ISLR 5.3 Lab: Cross-Validation and the Bootstrap

5.3.1 The Validation Set Approach

sample() function splits the set of observations into two halves, by selecting a random subset of 196 observations out of the original 392 observations.We refer to these observations as the training set.

> library (ISLR)
> set.seed (1)
> train=sample (392 ,196)

We then use the subset option in lm() to fit a linear regression using only the observations corresponding to the training set.

 lm.fit =lm(mpg∼horsepower ,data=Auto ,subset =train )

use the predict() function to estimate the response for all 392 observations, and we use the mean() function to calculate the MSE of the 196 observations in the validation set.

> attach (Auto)
> mean((mpg -predict (lm.fit ,Auto))[-train ]^2)
[1] 26.14142

use the poly() function to estimate the test error for the polynomial and cubic regressions.

> lm.fit2=lm(mpg∼poly(horsepower ,2) ,data=Auto ,subset =train )
> mean((mpg -predict (lm.fit2 ,Auto))[-train ]^2)
[1] 19.82259
> lm.fit3=lm(mpg∼poly(horsepower ,3) ,data=Auto ,subset =train )
> mean((mpg -predict (lm.fit3 ,Auto))[-train ]^2)
[1] 19.78252

5.3.2 Leave-One-Out Cross-Validation

In this lab, we will perform linear regression using the glm() function rather than the lm() function because
the latter can be used together with cv.glm(). The cv.glm() function is part of the boot library.

> library (boot)
> glm.fit=glm(mpg∼horsepower ,data=Auto)
> cv.err =cv.glm(Auto ,glm.fit)
> cv.err$delta
[1] 24.23151 24.23114

Our cross-validation estimate for the test error is approximately 24.23.

To automate the process, we use the for() function to initiate a for loop which iteratively fits polynomial regressions for polynomials of order i = 1 to i = 5, computes the associated cross-validation error, and stores it in the ith element of the vector cv.error. We begin by initializing the vector.

> 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
[1] 24.23151 19.24821 19.33498 19.42443 19.03321

the trend in cv.error indicates how cv is used for prm selection

5.3.3 k-Fold Cross-Validation

The cv.glm() function can also be used to implement k-fold CV.

> 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]
+ }

5.3.4 The Bootstrap

 Estimating the Accuracy of a Statistic of Interest

first create a function, alpha.fn(), which takes as input the (X, Y) data as well as a vector indicating which observations should be used to estimate α. The function then outputs the estimate for α based on the selected observations.

following command tells R to estimate α using all 100 observations.

> alpha.fn=function (data ,index){
+ X=data$X [index]
+ Y=data$Y [index]
+ return ((var(Y)-cov (X,Y))/(var(X)+var(Y) -2* cov(X,Y)))
+ }

The next command uses the sample() function to randomly select 100 observations from the range 1 to 100, with replacement. This is equivalent to constructing a new bootstrap data set and recomputing ˆα based on the new data set.

alpha.fn(Portfolio ,sample (100 ,100 , replace =T))

We can implement a bootstrap analysis by performing this command many times, recording all of the corresponding estimates for α, and computing the resulting standard deviation. However, the boot() function automates this approach. Below we produce R = 1, 000 bootstrap estimates for α.

> boot(Portfolio ,alpha.fn,R=1000)

ORDINARY NONPARAMETRIC BOOTSTRAP

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

Bootstrap Statistics :
     original        bias    std. error
t1* 0.5758321 -7.315422e-05  0.08861826

The final output shows that using the original data, ˆα = 0.5758, and that the bootstrap estimate for SE(ˆα) is 0.0886.

Estimating the Accuracy of a Linear Regression Model

We first create a simple function, boot.fn(), which takes in the Auto data set as well as a set of indices for the observations, and returns the intercept

and slope estimates for the linear regression model. We then apply this function to the full set of 392 observations in order to compute the estimates of β0 and β1 on the entire data set using the usual linear regression coefficient estimate formulas from Chapter 3.

> boot.fn=function (data ,index )
+ return (coef(lm(mpg∼horsepower ,data=data ,subset =index)))
> boot.fn(Auto ,1:392)
(Intercept)  horsepower
 39.9358610  -0.1578447 

Next, we use the boot() function to compute the standard errors of 1,000 bootstrap estimates for the intercept and slope terms

> boot(Auto ,boot.fn ,1000)

ORDINARY NONPARAMETRIC BOOTSTRAP

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

Bootstrap Statistics :
      original        bias    std. error
t1* 39.9358610  0.0269563085 0.859851825
t2* -0.1578447 -0.0002906457 0.007402954

Below we compute the bootstrap standard error estimates and the standard linear regression estimates that result from fitting the quadratic model to the data.

> boot.fn=function (data ,index )
+ coefficients(lm(mpg∼horsepower +I( horsepower ^2) ,data=data ,subset =index))
> set.seed (1)
> boot(Auto ,boot.fn ,1000)

ORDINARY NONPARAMETRIC BOOTSTRAP

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

Bootstrap Statistics :
        original        bias     std. error
t1* 56.900099702  6.098115e-03 2.0944855842
t2* -0.466189630 -1.777108e-04 0.0334123802
t3*  0.001230536  1.324315e-06 0.0001208339