3.1. Cross-validation: evaluating estimator performance¶

3.1.2.1.1. K-fold¶

KFold divides all the samples in \(k\) groups of samples, called folds (if \(k = n\), this is equivalent to the Leave One Out strategy), of equal sizes (if possible). The prediction function is learned using \(k - 1\) folds, and the fold left out is used for test.

Example of 2-fold cross-validation on a dataset with 4 samples:

>>> import numpy as np
>>> from sklearn.model_selection import KFold

>>> X = ["a", "b", "c", "d"]
>>> kf = KFold(n_splits=2)
>>> for train, test in kf.split(X):
...     print("%s %s" % (train, test))
[2 3] [0 1]
[0 1] [2 3]

Here is a visualization of the cross-validation behavior. Note that KFold is not affected by classes or groups.

Each fold is constituted by two arrays: the first one is related to the training set, and the second one to the test set. Thus, one can create the training/test sets using numpy indexing:

>>> X = np.array([[0., 0.], [1., 1.], [-1., -1.], [2., 2.]])
>>> y = np.array([0, 1, 0, 1])
>>> X_train, X_test, y_train, y_test = X[train], X[test], y[train], y[test]

3.1.2.1.2. Repeated K-Fold¶

RepeatedKFold repeats K-Fold n times. It can be used when one requires to run KFold n times, producing different splits in each repetition.

Example of 2-fold K-Fold repeated 2 times:

>>> import numpy as np
>>> from sklearn.model_selection import RepeatedKFold
>>> X = np.array([[1, 2], [3, 4], [1, 2], [3, 4]])
>>> random_state = 12883823
>>> rkf = RepeatedKFold(n_splits=2, n_repeats=2, random_state=random_state)
>>> for train, test in rkf.split(X):
...     print("%s %s" % (train, test))
...
[2 3] [0 1]
[0 1] [2 3]
[0 2] [1 3]
[1 3] [0 2]

Similarly, RepeatedStratifiedKFold repeats Stratified K-Fold n times with different randomization in each repetition.

3.1.2.1.3. Leave One Out (LOO)¶

LeaveOneOut (or LOO) is a simple cross-validation. Each learning set is created by taking all the samples except one, the test set being the sample left out. Thus, for \(n\) samples, we have \(n\) different training sets and \(n\) different tests set. This cross-validation procedure does not waste much data as only one sample is removed from the training set:

>>> from sklearn.model_selection import LeaveOneOut

>>> X = [1, 2, 3, 4]
>>> loo = LeaveOneOut()
>>> for train, test in loo.split(X):
...     print("%s %s" % (train, test))
[1 2 3] [0]
[0 2 3] [1]
[0 1 3] [2]
[0 1 2] [3]

Potential users of LOO for model selection should weigh a few known caveats. When compared with \(k\)-fold cross validation, one builds \(n\) models from \(n\) samples instead of \(k\) models, where \(n > k\). Moreover, each is trained on \(n - 1\) samples rather than \((k-1) n / k\). In both ways, assuming \(k\) is not too large and \(k < n\), LOO is more computationally expensive than \(k\)-fold cross validation.

In terms of accuracy, LOO often results in high variance as an estimator for the test error. Intuitively, since \(n - 1\) of the \(n\) samples are used to build each model, models constructed from folds are virtually identical to each other and to the model built from the entire training set.

However, if the learning curve is steep for the training size in question, then 5- or 10- fold cross validation can overestimate the generalization error.

As a general rule, most authors, and empirical evidence, suggest that 5- or 10- fold cross validation should be preferred to LOO.

3.1.2.1.4. Leave P Out (LPO)¶

LeavePOut is very similar to LeaveOneOut as it creates all the possible training/test sets by removing \(p\) samples from the complete set. For \(n\) samples, this produces \({n \choose p}\) train-test pairs. Unlike LeaveOneOut and KFold, the test sets will overlap for \(p > 1\).

Example of Leave-2-Out on a dataset with 4 samples:

>>> from sklearn.model_selection import LeavePOut

>>> X = np.ones(4)
>>> lpo = LeavePOut(p=2)
>>> for train, test in lpo.split(X):
...     print("%s %s" % (train, test))
[2 3] [0 1]
[1 3] [0 2]
[1 2] [0 3]
[0 3] [1 2]
[0 2] [1 3]
[0 1] [2 3]

3.1.2.1.5. Random permutations cross-validation a.k.a. Shuffle & Split¶

The ShuffleSplit iterator will generate a user defined number of independent train / test dataset splits. Samples are first shuffled and then split into a pair of train and test sets.

