.. currentmodule:: sklearn .. _model_evaluation: ======================================================== Model evaluation: quantifying the quality of predictions ======================================================== There are 3 different approaches to evaluate the quality of predictions of a model: * **Estimator score method**: Estimators have a ``score`` method providing a default evaluation criterion for the problem they are designed to solve. This is not discussed on this page, but in each estimator's documentation. * **Scoring parameter**: Model-evaluation tools using :ref:`cross-validation ` (such as :func:`cross_validation.cross_val_score` and :class:`grid_search.GridSearchCV`) rely on an internal *scoring* strategy. This is discussed on section :ref:`scoring_parameter`. * **Metric functions**: The :mod:`metrics` module implements functions assessing prediction errors for specific purposes. This is discussed in the section :ref:`prediction_error_metrics`. Finally, :ref:`dummy_estimators` are useful to get a baseline value of those metrics for random predictions. .. seealso:: For "pairwise" metrics, between *samples* and not estimators or predictions, see the :ref:`metrics` section. .. _scoring_parameter: The ``scoring`` parameter: defining model evaluation rules ========================================================== Model selection and evaluation using tools, such as :class:`grid_search.GridSearchCV` and :func:`cross_validation.cross_val_score`, take a ``scoring`` parameter that controls what metric they apply to estimators evaluated. Common cases: predefined values -------------------------------- For the most common usecases, you can simply provide a string as the ``scoring`` parameter. Possible values are: ====================== ================================================= Scoring Function ====================== ================================================= **Classification** 'accuracy' :func:`sklearn.metrics.accuracy_score` 'average_precision' :func:`sklearn.metrics.average_precision_score` 'f1' :func:`sklearn.metrics.f1_score` 'precision' :func:`sklearn.metrics.precision_score` 'recall' :func:`sklearn.metrics.recall_score` 'roc_auc' :func:`sklearn.metrics.roc_auc_score` **Clustering** 'adjusted_rand_score' :func:`sklearn.metrics.adjusted_rand_score` **Regression** 'mean_absolute_error' :func:`sklearn.metrics.mean_absolute_error` 'mean_squared_error' :func:`sklearn.metrics.mean_squared_error` 'r2' :func:`sklearn.metrics.r2_score` ====================== ================================================= Setting the ``scoring`` parameter to a wrong value should give you a list of acceptable values:: >>> from sklearn import svm, cross_validation, datasets >>> iris = datasets.load_iris() >>> X, y = iris.data, iris.target >>> model = svm.SVC() >>> cross_validation.cross_val_score(model, X, y, scoring='wrong_choice') Traceback (most recent call last): ValueError: 'wrong_choice' is not a valid scoring value. Valid options are ['accuracy', 'adjusted_rand_score', 'average_precision', 'f1', 'log_loss', 'mean_absolute_error', 'mean_squared_error', 'precision', 'r2', 'recall', 'roc_auc'] .. note:: The corresponding scorer objects are stored in the dictionary ``sklearn.metrics.SCORERS``. The above choices correspond to error-metric functions that can be applied to predicted values. These are detailed below, in the next sections. .. currentmodule:: sklearn.metrics .. _scoring: Defining your scoring strategy from score functions ----------------------------------------------------- The scoring parameter can be a callable that takes model predictions and ground truth. However, if you want to use a scoring function that takes additional parameters, such as :func:`fbeta_score`, you need to generate an appropriate scoring object. The simplest way to generate a callable object for scoring is by using :func:`make_scorer`. That function converts score functions (discussed below in :ref:`prediction_error_metrics`) into callables that can be used for model evaluation. One typical use case is to wrap an existing scoring function from the library with non default value for its parameters such as the ``beta`` parameter for the :func:`fbeta_score` function:: >>> from sklearn.metrics import fbeta_score, make_scorer >>> ftwo_scorer = make_scorer(fbeta_score, beta=2) >>> from sklearn.grid_search import GridSearchCV >>> from sklearn.svm import LinearSVC >>> grid = GridSearchCV(LinearSVC(), param_grid={'C': [1, 10]}, scoring=ftwo_scorer) The second use case is to build a completely new and custom scorer object from a simple python function:: >>> def my_custom_loss_func(ground_truth, predictions): ... diff = np.abs(ground_truth - predictions).max() ... return np.log(1 + diff) ... >>> my_custom_scorer = make_scorer(my_custom_loss_func, greater_is_better=False) >>> grid = GridSearchCV(LinearSVC(), param_grid={'C': [1, 10]}, scoring=my_custom_scorer) :func:`make_scorer` takes as parameters: * the function you want to use * whether it is a score (``greater_is_better=True``) or a loss (``greater_is_better=False``), * whether the function you provided takes predictions as input (``needs_threshold=False``) or needs confidence scores \ (``needs_threshold=True``) * any additional parameters, such as ``beta`` in an :func:`f1_score`. Implementing your own scoring object ------------------------------------ You can generate even more flexible model scores by constructing your own scoring object from scratch, without using the :func:`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 quality of ``estimator``'s predictions on ``X`` which reference to ``y``. Again, higher numbers are better. .. _prediction_error_metrics: Function for prediction-error metrics ====================================== The module :mod:`sklearn.metric` 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). .. _classification_metrics: Classification metrics ----------------------- .. currentmodule:: sklearn.metrics The :mod:`sklearn.metrics` implements several losses, scores and utility functions to measure classification performance. Some metrics might require probability estimates of the positive class, confidence values or binary decisions values. Some of these are restricted to the binary classification case: .. autosummary:: :template: function.rst hinge_loss matthews_corrcoef precision_recall_curve roc_curve Others also work in the multiclass case: .. autosummary:: :template: function.rst confusion_matrix And some also work in the multilabel case: .. autosummary:: :template: function.rst accuracy_score classification_report f1_score fbeta_score hamming_loss jaccard_similarity_score log_loss precision_recall_fscore_support precision_score recall_score zero_one_loss And some work with binary and multilabel indicator format: .. autosummary:: :template: function.rst average_precision_score roc_auc_score In the following sub-sections, we will describe each of those functions. Accuracy score .............. The :func:`accuracy_score` function computes the `accuracy `_, the fraction (default) or the number 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 :math:`\hat{y}_i` is the predicted value of the :math:`i`-th sample and :math:`y_i` is the corresponding true value, then the fraction of correct predictions over :math:`n_\text{samples}` is defined as .. math:: \texttt{accuracy}(y, \hat{y}) = \frac{1}{n_\text{samples}} \sum_{i=0}^{n_\text{samples}-1} 1(\hat{y}_i = y_i) where :math:`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([[0.0, 1.0], [1.0, 1.0]]), np.ones((2, 2))) 0.5 .. topic:: Example: * See :ref:`example_plot_permutation_test_for_classification.py` for an example of accuracy score usage using permutations of the dataset. Confusion matrix ................ The :func:`confusion_matrix` function computes the `confusion matrix `_ to evaluate the accuracy on a classification problem. By definition, a confusion matrix :math:`C` is such that :math:`C_{i, j}` is equal to the number of observations known to be in group :math:`i` but predicted to be in group :math:`j`. Here an example of such confusion matrix:: >>> from sklearn.metrics import confusion_matrix >>> y_true = [2, 0, 2, 2, 0, 1] >>> y_pred = [0, 0, 2, 2, 0, 2] >>> confusion_matrix(y_true, y_pred) array([[2, 0, 0], [0, 0, 1], [1, 0, 2]]) Here a visual representation of such confusion matrix (this figure comes from the :ref:`example_plot_confusion_matrix.py` example): .. image:: ../auto_examples/images/plot_confusion_matrix_001.png :target: ../auto_examples/plot_confusion_matrix.html :scale: 75 :align: center .. topic:: Example: * See :ref:`example_plot_confusion_matrix.py` for an example of confusion matrix usage to evaluate the quality of the output of a classifier. * See :ref:`example_plot_digits_classification.py` for an example of confusion matrix usage in the classification of hand-written digits. * See :ref:`example_document_classification_20newsgroups.py` for an example of confusion matrix usage in the classification of text documents. Classification report ...................... The :func:`classification_report` function builds a text report showing the main classification metrics. Here 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, 2, 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 1.00 1.00 2 avg / total 0.67 0.80 0.72 5 .. topic:: Example: * See :ref:`example_plot_digits_classification.py` for an example of classification report usage in the classification of the hand-written digits. * See :ref:`example_document_classification_20newsgroups.py` for an example of classification report usage in the classification of text documents. * See :ref:`example_grid_search_digits.py` for an example of classification report usage in parameter estimation using grid search with a nested cross-validation. Hamming loss ............. The :func:`hamming_loss` computes the average Hamming loss or `Hamming distance `_ between two sets of samples. If :math:`\hat{y}_j` is the predicted value for the :math:`j`-th labels of a given sample, :math:`y_j` is the corresponding true value and :math:`n_\text{labels}` is the number of class or labels, then the Hamming loss :math:`L_{Hamming}` between two samples is defined as: .. math:: L_{Hamming}(y, \hat{y}) = \frac{1}{n_\text{labels}} \sum_{j=0}^{n_\text{labels} - 1} 1(\hat{y}_j \not= y_j) where :math:`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([[0.0, 1.0], [1.0, 1.0]]), np.zeros((2, 2))) 0.75 .. note:: In multiclass classification, the Hamming loss correspond to the Hamming distance between ``y_true`` and ``y_pred`` which is equivalent to the :ref:`zero_one_loss` function. In multilabel classification, the Hamming loss is different from the zero-one loss. The zero-one loss penalizes any predictions that don't exactly match the true required set of labels, while Hamming loss will penalize the individual labels. So, predicting a subset or superset of the true labels will give a Hamming loss strictly between zero and one. The Hamming loss is upperbounded by the zero-one loss. When normalized over samples, the Hamming loss is always between zero and one. Jaccard similarity coefficient score ..................................... The :func:`jaccard_similarity_score` function computes the average (default) or sum of `Jaccard similarity coefficients `_, also called Jaccard index, between pairs of label sets. The Jaccard similarity coefficient of the :math:`i`-th samples with a ground truth label set :math:`y_i` and a predicted label set :math:`\hat{y}_i` is defined as .. math:: J(y_i, \hat{y}_i) = \frac{|y_i \cap \hat{y}_i|}{|y_i \cup \hat{y}_i|}. In binary and multiclass classification, the Jaccard similarity coefficient score is equal to the classification accuracy. :: >>> import numpy as np >>> from sklearn.metrics import jaccard_similarity_score >>> y_pred = [0, 2, 1, 3] >>> y_true = [0, 1, 2, 3] >>> jaccard_similarity_score(y_true, y_pred) 0.5 >>> jaccard_similarity_score(y_true, y_pred, normalize=False) 2 In the multilabel case with binary label indicators: :: >>> jaccard_similarity_score(np.array([[0.0, 1.0], [1.0, 1.0]]), np.ones((2, 2))) 0.75 .. _precision_recall_f_measure_metrics: Precision, recall and F-measures ................................. The `precision `_ is intuitively the ability of the classifier not to label as positive a sample that is negative. The `recall `_ is intuitively the ability of the classifier to find all the positive samples. The `F-measure `_ (:math:`F_\beta` and :math:`F_1` measures) can be interpreted as a weighted harmonic mean of the precision and recall. A :math:`F_\beta` measure reaches its best value at 1 and worst score at 0. With :math:`\beta = 1`, the :math:`F_\beta` measure leads to the :math:`F_1` measure, wheres the recall and the precision are equally important. The :func:`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 :func:`average_precision_score` function computes the average precision (AP) from prediction scores. This score corresponds to the area under the precision-recall curve. Several functions allow you to analyze the precision, recall and F-measures score: .. autosummary:: :template: function.rst average_precision_score f1_score fbeta_score precision_recall_curve precision_recall_fscore_support precision_score recall_score Note that the :func:`precision_recall_curve` function is restricted to the binary case. The :func:`average_precision_score` function works only in binary classification and multilabel indicator format. .. topic:: Examples: * See :ref:`example_document_classification_20newsgroups.py` for an example of :func:`f1_score` usage with classification of text documents. * See :ref:`example_grid_search_digits.py` for an example of :func:`precision_score` and :func:`recall_score` usage in parameter estimation using grid search with a nested cross-validation. * See :ref:`example_plot_precision_recall.py` for an example of precision-Recall metric to evaluate the quality of the output of a classifier with :func:`precision_recall_curve`. * See :ref:`example_linear_model_plot_sparse_recovery.py` for an example of :func:`precision_recall_curve` usage in feature selection for sparse linear models. Binary classification ^^^^^^^^^^^^^^^^^^^^^ In a binary classification task, the terms ''positive'' and ''negative'' refer to the classifier's prediction and the terms ''true'' and ''false'' refer to whether that prediction corresponds to the external judgment (sometimes known as the ''observation''). Given these definitions, we can formulate the following table: +-------------------+------------------------------------------------+ | | Actual class (observation) | +-------------------+---------------------+--------------------------+ | Predicted class | tp (true positive) | fp (false positive) | | (expectation) | Correct result | Unexpected result | | +---------------------+--------------------------+ | | fn (false negative) | tn (true negative) | | | Missing result | Correct absence of result| +-------------------+---------------------+--------------------------+ In this context, we can define the notions of precision, recall and F-measure: .. math:: \text{precision} = \frac{tp}{tp + fp}, .. math:: \text{recall} = \frac{tp}{tp + fn}, .. math:: F_\beta = (1 + \beta^2) \frac{\text{precision} \times \text{recall}}{\beta^2 \text{precision} + \text{recall}}. Here some small examples in binary classification:: >>> from sklearn import metrics >>> y_pred = [0, 1, 0, 0] >>> y_true = [0, 1, 0, 1] >>> metrics.precision_score(y_true, y_pred) 1.0 >>> metrics.recall_score(y_true, y_pred) 0.5 >>> metrics.f1_score(y_true, y_pred) # doctest: +ELLIPSIS 0.66... >>> metrics.fbeta_score(y_true, y_pred, beta=0.5) # doctest: +ELLIPSIS 0.83... >>> metrics.fbeta_score(y_true, y_pred, beta=1) # doctest: +ELLIPSIS 0.66... >>> metrics.fbeta_score(y_true, y_pred, beta=2) # doctest: +ELLIPSIS 0.55... >>> metrics.precision_recall_fscore_support(y_true, y_pred, beta=0.5) # doctest: +ELLIPSIS (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 # doctest: +ELLIPSIS 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) # doctest: +ELLIPSIS 0.79... Multiclass and multilabel classification ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In multiclass and multilabel classification task, the notions of precision, recall and F-measures can be applied to each label independently. There are a few ways to combine results across labels, specified by the ``average`` argument to the :func:`average_precision_score` (multilabel only), :func:`f1_score`, :func:`fbeta_score`, :func:`precision_recall_fscore_support`, :func:`precision_score` and :func:`recall_score` functions: * ``"micro"``: calculate metrics globally by counting the total true positives, false negatives and false positives. Except in the multi-label case this implies that precision, recall and :math:`F` are equal. * ``"samples"``: calculate metrics for each sample, comparing sets of labels assigned to each, and find the mean across all samples. This is only meaningful and available in the multilabel case. * ``"macro"``: calculate metrics for each label, and find their mean. This does not take label imbalance into account. * ``"weighted"``: calculate metrics for each label, and find their average weighted by the number of occurrences of the label in the true data. This alters ``"macro"`` to account for label imbalance; it may produce an F-score that is not between precision and recall. * ``None``: calculate metrics for each label and do not average them. To make this