Resampling Method

require(ISLR)

## Loading required package: ISLR

## Warning: package 'ISLR' was built under R version 3.1.2

#install.packages('boot')
require(boot)

## Loading required package: boot

## Warning: package 'boot' was built under R version 3.1.2

?cv.glm

## starting httpd help server ... done

plot(mpg~horsepower,data=Auto)

##LOOGV
glm.fit<- span=""> glm(mpg~horsepower,data=Auto)
cv.glm(Auto,glm.fit)$delta #pretty slow (doesnt use formula (5.2) on page 180)

## [1] 24.23151 24.23114

##Lets write a simple function to use formula (5.2)
loocv<- span=""> function(fit){
h<- span=""> lm.influence(fit)$h
mean((residuals(fit)/(1-h))^2)
}
## Now we try it out
loocv(glm.fit)

## [1] 24.23151

cv.error<- span=""> rep(0,5)
degree<- span=""> 1:5
for(d in degree){
glm.fit<- span=""> glm(mpg~poly(horsepower,d),data=Auto)
cv.error[d]<- span=""> loocv(glm.fit)
}
plot(degree,cv.error,type='b')
## 10-fold CV
cv.error10<- span=""> rep(0,5)
for(d in degree){
glm.fit<- span=""> glm(mpg~poly(horsepower,d),data=Auto)
cv.error10[d]<- span=""> cv.glm(Auto,glm.fit,K=10)$delta[1]
}
lines(degree,cv.error10,type='b',col='red')

## Bootstrap
## Minimum risk investment - Section 5.2
alpha<- span=""> function(x,y){
vx<- span=""> var(x)
vy<- span=""> var(y)
cxy<- span=""> cov(x,y)
(vy-cxy)/(vx+vy-2+cxy)
}
alpha(Portfolio$X,Portfolio$Y)

## [1] 0.6413231

## What is the standard error of alpha?
alpha.fn<- span=""> function(data,index){
with(data[index,],alpha(X,Y))
}
alpha.fn(Portfolio,1:100)

## [1] 0.6413231

set.seed(1)
alpha.fn(Portfolio,sample(1:100,100,replace=T))

## [1] 0.6727862

boot.out<- span=""> boot(Portfolio,alpha.fn,R=1000)
boot.out

## 
## ORDINARY NONPARAMETRIC BOOTSTRAP
## 
## 
## Call:
## boot(data = Portfolio, statistic = alpha.fn, R = 1000)
## 
## 
## Bootstrap Statistics :
##      original     bias    std. error
## t1* 0.6413231 0.08995096   0.4226578

plot(boot.out)