§4 Cross-validation and the Bootstrap

Training Error versus Test Error

• Recall the distinction between the test error and the training error:
• The test error is the average error that results from using a statistical learning method to predict the response on a new observation, one that was not used in training the method.
• In contrast, the training error can be easily calculated by applying the statistical learning method to the observations used in its training.
• But the training error rate often is quite different from the test error rate, and in particular the former can dramatically underestimate the latter.

More on prediction-error estimates

• Best solution: a large designated test set. Often not available.
• Some methods make a mathematical adjustment to the training error rate in order to estimate the test error rate. These include the $**C_p$ statistic**, $AIC$ and $BIC$. They are discussed elsewhere in this course.
• Here we instead consider a class of methods that estimate the test error by holding out a subset of the training observations from the fitting process, and then applying the statistical learning method to those held out observations.

Validation-set approach

• Here we randomly divide the available set of samples into two parts: a training set and a validation or hold-out set.
• The model is fit on the training set, and the fitted model is used to predict the responses for the observations in the validation set.
• The resulting validation-set error provides an estimate of the test error. This is typically assessed using MSE in the case of a quantitative response and misclassification rate in the case of a qualitative (discrete) response.

The validation process

• A random splitting into two halves: left part is training set, right part is validation set.

Drawbacks of validation set approach

• The validation estimate of the test error can be highly variable, depending on precisely which observations are included in the training set and which observations are included in the validation set.
• In the validation approach, only a subset of the observations — those that are included in the training set rather than in the validation set — are used to fit the model.
• This suggests that the validation set error may tend to overestimate the test error for the model fit on the entire data set.

$K$-fold Cross-validation

• Widely used approach for estimating test error.
• Estimates can be used to select best model, and to give an idea of the test error of the final chosen model.
• Idea is to randomly divide the data into $K$ equal-sized parts. We leave out part $k$ , fit the model to the other $K-1$ parts (combined), and then obtain predictions for the left-out $k$ th part.
• This is done in turn for each part $k=1,2, \ldots K$ , and then the results are combined.

The details

• Let the $K$ parts be $C_{1}, C_{2}, \ldots C_{K}$ , where $C_{k}$ denotes the indices of the observations in part $k$. There are $n_{k}$ observations in part $k$ : if $N$ is a multiple of $K$ , then $n_{k}=n / K$

• Compute

  $$\mathrm{CV}_{(K)}=\sum_{k=1}^{K} \frac{n_{k}}{n} \mathrm{MSE}_{k}$$

  where $\operatorname{MSE}_{k}=\sum_{i \in C_{k}}\left(y_{i}-\hat{y}_{i}\right)^{2} / n_{k}$ , and $\hat{y}_{i}$ is the fit for observation $i$ , obtained from the data with part $k$ removed.

• Setting $K=n$ yields $n$ -fold or leave-one out cross-validation (LOOCV).

A nice special case!

• With least-squares linear or polynomial regression, an amazing shortcut makes the cost of LOOCV the same as that of a single model fit! The following formula holds:

  $$\mathrm{CV}_{(n)}=\frac{1}{n} \sum_{i=1}^{n}\left(\frac{y_{i}-\hat{y}_{i}}{1-h_{i}}\right)^{2}$$

  where $\hat{y}_{i}$ is the $i$ th fitted value from the original least squares fit, and $h_{i}$ is the leverage (diagonal of the “hat” matrix; see book for details.) This is like the ordinary MSE, except the $i$ th residual is divided by $1-h_{i}$.

• LOOCV sometimes useful, but typically doesn’t shake up the data enough. The estimates from each fold are highly correlated and hence their average can have high variance.

• a better choice is $K=5$ or $10$.

Other issues with Cross-validation

• Since each training set is only $(K-1) / K$ as big as the original training set, the estimates of prediction error will typically be biased upward.
• This bias is minimized when $K=n(\mathrm{LOOCV})$ , but this estimate has high variance, as noted earlier.
• $K=5$ or $10$ provides a good compromise for this bias-variance tradeoff.

Cross-Validation for Classification Problems

• We divide the data into $K$ roughly equal-sized parts $C_{1}, C_{2}, \ldots C_{K}$. $C_{k}$ denotes the indices of the observations in part $k$. There are $n_{k}$ observations in part $k$ : if $n$ is a multiple of $K$ , then $n_{k}=n / K$.

• Compute

  $$\mathrm{CV}_{K}=\sum_{k=1}^{K} \frac{n_{k}}{n} \operatorname{Err}_{k}$$

  where $\operatorname{Err}_{k}=\sum_{i \in C_{k}} I\left(y_{i} \neq \hat{y}_{i}\right) / n_{k}$.

• The estimated standard deviation of $\mathrm{CV}_{K}$ is

  $$\widehat{\mathrm{SE}}\left(\mathrm{CV}_{K}\right)=\sqrt{\sum_{k=1}^{K}\left(\operatorname{Err}_{k}-\overline{\operatorname{Err}_{k}}\right)^{2} /(K-1)}$$

The Bootstrap

• The bootstrap is a flexible and powerful statistical tool that can be used to quantify the uncertainty associated with a given estimator or statistical learning method.
• For example, it can provide an estimate of the standard error of a coefficient, or a confidence interval for that coefficient.

A simple example

• Suppose that we wish to invest a fixed sum of money in two financial assets that yield returns of $X$ and $Y$ , respectively, where $X$ and $Y$ are random quantities.

• We will invest a fraction $\alpha$ of our money in $X$ , and will invest the remaining $1-\alpha$ in $Y$.

• We wish to choose $\alpha$ to minimize the total risk, or variance, of our investment. In other words, we want to minimize $\operatorname{Var}(\alpha X+(1-\alpha) Y)$.

