classification.py

"""Metrics to assess performance on classification task given classe prediction

Functions named as ``*_score`` return a scalar value to maximize: the higher
the better

Function named as ``*_error`` or ``*_loss`` return a scalar value to minimize:
the lower the better
"""

# Authors: Alexandre Gramfort <alexandre.gramfort@inria.fr>
#          Mathieu Blondel <mathieu@mblondel.org>
#          Olivier Grisel <olivier.grisel@ensta.org>
#          Arnaud Joly <a.joly@ulg.ac.be>
#          Jochen Wersdorfer <jochen@wersdoerfer.de>
#          Lars Buitinck <L.J.Buitinck@uva.nl>
#          Joel Nothman <joel.nothman@gmail.com>
#          Noel Dawe <noel@dawe.me>
#          Jatin Shah <jatindshah@gmail.com>
#          Saurabh Jha <saurabh.jhaa@gmail.com>
# License: BSD 3 clause

from __future__ import division

import warnings
import numpy as np

from scipy.sparse import coo_matrix
from scipy.sparse import csr_matrix
from scipy.spatial.distance import hamming as sp_hamming

from ..preprocessing import LabelBinarizer, label_binarize
from ..preprocessing import LabelEncoder
from ..utils import check_array
from ..utils import check_consistent_length
from ..utils import column_or_1d
from ..utils.multiclass import unique_labels
from ..utils.multiclass import type_of_target
from ..utils.validation import _num_samples
from ..utils.sparsefuncs import count_nonzero
from ..utils.fixes import bincount

from .base import UndefinedMetricWarning


def _check_targets(y_true, y_pred):
    """Check that y_true and y_pred belong to the same classification task

    This converts multiclass or binary types to a common shape, and raises a
    ValueError for a mix of multilabel and multiclass targets, a mix of
    multilabel formats, for the presence of continuous-valued or multioutput
    targets, or for targets of different lengths.

    Column vectors are squeezed to 1d, while multilabel formats are returned
    as CSR sparse label indicators.

    Parameters
    ----------
    y_true : array-like

    y_pred : array-like

    Returns
    -------
    type_true : one of {'multilabel-indicator', 'multiclass', 'binary'}
        The type of the true target data, as output by
        ``utils.multiclass.type_of_target``

    y_true : array or indicator matrix

    y_pred : array or indicator matrix
    """
    check_consistent_length(y_true, y_pred)
    type_true = type_of_target(y_true)
    type_pred = type_of_target(y_pred)

    y_type = set([type_true, type_pred])
    if y_type == set(["binary", "multiclass"]):
        y_type = set(["multiclass"])

    if len(y_type) > 1:
        raise ValueError("Can't handle mix of {0} and {1}"
                         "".format(type_true, type_pred))

    # We can't have more than one value on y_type => The set is no more needed
    y_type = y_type.pop()

    # No metrics support "multiclass-multioutput" format
    if (y_type not in ["binary", "multiclass", "multilabel-indicator"]):
        raise ValueError("{0} is not supported".format(y_type))

    if y_type in ["binary", "multiclass"]:
        y_true = column_or_1d(y_true)
        y_pred = column_or_1d(y_pred)

    if y_type.startswith('multilabel'):
        y_true = csr_matrix(y_true)
        y_pred = csr_matrix(y_pred)
        y_type = 'multilabel-indicator'

    return y_type, y_true, y_pred


def _weighted_sum(sample_score, sample_weight, normalize=False):
    if normalize:
        return np.average(sample_score, weights=sample_weight)
    elif sample_weight is not None:
        return np.dot(sample_score, sample_weight)
    else:
        return sample_score.sum()


def accuracy_score(y_true, y_pred, normalize=True, sample_weight=None):
    """Accuracy classification score.

    In multilabel classification, this function computes subset accuracy:
    the set of labels predicted for a sample must *exactly* match the
    corresponding set of labels in y_true.

    Read more in the :ref:`User Guide <accuracy_score>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) labels.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Predicted labels, as returned by a classifier.

    normalize : bool, optional (default=True)
        If ``False``, return the number of correctly classified samples.
        Otherwise, return the fraction of correctly classified samples.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    score : float
        If ``normalize == True``, return the correctly classified samples
        (float), else it returns the number of correctly classified samples
        (int).

        The best performance is 1 with ``normalize == True`` and the number
        of samples with ``normalize == False``.

    See also
    --------
    jaccard_similarity_score, hamming_loss, zero_one_loss

    Notes
    -----
    In binary and multiclass classification, this function is equal
    to the ``jaccard_similarity_score`` function.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.metrics import accuracy_score
    >>> y_pred = [0, 2, 1, 3]
    >>> y_true = [0, 1, 2, 3]
    >>> accuracy_score(y_true, y_pred)
    0.5
    >>> accuracy_score(y_true, y_pred, normalize=False)
    2

    In the multilabel case with binary label indicators:
    >>> accuracy_score(np.array([[0, 1], [1, 1]]), np.ones((2, 2)))
    0.5
    """

    # Compute accuracy for each possible representation
    y_type, y_true, y_pred = _check_targets(y_true, y_pred)
    if y_type.startswith('multilabel'):
        differing_labels = count_nonzero(y_true - y_pred, axis=1)
        score = differing_labels == 0
    else:
        score = y_true == y_pred

    return _weighted_sum(score, sample_weight, normalize)


def confusion_matrix(y_true, y_pred, labels=None):
    """Compute confusion matrix to evaluate the accuracy of a classification

    By definition a confusion matrix :math:`C` is such that :math:`C_{i, j}`
    is equal to the number of observations known to be in group :math:`i` but
    predicted to be in group :math:`j`.

    Read more in the :ref:`User Guide <confusion_matrix>`.

    Parameters
    ----------
    y_true : array, shape = [n_samples]
        Ground truth (correct) target values.

    y_pred : array, shape = [n_samples]
        Estimated targets as returned by a classifier.

    labels : array, shape = [n_classes], optional
        List of labels to index the matrix. This may be used to reorder
        or select a subset of labels.
        If none is given, those that appear at least once
        in ``y_true`` or ``y_pred`` are used in sorted order.

    Returns
    -------
    C : array, shape = [n_classes, n_classes]
        Confusion matrix

    References
    ----------
    .. [1] `Wikipedia entry for the Confusion matrix
           <http://en.wikipedia.org/wiki/Confusion_matrix>`_

    Examples
    --------
    >>> from sklearn.metrics import confusion_matrix
    >>> y_true = [2, 0, 2, 2, 0, 1]
    >>> y_pred = [0, 0, 2, 2, 0, 2]
    >>> confusion_matrix(y_true, y_pred)
    array([[2, 0, 0],
           [0, 0, 1],
           [1, 0, 2]])

    >>> y_true = ["cat", "ant", "cat", "cat", "ant", "bird"]
    >>> y_pred = ["ant", "ant", "cat", "cat", "ant", "cat"]
    >>> confusion_matrix(y_true, y_pred, labels=["ant", "bird", "cat"])
    array([[2, 0, 0],
           [0, 0, 1],
           [1, 0, 2]])

    """
    y_type, y_true, y_pred = _check_targets(y_true, y_pred)
    if y_type not in ("binary", "multiclass"):
        raise ValueError("%s is not supported" % y_type)

    if labels is None:
        labels = unique_labels(y_true, y_pred)
    else:
        labels = np.asarray(labels)

    n_labels = labels.size
    label_to_ind = dict((y, x) for x, y in enumerate(labels))
    # convert yt, yp into index
    y_pred = np.array([label_to_ind.get(x, n_labels + 1) for x in y_pred])
    y_true = np.array([label_to_ind.get(x, n_labels + 1) for x in y_true])

    # intersect y_pred, y_true with labels, eliminate items not in labels
    ind = np.logical_and(y_pred < n_labels, y_true < n_labels)
    y_pred = y_pred[ind]
    y_true = y_true[ind]

    CM = coo_matrix((np.ones(y_true.shape[0], dtype=np.int), (y_true, y_pred)),
                    shape=(n_labels, n_labels)
                    ).toarray()

    return CM


def cohen_kappa_score(y1, y2, labels=None):
    """Cohen's kappa: a statistic that measures inter-annotator agreement.

