3.3. Metrics and scoring: quantifying the quality of predictions¶
There are 3 different APIs for evaluating the quality of a model’s predictions:
Estimator score method: Estimators have a
score
method providing a default evaluation criterion for the problem they are designed to solve. This is not discussed on this page, but in each estimator’s documentation.Scoring parameter: Modelevaluation tools using crossvalidation (such as
model_selection.cross_val_score
andmodel_selection.GridSearchCV
) rely on an internal scoring strategy. This is discussed in the section The scoring parameter: defining model evaluation rules.Metric functions: The
metrics
module implements functions assessing prediction error for specific purposes. These metrics are detailed in sections on Classification metrics, Multilabel ranking metrics, Regression metrics and Clustering metrics.
Finally, Dummy estimators are useful to get a baseline value of those metrics for random predictions.
See also
For “pairwise” metrics, between samples and not estimators or predictions, see the Pairwise metrics, Affinities and Kernels section.
3.3.1. The scoring
parameter: defining model evaluation rules¶
Model selection and evaluation using tools, such as
model_selection.GridSearchCV
and
model_selection.cross_val_score
, take a scoring
parameter that
controls what metric they apply to the estimators evaluated.
3.3.1.1. Common cases: predefined values¶
For the most common use cases, you can designate a scorer object with the
scoring
parameter; the table below shows all possible values.
All scorer objects follow the convention that higher return values are better
than lower return values. Thus metrics which measure the distance between
the model and the data, like metrics.mean_squared_error
, are
available as neg_mean_squared_error which return the negated value
of the metric.
Scoring 
Function 
Comment 

Classification 

‘accuracy’ 

‘balanced_accuracy’ 

‘average_precision’ 

‘neg_brier_score’ 

‘f1’ 
for binary targets 

‘f1_micro’ 
microaveraged 

‘f1_macro’ 
macroaveraged 

‘f1_weighted’ 
weighted average 

‘f1_samples’ 
by multilabel sample 

‘neg_log_loss’ 
requires 

‘precision’ etc. 
suffixes apply as with ‘f1’ 

‘recall’ etc. 
suffixes apply as with ‘f1’ 

‘jaccard’ etc. 
suffixes apply as with ‘f1’ 

‘roc_auc’ 

‘roc_auc_ovr’ 

‘roc_auc_ovo’ 

‘roc_auc_ovr_weighted’ 

‘roc_auc_ovo_weighted’ 

Clustering 

‘adjusted_mutual_info_score’ 

‘adjusted_rand_score’ 

‘completeness_score’ 

‘fowlkes_mallows_score’ 

‘homogeneity_score’ 

‘mutual_info_score’ 

‘normalized_mutual_info_score’ 

‘v_measure_score’ 

Regression 

‘explained_variance’ 

‘max_error’ 

‘neg_mean_absolute_error’ 

‘neg_mean_squared_error’ 

‘neg_root_mean_squared_error’ 

‘neg_mean_squared_log_error’ 

‘neg_median_absolute_error’ 

‘r2’ 

‘neg_mean_poisson_deviance’ 

‘neg_mean_gamma_deviance’ 
Usage examples:
>>> from sklearn import svm, datasets
>>> from sklearn.model_selection import cross_val_score
>>> X, y = datasets.load_iris(return_X_y=True)
>>> clf = svm.SVC(random_state=0)
>>> cross_val_score(clf, X, y, cv=5, scoring='recall_macro')
array([0.96..., 0.96..., 0.96..., 0.93..., 1. ])
>>> model = svm.SVC()
>>> cross_val_score(model, X, y, cv=5, scoring='wrong_choice')
Traceback (most recent call last):
ValueError: 'wrong_choice' is not a valid scoring value. Use sorted(sklearn.metrics.SCORERS.keys()) to get valid options.
Note
The values listed by the ValueError exception correspond to the functions measuring
prediction accuracy described in the following sections.
The scorer objects for those functions are stored in the dictionary
sklearn.metrics.SCORERS
.
3.3.1.2. Defining your scoring strategy from metric functions¶
The module sklearn.metrics
also exposes a set of simple functions
measuring a prediction error given ground truth and prediction:
functions ending with
_score
return a value to maximize, the higher the better.functions ending with
_error
or_loss
return a value to minimize, the lower the better. When converting into a scorer object usingmake_scorer
, set thegreater_is_better
parameter to False (True by default; see the parameter description below).
Metrics available for various machine learning tasks are detailed in sections below.
Many metrics are not given names to be used as scoring
values,
sometimes because they require additional parameters, such as
fbeta_score
. In such cases, you need to generate an appropriate
scoring object. The simplest way to generate a callable object for scoring
is by using make_scorer
. That function converts metrics
into callables that can be used for model evaluation.
One typical use case is to wrap an existing metric function from the library
with nondefault values for its parameters, such as the beta
parameter for
the fbeta_score
function:
>>> from sklearn.metrics import fbeta_score, make_scorer
>>> ftwo_scorer = make_scorer(fbeta_score, beta=2)
>>> from sklearn.model_selection import GridSearchCV
>>> from sklearn.svm import LinearSVC
>>> grid = GridSearchCV(LinearSVC(), param_grid={'C': [1, 10]},
... scoring=ftwo_scorer, cv=5)
The second use case is to build a completely custom scorer object
from a simple python function using make_scorer
, which can
take several parameters:
the python function you want to use (
my_custom_loss_func
in the example below)whether the python function returns a score (
greater_is_better=True
, the default) or a loss (greater_is_better=False
). If a loss, the output of the python function is negated by the scorer object, conforming to the cross validation convention that scorers return higher values for better models.for classification metrics only: whether the python function you provided requires continuous decision certainties (
needs_threshold=True
). The default value is False.any additional parameters, such as
beta
orlabels
inf1_score
.
Here is an example of building custom scorers, and of using the
greater_is_better
parameter:
>>> import numpy as np
>>> def my_custom_loss_func(y_true, y_pred):
... diff = np.abs(y_true  y_pred).max()
... return np.log1p(diff)
...
>>> # score will negate the return value of my_custom_loss_func,
>>> # which will be np.log(2), 0.693, given the values for X
>>> # and y defined below.
>>> score = make_scorer(my_custom_loss_func, greater_is_better=False)
>>> X = [[1], [1]]
>>> y = [0, 1]
>>> from sklearn.dummy import DummyClassifier
>>> clf = DummyClassifier(strategy='most_frequent', random_state=0)
>>> clf = clf.fit(X, y)
>>> my_custom_loss_func(clf.predict(X), y)
0.69...
>>> score(clf, X, y)
0.69...
3.3.1.3. Implementing your own scoring object¶
You can generate even more flexible model scorers by constructing your own
scoring object from scratch, without using the make_scorer
factory.
For a callable to be a scorer, it needs to meet the protocol specified by
the following two rules:
It can be called with parameters
(estimator, X, y)
, whereestimator
is the model that should be evaluated,X
is validation data, andy
is the ground truth target forX
(in the supervised case) orNone
(in the unsupervised case).It returns a floating point number that quantifies the
estimator
prediction quality onX
, with reference toy
. Again, by convention higher numbers are better, so if your scorer returns loss, that value should be negated.
Note
Using custom scorers in functions where n_jobs > 1
While defining the custom scoring function alongside the calling function should work out of the box with the default joblib backend (loky), importing it from another module will be a more robust approach and work independently of the joblib backend.
For example, to use n_jobs
greater than 1 in the example below,
custom_scoring_function
function is saved in a usercreated module
(custom_scorer_module.py
) and imported:
>>> from custom_scorer_module import custom_scoring_function
>>> cross_val_score(model,
... X_train,
... y_train,
... scoring=make_scorer(custom_scoring_function, greater_is_better=False),
... cv=5,
... n_jobs=1)
3.3.1.4. Using multiple metric evaluation¶
Scikitlearn also permits evaluation of multiple metrics in GridSearchCV
,
RandomizedSearchCV
and cross_validate
.
There are two ways to specify multiple scoring metrics for the scoring
parameter:
 As an iterable of string metrics::
>>> scoring = ['accuracy', 'precision']
 As a
dict
mapping the scorer name to the scoring function:: >>> from sklearn.metrics import accuracy_score >>> from sklearn.metrics import make_scorer >>> scoring = {'accuracy': make_scorer(accuracy_score), ... 'prec': 'precision'}
 As a
Note that the dict values can either be scorer functions or one of the predefined metric strings.
Currently only those scorer functions that return a single score can be passed inside the dict. Scorer functions that return multiple values are not permitted and will require a wrapper to return a single metric:
>>> from sklearn.model_selection import cross_validate
>>> from sklearn.metrics import confusion_matrix
>>> # A sample toy binary classification dataset
>>> X, y = datasets.make_classification(n_classes=2, random_state=0)
>>> svm = LinearSVC(random_state=0)
>>> def tn(y_true, y_pred): return confusion_matrix(y_true, y_pred)[0, 0]
>>> def fp(y_true, y_pred): return confusion_matrix(y_true, y_pred)[0, 1]
>>> def fn(y_true, y_pred): return confusion_matrix(y_true, y_pred)[1, 0]
>>> def tp(y_true, y_pred): return confusion_matrix(y_true, y_pred)[1, 1]
>>> scoring = {'tp': make_scorer(tp), 'tn': make_scorer(tn),
... 'fp': make_scorer(fp), 'fn': make_scorer(fn)}
>>> cv_results = cross_validate(svm.fit(X, y), X, y, cv=5, scoring=scoring)
>>> # Getting the test set true positive scores
>>> print(cv_results['test_tp'])
[10 9 8 7 8]
>>> # Getting the test set false negative scores
>>> print(cv_results['test_fn'])
[0 1 2 3 2]
3.3.2. Classification metrics¶
The sklearn.metrics
module implements several loss, score, and utility
functions to measure classification performance.
Some metrics might require probability estimates of the positive class,
confidence values, or binary decisions values.
Most implementations allow each sample to provide a weighted contribution
to the overall score, through the sample_weight
parameter.
Some of these are restricted to the binary classification case:

