.. _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.
This situation is called **overfitting**.
To avoid it, it is common practice when performing
a (supervised) machine learning experiment
to hold out part of the available data as a **test set** ``X_test, y_test``.
Note that the word "experiment" is not intended
to denote academic use only,
because even in commercial settings
machine learning usually starts out experimentally.

In scikit-learn a random split into training and test sets
can be quickly computed with the :func:`train_test_split` helper function.
Let's load the iris data set to fit a linear support vector machine 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...

When evaluating different settings ("hyperparameters") for estimators,
such as the ``C`` setting that must be manually set for an SVM,
there is still a risk of overfitting *on the test set*
because the parameters can be tweaked until the estimator performs optimally.
This way, knowledge about the test set can "leak" into the model
and evaluation metrics no longer report on generalization performance.
To solve this problem, yet another part of the dataset can be held out
as a so-called "validation set": training proceeds on the training set,
after which evaluation is done on the validation set,
and when the experiment seems to be successful,
final evaluation can be done on the test set.

However, by partitioning the available data into three 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, validation) sets.

A solution to this problem is a procedure called
`cross-validation <http://en.wikipedia.org/wiki/Cross-validation_(statistics)>`_
(CV for short).
A test set should still be held out for final evaluation,
but the validation set is no longer needed when doing CV.
In the basic approach, called *k*-fold CV,
the training set is split into *k* smaller sets
(other approaches are described below,
but generally follow the same principles).
The following procedure is followed for each of the *k* "folds":

 * A model is trained using :math:`k-1` of the folds as training data;
 * the resulting model is validated on the remaining part of the data
   (i.e., it is used as a test set to compute a performance measure
   such as accuracy).

The performance measure reported by *k*-fold cross-validation
is then the average of the values computed in the loop.
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 cross-validation is 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, 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([ 0.96...,  1.  ...,  0.96...,  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: 0.98 (+/- 0.03)

By default, the score computed at each CV iteration is the ``score``
method of the estimator. It is possible to change this by using the
scoring parameter::

  >>> from sklearn import metrics
  >>> scores = cross_validation.cross_val_score(clf, iris.data, iris.target,
  ...     cv=5, scoring='f1')
  >>> scores                                              # doctest: +ELLIPSIS
  array([ 0.96...,  1.  ...,  0.96...,  0.96...,  1.        ])

See :ref:`scoring_parameter` for details.
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 other 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_iter=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 indices
that can be used to generate dataset splits according to different cross
validation strategies.


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 cross-validation on a dataset with 4 samples::

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

  >>> kf = KFold(4, n_folds=2)
  >>> for train, test in kf:
  ...     print("%s %s" % (train, test))
  [2 3] [0 1]
  [0 1] [2 3]

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]


Stratified k-fold
-----------------

:class:`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.

Example of stratified 2-fold cross-validation on a dataset with 10 samples from
two slightly unbalanced classes::

  >>> from sklearn.cross_validation import StratifiedKFold

  >>> labels = [0, 0, 0, 0, 1, 1, 1, 1, 1, 1]
  >>> skf = StratifiedKFold(labels, 3)
  >>> for train, test in skf:
  ...     print("%s %s" % (train, test))
  [2 3 6 7 8 9] [0 1 4 5]
  [0 1 3 4 5 8 9] [2 6 7]
  [0 1 2 4 5 6 7] [3 8 9]


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

  >>> loo = LeaveOneOut(4)
  >>> for train, test in loo:
  ...     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.


.. topic:: References:

 * http://www.faqs.org/faqs/ai-faq/neural-nets/part3/section-12.html
 * T. Hastie, R. Tibshirani, J. Friedman,  `The Elements of Statistical Learning
   <http://www-stat.stanford.edu/~tibs/ElemStatLearn>`_, Springer 2009
 * L. Breiman, P. Spector `Submodel selection and evaluation in regression: The X-random case
   <http://digitalassets.lib.berkeley.edu/sdtr/ucb/text/197.pdf>`_, International Statistical Review 1992
 * R. Kohavi, `A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection
   <http://www.cs.iastate.edu/~jtian/cs573/Papers/Kohavi-IJCAI-95.pdf>`_, Intl. Jnt. Conf. AI   
 * R. Bharat Rao, G. Fung, R. Rosales, `On the Dangers of Cross-Validation. An Experimental Evaluation
   <http://www.siam.org/proceedings/datamining/2008/dm08_54_Rao.pdf>`_, SIAM 2008
 * G. James, D. Witten, T. Hastie, R Tibshirani, `An Introduction to Statitical Learning
   <http://www-bcf.usc.edu/~gareth/ISL>`_, Springer 2013


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

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

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

  >>> from sklearn.cross_validation import LeavePOut

  >>> lpo = LeavePOut(4, p=2)
  >>> for train, test in lpo:
  ...     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]


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

:class:`LeaveOneLabelOut` (LOLO) is a cross-validation scheme which holds out
the samples according to a third-party provided array of integer labels. This
label 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 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

  >>> labels = [1, 1, 2, 2]
  >>> lolo = LeaveOneLabelOut(labels)
  >>> for train, test in lolo:
  ...     print("%s %s" % (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.

.. warning::

  Contrary to :class:`StratifiedKFold`, **the `labels` of
  :class:`LeaveOneLabelOut` should not encode the target class to predict**:
  the goal of :class:`StratifiedKFold` is to rebalance dataset classes across
  the train / test split to ensure that the train and test folds have
  approximately the same percentage of samples of each class while
  :class:`LeaveOneLabelOut` will do the opposite by ensuring that the samples
  of the train and test fold will not share the same label value.


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

  >>> labels = [1, 1, 2, 2, 3, 3]
  >>> lplo = LeavePLabelOut(labels, p=2)
  >>> for train, test in lplo:
  ...     print("%s %s" % (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 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::

  >>> ss = cross_validation.ShuffleSplit(5, n_iter=3, test_size=0.25,
  ...     random_state=0)
  >>> for train_index, test_index in ss:
  ...     print("%s %s" % (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.

A note on shuffling
===================

If the data ordering is not arbitrary (e.g. samples with the same label are
contiguous), shuffling it first may be essential to get a meaningful cross-
validation result. However, the opposite may be true if the samples are not
independently and identically distributed. For example, if samples correspond
to news articles, and are ordered by their time of publication, then shuffling
the data will likely lead to a model that is overfit and an inflated validation
score: it will be tested on samples that are artificially similar (close in
time) to training samples.

Some cross validation iterators, such as :class:`KFold`, have an inbuilt option
to shuffle the data indices before splitting them. Note that:

* This consumes less memory than shuffling the data directly.
* By default no shuffling occurs, including for the (stratified) K fold cross-
  validation performed by specifying ``cv=some_integer`` to
  :func:`cross_val_score`, grid search, etc. Keep in mind that
  :func:`train_test_split` still returns a random split.
* The ``random_state`` parameter defaults to ``None``, meaning that the
  shuffling will be different every time ``KFold(..., shuffle=True)`` is
  iterated. However, ``GridSearchCV`` will use the same shuffling for each set
  of parameters validated by a single call to its ``fit`` method.
* To ensure results are repeatable (*on the same platform*), use a fixed value
  for ``random_state``.

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`.