    This function computes Cohen's kappa [1], a score that expresses the level
    of agreement between two annotators on a classification problem. It is
    defined as

    .. math::
        \kappa = (p_o - p_e) / (1 - p_e)

    where :math:`p_o` is the empirical probability of agreement on the label
    assigned to any sample (the observed agreement ratio), and :math:`p_e` is
    the expected agreement when both annotators assign labels randomly.
    :math:`p_e` is estimated using a per-annotator empirical prior over the
    class labels [2].

    Parameters
    ----------
    y1 : array, shape = [n_samples]
        Labels assigned by the first annotator.

    y2 : array, shape = [n_samples]
        Labels assigned by the second annotator. The kappa statistic is
        symmetric, so swapping ``y1`` and ``y2`` doesn't change the value.

    labels : array, shape = [n_classes], optional
        List of labels to index the matrix. This may be used to select a
        subset of labels. If None, all labels that appear at least once in
        ``y1`` or ``y2`` are used.

    Returns
    -------
    kappa : float
        The kappa statistic, which is a number between -1 and 1. The maximum
        value means complete agreement; zero or lower means chance agreement.

    References
    ----------
    .. [1] J. Cohen (1960). "A coefficient of agreement for nominal scales".
           Educational and Psychological Measurement 20(1):37-46.
           doi:10.1177/001316446002000104.
    .. [2] R. Artstein and M. Poesio (2008). "Inter-coder agreement for
           computational linguistics". Computational Linguistic 34(4):555-596.
    """
    confusion = confusion_matrix(y1, y2, labels=labels)
    P = confusion / float(confusion.sum())
    p_observed = np.trace(P)
    p_expected = np.dot(P.sum(axis=0), P.sum(axis=1))
    return (p_observed - p_expected) / (1 - p_expected)


def jaccard_similarity_score(y_true, y_pred, normalize=True,
                             sample_weight=None):
    """Jaccard similarity coefficient score

    The Jaccard index [1], or Jaccard similarity coefficient, defined as
    the size of the intersection divided by the size of the union of two label
    sets, is used to compare set of predicted labels for a sample to the
    corresponding set of labels in ``y_true``.

    Read more in the :ref:`User Guide <jaccard_similarity_score>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) labels.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Predicted labels, as returned by a classifier.

    normalize : bool, optional (default=True)
        If ``False``, return the sum of the Jaccard similarity coefficient
        over the sample set. Otherwise, return the average of Jaccard
        similarity coefficient.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    score : float
        If ``normalize == True``, return the average Jaccard similarity
        coefficient, else it returns the sum of the Jaccard similarity
        coefficient over the sample set.

        The best performance is 1 with ``normalize == True`` and the number
        of samples with ``normalize == False``.

    See also
    --------
    accuracy_score, hamming_loss, zero_one_loss

    Notes
    -----
    In binary and multiclass classification, this function is equivalent
    to the ``accuracy_score``. It differs in the multilabel classification
    problem.

    References
    ----------
    .. [1] `Wikipedia entry for the Jaccard index
           <http://en.wikipedia.org/wiki/Jaccard_index>`_


    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.metrics import jaccard_similarity_score
    >>> y_pred = [0, 2, 1, 3]
    >>> y_true = [0, 1, 2, 3]
    >>> jaccard_similarity_score(y_true, y_pred)
    0.5
    >>> jaccard_similarity_score(y_true, y_pred, normalize=False)
    2

    In the multilabel case with binary label indicators:

    >>> jaccard_similarity_score(np.array([[0, 1], [1, 1]]),\
        np.ones((2, 2)))
    0.75
    """

    # Compute accuracy for each possible representation
    y_type, y_true, y_pred = _check_targets(y_true, y_pred)
    if y_type.startswith('multilabel'):
        with np.errstate(divide='ignore', invalid='ignore'):
            # oddly, we may get an "invalid" rather than a "divide" error here
            pred_or_true = count_nonzero(y_true + y_pred, axis=1)
            pred_and_true = count_nonzero(y_true.multiply(y_pred), axis=1)
            score = pred_and_true / pred_or_true

            # If there is no label, it results in a Nan instead, we set
            # the jaccard to 1: lim_{x->0} x/x = 1
            # Note with py2.6 and np 1.3: we can't check safely for nan.
            score[pred_or_true == 0.0] = 1.0
    else:
        score = y_true == y_pred

    return _weighted_sum(score, sample_weight, normalize)


def matthews_corrcoef(y_true, y_pred):
    """Compute the Matthews correlation coefficient (MCC) for binary classes

    The Matthews correlation coefficient is used in machine learning as a
    measure of the quality of binary (two-class) classifications. It takes into
    account true and false positives and negatives and is generally regarded as
    a balanced measure which can be used even if the classes are of very
    different sizes. The MCC is in essence a correlation coefficient value
    between -1 and +1. A coefficient of +1 represents a perfect prediction, 0
    an average random prediction and -1 an inverse prediction.  The statistic
    is also known as the phi coefficient. [source: Wikipedia]

    Only in the binary case does this relate to information about true and
    false positives and negatives. See references below.

    Read more in the :ref:`User Guide <matthews_corrcoef>`.

    Parameters
    ----------
    y_true : array, shape = [n_samples]
        Ground truth (correct) target values.

    y_pred : array, shape = [n_samples]
        Estimated targets as returned by a classifier.

    Returns
    -------
    mcc : float
        The Matthews correlation coefficient (+1 represents a perfect
        prediction, 0 an average random prediction and -1 and inverse
        prediction).

    References
    ----------
    .. [1] `Baldi, Brunak, Chauvin, Andersen and Nielsen, (2000). Assessing the
       accuracy of prediction algorithms for classification: an overview
       <http://dx.doi.org/10.1093/bioinformatics/16.5.412>`_

    .. [2] `Wikipedia entry for the Matthews Correlation Coefficient
       <http://en.wikipedia.org/wiki/Matthews_correlation_coefficient>`_

    Examples
    --------
    >>> from sklearn.metrics import matthews_corrcoef
    >>> y_true = [+1, +1, +1, -1]
    >>> y_pred = [+1, -1, +1, +1]
    >>> matthews_corrcoef(y_true, y_pred)  # doctest: +ELLIPSIS
    -0.33...

    """
    y_type, y_true, y_pred = _check_targets(y_true, y_pred)

    if y_type != "binary":
        raise ValueError("%s is not supported" % y_type)

    lb = LabelEncoder()
    lb.fit(np.hstack([y_true, y_pred]))
    y_true = lb.transform(y_true)
    y_pred = lb.transform(y_pred)
    with np.errstate(invalid='ignore'):
        mcc = np.corrcoef(y_true, y_pred)[0, 1]

    if np.isnan(mcc):
        return 0.
    else:
        return mcc


def zero_one_loss(y_true, y_pred, normalize=True, sample_weight=None):
    """Zero-one classification loss.