Compute precisionrecall pairs for different probability thresholds 

Compute Receiver operating characteristic (ROC) 
Others also work in the multiclass case:

Compute the balanced accuracy 

Cohen’s kappa: a statistic that measures interannotator agreement. 

Compute confusion matrix to evaluate the accuracy of a classification. 

Average hinge loss (nonregularized) 

Compute the Matthews correlation coefficient (MCC) 

Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores. 
Some also work in the multilabel case:

Accuracy classification score. 

Build a text report showing the main classification metrics 

Compute the F1 score, also known as balanced Fscore or Fmeasure 

Compute the Fbeta score 

Compute the average Hamming loss. 

Jaccard similarity coefficient score 

Log loss, aka logistic loss or crossentropy loss. 

Compute a confusion matrix for each class or sample 

Compute precision, recall, Fmeasure and support for each class 

Compute the precision 

Compute the recall 

Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores. 

Zeroone classification loss. 
And some work with binary and multilabel (but not multiclass) problems:

Compute average precision (AP) from prediction scores 
In the following subsections, we will describe each of those functions, preceded by some notes on common API and metric definition.
3.3.2.1. From binary to multiclass and multilabel¶
Some metrics are essentially defined for binary classification tasks (e.g.
f1_score
, roc_auc_score
). In these cases, by default
only the positive label is evaluated, assuming by default that the positive
class is labelled 1
(though this may be configurable through the
pos_label
parameter).
In extending a binary metric to multiclass or multilabel problems, the data
is treated as a collection of binary problems, one for each class.
There are then a number of ways to average binary metric calculations across
the set of classes, each of which may be useful in some scenario.
Where available, you should select among these using the average
parameter.
"macro"
simply calculates the mean of the binary metrics, giving equal weight to each class. In problems where infrequent classes are nonetheless important, macroaveraging may be a means of highlighting their performance. On the other hand, the assumption that all classes are equally important is often untrue, such that macroaveraging will overemphasize the typically low performance on an infrequent class."weighted"
accounts for class imbalance by computing the average of binary metrics in which each class’s score is weighted by its presence in the true data sample."micro"
gives each sampleclass pair an equal contribution to the overall metric (except as a result of sampleweight). Rather than summing the metric per class, this sums the dividends and divisors that make up the perclass metrics to calculate an overall quotient. Microaveraging may be preferred in multilabel settings, including multiclass classification where a majority class is to be ignored."samples"
applies only to multilabel problems. It does not calculate a perclass measure, instead calculating the metric over the true and predicted classes for each sample in the evaluation data, and returning their (sample_weight
weighted) average.Selecting
average=None
will return an array with the score for each class.
While multiclass data is provided to the metric, like binary targets, as an
array of class labels, multilabel data is specified as an indicator matrix,
in which cell [i, j]
has value 1 if sample i
has label j
and value
0 otherwise.
3.3.2.2. Accuracy score¶
The accuracy_score
function computes the
accuracy, either the fraction
(default) or the count (normalize=False) of correct predictions.
In multilabel classification, the function returns the subset accuracy. If the entire set of predicted labels for a sample strictly match with the true set of labels, then the subset accuracy is 1.0; otherwise it is 0.0.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample and \(y_i\) is the corresponding true value, then the fraction of correct predictions over \(n_\text{samples}\) is defined as
where \(1(x)\) is the indicator function.
>>> 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
Example:
See Test with permutations the significance of a classification score for an example of accuracy score usage using permutations of the dataset.
3.3.2.3. Balanced accuracy score¶
The balanced_accuracy_score
function computes the balanced accuracy, which avoids inflated
performance estimates on imbalanced datasets. It is the macroaverage of recall
scores per class or, equivalently, raw accuracy where each sample is weighted
according to the inverse prevalence of its true class.
Thus for balanced datasets, the score is equal to accuracy.
In the binary case, balanced accuracy is equal to the arithmetic mean of sensitivity (true positive rate) and specificity (true negative rate), or the area under the ROC curve with binary predictions rather than scores.
If the classifier performs equally well on either class, this term reduces to the conventional accuracy (i.e., the number of correct predictions divided by the total number of predictions).
In contrast, if the conventional accuracy is above chance only because the classifier takes advantage of an imbalanced test set, then the balanced accuracy, as appropriate, will drop to \(\frac{1}{n\_classes}\).
The score ranges from 0 to 1, or when adjusted=True
is used, it rescaled to
the range \(\frac{1}{1  n\_classes}\) to 1, inclusive, with
performance at random scoring 0.
If \(y_i\) is the true value of the \(i\)th sample, and \(w_i\) is the corresponding sample weight, then we adjust the sample weight to:
where \(1(x)\) is the indicator function. Given predicted \(\hat{y}_i\) for sample \(i\), balanced accuracy is defined as:
With adjusted=True
, balanced accuracy reports the relative increase from
\(\texttt{balancedaccuracy}(y, \mathbf{0}, w) =
\frac{1}{n\_classes}\). In the binary case, this is also known as
*Youden’s J statistic*,
or informedness.
Note
The multiclass definition here seems the most reasonable extension of the metric used in binary classification, though there is no certain consensus in the literature:
Our definition: [Mosley2013], [Kelleher2015] and [Guyon2015], where [Guyon2015] adopt the adjusted version to ensure that random predictions have a score of \(0\) and perfect predictions have a score of \(1\)..
Class balanced accuracy as described in [Mosley2013]: the minimum between the precision and the recall for each class is computed. Those values are then averaged over the total number of classes to get the balanced accuracy.
Balanced Accuracy as described in [Urbanowicz2015]: the average of sensitivity and specificity is computed for each class and then averaged over total number of classes.
References:
 Guyon2015(1,2)
I. Guyon, K. Bennett, G. Cawley, H.J. Escalante, S. Escalera, T.K. Ho, N. Macià, B. Ray, M. Saeed, A.R. Statnikov, E. Viegas, Design of the 2015 ChaLearn AutoML Challenge, IJCNN 2015.
 Mosley2013(1,2)
L. Mosley, A balanced approach to the multiclass imbalance problem, IJCV 2010.
 Kelleher2015
John. D. Kelleher, Brian Mac Namee, Aoife D’Arcy, Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies, 2015.
 Urbanowicz2015
Urbanowicz R.J., Moore, J.H. ExSTraCS 2.0: description and evaluation of a scalable learning classifier system, Evol. Intel. (2015) 8: 89.
3.3.2.4. Cohen’s kappa¶
The function cohen_kappa_score
computes Cohen’s kappa statistic.
This measure is intended to compare labelings by different human annotators,
not a classifier versus a ground truth.
