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<ul>
<li><a class="reference internal" href="#">3.3. Metrics and scoring: quantifying the quality of predictions</a><ul>
<li><a class="reference internal" href="#the-scoring-parameter-defining-model-evaluation-rules">3.3.1. The <code class="docutils literal notranslate"><span class="pre">scoring</span></code> parameter: defining model evaluation rules</a><ul>
<li><a class="reference internal" href="#common-cases-predefined-values">3.3.1.1. Common cases: predefined values</a></li>
<li><a class="reference internal" href="#defining-your-scoring-strategy-from-metric-functions">3.3.1.2. Defining your scoring strategy from metric functions</a></li>
<li><a class="reference internal" href="#implementing-your-own-scoring-object">3.3.1.3. Implementing your own scoring object</a></li>
<li><a class="reference internal" href="#using-multiple-metric-evaluation">3.3.1.4. Using multiple metric evaluation</a></li>
</ul>
</li>
<li><a class="reference internal" href="#classification-metrics">3.3.2. Classification metrics</a><ul>
<li><a class="reference internal" href="#from-binary-to-multiclass-and-multilabel">3.3.2.1. From binary to multiclass and multilabel</a></li>
<li><a class="reference internal" href="#accuracy-score">3.3.2.2. Accuracy score</a></li>
<li><a class="reference internal" href="#top-k-accuracy-score">3.3.2.3. Top-k accuracy score</a></li>
<li><a class="reference internal" href="#balanced-accuracy-score">3.3.2.4. Balanced accuracy score</a></li>
<li><a class="reference internal" href="#cohen-s-kappa">3.3.2.5. Cohen’s kappa</a></li>
<li><a class="reference internal" href="#confusion-matrix">3.3.2.6. Confusion matrix</a></li>
<li><a class="reference internal" href="#classification-report">3.3.2.7. Classification report</a></li>
<li><a class="reference internal" href="#hamming-loss">3.3.2.8. Hamming loss</a></li>
<li><a class="reference internal" href="#precision-recall-and-f-measures">3.3.2.9. Precision, recall and F-measures</a><ul>
<li><a class="reference internal" href="#binary-classification">3.3.2.9.1. Binary classification</a></li>
<li><a class="reference internal" href="#multiclass-and-multilabel-classification">3.3.2.9.2. Multiclass and multilabel classification</a></li>
</ul>
</li>
<li><a class="reference internal" href="#jaccard-similarity-coefficient-score">3.3.2.10. Jaccard similarity coefficient score</a></li>
<li><a class="reference internal" href="#hinge-loss">3.3.2.11. Hinge loss</a></li>
<li><a class="reference internal" href="#log-loss">3.3.2.12. Log loss</a></li>
<li><a class="reference internal" href="#matthews-correlation-coefficient">3.3.2.13. Matthews correlation coefficient</a></li>
<li><a class="reference internal" href="#multi-label-confusion-matrix">3.3.2.14. Multi-label confusion matrix</a></li>
<li><a class="reference internal" href="#receiver-operating-characteristic-roc">3.3.2.15. Receiver operating characteristic (ROC)</a><ul>
<li><a class="reference internal" href="#binary-case">3.3.2.15.1. Binary case</a></li>
<li><a class="reference internal" href="#multi-class-case">3.3.2.15.2. Multi-class case</a></li>
<li><a class="reference internal" href="#multi-label-case">3.3.2.15.3. Multi-label case</a></li>
</ul>
</li>
<li><a class="reference internal" href="#detection-error-tradeoff-det">3.3.2.16. Detection error tradeoff (DET)</a></li>
<li><a class="reference internal" href="#zero-one-loss">3.3.2.17. Zero one loss</a></li>
<li><a class="reference internal" href="#brier-score-loss">3.3.2.18. Brier score loss</a></li>
<li><a class="reference internal" href="#class-likelihood-ratios">3.3.2.19. Class likelihood ratios</a></li>
</ul>
</li>
<li><a class="reference internal" href="#multilabel-ranking-metrics">3.3.3. Multilabel ranking metrics</a><ul>
<li><a class="reference internal" href="#coverage-error">3.3.3.1. Coverage error</a></li>
<li><a class="reference internal" href="#label-ranking-average-precision">3.3.3.2. Label ranking average precision</a></li>
<li><a class="reference internal" href="#ranking-loss">3.3.3.3. Ranking loss</a></li>
<li><a class="reference internal" href="#normalized-discounted-cumulative-gain">3.3.3.4. Normalized Discounted Cumulative Gain</a></li>
</ul>
</li>
<li><a class="reference internal" href="#regression-metrics">3.3.4. Regression metrics</a><ul>
<li><a class="reference internal" href="#r2-score-the-coefficient-of-determination">3.3.4.1. R² score, the coefficient of determination</a></li>
<li><a class="reference internal" href="#mean-absolute-error">3.3.4.2. Mean absolute error</a></li>
<li><a class="reference internal" href="#mean-squared-error">3.3.4.3. Mean squared error</a></li>
<li><a class="reference internal" href="#mean-squared-logarithmic-error">3.3.4.4. Mean squared logarithmic error</a></li>
<li><a class="reference internal" href="#mean-absolute-percentage-error">3.3.4.5. Mean absolute percentage error</a></li>
<li><a class="reference internal" href="#median-absolute-error">3.3.4.6. Median absolute error</a></li>
<li><a class="reference internal" href="#max-error">3.3.4.7. Max error</a></li>
<li><a class="reference internal" href="#explained-variance-score">3.3.4.8. Explained variance score</a></li>
<li><a class="reference internal" href="#mean-poisson-gamma-and-tweedie-deviances">3.3.4.9. Mean Poisson, Gamma, and Tweedie deviances</a></li>
<li><a class="reference internal" href="#pinball-loss">3.3.4.10. Pinball loss</a></li>
<li><a class="reference internal" href="#d2-score">3.3.4.11. D² score</a><ul>
<li><a class="reference internal" href="#d2-tweedie-score">3.3.4.11.1. D² Tweedie score</a></li>
<li><a class="reference internal" href="#d2-pinball-score">3.3.4.11.2. D² pinball score</a></li>
<li><a class="reference internal" href="#d2-absolute-error-score">3.3.4.11.3. D² absolute error score</a></li>
</ul>
</li>
<li><a class="reference internal" href="#visual-evaluation-of-regression-models">3.3.4.12. Visual evaluation of regression models</a></li>
</ul>
</li>
<li><a class="reference internal" href="#clustering-metrics">3.3.5. Clustering metrics</a></li>
<li><a class="reference internal" href="#dummy-estimators">3.3.6. Dummy estimators</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<section id="metrics-and-scoring-quantifying-the-quality-of-predictions">
<span id="model-evaluation"></span><h1><span class="section-number">3.3. </span>Metrics and scoring: quantifying the quality of predictions<a class="headerlink" href="#metrics-and-scoring-quantifying-the-quality-of-predictions" title="Permalink to this heading">¶</a></h1>
<p>There are 3 different APIs for evaluating the quality of a model’s
predictions:</p>
<ul class="simple">
<li><p><strong>Estimator score method</strong>: Estimators have a <code class="docutils literal notranslate"><span class="pre">score</span></code> 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.</p></li>
<li><p><strong>Scoring parameter</strong>: Model-evaluation tools using
<a class="reference internal" href="cross_validation.html#cross-validation"><span class="std std-ref">cross-validation</span></a> (such as
<a class="reference internal" href="generated/sklearn.model_selection.cross_val_score.html#sklearn.model_selection.cross_val_score" title="sklearn.model_selection.cross_val_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">model_selection.cross_val_score</span></code></a> and
<a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">model_selection.GridSearchCV</span></code></a>) rely on an internal <em>scoring</em> strategy.
