---
layout: post
title: Non-linear Modeling
published: yes
---

## Hồi quy đa thức và Hàm bậc thang

```{r Polynomial Regression and Step Functions}
library(ISLR)
attach(Wage)

fit=lm(wage~poly(age,4),data=Wage)
coef(summary(fit))
fit2=lm(wage~poly(age,4,raw=T),data=Wage)
coef(summary(fit2))
fit2a=lm(wage~age+I(age^2)+I(age^3)+I(age^4),data=Wage)
coef(fit2a)
fit2b=lm(wage~cbind(age,age^2,age^3,age^4),data=Wage)
agelims=range(age)
age.grid=seq(from=agelims[1],to=agelims[2])
preds=predict(fit,newdata=list(age=age.grid),se=TRUE)
se.bands=cbind(preds$fit+2*preds$se.fit,preds$fit-2*preds$se.fit)
par(mfrow=c(1,2),mar=c(4.5,4.5,1,1),oma=c(0,0,4,0))
plot(age,wage,xlim=agelims,cex=.5,col="darkgrey")
title("Degree-4 Polynomial",outer=T)
lines(age.grid,preds$fit,lwd=2,col="blue")
matlines(age.grid,se.bands,lwd=1,col="blue",lty=3)
preds2=predict(fit2,newdata=list(age=age.grid),se=TRUE)
max(abs(preds$fit-preds2$fit))
fit.1=lm(wage~age,data=Wage)
fit.2=lm(wage~poly(age,2),data=Wage)
fit.3=lm(wage~poly(age,3),data=Wage)
fit.4=lm(wage~poly(age,4),data=Wage)
fit.5=lm(wage~poly(age,5),data=Wage)
anova(fit.1,fit.2,fit.3,fit.4,fit.5)
coef(summary(fit.5))
(-11.983)^2
fit.1=lm(wage~education+age,data=Wage)
fit.2=lm(wage~education+poly(age,2),data=Wage)
fit.3=lm(wage~education+poly(age,3),data=Wage)
anova(fit.1,fit.2,fit.3)
fit=glm(I(wage>250)~poly(age,4),data=Wage,family=binomial)
preds=predict(fit,newdata=list(age=age.grid),se=T)
pfit=exp(preds$fit)/(1+exp(preds$fit))
se.bands.logit = cbind(preds$fit+2*preds$se.fit, preds$fit-2*preds$se.fit)
se.bands = exp(se.bands.logit)/(1+exp(se.bands.logit))
preds=predict(fit,newdata=list(age=age.grid),type="response",se=T)
plot(age,I(wage>250),xlim=agelims,type="n",ylim=c(0,.2))
points(jitter(age), I((wage>250)/5),cex=.5,pch="|",col="darkgrey")
lines(age.grid,pfit,lwd=2, col="blue")
matlines(age.grid,se.bands,lwd=1,col="blue",lty=3)
table(cut(age,4))
fit=lm(wage~cut(age,4),data=Wage)
coef(summary(fit))
```

##  Splines

```{r Splines}
library(splines)
fit=lm(wage~bs(age,knots=c(25,40,60)),data=Wage)
pred=predict(fit,newdata=list(age=age.grid),se=T)
plot(age,wage,col="gray")
lines(age.grid,pred$fit,lwd=2)
lines(age.grid,pred$fit+2*pred$se,lty="dashed")
lines(age.grid,pred$fit-2*pred$se,lty="dashed")
dim(bs(age,knots=c(25,40,60)))
dim(bs(age,df=6))
attr(bs(age,df=6),"knots")
fit2=lm(wage~ns(age,df=4),data=Wage)
pred2=predict(fit2,newdata=list(age=age.grid),se=T)
lines(age.grid, pred2$fit,col="red",lwd=2)
plot(age,wage,xlim=agelims,cex=.5,col="darkgrey")
title("Smoothing Spline")
fit=smooth.spline(age,wage,df=16)
fit2=smooth.spline(age,wage,cv=TRUE)
fit2$df
lines(fit,col="red",lwd=2)
lines(fit2,col="blue",lwd=2)
legend("topright",legend=c("16 DF","6.8 DF"),col=c("red","blue"),lty=1,lwd=2,cex=.8)
plot(age,wage,xlim=agelims,cex=.5,col="darkgrey")
title("Local Regression")
fit=loess(wage~age,span=.2,data=Wage)
fit2=loess(wage~age,span=.5,data=Wage)
lines(age.grid,predict(fit,data.frame(age=age.grid)),col="red",lwd=2)
lines(age.grid,predict(fit2,data.frame(age=age.grid)),col="blue",lwd=2)
legend("topright",legend=c("Span=0.2","Span=0.5"),col=c("red","blue"),lty=1,lwd=2,cex=.8)
```

## GAMs

```{r GAMs}
gam1=lm(wage~ns(year,4)+ns(age,5)+education,data=Wage)
library(gam)
gam.m3=gam(wage~s(year,4)+s(age,5)+education,data=Wage)
par(mfrow=c(1,3))
plot(gam.m3, se=TRUE,col="blue")
plot.gam(gam1, se=TRUE, col="red")
gam.m1=gam(wage~s(age,5)+education,data=Wage)
gam.m2=gam(wage~year+s(age,5)+education,data=Wage)
anova(gam.m1,gam.m2,gam.m3,test="F")
summary(gam.m3)
preds=predict(gam.m2,newdata=Wage)
gam.lo=gam(wage~s(year,df=4)+lo(age,span=0.7)+education,data=Wage)
plot.gam(gam.lo, se=TRUE, col="green")
gam.lo.i=gam(wage~lo(year,age,span=0.5)+education,data=Wage)
library(akima)
plot(gam.lo.i)
gam.lr=gam(I(wage>250)~year+s(age,df=5)+education,family=binomial,data=Wage)
par(mfrow=c(1,3))
plot(gam.lr,se=T,col="green")
table(education,I(wage>250))
gam.lr.s=gam(I(wage>250)~year+s(age,df=5)+education,family=binomial,data=Wage,subset=(education!="1. < HS Grad"))
plot(gam.lr.s,se=T,col="green")
```