    If normalize is ``True``, return the fraction of misclassifications
    (float), else it returns the number of misclassifications (int). The best
    performance is 0.

    Read more in the :ref:`User Guide <zero_one_loss>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) labels.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Predicted labels, as returned by a classifier.

    normalize : bool, optional (default=True)
        If ``False``, return the number of misclassifications.
        Otherwise, return the fraction of misclassifications.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    loss : float or int,
        If ``normalize == True``, return the fraction of misclassifications
        (float), else it returns the number of misclassifications (int).

    Notes
    -----
    In multilabel classification, the zero_one_loss function corresponds to
    the subset zero-one loss: for each sample, the entire set of labels must be
    correctly predicted, otherwise the loss for that sample is equal to one.

    See also
    --------
    accuracy_score, hamming_loss, jaccard_similarity_score

    Examples
    --------
    >>> from sklearn.metrics import zero_one_loss
    >>> y_pred = [1, 2, 3, 4]
    >>> y_true = [2, 2, 3, 4]
    >>> zero_one_loss(y_true, y_pred)
    0.25
    >>> zero_one_loss(y_true, y_pred, normalize=False)
    1

    In the multilabel case with binary label indicators:

    >>> zero_one_loss(np.array([[0, 1], [1, 1]]), np.ones((2, 2)))
    0.5
    """
    score = accuracy_score(y_true, y_pred,
                           normalize=normalize,
                           sample_weight=sample_weight)

    if normalize:
        return 1 - score
    else:
        if sample_weight is not None:
            n_samples = np.sum(sample_weight)
        else:
            n_samples = _num_samples(y_true)
        return n_samples - score


def f1_score(y_true, y_pred, labels=None, pos_label=1, average='binary',
             sample_weight=None):
    """Compute the F1 score, also known as balanced F-score or F-measure

    The F1 score can be interpreted as a weighted average of the precision and
    recall, where an F1 score reaches its best value at 1 and worst score at 0.
    The relative contribution of precision and recall to the F1 score are
    equal. The formula for the F1 score is::

        F1 = 2 * (precision * recall) / (precision + recall)

    In the multi-class and multi-label case, this is the weighted average of
    the F1 score of each class.

    Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) target values.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Estimated targets as returned by a classifier.

    labels : list, optional
        The set of labels to include when ``average != 'binary'``, and their
        order if ``average is None``. Labels present in the data can be
        excluded, for example to calculate a multiclass average ignoring a
        majority negative class, while labels not present in the data will
        result in 0 components in a macro average. For multilabel targets,
        labels are column indices. By default, all labels in ``y_true`` and
        ``y_pred`` are used in sorted order.

        .. versionchanged:: 0.17
           parameter *labels* improved for multiclass problem.

    pos_label : str or int, 1 by default
        The class to report if ``average='binary'``. Until version 0.18 it is
        necessary to set ``pos_label=None`` if seeking to use another averaging
        method over binary targets.

    average : string, [None, 'binary' (default), 'micro', 'macro', 'samples', \
                       'weighted']
        This parameter is required for multiclass/multilabel targets.
        If ``None``, the scores for each class are returned. Otherwise, this
        determines the type of averaging performed on the data:

        ``'binary'``:
            Only report results for the class specified by ``pos_label``.
            This is applicable only if targets (``y_{true,pred}``) are binary.
        ``'micro'``:
            Calculate metrics globally by counting the total true positives,
            false negatives and false positives.
        ``'macro'``:
            Calculate metrics for each label, and find their unweighted
            mean.  This does not take label imbalance into account.
        ``'weighted'``:
            Calculate metrics for each label, and find their average, weighted
            by support (the number of true instances for each label). This
            alters 'macro' to account for label imbalance; it can result in an
            F-score that is not between precision and recall.
        ``'samples'``:
            Calculate metrics for each instance, and find their average (only
            meaningful for multilabel classification where this differs from
            :func:`accuracy_score`).

        Note that if ``pos_label`` is given in binary classification with
        `average != 'binary'`, only that positive class is reported. This
        behavior is deprecated and will change in version 0.18.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    f1_score : float or array of float, shape = [n_unique_labels]
        F1 score of the positive class in binary classification or weighted
        average of the F1 scores of each class for the multiclass task.

    References
    ----------
    .. [1] `Wikipedia entry for the F1-score <http://en.wikipedia.org/wiki/F1_score>`_

    Examples
    --------
    >>> from sklearn.metrics import f1_score
    >>> y_true = [0, 1, 2, 0, 1, 2]
    >>> y_pred = [0, 2, 1, 0, 0, 1]
    >>> f1_score(y_true, y_pred, average='macro')  # doctest: +ELLIPSIS
    0.26...
    >>> f1_score(y_true, y_pred, average='micro')  # doctest: +ELLIPSIS
    0.33...
    >>> f1_score(y_true, y_pred, average='weighted')  # doctest: +ELLIPSIS
    0.26...
    >>> f1_score(y_true, y_pred, average=None)
    array([ 0.8,  0. ,  0. ])


    """
    return fbeta_score(y_true, y_pred, 1, labels=labels,
                       pos_label=pos_label, average=average,
                       sample_weight=sample_weight)


def fbeta_score(y_true, y_pred, beta, labels=None, pos_label=1,
                average='binary', sample_weight=None):
    """Compute the F-beta score

    The F-beta score is the weighted harmonic mean of precision and recall,
    reaching its optimal value at 1 and its worst value at 0.

    The `beta` parameter determines the weight of precision in the combined
    score. ``beta < 1`` lends more weight to precision, while ``beta > 1``
    favors recall (``beta -> 0`` considers only precision, ``beta -> inf``
    only recall).

    Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) target values.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Estimated targets as returned by a classifier.

    beta: float
        Weight of precision in harmonic mean.

    labels : list, optional
        The set of labels to include when ``average != 'binary'``, and their
        order if ``average is None``. Labels present in the data can be
        excluded, for example to calculate a multiclass average ignoring a
        majority negative class, while labels not present in the data will
        result in 0 components in a macro average. For multilabel targets,
        labels are column indices. By default, all labels in ``y_true`` and
        ``y_pred`` are used in sorted order.

        .. versionchanged:: 0.17
           parameter *labels* improved for multiclass problem.

    pos_label : str or int, 1 by default
        The class to report if ``average='binary'``. Until version 0.18 it is
        necessary to set ``pos_label=None`` if seeking to use another averaging
        method over binary targets.

    average : string, [None, 'binary' (default), 'micro', 'macro', 'samples', \
                       'weighted']
        This parameter is required for multiclass/multilabel targets.
        If ``None``, the scores for each class are returned. Otherwise, this
        determines the type of averaging performed on the data:

        ``'binary'``:
            Only report results for the class specified by ``pos_label``.
            This is applicable only if targets (``y_{true,pred}``) are binary.
        ``'micro'``:
            Calculate metrics globally by counting the total true positives,
            false negatives and false positives.
        ``'macro'``:
            Calculate metrics for each label, and find their unweighted
            mean.  This does not take label imbalance into account.
        ``'weighted'``:
            Calculate metrics for each label, and find their average, weighted
            by support (the number of true instances for each label). This
            alters 'macro' to account for label imbalance; it can result in an
            F-score that is not between precision and recall.
        ``'samples'``:
            Calculate metrics for each instance, and find their average (only
            meaningful for multilabel classification where this differs from
            :func:`accuracy_score`).

