3.3. Metrics and scoring: quantifying the quality of predictions¶

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.

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.

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 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, 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 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(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(y, clf.predict(X))
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), 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).

• It returns a floating point number that quantifies the 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.

>>> 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¶

Scikit-learn also permits evaluation of multiple metrics in GridSearchCV, RandomizedSearchCV and cross_validate.

There are three 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'}
  

  Note that the dict values can either be scorer functions or one of the predefined metric strings.

• As a callable that returns a dictionary of scores:

  >>> 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 confusion_matrix_scorer(clf, X, y):
  ...      y_pred = clf.predict(X)
  ...      cm = confusion_matrix(y, y_pred)
  ...      return {'tn': cm[0, 0], 'fp': cm[0, 1],
  ...              'fn': cm[1, 0], 'tp': cm[1, 1]}
  >>> cv_results = cross_validate(svm, X, y, cv=5,
  ...                             scoring=confusion_matrix_scorer)
  >>> # 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]
  

3.3.2.1. 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, 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.

• 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

3.3.2.3. Top-k accuracy score¶

The top_k_accuracy_score function is a generalization of accuracy_score. The difference is that a prediction is considered correct as long as the true label is associated with one of the k highest predicted scores. accuracy_score is the special case of k = 1.

The function covers the binary and multiclass classification cases but not the multilabel case.

If \(\hat{f}_{i,j}\) is the predicted class for the \(i\)-th sample corresponding to the \(j\)-th largest predicted score and \(y_i\) is the corresponding true value, then the fraction of correct predictions over \(n_\text{samples}\) is defined as

where \(k\) is the number of guesses allowed and \(1(x)\) is the indicator function.

>>> import numpy as np
>>> from sklearn.metrics import top_k_accuracy_score
>>> y_true = np.array([0, 1, 2, 2])
>>> y_score = np.array([[0.5, 0.2, 0.2],
...                     [0.3, 0.4, 0.2],
...                     [0.2, 0.4, 0.3],
...                     [0.7, 0.2, 0.1]])
>>> top_k_accuracy_score(y_true, y_score, k=2)
0.75
>>> # Not normalizing gives the number of "correctly" classified samples
>>> top_k_accuracy_score(y_true, y_score, k=2, normalize=False)
3

3.3.2.4. Balanced accuracy score¶

The balanced_accuracy_score function computes the balanced accuracy, which avoids inflated performance estimates on imbalanced datasets. It is the macro-average 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 \(\texttt{balanced-accuracy}(y, \mathbf{0}, w) = \frac{1}{n\_classes}\). In the binary case, this is also known as *Youden’s J statistic*, or informedness.

3.3.2.5. 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 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

3.3.2.6. Confusion matrix¶

The confusion_matrix function evaluates classification accuracy by computing the confusion matrix with each row corresponding to the true class (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)

3.3.2.7. 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  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

    accuracy                           0.60         5
   macro avg       0.56      0.50      0.49         5
weighted avg       0.67      0.60      0.59         5

3.3.2.8. 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

3.3.2.9. Precision, recall and F-measures¶

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 (\(F_\beta\) 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 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 \(P_n\) 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 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:

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.

>>> 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...

>>> 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]...))

>>> metrics.recall_score(y_true, y_pred, labels=[1, 2], average='micro')
... # excluding 0, no labels were correctly recalled
0.0

>>> metrics.precision_score(y_true, y_pred, labels=[0, 1, 2, 3], average='macro')
0.166...

3.3.2.10. 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 set-wise 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...

3.3.2.11. Hinge loss¶

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, \(y\): 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...

3.3.2.12. Log loss¶

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 \(y \in \{0,1\}\) and a probability estimate \(p = \operatorname{Pr}(y = 1)\), 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 \(Y\), 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 \(p_{i,k} = \operatorname{Pr}(y_{i,k} = 1)\). 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 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.

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

In the binary (two-class) 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...