This is discussed in the section <a class="reference internal" href="#scoring-parameter"><span class="std std-ref">The scoring parameter: defining model evaluation rules</span></a>.</p></li>
<li><p><strong>Metric functions</strong>: The <a class="reference internal" href="classes.html#module-sklearn.metrics" title="sklearn.metrics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.metrics</span></code></a> module implements functions
assessing prediction error for specific purposes. These metrics are detailed
in sections on <a class="reference internal" href="#classification-metrics"><span class="std std-ref">Classification metrics</span></a>,
<a class="reference internal" href="#multilabel-ranking-metrics"><span class="std std-ref">Multilabel ranking metrics</span></a>, <a class="reference internal" href="#regression-metrics"><span class="std std-ref">Regression metrics</span></a> and
<a class="reference internal" href="#clustering-metrics"><span class="std std-ref">Clustering metrics</span></a>.</p></li>
</ul>
<p>Finally, <a class="reference internal" href="#dummy-estimators"><span class="std std-ref">Dummy estimators</span></a> are useful to get a baseline
value of those metrics for random predictions.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>For “pairwise” metrics, between <em>samples</em> and not estimators or
predictions, see the <a class="reference internal" href="metrics.html#metrics"><span class="std std-ref">Pairwise metrics, Affinities and Kernels</span></a> section.</p>
</div>
<section id="the-scoring-parameter-defining-model-evaluation-rules">
<span id="scoring-parameter"></span><h2><span class="section-number">3.3.1. </span>The <code class="docutils literal notranslate"><span class="pre">scoring</span></code> parameter: defining model evaluation rules<a class="headerlink" href="#the-scoring-parameter-defining-model-evaluation-rules" title="Permalink to this heading">¶</a></h2>
<p>Model selection and evaluation using tools, such as
<a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">model_selection.GridSearchCV</span></code></a> and
<a class="reference internal" href="generated/sklearn.model_selection.cross_val_score.html#sklearn.model_selection.cross_val_score" title="sklearn.model_selection.cross_val_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">model_selection.cross_val_score</span></code></a>, take a <code class="docutils literal notranslate"><span class="pre">scoring</span></code> parameter that
controls what metric they apply to the estimators evaluated.</p>
<section id="common-cases-predefined-values">
<h3><span class="section-number">3.3.1.1. </span>Common cases: predefined values<a class="headerlink" href="#common-cases-predefined-values" title="Permalink to this heading">¶</a></h3>
<p>For the most common use cases, you can designate a scorer object with the
<code class="docutils literal notranslate"><span class="pre">scoring</span></code> parameter; the table below shows all possible values.
All scorer objects follow the convention that <strong>higher return values are better
than lower return values</strong>. Thus metrics which measure the distance between
the model and the data, like <a class="reference internal" href="generated/sklearn.metrics.mean_squared_error.html#sklearn.metrics.mean_squared_error" title="sklearn.metrics.mean_squared_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_squared_error</span></code></a>, are
available as neg_mean_squared_error which return the negated value
of the metric.</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>Scoring</p></th>
<th class="head"><p>Function</p></th>
<th class="head"><p>Comment</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><strong>Classification</strong></p></td>
<td></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘accuracy’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.accuracy_score.html#sklearn.metrics.accuracy_score" title="sklearn.metrics.accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.accuracy_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘balanced_accuracy’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.balanced_accuracy_score.html#sklearn.metrics.balanced_accuracy_score" title="sklearn.metrics.balanced_accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.balanced_accuracy_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘top_k_accuracy’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.top_k_accuracy_score.html#sklearn.metrics.top_k_accuracy_score" title="sklearn.metrics.top_k_accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.top_k_accuracy_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘average_precision’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.average_precision_score.html#sklearn.metrics.average_precision_score" title="sklearn.metrics.average_precision_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.average_precision_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘neg_brier_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.brier_score_loss.html#sklearn.metrics.brier_score_loss" title="sklearn.metrics.brier_score_loss"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.brier_score_loss</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘f1’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.f1_score</span></code></a></p></td>
<td><p>for binary targets</p></td>
</tr>
<tr class="row-odd"><td><p>‘f1_micro’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.f1_score</span></code></a></p></td>
<td><p>micro-averaged</p></td>
</tr>
<tr class="row-even"><td><p>‘f1_macro’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.f1_score</span></code></a></p></td>
<td><p>macro-averaged</p></td>
</tr>
<tr class="row-odd"><td><p>‘f1_weighted’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.f1_score</span></code></a></p></td>
<td><p>weighted average</p></td>
</tr>
<tr class="row-even"><td><p>‘f1_samples’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.f1_score</span></code></a></p></td>
<td><p>by multilabel sample</p></td>
</tr>
<tr class="row-odd"><td><p>‘neg_log_loss’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.log_loss.html#sklearn.metrics.log_loss" title="sklearn.metrics.log_loss"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.log_loss</span></code></a></p></td>
<td><p>requires <code class="docutils literal notranslate"><span class="pre">predict_proba</span></code> support</p></td>
</tr>
<tr class="row-even"><td><p>‘precision’ etc.</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.precision_score.html#sklearn.metrics.precision_score" title="sklearn.metrics.precision_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.precision_score</span></code></a></p></td>
<td><p>suffixes apply as with ‘f1’</p></td>
</tr>
<tr class="row-odd"><td><p>‘recall’ etc.</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.recall_score.html#sklearn.metrics.recall_score" title="sklearn.metrics.recall_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.recall_score</span></code></a></p></td>
<td><p>suffixes apply as with ‘f1’</p></td>
</tr>
<tr class="row-even"><td><p>‘jaccard’ etc.</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.jaccard_score.html#sklearn.metrics.jaccard_score" title="sklearn.metrics.jaccard_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.jaccard_score</span></code></a></p></td>
<td><p>suffixes apply as with ‘f1’</p></td>
</tr>
<tr class="row-odd"><td><p>‘roc_auc’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.roc_auc_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘roc_auc_ovr’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.roc_auc_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘roc_auc_ovo’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.roc_auc_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘roc_auc_ovr_weighted’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.roc_auc_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘roc_auc_ovo_weighted’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.roc_auc_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p><strong>Clustering</strong></p></td>
