"""Metrics to assess performance on classification task given scores
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>
# License: BSD 3 clause
from __future__ import division
import warnings
import numpy as np
from scipy.sparse import csr_matrix
from ..utils import check_consistent_length
from ..utils import column_or_1d, check_array
from ..utils.multiclass import type_of_target
from ..utils.fixes import isclose
from ..utils.fixes import bincount
from ..utils.fixes import array_equal
from ..utils.stats import rankdata
from ..utils.sparsefuncs import count_nonzero
from .base import _average_binary_score
from .base import UndefinedMetricWarning
[docs]def auc(x, y, reorder=False):
"""Compute Area Under the Curve (AUC) using the trapezoidal rule
This is a general function, given points on a curve. For computing the
area under the ROC-curve, see :func:`roc_auc_score`.
Parameters
----------
x : array, shape = [n]
x coordinates.
y : array, shape = [n]
y coordinates.
reorder : boolean, optional (default=False)
If True, assume that the curve is ascending in the case of ties, as for
an ROC curve. If the curve is non-ascending, the result will be wrong.
Returns
-------
auc : float
Examples
--------
>>> import numpy as np
>>> from sklearn import metrics
>>> y = np.array([1, 1, 2, 2])
>>> pred = np.array([0.1, 0.4, 0.35, 0.8])
>>> fpr, tpr, thresholds = metrics.roc_curve(y, pred, pos_label=2)
>>> metrics.auc(fpr, tpr)
0.75
See also
--------
roc_auc_score : Computes the area under the ROC curve
precision_recall_curve :
Compute precision-recall pairs for different probability thresholds
"""
check_consistent_length(x, y)
x = column_or_1d(x)
y = column_or_1d(y)
if x.shape[0] < 2:
raise ValueError('At least 2 points are needed to compute'
' area under curve, but x.shape = %s' % x.shape)
direction = 1
if reorder:
# reorder the data points according to the x axis and using y to
# break ties
order = np.lexsort((y, x))
x, y = x[order], y[order]
else:
dx = np.diff(x)
if np.any(dx < 0):
if np.all(dx <= 0):
direction = -1
else:
raise ValueError("Reordering is not turned on, and "
"the x array is not increasing: %s" % x)
area = direction * np.trapz(y, x)
return area
def average_precision_score(y_true, y_score, average="macro",
sample_weight=None):
"""Compute average precision (AP) from prediction scores
This score corresponds to the area under the precision-recall curve.
Note: this implementation is restricted to the binary classification task
or multilabel classification task.
Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.
Parameters
----------
y_true : array, shape = [n_samples] or [n_samples, n_classes]
True binary labels in binary label indicators.
y_score : array, shape = [n_samples] or [n_samples, n_classes]
Target scores, can either be probability estimates of the positive
class, confidence values, or binary decisions.
average : string, [None, 'micro', 'macro' (default), 'samples', 'weighted']
If ``None``, the scores for each class are returned. Otherwise,
this determines the type of averaging performed on the data:
``'micro'``:
Calculate metrics globally by considering each element of the label
indicator matrix as a label.
``'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).
``'samples'``:
Calculate metrics for each instance, and find their average.
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
Returns
-------
average_precision : float
References
----------
.. [1] `Wikipedia entry for the Average precision
<http://en.wikipedia.org/wiki/Average_precision>`_
See also
--------
roc_auc_score : Area under the ROC curve
precision_recall_curve :
Compute precision-recall pairs for different probability thresholds
Examples
--------
>>> import numpy as np
>>> from sklearn.metrics import average_precision_score
>>> y_true = np.array([0, 0, 1, 1])
>>> y_scores = np.array([0.1, 0.4, 0.35, 0.8])
>>> average_precision_score(y_true, y_scores) # doctest: +ELLIPSIS
0.79...
"""
def _binary_average_precision(y_true, y_score, sample_weight=None):
precision, recall, thresholds = precision_recall_curve(
y_true, y_score, sample_weight=sample_weight)
return auc(recall, precision)
return _average_binary_score(_binary_average_precision, y_true, y_score,
average, sample_weight=sample_weight)
def roc_auc_score(y_true, y_score, average="macro", sample_weight=None):
"""Compute Area Under the Curve (AUC) from prediction scores
Note: this implementation is restricted to the binary classification task
or multilabel classification task in label indicator format.
Read more in the :ref:`User Guide <roc_metrics>`.