        Note that if ``pos_label`` is given in binary classification with
        `average != 'binary'`, only that positive class is reported. This
        behavior is deprecated and will change in version 0.18.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    fbeta_score : float (if average is not None) or array of float, shape =\
        [n_unique_labels]
        F-beta score of the positive class in binary classification or weighted
        average of the F-beta score of each class for the multiclass task.

    References
    ----------
    .. [1] R. Baeza-Yates and B. Ribeiro-Neto (2011).
           Modern Information Retrieval. Addison Wesley, pp. 327-328.

    .. [2] `Wikipedia entry for the F1-score
           <http://en.wikipedia.org/wiki/F1_score>`_

    Examples
    --------
    >>> from sklearn.metrics import fbeta_score
    >>> y_true = [0, 1, 2, 0, 1, 2]
    >>> y_pred = [0, 2, 1, 0, 0, 1]
    >>> fbeta_score(y_true, y_pred, average='macro', beta=0.5)
    ... # doctest: +ELLIPSIS
    0.23...
    >>> fbeta_score(y_true, y_pred, average='micro', beta=0.5)
    ... # doctest: +ELLIPSIS
    0.33...
    >>> fbeta_score(y_true, y_pred, average='weighted', beta=0.5)
    ... # doctest: +ELLIPSIS
    0.23...
    >>> fbeta_score(y_true, y_pred, average=None, beta=0.5)
    ... # doctest: +ELLIPSIS
    array([ 0.71...,  0.        ,  0.        ])

    """
    _, _, f, _ = precision_recall_fscore_support(y_true, y_pred,
                                                 beta=beta,
                                                 labels=labels,
                                                 pos_label=pos_label,
                                                 average=average,
                                                 warn_for=('f-score',),
                                                 sample_weight=sample_weight)
    return f


def _prf_divide(numerator, denominator, metric, modifier, average, warn_for):
    """Performs division and handles divide-by-zero.

    On zero-division, sets the corresponding result elements to zero
    and raises a warning.

    The metric, modifier and average arguments are used only for determining
    an appropriate warning.
    """
    result = numerator / denominator
    mask = denominator == 0.0
    if not np.any(mask):
        return result

    # remove infs
    result[mask] = 0.0

    # build appropriate warning
    # E.g. "Precision and F-score are ill-defined and being set to 0.0 in
    # labels with no predicted samples"
    axis0 = 'sample'
    axis1 = 'label'
    if average == 'samples':
        axis0, axis1 = axis1, axis0

    if metric in warn_for and 'f-score' in warn_for:
        msg_start = '{0} and F-score are'.format(metric.title())
    elif metric in warn_for:
        msg_start = '{0} is'.format(metric.title())
    elif 'f-score' in warn_for:
        msg_start = 'F-score is'
    else:
        return result

    msg = ('{0} ill-defined and being set to 0.0 {{0}} '
           'no {1} {2}s.'.format(msg_start, modifier, axis0))
    if len(mask) == 1:
        msg = msg.format('due to')
    else:
        msg = msg.format('in {0}s with'.format(axis1))
    warnings.warn(msg, UndefinedMetricWarning, stacklevel=2)
    return result


def precision_recall_fscore_support(y_true, y_pred, beta=1.0, labels=None,
                                    pos_label=1, average=None,
                                    warn_for=('precision', 'recall',
                                              'f-score'),
                                    sample_weight=None):
    """Compute precision, recall, F-measure and support for each class

    The precision is the ratio ``tp / (tp + fp)`` where ``tp`` is the number of
    true positives and ``fp`` the number of false positives. The precision is
    intuitively the ability of the classifier not to label as positive a sample
    that is negative.

    The recall is the ratio ``tp / (tp + fn)`` where ``tp`` is the number of
    true positives and ``fn`` the number of false negatives. The recall is
    intuitively the ability of the classifier to find all the positive samples.

    The F-beta score can be interpreted as a weighted harmonic mean of
    the precision and recall, where an F-beta score reaches its best
    value at 1 and worst score at 0.

    The F-beta score weights recall more than precision by a factor of
    ``beta``. ``beta == 1.0`` means recall and precision are equally important.

    The support is the number of occurrences of each class in ``y_true``.

    If ``pos_label is None`` and in binary classification, this function
    returns the average precision, recall and F-measure if ``average``
    is one of ``'micro'``, ``'macro'``, ``'weighted'`` or ``'samples'``.

    Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) target values.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Estimated targets as returned by a classifier.

    beta : float, 1.0 by default
        The strength of recall versus precision in the F-score.

    labels : list, optional
        The set of labels to include when ``average != 'binary'``, and their
        order if ``average is None``. Labels present in the data can be
        excluded, for example to calculate a multiclass average ignoring a
        majority negative class, while labels not present in the data will
        result in 0 components in a macro average. For multilabel targets,
        labels are column indices. By default, all labels in ``y_true`` and
        ``y_pred`` are used in sorted order.

    pos_label : str or int, 1 by default
        The class to report if ``average='binary'``. Until version 0.18 it is
        necessary to set ``pos_label=None`` if seeking to use another averaging
        method over binary targets.

    average : string, [None (default), 'binary', 'micro', 'macro', 'samples', \
                       'weighted']
        If ``None``, the scores for each class are returned. Otherwise, this
        determines the type of averaging performed on the data:

        ``'binary'``:
            Only report results for the class specified by ``pos_label``.
            This is applicable only if targets (``y_{true,pred}``) are binary.
        ``'micro'``:
            Calculate metrics globally by counting the total true positives,
            false negatives and false positives.
        ``'macro'``:
            Calculate metrics for each label, and find their unweighted
            mean.  This does not take label imbalance into account.
        ``'weighted'``:
            Calculate metrics for each label, and find their average, weighted
            by support (the number of true instances for each label). This
            alters 'macro' to account for label imbalance; it can result in an
            F-score that is not between precision and recall.
        ``'samples'``:
            Calculate metrics for each instance, and find their average (only
            meaningful for multilabel classification where this differs from
            :func:`accuracy_score`).

        Note that if ``pos_label`` is given in binary classification with
        `average != 'binary'`, only that positive class is reported. This
        behavior is deprecated and will change in version 0.18.

    warn_for : tuple or set, for internal use
        This determines which warnings will be made in the case that this
        function is being used to return only one of its metrics.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    precision: float (if average is not None) or array of float, shape =\
        [n_unique_labels]

    recall: float (if average is not None) or array of float, , shape =\
        [n_unique_labels]

    fbeta_score: float (if average is not None) or array of float, shape =\
        [n_unique_labels]

    support: int (if average is not None) or array of int, shape =\
        [n_unique_labels]
        The number of occurrences of each label in ``y_true``.