<td></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘adjusted_mutual_info_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.adjusted_mutual_info_score.html#sklearn.metrics.adjusted_mutual_info_score" title="sklearn.metrics.adjusted_mutual_info_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.adjusted_mutual_info_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘adjusted_rand_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.adjusted_rand_score.html#sklearn.metrics.adjusted_rand_score" title="sklearn.metrics.adjusted_rand_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.adjusted_rand_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘completeness_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.completeness_score.html#sklearn.metrics.completeness_score" title="sklearn.metrics.completeness_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.completeness_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘fowlkes_mallows_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.fowlkes_mallows_score.html#sklearn.metrics.fowlkes_mallows_score" title="sklearn.metrics.fowlkes_mallows_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.fowlkes_mallows_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘homogeneity_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.homogeneity_score.html#sklearn.metrics.homogeneity_score" title="sklearn.metrics.homogeneity_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.homogeneity_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘mutual_info_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mutual_info_score.html#sklearn.metrics.mutual_info_score" title="sklearn.metrics.mutual_info_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mutual_info_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘normalized_mutual_info_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.normalized_mutual_info_score.html#sklearn.metrics.normalized_mutual_info_score" title="sklearn.metrics.normalized_mutual_info_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.normalized_mutual_info_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘rand_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.rand_score.html#sklearn.metrics.rand_score" title="sklearn.metrics.rand_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.rand_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘v_measure_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.v_measure_score.html#sklearn.metrics.v_measure_score" title="sklearn.metrics.v_measure_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.v_measure_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p><strong>Regression</strong></p></td>
<td></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘explained_variance’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.explained_variance_score.html#sklearn.metrics.explained_variance_score" title="sklearn.metrics.explained_variance_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.explained_variance_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘max_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.max_error.html#sklearn.metrics.max_error" title="sklearn.metrics.max_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.max_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘neg_mean_absolute_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_absolute_error.html#sklearn.metrics.mean_absolute_error" title="sklearn.metrics.mean_absolute_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_absolute_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘neg_mean_squared_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_squared_error.html#sklearn.metrics.mean_squared_error" title="sklearn.metrics.mean_squared_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_squared_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘neg_root_mean_squared_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_squared_error.html#sklearn.metrics.mean_squared_error" title="sklearn.metrics.mean_squared_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_squared_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘neg_mean_squared_log_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_squared_log_error.html#sklearn.metrics.mean_squared_log_error" title="sklearn.metrics.mean_squared_log_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_squared_log_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘neg_median_absolute_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.median_absolute_error.html#sklearn.metrics.median_absolute_error" title="sklearn.metrics.median_absolute_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.median_absolute_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘r2’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.r2_score.html#sklearn.metrics.r2_score" title="sklearn.metrics.r2_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.r2_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘neg_mean_poisson_deviance’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_poisson_deviance.html#sklearn.metrics.mean_poisson_deviance" title="sklearn.metrics.mean_poisson_deviance"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_poisson_deviance</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘neg_mean_gamma_deviance’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_gamma_deviance.html#sklearn.metrics.mean_gamma_deviance" title="sklearn.metrics.mean_gamma_deviance"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_gamma_deviance</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘neg_mean_absolute_percentage_error’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.mean_absolute_percentage_error.html#sklearn.metrics.mean_absolute_percentage_error" title="sklearn.metrics.mean_absolute_percentage_error"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.mean_absolute_percentage_error</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘d2_absolute_error_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.d2_absolute_error_score.html#sklearn.metrics.d2_absolute_error_score" title="sklearn.metrics.d2_absolute_error_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.d2_absolute_error_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-odd"><td><p>‘d2_pinball_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.d2_pinball_score.html#sklearn.metrics.d2_pinball_score" title="sklearn.metrics.d2_pinball_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.d2_pinball_score</span></code></a></p></td>
<td></td>
</tr>
<tr class="row-even"><td><p>‘d2_tweedie_score’</p></td>
<td><p><a class="reference internal" href="generated/sklearn.metrics.d2_tweedie_score.html#sklearn.metrics.d2_tweedie_score" title="sklearn.metrics.d2_tweedie_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">metrics.d2_tweedie_score</span></code></a></p></td>
<td></td>
</tr>
</tbody>
</table>
<p>Usage examples:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span><span class="p">,</span> <span class="n">datasets</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_val_score</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_iris</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">cross_val_score</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">scoring</span><span class="o">=</span><span class="s1">'recall_macro'</span><span class="p">)</span>
<span class="go">array([0.96..., 0.96..., 0.96..., 0.93..., 1. ])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>If a wrong scoring name is passed, an <code class="docutils literal notranslate"><span class="pre">InvalidParameterError</span></code> is raised.