Parameters
----------
y_true : array, shape = [n_samples] or [n_samples, n_classes]
True binary labels in binary label indicators.
y_score : array, shape = [n_samples] or [n_samples, n_classes]
Target scores, can either be probability estimates of the positive
class, confidence values, or binary decisions.
average : string, [None, 'micro', 'macro' (default), 'samples', 'weighted']
If ``None``, the scores for each class are returned. Otherwise,
this determines the type of averaging performed on the data:
``'micro'``:
Calculate metrics globally by considering each element of the label
indicator matrix as a label.
``'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).
``'samples'``:
Calculate metrics for each instance, and find their average.
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
Returns
-------
auc : float
References
----------
.. [1] `Wikipedia entry for the Receiver operating characteristic
<http://en.wikipedia.org/wiki/Receiver_operating_characteristic>`_
See also
--------
average_precision_score : Area under the precision-recall curve
roc_curve : Compute Receiver operating characteristic (ROC)
Examples
--------
>>> import numpy as np
>>> from sklearn.metrics import roc_auc_score
>>> y_true = np.array([0, 0, 1, 1])
>>> y_scores = np.array([0.1, 0.4, 0.35, 0.8])
>>> roc_auc_score(y_true, y_scores)
0.75
"""
def _binary_roc_auc_score(y_true, y_score, sample_weight=None):
if len(np.unique(y_true)) != 2:
raise ValueError("Only one class present in y_true. ROC AUC score "
"is not defined in that case.")
fpr, tpr, tresholds = roc_curve(y_true, y_score,
sample_weight=sample_weight)
return auc(fpr, tpr, reorder=True)
return _average_binary_score(
_binary_roc_auc_score, y_true, y_score, average,
sample_weight=sample_weight)
def _binary_clf_curve(y_true, y_score, pos_label=None, sample_weight=None):
"""Calculate true and false positives per binary classification threshold.
Parameters
----------
y_true : array, shape = [n_samples]
True targets of binary classification
y_score : array, shape = [n_samples]
Estimated probabilities or decision function
pos_label : int, optional (default=None)
The label of the positive class
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
Returns
-------
fps : array, shape = [n_thresholds]
A count of false positives, at index i being the number of negative
samples assigned a score >= thresholds[i]. The total number of
negative samples is equal to fps[-1] (thus true negatives are given by
fps[-1] - fps).
tps : array, shape = [n_thresholds <= len(np.unique(y_score))]
An increasing count of true positives, at index i being the number
of positive samples assigned a score >= thresholds[i]. The total
number of positive samples is equal to tps[-1] (thus false negatives
are given by tps[-1] - tps).
thresholds : array, shape = [n_thresholds]
Decreasing score values.
"""
check_consistent_length(y_true, y_score)
y_true = column_or_1d(y_true)
y_score = column_or_1d(y_score)
if sample_weight is not None:
sample_weight = column_or_1d(sample_weight)
# ensure binary classification if pos_label is not specified
classes = np.unique(y_true)
if (pos_label is None and
not (array_equal(classes, [0, 1]) or
array_equal(classes, [-1, 1]) or
array_equal(classes, [0]) or
array_equal(classes, [-1]) or
array_equal(classes, [1]))):
raise ValueError("Data is not binary and pos_label is not specified")
elif pos_label is None:
pos_label = 1.
# make y_true a boolean vector
y_true = (y_true == pos_label)
# sort scores and corresponding truth values
desc_score_indices = np.argsort(y_score, kind="mergesort")[::-1]
y_score = y_score[desc_score_indices]
y_true = y_true[desc_score_indices]
if sample_weight is not None:
weight = sample_weight[desc_score_indices]
else:
weight = 1.
# y_score typically has many tied values. Here we extract
# the indices associated with the distinct values. We also
# concatenate a value for the end of the curve.
# We need to use isclose to avoid spurious repeated thresholds
# stemming from floating point roundoff errors.
distinct_value_indices = np.where(np.logical_not(isclose(
np.diff(y_score), 0)))[0]
threshold_idxs = np.r_[distinct_value_indices, y_true.size - 1]
# accumulate the true positives with decreasing threshold
tps = (y_true * weight).cumsum()[threshold_idxs]
if sample_weight is not None:
fps = weight.cumsum()[threshold_idxs] - tps
else:
fps = 1 + threshold_idxs - tps
return fps, tps, y_score[threshold_idxs]
def precision_recall_curve(y_true, probas_pred, pos_label=None,
sample_weight=None):
"""Compute precision-recall pairs for different probability thresholds
Note: this implementation is restricted to the binary classification task.
