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<li class="toctree-l1 current active has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="linear_model.html">1.1. Linear Models</a></li>
<li class="toctree-l2"><a class="reference internal" href="lda_qda.html">1.2. Linear and Quadratic Discriminant Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_ridge.html">1.3. Kernel ridge regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="svm.html">1.4. Support Vector Machines</a></li>
<li class="toctree-l2"><a class="reference internal" href="sgd.html">1.5. Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="neighbors.html">1.6. Nearest Neighbors</a></li>
<li class="toctree-l2"><a class="reference internal" href="gaussian_process.html">1.7. Gaussian Processes</a></li>
<li class="toctree-l2"><a class="reference internal" href="cross_decomposition.html">1.8. Cross decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="naive_bayes.html">1.9. Naive Bayes</a></li>
<li class="toctree-l2"><a class="reference internal" href="tree.html">1.10. Decision Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="ensemble.html">1.11. Ensembles: Gradient boosting, random forests, bagging, voting, stacking</a></li>
<li class="toctree-l2 current active"><a class="current reference internal" href="#">1.12. Multiclass and multioutput algorithms</a></li>
<li class="toctree-l2"><a class="reference internal" href="feature_selection.html">1.13. Feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="semi_supervised.html">1.14. Semi-supervised learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="isotonic.html">1.15. Isotonic regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="calibration.html">1.16. Probability calibration</a></li>
<li class="toctree-l2"><a class="reference internal" href="neural_networks_supervised.html">1.17. Neural network models (supervised)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../unsupervised_learning.html">2. Unsupervised learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="mixture.html">2.1. Gaussian mixture models</a></li>
<li class="toctree-l2"><a class="reference internal" href="manifold.html">2.2. Manifold learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="clustering.html">2.3. Clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="biclustering.html">2.4. Biclustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="decomposition.html">2.5. Decomposing signals in components (matrix factorization problems)</a></li>
<li class="toctree-l2"><a class="reference internal" href="covariance.html">2.6. Covariance estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="outlier_detection.html">2.7. Novelty and Outlier Detection</a></li>
<li class="toctree-l2"><a class="reference internal" href="density.html">2.8. Density Estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="neural_networks_unsupervised.html">2.9. Neural network models (unsupervised)</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../model_selection.html">3. Model selection and evaluation</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="cross_validation.html">3.1. Cross-validation: evaluating estimator performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="grid_search.html">3.2. Tuning the hyper-parameters of an estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="classification_threshold.html">3.3. Tuning the decision threshold for class prediction</a></li>
<li class="toctree-l2"><a class="reference internal" href="model_evaluation.html">3.4. Metrics and scoring: quantifying the quality of predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="learning_curve.html">3.5. Validation curves: plotting scores to evaluate models</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../metadata_routing.html">4. Metadata Routing</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../inspection.html">5. Inspection</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="partial_dependence.html">5.1. Partial Dependence and Individual Conditional Expectation plots</a></li>
<li class="toctree-l2"><a class="reference internal" href="permutation_importance.html">5.2. Permutation feature importance</a></li>
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</details></li>
<li class="toctree-l1"><a class="reference internal" href="../visualizations.html">6. Visualizations</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../data_transforms.html">7. Dataset transformations</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="compose.html">7.1. Pipelines and composite estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="feature_extraction.html">7.2. Feature extraction</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing.html">7.3. Preprocessing data</a></li>
<li class="toctree-l2"><a class="reference internal" href="impute.html">7.4. Imputation of missing values</a></li>
<li class="toctree-l2"><a class="reference internal" href="unsupervised_reduction.html">7.5. Unsupervised dimensionality reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="random_projection.html">7.6. Random Projection</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">7.7. Kernel Approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="metrics.html">7.8. Pairwise metrics, Affinities and Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">7.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../datasets.html">8. Dataset loading utilities</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../datasets/toy_dataset.html">8.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">8.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">8.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">8.4. Loading other datasets</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../computing.html">9. Computing with scikit-learn</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">9.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">9.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">9.3. Parallelism, resource management, and configuration</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">10. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">11. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">12. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="array_api.html">12.1. Array API support (experimental)</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">13. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">14. External Resources, Videos and Talks</a></li>
