.. _cross_validation:

===================================================
Cross-Validation: evaluating estimator performance
===================================================

.. currentmodule:: sklearn.cross_validation

Learning the parameters of a prediction function and testing it on the
same data is a methodological mistake: a model that would just repeat
the labels of the samples that it has just seen would have a perfect
score but would fail to predict anything useful on yet-unseen data.

To **avoid over-fitting**, we have to define two different sets :
a **training set** ``X_train, y_train`` which is used for learning
the parameters of a predictive model, and a **testing set** ``X_test,
y_test`` which is used for evaluating the fitted predictive model.

In scikit-learn such a random split can be quickly computed with the
:func:`train_test_split` helper function. Let load the iris data set to
fit a linear Support Vector Machine model on it::

  >>> import numpy as np
  >>> from sklearn import cross_validation
  >>> from sklearn import datasets
  >>> from sklearn import svm

  >>> iris = datasets.load_iris()
  >>> iris.data.shape, iris.target.shape
  ((150, 4), (150,))

We can now quickly sample a training set while holding out 40% of the
data for testing (evaluating) our classifier::

  >>> X_train, X_test, y_train, y_test = cross_validation.train_test_split(
  ...     iris.data, iris.target, test_size=0.4, random_state=0)

  >>> X_train.shape, y_train.shape
  ((90, 4), (90,))
  >>> X_test.shape, y_test.shape
  ((60, 4), (60,))

  >>> clf = svm.SVC(kernel='linear', C=1).fit(X_train, y_train)
  >>> clf.score(X_test, y_test)                           # doctest: +ELLIPSIS
  0.96...

However, by defining these two sets, we drastically reduce the number
of samples which can be used for learning the model, and the results can
depend on a particular random choice for the pair of (train, test) sets.

A solution is to **split the whole data several consecutive times in
different train set and test set**, and to return the averaged value of
the prediction scores obtained with the different sets. Such a procedure
is called **cross-validation**. This approach can be **computationally
expensive, but does not waste too much data** (as it is the case when
fixing an arbitrary test set), which is a major advantage in problem
such as inverse inference where the number of samples is very small.


Computing cross-validated metrics
=================================

The simplest way to use perform cross-validation in to call the
:func:`cross_val_score` helper function on the estimator and the dataset.

The following example demonstrates how to estimate the accuracy of a
linear kernel Support Vector Machine on the iris dataset by splitting
the data and fitting a model and computing the score 5 consecutive times
(with different splits each time)::

  >>> clf = svm.SVC(kernel='linear', C=1)
  >>> scores = cross_validation.cross_val_score(
  ...    clf, iris.data, iris.target, cv=5)
  ...
  >>> scores                                            # doctest: +ELLIPSIS
  array([ 1.  ...,  0.96...,  0.9 ...,  0.96...,  1.        ])

The mean score and the standard deviation of the score estimate are hence given
by::

  >>> print "Accuracy: %0.2f (+/- %0.2f)" % (scores.mean(), scores.std() / 2)
  Accuracy: 0.97 (+/- 0.02)

By default, the score computed at each CV iteration is the ``score``
method of the estimator. It is possible to change this by passing a custom
scoring function, e.g. from the metrics module::

  >>> from sklearn import metrics
  >>> cross_validation.cross_val_score(clf, iris.data, iris.target, cv=5,
  ...     score_func=metrics.f1_score)
  ...                                                     # doctest: +ELLIPSIS
  array([ 1.  ...,  0.96...,  0.89...,  0.96...,  1.        ])

In the case of the Iris dataset, the samples are balanced across target
classes hence the accuracy and the F1-score are almost equal.

When the ``cv`` argument is an integer, :func:`cross_val_score` uses the
:class:`KFold` or :class:`StratifiedKFold` strategies by default (depending on
the absence or presence of the target array).

It is also possible to use othe cross validation strategies by passing a cross
validation iterator instead, for instance::

  >>> n_samples = iris.data.shape[0]
  >>> cv = cross_validation.ShuffleSplit(n_samples, n_iterations=3,
  ...     test_size=0.3, random_state=0)

  >>> cross_validation.cross_val_score(clf, iris.data, iris.target, cv=cv)
  ...                                                     # doctest: +ELLIPSIS
  array([ 0.97...,  0.97...,  1.        ])

The available cross validation iterators are introduced in the following.