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 last precision and recall values are 1. and 0. respectively and do not
have a corresponding threshold. This ensures that the graph starts on the
x axis.
Read more in the :ref:`User Guide <precision_recall_f_measure_metrics>`.
Parameters
----------
y_true : array, shape = [n_samples]
True targets of binary classification in range {-1, 1} or {0, 1}.
probas_pred : array, shape = [n_samples]
Estimated probabilities or decision function.
pos_label : int, optional (default=None)
The label of the positive class
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
Returns
-------
precision : array, shape = [n_thresholds + 1]
Precision values such that element i is the precision of
predictions with score >= thresholds[i] and the last element is 1.
recall : array, shape = [n_thresholds + 1]
Decreasing recall values such that element i is the recall of
predictions with score >= thresholds[i] and the last element is 0.
thresholds : array, shape = [n_thresholds <= len(np.unique(probas_pred))]
Increasing thresholds on the decision function used to compute
precision and recall.
Examples
--------
>>> import numpy as np
>>> from sklearn.metrics import precision_recall_curve
>>> y_true = np.array([0, 0, 1, 1])
>>> y_scores = np.array([0.1, 0.4, 0.35, 0.8])
>>> precision, recall, thresholds = precision_recall_curve(
... y_true, y_scores)
>>> precision # doctest: +ELLIPSIS
array([ 0.66..., 0.5 , 1. , 1. ])
>>> recall
array([ 1. , 0.5, 0.5, 0. ])
>>> thresholds
array([ 0.35, 0.4 , 0.8 ])
"""
fps, tps, thresholds = _binary_clf_curve(y_true, probas_pred,
pos_label=pos_label,
sample_weight=sample_weight)
precision = tps / (tps + fps)
recall = tps / tps[-1]
# stop when full recall attained
# and reverse the outputs so recall is decreasing
last_ind = tps.searchsorted(tps[-1])
sl = slice(last_ind, None, -1)
return np.r_[precision[sl], 1], np.r_[recall[sl], 0], thresholds[sl]
def roc_curve(y_true, y_score, pos_label=None, sample_weight=None,
drop_intermediate=True):
"""Compute Receiver operating characteristic (ROC)
Note: this implementation is restricted to the binary classification task.
Read more in the :ref:`User Guide <roc_metrics>`.
Parameters
----------
y_true : array, shape = [n_samples]
True binary labels in range {0, 1} or {-1, 1}. If labels are not
binary, pos_label should be explicitly given.
y_score : array, shape = [n_samples]
Target scores, can either be probability estimates of the positive
class or confidence values.
pos_label : int
Label considered as positive and others are considered negative.
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
drop_intermediate : boolean, optional (default=True)
Whether to drop some suboptimal thresholds which would not appear
on a plotted ROC curve. This is useful in order to create lighter
ROC curves.
.. versionadded:: 0.17
parameter *drop_intermediate*.
Returns
-------
fpr : array, shape = [>2]
Increasing false positive rates such that element i is the false
positive rate of predictions with score >= thresholds[i].
tpr : array, shape = [>2]
Increasing true positive rates such that element i is the true
positive rate of predictions with score >= thresholds[i].
thresholds : array, shape = [n_thresholds]
Decreasing thresholds on the decision function used to compute
fpr and tpr. `thresholds[0]` represents no instances being predicted
and is arbitrarily set to `max(y_score) + 1`.
See also
--------
roc_auc_score : Compute Area Under the Curve (AUC) from prediction scores
Notes
-----
Since the thresholds are sorted from low to high values, they
are reversed upon returning them to ensure they correspond to both ``fpr``
and ``tpr``, which are sorted in reversed order during their calculation.