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<section id="multiclass-and-multioutput-algorithms">
<span id="multiclass"></span><h1><span class="section-number">1.12. </span>Multiclass and multioutput algorithms<a class="headerlink" href="#multiclass-and-multioutput-algorithms" title="Link to this heading">#</a></h1>
<p>This section of the user guide covers functionality related to multi-learning
problems, including <a class="reference internal" href="../glossary.html#term-multiclass"><span class="xref std std-term">multiclass</span></a>, <a class="reference internal" href="../glossary.html#term-multilabel"><span class="xref std std-term">multilabel</span></a>, and
<a class="reference internal" href="../glossary.html#term-multioutput"><span class="xref std std-term">multioutput</span></a> classification and regression.</p>
<p>The modules in this section implement <a class="reference internal" href="../glossary.html#term-meta-estimators"><span class="xref std std-term">meta-estimators</span></a>, which require a
base estimator to be provided in their constructor. Meta-estimators extend the
functionality of the base estimator to support multi-learning problems, which
is accomplished by transforming the multi-learning problem into a set of
simpler problems, then fitting one estimator per problem.</p>
<p>This section covers two modules: <a class="reference internal" href="../api/sklearn.multiclass.html#module-sklearn.multiclass" title="sklearn.multiclass"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.multiclass</span></code></a> and
<a class="reference internal" href="../api/sklearn.multioutput.html#module-sklearn.multioutput" title="sklearn.multioutput"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.multioutput</span></code></a>. The chart below demonstrates the problem types
that each module is responsible for, and the corresponding meta-estimators
that each module provides.</p>
<img alt="../_images/multi_org_chart.png" class="align-center" src="../_images/multi_org_chart.png" />
<p>The table below provides a quick reference on the differences between problem
types. More detailed explanations can be found in subsequent sections of this
guide.</p>
<div class="pst-scrollable-table-container"><table class="table">
<thead>
<tr class="row-odd"><th class="head"></th>
<th class="head"><p>Number of targets</p></th>
<th class="head"><p>Target cardinality</p></th>
<th class="head"><p>Valid
<a class="reference internal" href="generated/sklearn.utils.multiclass.type_of_target.html#sklearn.utils.multiclass.type_of_target" title="sklearn.utils.multiclass.type_of_target"><code class="xref py py-func docutils literal notranslate"><span class="pre">type_of_target</span></code></a></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Multiclass
classification</p></td>
<td><p>1</p></td>
<td><p>>2</p></td>
<td><p>‘multiclass’</p></td>
</tr>
<tr class="row-odd"><td><p>Multilabel
classification</p></td>
<td><p>>1</p></td>
<td><p>2 (0 or 1)</p></td>
<td><p>‘multilabel-indicator’</p></td>
</tr>
<tr class="row-even"><td><p>Multiclass-multioutput
classification</p></td>
<td><p>>1</p></td>
<td><p>>2</p></td>
<td><p>‘multiclass-multioutput’</p></td>
</tr>
<tr class="row-odd"><td><p>Multioutput
regression</p></td>
<td><p>>1</p></td>
<td><p>Continuous</p></td>
<td><p>‘continuous-multioutput’</p></td>
</tr>
</tbody>
</table>
</div>
<p>Below is a summary of scikit-learn estimators that have multi-learning support
built-in, grouped by strategy. You don’t need the meta-estimators provided by
this section if you’re using one of these estimators. However, meta-estimators
can provide additional strategies beyond what is built-in:</p>
<ul class="simple">
<li><p><strong>Inherently multiclass:</strong></p>
<ul>
<li><p><a class="reference internal" href="generated/sklearn.naive_bayes.BernoulliNB.html#sklearn.naive_bayes.BernoulliNB" title="sklearn.naive_bayes.BernoulliNB"><code class="xref py py-class docutils literal notranslate"><span class="pre">naive_bayes.BernoulliNB</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.tree.DecisionTreeClassifier.html#sklearn.tree.DecisionTreeClassifier" title="sklearn.tree.DecisionTreeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">tree.DecisionTreeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.tree.ExtraTreeClassifier.html#sklearn.tree.ExtraTreeClassifier" title="sklearn.tree.ExtraTreeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">tree.ExtraTreeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.ExtraTreesClassifier.html#sklearn.ensemble.ExtraTreesClassifier" title="sklearn.ensemble.ExtraTreesClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.ExtraTreesClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB"><code class="xref py py-class docutils literal notranslate"><span