The kappa score (see docstring) is a number between 1 and 1. Scores above .8 are generally considered good agreement; zero or lower means no agreement (practically random labels).
Kappa scores can be computed for binary or multiclass problems, but not for multilabel problems (except by manually computing a perlabel score) and not for more than two annotators.
>>> from sklearn.metrics import cohen_kappa_score
>>> y_true = [2, 0, 2, 2, 0, 1]
>>> y_pred = [0, 0, 2, 2, 0, 2]
>>> cohen_kappa_score(y_true, y_pred)
0.4285714285714286
3.3.2.5. Confusion matrix¶
The confusion_matrix
function evaluates
classification accuracy by computing the confusion matrix
with each row corresponding to the true class
<https://en.wikipedia.org/wiki/Confusion_matrix>`_.
(Wikipedia and other references may use different convention for axes.)
By definition, entry \(i, j\) in a confusion matrix is the number of observations actually in group \(i\), but predicted to be in group \(j\). Here is an example:
>>> 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]])
plot_confusion_matrix
can be used to visually represent a confusion
matrix as shown in the
Confusion matrix
example, which creates the following figure:
The parameter normalize
allows to report ratios instead of counts. The
confusion matrix can be normalized in 3 different ways: 'pred'
, 'true'
,
and 'all'
which will divide the counts by the sum of each columns, rows, or
the entire matrix, respectively.
>>> y_true = [0, 0, 0, 1, 1, 1, 1, 1]
>>> y_pred = [0, 1, 0, 1, 0, 1, 0, 1]
>>> confusion_matrix(y_true, y_pred, normalize='all')
array([[0.25 , 0.125],
[0.25 , 0.375]])
For binary problems, we can get counts of true negatives, false positives, false negatives and true positives as follows:
>>> y_true = [0, 0, 0, 1, 1, 1, 1, 1]
>>> y_pred = [0, 1, 0, 1, 0, 1, 0, 1]
>>> tn, fp, fn, tp = confusion_matrix(y_true, y_pred).ravel()
>>> tn, fp, fn, tp
(2, 1, 2, 3)
Example:
See Confusion matrix for an example of using a confusion matrix to evaluate classifier output quality.
See Recognizing handwritten digits for an example of using a confusion matrix to classify handwritten digits.
See Classification of text documents using sparse features for an example of using a confusion matrix to classify text documents.
3.3.2.6. Classification report¶
The classification_report
function builds a text report showing the
main classification metrics. Here is a small example with custom target_names
and inferred labels:
>>> from sklearn.metrics import classification_report
>>> y_true = [0, 1, 2, 2, 0]
>>> y_pred = [0, 0, 2, 1, 0]
>>> target_names = ['class 0', 'class 1', 'class 2']
>>> print(classification_report(y_true, y_pred, target_names=target_names))
precision recall f1score support
class 0 0.67 1.00 0.80 2
class 1 0.00 0.00 0.00 1
class 2 1.00 0.50 0.67 2
accuracy 0.60 5
macro avg 0.56 0.50 0.49 5
weighted avg 0.67 0.60 0.59 5
Example:
See Recognizing handwritten digits for an example of classification report usage for handwritten digits.
See Classification of text documents using sparse features for an example of classification report usage for text documents.
See Parameter estimation using grid search with crossvalidation for an example of classification report usage for grid search with nested crossvalidation.
3.3.2.7. Hamming loss¶
The hamming_loss
computes the average Hamming loss or Hamming
distance between two sets
of samples.
If \(\hat{y}_j\) is the predicted value for the \(j\)th label of a given sample, \(y_j\) is the corresponding true value, and \(n_\text{labels}\) is the number of classes or labels, then the Hamming loss \(L_{Hamming}\) between two samples is defined as:
where \(1(x)\) is the indicator function.
>>> 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
Note
In multiclass classification, the Hamming loss corresponds to the Hamming
distance between y_true
and y_pred
which is similar to the
Zero one loss function. However, while zeroone loss penalizes
prediction sets that do not strictly match true sets, the Hamming loss
penalizes individual labels. Thus the Hamming loss, upper bounded by the zeroone
loss, is always between zero and one, inclusive; and predicting a proper subset
or superset of the true labels will give a Hamming loss between
zero and one, exclusive.
3.3.2.8. Precision, recall and Fmeasures¶
Intuitively, precision is the ability of the classifier not to label as positive a sample that is negative, and recall is the ability of the classifier to find all the positive samples.
The Fmeasure (\(F_\beta\) and \(F_1\) measures) can be interpreted as a weighted harmonic mean of the precision and recall. A \(F_\beta\) measure reaches its best value at 1 and its worst score at 0. With \(\beta = 1\), \(F_\beta\) and \(F_1\) are equivalent, and the recall and the precision are equally important.
The precision_recall_curve
computes a precisionrecall curve
from the ground truth label and a score given by the classifier
by varying a decision threshold.
The average_precision_score
function computes the
average precision
(AP) from prediction scores. The value is between 0 and 1 and higher is better.
AP is defined as
where \(P_n\) and \(R_n\) are the precision and recall at the nth threshold. With random predictions, the AP is the fraction of positive samples.
References [Manning2008] and [Everingham2010] present alternative variants of
AP that interpolate the precisionrecall curve. Currently,
average_precision_score
does not implement any interpolated variant.
References [Davis2006] and [Flach2015] describe why a linear interpolation of
points on the precisionrecall curve provides an overlyoptimistic measure of
classifier performance. This linear interpolation is used when computing area
under the curve with the trapezoidal rule in auc
.
Several functions allow you to analyze the precision, recall and Fmeasures score:

Compute average precision (AP) from prediction scores 

Compute the F1 score, also known as balanced Fscore or Fmeasure 

Compute the Fbeta score 

Compute precisionrecall pairs for different probability thresholds 

Compute precision, recall, Fmeasure and support for each class 

Compute the precision 

Compute the recall 
Note that the precision_recall_curve
function is restricted to the
binary case. The average_precision_score
function works only in
binary classification and multilabel indicator format. The
plot_precision_recall_curve
function plots the precision recall as
follows.
Examples:
See Classification of text documents using sparse features for an example of
f1_score
usage to classify text documents.See Parameter estimation using grid search with crossvalidation for an example of
precision_score
andrecall_score
usage to estimate parameters using grid search with nested crossvalidation.See PrecisionRecall for an example of
precision_recall_curve
usage to evaluate classifier output quality.
References:
 Manning2008
C.D. Manning, P. Raghavan, H. Schütze, Introduction to Information Retrieval, 2008.
 Everingham2010
M. Everingham, L. Van Gool, C.K.I. Williams, J. Winn, A. Zisserman, The Pascal Visual Object Classes (VOC) Challenge, IJCV 2010.
 Davis2006
J. Davis, M. Goadrich, The Relationship Between PrecisionRecall and ROC Curves, ICML 2006.
 Flach2015
P.A. Flach, M. Kull, PrecisionRecallGain Curves: PR Analysis Done Right, NIPS 2015.
3.3.2.8.1. Binary classification¶
In a binary classification task, the terms ‘’positive’’ and ‘’negative’’ refer to the classifier’s prediction, and the terms ‘’true’’ and ‘’false’’ refer to whether that prediction corresponds to the external judgment (sometimes known as the ‘’observation’’). Given these definitions, we can formulate the following table:
Actual class (observation) 

