###### 단순선형회귀 ########
women
plot(weight~height, data=women)
fit <- lm(weight~height, data=women)
abline(fit, col="blue")

summary(fit)
cor.test(women$weight, women$height)
0.9954948^2

plot(fit)

par(mfrow=c(2,2))
plot(fit)
par(mfrow=c(1,1))

### 다항회귀(polynomial regression)
fit2 <- lm(weight~height+ I(height^2), data = women) 
summary(fit2)
lines(women$height, fitted(fit2), col="red")
plot(fit2)

fit3 <- lm(weight~height + I(height^2) + I(height^3), data = women)
plot(fit3)

install.packages("gvlma")
library(gvlma)
gvmodel <- gvlma(fit)
summary(gvmodel)

########  다중회귀분석 ##########
state.x77
states <- as.data.frame(state.x77[,c("Murder","Population","Illiteracy","Income","Frost")])
states

fit = lm(Murder~Population +Illiteracy + Income + Frost , data=states)
plot(fit)
summary(fit)

install.packages("car", dependencies = TRUE)
library(car)
vif(fit)
sqrt(vif(fit))

###### 이상관측치 #######
influencePlot(fit, id.method = "identify")

states["Nevada",]
fitted(fit)["Nevada"]
residuals(fit)["Nevada"]

##### 회귀모형의 교정 ######
states
summary(powerTransform(states$Murder))

boxTidwell(Murder~ Population + Illiteracy , data= states)

ncvTest(fit)
spreadLevelPlot(fit)

##### 예측 변수 선택 #####

fit1 <- lm(Murder~ ., data=states)
summary(fit1)

fit2 <- lm(Murder ~ Population + Illiteracy , data=states)
summary(fit2)

#AIC (Akaite's An information Criterion)
AIC(fit1, fit2)

###### stepwise regression (Backward stepwise regression, Forward stepwise regression)
#Backward stepwise regression
full.model = lm(Murder~. , data = states)
reduced.model = step(full.model, direction = "backward")
summary(reduced.model)

#Forward stepwise regression
min.model = lm(Murder~1, data = states)
fwd.model  <- step(min.model, direction="forward", scope = (Murder~Population+ Illiteracy + Income
                                                           + Frost), trace=0)
summary(fwd.model)

## all subset regression
library(leaps)
leaps <- regsubsets(Murder~ Population + Illiteracy + Income + Frost, data= states, nbest=4)
plot(leaps, scale="adjr2")