.. topic:: Examples

    * :ref:`example_plot_roc_crossval.py`,
    * :ref:`example_plot_rfe_with_cross_validation.py`,
    * :ref:`example_grid_search_digits.py`,
    * :ref:`example_grid_search_text_feature_extraction.py`,


Cross validation iterators
==========================

The following sections list utilities to generate boolean masks or indices
that can be used to generate dataset splits according to different cross
validation strategies.


.. topic:: Boolean mask vs integer indices

   Most cross validators support generating both boolean masks or integer
   indices to select the samples from a given fold.

   When the data matrix is sparse, only the integer indices will work as
   expected. Integer indexing is hence the default behavior (since version
   0.10).

   You can explicitly pass ``indices=False`` to the constructor of the
   CV object (when supported) to use the boolean mask method instead.


K-fold
------

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

Example of 2-fold::

  >>> import numpy as np
  >>> from sklearn.cross_validation import KFold
  >>> X = np.array([[0., 0.], [1., 1.], [-1., -1.], [2., 2.]])
  >>> Y = np.array([0, 1, 0, 1])

  >>> kf = KFold(len(Y), 2, indices=False)
  >>> print kf
  sklearn.cross_validation.KFold(n=4, k=2)

  >>> for train, test in kf:
  ...     print train, test
  [False False  True  True] [ True  True False False]
  [ True  True False False] [False False  True  True]

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::

  >>> X_train, X_test, y_train, y_test = X[train], X[test], Y[train], Y[test]

If X or Y are `scipy.sparse` matrices, train and test need to be integer
indices. It can be obtained by setting the parameter indices to True
when creating the cross-validation procedure::

  >>> X = np.array([[0., 0.], [1., 1.], [-1., -1.], [2., 2.]])
  >>> Y = np.array([0, 1, 0, 1])

  >>> kf = KFold(len(Y), 2, indices=True)
  >>> for train, test in kf:
  ...    print train, test
  [2 3] [0 1]
  [0 1] [2 3]


Stratified K-Fold
-----------------

:class:`StratifiedKFold` is a variation of *K-fold*, which returns
stratified folds, *i.e* which creates folds by preserving the same
percentage for each target class as in the complete set.

Example of stratified 2-fold::

  >>> from sklearn.cross_validation import StratifiedKFold
  >>> X = [[0., 0.],
  ...      [1., 1.],
  ...      [-1., -1.],
  ...      [2., 2.],
  ...      [3., 3.],
  ...      [4., 4.],
  ...      [0., 1.]]
  >>> Y = [0, 0, 0, 1, 1, 1, 0]

  >>> skf = StratifiedKFold(Y, 2)
  >>> print skf
  sklearn.cross_validation.StratifiedKFold(labels=[0 0 0 1 1 1 0], k=2)

  >>> for train, test in skf:
  ...     print train, test
  [1 4 6] [0 2 3 5]
  [0 2 3 5] [1 4 6]


Leave-One-Out - LOO
-------------------

:class:`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 learning
sets and `n` different tests set. This cross-validation procedure does
not waste much data as only one sample is removed from the learning set::

  >>> from sklearn.cross_validation import LeaveOneOut
  >>> X = np.array([[0., 0.], [1., 1.], [-1., -1.], [2., 2.]])
  >>> Y = np.array([0, 1, 0, 1])

  >>> loo = LeaveOneOut(len(Y))
  >>> print loo
  sklearn.cross_validation.LeaveOneOut(n=4)

  >>> for train, test in loo:
  ...    print train, test
  [1 2 3] [0]
  [0 2 3] [1]
  [0 1 3] [2]
  [0 1 2] [3]


Leave-P-Out - LPO
-----------------

:class:`LeavePOut` is very similar to *Leave-One-Out*, as it creates all the
possible training/test sets by removing :math:`P` samples from the complete set.