Predicted class (expectation) 
tp (true positive) Correct result 
fp (false positive) Unexpected result 
fn (false negative) Missing result 
tn (true negative) Correct absence of result 
In this context, we can define the notions of precision, recall and Fmeasure:
Here are some small examples in binary classification:
>>> from sklearn import metrics
>>> y_pred = [0, 1, 0, 0]
>>> y_true = [0, 1, 0, 1]
>>> metrics.precision_score(y_true, y_pred)
1.0
>>> metrics.recall_score(y_true, y_pred)
0.5
>>> metrics.f1_score(y_true, y_pred)
0.66...
>>> metrics.fbeta_score(y_true, y_pred, beta=0.5)
0.83...
>>> metrics.fbeta_score(y_true, y_pred, beta=1)
0.66...
>>> metrics.fbeta_score(y_true, y_pred, beta=2)
0.55...
>>> metrics.precision_recall_fscore_support(y_true, y_pred, beta=0.5)
(array([0.66..., 1. ]), array([1. , 0.5]), array([0.71..., 0.83...]), array([2, 2]))
>>> import numpy as np
>>> from sklearn.metrics import precision_recall_curve
>>> 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])
>>> precision, recall, threshold = precision_recall_curve(y_true, y_scores)
>>> precision
array([0.66..., 0.5 , 1. , 1. ])
>>> recall
array([1. , 0.5, 0.5, 0. ])
>>> threshold
array([0.35, 0.4 , 0.8 ])
>>> average_precision_score(y_true, y_scores)
0.83...
3.3.2.8.2. Multiclass and multilabel classification¶
In multiclass and multilabel classification task, the notions of precision,
recall, and Fmeasures can be applied to each label independently.
There are a few ways to combine results across labels,
specified by the average
argument to the
average_precision_score
(multilabel only), f1_score
,
fbeta_score
, precision_recall_fscore_support
,
precision_score
and recall_score
functions, as described
above. Note that if all labels are included, “micro”averaging
in a multiclass setting will produce precision, recall and \(F\)
that are all identical to accuracy. Also note that “weighted” averaging may
produce an Fscore that is not between precision and recall.
To make this more explicit, consider the following notation:
\(y\) the set of predicted \((sample, label)\) pairs
\(\hat{y}\) the set of true \((sample, label)\) pairs
\(L\) the set of labels
\(S\) the set of samples
\(y_s\) the subset of \(y\) with sample \(s\), i.e. \(y_s := \left\{(s', l) \in y  s' = s\right\}\)
\(y_l\) the subset of \(y\) with label \(l\)
similarly, \(\hat{y}_s\) and \(\hat{y}_l\) are subsets of \(\hat{y}\)
\(P(A, B) := \frac{\left A \cap B \right}{\leftA\right}\) for some sets \(A\) and \(B\)
\(R(A, B) := \frac{\left A \cap B \right}{\leftB\right}\) (Conventions vary on handling \(B = \emptyset\); this implementation uses \(R(A, B):=0\), and similar for \(P\).)
\(F_\beta(A, B) := \left(1 + \beta^2\right) \frac{P(A, B) \times R(A, B)}{\beta^2 P(A, B) + R(A, B)}\)
Then the metrics are defined as:

Precision 
Recall 
F_beta 


\(P(y, \hat{y})\) 
\(R(y, \hat{y})\) 
\(F_\beta(y, \hat{y})\) 

\(\frac{1}{\leftS\right} \sum_{s \in S} P(y_s, \hat{y}_s)\) 
\(\frac{1}{\leftS\right} \sum_{s \in S} R(y_s, \hat{y}_s)\) 
\(\frac{1}{\leftS\right} \sum_{s \in S} F_\beta(y_s, \hat{y}_s)\) 

\(\frac{1}{\leftL\right} \sum_{l \in L} P(y_l, \hat{y}_l)\) 
\(\frac{1}{\leftL\right} \sum_{l \in L} R(y_l, \hat{y}_l)\) 
\(\frac{1}{\leftL\right} \sum_{l \in L} F_\beta(y_l, \hat{y}_l)\) 

\(\frac{1}{\sum_{l \in L} \left\hat{y}_l\right} \sum_{l \in L} \left\hat{y}_l\right P(y_l, \hat{y}_l)\) 
\(\frac{1}{\sum_{l \in L} \left\hat{y}_l\right} \sum_{l \in L} \left\hat{y}_l\right R(y_l, \hat{y}_l)\) 
\(\frac{1}{\sum_{l \in L} \left\hat{y}_l\right} \sum_{l \in L} \left\hat{y}_l\right F_\beta(y_l, \hat{y}_l)\) 