You can retrieve the names of all available scorers by calling
<a class="reference internal" href="generated/sklearn.metrics.get_scorer_names.html#sklearn.metrics.get_scorer_names" title="sklearn.metrics.get_scorer_names"><code class="xref py py-func docutils literal notranslate"><span class="pre">get_scorer_names</span></code></a>.</p>
</div>
</section>
<section id="defining-your-scoring-strategy-from-metric-functions">
<span id="scoring"></span><h3><span class="section-number">3.3.1.2. </span>Defining your scoring strategy from metric functions<a class="headerlink" href="#defining-your-scoring-strategy-from-metric-functions" title="Permalink to this heading">¶</a></h3>
<p>The module <a class="reference internal" href="classes.html#module-sklearn.metrics" title="sklearn.metrics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.metrics</span></code></a> also exposes a set of simple functions
measuring a prediction error given ground truth and prediction:</p>
<ul class="simple">
<li><p>functions ending with <code class="docutils literal notranslate"><span class="pre">_score</span></code> return a value to
maximize, the higher the better.</p></li>
<li><p>functions ending with <code class="docutils literal notranslate"><span class="pre">_error</span></code> or <code class="docutils literal notranslate"><span class="pre">_loss</span></code> return a
value to minimize, the lower the better. When converting
into a scorer object using <a class="reference internal" href="generated/sklearn.metrics.make_scorer.html#sklearn.metrics.make_scorer" title="sklearn.metrics.make_scorer"><code class="xref py py-func docutils literal notranslate"><span class="pre">make_scorer</span></code></a>, set
the <code class="docutils literal notranslate"><span class="pre">greater_is_better</span></code> parameter to <code class="docutils literal notranslate"><span class="pre">False</span></code> (<code class="docutils literal notranslate"><span class="pre">True</span></code> by default; see the
parameter description below).</p></li>
</ul>
<p>Metrics available for various machine learning tasks are detailed in sections
below.</p>
<p>Many metrics are not given names to be used as <code class="docutils literal notranslate"><span class="pre">scoring</span></code> values,
sometimes because they require additional parameters, such as
<a class="reference internal" href="generated/sklearn.metrics.fbeta_score.html#sklearn.metrics.fbeta_score" title="sklearn.metrics.fbeta_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">fbeta_score</span></code></a>. In such cases, you need to generate an appropriate
scoring object. The simplest way to generate a callable object for scoring
is by using <a class="reference internal" href="generated/sklearn.metrics.make_scorer.html#sklearn.metrics.make_scorer" title="sklearn.metrics.make_scorer"><code class="xref py py-func docutils literal notranslate"><span class="pre">make_scorer</span></code></a>. That function converts metrics
into callables that can be used for model evaluation.</p>
<p>One typical use case is to wrap an existing metric function from the library
with non-default values for its parameters, such as the <code class="docutils literal notranslate"><span class="pre">beta</span></code> parameter for
the <a class="reference internal" href="generated/sklearn.metrics.fbeta_score.html#sklearn.metrics.fbeta_score" title="sklearn.metrics.fbeta_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">fbeta_score</span></code></a> function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">fbeta_score</span><span class="p">,</span> <span class="n">make_scorer</span>
<span class="gp">>>> </span><span class="n">ftwo_scorer</span> <span class="o">=</span> <span class="n">make_scorer</span><span class="p">(</span><span class="n">fbeta_score</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">GridSearchCV</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">LinearSVC</span>
<span class="gp">>>> </span><span class="n">grid</span> <span class="o">=</span> <span class="n">GridSearchCV</span><span class="p">(</span><span class="n">LinearSVC</span><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">),</span> <span class="n">param_grid</span><span class="o">=</span><span class="p">{</span><span class="s1">'C'</span><span class="p">:</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">]},</span>
<span class="gp">... </span> <span class="n">scoring</span><span class="o">=</span><span class="n">ftwo_scorer</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
</pre></div>
</div>
<p><details>
<summary class="btn btn-light">
<strong>Custom scorer objects</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>The second use case is to build a completely custom scorer object
from a simple python function using <a class="reference internal" href="generated/sklearn.metrics.make_scorer.html#sklearn.metrics.make_scorer" title="sklearn.metrics.make_scorer"><code class="xref py py-func docutils literal notranslate"><span class="pre">make_scorer</span></code></a>, which can
take several parameters:</p>
<ul class="simple">
<li><p>the python function you want to use (<code class="docutils literal notranslate"><span class="pre">my_custom_loss_func</span></code>
in the example below)</p></li>
<li><p>whether the python function returns a score (<code class="docutils literal notranslate"><span class="pre">greater_is_better=True</span></code>,
the default) or a loss (<code class="docutils literal notranslate"><span class="pre">greater_is_better=False</span></code>). 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.</p></li>
<li><p>for classification metrics only: whether the python function you provided requires continuous decision
certainties (<code class="docutils literal notranslate"><span class="pre">needs_threshold=True</span></code>). The default value is
False.</p></li>
<li><p>any additional parameters, such as <code class="docutils literal notranslate"><span class="pre">beta</span></code> or <code class="docutils literal notranslate"><span class="pre">labels</span></code> in <a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">f1_score</span></code></a>.</p></li>
</ul>
<p>Here is an example of building custom scorers, and of using the
<code class="docutils literal notranslate"><span class="pre">greater_is_better</span></code> parameter:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">my_custom_loss_func</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">diff</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">y_true</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="n">diff</span><span class="p">)</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="c1"># score will negate the return value of my_custom_loss_func,</span>
<span class="gp">>>> </span><span class="c1"># which will be np.log(2), 0.693, given the values for X</span>
<span class="gp">>>> </span><span class="c1"># and y defined below.</span>
<span class="gp">>>> </span><span class="n">score</span> <span class="o">=</span> <span class="n">make_scorer</span><span class="p">(</span><span class="n">my_custom_loss_func</span><span class="p">,</span> <span class="n">greater_is_better</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.dummy</span> <span class="kn">import</span> <span class="n">DummyClassifier</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">DummyClassifier</span><span class="p">(</span><span class="n">strategy</span><span class="o">=</span><span class="s1">'most_frequent'</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">my_custom_loss_func</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
<span class="go">0.69...</span>
<span class="gp">>>> </span><span class="n">score</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">-0.69...</span>
</pre></div>
</div>
<p></div>
</details></p>
</section>
<section id="implementing-your-own-scoring-object">
<span id="diy-scoring"></span><h3><span class="section-number">3.3.1.3. </span>Implementing your own scoring object<a class="headerlink" href="#implementing-your-own-scoring-object" title="Permalink to this heading">¶</a></h3>
<p>You can generate even more flexible model scorers by constructing your own
scoring object from scratch, without using the <a class="reference internal" href="generated/sklearn.metrics.make_scorer.html#sklearn.metrics.make_scorer" title="sklearn.metrics.make_scorer"><code class="xref py py-func docutils literal notranslate"><span class="pre">make_scorer</span></code></a> factory.</p>
<p><details>
<summary class="btn btn-light">
<strong>How to build a scorer from scratch</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>For a callable to be a scorer, it needs to meet the protocol specified by
the following two rules:</p>
<ul class="simple">
<li><p>It can be called with parameters <code class="docutils literal notranslate"><span class="pre">(estimator,</span> <span class="pre">X,</span> <span class="pre">y)</span></code>, where <code class="docutils literal notranslate"><span class="pre">estimator</span></code>
is the model that should be evaluated, <code class="docutils literal notranslate"><span class="pre">X</span></code> is validation data, and <code class="docutils literal notranslate"><span class="pre">y</span></code> is
the ground truth target for <code class="docutils literal notranslate"><span class="pre">X</span></code> (in the supervised case) or <code class="docutils literal notranslate"><span class="pre">None</span></code> (in the
unsupervised case).</p></li>
<li><p>It returns a floating point number that quantifies the
<code class="docutils literal notranslate"><span class="pre">estimator</span></code> prediction quality on <code class="docutils literal notranslate"><span class="pre">X</span></code>, with reference to <code class="docutils literal notranslate"><span class="pre">y</span></code>.