3.3.2.14. Multi-label confusion matrix¶

The multilabel_confusion_matrix function computes class-wise (default) or sample-wise (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 class-wise 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]],

       [[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. ])

3.3.2.15. 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.

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.

>>> from sklearn.datasets import load_breast_cancer
>>> from sklearn.linear_model import LogisticRegression
>>> from sklearn.metrics import roc_auc_score
>>> X, y = load_breast_cancer(return_X_y=True)
>>> clf = LogisticRegression(solver="liblinear").fit(X, y)
>>> clf.classes_
array([0, 1])

>>> y_score = clf.predict_proba(X)[:, 1]
>>> roc_auc_score(y, y_score)
0.99...

>>> roc_auc_score(y, clf.decision_function(X))
0.99...

3.3.2.15.2. Multi-class case¶

The roc_auc_score function can also be used in multi-class classification. Two averaging strategies are currently supported: the one-vs-one algorithm computes the average of the pairwise ROC AUC scores, and the one-vs-rest 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 support weighting uniformly (average='macro') and by prevalence (average='weighted').

One-vs-one Algorithm: Computes the average AUC of all possible pairwise combinations of classes. [HT2001] defines a multiclass AUC metric weighted uniformly:

\[\frac{1}{c(c-1)}\sum_{j=1}^{c}\sum_{k > j}^c (\text{AUC}(j | k) + \text{AUC}(k | j))\]

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:

\[\frac{1}{c(c-1)}\sum_{j=1}^{c}\sum_{k > j}^c p(j \cup k)( \text{AUC}(j | k) + \text{AUC}(k | j))\]

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 prevalence-weighted average as described in [FC2009].

One-vs-rest Algorithm: Computes the AUC of each class against the rest [PD2000]. The algorithm is functionally the same as the multilabel case. To enable this algorithm set the keyword argument multiclass to 'ovr'. Like 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.

>>> from sklearn.datasets import make_multilabel_classification
>>> from sklearn.multioutput import MultiOutputClassifier
>>> X, y = make_multilabel_classification(random_state=0)
>>> inner_clf = LogisticRegression(solver="liblinear", random_state=0)
>>> clf = MultiOutputClassifier(inner_clf).fit(X, y)
>>> y_score = np.transpose([y_pred[:, 1] for y_pred in clf.predict_proba(X)])
>>> roc_auc_score(y, y_score, average=None)
array([0.82..., 0.86..., 0.94..., 0.85... , 0.94...])

>>> from sklearn.linear_model import RidgeClassifierCV
>>> clf = RidgeClassifierCV().fit(X, y)
>>> y_score = clf.decision_function(X)
>>> roc_auc_score(y, y_score, average=None)
array([0.81..., 0.84... , 0.93..., 0.87..., 0.94...])

3.3.2.16. Detection error tradeoff (DET)¶

The function det_curve computes the detection error tradeoff curve (DET) curve [WikipediaDET2017]. Quoting Wikipedia:

“A detection error tradeoff (DET) graph is a graphical plot of error rates for binary classification systems, plotting false reject rate vs. false accept rate. The x- and y-axes are scaled non-linearly by their standard normal deviates (or just by logarithmic transformation), yielding tradeoff curves that are more linear than ROC curves, and use most of the image area to highlight the differences of importance in the critical operating region.”

DET curves are a variation of receiver operating characteristic (ROC) curves where False Negative Rate is plotted on the y-axis instead of True Positive Rate. DET curves are commonly plotted in normal deviate scale by transformation with \(\phi^{-1}\) (with \(\phi\) being the cumulative distribution function). The resulting performance curves explicitly visualize the tradeoff of error types for given classification algorithms. See [Martin1997] for examples and further motivation.

This figure compares the ROC and DET curves of two example classifiers on the same classification task:

Properties:

• DET curves form a linear curve in normal deviate scale if the detection scores are normally (or close-to normally) distributed. It was shown by [Navratil2007] that the reverse it not necessarily true and even more general distributions are able produce linear DET curves.