It is possible to control the randomness for reproducibility of the results by explicitly seeding the random_state pseudo random number generator.

Here is a usage example:

>>> from sklearn.model_selection import ShuffleSplit
>>> X = np.arange(10)
>>> ss = ShuffleSplit(n_splits=5, test_size=0.25, random_state=0)
>>> for train_index, test_index in ss.split(X):
...     print("%s %s" % (train_index, test_index))
[9 1 6 7 3 0 5] [2 8 4]
[2 9 8 0 6 7 4] [3 5 1]
[4 5 1 0 6 9 7] [2 3 8]
[2 7 5 8 0 3 4] [6 1 9]
[4 1 0 6 8 9 3] [5 2 7]

Here is a visualization of the cross-validation behavior. Note that ShuffleSplit is not affected by classes or groups.

ShuffleSplit is thus a good alternative to KFold cross validation that allows a finer control on the number of iterations and the proportion of samples on each side of the train / test split.

3.1.2.2.1. Stratified k-fold¶

StratifiedKFold is a variation of k-fold which returns stratified folds: each set contains approximately the same percentage of samples of each target class as the complete set.

Here is an example of stratified 3-fold cross-validation on a dataset with 50 samples from two unbalanced classes. We show the number of samples in each class and compare with KFold.

>>> from sklearn.model_selection import StratifiedKFold, KFold
>>> import numpy as np
>>> X, y = np.ones((50, 1)), np.hstack(([0] * 45, [1] * 5))
>>> skf = StratifiedKFold(n_splits=3)
>>> for train, test in skf.split(X, y):
...     print('train -  {}   |   test -  {}'.format(
...         np.bincount(y[train]), np.bincount(y[test])))
train -  [30  3]   |   test -  [15  2]
train -  [30  3]   |   test -  [15  2]
train -  [30  4]   |   test -  [15  1]
>>> kf = KFold(n_splits=3)
>>> for train, test in kf.split(X, y):
...     print('train -  {}   |   test -  {}'.format(
...         np.bincount(y[train]), np.bincount(y[test])))
train -  [28  5]   |   test -  [17]
train -  [28  5]   |   test -  [17]
train -  [34]   |   test -  [11  5]

We can see that StratifiedKFold preserves the class ratios (approximately 1 / 10) in both train and test dataset.

Here is a visualization of the cross-validation behavior.

RepeatedStratifiedKFold can be used to repeat Stratified K-Fold n times with different randomization in each repetition.

3.1.2.2.2. Stratified Shuffle Split¶

StratifiedShuffleSplit is a variation of ShuffleSplit, which returns stratified splits, i.e which creates splits by preserving the same percentage for each target class as in the complete set.

Here is a visualization of the cross-validation behavior.

3.1.2.3.1. Group k-fold¶

GroupKFold is a variation of k-fold which ensures that the same group is not represented in both testing and training sets. For example if the data is obtained from different subjects with several samples per-subject and if the model is flexible enough to learn from highly person specific features it could fail to generalize to new subjects. GroupKFold makes it possible to detect this kind of overfitting situations.

Imagine you have three subjects, each with an associated number from 1 to 3:

>>> from sklearn.model_selection import GroupKFold

>>> X = [0.1, 0.2, 2.2, 2.4, 2.3, 4.55, 5.8, 8.8, 9, 10]
>>> y = ["a", "b", "b", "b", "c", "c", "c", "d", "d", "d"]
>>> groups = [1, 1, 1, 2, 2, 2, 3, 3, 3, 3]

>>> gkf = GroupKFold(n_splits=3)
>>> for train, test in gkf.split(X, y, groups=groups):
...     print("%s %s" % (train, test))
[0 1 2 3 4 5] [6 7 8 9]
[0 1 2 6 7 8 9] [3 4 5]
[3 4 5 6 7 8 9] [0 1 2]

Each subject is in a different testing fold, and the same subject is never in both testing and training. Notice that the folds do not have exactly the same size due to the imbalance in the data.

Here is a visualization of the cross-validation behavior.

