.. _cross_validation:

===================================================
Cross-validation: evaluating estimator performance
===================================================

.. currentmodule:: sklearn.model_selection

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.model_selection import train_test_split
  >>> 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 = 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 <https://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)::

  >>> from sklearn.model_selection import cross_val_score
  >>> clf = svm.SVC(kernel='linear', C=1)
  >>> scores = 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 95\% confidence interval 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_val_score(
  ...     clf, iris.data, iris.target, cv=5, scoring='f1_macro')
  >>> 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, the latter
being used if the estimator derives from :class:`ClassifierMixin
<sklearn.base.ClassifierMixin>`.

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

  >>> from sklearn.model_selection import ShuffleSplit
  >>> n_samples = iris.data.shape[0]
  >>> cv = ShuffleSplit(n_splits=3, test_size=0.3, random_state=0)
  >>> cross_val_score(clf, iris.data, iris.target, cv=cv)
  ...                                                     # doctest: +ELLIPSIS
  array([ 0.97...,  0.97...,  1.        ])


.. topic:: Data transformation with held out data

    Just as it is important to test a predictor on data held-out from
    training, preprocessing (such as standardization, feature selection, etc.)
    and similar :ref:`data transformations <data-transforms>` similarly should
    be learnt from a training set and applied to held-out data for prediction::

      >>> from sklearn import preprocessing
      >>> X_train, X_test, y_train, y_test = train_test_split(
      ...     iris.data, iris.target, test_size=0.4, random_state=0)
      >>> scaler = preprocessing.StandardScaler().fit(X_train)
      >>> X_train_transformed = scaler.transform(X_train)
      >>> clf = svm.SVC(C=1).fit(X_train_transformed, y_train)
      >>> X_test_transformed = scaler.transform(X_test)
      >>> clf.score(X_test_transformed, y_test)  # doctest: +ELLIPSIS
      0.9333...

    A :class:`Pipeline <sklearn.pipeline.Pipeline>` makes it easier to compose
    estimators, providing this behavior under cross-validation::

      >>> from sklearn.pipeline import make_pipeline
      >>> clf = make_pipeline(preprocessing.StandardScaler(), svm.SVC(C=1))
      >>> cross_val_score(clf, iris.data, iris.target, cv=cv)
      ...                                                 # doctest: +ELLIPSIS
      array([ 0.97...,  0.93...,  0.95...])

    See :ref:`combining_estimators`.

Obtaining predictions by cross-validation
-----------------------------------------

The function :func:`cross_val_predict` has a similar interface to
:func:`cross_val_score`, but returns, for each element in the input, the
prediction that was obtained for that element when it was in the test set. Only
cross-validation strategies that assign all elements to a test set exactly once
can be used (otherwise, an exception is raised).

These prediction can then be used to evaluate the classifier::

  >>> from sklearn.model_selection import cross_val_predict
  >>> predicted = cross_val_predict(clf, iris.data, iris.target, cv=10)
  >>> metrics.accuracy_score(iris.target, predicted) # doctest: +ELLIPSIS
  0.966...

Note that the result of this computation may be slightly different from those
obtained using :func:`cross_val_score` as the elements are grouped in different
ways.

The available cross validation iterators are introduced in the following
section.

.. topic:: Examples

    * :ref:`sphx_glr_auto_examples_model_selection_plot_roc_crossval.py`,
    * :ref:`sphx_glr_auto_examples_feature_selection_plot_rfe_with_cross_validation.py`,
    * :ref:`sphx_glr_auto_examples_model_selection_grid_search_digits.py`,
    * :ref:`sphx_glr_auto_examples_model_selection_grid_search_text_feature_extraction.py`,
    * :ref:`sphx_glr_auto_examples_plot_cv_predict.py`,
    * :ref:`sphx_glr_auto_examples_model_selection_plot_nested_cross_validation_iris.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.

.. _iid_cv

Cross-validation iterators for i.i.d. data
==========================================

Assuming that some data is Independent Identically Distributed (i.i.d.) is
making the assumption that all samples stem from the same generative process
and that the generative process is assumed to have no memory of past generated
samples.

The following cross-validators can be used in such cases.

**NOTE**

While i.i.d. data is a common assumption in machine learning theory, it rarely
holds in practice. If one knows that the samples have been generated using a
time-dependent process, it's safer to
use a `time-series aware cross-validation scheme <time_series_cv>`
Similarly if we know that the generative process has a group structure
(samples from collected from different subjects, experiments, measurement
devices) it safer to use `group-wise cross-validation <group_cv>`.