• The normal deviate scale transformation spreads out the points such that a comparatively larger space of plot is occupied. Therefore curves with similar classification performance might be easier to distinguish on a DET plot.

• With False Negative Rate being “inverse” to True Positive Rate the point of perfection for DET curves is the origin (in contrast to the top left corner for ROC curves).

Applications and limitations:

DET curves are intuitive to read and hence allow quick visual assessment of a classifier’s performance. Additionally DET curves can be consulted for threshold analysis and operating point selection. This is particularly helpful if a comparison of error types is required.

One the other hand DET curves do not provide their metric as a single number. Therefore for either automated evaluation or comparison to other classification tasks metrics like the derived area under ROC curve might be better suited.

3.3.2.17. Zero one loss¶

The zero_one_loss function computes the sum or the average of the 0-1 classification loss (\(L_{0-1}\)) over \(n_{\text{samples}}\). By default, the function normalizes over the sample. To get the sum of the \(L_{0-1}\), 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 0-1 loss \(L_{0-1}\) 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

3.3.2.18. Brier score loss¶

The brier_score_loss function computes the Brier score for binary classes [Brier1950]. 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 the mean squared error of the actual outcome \(y \in \{0,1\}\) and the predicted probability estimate \(p = \operatorname{Pr}(y = 1)\) (predict_proba) as outputted by:

The Brier score loss is also between 0 to 1 and the lower the value (the mean square difference is smaller), the more accurate the prediction is.

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

The Brier score can be used to assess how well a classifier is calibrated. However, a lower Brier score loss does not always mean a better calibration. This is because, by analogy with the bias-variance decomposition of the mean squared error, the Brier score loss can be decomposed as the sum of calibration loss and refinement loss [Bella2012]. Calibration loss is defined as the mean squared deviation from empirical probabilities derived from the slope of ROC segments. Refinement loss can be defined as the expected optimal loss as measured by the area under the optimal cost curve. Refinement loss can change independently from calibration loss, thus a lower Brier score loss does not necessarily mean a better calibrated model. “Only when refinement loss remains the same does a lower Brier score loss always mean better calibration” [Bella2012], [Flach2008].

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 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.

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 higher-ranked 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

3.3.3.4. Normalized Discounted Cumulative Gain¶

Discounted Cumulative Gain (DCG) and Normalized Discounted Cumulative Gain (NDCG) are ranking metrics implemented in dcg_score and ndcg_score ; they compare a predicted order to ground-truth 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 search-engine 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 ground-truth ranking. So if the ground-truth consists only of an ordering, the ranking loss should be preferred; if the ground-truth 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 ground-truth 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 \(y\).

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...

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 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...

3.3.4.6. Mean absolute percentage error¶

The mean_absolute_percentage_error (MAPE), also known as mean absolute percentage deviation (MAPD), is an evaluation metric for regression problems. The idea of this metric is to be sensitive to relative errors. It is for example not changed by a global scaling of the target variable.

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 percentage error (MAPE) estimated over \(n_{\text{samples}}\) is defined as

where \(\epsilon\) is an arbitrary small yet strictly positive number to avoid undefined results when y is zero.

The mean_absolute_percentage_error function supports multioutput.

Here is a small example of usage of the mean_absolute_percentage_error function:

>>> from sklearn.metrics import mean_absolute_percentage_error
>>> y_true = [1, 10, 1e6]
>>> y_pred = [0.9, 15, 1.2e6]
>>> mean_absolute_percentage_error(y_true, y_pred)
0.2666...

In above example, if we had used mean_absolute_error, it would have ignored the small magnitude values and only reflected the error in prediction of highest magnitude value. But that problem is resolved in case of MAPE because it calculates relative percentage error with respect to actual output.

3.3.4.7. 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

3.3.4.8. 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...

3.3.4.9. 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 to mean_squared_error.

• when power=1 it is equivalent to mean_poisson_deviance.

• when power=2 it is equivalent to mean_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 2-power. 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.