There are 3 different APIs for evaluating the quality of a model’s predictions:
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.model_selection.cross_val_score
and model_selection.GridSearchCV
) rely on an internal scoring strategy. This is discussed in the section The scoring parameter: defining model evaluation rules.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.
scoring
parameter: defining model evaluation rulesModel 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.
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’ | metrics.accuracy_score | |
‘average_precision’ | metrics.average_precision_score | |
‘f1’ | metrics.f1_score | for binary targets |
‘f1_micro’ | metrics.f1_score | micro-averaged |
‘f1_macro’ | metrics.f1_score | macro-averaged |
‘f1_weighted’ | metrics.f1_score | weighted average |
‘f1_samples’ | metrics.f1_score | by multilabel sample |
‘neg_log_loss’ | metrics.log_loss | requires predict_proba support |
‘precision’ etc. | metrics.precision_score | suffixes apply as with ‘f1’ |
‘recall’ etc. | metrics.recall_score | suffixes apply as with ‘f1’ |
‘roc_auc’ | metrics.roc_auc_score | |
Clustering | ||
‘adjusted_mutual_info_score’ | metrics.adjusted_mutual_info_score | |
‘adjusted_rand_score’ | metrics.adjusted_rand_score | |
‘completeness_score’ | metrics.completeness_score | |
‘fowlkes_mallows_score’ | metrics.fowlkes_mallows_score | |
‘homogeneity_score’ | metrics.homogeneity_score | |
‘mutual_info_score’ | metrics.mutual_info_score | |
‘normalized_mutual_info_score’ | metrics.normalized_mutual_info_score | |
‘v_measure_score’ | metrics.v_measure_score | |
Regression | ||
‘explained_variance’ | metrics.explained_variance_score | |
‘neg_mean_absolute_error’ | metrics.mean_absolute_error | |
‘neg_mean_squared_error’ | metrics.mean_squared_error | |
‘neg_mean_squared_log_error’ | metrics.mean_squared_log_error | |
‘neg_median_absolute_error’ | metrics.median_absolute_error | |
‘r2’ | metrics.r2_score |
Usage examples:
>>> from sklearn import svm, datasets >>> from sklearn.model_selection import cross_val_score >>> iris = datasets.load_iris() >>> X, y = iris.data, iris.target >>> clf = svm.SVC(probability=True, random_state=0) >>> cross_val_score(clf, X, y, scoring='neg_log_loss') array([-0.07..., -0.16..., -0.06...]) >>> model = svm.SVC() >>> cross_val_score(model, X, y, scoring='wrong_choice') Traceback (most recent call last): ValueError: 'wrong_choice' is not a valid scoring value. Valid options are ['accuracy', 'adjusted_mutual_info_score', 'adjusted_rand_score', 'average_precision', 'completeness_score', 'explained_variance', 'f1', 'f1_macro', 'f1_micro', 'f1_samples', 'f1_weighted', 'fowlkes_mallows_score', 'homogeneity_score', 'mutual_info_score', 'neg_log_loss', 'neg_mean_absolute_error', 'neg_mean_squared_error', 'neg_mean_squared_log_error', 'neg_median_absolute_error', 'normalized_mutual_info_score', 'precision', 'precision_macro', 'precision_micro', 'precision_samples', 'precision_weighted', 'r2', 'recall', 'recall_macro', 'recall_micro', 'recall_samples', 'recall_weighted', 'roc_auc', 'v_measure_score']
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
.
