# Chapter 7 Lab: Non-linear Modeling library(ISLR) attach(Wage) # Polynomial Regression and Step Functions 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 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 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")