# Bootstrap aggregating

Bootstrap aggregating, also called bagging, is a machine learning ensemble meta-algorithm designed to improve the stability and accuracy of machine learning algorithms used in statistical classification and regression. It also reduces variance and helps to avoid overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the model averaging approach.

## Description of the technique

Given a standard training set D of size n, bagging generates m new training sets $D_i$, each of size n′, by sampling from D uniformly and with replacement. By sampling with replacement, some observations may be repeated in each $D_i$. If n=n, then for large n the set $D_i$ is expected to have the fraction (1 - 1/e) (≈63.2%) of the unique examples of D, the rest being duplicates.[1] This kind of sample is known as a bootstrap sample. The m models are fitted using the above m bootstrap samples and combined by averaging the output (for regression) or voting (for classification).

Bagging leads to "improvements for unstable procedures" (Breiman, 1996), which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression (Breiman, 1994). An interesting application of bagging showing improvement in preimage learning is provided here.[2][3] On the other hand, it can mildly degrade the performance of stable methods such as K-nearest neighbors (Breiman, 1996).

## Example: Ozone data

To illustrate the basic principles of bagging, below is an analysis on the relationship between ozone and temperature (data from Rousseeuw and Leroy (1986), available at classic data sets, analysis done in R).

The relationship between temperature and ozone in this data set is apparently non-linear, based on the scatter plot. To mathematically describe this relationship, LOESS smoothers (with span 0.5) are used. Instead of building a single smoother from the complete data set, 100 bootstrap samples of the data were drawn. Each sample is different from the original data set, yet resembles it in distribution and variability. For each bootstrap sample, a LOESS smoother was fit. Predictions from these 100 smoothers were then made across the range of the data. The first 10 predicted smooth fits appear as grey lines in the figure below. The lines are clearly very wiggly and they overfit the data - a result of the span being too low.

By taking the average of 100 smoothers, each fitted to a subset of the original data set, we arrive at one bagged predictor (red line). Clearly, the mean is more stable and there is less overfit.

## Bagging for nearest neighbour classifiers

The risk of a 1 nearest neighbour (1NN) classifier is at most twice the risk of the Bayes classifier,[4] but there are no guarantees that this classifier will be consistent. By careful choice of the size of the resamples, bagging can lead to substantial improvements of the performance of the 1NN classifier. By taking a large number of resamples of the data of size $n'$, the bagged nearest neighbour classifier will be consistent provided $n' \to \infty$ diverges but $n'/n \to 0$ as the sample size $n \to \infty$.

Under infinite simulation, the bagged nearest neighbour classifier can be viewed as a weighted nearest neighbour classifier. Suppose that the feature space is $d$ dimensional and denote by $C^{bnn}_{n,n'}$ the bagged nearest neighbour classifier based on a training set of size $n$, with resamples of size $n'$. In the infinite sampling case, under certain regularity conditions on the class distributions, the excess risk has the following asymptotic expansion[5]

$\mathcal{R}_\mathcal{R}(C^{bnn}_{n,n'}) - \mathcal{R}_{\mathcal{R}}(C^{Bayes}) = \left(B_1 \frac{n'}{n} + B_2 \frac 1 {(n')^{4/d}}\right) \{1+o(1)\},$

for some constants $B_1$ and $B_2$. The optimal choice of $n'$, that balances the two terms in the asymptotic expansion, is given by $n' = B n^{d/(d+4)}$ for some constant $B$.

## History

Bagging (Bootstrap aggregating) was proposed by Leo Breiman in 1994 to improve the classification by combining classifications of randomly generated training sets. See Breiman, 1994. Technical Report No. 421.

1. ^ Aslam, Javed A.; Popa, Raluca A.; and Rivest, Ronald L. (2007); On Estimating the Size and Confidence of a Statistical Audit, Proceedings of the Electronic Voting Technology Workshop (EVT '07), Boston, MA, August 6, 2007. More generally, when drawing with replacement n′ values out of a set of n (different and equally likely), the expected number of unique draws is $n(1 - e^{-n'/n})$.