Mean squared error example

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
sklearn.metrics.get_scorer_names() 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

[texttt{accuracy}(y, hat{y}) = frac{1}{n_text{samples}} sum_{i=0}^{n_text{samples}-1} 1(hat{y}_i = y_i)]

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

[texttt{top-k accuracy}(y, hat{f}) = frac{1}{n_text{samples}} sum_{i=0}^{n_text{samples}-1} sum_{j=1}^{k} 1(hat{f}_{i,j} = y_i)]

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:

[texttt{balanced-accuracy} = frac{1}{2}left( frac{TP}{TP + FN} + frac{TN}{TN + FP}right )]

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:

[hat{w}_i = frac{w_i}{sum_j{1(y_j = y_i) w_j}}]

where (1(x)) is the indicator function.
Given predicted (hat{y}_i) for sample (i), balanced accuracy is
defined as:

[texttt{balanced-accuracy}(y, hat{y}, w) = frac{1}{sum{hat{w}_i}} sum_i 1(hat{y}_i = y_i) hat{w}_i]

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]])

ConfusionMatrixDisplay 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}_{i,j}) is the predicted value for the (j)-th label of a
given sample (i), (y_{i,j}) is the corresponding true value,
(n_text{samples}) is the number of samples and (n_text{labels})
is the number of labels, then the Hamming loss (L_{Hamming}) is defined
as:

[L_{Hamming}(y, hat{y}) = frac{1}{n_text{samples} * n_text{labels}} sum_{i=0}^{n_text{samples}-1} sum_{j=0}^{n_text{labels} — 1} 1(hat{y}_{i,j} not= y_{i,j})]

where (1(x)) is the indicator function.

The equation above does not hold true in the case of multiclass classification.
Please refer to the note below for more information.

>>> 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

[text{AP} = sum_n (R_n — R_{n-1}) P_n]

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 PredictionRecallDisplay.from_estimator and
PredictionRecallDisplay.from_predictions functions will plot the
precision-recall curve 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([0.5       , 0.66..., 0.5       , 1.        , 1.        ])
>>> recall
array([1. , 1. , 0.5, 0.5, 0. ])
>>> threshold
array([0.1 , 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 with a ground truth label set (y) and
predicted label set (hat{y}), is defined as

[J(y, hat{y}) = frac{|y cap hat{y}|}{|y cup hat{y}|}.]

The jaccard_score (like precision_recall_fscore_support) applies
natively to binary targets. By computing it set-wise it can be 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 2D comparison case (e.g. image similarity):

>>> jaccard_score(y_true, y_pred, average="micro")
0.6

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 true label (y_i) of a binary classification task is encoded as
(y_i=left{-1, +1right}) for every sample (i); and (w_i)
is the corresponding predicted decision (an array of shape (n_samples,) as
output by the decision_function method), then the hinge loss is defined as:

[L_text{Hinge}(y, w) = frac{1}{n_text{samples}} sum_{i=0}^{n_text{samples}-1} maxleft{1 — w_i y_i, 0right}]

If there are more than two labels, hinge_loss uses a multiclass variant
due to Crammer & Singer.
Here is
the paper describing it.

In this case the predicted decision is an array of shape (n_samples,
n_labels). If (w_{i, y_i}) is the predicted decision for the true label
(y_i) of the (i)-th sample; and
(hat{w}_{i, y_i} = maxleft{w_{i, y_j}~|~y_j ne y_i right})
is the maximum of the
predicted decisions for all the other labels, then the multi-class hinge loss
is defined by:

[L_text{Hinge}(y, w) = frac{1}{n_text{samples}}
sum_{i=0}^{n_text{samples}-1} maxleft{1 + hat{w}_{i, y_i}
— w_{i, y_i}, 0right}]

Here is 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=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:

[L_{log}(y, p) = -log operatorname{Pr}(y|p) = -(y log (p) + (1 — y) log (1 — p))]

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

[L_{log}(Y, P) = -log operatorname{Pr}(Y|P) = — frac{1}{N} sum_{i=0}^{N-1} sum_{k=0}^{K-1} y_{i,k} log p_{i,k}]

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

[MCC = frac{tp times tn — fp times fn}{sqrt{(tp + fp)(tp + fn)(tn + fp)(tn + fn)}}.]