Again, by convention higher numbers are better, so if your scorer
returns loss, that value should be negated.</p></li>
<li><p>Advanced: If it requires extra metadata to be passed to it, it should expose
a <code class="docutils literal notranslate"><span class="pre">get_metadata_routing</span></code> method returning the requested metadata. The user
should be able to set the requested metadata via a <code class="docutils literal notranslate"><span class="pre">set_score_request</span></code>
method. Please see <a class="reference internal" href="../metadata_routing.html#metadata-routing"><span class="std std-ref">User Guide</span></a> and <a class="reference internal" href="../auto_examples/miscellaneous/plot_metadata_routing.html#sphx-glr-auto-examples-miscellaneous-plot-metadata-routing-py"><span class="std std-ref">Developer
Guide</span></a> for
more details.</p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><strong>Using custom scorers in functions where n_jobs > 1</strong></p>
<p>While defining the custom scoring function alongside the calling function
should work out of the box with the default joblib backend (loky),
importing it from another module will be a more robust approach and work
independently of the joblib backend.</p>
<p>For example, to use <code class="docutils literal notranslate"><span class="pre">n_jobs</span></code> greater than 1 in the example below,
<code class="docutils literal notranslate"><span class="pre">custom_scoring_function</span></code> function is saved in a user-created module
(<code class="docutils literal notranslate"><span class="pre">custom_scorer_module.py</span></code>) and imported:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">custom_scorer_module</span> <span class="kn">import</span> <span class="n">custom_scoring_function</span>
<span class="gp">>>> </span><span class="n">cross_val_score</span><span class="p">(</span><span class="n">model</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">X_train</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">y_train</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">scoring</span><span class="o">=</span><span class="n">make_scorer</span><span class="p">(</span><span class="n">custom_scoring_function</span><span class="p">,</span> <span class="n">greater_is_better</span><span class="o">=</span><span class="kc">False</span><span class="p">),</span>
<span class="gp">... </span> <span class="n">cv</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">n_jobs</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
</div>
<p></div>
</details></p>
</section>
<section id="using-multiple-metric-evaluation">
<span id="multimetric-scoring"></span><h3><span class="section-number">3.3.1.4. </span>Using multiple metric evaluation<a class="headerlink" href="#using-multiple-metric-evaluation" title="Permalink to this heading">¶</a></h3>
<p>Scikit-learn also permits evaluation of multiple metrics in <code class="docutils literal notranslate"><span class="pre">GridSearchCV</span></code>,
<code class="docutils literal notranslate"><span class="pre">RandomizedSearchCV</span></code> and <code class="docutils literal notranslate"><span class="pre">cross_validate</span></code>.</p>
<p>There are three ways to specify multiple scoring metrics for the <code class="docutils literal notranslate"><span class="pre">scoring</span></code>
parameter:</p>
<ul>
<li><dl>
<dt>As an iterable of string metrics::</dt><dd><div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">scoring</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'accuracy'</span><span class="p">,</span> <span class="s1">'precision'</span><span class="p">]</span>
</pre></div>
</div>
</dd>
</dl>
</li>
<li><dl>
<dt>As a <code class="docutils literal notranslate"><span class="pre">dict</span></code> mapping the scorer name to the scoring function::</dt><dd><div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">make_scorer</span>
<span class="gp">>>> </span><span class="n">scoring</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'accuracy'</span><span class="p">:</span> <span class="n">make_scorer</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">),</span>
<span class="gp">... </span> <span class="s1">'prec'</span><span class="p">:</span> <span class="s1">'precision'</span><span class="p">}</span>
</pre></div>
</div>
</dd>
</dl>
<p>Note that the dict values can either be scorer functions or one of the
predefined metric strings.</p>
</li>
<li><p>As a callable that returns a dictionary of scores:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>
<span class="gp">>>> </span><span class="c1"># A sample toy binary classification dataset</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">make_classification</span><span class="p">(</span><span class="n">n_classes</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">svm</span> <span class="o">=</span> <span class="n">LinearSVC</span><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">confusion_matrix_scorer</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">... </span> <span class="n">cm</span> <span class="o">=</span> <span class="n">confusion_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="gp">... </span> <span class="k">return</span> <span class="p">{</span><span class="s1">'tn'</span><span class="p">:</span> <span class="n">cm</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s1">'fp'</span><span class="p">:</span> <span class="n">cm</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="gp">... </span> <span class="s1">'fn'</span><span class="p">:</span> <span class="n">cm</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s1">'tp'</span><span class="p">:</span> <span class="n">cm</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]}</span>
<span class="gp">>>> </span><span class="n">cv_results</span> <span class="o">=</span> <span class="n">cross_validate</span><span class="p">(</span><span class="n">svm</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">scoring</span><span class="o">=</span><span class="n">confusion_matrix_scorer</span><span class="p">)</span>
<span class="gp">>>> </span><span class="c1"># Getting the test set true positive scores</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">cv_results</span><span class="p">[</span><span class="s1">'test_tp'</span><span class="p">])</span>
<span class="go">[10 9 8 7 8]</span>
<span class="gp">>>> </span><span class="c1"># Getting the test set false negative scores</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">cv_results</span><span class="p">[</span><span class="s1">'test_fn'</span><span class="p">])</span>
<span class="go">[0 1 2 3 2]</span>
</pre></div>
</div>
</li>
</ul>
</section>
</section>
<section id="classification-metrics">
<span id="id1"></span><h2><span class="section-number">3.3.2. </span>Classification metrics<a class="headerlink" href="#classification-metrics" title="Permalink to this heading">¶</a></h2>
<p>The <a class="reference internal" href="classes.html#module-sklearn.metrics" title="sklearn.metrics"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.metrics</span></code></a> module implements several loss, score, and utility
functions to measure classification performance.
Some metrics might require probability estimates of the positive class,
confidence values, or binary decisions values.