Example of Leave-2-Out::

  >>> from sklearn.cross_validation import LeavePOut
  >>> X = [[0., 0.], [1., 1.], [-1., -1.], [2., 2.]]
  >>> Y = [0, 1, 0, 1]

  >>> lpo = LeavePOut(len(Y), 2)
  >>> print lpo
  sklearn.cross_validation.LeavePOut(n=4, p=2)

  >>> for train, test in lpo:
  ...     print 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]


Leave-One-Label-Out - LOLO
--------------------------

:class:`LeaveOneLabelOut` (LOLO) is a cross-validation scheme which
holds out the samples according to a third-party provided label. This
label information can be used to encode arbitrary domain specific
stratifications of the samples as integers.

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

For example, in the cases of multiple experiments, *LOLO* 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.cross_validation import LeaveOneLabelOut
  >>> X = [[0., 0.], [1., 1.], [-1., -1.], [2., 2.]]
  >>> Y = [0, 1, 0, 1]
  >>> labels = [1, 1, 2, 2]

  >>> lolo = LeaveOneLabelOut(labels)
  >>> print lolo
  sklearn.cross_validation.LeaveOneLabelOut(labels=[1, 1, 2, 2])

  >>> for train, test in lolo:
  ...     print train, test
  [2 3] [0 1]
  [0 1] [2 3]

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


Leave-P-Label-Out
-----------------

:class:`LeavePLabelOut` is similar as *Leave-One-Label-Out*, but removes
samples related to :math:`P` labels for each training/test set.

Example of Leave-2-Label Out::

  >>> from sklearn.cross_validation import LeavePLabelOut
  >>> X = [[0., 0.], [1., 1.], [-1., -1.], [2., 2.], [3., 3.], [4., 4.]]
  >>> Y = [0, 1, 0, 1, 0, 1]
  >>> labels = [1, 1, 2, 2, 3, 3]

  >>> lplo = LeavePLabelOut(labels, 2)
  >>> print lplo
  sklearn.cross_validation.LeavePLabelOut(labels=[1, 1, 2, 2, 3, 3], p=2)

  >>> for train, test in lplo:
  ...     print train, test
  [4 5] [0 1 2 3]
  [2 3] [0 1 4 5]
  [0 1] [2 3 4 5]

.. _ShuffleSplit:

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

:class:`ShuffleSplit`

The :class:`ShuffleSplit` iterator will generate a user defined number of
independent train / test dataset splits. Samples are first shuffled and
then splitted 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::

  >>> ss = cross_validation.ShuffleSplit(5, n_iterations=3, test_size=0.25,
  ...     random_state=0)
  >>> len(ss)
  3
  >>> print ss                                            # doctest: +ELLIPSIS
  ShuffleSplit(5, n_iterations=3, test_size=0.25, indices=True, ...)

  >>> for train_index, test_index in ss:
  ...    print train_index, test_index
  ...
  [1 3 4] [2 0]
  [1 4 3] [0 2]
  [4 0 2] [1 3]

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

See also
--------
:class:`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.

.. _Bootstrap:

Bootstrapping cross-validation
------------------------------

:class:`Bootstrap`

Bootstrapping_ is a general statistics technique that iterates the
computation of an estimator on a resampled dataset.

The :class:`Bootstrap` iterator will generate a user defined number
of independent train / test dataset splits. Samples are then drawn
(with replacement) on each side of the split. It furthermore possible
to control the size of the train and test subset to make their union
smaller than the total dataset if it is very large.

.. note::

  Contrary to other cross-validation strategies, bootstrapping
  will allow some samples to occur several times in each splits.

.. _Bootstrapping: http://en.wikipedia.org/wiki/Bootstrapping_%28statistics%29

  >>> bs = cross_validation.Bootstrap(9, random_state=0)
  >>> len(bs)
  3
  >>> print bs
  Bootstrap(9, n_bootstraps=3, train_size=5, test_size=4, random_state=0)

  >>> for train_index, test_index in bs:
  ...    print train_index, test_index
  ...
  [1 8 7 7 8] [0 3 0 5]
  [5 4 2 4 2] [6 7 1 0]
  [4 7 0 1 1] [5 3 6 5]


Cross validation and model selection
====================================

Cross validation iterators can also be used to directly perform model
selection using Grid Search for the optimal hyperparameters of the
model. This is the topic if the next section: :ref:`grid_search`.