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:

[MCC = frac{
c times s — sum_{k}^{K} p_k times t_k
}{sqrt{
(s^2 — sum_{k}^{K} p_k^2) times
(s^2 — sum_{k}^{K} t_k^2)
}}]

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 ])

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, denoted by ROC-AUC or AUROC, computes the
area under the ROC curve. By doing so, the curve information is summarized in
one number.

The following figure shows the ROC curve and ROC-AUC score for a classifier
aimed to distinguish the virginica flower from the rest of the species in the
Iris plants dataset:

For more information see the Wikipedia article on AUC.

>>> 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...

>>> 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 is not necessarily true and
  even more general distributions are able to 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.

On 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:

[L_{0-1}(y, hat{y}) = frac{1}{n_text{samples}} sum_{i=0}^{n_text{samples}-1} 1(hat{y}_i not= y_i)]

where (1(x)) is the indicator function. The zero one
loss can also be computed as (zero-one loss = 1 — accuracy).

>>> 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:

[BS = frac{1}{n_{text{samples}}} sum_{i=0}^{n_{text{samples}} — 1}(y_i — p_i)^2]

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].

3.3.2.19. Class likelihood ratios¶

The class_likelihood_ratios function computes the positive and negative
likelihood ratios
(LR_pm) for binary classes, which can be interpreted as the ratio of
post-test to pre-test odds as explained below. As a consequence, this metric is
invariant w.r.t. the class prevalence (the number of samples in the positive
class divided by the total number of samples) and can be extrapolated between
populations regardless of any possible class imbalance.

The (LR_pm) metrics are therefore very useful in settings where the data
available to learn and evaluate a classifier is a study population with nearly
balanced classes, such as a case-control study, while the target application,
i.e. the general population, has very low prevalence.

The positive likelihood ratio (LR_+) is the probability of a classifier to
correctly predict that a sample belongs to the positive class divided by the
probability of predicting the positive class for a sample belonging to the
negative class:

[LR_+ = frac{text{PR}(P+|T+)}{text{PR}(P+|T-)}.]

The notation here refers to predicted ((P)) or true ((T)) label and
the sign (+) and (-) refer to the positive and negative class,
respectively, e.g. (P+) stands for “predicted positive”.

Analogously, the negative likelihood ratio (LR_-) is the probability of a
sample of the positive class being classified as belonging to the negative class
divided by the probability of a sample of the negative class being correctly
classified:

[LR_- = frac{text{PR}(P-|T+)}{text{PR}(P-|T-)}.]

For classifiers above chance (LR_+) above 1 higher is better, while
(LR_-) ranges from 0 to 1 and lower is better.
Values of (LR_pmapprox 1) correspond to chance level.

Notice that probabilities differ from counts, for instance
(operatorname{PR}(P+|T+)) is not equal to the number of true positive
counts tp (see the wikipedia page for
the actual formulas).

Interpretation across varying prevalence:

Both class likelihood ratios are interpretable in terms of an odds ratio
(pre-test and post-tests):

[text{post-test odds} = text{Likelihood ratio} times text{pre-test odds}.]

Odds are in general related to probabilities via

[text{odds} = frac{text{probability}}{1 — text{probability}},]

or equivalently

[text{probability} = frac{text{odds}}{1 + text{odds}}.]

On a given population, the pre-test probability is given by the prevalence. By
converting odds to probabilities, the likelihood ratios can be translated into a
probability of truly belonging to either class before and after a classifier
prediction:

[text{post-test odds} = text{Likelihood ratio} times
frac{text{pre-test probability}}{1 — text{pre-test probability}},]

[text{post-test probability} = frac{text{post-test odds}}{1 + text{post-test odds}}.]

Mathematical divergences:

The positive likelihood ratio is undefined when (fp = 0), which can be
interpreted as the classifier perfectly identifying positive cases. If (fp
= 0) and additionally (tp = 0), this leads to a zero/zero division. This
happens, for instance, when using a DummyClassifier that always predicts the
negative class and therefore the interpretation as a perfect classifier is lost.