The module sklearn.metrics
also exposes a set of simple functions measuring a prediction error given ground truth and prediction:
_score
return a value to maximize, the higher the better._error
or _loss
return a value to minimize, the lower the better. When converting into a scorer object using make_scorer
, set the greater_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 non-default 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)
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:
my_custom_loss_func
in the example below)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.needs_threshold=True
). The default value is False.beta
or labels
in f1_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(ground_truth, predictions): ... diff = np.abs(ground_truth - predictions).max() ... return np.log(1 + diff) ... >>> # loss_func will negate the return value of my_custom_loss_func, >>> # which will be np.log(2), 0.693, given the values for ground_truth >>> # and predictions defined below. >>> loss = make_scorer(my_custom_loss_func, greater_is_better=False) >>> score = make_scorer(my_custom_loss_func, greater_is_better=True) >>> ground_truth = [[1], [1]] >>> predictions = [0, 1] >>> from sklearn.dummy import DummyClassifier >>> clf = DummyClassifier(strategy='most_frequent', random_state=0) >>> clf = clf.fit(ground_truth, predictions) >>> loss(clf,ground_truth, predictions) -0.69... >>> score(clf,ground_truth, predictions) 0.69...
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:
(estimator, X, y)
, where estimator
is the model that should be evaluated, X
is validation data, and y
is the ground truth target for X
(in the supervised case) or None
(in the unsupervised case).estimator
prediction quality on X
, with reference to y
. Again, by convention higher numbers are better, so if your scorer returns loss, that value should be negated.Scikit-learn 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:
>>> 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'}
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 tp(y_true, y_pred): return confusion_matrix(y_true, y_pred)[0, 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)[1, 0] >>> def fn(y_true, y_pred): return confusion_matrix(y_true, y_pred)[0, 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, scoring=scoring) >>> # Getting the test set true positive scores >>> print(cv_results['test_tp']) [12 13 15] >>> # Getting the test set false negative scores >>> print(cv_results['test_fn']) [5 4 1]
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:
precision_recall_curve (y_true, probas_pred) | Compute precision-recall pairs for different probability thresholds |
roc_curve (y_true, y_score[, pos_label, …]) | Compute Receiver operating characteristic (ROC) |
Others also work in the multiclass case:
cohen_kappa_score (y1, y2[, labels, weights, …]) | Cohen’s kappa: a statistic that measures inter-annotator agreement. |
confusion_matrix (y_true, y_pred[, labels, …]) | Compute confusion matrix to evaluate the accuracy of a classification |
hinge_loss (y_true, pred_decision[, labels, …]) | Average hinge loss (non-regularized) |
matthews_corrcoef (y_true, y_pred[, …]) | Compute the Matthews correlation coefficient (MCC) |
Some also work in the multilabel case:
accuracy_score (y_true, y_pred[, normalize, …]) | Accuracy classification score. |
classification_report (y_true, y_pred[, …]) | Build a text report showing the main classification metrics |
f1_score (y_true, y_pred[, labels, …]) | Compute the F1 score, also known as balanced F-score or F-measure |
fbeta_score (y_true, y_pred, beta[, labels, …]) | Compute the F-beta score |
hamming_loss (y_true, y_pred[, labels, …]) | Compute the average Hamming loss. |
jaccard_similarity_score (y_true, y_pred[, …]) | Jaccard similarity coefficient score |
log_loss (y_true, y_pred[, eps, normalize, …]) | Log loss, aka logistic loss or cross-entropy loss. |
precision_recall_fscore_support (y_true, y_pred) | Compute precision, recall, F-measure and support for each class |
precision_score (y_true, y_pred[, labels, …]) | Compute the precision |
recall_score (y_true, y_pred[, labels, …]) | Compute the recall |
zero_one_loss (y_true, y_pred[, normalize, …]) | Zero-one classification loss. |
And some work with binary and multilabel (but not multiclass) problems:
average_precision_score (y_true, y_score[, …]) | Compute average precision (AP) from prediction scores |
roc_auc_score (y_true, y_score[, average, …]) | Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores. |
In the following sub-sections, we will describe each of those functions, preceded by some notes on common API and metric definition.
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, macro-averaging 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 macro-averaging will over-emphasize 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 sample-class pair an equal contribution to the overall metric (except as a result of sample-weight). Rather than summing the metric per class, this sums the dividends and divisors that make up the per-class metrics to calculate an overall quotient. Micro-averaging 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 per-class 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.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.
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 is the predicted value of the -th sample and is the corresponding true value, then the fraction of correct predictions over is defined as
where 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:
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 per-label 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
The confusion_matrix
function evaluates classification accuracy by computing the confusion matrix.
