,2,21.5)
y<- c(2158.7,1678.15,2316,2061.3,2207.5,1708.3,1784.7,2575,2357.9,2265.7,2156.2,2399.55,1779.8,2336.75,1765.3,2053.5,2414.4,2200.5,2654.2,1753.7)
cbind(x,y)
plot(x,y,pch=20)
cor.test(x,y)
#?cor.test#
###spearmanTest Baraye Halat Naparametri###
cor.test(x, y,method ="spearman")
sum(x)
sum(y)
sum(x)^2
sum(x^2)
sum(x*y)
mean(y)
mean(x)
M<-lm(y~x)
M
plot(x,y,pch=20)
abline(M,lty=2,col=2)
abline(a=2628.04,b=-37.17,col=3) #y=a+bx
points(mean(x),mean(y),pch=3,col=4,cex=2)
text(13.5,2080,"(xbar,ybar)",col=4)
title("scatter plot of x,y")
text(15,2600," Pearson correlation=-0.949",col=4)


### Barresi Monasebat Model ###

#normal bodan khataha
R<-resid(M)
R
qqnorm(R)
qqline(R)
shapiro.test(R)
ks.test(R,"pnorm",mean(R),sd(R))
n=length(x)
n
MSE<-sum(R^2)/(n-2)
MSE
R.standard<-R/sqrt(MSE)
R.standard
plot(R.standard,ylim=c(-3,3))
abline(h=c(-3,3),col=2,lty=2)
shapiro.test(R.standard)
ks.test(R.standard,"pnorm")
rstandard(M)
rstudent(M)
plot(rstudent(M),ylim=c(-3,3))
abline(h=c(-2,2),col=2,lty=2)
points(5,rstudent(M)[5],pch=20,col=2,cex=0.8)
text(3,-2.7,"outlier",col=2)
car::outlierTest(M)


## Barazesh Model bedon Dade Part ##
x1<-x[-5]
x1
y1<-y[-5]
y1
M1<-lm(y1~x1)
M1
M
plot(x,y,pch=20)
abline(M,lty=2,col=2)
abline(M1,lty=4,col=4)
R1<-resid(M1)
R1
shapiro.test(R1)


### Ham Variansi ###

R<-resid(M)
yhat<-fitted(M)
plot(yhat,R)
abline(h=c(110,-220),col="red")
library(lmtest)
?bptest
bptest(M)
library(car)
ncvTest(M)

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

###########
##Jalase5##
###########


## Azmon Baresi NaHambaste Bodan Khata ha ##

M<-lm(y~x)
M
R<-resid(M)
R
ts.plot(R)
abline(h=0,lty=2,col="Red")
points(R,pch=20,col="blue")
library(lmtest)
dwtest(M)


### Azmon Farz ###

summary(M)

## Mohasebe Fasele Etminan Zarayeb ##

confint(M)
confint(M, level = 0.99)

## Ravesh Digar Azmon Kardan Zarayeb ##
# H0:Beta1=-30

M0=lm(y~offset(-30*x))
M0
anova(M0,M)

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

###########
##Jalase6##
###########


predict(M)
x0 <- data.frame(x = 5.5)
predict(M,x0, interval = "confidence",level=0.95)
predict(M, x0, se.fit = TRUE)
xnew<-data.frame(x=20)
predict(M,xnew, interval = "prediction",level=0.95)
anova(M)
sqrt(9263)


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

##############
## Jalase 7 ##
##############

x<-c(1,1,2,3.3,3.3,4,4,4,4.7,5,5.6,5.6,5.6,6,6,6.5,6.9)
y<-c(10.84,9.30,16.35,22.88,24.35,24.56,25.86,29.16,24.59,22.25,25.9,27.20,25.61,25.45,26.56,21.03,21.46)
cbind(x,y)
table(x)
plot(x,y,pch=20)
cor.test(x,y)
cor.test(x,y,method = "spearman")
shapiro.test(x)
shapiro.test(y)
qqnorm(x)
qqline(x)
qqnorm(y)
qqline(y)
#Azmon Lack of fit
M<-lm(y~x)
M
x1<-as.factor(x)
M1<-lm(y~x1)
M1
anova(M,M1)
plot(x,y,pch=20)
abline(M,lty=2,col=2)