\(\langle P(y_l, \hat{y}_l)  l \in L \rangle\) 
\(\langle R(y_l, \hat{y}_l)  l \in L \rangle\) 
\(\langle F_\beta(y_l, \hat{y}_l)  l \in L \rangle\) 
>>> from sklearn import metrics
>>> y_true = [0, 1, 2, 0, 1, 2]
>>> y_pred = [0, 2, 1, 0, 0, 1]
>>> metrics.precision_score(y_true, y_pred, average='macro')
0.22...
>>> metrics.recall_score(y_true, y_pred, average='micro')
0.33...
>>> metrics.f1_score(y_true, y_pred, average='weighted')
0.26...
>>> metrics.fbeta_score(y_true, y_pred, average='macro', beta=0.5)
0.23...
>>> metrics.precision_recall_fscore_support(y_true, y_pred, beta=0.5, average=None)
(array([0.66..., 0. , 0. ]), array([1., 0., 0.]), array([0.71..., 0. , 0. ]), array([2, 2, 2]...))
For multiclass classification with a “negative class”, it is possible to exclude some labels:
>>> metrics.recall_score(y_true, y_pred, labels=[1, 2], average='micro')
... # excluding 0, no labels were correctly recalled
0.0
Similarly, labels not present in the data sample may be accounted for in macroaveraging.
>>> metrics.precision_score(y_true, y_pred, labels=[0, 1, 2, 3], average='macro')
0.166...
3.3.2.9. Jaccard similarity coefficient score¶
The jaccard_score
function computes the average of Jaccard similarity
coefficients, also called the
Jaccard index, between pairs of label sets.
The Jaccard similarity coefficient of the \(i\)th samples, with a ground truth label set \(y_i\) and predicted label set \(\hat{y}_i\), is defined as
jaccard_score
works like precision_recall_fscore_support
as a
naively setwise measure applying natively to binary targets, and extended to
apply to multilabel and multiclass through the use of average
(see
above).
In the binary case:
>>> import numpy as np
>>> from sklearn.metrics import jaccard_score
>>> y_true = np.array([[0, 1, 1],
... [1, 1, 0]])
>>> y_pred = np.array([[1, 1, 1],
... [1, 0, 0]])
>>> jaccard_score(y_true[0], y_pred[0])
0.6666...
In the multilabel case with binary label indicators:
>>> jaccard_score(y_true, y_pred, average='samples')
0.5833...
>>> jaccard_score(y_true, y_pred, average='macro')
0.6666...
>>> jaccard_score(y_true, y_pred, average=None)
array([0.5, 0.5, 1. ])
Multiclass problems are binarized and treated like the corresponding multilabel problem:
>>> y_pred = [0, 2, 1, 2]
>>> y_true = [0, 1, 2, 2]
>>> jaccard_score(y_true, y_pred, average=None)
array([1. , 0. , 0.33...])
>>> jaccard_score(y_true, y_pred, average='macro')
0.44...
>>> jaccard_score(y_true, y_pred, average='micro')
0.33...
3.3.2.10. Hinge loss¶
The hinge_loss
function computes the average distance between
the model and the data using
hinge loss, a onesided metric
that considers only prediction errors. (Hinge
loss is used in maximal margin classifiers such as support vector machines.)
If the labels are encoded with +1 and 1, \(y\): is the true
value, and \(w\) is the predicted decisions as output by
decision_function
, then the hinge loss is defined as:
If there are more than two labels, hinge_loss
uses a multiclass variant
due to Crammer & Singer.
Here is
the paper describing it.
If \(y_w\) is the predicted decision for true label and \(y_t\) is the maximum of the predicted decisions for all other labels, where predicted decisions are output by decision function, then multiclass hinge loss is defined by:
Here a small example demonstrating the use of the hinge_loss
function
with a svm classifier in a binary class problem:
>>> 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(random_state=0)
>>> pred_decision = est.decision_function([[2], [3], [0.5]])
>>> pred_decision
array([2.18..., 2.36..., 0.09...])
>>> hinge_loss([1, 1, 1], pred_decision)
0.3...
Here is an example demonstrating the use of the hinge_loss
function
with a svm classifier in a multiclass problem:
>>> 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()
>>> pred_decision = est.decision_function([[1], [2], [3]])
>>> y_true = [0, 2, 3]
>>> hinge_loss(y_true, pred_decision, labels)
0.56...
3.3.2.11. Log loss¶
Log loss, also called logistic regression loss or
crossentropy loss, is defined on probability estimates. It is
commonly used in (multinomial) logistic regression and neural networks, as well
as in some variants of expectationmaximization, and can be used to evaluate the
probability outputs (predict_proba
) of a classifier instead of its
discrete predictions.
For binary classification with a true label \(y \in \{0,1\}\) and a probability estimate \(p = \operatorname{Pr}(y = 1)\), the log loss per sample is the negative loglikelihood of the classifier given the true label:
This extends to the multiclass case as follows. Let the true labels for a set of samples be encoded as a 1ofK binary indicator matrix \(Y\), i.e., \(y_{i,k} = 1\) if sample \(i\) has label \(k\) taken from a set of \(K\) labels. Let \(P\) be a matrix of probability estimates, with \(p_{i,k} = \operatorname{Pr}(t_{i,k} = 1)\). Then the log loss of the whole set is
To see how this generalizes the binary log loss given above, note that in the binary case, \(p_{i,0} = 1  p_{i,1}\) and \(y_{i,0} = 1  y_{i,1}\), so expanding the inner sum over \(y_{i,k} \in \{0,1\}\) gives the binary log loss.
The log_loss
function computes log loss given a list of groundtruth
labels and a probability matrix, as returned by an estimator’s predict_proba
method.
>>> from sklearn.metrics import log_loss
>>> y_true = [0, 0, 1, 1]
>>> y_pred = [[.9, .1], [.8, .2], [.3, .7], [.01, .99]]
>>> log_loss(y_true, y_pred)
0.1738...
The first [.9, .1]
in y_pred
denotes 90% probability that the first
sample has label 0. The log loss is nonnegative.
3.3.2.12. Matthews correlation coefficient¶
The matthews_corrcoef
function computes the
Matthew’s correlation coefficient (MCC)
for binary classes. Quoting Wikipedia:
“The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (twoclass) 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.”
In the binary (twoclass) case, \(tp\), \(tn\), \(fp\) and \(fn\) are respectively the number of true positives, true negatives, false positives and false negatives, the MCC is defined as
In the multiclass case, the Matthews correlation coefficient can be defined in terms of a
confusion_matrix
\(C\) for \(K\) classes. To simplify the
definition consider the following intermediate variables:
\(t_k=\sum_{i}^{K} C_{ik}\) the number of times class \(k\) truly occurred,
\(p_k=\sum_{i}^{K} C_{ki}\) the number of times class \(k\) was predicted,
\(c=\sum_{k}^{K} C_{kk}\) the total number of samples correctly predicted,
\(s=\sum_{i}^{K} \sum_{j}^{K} C_{ij}\) the total number of samples.
Then the multiclass MCC is defined as:
When there are more than two labels, the value of the MCC will no longer range between 1 and +1. Instead the minimum value will be somewhere between 1 and 0 depending on the number and distribution of ground true labels. The maximum value is always +1.
Here is a small example illustrating the usage of the matthews_corrcoef
function:
>>> from sklearn.metrics import matthews_corrcoef
>>> y_true = [+1, +1, +1, 1]
>>> y_pred = [+1, 1, +1, +1]
>>> matthews_corrcoef(y_true, y_pred)
0.33...
3.3.2.13. Multilabel confusion matrix¶
The multilabel_confusion_matrix
function computes classwise (default)
or samplewise (samplewise=True) multilabel confusion matrix to evaluate
the accuracy of a classification. multilabel_confusion_matrix also treats
multiclass data as if it were multilabel, as this is a transformation commonly
applied to evaluate multiclass problems with binary classification metrics
(such as precision, recall, etc.).
When calculating classwise multilabel confusion matrix \(C\), the count of true negatives for class \(i\) is \(C_{i,0,0}\), false negatives is \(C_{i,1,0}\), true positives is \(C_{i,1,1}\) and false positives is \(C_{i,0,1}\).
Here is an example demonstrating the use of the
multilabel_confusion_matrix
function with
multilabel indicator matrix input:
>>> import numpy as np
>>> from sklearn.metrics import multilabel_confusion_matrix
>>> y_true = np.array([[1, 0, 1],
... [0, 1, 0]])
>>> y_pred = np.array([[1, 0, 0],
... [0, 1, 1]])
>>> multilabel_confusion_matrix(y_true, y_pred)
array([[[1, 0],
[0, 1]],
[[1, 0],
[0, 1]],
[[0, 1],
[1, 0]]])
Or a confusion matrix can be constructed for each sample’s labels:
>>> multilabel_confusion_matrix(y_true, y_pred, samplewise=True)
array([[[1, 0],
[1, 1]],
<BLANKLINE>
[[1, 1],
[0, 1]]])
Here is an example demonstrating the use of the
multilabel_confusion_matrix
function with
multiclass input:
>>> y_true = ["cat", "ant", "cat", "cat", "ant", "bird"]
>>> y_pred = ["ant", "ant", "cat", "cat", "ant", "cat"]
>>> multilabel_confusion_matrix(y_true, y_pred,
... labels=["ant", "bird", "cat"])
array([[[3, 1],
[0, 2]],
[[5, 0],
[1, 0]],
[[2, 1],
[1, 2]]])
Here are some examples demonstrating the use of the
multilabel_confusion_matrix
function to calculate recall
(or sensitivity), specificity, fall out and miss rate for each class in a
problem with multilabel indicator matrix input.
Calculating recall (also called the true positive rate or the sensitivity) for each class:
>>> y_true = np.array([[0, 0, 1],
... [0, 1, 0],
... [1, 1, 0]])
>>> y_pred = np.array([[0, 1, 0],
... [0, 0, 1],
... [1, 1, 0]])
>>> mcm = multilabel_confusion_matrix(y_true, y_pred)
>>> tn = mcm[:, 0, 0]
>>> tp = mcm[:, 1, 1]
>>> fn = mcm[:, 1, 0]
>>> fp = mcm[:, 0, 1]
>>> tp / (tp + fn)
array([1. , 0.5, 0. ])
Calculating specificity (also called the true negative rate) for each class:
>>> tn / (tn + fp)
array([1. , 0. , 0.5])
Calculating fall out (also called the false positive rate) for each class:
>>> fp / (fp + tn)
array([0. , 1. , 0.5])
Calculating miss rate (also called the false negative rate) for each class:
>>> fn / (fn + tp)
array([0. , 0.5, 1. ])
3.3.2.14. Receiver operating characteristic (ROC)¶
The function roc_curve
computes the
receiver operating characteristic curve, or ROC curve.
Quoting Wikipedia :
“A receiver operating characteristic (ROC), or simply ROC curve, is a graphical plot which illustrates the performance of a binary classifier system as its discrimination threshold is varied. It is created by plotting the fraction of true positives out of the positives (TPR = true positive rate) vs. the fraction of false positives out of the negatives (FPR = false positive rate), at various threshold settings. TPR is also known as sensitivity, and FPR is one minus the specificity or true negative rate.”
This function requires the true binary
value and the target scores, which can either be probability estimates of the
positive class, confidence values, or binary decisions.
Here is a small example of how to use the roc_curve
function:
>>> import numpy as np
>>> from sklearn.metrics import roc_curve
>>> y = np.array([1, 1, 2, 2])
>>> scores = np.array([0.1, 0.4, 0.35, 0.8])
>>> fpr, tpr, thresholds = roc_curve(y, scores, pos_label=2)
>>> fpr
array([0. , 0. , 0.5, 0.5, 1. ])
>>> tpr
array([0. , 0.5, 0.5, 1. , 1. ])
>>> thresholds
array([1.8 , 0.8 , 0.4 , 0.35, 0.1 ])
This figure shows an example of such an ROC curve:
The roc_auc_score
function computes the area under the receiver
operating characteristic (ROC) curve, which is also denoted by
AUC or AUROC. By computing the
area under the roc curve, the curve information is summarized in one number.
For more information see the Wikipedia article on AUC.
>>> 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
In multilabel classification, the roc_auc_score
function is
extended by averaging over the labels as above.
Compared to metrics such as the subset accuracy, the Hamming loss, or the F1 score, ROC doesn’t require optimizing a threshold for each label.
The roc_auc_score
function can also be used in multiclass
classification. Two averaging strategies are currently supported: the
onevsone algorithm computes the average of the pairwise ROC AUC scores, and
the onevsrest algorithm computes the average of the ROC AUC scores for each
class against all other classes. In both cases, the predicted labels are
provided in an array with values from 0 to n_classes
, and the scores
correspond to the probability estimates that a sample belongs to a particular
class. The OvO and OvR algorithms supports weighting uniformly
(average='macro'
) and weighting by the prevalence (average='weighted'
).
Onevsone Algorithm: Computes the average AUC of all possible pairwise combinations of classes. [HT2001] defines a multiclass AUC metric weighted uniformly:
where \(c\) is the number of classes and \(\text{AUC}(j  k)\) is the
AUC with class \(j\) as the positive class and class \(k\) as the
negative class. In general,
\(\text{AUC}(j  k) \neq \text{AUC}(k  j))\) in the multiclass
case. This algorithm is used by setting the keyword argument multiclass
to 'ovo'
and average
to 'macro'
.
The [HT2001] multiclass AUC metric can be extended to be weighted by the prevalence:
where \(c\) is the number of classes. This algorithm is used by setting
the keyword argument multiclass
to 'ovo'
and average
to
'weighted'
. The 'weighted'
option returns a prevalenceweighted average
as described in [FC2009].
Onevsrest Algorithm: Computes the AUC of each class against the rest.
The algorithm is functionally the same as the multilabel case. To enable this
algorithm set the keyword argument multiclass
to 'ovr'
. Similar to
OvO, OvR supports two types of averaging: 'macro'
[F2006] and
'weighted'
[F2001].
In applications where a high false positive rate is not tolerable the parameter
max_fpr
of roc_auc_score
can be used to summarize the ROC curve up
to the given limit.
Examples:
See Receiver Operating Characteristic (ROC) for an example of using ROC to evaluate the quality of the output of a classifier.
See Receiver Operating Characteristic (ROC) with cross validation for an example of using ROC to evaluate classifier output quality, using crossvalidation.
See Species distribution modeling for an example of using ROC to model species distribution.
References:
 HT2001(1,2)
Hand, D.J. and Till, R.J., (2001). A simple generalisation of the area under the ROC curve for multiple class classification problems. Machine learning, 45(2), pp.171186.
 FC2009
Ferri, Cèsar & HernandezOrallo, Jose & Modroiu, R. (2009). An Experimental Comparison of Performance Measures for Classification. Pattern Recognition Letters. 30. 2738.
 F2006
Fawcett, T., 2006. An introduction to ROC analysis. Pattern Recognition Letters, 27(8), pp. 861874.
 F2001
Fawcett, T., 2001. Using rule sets to maximize ROC performance In Data Mining, 2001. Proceedings IEEE International Conference, pp. 131138.
3.3.2.15. Zero one loss¶
The zero_one_loss
function computes the sum or the average of the 01
classification loss (\(L_{01}\)) over \(n_{\text{samples}}\). By
default, the function normalizes over the sample. To get the sum of the
\(L_{01}\), set normalize
to False
.
In multilabel classification, the zero_one_loss
scores a subset as
one if its labels strictly match the predictions, and as a zero if there
are any errors. By default, the function returns the percentage of imperfectly
predicted subsets. To get the count of such subsets instead, set
normalize
to False
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample and \(y_i\) is the corresponding true value, then the 01 loss \(L_{01}\) is defined as:
where \(1(x)\) is the indicator function.
>>> 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, where the first label set [0,1] has an error:
>>> zero_one_loss(np.array([[0, 1], [1, 1]]), np.ones((2, 2)))
0.5
>>> zero_one_loss(np.array([[0, 1], [1, 1]]), np.ones((2, 2)), normalize=False)
1
Example:
See Recursive feature elimination with crossvalidation for an example of zero one loss usage to perform recursive feature elimination with crossvalidation.
3.3.2.16. Brier score loss¶
The brier_score_loss
function computes the
Brier score
for binary classes. Quoting Wikipedia:
“The Brier score is a proper score function that measures the accuracy of probabilistic predictions. It is applicable to tasks in which predictions must assign probabilities to a set of mutually exclusive discrete outcomes.”
This function returns a score of the mean square difference between the actual outcome and the predicted probability of the possible outcome. The actual outcome has to be 1 or 0 (true or false), while the predicted probability of the actual outcome can be a value between 0 and 1.
The brier score loss is also between 0 to 1 and the lower the score (the mean square difference is smaller), the more accurate the prediction is. It can be thought of as a measure of the “calibration” of a set of probabilistic predictions.
where : \(N\) is the total number of predictions, \(f_t\) is the predicted probability of the actual outcome \(o_t\).
Here is a small example of usage of this function::
>>> 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, 0.4])
>>> y_pred = np.array([0, 1, 1, 0])
>>> brier_score_loss(y_true, y_prob)
0.055
>>> brier_score_loss(y_true, 1  y_prob, pos_label=0)
0.055
>>> brier_score_loss(y_true_categorical, y_prob, pos_label="ham")
0.055
>>> brier_score_loss(y_true, y_prob > 0.5)
0.0
Example:
See Probability calibration of classifiers for an example of Brier score loss usage to perform probability calibration of classifiers.
References:
G. Brier, Verification of forecasts expressed in terms of probability, Monthly weather review 78.1 (1950)
3.3.3. Multilabel ranking metrics¶
In multilabel learning, each sample can have any number of ground truth labels associated with it. The goal is to give high scores and better rank to the ground truth labels.
3.3.3.1. Coverage error¶
The coverage_error
function computes the average number of labels that
have to be included in the final prediction such that all true labels
are predicted. This is useful if you want to know how many topscoredlabels
you have to predict in average without missing any true one. The best value
of this metrics is thus the average number of true labels.
Note
Our implementation’s score is 1 greater than the one given in Tsoumakas et al., 2010. This extends it to handle the degenerate case in which an instance has 0 true labels.
Formally, given a binary indicator matrix of the ground truth labels \(y \in \left\{0, 1\right\}^{n_\text{samples} \times n_\text{labels}}\) and the score associated with each label \(\hat{f} \in \mathbb{R}^{n_\text{samples} \times n_\text{labels}}\), the coverage is defined as
with \(\text{rank}_{ij} = \left\left\{k: \hat{f}_{ik} \geq \hat{f}_{ij} \right\}\right\).
Given the rank definition, ties in y_scores
are broken by giving the
maximal rank that would have been assigned to all tied values.
Here is a small example of usage of this function:
>>> import numpy as np
>>> from sklearn.metrics import coverage_error
>>> y_true = np.array([[1, 0, 0], [0, 0, 1]])
>>> y_score = np.array([[0.75, 0.5, 1], [1, 0.2, 0.1]])
>>> coverage_error(y_true, y_score)
