5.3.3 The Bootstrap
We illustrate the use of the bootstrap in the simple example of Section 5.2, as well as on an example involving estimating the accuracy of the linear regression model on the Auto data set.
Estimating the Accuracy of a Statistic of Interest
One of the great advantages of the bootstrap approach is that it can be applied in almost all situations. No complicated mathematical calculations are required. While there are several implementations of the bootstrap in Python, its use for estimating standard error is simple enough that we write our own function below for the case when our data is stored in a dataframe.
To illustrate the bootstrap, we start with a simple example. The Portfolio data set in the ISLP package is described in Section 5.2. The goal is to estimate the sampling variance of the parameter α given in formula (5.7). We will create a function alpha_func() , which takes as input a dataframe D assumed to have columns X and Y , as well as a vector idx indicating which observations should be used to estimate α . The function then outputs the estimate for α based on the selected observations.
In [15]:Portfolio=load_data('Portfolio')
defalpha_func(D,idx):
cov_=np.cov(D[['X','Y']].loc[idx],rowvar=False)
return((cov_[1,1]-cov_[0,1])/
(cov_[0,0]+cov_[1,1]-2*cov_[0,1]))
This function returns an estimate for α based on applying the minimum variance formula (5.7) to the observations indexed by the argument idx . For instance, the following command estimates α using all 100 observations.
In [16]:alpha_func(Portfolio,range(100))
5.3 Lab: Cross-Validation and the Bootstrap 221
Out[16]:0.5758
Next we randomly select 100 observations from range(100) , with replacement. This is equivalent to constructing a new bootstrap data set and recomputing α ˆ based on the new data set.
In [17]:rng=np.random.default_rng(0)
alpha_func(Portfolio,
rng.choice(100,
100,
replace=True))
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