class="pre">naive_bayes.GaussianNB</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.KNeighborsClassifier.html#sklearn.neighbors.KNeighborsClassifier" title="sklearn.neighbors.KNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.KNeighborsClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.semi_supervised.LabelPropagation.html#sklearn.semi_supervised.LabelPropagation" title="sklearn.semi_supervised.LabelPropagation"><code class="xref py py-class docutils literal notranslate"><span class="pre">semi_supervised.LabelPropagation</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.semi_supervised.LabelSpreading.html#sklearn.semi_supervised.LabelSpreading" title="sklearn.semi_supervised.LabelSpreading"><code class="xref py py-class docutils literal notranslate"><span class="pre">semi_supervised.LabelSpreading</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html#sklearn.discriminant_analysis.LinearDiscriminantAnalysis" title="sklearn.discriminant_analysis.LinearDiscriminantAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">discriminant_analysis.LinearDiscriminantAnalysis</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">svm.LinearSVC</span></code></a> (setting multi_class=”crammer_singer”)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.LogisticRegression</span></code></a> (with most solvers)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.LogisticRegressionCV.html#sklearn.linear_model.LogisticRegressionCV" title="sklearn.linear_model.LogisticRegressionCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.LogisticRegressionCV</span></code></a> (with most solvers)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier" title="sklearn.neural_network.MLPClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neural_network.MLPClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.NearestCentroid.html#sklearn.neighbors.NearestCentroid" title="sklearn.neighbors.NearestCentroid"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.NearestCentroid</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis.html#sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis" title="sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis"><code class="xref py py-class docutils literal notranslate"><span class="pre">discriminant_analysis.QuadraticDiscriminantAnalysis</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.RadiusNeighborsClassifier.html#sklearn.neighbors.RadiusNeighborsClassifier" title="sklearn.neighbors.RadiusNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.RadiusNeighborsClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.RandomForestClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.RidgeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifierCV.html#sklearn.linear_model.RidgeClassifierCV" title="sklearn.linear_model.RidgeClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.RidgeClassifierCV</span></code></a></p></li>
</ul>
</li>
<li><p><strong>Multiclass as One-Vs-One:</strong></p>
<ul>
<li><p><a class="reference internal" href="generated/sklearn.svm.NuSVC.html#sklearn.svm.NuSVC" title="sklearn.svm.NuSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">svm.NuSVC</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">svm.SVC</span></code></a>.</p></li>
<li><p><a class="reference internal" href="generated/sklearn.gaussian_process.GaussianProcessClassifier.html#sklearn.gaussian_process.GaussianProcessClassifier" title="sklearn.gaussian_process.GaussianProcessClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">gaussian_process.GaussianProcessClassifier</span></code></a> (setting multi_class = “one_vs_one”)</p></li>
</ul>
</li>
<li><p><strong>Multiclass as One-Vs-The-Rest:</strong></p>
<ul>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.GradientBoostingClassifier.html#sklearn.ensemble.GradientBoostingClassifier" title="sklearn.ensemble.GradientBoostingClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.GradientBoostingClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.gaussian_process.GaussianProcessClassifier.html#sklearn.gaussian_process.GaussianProcessClassifier" title="sklearn.gaussian_process.GaussianProcessClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">gaussian_process.GaussianProcessClassifier</span></code></a> (setting multi_class = “one_vs_rest”)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">svm.LinearSVC</span></code></a> (setting multi_class=”ovr”)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.LogisticRegression</span></code></a> (most solvers)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.LogisticRegressionCV.html#sklearn.linear_model.LogisticRegressionCV" title="sklearn.linear_model.LogisticRegressionCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.LogisticRegressionCV</span></code></a> (most solvers)</p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.SGDClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.Perceptron.html#sklearn.linear_model.Perceptron" title="sklearn.linear_model.Perceptron"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.Perceptron</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.PassiveAggressiveClassifier.html#sklearn.linear_model.PassiveAggressiveClassifier" title="sklearn.linear_model.PassiveAggressiveClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.PassiveAggressiveClassifier</span></code></a></p></li>
</ul>
</li>
<li><p><strong>Support multilabel:</strong></p>
<ul>