By definition, entry in a confusion matrix is the number of observations actually in group , but predicted to be in group . 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]])
Here is a visual representation of such a confusion matrix (this figure comes from the Confusion matrix example):
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:
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 f1-score 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 avg / total 0.67 0.60 0.59 5
Example:
The hamming_loss
computes the average Hamming loss or Hamming distance between two sets of samples.
If is the predicted value for the -th label of a given sample, is the corresponding true value, and is the number of classes or labels, then the Hamming loss between two samples is defined as:
where 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 zero-one 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 zero-one 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.
The jaccard_similarity_score
function computes the average (default) or sum of Jaccard similarity coefficients, also called the Jaccard index, between pairs of label sets.
The Jaccard similarity coefficient of the -th samples, with a ground truth label set and predicted label set , is defined as
In binary and multiclass classification, the Jaccard similarity coefficient score is equal to the classification accuracy.
>>> import numpy as np >>> from sklearn.metrics import jaccard_similarity_score >>> y_pred = [0, 2, 1, 3] >>> y_true = [0, 1, 2, 3] >>> jaccard_similarity_score(y_true, y_pred) 0.5 >>> jaccard_similarity_score(y_true, y_pred, normalize=False) 2
In the multilabel case with binary label indicators:
>>> jaccard_similarity_score(np.array([[0, 1], [1, 1]]), np.ones((2, 2))) 0.75
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 F-measure ( and measures) can be interpreted as a weighted harmonic mean of the precision and recall. A measure reaches its best value at 1 and its worst score at 0. With , and are equivalent, and the recall and the precision are equally important.
The precision_recall_curve
computes a precision-recall 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 and 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 precision-recall curve. Currently, average_precision_score
does not implement any interpolated variant. References [Davis2006] and [Flach2015] describe why a linear interpolation of points on the precision-recall curve provides an overly-optimistic 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 F-measures score:
average_precision_score (y_true, y_score[, …]) | Compute average precision (AP) from prediction scores |
f1_score (y_true, y_pred[, labels, …]) | Compute the F1 score, also known as balanced F-score or F-measure |
fbeta_score (y_true, y_pred, beta[, labels, …]) | Compute the F-beta score |
precision_recall_curve (y_true, probas_pred) | Compute precision-recall pairs for different probability thresholds |
precision_recall_fscore_support (y_true, y_pred) | Compute precision, recall, F-measure and support for each class |
precision_score (y_true, y_pred[, labels, …]) | Compute the precision |
recall_score (y_true, y_pred[, labels, …]) | 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.
Examples:
f1_score
usage to classify text documents.precision_score
and recall_score
usage to estimate parameters using grid search with nested cross-validation.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 Precision-Recall and ROC Curves, ICML 2006. |
[Flach2015] | P.A. Flach, M. Kull, Precision-Recall-Gain Curves: PR Analysis Done Right, NIPS 2015. |
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 F-measure:
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...
In multiclass and multilabel classification task, the notions of precision, recall, and F-measures 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 for “micro”-averaging in a multiclass setting with all labels included will produce equal precision, recall and , while “weighted” averaging may produce an F-score that is not between precision and recall.
To make this more explicit, consider the following notation:
Then the metrics are defined as:
average | Precision | Recall | F_beta |
---|---|---|---|
"micro" | |||
"samples" | |||
"macro" | |||
"weighted" | |||
None |
>>> 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 macro-averaging.
>>> metrics.precision_score(y_true, y_pred, labels=[0, 1, 2, 3], average='macro') ... 0.166...
The hinge_loss
function computes the average distance between the model and the data using hinge loss, a one-sided 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, : is the true value, and 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 is the predicted decision for true label and 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(C=1.0, class_weight=None, dual=True, fit_intercept=True, intercept_scaling=1, loss='squared_hinge', max_iter=1000, multi_class='ovr', penalty='l2', random_state=0, tol=0.0001, verbose=0) >>> pred_decision = est.decision_function([[-2], [3], [0.5]]) >>> pred_decision 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(C=1.0, class_weight=None, dual=True, fit_intercept=True, intercept_scaling=1, loss='squared_hinge', max_iter=1000, multi_class='ovr', penalty='l2', random_state=None, tol=0.0001, verbose=0) >>> pred_decision = est.decision_function([[-1], [2], [3]]) >>> y_true = [0, 2, 3] >>> hinge_loss(y_true, pred_decision, labels) 0.56...