3.1.2.3.2. Leave One Group Out¶

LeaveOneGroupOut is a cross-validation scheme which holds out the samples according to a third-party provided array of integer groups. This group information can be used to encode arbitrary domain specific pre-defined cross-validation folds.

Each training set is thus constituted by all the samples except the ones related to a specific group.

For example, in the cases of multiple experiments, LeaveOneGroupOut can be used to create a cross-validation based on the different experiments: we create a training set using the samples of all the experiments except one:

>>> from sklearn.model_selection import LeaveOneGroupOut

>>> X = [1, 5, 10, 50, 60, 70, 80]
>>> y = [0, 1, 1, 2, 2, 2, 2]
>>> groups = [1, 1, 2, 2, 3, 3, 3]
>>> logo = LeaveOneGroupOut()
>>> for train, test in logo.split(X, y, groups=groups):
...     print("%s %s" % (train, test))
[2 3 4 5 6] [0 1]
[0 1 4 5 6] [2 3]
[0 1 2 3] [4 5 6]

Another common application is to use time information: for instance the groups could be the year of collection of the samples and thus allow for cross-validation against time-based splits.

3.1.2.3.3. Leave P Groups Out¶

LeavePGroupsOut is similar as LeaveOneGroupOut, but removes samples related to \(P\) groups for each training/test set.

Example of Leave-2-Group Out:

>>> from sklearn.model_selection import LeavePGroupsOut

>>> X = np.arange(6)
>>> y = [1, 1, 1, 2, 2, 2]
>>> groups = [1, 1, 2, 2, 3, 3]
>>> lpgo = LeavePGroupsOut(n_groups=2)
>>> for train, test in lpgo.split(X, y, groups=groups):
...     print("%s %s" % (train, test))
[4 5] [0 1 2 3]
[2 3] [0 1 4 5]
[0 1] [2 3 4 5]

3.1.2.3.4. Group Shuffle Split¶

The GroupShuffleSplit iterator behaves as a combination of ShuffleSplit and LeavePGroupsOut, and generates a sequence of randomized partitions in which a subset of groups are held out for each split.

Here is a usage example:

>>> from sklearn.model_selection import GroupShuffleSplit

>>> X = [0.1, 0.2, 2.2, 2.4, 2.3, 4.55, 5.8, 0.001]
>>> y = ["a", "b", "b", "b", "c", "c", "c", "a"]
>>> groups = [1, 1, 2, 2, 3, 3, 4, 4]
>>> gss = GroupShuffleSplit(n_splits=4, test_size=0.5, random_state=0)
>>> for train, test in gss.split(X, y, groups=groups):
...     print("%s %s" % (train, test))
...
[0 1 2 3] [4 5 6 7]
[2 3 6 7] [0 1 4 5]
[2 3 4 5] [0 1 6 7]
[4 5 6 7] [0 1 2 3]

Here is a visualization of the cross-validation behavior.

This class is useful when the behavior of LeavePGroupsOut is desired, but the number of groups is large enough that generating all possible partitions with \(P\) groups withheld would be prohibitively expensive. In such a scenario, GroupShuffleSplit provides a random sample (with replacement) of the train / test splits generated by LeavePGroupsOut.

3.1.2.6.1. Time Series Split¶

TimeSeriesSplit is a variation of k-fold which returns first \(k\) folds as train set and the \((k+1)\) th fold as test set. Note that unlike standard cross-validation methods, successive training sets are supersets of those that come before them. Also, it adds all surplus data to the first training partition, which is always used to train the model.

This class can be used to cross-validate time series data samples that are observed at fixed time intervals.

Example of 3-split time series cross-validation on a dataset with 6 samples:

>>> from sklearn.model_selection import TimeSeriesSplit

>>> X = np.array([[1, 2], [3, 4], [1, 2], [3, 4], [1, 2], [3, 4]])
>>> y = np.array([1, 2, 3, 4, 5, 6])
>>> tscv = TimeSeriesSplit(n_splits=3)
>>> print(tscv)
TimeSeriesSplit(gap=0, max_train_size=None, n_splits=3, test_size=None)
>>> for train, test in tscv.split(X):
...     print("%s %s" % (train, test))
[0 1 2] [3]
[0 1 2 3] [4]
[0 1 2 3 4] [5]

Here is a visualization of the cross-validation behavior.