#Tabdilat Khati Kardan Model
#example 7-3 Page(156)
x<-c(5,6,3.4,2.7,10,9.7,9.55,3.05,8.15,6.20,2.9,6.35,4.6,5.8,7.4,3.6,7.58,8.8,7,5.45,9.1,10.2,4.1,3.95,2.45)
y<-c(1.582,1.822,1.057,0.5,2.236,2.386,2.294,0.558,2.166,1.866,0.653,1.930,1.562,1.737,2.088,1.137,2.179,2.112,1.8,1.501,2.303,2.310,1.194,1.144,0.123)
cbind(x,y)
plot(x,y, pch=20,col=2)
cor.test(x,y)
cor.test(x,y, method="spearman")
M<-lm(y~x)
M
plot(x,y,pch=20,col=2)
abline(M,lty=2,col=4)
yhat<-fitted(M)
R<-resid(M)
plot(yhat,R,pch=20,col=6)
x2<-x^2
M2<-lm(y~x+x2)
M2
yhat<-fitted(M2)
R<-resid(M2)
plot(yhat,R,pch=20,col=6)
M3<-lm(y~I(log(x)))
M3
yhat<-fitted(M3)
R<-resid(M3)
plot(yhat,R,pch=20,col=6)
M4<-lm(y~I(1/x))
M4
yhat<-fitted(M4)
R<-resid(M4)
plot(yhat,R,pch=20,col=6)
plot(1/x,y,pch=20,col=2)
abline(M4,lty=2,col=4)
anova(M4)
anova(M)


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

##############
## Jalase 8 ##
##############

#Regression ChandGane

#example 1-4 page202
x1<- c(7,3,3,4,6,7,2,7,30,5,16,10,4,6,9,10,6,7,3,17,10,26,9,8,4)
x2<- c(560,220,340,80,150,330,110,210,1460,605,688,215,255,462,448,776,200,132,36,770,140,810,450,635,150)
y<- c(16.68,11.5,12.03,14.88,13.75,18.11,8,17.83,79.24,21.5,40.33,21,13.5,19.75,24,29,15.35,19,9.5,35.10,17.9,52.32,18.75,19.83,10.75)
cbind(x1,x2,y)
X<- cbind(1,x1,x2)
X
t(X)%*%X
sum(x1)
sum(x2)
sum(x1^2)
sum(x1*x2)
sum(x2^2)
solve(t(X)%*%X)
t(X)%*%y
sum(y)
sum(x1*y)
sum(x2*y)
Betahat<-solve(t(X)%*%X)%*%t(X)%*%y
Betahat

M<-lm(y~x1+x2)
M

anova(M)
summary(M)
vcov(M) 
pairs(cbind(y,x1,x2))
matplot(cbind(scale(x1),scale(x2)),y)
car::scatterplotMatrix(cbind(y,x1,x2))
install.packages("scatterplot3d")
library("scatterplot3d")
scatterplot3d(x1,x2,y, pch = 16, color="steelblue")
s3d<-scatterplot3d(x1,x2,y, pch = 16, color="steelblue")
s3d$plane3d(M)
cor(cbind(x1,x2,y))
cor.test(x1,x2)
cor.test(x1,y)
cor.test(x2,y)
confint(M)
#Monasebat Model
R<-resid(M)
R
qqnorm(R)
qqline(R)
shapiro.test(R)
ks.test(R,"pnorm")
yhat<-fitted(M)
plot(R,yhat)
lmtest::bptest(M)
car::ncvTest(M)

y1<-log(y)
M1<-lm(y1~x1+x2)
M1
anova(M1)
R1<-resid(M1)
qqnorm(R1)
qqline(R1)
shapiro.test(R1)

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

##############
## Jalase 9 ##
##############

#Example page212
y<-c(10,17,48,27,55,26,9,16)
x1<-c(2,3,4,1,5,6,7,8)
x2<-c(1,2,5,2,6,4,3,4)
cbind(x1,x2,y)
M<-lm(y~x1+x2)
M
anova(M)
summary(M)
vcov(M) 
pairs(cbind(y,x1,x2))
matplot(cbind(scale(x1),scale(x2)),y)
car::scatterplotMatrix(cbind(y,x1,x2))
library("scatterplot3d")
scatterplot3d(x1,x2,y, pch = 16, color="steelblue")
s3d<-scatterplot3d(x1,x2,y, pch = 16, color="steelblue")
s3d$plane3d(M)

#Monasebat Model
R<-resid(M)
R
qqnorm(R)
qqline(R)
shapiro.test(R)
ks.test(rstandard(M),"pnorm")
yhat<-fitted(M)
plot(R,yhat)
lmtest::bptest(M)
car::ncvTest(M)
ts.plot(R)
points(R,col=2,pch=20)
abline(h=0,lty=2)
lmtest::dwtest(M)

##Azmon Farz
#H0:beta1=beta2
FM<-lm(y~x1+x2)
RM<-lm(y~I(x1+x2))
anova(RM,FM)
#H0:beta1=-4
RM<-lm(y~offset(-4*x1)+x2)
anova(RM,FM)
confint(FM)
#H0:beta1=-5
RM<-lm(y~offset(-5*x1)+x2)
anova(RM,FM)