    References
    ----------
    .. [1] `Wikipedia entry for the Precision and recall
           <http://en.wikipedia.org/wiki/Precision_and_recall>`_

    .. [2] `Wikipedia entry for the F1-score
           <http://en.wikipedia.org/wiki/F1_score>`_

    .. [3] `Discriminative Methods for Multi-labeled Classification Advances
           in Knowledge Discovery and Data Mining (2004), pp. 22-30 by Shantanu
           Godbole, Sunita Sarawagi
           <http://www.godbole.net/shantanu/pubs/multilabelsvm-pakdd04.pdf>`

    Examples
    --------
    >>> from sklearn.metrics import precision_recall_fscore_support
    >>> y_true = np.array(['cat', 'dog', 'pig', 'cat', 'dog', 'pig'])
    >>> y_pred = np.array(['cat', 'pig', 'dog', 'cat', 'cat', 'dog'])
    >>> precision_recall_fscore_support(y_true, y_pred, average='macro')
    ... # doctest: +ELLIPSIS
    (0.22..., 0.33..., 0.26..., None)
    >>> precision_recall_fscore_support(y_true, y_pred, average='micro')
    ... # doctest: +ELLIPSIS
    (0.33..., 0.33..., 0.33..., None)
    >>> precision_recall_fscore_support(y_true, y_pred, average='weighted')
    ... # doctest: +ELLIPSIS
    (0.22..., 0.33..., 0.26..., None)

    It is possible to compute per-label precisions, recalls, F1-scores and
    supports instead of averaging:
    >>> precision_recall_fscore_support(y_true, y_pred, average=None,
    ... labels=['pig', 'dog', 'cat'])
    ... # doctest: +ELLIPSIS,+NORMALIZE_WHITESPACE
    (array([ 0. ,  0. ,  0.66...]),
     array([ 0.,  0.,  1.]),
     array([ 0. ,  0. ,  0.8]),
     array([2, 2, 2]))

    """
    average_options = (None, 'micro', 'macro', 'weighted', 'samples')
    if average not in average_options and average != 'binary':
        raise ValueError('average has to be one of ' +
                         str(average_options))
    if beta <= 0:
        raise ValueError("beta should be >0 in the F-beta score")

    y_type, y_true, y_pred = _check_targets(y_true, y_pred)
    present_labels = unique_labels(y_true, y_pred)

    if average == 'binary' and (y_type != 'binary' or pos_label is None):
        warnings.warn('The default `weighted` averaging is deprecated, '
                      'and from version 0.18, use of precision, recall or '
                      'F-score with multiclass or multilabel data or '
                      'pos_label=None will result in an exception. '
                      'Please set an explicit value for `average`, one of '
                      '%s. In cross validation use, for instance, '
                      'scoring="f1_weighted" instead of scoring="f1".'
                      % str(average_options), DeprecationWarning, stacklevel=2)
        average = 'weighted'

    if y_type == 'binary' and pos_label is not None and average is not None:
        if average != 'binary':
            warnings.warn('From version 0.18, binary input will not be '
                          'handled specially when using averaged '
                          'precision/recall/F-score. '
                          'Please use average=\'binary\' to report only the '
                          'positive class performance.', DeprecationWarning)
        if labels is None or len(labels) <= 2:
            if pos_label not in present_labels:
                if len(present_labels) < 2:
                    # Only negative labels
                    return (0., 0., 0., 0)
                else:
                    raise ValueError("pos_label=%r is not a valid label: %r" %
                                     (pos_label, present_labels))
            labels = [pos_label]
    if labels is None:
        labels = present_labels
        n_labels = None
    else:
        n_labels = len(labels)
        labels = np.hstack([labels, np.setdiff1d(present_labels, labels,
                                                 assume_unique=True)])

    # Calculate tp_sum, pred_sum, true_sum ###

    if y_type.startswith('multilabel'):
        sum_axis = 1 if average == 'samples' else 0

        # All labels are index integers for multilabel.
        # Select labels:
        if not np.all(labels == present_labels):
            if np.max(labels) > np.max(present_labels):
                raise ValueError('All labels must be in [0, n labels). '
                                 'Got %d > %d' %
                                 (np.max(labels), np.max(present_labels)))
            if np.min(labels) < 0:
                raise ValueError('All labels must be in [0, n labels). '
                                 'Got %d < 0' % np.min(labels))

            y_true = y_true[:, labels[:n_labels]]
            y_pred = y_pred[:, labels[:n_labels]]

        # calculate weighted counts
        true_and_pred = y_true.multiply(y_pred)
        tp_sum = count_nonzero(true_and_pred, axis=sum_axis,
                               sample_weight=sample_weight)
        pred_sum = count_nonzero(y_pred, axis=sum_axis,
                                 sample_weight=sample_weight)
        true_sum = count_nonzero(y_true, axis=sum_axis,
                                 sample_weight=sample_weight)

    elif average == 'samples':
        raise ValueError("Sample-based precision, recall, fscore is "
                         "not meaningful outside multilabel "
                         "classification. See the accuracy_score instead.")
    else:
        le = LabelEncoder()
        le.fit(labels)
        y_true = le.transform(y_true)
        y_pred = le.transform(y_pred)
        sorted_labels = le.classes_

        # labels are now from 0 to len(labels) - 1 -> use bincount
        tp = y_true == y_pred
        tp_bins = y_true[tp]
        if sample_weight is not None:
            tp_bins_weights = np.asarray(sample_weight)[tp]
        else:
            tp_bins_weights = None

        if len(tp_bins):
            tp_sum = bincount(tp_bins, weights=tp_bins_weights,
                              minlength=len(labels))
        else:
            # Pathological case
            true_sum = pred_sum = tp_sum = np.zeros(len(labels))
        if len(y_pred):
            pred_sum = bincount(y_pred, weights=sample_weight,
                                minlength=len(labels))
        if len(y_true):
            true_sum = bincount(y_true, weights=sample_weight,
                                minlength=len(labels))

        # Retain only selected labels
        indices = np.searchsorted(sorted_labels, labels[:n_labels])
        tp_sum = tp_sum[indices]
        true_sum = true_sum[indices]
        pred_sum = pred_sum[indices]

    if average == 'micro':
        tp_sum = np.array([tp_sum.sum()])
        pred_sum = np.array([pred_sum.sum()])
        true_sum = np.array([true_sum.sum()])

    # Finally, we have all our sufficient statistics. Divide! #

    beta2 = beta ** 2
    with np.errstate(divide='ignore', invalid='ignore'):
        # Divide, and on zero-division, set scores to 0 and warn:

        # Oddly, we may get an "invalid" rather than a "divide" error
        # here.
        precision = _prf_divide(tp_sum, pred_sum,
                                'precision', 'predicted', average, warn_for)
        recall = _prf_divide(tp_sum, true_sum,
                             'recall', 'true', average, warn_for)
        # Don't need to warn for F: either P or R warned, or tp == 0 where pos
        # and true are nonzero, in which case, F is well-defined and zero
        f_score = ((1 + beta2) * precision * recall /
                   (beta2 * precision + recall))
        f_score[tp_sum == 0] = 0.0

    # Average the results

    if average == 'weighted':
        weights = true_sum
        if weights.sum() == 0:
            return 0, 0, 0, None
    elif average == 'samples':
        weights = sample_weight
    else:
        weights = None

    if average is not None:
        assert average != 'binary' or len(precision) == 1
        precision = np.average(precision, weights=weights)
        recall = np.average(recall, weights=weights)
        f_score = np.average(f_score, weights=weights)
        true_sum = None  # return no support

    return precision, recall, f_score, true_sum


def precision_score(y_true, y_pred, labels=None, pos_label=1,
                    average='binary', sample_weight=None):
    """Compute the precision

    The precision is the ratio ``tp / (tp + fp)`` where ``tp`` is the number of
    true positives and ``fp`` the number of false positives. The precision is
    intuitively the ability of the classifier not to label as positive a sample
    that is negative.

    The best value is 1 and the worst value is 0.

    Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) target values.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Estimated targets as returned by a classifier.

    labels : list, optional
        The set of labels to include when ``average != 'binary'``, and their
        order if ``average is None``. Labels present in the data can be
        excluded, for example to calculate a multiclass average ignoring a
        majority negative class, while labels not present in the data will
        result in 0 components in a macro average. For multilabel targets,
        labels are column indices. By default, all labels in ``y_true`` and
        ``y_pred`` are used in sorted order.

        .. versionchanged:: 0.17
           parameter *labels* improved for multiclass problem.

    pos_label : str or int, 1 by default
        The class to report if ``average='binary'``. Until version 0.18 it is
        necessary to set ``pos_label=None`` if seeking to use another averaging
        method over binary targets.

    average : string, [None, 'binary' (default), 'micro', 'macro', 'samples', \
                       'weighted']
        This parameter is required for multiclass/multilabel targets.
        If ``None``, the scores for each class are returned. Otherwise, this
        determines the type of averaging performed on the data:

        ``'binary'``:
            Only report results for the class specified by ``pos_label``.
            This is applicable only if targets (``y_{true,pred}``) are binary.
        ``'micro'``:
            Calculate metrics globally by counting the total true positives,
            false negatives and false positives.
        ``'macro'``:
            Calculate metrics for each label, and find their unweighted
            mean.  This does not take label imbalance into account.
        ``'weighted'``:
            Calculate metrics for each label, and find their average, weighted
            by support (the number of true instances for each label). This
            alters 'macro' to account for label imbalance; it can result in an
            F-score that is not between precision and recall.
        ``'samples'``:
            Calculate metrics for each instance, and find their average (only
            meaningful for multilabel classification where this differs from
            :func:`accuracy_score`).

        Note that if ``pos_label`` is given in binary classification with
        `average != 'binary'`, only that positive class is reported. This
        behavior is deprecated and will change in version 0.18.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    precision : float (if average is not None) or array of float, shape =\
        [n_unique_labels]
        Precision of the positive class in binary classification or weighted
        average of the precision of each class for the multiclass task.

    Examples
    --------

    >>> from sklearn.metrics import precision_score
    >>> y_true = [0, 1, 2, 0, 1, 2]
    >>> y_pred = [0, 2, 1, 0, 0, 1]
    >>> precision_score(y_true, y_pred, average='macro')  # doctest: +ELLIPSIS
    0.22...
    >>> precision_score(y_true, y_pred, average='micro')  # doctest: +ELLIPSIS
    0.33...
    >>> precision_score(y_true, y_pred, average='weighted')
    ... # doctest: +ELLIPSIS
    0.22...
    >>> precision_score(y_true, y_pred, average=None)  # doctest: +ELLIPSIS
    array([ 0.66...,  0.        ,  0.        ])

    """
    p, _, _, _ = precision_recall_fscore_support(y_true, y_pred,
                                                 labels=labels,
                                                 pos_label=pos_label,
                                                 average=average,
                                                 warn_for=('precision',),
                                                 sample_weight=sample_weight)
    return p


def recall_score(y_true, y_pred, labels=None, pos_label=1, average='binary',
                 sample_weight=None):
    """Compute the recall

    The recall is the ratio ``tp / (tp + fn)`` where ``tp`` is the number of
    true positives and ``fn`` the number of false negatives. The recall is
    intuitively the ability of the classifier to find all the positive samples.

    The best value is 1 and the worst value is 0.

    Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) target values.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Estimated targets as returned by a classifier.

    labels : list, optional
        The set of labels to include when ``average != 'binary'``, and their
        order if ``average is None``. Labels present in the data can be
        excluded, for example to calculate a multiclass average ignoring a
        majority negative class, while labels not present in the data will
        result in 0 components in a macro average. For multilabel targets,
        labels are column indices. By default, all labels in ``y_true`` and
        ``y_pred`` are used in sorted order.

        .. versionchanged:: 0.17
           parameter *labels* improved for multiclass problem.

    pos_label : str or int, 1 by default
        The class to report if ``average='binary'``. Until version 0.18 it is
        necessary to set ``pos_label=None`` if seeking to use another averaging
        method over binary targets.

    average : string, [None, 'binary' (default), 'micro', 'macro', 'samples', \
                       'weighted']
        This parameter is required for multiclass/multilabel targets.
        If ``None``, the scores for each class are returned. Otherwise, this
        determines the type of averaging performed on the data:

        ``'binary'``:
            Only report results for the class specified by ``pos_label``.
            This is applicable only if targets (``y_{true,pred}``) are binary.
        ``'micro'``:
            Calculate metrics globally by counting the total true positives,
            false negatives and false positives.
        ``'macro'``:
            Calculate metrics for each label, and find their unweighted
            mean.  This does not take label imbalance into account.
        ``'weighted'``:
            Calculate metrics for each label, and find their average, weighted
            by support (the number of true instances for each label). This
            alters 'macro' to account for label imbalance; it can result in an
            F-score that is not between precision and recall.
        ``'samples'``:
            Calculate metrics for each instance, and find their average (only
            meaningful for multilabel classification where this differs from
            :func:`accuracy_score`).

        Note that if ``pos_label`` is given in binary classification with
        `average != 'binary'`, only that positive class is reported. This
        behavior is deprecated and will change in version 0.18.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    recall : float (if average is not None) or array of float, shape =\
        [n_unique_labels]
        Recall of the positive class in binary classification or weighted
        average of the recall of each class for the multiclass task.

    Examples
    --------
    >>> from sklearn.metrics import recall_score
    >>> y_true = [0, 1, 2, 0, 1, 2]
    >>> y_pred = [0, 2, 1, 0, 0, 1]
    >>> recall_score(y_true, y_pred, average='macro')  # doctest: +ELLIPSIS
    0.33...
    >>> recall_score(y_true, y_pred, average='micro')  # doctest: +ELLIPSIS
    0.33...
    >>> recall_score(y_true, y_pred, average='weighted')  # doctest: +ELLIPSIS
    0.33...
    >>> recall_score(y_true, y_pred, average=None)
    array([ 1.,  0.,  0.])


    """
    _, r, _, _ = precision_recall_fscore_support(y_true, y_pred,
                                                 labels=labels,
                                                 pos_label=pos_label,
                                                 average=average,
                                                 warn_for=('recall',),
                                                 sample_weight=sample_weight)
    return r


def classification_report(y_true, y_pred, labels=None, target_names=None,
                          sample_weight=None, digits=2):
    """Build a text report showing the main classification metrics

    Read more in the :ref:`User Guide <classification_report>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) target values.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Estimated targets as returned by a classifier.

    labels : array, shape = [n_labels]
        Optional list of label indices to include in the report.

    target_names : list of strings
        Optional display names matching the labels (same order).

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    digits : int
        Number of digits for formatting output floating point values

    Returns
    -------
    report : string
        Text summary of the precision, recall, F1 score for each class.