2.5
3.3.3.2. Label ranking average precision¶
The label_ranking_average_precision_score
function
implements label ranking average precision (LRAP). This metric is linked to
the average_precision_score
function, but is based on the notion of
label ranking instead of precision and recall.
Label ranking average precision (LRAP) averages over the samples the answer to the following question: for each ground truth label, what fraction of higherranked labels were true labels? This performance measure will be higher if you are able to give better rank to the labels associated with each sample. The obtained score is always strictly greater than 0, and the best value is 1. If there is exactly one relevant label per sample, label ranking average precision is equivalent to the mean reciprocal rank.
Formally, given a binary indicator matrix of the ground truth labels \(y \in \left\{0, 1\right\}^{n_\text{samples} \times n_\text{labels}}\) and the score associated with each label \(\hat{f} \in \mathbb{R}^{n_\text{samples} \times n_\text{labels}}\), the average precision is defined as
where \(\mathcal{L}_{ij} = \left\{k: y_{ik} = 1, \hat{f}_{ik} \geq \hat{f}_{ij} \right\}\), \(\text{rank}_{ij} = \left\left\{k: \hat{f}_{ik} \geq \hat{f}_{ij} \right\}\right\), \(\cdot\) computes the cardinality of the set (i.e., the number of elements in the set), and \(\cdot_0\) is the \(\ell_0\) “norm” (which computes the number of nonzero elements in a vector).
Here is a small example of usage of this function:
>>> 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)
0.416...
3.3.3.3. Ranking loss¶
The label_ranking_loss
function computes the ranking loss which
averages over the samples the number of label pairs that are incorrectly
ordered, i.e. true labels have a lower score than false labels, weighted by
the inverse of the number of ordered pairs of false and true labels.
The lowest achievable ranking loss is zero.
Formally, given a binary indicator matrix of the ground truth labels \(y \in \left\{0, 1\right\}^{n_\text{samples} \times n_\text{labels}}\) and the score associated with each label \(\hat{f} \in \mathbb{R}^{n_\text{samples} \times n_\text{labels}}\), the ranking loss is defined as
where \(\cdot\) computes the cardinality of the set (i.e., the number of elements in the set) and \(\cdot_0\) is the \(\ell_0\) “norm” (which computes the number of nonzero elements in a vector).
Here is a small example of usage of this function:
>>> import numpy as np
>>> from sklearn.metrics import label_ranking_loss
>>> 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_loss(y_true, y_score)
0.75...
>>> # With the following prediction, we have perfect and minimal loss
>>> y_score = np.array([[1.0, 0.1, 0.2], [0.1, 0.2, 0.9]])
>>> label_ranking_loss(y_true, y_score)
0.0
References:
Tsoumakas, G., Katakis, I., & Vlahavas, I. (2010). Mining multilabel data. In Data mining and knowledge discovery handbook (pp. 667685). Springer US.
3.3.3.4. Normalized Discounted Cumulative Gain¶
Discounted Cumulative Gain (DCG) and Normalized Discounted Cumulative Gain (NDCG) are ranking metrics; they compare a predicted order to groundtruth scores, such as the relevance of answers to a query.
from the Wikipedia page for Discounted Cumulative Gain:
“Discounted cumulative gain (DCG) is a measure of ranking quality. In information retrieval, it is often used to measure effectiveness of web search engine algorithms or related applications. Using a graded relevance scale of documents in a searchengine result set, DCG measures the usefulness, or gain, of a document based on its position in the result list. The gain is accumulated from the top of the result list to the bottom, with the gain of each result discounted at lower ranks”
DCG orders the true targets (e.g. relevance of query answers) in the predicted order, then multiplies them by a logarithmic decay and sums the result. The sum can be truncated after the first \(K\) results, in which case we call it DCG@K. NDCG, or NDCG@K is DCG divided by the DCG obtained by a perfect prediction, so that it is always between 0 and 1. Usually, NDCG is preferred to DCG.
Compared with the ranking loss, NDCG can take into account relevance scores, rather than a groundtruth ranking. So if the groundtruth consists only of an ordering, the ranking loss should be preferred; if the groundtruth consists of actual usefulness scores (e.g. 0 for irrelevant, 1 for relevant, 2 for very relevant), NDCG can be used.
For one sample, given the vector of continuous groundtruth values for each target \(y \in \mathbb{R}^{M}\), where \(M\) is the number of outputs, and the prediction \(\hat{y}\), which induces the ranking function \(f\), the DCG score is
and the NDCG score is the DCG score divided by the DCG score obtained for \(y\).
References:
Wikipedia entry for Discounted Cumulative Gain: https://en.wikipedia.org/wiki/Discounted_cumulative_gain
Jarvelin, K., & Kekalainen, J. (2002). Cumulated gainbased evaluation of IR techniques. ACM Transactions on Information Systems (TOIS), 20(4), 422446.
Wang, Y., Wang, L., Li, Y., He, D., Chen, W., & Liu, T. Y. (2013, May). A theoretical analysis of NDCG ranking measures. In Proceedings of the 26th Annual Conference on Learning Theory (COLT 2013)
McSherry, F., & Najork, M. (2008, March). Computing information retrieval performance measures efficiently in the presence of tied scores. In European conference on information retrieval (pp. 414421). Springer, Berlin, Heidelberg.
3.3.4. Regression metrics¶
The sklearn.metrics
module implements several loss, score, and utility
functions to measure regression performance. Some of those have been enhanced
to handle the multioutput case: mean_squared_error
,
mean_absolute_error
, explained_variance_score
and
r2_score
.
These functions have an multioutput
keyword argument which specifies the
way the scores or losses for each individual target should be averaged. The
default is 'uniform_average'
, which specifies a uniformly weighted mean
over outputs. If an ndarray
of shape (n_outputs,)
is passed, then its
entries are interpreted as weights and an according weighted average is
returned. If multioutput
is 'raw_values'
is specified, then all
unaltered individual scores or losses will be returned in an array of shape
(n_outputs,)
.
The r2_score
and explained_variance_score
accept an additional
value 'variance_weighted'
for the multioutput
parameter. This option
leads to a weighting of each individual score by the variance of the
corresponding target variable. This setting quantifies the globally captured
unscaled variance. If the target variables are of different scale, then this
score puts more importance on well explaining the higher variance variables.
multioutput='variance_weighted'
is the default value for r2_score
for backward compatibility. This will be changed to uniform_average
in the
future.
3.3.4.1. Explained variance score¶
The explained_variance_score
computes the explained variance
regression score.
If \(\hat{y}\) is the estimated target output, \(y\) the corresponding (correct) target output, and \(Var\) is Variance, the square of the standard deviation, then the explained variance is estimated as follow:
The best possible score is 1.0, lower values are worse.
Here is a small example of usage of the explained_variance_score
function:
>>> from sklearn.metrics import explained_variance_score
>>> y_true = [3, 0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> explained_variance_score(y_true, y_pred)
0.957...
>>> y_true = [[0.5, 1], [1, 1], [7, 6]]
>>> y_pred = [[0, 2], [1, 2], [8, 5]]
>>> explained_variance_score(y_true, y_pred, multioutput='raw_values')
array([0.967..., 1. ])
>>> explained_variance_score(y_true, y_pred, multioutput=[0.3, 0.7])
0.990...
3.3.4.2. Max error¶
The max_error
function computes the maximum residual error , a metric
that captures the worst case error between the predicted value and
the true value. In a perfectly fitted single output regression
model, max_error
would be 0
on the training set and though this
would be highly unlikely in the real world, this metric shows the
extent of error that the model had when it was fitted.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample, and \(y_i\) is the corresponding true value, then the max error is defined as
Here is a small example of usage of the max_error
function:
>>> from sklearn.metrics import max_error
>>> y_true = [3, 2, 7, 1]
>>> y_pred = [9, 2, 7, 1]
>>> max_error(y_true, y_pred)
6
The max_error
does not support multioutput.
3.3.4.3. Mean absolute error¶
The mean_absolute_error
function computes mean absolute
error, a risk
metric corresponding to the expected value of the absolute error loss or
\(l1\)norm loss.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample, and \(y_i\) is the corresponding true value, then the mean absolute error (MAE) estimated over \(n_{\text{samples}}\) is defined as
Here is a small example of usage of the mean_absolute_error
function:
>>> from sklearn.metrics import mean_absolute_error
>>> y_true = [3, 0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> mean_absolute_error(y_true, y_pred)
0.5
>>> y_true = [[0.5, 1], [1, 1], [7, 6]]
>>> y_pred = [[0, 2], [1, 2], [8, 5]]
>>> mean_absolute_error(y_true, y_pred)
0.75
>>> mean_absolute_error(y_true, y_pred, multioutput='raw_values')
array([0.5, 1. ])
>>> mean_absolute_error(y_true, y_pred, multioutput=[0.3, 0.7])