Most implementations allow each sample to provide a weighted contribution
to the overall score, through the <code class="docutils literal notranslate"><span class="pre">sample_weight</span></code> parameter.</p>
<p>Some of these are restricted to the binary classification case:</p>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.precision_recall_curve.html#sklearn.metrics.precision_recall_curve" title="sklearn.metrics.precision_recall_curve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">precision_recall_curve</span></code></a>(y_true, probas_pred, *)</p></td>
<td><p>Compute precision-recall pairs for different probability thresholds.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.roc_curve.html#sklearn.metrics.roc_curve" title="sklearn.metrics.roc_curve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">roc_curve</span></code></a>(y_true, y_score, *[, pos_label, ...])</p></td>
<td><p>Compute Receiver operating characteristic (ROC).</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.class_likelihood_ratios.html#sklearn.metrics.class_likelihood_ratios" title="sklearn.metrics.class_likelihood_ratios"><code class="xref py py-obj docutils literal notranslate"><span class="pre">class_likelihood_ratios</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Compute binary classification positive and negative likelihood ratios.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.det_curve.html#sklearn.metrics.det_curve" title="sklearn.metrics.det_curve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">det_curve</span></code></a>(y_true, y_score[, pos_label, ...])</p></td>
<td><p>Compute error rates for different probability thresholds.</p></td>
</tr>
</tbody>
</table>
<p>Others also work in the multiclass case:</p>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.balanced_accuracy_score.html#sklearn.metrics.balanced_accuracy_score" title="sklearn.metrics.balanced_accuracy_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">balanced_accuracy_score</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Compute the balanced accuracy.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.cohen_kappa_score.html#sklearn.metrics.cohen_kappa_score" title="sklearn.metrics.cohen_kappa_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cohen_kappa_score</span></code></a>(y1, y2, *[, labels, ...])</p></td>
<td><p>Compute Cohen's kappa: a statistic that measures inter-annotator agreement.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.confusion_matrix.html#sklearn.metrics.confusion_matrix" title="sklearn.metrics.confusion_matrix"><code class="xref py py-obj docutils literal notranslate"><span class="pre">confusion_matrix</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Compute confusion matrix to evaluate the accuracy of a classification.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.hinge_loss.html#sklearn.metrics.hinge_loss" title="sklearn.metrics.hinge_loss"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hinge_loss</span></code></a>(y_true, pred_decision, *[, ...])</p></td>
<td><p>Average hinge loss (non-regularized).</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.matthews_corrcoef.html#sklearn.metrics.matthews_corrcoef" title="sklearn.metrics.matthews_corrcoef"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matthews_corrcoef</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Compute the Matthews correlation coefficient (MCC).</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">roc_auc_score</span></code></a>(y_true, y_score, *[, average, ...])</p></td>
<td><p>Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.top_k_accuracy_score.html#sklearn.metrics.top_k_accuracy_score" title="sklearn.metrics.top_k_accuracy_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">top_k_accuracy_score</span></code></a>(y_true, y_score, *[, ...])</p></td>
<td><p>Top-k Accuracy classification score.</p></td>
</tr>
</tbody>
</table>
<p>Some also work in the multilabel case:</p>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.accuracy_score.html#sklearn.metrics.accuracy_score" title="sklearn.metrics.accuracy_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">accuracy_score</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Accuracy classification score.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.classification_report.html#sklearn.metrics.classification_report" title="sklearn.metrics.classification_report"><code class="xref py py-obj docutils literal notranslate"><span class="pre">classification_report</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Build a text report showing the main classification metrics.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">f1_score</span></code></a>(y_true, y_pred, *[, labels, ...])</p></td>
<td><p>Compute the F1 score, also known as balanced F-score or F-measure.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.fbeta_score.html#sklearn.metrics.fbeta_score" title="sklearn.metrics.fbeta_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fbeta_score</span></code></a>(y_true, y_pred, *, beta[, ...])</p></td>
<td><p>Compute the F-beta score.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.hamming_loss.html#sklearn.metrics.hamming_loss" title="sklearn.metrics.hamming_loss"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hamming_loss</span></code></a>(y_true, y_pred, *[, sample_weight])</p></td>
<td><p>Compute the average Hamming loss.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.jaccard_score.html#sklearn.metrics.jaccard_score" title="sklearn.metrics.jaccard_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">jaccard_score</span></code></a>(y_true, y_pred, *[, labels, ...])</p></td>
<td><p>Jaccard similarity coefficient score.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.log_loss.html#sklearn.metrics.log_loss" title="sklearn.metrics.log_loss"><code class="xref py py-obj docutils literal notranslate"><span class="pre">log_loss</span></code></a>(y_true, y_pred, *[, eps, ...])</p></td>
<td><p>Log loss, aka logistic loss or cross-entropy loss.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.multilabel_confusion_matrix.html#sklearn.metrics.multilabel_confusion_matrix" title="sklearn.metrics.multilabel_confusion_matrix"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multilabel_confusion_matrix</span></code></a>(y_true, y_pred, *)</p></td>
<td><p>Compute a confusion matrix for each class or sample.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.precision_recall_fscore_support.html#sklearn.metrics.precision_recall_fscore_support" title="sklearn.metrics.precision_recall_fscore_support"><code class="xref py py-obj docutils literal notranslate"><span class="pre">precision_recall_fscore_support</span></code></a>(y_true, ...)</p></td>
<td><p>Compute precision, recall, F-measure and support for each class.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.precision_score.html#sklearn.metrics.precision_score" title="sklearn.metrics.precision_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">precision_score</span></code></a>(y_true, y_pred, *[, labels, ...])</p></td>
<td><p>Compute the precision.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.recall_score.html#sklearn.metrics.recall_score" title="sklearn.metrics.recall_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">recall_score</span></code></a>(y_true, y_pred, *[, labels, ...])</p></td>
<td><p>Compute the recall.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">roc_auc_score</span></code></a>(y_true, y_score, *[, average, ...])</p></td>
<td><p>Compute Area Under the Receiver Operating Characteristic Curve (ROC AUC) from prediction scores.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.zero_one_loss.html#sklearn.metrics.zero_one_loss" title="sklearn.metrics.zero_one_loss"><code class="xref py py-obj docutils literal notranslate"><span class="pre">zero_one_loss</span></code></a>(y_true, y_pred, *[, ...])</p></td>
<td><p>Zero-one classification loss.</p></td>
</tr>
</tbody>
</table>
<p>And some work with binary and multilabel (but not multiclass) problems:</p>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/sklearn.metrics.average_precision_score.html#sklearn.metrics.average_precision_score" title="sklearn.metrics.average_precision_score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">average_precision_score</span></code></a>(y_true, y_score, *)</p></td>
<td><p>Compute average precision (AP) from prediction scores.</p></td>
</tr>
</tbody>
</table>
<p>In the following sub-sections, we will describe each of those functions,
preceded by some notes on common API and metric definition.</p>
<section id="from-binary-to-multiclass-and-multilabel">
<span id="average"></span><h3><span class="section-number">3.3.2.1. </span>From binary to multiclass and multilabel<a class="headerlink" href="#from-binary-to-multiclass-and-multilabel" title="Permalink to this heading">¶</a></h3>
<p>Some metrics are essentially defined for binary classification tasks (e.g.