    Examples
    --------
    >>> from sklearn.metrics import classification_report
    >>> y_true = [0, 1, 2, 2, 2]
    >>> y_pred = [0, 0, 2, 2, 1]
    >>> target_names = ['class 0', 'class 1', 'class 2']
    >>> print(classification_report(y_true, y_pred, target_names=target_names))
                 precision    recall  f1-score   support
    <BLANKLINE>
        class 0       0.50      1.00      0.67         1
        class 1       0.00      0.00      0.00         1
        class 2       1.00      0.67      0.80         3
    <BLANKLINE>
    avg / total       0.70      0.60      0.61         5
    <BLANKLINE>

    """

    if labels is None:
        labels = unique_labels(y_true, y_pred)
    else:
        labels = np.asarray(labels)

    last_line_heading = 'avg / total'

    if target_names is None:
        width = len(last_line_heading)
        target_names = ['%s' % l for l in labels]
    else:
        width = max(len(cn) for cn in target_names)
        width = max(width, len(last_line_heading), digits)

    headers = ["precision", "recall", "f1-score", "support"]
    fmt = '%% %ds' % width  # first column: class name
    fmt += '  '
    fmt += ' '.join(['% 9s' for _ in headers])
    fmt += '\n'

    headers = [""] + headers
    report = fmt % tuple(headers)
    report += '\n'

    p, r, f1, s = precision_recall_fscore_support(y_true, y_pred,
                                                  labels=labels,
                                                  average=None,
                                                  sample_weight=sample_weight)

    for i, label in enumerate(labels):
        values = [target_names[i]]
        for v in (p[i], r[i], f1[i]):
            values += ["{0:0.{1}f}".format(v, digits)]
        values += ["{0}".format(s[i])]
        report += fmt % tuple(values)

    report += '\n'

    # compute averages
    values = [last_line_heading]
    for v in (np.average(p, weights=s),
              np.average(r, weights=s),
              np.average(f1, weights=s)):
        values += ["{0:0.{1}f}".format(v, digits)]
    values += ['{0}'.format(np.sum(s))]
    report += fmt % tuple(values)
    return report


def hamming_loss(y_true, y_pred, classes=None):
    """Compute the average Hamming loss.

    The Hamming loss is the fraction of labels that are incorrectly predicted.

    Read more in the :ref:`User Guide <hamming_loss>`.

    Parameters
    ----------
    y_true : 1d array-like, or label indicator array / sparse matrix
        Ground truth (correct) labels.

    y_pred : 1d array-like, or label indicator array / sparse matrix
        Predicted labels, as returned by a classifier.

    classes : array, shape = [n_labels], optional
        Integer array of labels.

    Returns
    -------
    loss : float or int,
        Return the average Hamming loss between element of ``y_true`` and
        ``y_pred``.

    See Also
    --------
    accuracy_score, jaccard_similarity_score, zero_one_loss

    Notes
    -----
    In multiclass classification, the Hamming loss correspond to the Hamming
    distance between ``y_true`` and ``y_pred`` which is equivalent to the
    subset ``zero_one_loss`` function.

    In multilabel classification, the Hamming loss is different from the
    subset zero-one loss. The zero-one loss considers the entire set of labels
    for a given sample incorrect if it does entirely match the true set of
    labels. Hamming loss is more forgiving in that it penalizes the individual
    labels.

    The Hamming loss is upperbounded by the subset zero-one loss. When
    normalized over samples, the Hamming loss is always between 0 and 1.

    References
    ----------
    .. [1] Grigorios Tsoumakas, Ioannis Katakis. Multi-Label Classification:
           An Overview. International Journal of Data Warehousing & Mining,
           3(3), 1-13, July-September 2007.

    .. [2] `Wikipedia entry on the Hamming distance
           <http://en.wikipedia.org/wiki/Hamming_distance>`_

    Examples
    --------
    >>> from sklearn.metrics import hamming_loss
    >>> y_pred = [1, 2, 3, 4]
    >>> y_true = [2, 2, 3, 4]
    >>> hamming_loss(y_true, y_pred)
    0.25

    In the multilabel case with binary label indicators:

    >>> hamming_loss(np.array([[0, 1], [1, 1]]), np.zeros((2, 2)))
    0.75
    """
    y_type, y_true, y_pred = _check_targets(y_true, y_pred)

    if classes is None:
        classes = unique_labels(y_true, y_pred)
    else:
        classes = np.asarray(classes)

    if y_type.startswith('multilabel'):
        n_differences = count_nonzero(y_true - y_pred)
        return (n_differences / (y_true.shape[0] * len(classes)))

    elif y_type in ["binary", "multiclass"]:
        return sp_hamming(y_true, y_pred)
    else:
        raise ValueError("{0} is not supported".format(y_type))


def log_loss(y_true, y_pred, eps=1e-15, normalize=True, sample_weight=None):
    """Log loss, aka logistic loss or cross-entropy loss.

    This is the loss function used in (multinomial) logistic regression
    and extensions of it such as neural networks, defined as the negative
    log-likelihood of the true labels given a probabilistic classifier's
    predictions. For a single sample with true label yt in {0,1} and
    estimated probability yp that yt = 1, the log loss is

        -log P(yt|yp) = -(yt log(yp) + (1 - yt) log(1 - yp))

    Read more in the :ref:`User Guide <log_loss>`.

    Parameters
    ----------
    y_true : array-like or label indicator matrix
        Ground truth (correct) labels for n_samples samples.

    y_pred : array-like of float, shape = (n_samples, n_classes)
        Predicted probabilities, as returned by a classifier's
        predict_proba method.

    eps : float
        Log loss is undefined for p=0 or p=1, so probabilities are
        clipped to max(eps, min(1 - eps, p)).

    normalize : bool, optional (default=True)
        If true, return the mean loss per sample.
        Otherwise, return the sum of the per-sample losses.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    loss : float

    Examples
    --------
    >>> log_loss(["spam", "ham", "ham", "spam"],  # doctest: +ELLIPSIS
    ...          [[.1, .9], [.9, .1], [.8, .2], [.35, .65]])
    0.21616...

    References
    ----------
    C.M. Bishop (2006). Pattern Recognition and Machine Learning. Springer,
    p. 209.

    Notes
    -----
    The logarithm used is the natural logarithm (base-e).
    """
    lb = LabelBinarizer()
    T = lb.fit_transform(y_true)
    if T.shape[1] == 1:
        T = np.append(1 - T, T, axis=1)

    # Clipping
    Y = np.clip(y_pred, eps, 1 - eps)

    # This happens in cases when elements in y_pred have type "str".
    if not isinstance(Y, np.ndarray):
        raise ValueError("y_pred should be an array of floats.")

    # If y_pred is of single dimension, assume y_true to be binary
    # and then check.
    if Y.ndim == 1:
        Y = Y[:, np.newaxis]
    if Y.shape[1] == 1:
        Y = np.append(1 - Y, Y, axis=1)

    # Check if dimensions are consistent.
    check_consistent_length(T, Y)
    T = check_array(T)
    Y = check_array(Y)
    if T.shape[1] != Y.shape[1]:
        raise ValueError("y_true and y_pred have different number of classes "
                         "%d, %d" % (T.shape[1], Y.shape[1]))

    # Renormalize
    Y /= Y.sum(axis=1)[:, np.newaxis]
    loss = -(T * np.log(Y)).sum(axis=1)

    return _weighted_sum(loss, sample_weight, normalize)


def hinge_loss(y_true, pred_decision, labels=None, sample_weight=None):
    """Average hinge loss (non-regularized)

    In binary class case, assuming labels in y_true are encoded with +1 and -1,
    when a prediction mistake is made, ``margin = y_true * pred_decision`` is
    always negative (since the signs disagree), implying ``1 - margin`` is
    always greater than 1.  The cumulated hinge loss is therefore an upper
    bound of the number of mistakes made by the classifier.

    In multiclass case, the function expects that either all the labels are
    included in y_true or an optional labels argument is provided which
    contains all the labels. The multilabel margin is calculated according
    to Crammer-Singer's method. As in the binary case, the cumulated hinge loss
    is an upper bound of the number of mistakes made by the classifier.