#example 1-4 page202
x1<- c(7,3,3,4,6,7,2,7,30,5,16,10,4,6,9,10,6,7,3,17,10,26,9,8,4)
x2<- c(560,220,340,80,150,330,110,210,1460,605,688,215,255,462,448,776,200,132,36,770,140,810,450,635,150)
y<- c(16.68,11.5,12.03,14.88,13.75,18.11,8,17.83,79.24,21.5,40.33,21,13.5,19.75,24,29,15.35,19,9.5,35.10,17.9,52.32,18.75,19.83,10.75)
M<-lm(y~x1+x2)
M
confint(M)
library(car)
linearHypothesis(M, c("x1 = -4"))
linearHypothesis(M, c("x1 = 1.5"))
linearHypothesis(M, c("x1 = x2"))
linearHypothesis(M, c("x1 = 2","x2 = 3"))
RM<-lm(y~offset(2*x1)+offset(3*x2))
anova(RM,M)

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

###############
## Jalase 10 ##
###############

#example 1-4 page202
x1<- c(7,3,3,4,6,7,2,7,30,5,16,10,4,6,9,10,6,7,3,17,10,26,9,8,4)
x2<- c(560,220,340,80,150,330,110,210,1460,605,688,215,255,462,448,776,200,132,36,770,140,810,450,635,150)
y<- c(16.68,11.5,12.03,14.88,13.75,18.11,8,17.83,79.24,21.5,40.33,21,13.5,19.75,24,29,15.35,19,9.5,35.10,17.9,52.32,18.75,19.83,10.75)
cbind(x1,x2,y)
M<-lm(y~x1+x2)
M
plot(x1,x2)
xnew<-data.frame(x1=5,x2=200)
predict(M, xnew, se.fit = TRUE,interval = "confidence")
predict(M, xnew, se.fit = TRUE,interval = "prediction")


R<-resid(M)
rstandard(M)
rstudent(M)
plot(rstandard(M))
abline(h=c(0,-2,2),lty=3,col=4)
plot(rstudent(M))
abline(h=c(0,-2,2),lty=3,col=4)
car::outlierTest(M)
#hatMatrix
X<- cbind(1,x1,x2)
X
H<-X%*%solve(t(X)%*%X)%*%t(X) 
hii<-hat(X)
diag(H)
hii
sum(hii)
sum(H[1,])
sum(H[2,])
sum(H[5,])
H
plot(hii)
p<-sum(hii)
n<-length(y)
abline(h=2*p/n,lty=2,col=2)
library(car)
cook<-cooks.distance(M)
plot(cook)
abline(h=4/(n-p-1),lty=2,col=2)
dfbeta(M)
plot(dfbeta(M)[,2])
plot(dfbeta(M)[,3])
abline(h=c(2/sqrt(n),-2/sqrt(n)),lty=2,col=2)
par(mfrow=c(2,2))
plot(M)
vif(M) #HamKhati

#Entekhab Motaghayer Dar Model
#Backward
FM<-lm(y~x1+x2) #full Model
NM<-lm(y~1)  #None Model
step(FM,direction = "backward")
#forward
step(NM,scope=list(upper=FM),direction = "forward")
#cp malouz
A<-summary(FM)
names(A)
MSE<-A$sigma^2
step(NM,scope=list(upper=FM),direction = "forward",scale=MSE)
step(FM,scope=list(upper=FM),direction = "backward",scale=MSE)

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

###############
## Jalase 11 ##
###############

#Example7-1 Page426
x1<-c(7,1,11,11,7,11,3,1,2,21,1,11,10)
x2<-c(26,29,56,31,52,55,71,31,54,47,40,66,68)
x3<-c(6,15,8,8,6,9,17,22,18,4,23,9,8)
x4<-c(60,52,20,47,33,22,6,44,22,26,34,12,12)
y<-c(78.5,74.3,104.3,87.6,95.9,109.2,102.7,72.5,93.1,115.9,83.8,113.3,109.4)
cbind(x1,x2,x3,x4,y)
pairs(cbind(y,x1,x2,x3,x4))
car::scatterplotMatrix(cbind(y,x1,x2,x3,x4))
M<-lm(y~x1+x2+x3+x4)
M
#Backward
FM<-lm(y~x1+x2+x3+x4) #full Model
NM<-lm(y~1)  #None Model
step(FM,direction = "backward")
#forward
step(NM,scope=list(upper=FM),direction = "forward")
#stepwise
step(NM,scope=list(upper=FM),direction = "both")
#cp malouz
A<-summary(FM)
names(A)
MSE<-A$sigma^2
step(NM,scope=list(upper=FM),direction = "forward",scale=MSE)
step(FM,scope=list(upper=FM),direction = "backward",scale=MSE)
car::vif(M)

#Example page212
x1<-c(2,3,4,1,5,6,7,8)
x2<-c(1,2,5,2,6,4,3,4)
y<-c(10,17,48,27,55,26,9,16)
cbind(x1,x2,y)
X<-cbind(1,x1,x2)
X
t(X)%*%X
sum(x1)
sum(x2)
sum(x1^2)
sum(x1*x2)
sum(x2^2)
sum(y)
sum(x1*y)
sum(x2*y)
det(t(X)%*%X)
solve(t(X)%*%X)
t(X)%*%y
Betahat<-solve(t(X)%*%X)%*%t(X)%*%y
Betahat

M<-lm(y~x1+x2)
M