more explicit, consider the following notation: * :math:`y` the set of *predicted* :math:`(sample, label)` pairs * :math:`\hat{y}` the set of *true* :math:`(sample, label)` pairs * :math:`L` the set of labels * :math:`S` the set of samples * :math:`y_s` the subset of :math:`y` with sample :math:`s`, i.e. :math:`y_s := \left\{(s', l) \in y | s' = s\right\}` * :math:`y_l` the subset of :math:`y` with label :math:`l` * similarly, :math:`\hat{y}_s` and :math:`\hat{y}_l` are subsets of :math:`\hat{y}` * :math:`P(A, B) := \frac{\left| A \cap B \right|}{\left|A\right|}` * :math:`R(A, B) := \frac{\left| A \cap B \right|}{\left|B\right|}` (Conventions vary on handling :math:`B = \emptyset`; this implementation uses :math:`R(A, B):=0`, and similar for :math:`P`.) * :math:`F_\beta(A, B) := \left(1 + \beta^2\right) \frac{P(A, B) \times R(A, B)}{\beta^2 P(A, B) + R(A, B)}` Then the metrics are defined as: +---------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------+ |``average`` | Precision | Recall | F\_beta | +===============+==================================================================================================================+==================================================================================================================+======================================================================================================================+ |``"micro"`` | :math:`P(y, \hat{y})` | :math:`R(y, \hat{y})` | :math:`F_\beta(y, \hat{y})` | +---------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------+ |``"samples"`` | :math:`\frac{1}{\left|S\right|} \sum_{s \in S} P(y_s, \hat{y}_s)` | :math:`\frac{1}{\left|S\right|} \sum_{s \in S} R(y_s, \hat{y}_s)` | :math:`\frac{1}{\left|S\right|} \sum_{s \in S} F_\beta(y_s, \hat{y}_s)` | +---------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------+ |``"macro"`` | :math:`\frac{1}{\left|L\right|} \sum_{l \in L} P(y_l, \hat{y}_l)` | :math:`\frac{1}{\left|L\right|} \sum_{l \in L} R(y_l, \hat{y}_l)` | :math:`\frac{1}{\left|L\right|} \sum_{l \in L} F_\beta(y_l, \hat{y}_l)` | +---------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------+ |``"weighted"`` | :math:`\frac{1}{\sum_{l \in L} \left|\hat{y}_l\right|} \sum_{l \in L} \left|\hat{y}_l\right| P(y_l, \hat{y}_l)` | :math:`\frac{1}{\sum_{l \in L} \left|\hat{y}_l\right|} \sum_{l \in L} \left|\hat{y}_l\right| R(y_l, \hat{y}_l)` | :math:`\frac{1}{\sum_{l \in L} \left|\hat{y}_l\right|} \sum_{l \in L} \left|\hat{y}_l\right| F_\beta(y_l, \hat{y}_l)`| +---------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------+ |``None`` | :math:`\langle P(y_l, \hat{y}_l) | l \in L \rangle` | :math:`\langle R(y_l, \hat{y}_l) | l \in L \rangle` | :math:`\langle F_\beta(y_l, \hat{y}_l) | l \in L \rangle` | +---------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------+ >>> from sklearn import metrics >>> y_true = [0, 1, 2, 0, 1, 2] >>> y_pred = [0, 2, 1, 0, 0, 1] >>> metrics.precision_score(y_true, y_pred, average='macro') # doctest: +ELLIPSIS 0.22... >>> metrics.recall_score(y_true, y_pred, average='micro') ... # doctest: +ELLIPSIS 0.33... >>> metrics.f1_score(y_true, y_pred, average='weighted') # doctest: +ELLIPSIS 0.26... >>> metrics.fbeta_score(y_true, y_pred, average='macro', beta=0.5) # doctest: +ELLIPSIS 0.23... >>> metrics.precision_recall_fscore_support(y_true, y_pred, beta=0.5, average=None) ... # doctest: +ELLIPSIS (array([ 0.66..., 0. , 0. ]), array([ 1., 0., 0.]), array([ 0.71..., 0. , 0. ]), array([2, 2, 2]...)) Hinge loss ........... The :func:`hinge_loss` function computes the average `hinge loss function `_. The hinge loss is used in maximal margin classification as support vector machines. If the labels are encoded with +1 and -1, :math:`y`: is the true value and :math:`w` is the predicted decisions as output by ``decision_function``, then the hinge loss is defined as: .. math:: L_\text{Hinge}(y, w) = \max\left\{1 - wy, 0\right\} = \left|1 - wy\right|_+ Here a small example demonstrating the use of the :func:`hinge_loss` function with a svm classifier:: >>> from sklearn import svm >>> from sklearn.metrics import hinge_loss >>> X = [[0], [1]] >>> y = [-1, 1] >>> est = svm.LinearSVC(random_state=0) >>> est.fit(X, y) LinearSVC(C=1.0, class_weight=None, dual=True, fit_intercept=True, intercept_scaling=1, loss='l2', multi_class='ovr', penalty='l2', random_state=0, tol=0.0001, verbose=0) >>> pred_decision = est.decision_function([[-2], [3], [0.5]]) >>> pred_decision # doctest: +ELLIPSIS array([-2.18..., 2.36..., 0.09...]) >>> hinge_loss([-1, 1, 1], pred_decision) # doctest: +ELLIPSIS 0.3... Log loss ........ The log loss, also called logistic regression loss or cross-entropy loss, is a loss function defined on probability estimates. It is commonly used in (multinomial) logistic regression and neural networks, as well as some variants of expectation-maximization, and can be used to evaluate the probability outputs (``predict_proba``) of a classifier, rather than its discrete predictions. For binary classification with a true label :math:`y \in \{0,1\}` and a probability estimate :math:`p = \operatorname{Pr}(y = 1)`, the log loss per sample is the negative log-likelihood of the classifier given the true label: .. math:: 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 :math:`Y`, i.e. :math:`y_{i,k} = 1` if sample :math:`i` has label :math:`k` taken from a set of :math:`K` labels. Let :math:`P` be a matrix of probability estimates, with :math:`p_{i,k} = \operatorname{Pr}(t_{i,k} = 1)`. Then the log loss of the whole set is .. math:: 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, :math:`p_{i,0} = 1 - p_{i,1}` and :math:`y_{i,0} = 1 - y_{i,1}`, so expanding the inner sum over :math:`y_{i,k} \in \{0,1\}` gives the binary log loss. The function :func:`log_loss` 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) # doctest: +ELLIPSIS 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. Matthews correlation coefficient ................................. The :func:`matthews_corrcoef` function computes the Matthew's correlation coefficient (MCC) for binary classes (quoting the `Wikipedia article on the Matthew's correlation coefficient `_): "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." If :math:`tp`, :math:`tn`, :math:`fp` and :math:`fn` are respectively the number of true positives, true negatives, false positives ans false negatives, the MCC coefficient is defined as .. math:: MCC = \frac{tp \times tn - fp \times fn}{\sqrt{(tp + fp)(tp + fn)(tn + fp)(tn + fn)}}. Here a small example illustrating the usage of the :func:`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) # doctest: +ELLIPSIS -0.33... .. _roc_metrics: Receiver operating characteristic (ROC) ....................................... The function :func:`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 a small example of how to use the :func:`roc_curve` function:: >>> import numpy as np >>> from sklearn.metrics import roc_curve >>> y = np.array([1, 1, 2, 2]) >>> scores = np.array([0.1, 0.4, 0.35, 0.8]) >>> fpr, tpr, thresholds = roc_curve(y, scores, pos_label=2) >>> fpr array([ 0. , 0.5, 0.5, 1. ]) >>> tpr array([ 0.5, 0.5, 1. , 1. ]) >>> thresholds array([ 0.8 , 0.4 , 0.35, 0.1 ]) The following figure shows an example of such ROC curve. .. image:: ../auto_examples/images/plot_roc_001.png :target: ../auto_examples/plot_roc.html :scale: 75 :align: center The :func:`roc_auc_score` function computes the area under the receiver operating characteristic (ROC) curve, which is also denoted by AUC or AUROC. By computing the area under the roc curve, the curve information is summarized in one number. For more information see the `Wikipedia article on AUC `_. >>> import numpy as np >>> from sklearn.metrics import roc_auc_score >>> y_true = np.array([0, 0, 1, 1]) >>> y_scores = np.array([0.1, 0.4, 0.35, 0.8]) >>> roc_auc_score(y_true, y_scores) 0.75 In multi-label classification, the :func:`roc_auc_score` function is extended by averaging over the labels: * ``"micro"``: computes the area under the ROC curve globally obtained by considering each element of the label indicator matrix as a label. * ``"samples"``: computes the area under the ROC curve on each sample, comparing the set of labels and scores assigned to each, and find the mean across all samples. * ``"macro"``: computes the area under the ROC curve for each label, and find their mean. * ``"weighted"``: computes the area under the ROC curve for each label, and find their average weighted by the number of occurrences of the label in the true data. * ``None``: this returns an array of scores with scores with shape (n_classes,) instead of an aggregate scalar score. Compared to metrics such as the subset accuracy, the hamming loss or the F1 score, ROC AUC