R Notebook

require(ISLR)

Loading required package: ISLR

require(boot)

Loading required package: boot

## LOOCV
glm.fit=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

We can do LOOCV without actually performing the model

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

[1] 24.23151

Fitting polynomials 1 to 5 on the data

cv.error=rep(0,5)
degree=1:5
for(d in degree){
  glm.fit=glm(mpg~poly(horsepower,d), data=Auto)
  cv.error[d]=loocv(glm.fit)
}
plot(degree,cv.error,type="b")

10 - fold cross validation

Bootstrap

Minimum risk investment - Section 5.2

alpha=function(x,y){
  vx=var(x)
  vy=var(y)
  cxy=cov(x,y)
  (vy-cxy)/(vx+vy-2*cxy)
}
alpha(Portfolio$X,Portfolio$Y)

[1] 0.5758321

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

[1] 0.5758321

Bootstrap resampling

boot.out

ORDINARY NONPARAMETRIC BOOTSTRAP


Call:
boot(data = Portfolio, statistic = alpha.fn, R = 1000)


Bootstrap Statistics :
     original       bias    std. error
t1* 0.5758321 -0.001695873  0.09366347

boot.out

ORDINARY NONPARAMETRIC BOOTSTRAP


Call:
boot(data = Portfolio, statistic = alpha.fn, R = 1000)


Bootstrap Statistics :
     original       bias    std. error
t1* 0.5758321 -0.001695873  0.09366347

plot(boot.out)