0.85...
3.3.4.4. Mean squared error¶
The mean_squared_error
function computes mean square
error, a risk
metric corresponding to the expected value of the squared (quadratic) error or
loss.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample, and \(y_i\) is the corresponding true value, then the mean squared error (MSE) estimated over \(n_{\text{samples}}\) is defined as
Here is a small example of usage of the mean_squared_error
function:
>>> from sklearn.metrics import mean_squared_error
>>> y_true = [3, 0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> mean_squared_error(y_true, y_pred)
0.375
>>> y_true = [[0.5, 1], [1, 1], [7, 6]]
>>> y_pred = [[0, 2], [1, 2], [8, 5]]
>>> mean_squared_error(y_true, y_pred)
0.7083...
Examples:
See Gradient Boosting regression for an example of mean squared error usage to evaluate gradient boosting regression.
3.3.4.5. Mean squared logarithmic error¶
The mean_squared_log_error
function computes a risk metric
corresponding to the expected value of the squared logarithmic (quadratic)
error or loss.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample, and \(y_i\) is the corresponding true value, then the mean squared logarithmic error (MSLE) estimated over \(n_{\text{samples}}\) is defined as
Where \(\log_e (x)\) means the natural logarithm of \(x\). This metric is best to use when targets having exponential growth, such as population counts, average sales of a commodity over a span of years etc. Note that this metric penalizes an underpredicted estimate greater than an overpredicted estimate.
Here is a small example of usage of the mean_squared_log_error
function:
>>> from sklearn.metrics import mean_squared_log_error
>>> y_true = [3, 5, 2.5, 7]
>>> y_pred = [2.5, 5, 4, 8]
>>> mean_squared_log_error(y_true, y_pred)
0.039...
>>> y_true = [[0.5, 1], [1, 2], [7, 6]]
>>> y_pred = [[0.5, 2], [1, 2.5], [8, 8]]
>>> mean_squared_log_error(y_true, y_pred)
0.044...
3.3.4.6. Median absolute error¶
The median_absolute_error
is particularly interesting because it is
robust to outliers. The loss is calculated by taking the median of all absolute
differences between the target and the prediction.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample and \(y_i\) is the corresponding true value, then the median absolute error (MedAE) estimated over \(n_{\text{samples}}\) is defined as
The median_absolute_error
does not support multioutput.
Here is a small example of usage of the median_absolute_error
function:
>>> from sklearn.metrics import median_absolute_error
>>> y_true = [3, 0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> median_absolute_error(y_true, y_pred)
0.5
3.3.4.7. R² score, the coefficient of determination¶
The r2_score
function computes the coefficient of
determination,
usually denoted as R².
It represents the proportion of variance (of y) that has been explained by the independent variables in the model. It provides an indication of goodness of fit and therefore a measure of how well unseen samples are likely to be predicted by the model, through the proportion of explained variance.
As such variance is dataset dependent, R² may not be meaningfully comparable across different datasets. Best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a R² score of 0.0.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample and \(y_i\) is the corresponding true value for total \(n\) samples, the estimated R² is defined as:
where \(\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i\) and \(\sum_{i=1}^{n} (y_i  \hat{y}_i)^2 = \sum_{i=1}^{n} \epsilon_i^2\).
Note that r2_score
calculates unadjusted R² without correcting for
bias in sample variance of y.
Here is a small example of usage of the r2_score
function:
>>> from sklearn.metrics import r2_score
>>> y_true = [3, 0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> r2_score(y_true, y_pred)
0.948...
>>> y_true = [[0.5, 1], [1, 1], [7, 6]]
>>> y_pred = [[0, 2], [1, 2], [8, 5]]
>>> r2_score(y_true, y_pred, multioutput='variance_weighted')
0.938...
>>> y_true = [[0.5, 1], [1, 1], [7, 6]]
>>> y_pred = [[0, 2], [1, 2], [8, 5]]
>>> r2_score(y_true, y_pred, multioutput='uniform_average')
0.936...
>>> r2_score(y_true, y_pred, multioutput='raw_values')
array([0.965..., 0.908...])
>>> r2_score(y_true, y_pred, multioutput=[0.3, 0.7])
0.925...
Example:
See Lasso and Elastic Net for Sparse Signals for an example of R² score usage to evaluate Lasso and Elastic Net on sparse signals.
3.3.4.8. Mean Poisson, Gamma, and Tweedie deviances¶
The mean_tweedie_deviance
function computes the mean Tweedie
deviance error
with a power
parameter (\(p\)). This is a metric that elicits
predicted expectation values of regression targets.
Following special cases exist,
when
power=0
it is equivalent tomean_squared_error
.when
power=1
it is equivalent tomean_poisson_deviance
.when
power=2
it is equivalent tomean_gamma_deviance
.
If \(\hat{y}_i\) is the predicted value of the \(i\)th sample, and \(y_i\) is the corresponding true value, then the mean Tweedie deviance error (D) for power \(p\), estimated over \(n_{\text{samples}}\) is defined as
Tweedie deviance is a homogeneous function of degree 2power
.
Thus, Gamma distribution with power=2
means that simultaneously scaling
y_true
and y_pred
has no effect on the deviance. For Poisson
distribution power=1
the deviance scales linearly, and for Normal
distribution (power=0
), quadratically. In general, the higher
power
the less weight is given to extreme deviations between true
and predicted targets.
For instance, let’s compare the two predictions 1.0 and 100 that are both 50% of their corresponding true value.
The mean squared error (power=0
) is very sensitive to the
prediction difference of the second point,:
>>> from sklearn.metrics import mean_tweedie_deviance
>>> mean_tweedie_deviance([1.0], [1.5], power=0)
0.25
>>> mean_tweedie_deviance([100.], [150.], power=0)
2500.0
If we increase power
to 1,:
>>> mean_tweedie_deviance([1.0], [1.5], power=1)
0.18...
>>> mean_tweedie_deviance([100.], [150.], power=1)
18.9...
the difference in errors decreases. Finally, by setting, power=2
:
>>> mean_tweedie_deviance([1.0], [1.5], power=2)
0.14...
>>> mean_tweedie_deviance([100.], [150.], power=2)
0.14...
we would get identical errors. The deviance when power=2
is thus only
sensitive to relative errors.
3.3.5. Clustering metrics¶
The sklearn.metrics
module implements several loss, score, and utility
functions. For more information see the Clustering performance evaluation
section for instance clustering, and Biclustering evaluation for
biclustering.
3.3.6. Dummy estimators¶
When doing supervised learning, a simple sanity check consists of comparing
one’s estimator against simple rules of thumb. DummyClassifier
implements several such simple strategies for classification:
stratified
generates random predictions by respecting the training set class distribution.most_frequent
always predicts the most frequent label in the training set.prior
always predicts the class that maximizes the class prior (likemost_frequent
) andpredict_proba
returns the class prior.uniform
generates predictions uniformly at random.constant
always predicts a constant label that is provided by the user.A major motivation of this method is F1scoring, when the positive class is in the minority.
Note that with all these strategies, the predict
method completely ignores
the input data!
To illustrate DummyClassifier
, first let’s create an imbalanced
dataset:
>>> from sklearn.datasets import load_iris
>>> from sklearn.model_selection import train_test_split
>>> X, y = load_iris(return_X_y=True)
>>> y[y != 1] = 1
>>> X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
Next, let’s compare the accuracy of SVC
and most_frequent
:
>>> from sklearn.dummy import DummyClassifier
>>> from sklearn.svm import SVC
>>> clf = SVC(kernel='linear', C=1).fit(X_train, y_train)
>>> clf.score(X_test, y_test)
0.63...
>>> clf = DummyClassifier(strategy='most_frequent', random_state=0)
>>> clf.fit(X_train, y_train)
DummyClassifier(random_state=0, strategy='most_frequent')
>>> clf.score(X_test, y_test)
0.57...
We see that SVC
doesn’t do much better than a dummy classifier. Now, let’s
change the kernel:
>>> clf = SVC(kernel='rbf', C=1).fit(X_train, y_train)
>>> clf.score(X_test, y_test)
0.94...
We see that the accuracy was boosted to almost 100%. A cross validation strategy is recommended for a better estimate of the accuracy, if it is not too CPU costly. For more information see the Crossvalidation: evaluating estimator performance section. Moreover if you want to optimize over the parameter space, it is highly recommended to use an appropriate methodology; see the Tuning the hyperparameters of an estimator section for details.
More generally, when the accuracy of a classifier is too close to random, it probably means that something went wrong: features are not helpful, a hyperparameter is not correctly tuned, the classifier is suffering from class imbalance, etc…
DummyRegressor
also implements four simple rules of thumb for regression:
mean
always predicts the mean of the training targets.median
always predicts the median of the training targets.quantile
always predicts a user provided quantile of the training targets.constant
always predicts a constant value that is provided by the user.
In all these strategies, the predict
method completely ignores
the input data.