The negative likelihood ratio is undefined when (tn = 0). Such divergence
is invalid, as (LR_- > 1) would indicate an increase in the odds of a
sample belonging to the positive class after being classified as negative, as if
the act of classifying caused the positive condition. This includes the case of
a DummyClassifier that always predicts the positive class (i.e. when
(tn=fn=0)).

Both class likelihood ratios are undefined when (tp=fn=0), which means
that no samples of the positive class were present in the testing set. This can
also happen when cross-validating highly imbalanced data.

In all the previous cases the class_likelihood_ratios function raises by
default an appropriate warning message and returns nan to avoid pollution when
averaging over cross-validation folds.

For a worked-out demonstration of the class_likelihood_ratios function,
see the example below.

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, 1right}^{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

[coverage(y, hat{f}) = frac{1}{n_{text{samples}}}
sum_{i=0}^{n_{text{samples}} — 1} max_{j:y_{ij} = 1} text{rank}_{ij}]

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, 1right}^{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

[LRAP(y, hat{f}) = frac{1}{n_{text{samples}}}
sum_{i=0}^{n_{text{samples}} — 1} frac{1}{||y_i||_0}
sum_{j:y_{ij} = 1} frac{|mathcal{L}_{ij}|}{text{rank}_{ij}}]

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, 1right}^{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

[ranking_loss(y, hat{f}) = frac{1}{n_{text{samples}}}
sum_{i=0}^{n_{text{samples}} — 1} frac{1}{||y_i||_0(n_text{labels} — ||y_i||_0)}
left|left{(k, l): hat{f}_{ik} leq hat{f}_{il}, y_{ik} = 1, y_{il} = 0 right}right|]

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

[sum_{r=1}^{min(K, M)}frac{y_{f(r)}}{log(1 + r)}]

and the NDCG score is the DCG score divided by the DCG score obtained for
(y).

3.3.4.1. R² score, the coefficient of determination¶

The r2_score function computes the coefficient of
determination,
usually denoted as (R^2).

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, (R^2) 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 (average) value of y, disregarding the input features,
would get an (R^2) score of 0.0.

Note: when the prediction residuals have zero mean, the (R^2) score and
the Explained variance score are identical.

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 (R^2) is defined as:

[R^2(y, hat{y}) = 1 — frac{sum_{i=1}^{n} (y_i — hat{y}_i)^2}{sum_{i=1}^{n} (y_i — bar{y})^2}]

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 (R^2) without correcting for
bias in sample variance of y.

In the particular case where the true target is constant, the (R^2) score is
not finite: it is either NaN (perfect predictions) or -Inf (imperfect
predictions). Such non-finite scores may prevent correct model optimization
such as grid-search cross-validation to be performed correctly. For this reason
the default behaviour of r2_score is to replace them with 1.0 (perfect
predictions) or 0.0 (imperfect predictions). If force_finite
is set to False, this score falls back on the original (R^2) definition.

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...
>>> y_true = [-2, -2, -2]
>>> y_pred = [-2, -2, -2]
>>> r2_score(y_true, y_pred)
1.0
>>> r2_score(y_true, y_pred, force_finite=False)
nan
>>> y_true = [-2, -2, -2]
>>> y_pred = [-2, -2, -2 + 1e-8]
>>> r2_score(y_true, y_pred)
0.0
>>> r2_score(y_true, y_pred, force_finite=False)
-inf

3.3.4.2. 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

[text{MAE}(y, hat{y}) = frac{1}{n_{text{samples}}} sum_{i=0}^{n_{text{samples}}-1} left| y_i — hat{y}_i right|.]

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...

3.3.4.3. 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

[text{MSE}(y, hat{y}) = frac{1}{n_text{samples}} sum_{i=0}^{n_text{samples} — 1} (y_i — hat{y}_i)^2.]

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...

3.3.4.4. 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

[text{MSLE}(y, hat{y}) = frac{1}{n_text{samples}} sum_{i=0}^{n_text{samples} — 1} (log_e (1 + y_i) — log_e (1 + hat{y}_i) )^2.]

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...