    Read more in the :ref:`User Guide <hinge_loss>`.

    Parameters
    ----------
    y_true : array, shape = [n_samples]
        True target, consisting of integers of two values. The positive label
        must be greater than the negative label.

    pred_decision : array, shape = [n_samples] or [n_samples, n_classes]
        Predicted decisions, as output by decision_function (floats).

    labels : array, optional, default None
        Contains all the labels for the problem. Used in multiclass hinge loss.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    Returns
    -------
    loss : float

    References
    ----------
    .. [1] `Wikipedia entry on the Hinge loss
           <http://en.wikipedia.org/wiki/Hinge_loss>`_

    .. [2] Koby Crammer, Yoram Singer. On the Algorithmic
           Implementation of Multiclass Kernel-based Vector
           Machines. Journal of Machine Learning Research 2,
           (2001), 265-292

    .. [3] `L1 AND L2 Regularization for Multiclass Hinge Loss Models
           by Robert C. Moore, John DeNero.
           <http://www.ttic.edu/sigml/symposium2011/papers/
           Moore+DeNero_Regularization.pdf>`_

    Examples
    --------
    >>> from sklearn import svm
    >>> from sklearn.metrics import hinge_loss
    >>> X = [[0], [1]]
    >>> y = [-1, 1]
    >>> est = svm.LinearSVC(random_state=0)
    >>> est.fit(X, y)
    LinearSVC(C=1.0, class_weight=None, dual=True, fit_intercept=True,
         intercept_scaling=1, loss='squared_hinge', max_iter=1000,
         multi_class='ovr', penalty='l2', random_state=0, tol=0.0001,
         verbose=0)
    >>> pred_decision = est.decision_function([[-2], [3], [0.5]])
    >>> pred_decision  # doctest: +ELLIPSIS
    array([-2.18...,  2.36...,  0.09...])
    >>> hinge_loss([-1, 1, 1], pred_decision)  # doctest: +ELLIPSIS
    0.30...

    In the multiclass case:

    >>> X = np.array([[0], [1], [2], [3]])
    >>> Y = np.array([0, 1, 2, 3])
    >>> labels = np.array([0, 1, 2, 3])
    >>> est = svm.LinearSVC()
    >>> est.fit(X, Y)
    LinearSVC(C=1.0, class_weight=None, dual=True, fit_intercept=True,
         intercept_scaling=1, loss='squared_hinge', max_iter=1000,
         multi_class='ovr', penalty='l2', random_state=None, tol=0.0001,
         verbose=0)
    >>> pred_decision = est.decision_function([[-1], [2], [3]])
    >>> y_true = [0, 2, 3]
    >>> hinge_loss(y_true, pred_decision, labels)  #doctest: +ELLIPSIS
    0.56...
    """
    check_consistent_length(y_true, pred_decision, sample_weight)
    pred_decision = check_array(pred_decision, ensure_2d=False)
    y_true = column_or_1d(y_true)
    y_true_unique = np.unique(y_true)
    if y_true_unique.size > 2:
        if (labels is None and pred_decision.ndim > 1 and
                (np.size(y_true_unique) != pred_decision.shape[1])):
            raise ValueError("Please include all labels in y_true "
                             "or pass labels as third argument")
        if labels is None:
            labels = y_true_unique
        le = LabelEncoder()
        le.fit(labels)
        y_true = le.transform(y_true)
        mask = np.ones_like(pred_decision, dtype=bool)
        mask[np.arange(y_true.shape[0]), y_true] = False
        margin = pred_decision[~mask]
        margin -= np.max(pred_decision[mask].reshape(y_true.shape[0], -1),
                         axis=1)

    else:
        # Handles binary class case
        # this code assumes that positive and negative labels
        # are encoded as +1 and -1 respectively
        pred_decision = column_or_1d(pred_decision)
        pred_decision = np.ravel(pred_decision)

        lbin = LabelBinarizer(neg_label=-1)
        y_true = lbin.fit_transform(y_true)[:, 0]

        try:
            margin = y_true * pred_decision
        except TypeError:
            raise TypeError("pred_decision should be an array of floats.")

    losses = 1 - margin
    # The hinge_loss doesn't penalize good enough predictions.
    losses[losses <= 0] = 0
    return np.average(losses, weights=sample_weight)


def _check_binary_probabilistic_predictions(y_true, y_prob):
    """Check that y_true is binary and y_prob contains valid probabilities"""
    check_consistent_length(y_true, y_prob)

    labels = np.unique(y_true)

    if len(labels) != 2:
        raise ValueError("Only binary classification is supported. "
                         "Provided labels %s." % labels)

    if y_prob.max() > 1:
        raise ValueError("y_prob contains values greater than 1.")

    if y_prob.min() < 0:
        raise ValueError("y_prob contains values less than 0.")

    return label_binarize(y_true, labels)[:, 0]


def brier_score_loss(y_true, y_prob, sample_weight=None, pos_label=None):
    """Compute the Brier score.

    The smaller the Brier score, the better, hence the naming with "loss".

    Across all items in a set N predictions, the Brier score measures the
    mean squared difference between (1) the predicted probability assigned
    to the possible outcomes for item i, and (2) the actual outcome.
    Therefore, the lower the Brier score is for a set of predictions, the
    better the predictions are calibrated. Note that the Brier score always
    takes on a value between zero and one, since this is the largest
    possible difference between a predicted probability (which must be
    between zero and one) and the actual outcome (which can take on values
    of only 0 and 1).

    The Brier score is appropriate for binary and categorical outcomes that
    can be structured as true or false, but is inappropriate for ordinal
    variables which can take on three or more values (this is because the
    Brier score assumes that all possible outcomes are equivalently
    "distant" from one another). Which label is considered to be the positive
    label is controlled via the parameter pos_label, which defaults to 1.

    Read more in the :ref:`User Guide <calibration>`.

    Parameters
    ----------
    y_true : array, shape (n_samples,)
        True targets.

    y_prob : array, shape (n_samples,)
        Probabilities of the positive class.

    sample_weight : array-like of shape = [n_samples], optional
        Sample weights.

    pos_label : int (default: None)
        Label of the positive class. If None, the maximum label is used as
        positive class

    Returns
    -------
    score : float
        Brier score

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.metrics import brier_score_loss
    >>> y_true = np.array([0, 1, 1, 0])
    >>> y_true_categorical = np.array(["spam", "ham", "ham", "spam"])
    >>> y_prob = np.array([0.1, 0.9, 0.8, 0.3])
    >>> brier_score_loss(y_true, y_prob)  # doctest: +ELLIPSIS
    0.037...
    >>> brier_score_loss(y_true, 1-y_prob, pos_label=0)  # doctest: +ELLIPSIS
    0.037...
    >>> brier_score_loss(y_true_categorical, y_prob, \
                         pos_label="ham")  # doctest: +ELLIPSIS
    0.037...
    >>> brier_score_loss(y_true, np.array(y_prob) > 0.5)
    0.0

    References
    ----------
    http://en.wikipedia.org/wiki/Brier_score
    """
    y_true = column_or_1d(y_true)
    y_prob = column_or_1d(y_prob)
    if pos_label is None:
        pos_label = y_true.max()
    y_true = np.array(y_true == pos_label, int)
    y_true = _check_binary_probabilistic_predictions(y_true, y_prob)
    return np.average((y_true - y_prob) ** 2, weights=sample_weight)