doesn't require to optimize a threshold for each label. The :func:`roc_auc_score` function can also be used in multi-class classification if predicted outputs have been binarized. .. image:: ../auto_examples/images/plot_roc_002.png :target: ../auto_examples/plot_roc.html :scale: 75 :align: center .. topic:: Examples: * See :ref:`example_plot_roc.py` for an example of receiver operating characteristic (ROC) metric to evaluate the quality of the output of a classifier. * See :ref:`example_plot_roc_crossval.py` for an example of receiver operating characteristic (ROC) metric to evaluate the quality of the output of a classifier using cross-validation. * See :ref:`example_applications_plot_species_distribution_modeling.py` for an example of receiver operating characteristic (ROC) metric to model species distribution. .. _zero_one_loss: Zero one loss .............. The :func:`zero_one_loss` function computes the sum or the average of the 0-1 classification loss (:math:`L_{0-1}`) over :math:`n_{\text{samples}}`. By defaults, the function normalizes over the sample. To get the sum of the :math:`L_{0-1}`, set ``normalize`` to ``False``. In multilabel classification, the :func:`zero_one_loss` function corresponds to the subset zero-one loss: the subset of labels must be correctly predict. If :math:`\hat{y}_i` is the predicted value of the :math:`i`-th sample and :math:`y_i` is the corresponding true value, then the 0-1 loss :math:`L_{0-1}` is defined as: .. math:: L_{0-1}(y_i, \hat{y}_i) = 1(\hat{y}_i \not= y_i) where :math:`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: :: >>> zero_one_loss(np.array([[0.0, 1.0], [1.0, 1.0]]), np.ones((2, 2))) 0.5 .. topic:: Example: * See :ref:`example_plot_rfe_with_cross_validation.py` for an example of the zero one loss usage to perform recursive feature elimination with cross-validation. .. _regression_metrics: Regression metrics ------------------- .. currentmodule:: sklearn.metrics The :mod:`sklearn.metrics` implements several losses, scores and utility functions to measure regression performance. Some of those have been enhanced to handle the multioutput case: :func:`mean_absolute_error`, :func:`mean_absolute_error` and :func:`r2_score`. Explained variance score ......................... The :func:`explained_variance_score` computes the `explained variance regression score `_. If :math:`\hat{y}` is the estimated target output and :math:`y` is the corresponding (correct) target output, then the explained variance is estimated as follow: .. math:: \texttt{explained\_{}variance}(y, \hat{y}) = 1 - \frac{Var\{ y - \hat{y}\}}{Var\{y\}} The best possible score is 1.0, lower values are worse. Here a small example of usage of the :func:`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) # doctest: +ELLIPSIS 0.957... Mean absolute error ................... The :func:`mean_absolute_error` function computes the `mean absolute error `_, which is a risk function corresponding to the expected value of the absolute error loss or :math:`l1`-norm loss. If :math:`\hat{y}_i` is the predicted value of the :math:`i`-th sample and :math:`y_i` is the corresponding true value, then the mean absolute error (MAE) estimated over :math:`n_{\text{samples}}` is defined as .. math:: \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 a small example of usage of the :func:`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 squared error ................... The :func:`mean_squared_error` function computes the `mean square error `_, which is a risk function corresponding to the expected value of the squared error loss or quadratic loss. If :math:`\hat{y}_i` is the predicted value of the :math:`i`-th sample and :math:`y_i` is the corresponding true value, then the mean squared error (MSE) estimated over :math:`n_{\text{samples}}` is defined as .. math:: \text{MSE}(y, \hat{y}) = \frac{1}{n_\text{samples}} \sum_{i=0}^{n_\text{samples} - 1} (y_i - \hat{y}_i)^2. Here a small example of usage of the :func:`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) # doctest: +ELLIPSIS 0.7083... .. topic:: Examples: * See :ref:`example_ensemble_plot_gradient_boosting_regression.py` for an example of mean squared error usage to evaluate gradient boosting regression. R² score, the coefficient of determination ........................................... The :func:`r2_score` function computes R², the `coefficient