3.3.4.5. 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

[text{MAPE}(y, hat{y}) = frac{1}{n_{text{samples}}} sum_{i=0}^{n_{text{samples}}-1} frac{{}left| y_i — hat{y}_i right|}{max(epsilon, left| y_i right|)}]

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.

3.3.4.6. Median absolute error¶

The median_absolute_error is particularly interesting because it is
robust to outliers. The loss is calculated by taking the median of all absolute
differences between the target and the prediction.

If (hat{y}_i) is the predicted value of the (i)-th sample
and (y_i) is the corresponding true value, then the median absolute error
(MedAE) estimated over (n_{text{samples}}) is defined as

[text{MedAE}(y, hat{y}) = text{median}(mid y_1 — hat{y}_1 mid, ldots, mid y_n — hat{y}_n mid).]

The median_absolute_error does not support multioutput.

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

>>> from sklearn.metrics import median_absolute_error
>>> y_true = [3, -0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> median_absolute_error(y_true, y_pred)
0.5

3.3.4.7. 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

[text{Max Error}(y, hat{y}) = max(| y_i — hat{y}_i |)]

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.

3.3.4.8. 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:

[explained_{}variance(y, hat{y}) = 1 — frac{Var{ y — hat{y}}}{Var{y}}]

The best possible score is 1.0, lower values are worse.

In the particular case where the true target is constant, the Explained
Variance score is not finite: it is either NaN (perfect predictions) or
-Inf (imperfect predictions). Such non-finite scores may prevent correct
model optimization such as grid-search cross-validation to be performed
correctly. For this reason the default behaviour of
explained_variance_score is to replace them with 1.0 (perfect
predictions) or 0.0 (imperfect predictions). You can set the force_finite
parameter to False to prevent this fix from happening and fallback on the
original Explained Variance score.

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...
>>> y_true = [-2, -2, -2]
>>> y_pred = [-2, -2, -2]
>>> explained_variance_score(y_true, y_pred)
1.0
>>> explained_variance_score(y_true, y_pred, force_finite=False)
nan
>>> y_true = [-2, -2, -2]
>>> y_pred = [-2, -2, -2 + 1e-8]
>>> explained_variance_score(y_true, y_pred)
0.0
>>> explained_variance_score(y_true, y_pred, force_finite=False)
-inf

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

[begin{split}text{D}(y, hat{y}) = frac{1}{n_text{samples}}
sum_{i=0}^{n_text{samples} — 1}
begin{cases}
(y_i-hat{y}_i)^2, & text{for }p=0text{ (Normal)}\
2(y_i log(y_i/hat{y}_i) + hat{y}_i — y_i), & text{for }p=1text{ (Poisson)}\
2(log(hat{y}_i/y_i) + y_i/hat{y}_i — 1), & text{for }p=2text{ (Gamma)}\
2left(frac{max(y_i,0)^{2-p}}{(1-p)(2-p)}-
frac{y_i,hat{y}_i^{1-p}}{1-p}+frac{hat{y}_i^{2-p}}{2-p}right),
& text{otherwise}
end{cases}end{split}]

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 1.5 and 150 that are both
50% larger than their corresponding true value.

The mean squared error (power=0) is very sensitive to the
prediction difference of the second point,:

>>> from sklearn.metrics import mean_tweedie_deviance
>>> mean_tweedie_deviance([1.0], [1.5], power=0)
0.25
>>> mean_tweedie_deviance([100.], [150.], power=0)
2500.0

If we increase power to 1,:

>>> mean_tweedie_deviance([1.0], [1.5], power=1)
0.18...
>>> mean_tweedie_deviance([100.], [150.], power=1)
18.9...

the difference in errors decreases. Finally, by setting, power=2:

>>> mean_tweedie_deviance([1.0], [1.5], power=2)
0.14...
>>> mean_tweedie_deviance([100.], [150.], power=2)
0.14...

we would get identical errors. The deviance when power=2 is thus only
sensitive to relative errors.

3.3.4.10. Pinball loss¶

The mean_pinball_loss function is used to evaluate the predictive
performance of quantile regression models.