<li><p><a class="reference internal" href="generated/sklearn.tree.DecisionTreeClassifier.html#sklearn.tree.DecisionTreeClassifier" title="sklearn.tree.DecisionTreeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">tree.DecisionTreeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.tree.ExtraTreeClassifier.html#sklearn.tree.ExtraTreeClassifier" title="sklearn.tree.ExtraTreeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">tree.ExtraTreeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.ExtraTreesClassifier.html#sklearn.ensemble.ExtraTreesClassifier" title="sklearn.ensemble.ExtraTreesClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.ExtraTreesClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.KNeighborsClassifier.html#sklearn.neighbors.KNeighborsClassifier" title="sklearn.neighbors.KNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.KNeighborsClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier" title="sklearn.neural_network.MLPClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neural_network.MLPClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.RadiusNeighborsClassifier.html#sklearn.neighbors.RadiusNeighborsClassifier" title="sklearn.neighbors.RadiusNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.RadiusNeighborsClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.RandomForestClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.RidgeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifierCV.html#sklearn.linear_model.RidgeClassifierCV" title="sklearn.linear_model.RidgeClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">linear_model.RidgeClassifierCV</span></code></a></p></li>
</ul>
</li>
<li><p><strong>Support multiclass-multioutput:</strong></p>
<ul>
<li><p><a class="reference internal" href="generated/sklearn.tree.DecisionTreeClassifier.html#sklearn.tree.DecisionTreeClassifier" title="sklearn.tree.DecisionTreeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">tree.DecisionTreeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.tree.ExtraTreeClassifier.html#sklearn.tree.ExtraTreeClassifier" title="sklearn.tree.ExtraTreeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">tree.ExtraTreeClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.ExtraTreesClassifier.html#sklearn.ensemble.ExtraTreesClassifier" title="sklearn.ensemble.ExtraTreesClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.ExtraTreesClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.KNeighborsClassifier.html#sklearn.neighbors.KNeighborsClassifier" title="sklearn.neighbors.KNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.KNeighborsClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.neighbors.RadiusNeighborsClassifier.html#sklearn.neighbors.RadiusNeighborsClassifier" title="sklearn.neighbors.RadiusNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">neighbors.RadiusNeighborsClassifier</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">ensemble.RandomForestClassifier</span></code></a></p></li>
</ul>
</li>
</ul>
<section id="multiclass-classification">
<span id="id1"></span><h2><span class="section-number">1.12.1. </span>Multiclass classification<a class="headerlink" href="#multiclass-classification" title="Link to this heading">#</a></h2>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>All classifiers in scikit-learn do multiclass classification
out-of-the-box. You don’t need to use the <a class="reference internal" href="../api/sklearn.multiclass.html#module-sklearn.multiclass" title="sklearn.multiclass"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.multiclass</span></code></a> module
unless you want to experiment with different multiclass strategies.</p>
</div>
<p><strong>Multiclass classification</strong> is a classification task with more than two
classes. Each sample can only be labeled as one class.</p>
<p>For example, classification using features extracted from a set of images of
fruit, where each image may either be of an orange, an apple, or a pear.
Each image is one sample and is labeled as one of the 3 possible classes.
Multiclass classification makes the assumption that each sample is assigned
to one and only one label - one sample cannot, for example, be both a pear
and an apple.</p>
<p>While all scikit-learn classifiers are capable of multiclass classification,
the meta-estimators offered by <a class="reference internal" href="../api/sklearn.multiclass.html#module-sklearn.multiclass" title="sklearn.multiclass"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.multiclass</span></code></a>
permit changing the way they handle more than two classes
because this may have an effect on classifier performance
(either in terms of generalization error or required computational resources).</p>
<section id="target-format">
<h3><span class="section-number">1.12.1.1. </span>Target format<a class="headerlink" href="#target-format" title="Link to this heading">#</a></h3>
<p>Valid <a class="reference internal" href="../glossary.html#term-multiclass"><span class="xref std std-term">multiclass</span></a> representations for
<a class="reference internal" href="generated/sklearn.utils.multiclass.type_of_target.html#sklearn.utils.multiclass.type_of_target" title="sklearn.utils.multiclass.type_of_target"><code class="xref py py-func docutils literal notranslate"><span class="pre">type_of_target</span></code></a> (<code class="docutils literal notranslate"><span class="pre">y</span></code>) are:</p>
<ul>
<li><p>1d or column vector containing more than two discrete values. An
example of a vector <code class="docutils literal notranslate"><span class="pre">y</span></code> for 4 samples:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">y</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="s1">'apple'</span><span class="p">,</span> <span class="s1">'pear'</span><span class="p">,</span> <span class="s1">'apple'</span><span class="p">,</span> <span class="s1">'orange'</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="go">['apple' 'pear' 'apple' 'orange']</span>
</pre></div>
</div>
</li>