Log loss, also called logistic regression loss or cross-entropy loss, is defined on probability estimates. It is commonly used in (multinomial) logistic regression and neural networks, as well as in some variants of expectation-maximization, 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 and a probability estimate , the log loss per sample is the negative log-likelihood 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 1-of-K binary indicator matrix , i.e., if sample has label taken from a set of labels. Let be a matrix of probability estimates, with . 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, and , so expanding the inner sum over gives the binary log loss.
The log_loss
function computes log loss given a list of ground-truth 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 non-negative.
The matthews_corrcoef
function computes the Matthew’s correlation coefficient (MCC) for binary classes. Quoting Wikipedia:
In the binary (two-class) case, , , and 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
for classes. To simplify the definition consider the following intermediate variables:
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...
The function roc_curve
computes the receiver operating characteristic curve, or ROC curve. Quoting Wikipedia :
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. , 0.5, 0.5, 1. ]) >>> tpr array([ 0.5, 0.5, 1. , 1. ]) >>> thresholds array([ 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 multi-label 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 multi-class classification, if the predicted outputs have been binarized.
Examples:
The zero_one_loss
function computes the sum or the average of the 0-1 classification loss () over . By default, the function normalizes over the sample. To get the sum of the , 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 is the predicted value of the -th sample and is the corresponding true value, then the 0-1 loss is defined as:
where 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:
The brier_score_loss
function computes the Brier score for binary classes. Quoting Wikipedia:
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 : is the total number of predictions, is the predicted probability of the actual outcome .
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:
References:
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.
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 top-scored-labels 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 and the score associated with each label , the coverage is defined as
with . 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
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) 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 will yield better scores 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 and the score associated with each label , the average precision is defined as
with , and is the l0 norm or the cardinality of the set.
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...
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 number of false and true labels. The lowest achievable ranking loss is zero.
Formally, given a binary indicator matrix of the ground truth labels and the score associated with each label , the ranking loss is defined as
where is the norm or the cardinality of the set.
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:
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.
The explained_variance_score
computes the explained variance regression score.
If is the estimated target output, the corresponding (correct) target output, and 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...
The mean_absolute_error
function computes mean absolute error, a risk metric corresponding to the expected value of the absolute error loss or -norm loss.
If is the predicted value of the -th sample, and is the corresponding true value, then the mean absolute error (MAE) estimated over 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]) ... 0.849...
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 is the predicted value of the -th sample, and is the corresponding true value, then the mean squared error (MSE) estimated over 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:
The mean_squared_log_error
function computes a risk metric corresponding to the expected value of the squared logarithmic (quadratic) error or loss.
If is the predicted value of the -th sample, and is the corresponding true value, then the mean squared logarithmic error (MSLE) estimated over is defined as
Where means the natural logarithm of . 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 under-predicted estimate greater than an over-predicted 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...
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 is the predicted value of the -th sample and is the corresponding true value, then the median absolute error (MedAE) estimated over 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
The r2_score
function computes R², the coefficient of determination. It provides a measure of how well future samples are likely to be predicted by the model. 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 R^2 score of 0.0.
If is the predicted value of the -th sample and is the corresponding true value, then the score R² estimated over is defined as
where .
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:
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.
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 (like most_frequent`) and ``predict_proba
returns the class prior.uniform
generates predictions uniformly at random.constant always predicts a constant label that is provided by the user.
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 >>> iris = load_iris() >>> X, y = iris.data, iris.target >>> 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(constant=None, 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) 0.97...
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 Cross-validation: 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 hyper-parameters 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.
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Licensed under the 3-clause BSD License.
http://scikit-learn.org/stable/modules/model_evaluation.html