<a class="reference internal" href="generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">f1_score</span></code></a>, <a class="reference internal" href="generated/sklearn.metrics.roc_auc_score.html#sklearn.metrics.roc_auc_score" title="sklearn.metrics.roc_auc_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">roc_auc_score</span></code></a>). In these cases, by default
only the positive label is evaluated, assuming by default that the positive
class is labelled <code class="docutils literal notranslate"><span class="pre">1</span></code> (though this may be configurable through the
<code class="docutils literal notranslate"><span class="pre">pos_label</span></code> parameter).</p>
<p>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 <code class="docutils literal notranslate"><span class="pre">average</span></code> parameter.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">"macro"</span></code> 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.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"weighted"</span></code> 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.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"micro"</span></code> 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.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"samples"</span></code> 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
(<code class="docutils literal notranslate"><span class="pre">sample_weight</span></code>-weighted) average.</p></li>
<li><p>Selecting <code class="docutils literal notranslate"><span class="pre">average=None</span></code> will return an array with the score for each
class.</p></li>
</ul>
<p>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 <code class="docutils literal notranslate"><span class="pre">[i,</span> <span class="pre">j]</span></code> has value 1 if sample <code class="docutils literal notranslate"><span class="pre">i</span></code> has label <code class="docutils literal notranslate"><span class="pre">j</span></code> and value
0 otherwise.</p>
</section>
<section id="accuracy-score">
<span id="id2"></span><h3><span class="section-number">3.3.2.2. </span>Accuracy score<a class="headerlink" href="#accuracy-score" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.metrics.accuracy_score.html#sklearn.metrics.accuracy_score" title="sklearn.metrics.accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">accuracy_score</span></code></a> function computes the
<a class="reference external" href="https://en.wikipedia.org/wiki/Accuracy_and_precision">accuracy</a>, either the fraction
(default) or the count (normalize=False) of correct predictions.</p>
<p>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.</p>
<p>If <span class="math notranslate nohighlight">\(\hat{y}_i\)</span> is the predicted value of
the <span class="math notranslate nohighlight">\(i\)</span>-th sample and <span class="math notranslate nohighlight">\(y_i\)</span> is the corresponding true value,
then the fraction of correct predictions over <span class="math notranslate nohighlight">\(n_\text{samples}\)</span> is
defined as</p>
<div class="math notranslate nohighlight">
\[\texttt{accuracy}(y, \hat{y}) = \frac{1}{n_\text{samples}} \sum_{i=0}^{n_\text{samples}-1} 1(\hat{y}_i = y_i)\]</div>
<p>where <span class="math notranslate nohighlight">\(1(x)\)</span> is the <a class="reference external" href="https://en.wikipedia.org/wiki/Indicator_function">indicator function</a>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
<span class="gp">>>> </span><span class="n">y_pred</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">y_true</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="go">0.5</span>
<span class="gp">>>> </span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="go">2</span>
</pre></div>
</div>
<p>In the multilabel case with binary label indicators:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
<span class="go">0.5</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title">Example:</p>
<ul class="simple">
<li><p>See <a class="reference internal" href="../auto_examples/model_selection/plot_permutation_tests_for_classification.html#sphx-glr-auto-examples-model-selection-plot-permutation-tests-for-classification-py"><span class="std std-ref">Test with permutations the significance of a classification score</span></a>
for an example of accuracy score usage using permutations of
the dataset.</p></li>
</ul>
</aside>
</section>
<section id="top-k-accuracy-score">
<span id="id3"></span><h3><span class="section-number">3.3.2.3. </span>Top-k accuracy score<a class="headerlink" href="#top-k-accuracy-score" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.metrics.top_k_accuracy_score.html#sklearn.metrics.top_k_accuracy_score" title="sklearn.metrics.top_k_accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">top_k_accuracy_score</span></code></a> function is a generalization of
<a class="reference internal" href="generated/sklearn.metrics.accuracy_score.html#sklearn.metrics.accuracy_score" title="sklearn.metrics.accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">accuracy_score</span></code></a>. The difference is that a prediction is considered
correct as long as the true label is associated with one of the <code class="docutils literal notranslate"><span class="pre">k</span></code> highest
predicted scores. <a class="reference internal" href="generated/sklearn.metrics.accuracy_score.html#sklearn.metrics.accuracy_score" title="sklearn.metrics.accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">accuracy_score</span></code></a> is the special case of <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">=</span> <span class="pre">1</span></code>.</p>
<p>The function covers the binary and multiclass classification cases but not the
multilabel case.</p>
<p>If <span class="math notranslate nohighlight">\(\hat{f}_{i,j}\)</span> is the predicted class for the <span class="math notranslate nohighlight">\(i\)</span>-th sample
corresponding to the <span class="math notranslate nohighlight">\(j\)</span>-th largest predicted score and <span class="math notranslate nohighlight">\(y_i\)</span> is the
corresponding true value, then the fraction of correct predictions over
<span class="math notranslate nohighlight">\(n_\text{samples}\)</span> is defined as</p>
<div class="math notranslate nohighlight">
\[\texttt{top-k accuracy}(y, \hat{f}) = \frac{1}{n_\text{samples}} \sum_{i=0}^{n_\text{samples}-1} \sum_{j=1}^{k} 1(\hat{f}_{i,j} = y_i)\]</div>
<p>where <span class="math notranslate nohighlight">\(k\)</span> is the number of guesses allowed and <span class="math notranslate nohighlight">\(1(x)\)</span> is the
<a class="reference external" href="https://en.wikipedia.org/wiki/Indicator_function">indicator function</a>.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">top_k_accuracy_score</span>
<span class="gp">>>> </span><span class="n">y_true</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">y_score</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">top_k_accuracy_score</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_score</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="go">0.75</span>
<span class="gp">>>> </span><span class="c1"># Not normalizing gives the number of "correctly" classified samples</span>
<span class="gp">>>> </span><span class="n">top_k_accuracy_score</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_score</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="go">3</span>
</pre></div>
</div>
</section>
<section id="balanced-accuracy-score">
<span id="id5"></span><h3><span class="section-number">3.3.2.4. </span>Balanced accuracy score<a class="headerlink" href="#balanced-accuracy-score" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.metrics.balanced_accuracy_score.html#sklearn.metrics.balanced_accuracy_score" title="sklearn.metrics.balanced_accuracy_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">balanced_accuracy_score</span></code></a> function computes the <a class="reference external" href="https://en.wikipedia.org/wiki/Accuracy_and_precision">balanced accuracy</a>, 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.</p>
<p>In the binary case, balanced accuracy is equal to the arithmetic mean of
<a class="reference external" href="https://en.wikipedia.org/wiki/Sensitivity_and_specificity">sensitivity</a>
(true positive rate) and <a class="reference external" href="https://en.wikipedia.org/wiki/Sensitivity_and_specificity">specificity</a> (true negative
rate), or the area under the ROC curve with binary predictions rather than
scores:</p>
<div class="math notranslate nohighlight">
\[\texttt{balanced-accuracy} = \frac{1}{2}\left( \frac{TP}{TP + FN} + \frac{TN}{TN + FP}\right )\]</div>
<p>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).</p>
<p>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 <span class="math notranslate nohighlight">\(\frac{1}{n\_classes}\)</span>.</p>
<p>The score ranges from 0 to 1, or when <code class="docutils literal notranslate"><span class="pre">adjusted=True</span></code> is used, it rescaled to
the range <span class="math notranslate nohighlight">\(\frac{1}{1 - n\_classes}\)</span> to 1, inclusive, with
performance at random scoring 0.</p>
<p>If <span class="math notranslate nohighlight">\(y_i\)</span> is the true value of the <span class="math notranslate nohighlight">\(i\)</span>-th sample, and <span class="math notranslate nohighlight">\(w_i\)</span>
is the corresponding sample weight, then we adjust the sample weight to:</p>
<div class="math notranslate nohighlight">
\[\hat{w}_i = \frac{w_i}{\sum_j{1(y_j = y_i) w_j}}\]</div>
<p>where <span class="math notranslate nohighlight">\(1(x)\)</span> is the <a class="reference external" href="https://en.wikipedia.org/wiki/Indicator_function">indicator function</a>.
