# bootstrap_point632_score

An implementation of the .632 bootstrap to evaluate supervised learning algorithms.

from mlxtend.evaluate import bootstrap_point632_score

## Overview

Originally, the bootstrap method aims to determine the statistical properties of an estimator when the underlying distribution was unknown and additional samples are not available. Now, in order to exploit this method for the evaluation of predictive models, such as hypotheses for classification and regression, we may prefer a slightly different approach to bootstrapping using the so-called Out-Of-Bag (OOB) or Leave-One-Out Bootstrap (LOOB) technique. Here, we use out-of-bag samples as test sets for evaluation instead of evaluating the model on the training data. Out-of-bag samples are the unique sets of instances that are not used for model fitting as shown in the figure below [1].

The figure above illustrates how three random bootstrap samples drawn from an exemplary ten-sample dataset ($X_1,X_2, ..., X_{10}$) and their out-of-bag sample for testing may look like. In practice, Bradley Efron and Robert Tibshirani recommend drawing 50 to 200 bootstrap samples as being sufficient for reliable estimates [2].

In 1983, Bradley Efron described the .632 Estimate, a further improvement to address the pessimistic bias of the bootstrap cross-validation approach described above (Efron, 1983). The pessimistic bias in the "classic" bootstrap method can be attributed to the fact that the bootstrap samples only contain approximately 63.2% of the unique samples from the original dataset. For instance, we can compute the probability that a given sample from a dataset of size n is not drawn as a bootstrap sample as

which is asymptotically equivalent to $\frac{1}{e} \approx 0.368$ as $n \rightarrow \infty.$

Vice versa, we can then compute the probability that a sample is chosen as $P (\text{chosen}) = 1 - \bigg(1 - \frac{1}{n}\bigg)^n \approx 0.632$ for reasonably large datasets, so that we'd select approximately $0.632 \times n$ uniques samples as bootstrap training sets and reserve $0.368 \times n$ out-of-bag samples for testing in each iteration.

Now, to address the bias that is due to this the sampling with replacement, Bradley Efron proposed the .632 Estimate that we mentioned earlier, which is computed via the following equation:

where $\text{ACC}_{r, i}$ is the resubstitution accuracy, and $\text{ACC}_{h, i}$ is the accuracy on the out-of-bag sample.

### References

• [1] https://sebastianraschka.com/blog/2016/model-evaluation-selection-part2.html
• [2] Efron, Bradley, and Robert J. Tibshirani. An introduction to the bootstrap. CRC press, 1994. Management of Data (ACM SIGMOD '97), pages 265-276, 1997.

## Example 1 -- Evaluating the predictive performance of a model

The bootstrap_point632_score function mimics the behavior of scikit-learn's cross_val_score, and a typically usage example is shown below:

from sklearn import datasets, linear_model
from mlxtend.evaluate import bootstrap_point632_score
import numpy as np

X = iris.data
y = iris.target
lr = linear_model.LogisticRegression()

# Model accuracy
scores = bootstrap_point632_score(lr, X, y)
acc = np.mean(scores)
print('Accuracy: %.2f%%' % (100*acc))

# Confidence interval
lower = np.percentile(scores, 2.5)
upper = np.percentile(scores, 97.5)
print('95%% Confidence interval: [%.2f, %.2f]' % (100*lower, 100*upper))

Accuracy: 94.99%
95% Confidence interval: [90.76, 98.28]


## API

bootstrap_point632_score(estimator, X, y, n_splits=200, method='.632', scoring=None, random_seed=None)

Implementation of the 0.632 bootstrap for supervised learning

Parameters

• estimator : object

An estimator for classification or regression that follows the scikit-learn API and implements "fit" and "predict" methods.

• X : array-like

The data to fit. Can be, for example a list, or an array at least 2d.

• y : array-like, optional, default: None

The target variable to try to predict in the case of supervised learning.

• n_splits : int (default=200)

Number of bootstrap iterations. Must be larger than 1.

• method : str (default='.632')

The bootstrap method, which can be either the regular '.632' bootstrap (default) or the '.632+' bootstrap (not implemented, yet).

• scoring : str, callable, or None (default: None)

If None (default), uses 'accuracy' for sklearn classifiers and 'r2' for sklearn regressors. If str, uses a sklearn scoring metric string identifier, for example {'accuracy', 'f1', 'precision', 'recall', 'roc_auc', etc.} for classifiers, {'mean_absolute_error', 'mean_squared_error'/'neg_mean_squared_error', 'median_absolute_error', 'r2', etc.} for regressors. If a callable object or function is provided, it has to be conform with sklearn's signature scorer(estimator, X, y); see http://scikit-learn.org/stable/modules/generated/sklearn.metrics.make_scorer.html for more information.

• random_seed : int (default=None)

If int, random_seed is the seed used by the random number generator.

Returns

• scores : array of float, shape=(len(list(n_splits)),)

Array of scores of the estimator for each bootstrap replicate.

Examples

>>> from sklearn import datasets, linear_model
>>> from mlxtend.evaluate import bootstrap_point632_score
>>> X = iris.data
>>> y = iris.target
>>> lr = linear_model.LogisticRegression()
>>> scores = bootstrap_point632_score(lr, X, y)
>>> acc = np.mean(scores)
>>> print('Accuracy:', acc)
0.953023146884
>>> lower = np.percentile(scores, 2.5)
>>> upper = np.percentile(scores, 97.5)
>>> print('95%% Confidence interval: [%.2f, %.2f]' % (lower, upper))
95% Confidence interval: [0.90, 0.98]
`

For more usage examples, please see http://rasbt.github.io/mlxtend/user_guide/evaluate/bootstrap_point632_score/