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.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]

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]


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 :math:`n` samples, we have :math:`n` different
training sets and :math:`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 :math:`k`-fold cross validation, one builds :math:`n` models
from :math:`n` samples instead of :math:`k` models, where :math:`n > k`.
Moreover, each is trained on :math:`n - 1` samples rather than
:math:`(k-1) n / k`. In both ways, assuming :math:`k` is not too large
and :math:`k < n`, LOO is more computationally expensive than :math:`k`-fold
cross validation.

In terms of accuracy, LOO often results in high variance as an estimator for the
test error. Intuitively, since :math:`n - 1` of
the :math:`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://statweb.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://web.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://people.csail.mit.edu/romer/papers/CrossVal_SDM08.pdf>`_, SIAM 2008;
 * G. James, D. Witten, T. Hastie, R Tibshirani, `An Introduction to
   Statistical 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.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]


.. _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::

  >>> from sklearn.model_selection import ShuffleSplit
  >>> X = np.arange(5)
  >>> ss = ShuffleSplit(n_splits=3, test_size=0.25,
  ...     random_state=0)
  >>> for train_index, test_index in ss.split(X):
  ...     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 on each side of the train / test split.

Cross-validation iterators with stratification based on class labels.
=====================================================================

Some classification problems can exhibit a large imbalance in the distribution
of the target classes: for instance there could be several times more negative
samples than positive samples. In such cases it is recommended to use
stratified sampling as implemented in :class:`StratifiedKFold` and
:class:`StratifiedShuffleSplit` to ensure that relative class frequencies is
approximately preserved in each train and validation fold.

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 3-fold cross-validation on a dataset with 10 samples from
two slightly unbalanced classes::

  >>> from sklearn.model_selection import StratifiedKFold

  >>> X = np.ones(10)
  >>> y = [0, 0, 0, 0, 1, 1, 1, 1, 1, 1]
  >>> skf = StratifiedKFold(n_splits=3)
  >>> for train, test in skf.split(X, y):
  ...     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]

Stratified Shuffle Split
------------------------

: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.

.. _group_cv

Cross-validation iterators for grouped data.
============================================

The i.i.d. assumption is broken if the underlying generative process yield
groups of dependent samples.

Such a grouping of data is domain specific. An example would be when there is
medical data collected from multiple patients, with multiple samples taken from
each patient. And such data is likely to be dependent on the individual group.
In our example, the patient id for each sample will be its group identifier.

In this case we would like to know if a model trained on a particular set of
groups generalizes well to the unseen groups. To measure this, we need to
ensure that all the samples in the validation fold come from groups that are
not represented at all in the paired training fold.
 
The following cross-validation splitters can be used to do that.
The grouping identifier for the samples is specified via the ``groups``
parameter.


Group k-fold
------------

class: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. class: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.


Leave One Group Out
-------------------

:class:`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, :class:`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.

Leave P Groups Out
------------------

:class:`LeavePGroupsOut` is similar as :class:`LeaveOneGroupOut`, but removes
samples related to :math:`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]

Group Shuffle Split
-------------------

:class:`GroupShuffleSplit`

The :class:`GroupShuffleSplit` iterator behaves as a combination of
:class:`ShuffleSplit` and :class:`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]

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


Predefined Fold-Splits / Validation-Sets
========================================

For some datasets, a pre-defined split of the data into training- and
validation fold or into several cross-validation folds already
exists. Using :class:`PredefinedSplit` it is possible to use these folds
e.g. when searching for hyperparameters.

For example, when using a validation set, set the ``test_fold`` to 0 for all
samples that are part of the validation set, and to -1 for all other samples.

.. _timeseries_cv

Cross validation of time series data
====================================

Time series data is characterised by the correlation between observations 
that are near in time (*autocorrelation*). However, classical 
cross-validation techniques such as :class:`KFold` and 
:class:`ShuffleSplit` assume the samples are independent and 
identically distributed, and would result in unreasonable correlation 
between training and testing instances (yielding poor estimates of 
generalisation error) on time series data. Therefore, it is very important 
to evaluate our model for time series data on the "future" observations 
least like those that are used to train the model. To achieve this, one 
solution is provided by :class:`TimeSeriesSplit`.


Time Series Split
-----------------

:class:`TimeSeriesSplit` is a variation of *k-fold* which 
returns first :math:`k` folds as train set and the :math:`(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)  # doctest: +NORMALIZE_WHITESPACE
  TimeSeriesSplit(n_splits=3)
  >>> 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]


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

If the data ordering is not arbitrary (e.g. samples with the same class 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 of the next section: :ref:`grid_search`.