of determination `_. It provides a measure of how well future samples are likely to be predicted by the model. If :math:`\hat{y}_i` is the predicted value of the :math:`i`-th sample and :math:`y_i` is the corresponding true value, then the score R² estimated over :math:`n_{\text{samples}}` is defined as .. math:: R^2(y, \hat{y}) = 1 - \frac{\sum_{i=0}^{n_{\text{samples}} - 1} (y_i - \hat{y}_i)^2}{\sum_{i=0}^{n_\text{samples} - 1} (y_i - \bar{y})^2} where :math:`\bar{y} = \frac{1}{n_{\text{samples}}} \sum_{i=0}^{n_{\text{samples}} - 1} y_i`. Here a small example of usage of the :func:`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) # doctest: +ELLIPSIS 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) # doctest: +ELLIPSIS 0.938... .. topic:: Example: * See :ref:`example_linear_model_plot_lasso_and_elasticnet.py` for an example of R² score usage to evaluate Lasso and Elastic Net on sparse signals. Clustering metrics ====================== The :mod:`sklearn.metrics` implements several losses, scores and utility function for more information see the :ref:`clustering_evaluation` section. Biclustering metrics ==================== The :mod:`sklearn.metrics` module implements bicluster scoring metrics. For more information see the :ref:`biclustering_evaluation` section. .. currentmodule:: sklearn.metrics .. _clustering_metrics: Clustering metrics ------------------- The :mod:`sklearn.metrics` implements several losses, scores and utility functions. For more information see the :ref:`clustering_evaluation` section. .. _dummy_estimators: Dummy estimators ================= .. currentmodule:: sklearn.dummy When doing supervised learning, a simple sanity check consists in comparing one's estimator against simple rules of thumb. :class:`DummyClassifier` implements three such simple strategies for classification: - ``stratified`` generates randomly predictions by respecting the training set's class distribution, - ``most_frequent`` always predicts the most frequent label in the training set, - ``uniform`` generates predictions uniformly at random. - ``constant`` always predicts a constant label that is provided by the user. A major motivation of this method is F1-scoring when the positive class is in the minority. Note that with all these strategies, the ``predict`` method completely ignores the input data! To illustrate :class:`DummyClassifier`, first let's create an imbalanced dataset:: >>> from sklearn.datasets import load_iris >>> from sklearn.cross_validation import train_test_split >>> iris = load_iris() >>> X, y = iris.data, iris.target >>> y[y != 1] = -1 >>> X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0) Next, let's compare the accuracy of ``SVC`` and ``most_frequent``:: >>> from sklearn.dummy import DummyClassifier >>> from sklearn.svm import SVC >>> clf = SVC(kernel='linear', C=1).fit(X_train, y_train) >>> clf.score(X_test, y_test) # doctest: +ELLIPSIS 0.63... >>> clf = DummyClassifier(strategy='most_frequent',random_state=0) >>> clf.fit(X_train, y_train) DummyClassifier(constant=None, random_state=0, strategy='most_frequent') >>> clf.score(X_test, y_test) # doctest: +ELLIPSIS 0.57... We see that ``SVC`` doesn't do much better than a dummy classifier. Now, let's change the kernel:: >>> clf = SVC(kernel='rbf', C=1).fit(X_train, y_train) >>> clf.score(X_test, y_test) # doctest: +ELLIPSIS 0.97... We see that the accuracy was boosted to almost 100%. For a better estimate of the accuracy, it is recommended to use a cross validation strategy, if it is not too CPU costly. For more information see the :ref:`cross_validation` section. Moreover if you want to optimize over the parameter space, it is highly recommended to use an appropriate methodology see the :ref:`grid_search` section. More generally, when the accuracy of a classifier is too close to random classification, it probably means that something went wrong: features are not helpful, a hyper parameter is not correctly tuned, the classifier is suffering from class imbalance, etc... :class:`DummyRegressor` also implements three simple rules of thumb for regression: - ``mean`` always predicts the mean of the training targets. - ``median`` always predicts the median of the training targests. - ``constant`` always predicts a constant value that is provided by the user. In all these strategies, the ``predict`` method completely ignores the input data.