<li><p>Dense or sparse <a class="reference internal" href="../glossary.html#term-binary"><span class="xref std std-term">binary</span></a> matrix of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_classes)</span></code>
with a single sample per row, where each column represents one class. An
example of both a dense and sparse <a class="reference internal" href="../glossary.html#term-binary"><span class="xref std std-term">binary</span></a> matrix <code class="docutils literal notranslate"><span class="pre">y</span></code> for 4
samples, where the columns, in order, are apple, orange, and pear:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.preprocessing</span><span class="w"> </span><span class="kn">import</span> <span class="n">LabelBinarizer</span>
<span class="gp">>>> </span><span class="n">y</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="s1">'apple'</span><span class="p">,</span> <span class="s1">'pear'</span><span class="p">,</span> <span class="s1">'apple'</span><span class="p">,</span> <span class="s1">'orange'</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">y_dense</span> <span class="o">=</span> <span class="n">LabelBinarizer</span><span class="p">()</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">y_dense</span><span class="p">)</span>
<span class="go">[[1 0 0]</span>
<span class="go"> [0 0 1]</span>
<span class="go"> [1 0 0]</span>
<span class="go"> [0 1 0]]</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">scipy</span><span class="w"> </span><span class="kn">import</span> <span class="n">sparse</span>
<span class="gp">>>> </span><span class="n">y_sparse</span> <span class="o">=</span> <span class="n">sparse</span><span class="o">.</span><span class="n">csr_matrix</span><span class="p">(</span><span class="n">y_dense</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">y_sparse</span><span class="p">)</span>
<span class="go"><Compressed Sparse Row sparse matrix of dtype 'int64'</span>
<span class="go"> with 4 stored elements and shape (4, 3)></span>
<span class="go"> Coords Values</span>
<span class="go"> (0, 0) 1</span>
<span class="go"> (1, 2) 1</span>
<span class="go"> (2, 0) 1</span>
<span class="go"> (3, 1) 1</span>
</pre></div>
</div>
</li>
</ul>
<p>For more information about <a class="reference internal" href="generated/sklearn.preprocessing.LabelBinarizer.html#sklearn.preprocessing.LabelBinarizer" title="sklearn.preprocessing.LabelBinarizer"><code class="xref py py-class docutils literal notranslate"><span class="pre">LabelBinarizer</span></code></a>,
refer to <a class="reference internal" href="preprocessing_targets.html#preprocessing-targets"><span class="std std-ref">Transforming the prediction target (y)</span></a>.</p>
</section>
<section id="onevsrestclassifier">
<span id="ovr-classification"></span><h3><span class="section-number">1.12.1.2. </span>OneVsRestClassifier<a class="headerlink" href="#onevsrestclassifier" title="Link to this heading">#</a></h3>
<p>The <strong>one-vs-rest</strong> strategy, also known as <strong>one-vs-all</strong>, is implemented in
<a class="reference internal" href="generated/sklearn.multiclass.OneVsRestClassifier.html#sklearn.multiclass.OneVsRestClassifier" title="sklearn.multiclass.OneVsRestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneVsRestClassifier</span></code></a>. The strategy consists in
fitting one classifier per class. For each classifier, the class is fitted
against all the other classes. In addition to its computational efficiency
(only <code class="docutils literal notranslate"><span class="pre">n_classes</span></code> classifiers are needed), one advantage of this approach is
its interpretability. Since each class is represented by one and only one
classifier, it is possible to gain knowledge about the class by inspecting its
corresponding classifier. This is the most commonly used strategy and is a fair
default choice.</p>
<p>Below is an example of multiclass learning using OvR:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">datasets</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.multiclass</span><span class="w"> </span><span class="kn">import</span> <span class="n">OneVsRestClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.svm</span><span class="w"> </span><span class="kn">import</span> <span class="n">LinearSVC</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">OneVsRestClassifier</span><span class="p">(</span><span class="n">LinearSVC</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="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="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,</span>
<span class="go"> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,</span>
<span class="go"> 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,</span>
<span class="go"> 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1,</span>
<span class="go"> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,</span>
<span class="go"> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2,</span>
<span class="go"> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])</span>
</pre></div>
</div>
<p><a class="reference internal" href="generated/sklearn.multiclass.OneVsRestClassifier.html#sklearn.multiclass.OneVsRestClassifier" title="sklearn.multiclass.OneVsRestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneVsRestClassifier</span></code></a> also supports multilabel
classification. To use this feature, feed the classifier an indicator matrix,