Given predicted <span class="math notranslate nohighlight">\(\hat{y}_i\)</span> for sample <span class="math notranslate nohighlight">\(i\)</span>, balanced accuracy is
defined as:</p>
<div class="math notranslate nohighlight">
\[\texttt{balanced-accuracy}(y, \hat{y}, w) = \frac{1}{\sum{\hat{w}_i}} \sum_i 1(\hat{y}_i = y_i) \hat{w}_i\]</div>
<p>With <code class="docutils literal notranslate"><span class="pre">adjusted=True</span></code>, balanced accuracy reports the relative increase from
<span class="math notranslate nohighlight">\(\texttt{balanced-accuracy}(y, \mathbf{0}, w) =
\frac{1}{n\_classes}\)</span>. In the binary case, this is also known as
<a class="reference external" href="https://en.wikipedia.org/wiki/Youden%27s_J_statistic">*Youden’s J statistic*</a>,
or <em>informedness</em>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The multiclass definition here seems the most reasonable extension of the
metric used in binary classification, though there is no certain consensus
in the literature:</p>
<ul class="simple">
<li><p>Our definition: <a class="reference internal" href="#mosley2013" id="id7"><span>[Mosley2013]</span></a>, <a class="reference internal" href="#kelleher2015" id="id8"><span>[Kelleher2015]</span></a> and <a class="reference internal" href="#guyon2015" id="id9"><span>[Guyon2015]</span></a>, where
<a class="reference internal" href="#guyon2015" id="id10"><span>[Guyon2015]</span></a> adopt the adjusted version to ensure that random predictions
have a score of <span class="math notranslate nohighlight">\(0\)</span> and perfect predictions have a score of <span class="math notranslate nohighlight">\(1\)</span>..</p></li>
<li><p>Class balanced accuracy as described in <a class="reference internal" href="#mosley2013" id="id11"><span>[Mosley2013]</span></a>: the minimum between the precision
and the recall for each class is computed. Those values are then averaged over the total
number of classes to get the balanced accuracy.</p></li>
<li><p>Balanced Accuracy as described in <a class="reference internal" href="#urbanowicz2015" id="id12"><span>[Urbanowicz2015]</span></a>: the average of sensitivity and specificity
is computed for each class and then averaged over total number of classes.</p></li>
</ul>
</div>
<aside class="topic">
<p class="topic-title">References:</p>
<div role="list" class="citation-list">
<div class="citation" id="guyon2015" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>Guyon2015<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id9">1</a>,<a role="doc-backlink" href="#id10">2</a>)</span>
<p>I. Guyon, K. Bennett, G. Cawley, H.J. Escalante, S. Escalera, T.K. Ho, N. Macià,
B. Ray, M. Saeed, A.R. Statnikov, E. Viegas, <a class="reference external" href="https://ieeexplore.ieee.org/document/7280767">Design of the 2015 ChaLearn AutoML Challenge</a>,
IJCNN 2015.</p>
</div>
<div class="citation" id="mosley2013" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>Mosley2013<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id7">1</a>,<a role="doc-backlink" href="#id11">2</a>)</span>
<p>L. Mosley, <a class="reference external" href="https://lib.dr.iastate.edu/etd/13537/">A balanced approach to the multi-class imbalance problem</a>,
IJCV 2010.</p>
</div>
<div class="citation" id="kelleher2015" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">Kelleher2015</a><span class="fn-bracket">]</span></span>
<p>John. D. Kelleher, Brian Mac Namee, Aoife D’Arcy, <a class="reference external" href="https://mitpress.mit.edu/books/fundamentals-machine-learning-predictive-data-analytics">Fundamentals of
Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples,
and Case Studies</a>,
2015.</p>
</div>
<div class="citation" id="urbanowicz2015" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id12">Urbanowicz2015</a><span class="fn-bracket">]</span></span>
<p>Urbanowicz R.J., Moore, J.H. <a class="reference external" href="https://doi.org/10.1007/s12065-015-0128-8">ExSTraCS 2.0: description
and evaluation of a scalable learning classifier
system</a>, Evol. Intel. (2015) 8: 89.</p>
</div>
</div>
</aside>
</section>
<section id="cohen-s-kappa">
<span id="cohen-kappa"></span><h3><span class="section-number">3.3.2.5. </span>Cohen’s kappa<a class="headerlink" href="#cohen-s-kappa" title="Permalink to this heading">¶</a></h3>
<p>The function <a class="reference internal" href="generated/sklearn.metrics.cohen_kappa_score.html#sklearn.metrics.cohen_kappa_score" title="sklearn.metrics.cohen_kappa_score"><code class="xref py py-func docutils literal notranslate"><span class="pre">cohen_kappa_score</span></code></a> computes <a class="reference external" href="https://en.wikipedia.org/wiki/Cohen%27s_kappa">Cohen’s kappa</a> statistic.
This measure is intended to compare labelings by different human annotators,
not a classifier versus a ground truth.</p>
<p>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).</p>
<p>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.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">cohen_kappa_score</span>
<span class="gp">>>> </span><span class="n">y_true</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">y_pred</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">cohen_kappa_score</span><span class="p">(</span><span class="n">y_true</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">)</span>
<span class="go">0.4285714285714286</span>