• One can show that the value that minimizes the risk is given by

  $$\alpha=\frac{\sigma_{Y}^{2}-\sigma_{X Y}}{\sigma_{X}^{2}+\sigma_{Y}^{2}-2 \sigma_{X Y}}$$

  where $\sigma_{X}^{2}=\operatorname{Var}(X), \sigma_{Y}^{2}=\operatorname{Var}(Y)$ , and $\sigma_{X Y}=\operatorname{Cov}(X, Y).$

• But the values of $\sigma_{X}^{2}, \sigma_{Y}^{2}$ , and $\sigma_{X Y}$ are unknown.

• We can compute estimates for these quantities, $\hat{\sigma}_{X}^{2}, \hat{\sigma}_{Y}^{2}$ , and $\hat{\sigma}_{X Y}$ , using a data set that contains measurements for $X$ and $Y$.

• We can then estimate the value of $\alpha$ that minimizes the variance of our investment using

  $$\hat{\alpha}=\frac{\hat{\sigma}_{Y}^{2}-\hat{\sigma}_{X Y}}{\hat{\sigma}_{X}^{2}+\hat{\sigma}_{Y}^{2}-2 \hat{\sigma}_{X Y}}$$

• To estimate the standard deviation of $\hat{\alpha}$ , we repeated the process of simulating 100 paired observations of $X$ and $Y$ , and estimating $\alpha$ 1,000 times.

• We thereby obtained 1,000 estimates for $\alpha$ , which we can call $\hat{\alpha}_{1}, \hat{\alpha}_{2}, \ldots, \hat{\alpha}_{1000}$

• For these simulations the parameters were set to $\sigma_{X}^{2}=1, \sigma_{Y}^{2}=1.25$ , and $\sigma_{X Y}=0.5$ , and so we know that the true value of $\alpha$ is $0.6$.

• The mean over all 1,000 estimates for $\alpha$ is

  $$\bar{\alpha}=\frac{1}{1000} \sum_{r=1}^{1000} \hat{\alpha}_{r}=0.5996$$

  very close to $\alpha=0.6$ , and the standard deviation of the estimates is

  $$\sqrt{\frac{1}{1000-1} \sum_{r=1}^{1000}\left(\hat{\alpha}_{r}-\bar{\alpha}\right)^{2}}=0.083$$

• This gives us a very good idea of the accuracy of $\hat{\alpha}$ : $\mathrm{SE}(\hat{\alpha}) \approx 0.083$

• So roughly speaking, for a random sample from the population, we would expect $\hat{\alpha}$ to differ from $\alpha$ by approximately $0.08$ , on average.

Now back to the real world

• The procedure outlined above cannot be applied, because for real data we cannot generate new samples from the original population.

• However, the bootstrap approach allows us to use a computer to mimic the process of obtaining new data sets, so that we can estimate the variability of our estimate without generating additional samples.
  • Rather than repeatedly obtaining independent data sets from the population, we instead obtain distinct data sets by repeatedly sampling observations from the original data set with replacement.

• Each of these “bootstrap data sets” is created by sampling with replacement, and is the same size as our original dataset. As a result some observations may appear more than once in a given bootstrap data set and some not at all.

• Denoting the first bootstrap data set by $Z^{* 1}$ , we use $Z^{* 1}$ to produce a new bootstrap estimate for $\alpha$ , which we call $\hat{\alpha}^{* 1}$.

• This procedure is repeated $B$ times for some large value of $B$ (say 100 or 1000 ), in order to produce $B$ different bootstrap data sets, $Z^{* 1}, Z^{* 2}, \ldots, Z^{* B}$ , and $B$ corresponding $\alpha$ estimates, $\hat{\alpha}^{* 1}, \hat{\alpha}^{* 2}, \ldots, \hat{\alpha}^{* B}$.

• We estimate the standard error of these bootstrap estimates using the formula

  $$\operatorname{SE}_{B}(\hat{\alpha})=\sqrt{\frac{1}{B-1} \sum_{r=1}^{B}\left(\hat{\alpha}^{* r}-\overline{\hat{\alpha}}^{*}\right)^{2}}$$

• This serves as an estimate of the standard error of $\hat{\alpha}$ estimated from the original data set. For this example $\operatorname{SE}_{B}(\hat{\alpha})=0.087$.

The bootstrap in general

• In more complex data situations, figuring out the appropriate way to generate bootstrap samples can require some thought.
• For example, if the data is a time series, we can’t simply sample the observations with replacement. (why not?).
• We can instead create blocks of consecutive observations, and sample those with replacements. Then we paste together sampled blocks to obtain a bootstrap dataset.

Other uses of the bootstrap

• Primarily used to obtain standard errors of an estimate.
• Also provides approximate confidence intervals for a population parameter.
• The above interval is called a Bootstrap Percentile confidence interval. It is the simplest method (among many approaches) for obtaining a confidence interval from the bootstrap.

Can the bootstrap estimate prediction error?

• In cross-validation, each of the $K$ validation folds is distinct from the other $K-1$ folds used for training: there is no overlap. This is crucial for its success.
• To estimate prediction error using the bootstrap, we could think about using each bootstrap dataset as our training sample, and the original sample as our validation sample.
• But each bootstrap sample has significant overlap with the original data. About two-thirds of the original data points appear in each bootstrap sample.
• This will cause the bootstrap to seriously underestimate the true prediction error.
• The other way around - with original sample = training sample, bootstrap dataset = validation sample - is worse!

Removing the overlap

• Can partly fix this problem by only using predictions for those observations that did not (by chance) occur in the current bootstrap sample.
• But the method gets complicated, and in the end, cross-validation provides a simpler, more attractive approach for estimating prediction error.

— Jul 15, 2022

§4 Cross-validation and the Bootstrap by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.