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unit7.R
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####################################################
# The following code is for section 7.1
####################################################
# The Validation Set Approach
library(ISLR)
set.seed(1)
train=sample(392,196)
lm.fit=lm(mpg~horsepower,data=Auto,subset=train)
attach(Auto)
mean((mpg-predict(lm.fit,Auto))[-train]^2)
lm.fit2=lm(mpg~poly(horsepower,2),data=Auto,subset=train)
mean((mpg-predict(lm.fit2,Auto))[-train]^2)
lm.fit3=lm(mpg~poly(horsepower,3),data=Auto,subset=train)
mean((mpg-predict(lm.fit3,Auto))[-train]^2)
set.seed(2)
train=sample(392,196)
lm.fit=lm(mpg~horsepower,subset=train)
mean((mpg-predict(lm.fit,Auto))[-train]^2)
lm.fit2=lm(mpg~poly(horsepower,2),data=Auto,subset=train)
mean((mpg-predict(lm.fit2,Auto))[-train]^2)
lm.fit3=lm(mpg~poly(horsepower,3),data=Auto,subset=train)
mean((mpg-predict(lm.fit3,Auto))[-train]^2)
# Leave-One-Out Cross-Validation
glm.fit=glm(mpg~horsepower,data=Auto)
coef(glm.fit)
lm.fit=lm(mpg~horsepower,data=Auto)
coef(lm.fit)
library(boot)
glm.fit=glm(mpg~horsepower,data=Auto)
cv.err=cv.glm(Auto,glm.fit)
cv.err$delta
cv.err2 = NULL
indx = 1:length(mpg)
for(i in 1:length(mpg))
{
fit = lm(mpg~horsepower, data = Auto, subset = indx[-i])
cv.err2 = c(cv.err2, ( mpg[i] - predict(fit, new = data.frame(horsepower= horsepower[i])))^2)
}
cv.error=rep(0,5)
for (i in 1:5){
glm.fit=glm(mpg~poly(horsepower,i),data=Auto)
cv.error[i]=cv.glm(Auto,glm.fit)$delta[1]
}
cv.error
# k-Fold Cross-Validation
set.seed(17)
cv.error.10=rep(0,10)
for (i in 1:10){
glm.fit=glm(mpg~poly(horsepower,i),data=Auto)
cv.error.10[i]=cv.glm(Auto,glm.fit,K=10)$delta[1]
}
cv.error.10
cv.err.slr = NULL
indx = sample(1:length(mpg),392)
for(i in 1:9)
{
fit = lm(mpg~horsepower, data = Auto, subset = indx[- (((i-1)*39+1):(i*39))])
cv.err.slr = c(cv.err.slr, ( mpg[((i-1)*39+1):(i*39)] - predict(fit, new = data.frame(horsepower= horsepower[((i-1)*39+1):(i*39)])))^2)
}
fit = lm(mpg~horsepower, data = Auto, subset = indx[- (((10-1)*39+1):(392))])
cv.err.slr = c(cv.err.slr, ( mpg[((10-1)*39+1):(392)] - predict(fit, new = data.frame(horsepower= horsepower[((10-1)*39+1):(392)])))^2)
mean(cv.err.slr)
####################################################
# The following code is for section 7.2
####################################################
# simulate random variables
set.seed(999)
n<-20
rnorm(n,10,3) # normal distribution
runif(n,2,3) # uniform distribution between 2 and 3
rbeta(n,2,5) # beta distribution
hist(rnorm(1000,10,4))
hist(runif(1000,2,3))
hist(rbeta(1000,2,5))
mean(rbeta(10000,2,5))
sd(rnorm(10000,10,4))
####################################################