References
----------
.. [1] `Wikipedia entry for the Receiver operating characteristic
<http://en.wikipedia.org/wiki/Receiver_operating_characteristic>`_
Examples
--------
>>> import numpy as np
>>> from sklearn import metrics
>>> y = np.array([1, 1, 2, 2])
>>> scores = np.array([0.1, 0.4, 0.35, 0.8])
>>> fpr, tpr, thresholds = metrics.roc_curve(y, scores, pos_label=2)
>>> fpr
array([ 0. , 0.5, 0.5, 1. ])
>>> tpr
array([ 0.5, 0.5, 1. , 1. ])
>>> thresholds
array([ 0.8 , 0.4 , 0.35, 0.1 ])
"""
fps, tps, thresholds = _binary_clf_curve(
y_true, y_score, pos_label=pos_label, sample_weight=sample_weight)
# Attempt to drop thresholds corresponding to points in between and
# collinear with other points. These are always suboptimal and do not
# appear on a plotted ROC curve (and thus do not affect the AUC).
# Here np.diff(_, 2) is used as a "second derivative" to tell if there
# is a corner at the point. Both fps and tps must be tested to handle
# thresholds with multiple data points (which are combined in
# _binary_clf_curve). This keeps all cases where the point should be kept,
# but does not drop more complicated cases like fps = [1, 3, 7],
# tps = [1, 2, 4]; there is no harm in keeping too many thresholds.
if drop_intermediate and len(fps) > 2:
optimal_idxs = np.where(np.r_[True,
np.logical_or(np.diff(fps, 2),
np.diff(tps, 2)),
True])[0]
fps = fps[optimal_idxs]
tps = tps[optimal_idxs]
thresholds = thresholds[optimal_idxs]
if tps.size == 0 or fps[0] != 0:
# Add an extra threshold position if necessary
tps = np.r_[0, tps]
fps = np.r_[0, fps]
thresholds = np.r_[thresholds[0] + 1, thresholds]
if fps[-1] <= 0:
warnings.warn("No negative samples in y_true, "
"false positive value should be meaningless",
UndefinedMetricWarning)
fpr = np.repeat(np.nan, fps.shape)
else:
fpr = fps / fps[-1]
if tps[-1] <= 0:
warnings.warn("No positive samples in y_true, "
"true positive value should be meaningless",
UndefinedMetricWarning)
tpr = np.repeat(np.nan, tps.shape)
else:
tpr = tps / tps[-1]
return fpr, tpr, thresholds
def label_ranking_average_precision_score(y_true, y_score):
"""Compute ranking-based average precision
Label ranking average precision (LRAP) is the average over each ground
truth label assigned to each sample, of the ratio of true vs. total
labels with lower score.
This metric is used in multilabel ranking problem, where the goal
is to give better rank to the labels associated to each sample.
The obtained score is always strictly greater than 0 and
the best value is 1.
Read more in the :ref:`User Guide <label_ranking_average_precision>`.
Parameters
----------
y_true : array or sparse matrix, shape = [n_samples, n_labels]
True binary labels in binary indicator format.
y_score : array, shape = [n_samples, n_labels]
Target scores, can either be probability estimates of the positive
class, confidence values, or binary decisions.
Returns
-------
score : float
Examples
--------
>>> import numpy as np
>>> from sklearn.metrics import label_ranking_average_precision_score
>>> y_true = np.array([[1, 0, 0], [0, 0, 1]])
>>> y_score = np.array([[0.75, 0.5, 1], [1, 0.2, 0.1]])
>>> label_ranking_average_precision_score(y_true, y_score) \
# doctest: +ELLIPSIS
0.416...
"""
check_consistent_length(y_true, y_score)
y_true = check_array(y_true, ensure_2d=False)
y_score = check_array(y_score, ensure_2d=False)
if y_true.shape != y_score.shape:
raise ValueError("y_true and y_score have different shape")
# Handle badly formated array and the degenerate case with one label
y_type = type_of_target(y_true)
if (y_type != "multilabel-indicator" and
not (y_type == "binary" and y_true.ndim == 2)):
raise ValueError("{0} format is not supported".format(y_type))
y_true = csr_matrix(y_true)
y_score = -y_score
n_samples, n_labels = y_true.shape
out = 0.
for i, (start, stop) in enumerate(zip(y_true.indptr, y_true.indptr[1:])):
relevant = y_true.indices[start:stop]
if (relevant.size == 0 or relevant.size == n_labels):
# If all labels are relevant or unrelevant, the score is also
# equal to 1. The label ranking has no meaning.
out += 1.
continue
scores_i = y_score[i]
rank = rankdata(scores_i, 'max')[relevant]
L = rankdata(scores_i[relevant], 'max')
out += (L / rank).mean()
return out / n_samples
def coverage_error(y_true, y_score, sample_weight=None):
"""Coverage error measure
Compute how far we need to go through the ranked scores to cover all
true labels. The best value is equal to the average number
of labels in ``y_true`` per sample.