[text{pinball}(y, hat{y}) = frac{1}{n_{text{samples}}} sum_{i=0}^{n_{text{samples}}-1} alpha max(y_i — hat{y}_i, 0) + (1 — alpha) max(hat{y}_i — y_i, 0)]

The value of pinball loss is equivalent to half of mean_absolute_error when the quantile
parameter alpha is set to 0.5.

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

>>> from sklearn.metrics import mean_pinball_loss
>>> y_true = [1, 2, 3]
>>> mean_pinball_loss(y_true, [0, 2, 3], alpha=0.1)
0.03...
>>> mean_pinball_loss(y_true, [1, 2, 4], alpha=0.1)
0.3...
>>> mean_pinball_loss(y_true, [0, 2, 3], alpha=0.9)
0.3...
>>> mean_pinball_loss(y_true, [1, 2, 4], alpha=0.9)
0.03...
>>> mean_pinball_loss(y_true, y_true, alpha=0.1)
0.0
>>> mean_pinball_loss(y_true, y_true, alpha=0.9)
0.0

It is possible to build a scorer object with a specific choice of alpha:

>>> from sklearn.metrics import make_scorer
>>> mean_pinball_loss_95p = make_scorer(mean_pinball_loss, alpha=0.95)

Such a scorer can be used to evaluate the generalization performance of a
quantile regressor via cross-validation:

>>> from sklearn.datasets import make_regression
>>> from sklearn.model_selection import cross_val_score
>>> from sklearn.ensemble import GradientBoostingRegressor
>>>
>>> X, y = make_regression(n_samples=100, random_state=0)
>>> estimator = GradientBoostingRegressor(
...     loss="quantile",
...     alpha=0.95,
...     random_state=0,
... )
>>> cross_val_score(estimator, X, y, cv=5, scoring=mean_pinball_loss_95p)
array([13.6..., 9.7..., 23.3..., 9.5..., 10.4...])

It is also possible to build scorer objects for hyper-parameter tuning. The
sign of the loss must be switched to ensure that greater means better as
explained in the example linked below.

3.3.4.11. D² score¶

The D² score computes the fraction of deviance explained.
It is a generalization of R², where the squared error is generalized and replaced
by a deviance of choice (text{dev}(y, hat{y}))
(e.g., Tweedie, pinball or mean absolute error). D² is a form of a skill score.
It is calculated as

[D^2(y, hat{y}) = 1 — frac{text{dev}(y, hat{y})}{text{dev}(y, y_{text{null}})} ,.]

Where (y_{text{null}}) is the optimal prediction of an intercept-only model
(e.g., the mean of y_true for the Tweedie case, the median for absolute
error and the alpha-quantile for pinball loss).

Like R², the best possible score is 1.0 and it can be negative (because the
model can be arbitrarily worse). A constant model that always predicts
(y_{text{null}}), disregarding the input features, would get a D² score
of 0.0.

>>> from sklearn.metrics import d2_tweedie_score, make_scorer
>>> d2_tweedie_score_15 = make_scorer(d2_tweedie_score, power=1.5)

>>> from sklearn.metrics import d2_pinball_score, make_scorer
>>> d2_pinball_score_08 = make_scorer(d2_pinball_score, alpha=0.8)

>>> from sklearn.metrics import d2_absolute_error_score
>>> y_true = [3, -0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> d2_absolute_error_score(y_true, y_pred)
0.764...
>>> y_true = [1, 2, 3]
>>> y_pred = [1, 2, 3]
>>> d2_absolute_error_score(y_true, y_pred)
1.0
>>> y_true = [1, 2, 3]
>>> y_pred = [2, 2, 2]
>>> d2_absolute_error_score(y_true, y_pred)
0.0

3.3.4.12. Visual evaluation of regression models¶

Among methods to assess the quality of regression models, scikit-learn provides
the PredictionErrorDisplay class. It allows to
visually inspect the prediction errors of a model in two different manners.

The plot on the left shows the actual values vs predicted values. For a
noise-free regression task aiming to predict the (conditional) expectation of
y, a perfect regression model would display data points on the diagonal
defined by predicted equal to actual values. The further away from this optimal
line, the larger the error of the model. In a more realistic setting with
irreducible noise, that is, when not all the variations of y can be explained
by features in X, then the best model would lead to a cloud of points densely
arranged around the diagonal.