# The following code is for Section 7.3
####################################################
# population and sample
set.seed(1000)
N<-10000
X<-rnorm(N,10,3)
mean(X) # the population mean
var(X) # the population variance
n<-500
x<-sample(X,n,replace=F)
mean(x) # the sample mean
var(x) # the sample variance
# sample mean and sample variance
fsample<-function(mu,sigma,n,nsim)
{ xbar<-rep(0,nsim);xvar<-rep(0,nsim)
for(i in 1:nsim)
{set.seed(i)
# x contains a random sample of size n of the variable X
x<-rnorm(n,mu,sigma)
xbar[i]<-mean(x)
xvar[i]<-var(x)
}
cat('sample mean:',mean(xbar),"\n")
cat('sample variance:',mean(xvar),"\n")
# plot the samples
par(mfrow=c(2,2))
hist(xbar);qqnorm(xbar);qqline(xbar);
hist(xvar);qqnorm(xvar);qqline(xvar);
}
# sample mean and sample variance of samples of size 10
# from a normal distribution with mean 2 and standard deviation 2
fsample(2,2,10,10)
# increase simulation times to 100, 1000, 10000
fsample(2,2,10,100)
fsample(2,2,10,1000)
fsample(2,2,10,10000)
# change sample size
mu<-2;sigma<-2;n<-10;nsim<-1000
xbar<-rep(0,nsim);xbar2<-rep(0,nsim);xbar3<-rep(0,nsim)
xvar<-rep(0,nsim);xvar2<-rep(0,nsim);xvar3<-rep(0,nsim)
for(i in 1:nsim)
{set.seed(i)
x<-rnorm(n,mu,sigma);x2<-rnorm(n*10,mu,sigma)
x3<-rnorm(n*100,mu,sigma)
xbar[i]<-mean(x);xbar2[i]<-mean(x2);xbar3[i]<-mean(x3)
xvar[i]<-var(x);xvar2[i]<-var(x2);xvar3[i]<-var(x3)}
c(mean(xbar),mean(xbar2),mean(xbar3))
c(var(xbar),var(xbar2),var(xbar3))
c(mean(xvar),mean(xvar2),mean(xvar3))
# histogram and QQ-plot for xbar
par(mfcol=c(3,2))
hist(xbar);hist(xbar2);hist(xbar3)
qqnorm(xbar);qqline(xbar)
qqnorm(xbar2);qqline(xbar2)
qqnorm(xbar3);qqline(xbar3)
# histogram and QQ-plot for xvar
par(mfcol=c(3,2))
hist(xvar);hist(xvar2);hist(xvar3)
qqnorm(xvar);qqline(xvar)
qqnorm(xvar2);qqline(xvar2)
qqnorm(xvar3);qqline(xvar3)
# 95% confidence interval
alpha<-0.05
c(mean(x)-qt(1-alpha/2,n-1)*sd(x)/sqrt(n),mean(x)+qt(1-alpha/2,n-1)*sd(x)/sqrt(n))
# now repeat the above simulation for beta distribuiton
alpha<-2;beta<-4;n<-10;nsim<-1000
xbar<-rep(NA,nsim);xbar2<-rep(NA,nsim)
xbar3<-rep(NA,nsim)
xvar<-rep(NA,nsim);xvar2<-rep(NA,nsim)
xvar3<-rep(NA,nsim)
for(i in 1:nsim)
{set.seed(i)
x<-rbeta(n,alpha,beta);x2<-rbeta(n*10,alpha,beta)
x3<-rbeta(n*100,alpha,beta)
xbar[i]<-mean(x);xbar2[i]<-mean(x2);xbar3[i]<-mean(x3)
xvar[i]<-var(x);xvar2[i]<-var(x2);xvar3[i]<-var(x3)}
c(mean(xbar),mean(xbar2),mean(xbar3))
c(var(xbar),var(xbar2),var(xbar3))
c(mean(xvar),mean(xvar2),mean(xvar3))
# histogram and QQ-plot for xbar
par(mfcol=c(3,2))
hist(xbar);hist(xbar2);hist(xbar3)
qqnorm(xbar);qqline(xbar)
qqnorm(xbar2);qqline(xbar2)
qqnorm(xbar3);qqline(xbar3)
# histogram and QQ-plot for xvar
par(mfcol=c(3,2))