Ties in ``y_scores`` are broken by giving maximal rank that would have
been assigned to all tied values.
Read more in the :ref:`User Guide <coverage_error>`.
Parameters
----------
y_true : array, shape = [n_samples, n_labels]
True binary labels in binary indicator format.
y_score : array, shape = [n_samples, n_labels]
Target scores, can either be probability estimates of the positive
class, confidence values, or binary decisions.
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
Returns
-------
coverage_error : float
References
----------
.. [1] Tsoumakas, G., Katakis, I., & Vlahavas, I. (2010).
Mining multi-label data. In Data mining and knowledge discovery
handbook (pp. 667-685). Springer US.
"""
y_true = check_array(y_true, ensure_2d=False)
y_score = check_array(y_score, ensure_2d=False)
check_consistent_length(y_true, y_score, sample_weight)
y_type = type_of_target(y_true)
if y_type != "multilabel-indicator":
raise ValueError("{0} format is not supported".format(y_type))
if y_true.shape != y_score.shape:
raise ValueError("y_true and y_score have different shape")
y_score_mask = np.ma.masked_array(y_score, mask=np.logical_not(y_true))
y_min_relevant = y_score_mask.min(axis=1).reshape((-1, 1))
coverage = (y_score >= y_min_relevant).sum(axis=1)
coverage = coverage.filled(0)
return np.average(coverage, weights=sample_weight)
def label_ranking_loss(y_true, y_score, sample_weight=None):
"""Compute Ranking loss measure
Compute the average number of label pairs that are incorrectly ordered
given y_score weighted by the size of the label set and the number of
labels not in the label set.
This is similar to the error set size, but weighted by the number of
relevant and irrelevant labels. The best performance is achieved with
a ranking loss of zero.
Read more in the :ref:`User Guide <label_ranking_loss>`.
.. versionadded:: 0.17
A function *label_ranking_loss*
Parameters
----------
y_true : array or sparse matrix, shape = [n_samples, n_labels]
True binary labels in binary indicator format.
y_score : array, shape = [n_samples, n_labels]
Target scores, can either be probability estimates of the positive
class, confidence values, or binary decisions.
sample_weight : array-like of shape = [n_samples], optional
Sample weights.
Returns
-------
loss : float
References
----------
.. [1] Tsoumakas, G., Katakis, I., & Vlahavas, I. (2010).
Mining multi-label data. In Data mining and knowledge discovery
handbook (pp. 667-685). Springer US.
"""
y_true = check_array(y_true, ensure_2d=False, accept_sparse='csr')
y_score = check_array(y_score, ensure_2d=False)
check_consistent_length(y_true, y_score, sample_weight)
y_type = type_of_target(y_true)
if y_type not in ("multilabel-indicator",):
raise ValueError("{0} format is not supported".format(y_type))
if y_true.shape != y_score.shape:
raise ValueError("y_true and y_score have different shape")
n_samples, n_labels = y_true.shape
y_true = csr_matrix(y_true)
loss = np.zeros(n_samples)
for i, (start, stop) in enumerate(zip(y_true.indptr, y_true.indptr[1:])):
# Sort and bin the label scores
unique_scores, unique_inverse = np.unique(y_score[i],
return_inverse=True)
true_at_reversed_rank = bincount(
unique_inverse[y_true.indices[start:stop]],
minlength=len(unique_scores))
all_at_reversed_rank = bincount(unique_inverse,
minlength=len(unique_scores))
false_at_reversed_rank = all_at_reversed_rank - true_at_reversed_rank
# if the scores are ordered, it's possible to count the number of
# incorrectly ordered paires in linear time by cumulatively counting
# how many false labels of a given score have a score higher than the
# accumulated true labels with lower score.
loss[i] = np.dot(true_at_reversed_rank.cumsum(),
false_at_reversed_rank)
n_positives = count_nonzero(y_true, axis=1)
with np.errstate(divide="ignore", invalid="ignore"):
loss /= ((n_labels - n_positives) * n_positives)
# When there is no positive or no negative labels, those values should
# be consider as correct, i.e. the ranking doesn't matter.
loss[np.logical_or(n_positives == 0, n_positives == n_labels)] = 0.
return np.average(loss, weights=sample_weight)