bootstrap_point632_score: The .632 and .632+ boostrap for classifier evaluation
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 socalled OutOfBag (OOB) or LeaveOneOut Bootstrap (LOOB) technique. Here, we use outofbag samples as test sets for evaluation instead of evaluating the model on the training data. Outofbag 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 tensample dataset () and their outofbag 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].
.632 Bootstrap
In 1983, Bradley Efron described the .632 Estimate, a further improvement to address the pessimistic bias of the bootstrap crossvalidation approach described above [3]. 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 as
Vice versa, we can then compute the probability that a sample is chosen as for reasonably large datasets, so that we'd select approximately uniques samples as bootstrap training sets and reserve outofbag 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 is the accuracy computed on the whole training set, and is the accuracy on the outofbag sample.
.632+ Bootstrap
Now, while the .632 Boostrap attempts to address the pessimistic bias of the estimate, an optimistic bias may occur with models that tend to overfit so that Bradley Efron and Robert Tibshirani proposed the The .632+ Bootstrap Method (Efron and Tibshirani, 1997). Instead of using a fixed "weight" in
we compute the weight as
where R is the relative overfitting rate
(Since we are plugging into the equation for computing that we defined above, and still refer to the outofbag accuracy in the ith bootstrap round and the whole training set accuracy, respectively.)
Further, we need to determine the noinformation rate in order to compute R. For instance, we can compute by fitting a model to a dataset that contains all possible combinations between samples and target class labels — we pretend that the observations and class labels are independent:
Alternatively, we can estimate the noinformation rate as follows:
where is the proportion of class samples observed in the dataset, and is the proportion of class samples that the classifier predicts in the dataset.
References
 [1] https://sebastianraschka.com/blog/2016/modelevaluationselectionpart2.html
 [2] Efron, Bradley, and Robert J. Tibshirani. An introduction to the bootstrap. CRC press, 1994. Management of Data (ACM SIGMOD '97), pages 265276, 1997. [3] Efron, Bradley. 1983. “Estimating the Error Rate of a Prediction Rule: Improvement on CrossValidation.” Journal of the American Statistical Association 78 (382): 316. doi:10.2307/2288636.
 [4] Efron, Bradley, and Robert Tibshirani. 1997. “Improvements on CrossValidation: The .632+ Bootstrap Method.” Journal of the American Statistical Association 92 (438): 548. doi:10.2307/2965703.
Example 1  Evaluating the predictive performance of a model via the classic outofbag Bootstrap
The bootstrap_point632_score
function mimics the behavior of scikitlearn's `cross_val_score, and a typically usage example is shown below:
from sklearn import datasets
from sklearn.tree import DecisionTreeClassifier
from mlxtend.evaluate import bootstrap_point632_score
import numpy as np
iris = datasets.load_iris()
X = iris.data
y = iris.target
tree = DecisionTreeClassifier(random_state=123)
# Model accuracy
scores = bootstrap_point632_score(tree, X, y, method='oob')
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: 94.45%
95% Confidence interval: [87.71, 100.00]
Example 2  Evaluating the predictive performance of a model via the .632 Bootstrap
from sklearn import datasets
from sklearn.tree import DecisionTreeClassifier
from mlxtend.evaluate import bootstrap_point632_score
import numpy as np
iris = datasets.load_iris()
X = iris.data
y = iris.target
tree = DecisionTreeClassifier(random_state=123)
# Model accuracy
scores = bootstrap_point632_score(tree, 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: 96.42%
95% Confidence interval: [92.41, 100.00]
Example 3  Evaluating the predictive performance of a model via the .632+ Bootstrap
from sklearn import datasets
from sklearn.tree import DecisionTreeClassifier
from mlxtend.evaluate import bootstrap_point632_score
import numpy as np
iris = datasets.load_iris()
X = iris.data
y = iris.target
tree = DecisionTreeClassifier(random_state=123)
# Model accuracy
scores = bootstrap_point632_score(tree, X, y, method='.632+')
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: 96.29%
95% Confidence interval: [91.86, 98.92]
API
bootstrap_point632_score(estimator, X, y, n_splits=200, method='.632', scoring_func=None, predict_proba=False, random_seed=None, clone_estimator=True)
Implementation of the .632 [1] and .632+ [2] bootstrap for supervised learning
References:
 [1] Efron, Bradley. 1983. "Estimating the Error Rate
of a Prediction Rule: Improvement on CrossValidation."
Journal of the American Statistical Association
78 (382): 316. doi:10.2307/2288636.
 [2] Efron, Bradley, and Robert Tibshirani. 1997.
"Improvements on CrossValidation: The .632+ Bootstrap Method."
Journal of the American Statistical Association
92 (438): 548. doi:10.2307/2965703.
Parameters

estimator
: objectAn estimator for classification or regression that follows the scikitlearn API and implements "fit" and "predict" methods.

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

y
: arraylike, optional, default: NoneThe 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  1) '.632' bootstrap (default)  2) '.632+' bootstrap  3) 'oob' (regular outofbag, no weighting) for comparison studies.

scoring_func
: callable,Score function (or loss function) with signature
scoring_func(y, y_pred, **kwargs)
. If none, uses classification accuracy if the
estimator is a classifier and mean squared error if the estimator is a regressor.

predict_proba
: boolWhether to use the
predict_proba
function for theestimator
argument. This is to be used in conjunction withscoring_func
which takes in probability values instead of actual predictions. For example, if the scoring_func is :meth:sklearn.metrics.roc_auc_score
, then usepredict_proba=True
. Note that this requiresestimator
to havepredict_proba
method implemented. 
random_seed
: int (default=None)If int, random_seed is the seed used by the random number generator.

clone_estimator
: bool (default=True)Clones the estimator if true, otherwise fits the original.
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
>>> iris = datasets.load_iris()
>>> 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/