hist(xvar);hist(xvar2);hist(xvar3)
qqnorm(xvar);qqline(xvar)
qqnorm(xvar2);qqline(xvar2)
qqnorm(xvar3);qqline(xvar3)
# simulated value of the mean squared error of an estimator
# try it for n=10,50, 100
fMSE<-function(mu,sigma,n,nsim)
{xbar<-rep(NA,nsim)
for(i in 1:nsim)
{set.seed(i)
# x contains a random sample of size n of the variable X
x<-rnorm(n,mu,sigma);xbar[i]<-mean(x)};mse<-mean((xbar-mu)^2)
xvar<-var(xbar);bias<-mean(xbar)-mu
cat('case n=',n, ': mse=', mse, ', var=', xvar, ', bias=', bias, "\n")}
fMSE(2,2,10,1000)
fMSE(2,2,50,1000)
fMSE(2,2,100,1000)
################### A case study
# function to generate data
generateData <- function(n,p,beta)
{
X <- matrix(runif(n*p),n,p)
y <- rnorm(n,X%*%beta)
return(list(X=X,y=y))
}
# best subset selection given data
bestsubset <- function(Data)
{
require(leaps)
fit = regsubsets(y~X,Data)
b = numeric(ncol(Data$X)+1)
names(b) = fit$xnames
bb = coef(fit,which.min(summary(fit)[["cp"]]))
b[names(bb)] = bb
return(b)
}
# ridge regression
ridge <- function(Data,n.lam=501)
{
require(MASS)
lam = c(0,exp(seq(0,20,len=n.lam)))
fit = lm.ridge(y~X,Data,lambda=lam)
b = coef(fit)[which.min(fit$GCV),]
return(b)
}
# lasso variable selection
lasso <- function(Data)
{
require(glmnet)
cvfit = cv.glmnet(Data$X,Data$y)
b = as.numeric(coef(cvfit,s=cvfit$lambda.min))
return(b)
}
# the main body of simulation
N = 1000
n = 50
beta = c(3,-3,rep(0,8))
p = length(beta)
out = array(NA,dim=c(N,p,4),
dimnames=list(1:N,1:p,c("Subset","Lasso","Ridge","OLS")))
for (i in 1:N)
{
Data = generateData(n,p,beta)
out[i,,1] = bestsubset(Data)[-1]
out[i,,2] = lasso(Data)[-1]
out[i,,3] = ridge(Data)[-1]
out[i,,4] = coef(lm(y~X,Data))[-1]
}
bias <- apply(out,2:3,mean)-beta
variance <- apply(out,2:3,var)
MSE <- bias^2+variance
apply(MSE,2,sum)
par(mfrow=c(2,2))
ylim = range(out)
boxplot(out[,,1],col="blue",ylim=ylim,main="Subset")
boxplot(out[,,2],col="blue",ylim=ylim,main="Lasso")
boxplot(out[,,3],col="blue",ylim=ylim,main="Ridge")
boxplot(out[,,4],col="blue",ylim=ylim,main="OLS")
####################################################
# The following code is for Section 7.4
####################################################
# Estimating the Accuracy of a Linear Regression Model
boot.fn=function(data,index)
return(coef(lm(mpg~horsepower,data=data,subset=index)))
boot.fn(Auto,1:392)
set.seed(1)
boot.fn(Auto,sample(392,392,replace=T))
boot.fn(Auto,sample(392,392,replace=T))
boot(Auto,boot.fn,1000)
summary(lm(mpg~horsepower,data=Auto))$coef
boot.fn=function(data,index)
coefficients(lm(mpg~horsepower+I(horsepower^2),data=data,subset=index))
set.seed(1)
boot(Auto,boot.fn,1000)
summary(lm(mpg~horsepower+I(horsepower^2),data=Auto))$coef