in which cell [i, j] indicates the presence of label j in sample i.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/miscellaneous/plot_multilabel.html"><img alt="../_images/sphx_glr_plot_multilabel_001.png" src="../_images/sphx_glr_plot_multilabel_001.png" style="width: 600.0px; height: 450.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/miscellaneous/plot_multilabel.html#sphx-glr-auto-examples-miscellaneous-plot-multilabel-py"><span class="std std-ref">Multilabel classification</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/classification/plot_classification_probability.html#sphx-glr-auto-examples-classification-plot-classification-probability-py"><span class="std std-ref">Plot classification probability</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_logistic_multinomial.html#sphx-glr-auto-examples-linear-model-plot-logistic-multinomial-py"><span class="std std-ref">Decision Boundaries of Multinomial and One-vs-Rest Logistic Regression</span></a></p></li>
</ul>
</section>
<section id="onevsoneclassifier">
<span id="ovo-classification"></span><h3><span class="section-number">1.12.1.3. </span>OneVsOneClassifier<a class="headerlink" href="#onevsoneclassifier" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.multiclass.OneVsOneClassifier.html#sklearn.multiclass.OneVsOneClassifier" title="sklearn.multiclass.OneVsOneClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">OneVsOneClassifier</span></code></a> constructs one classifier per
pair of classes. At prediction time, the class which received the most votes
is selected. In the event of a tie (among two classes with an equal number of
votes), it selects the class with the highest aggregate classification
confidence by summing over the pair-wise classification confidence levels
computed by the underlying binary classifiers.</p>
<p>Since it requires to fit <code class="docutils literal notranslate"><span class="pre">n_classes</span> <span class="pre">*</span> <span class="pre">(n_classes</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">/</span> <span class="pre">2</span></code> classifiers,
this method is usually slower than one-vs-the-rest, due to its
O(n_classes^2) complexity. However, this method may be advantageous for
algorithms such as kernel algorithms which don’t scale well with
<code class="docutils literal notranslate"><span class="pre">n_samples</span></code>. This is because each individual learning problem only involves
a small subset of the data whereas, with one-vs-the-rest, the complete
dataset is used <code class="docutils literal notranslate"><span class="pre">n_classes</span></code> times. The decision function is the result
of a monotonic transformation of the one-versus-one classification.</p>
<p>Below is an example of multiclass learning using OvO:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">datasets</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.multiclass</span><span class="w"> </span><span class="kn">import</span> <span class="n">OneVsOneClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.svm</span><span class="w"> </span><span class="kn">import</span> <span class="n">LinearSVC</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">OneVsOneClassifier</span><span class="p">(</span><span class="n">LinearSVC</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="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="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,</span>
<span class="go"> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,</span>
<span class="go"> 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,</span>
<span class="go"> 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,</span>
<span class="go"> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,</span>
<span class="go"> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,</span>
<span class="go"> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])</span>
</pre></div>
</div>
<p class="rubric">References</p>
<ul class="simple">
<li><p>“Pattern Recognition and Machine Learning. Springer”,
Christopher M. Bishop, page 183, (First Edition)</p></li>
</ul>
</section>
<section id="outputcodeclassifier">
<span id="ecoc"></span><h3><span class="section-number">1.12.1.4. </span>OutputCodeClassifier<a class="headerlink" href="#outputcodeclassifier" title="Link to this heading">#</a></h3>
<p>Error-Correcting Output Code-based strategies are fairly different from
one-vs-the-rest and one-vs-one. With these strategies, each class is
represented in a Euclidean space, where each dimension can only be 0 or 1.
Another way to put it is that each class is represented by a binary code (an
array of 0 and 1). The matrix which keeps track of the location/code of each
class is called the code book. The code size is the dimensionality of the
aforementioned space. Intuitively, each class should be represented by a code
as unique as possible and a good code book should be designed to optimize
classification accuracy. In this implementation, we simply use a
randomly-generated code book as advocated in <a class="footnote-reference brackets" href="#id3" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> although more elaborate
methods may be added in the future.</p>
<p>At fitting time, one binary classifier per bit in the code book is fitted.
At prediction time, the classifiers are used to project new points in the
class space and the class closest to the points is chosen.</p>
<p>In <a class="reference internal" href="generated/sklearn.multiclass.OutputCodeClassifier.html#sklearn.multiclass.OutputCodeClassifier" title="sklearn.multiclass.OutputCodeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">OutputCodeClassifier</span></code></a>, the <code class="docutils literal notranslate"><span class="pre">code_size</span></code>
attribute allows the user to control the number of classifiers which will be
used. It is a percentage of the total number of classes.</p>
<p>A number between 0 and 1 will require fewer classifiers than
one-vs-the-rest. In theory, <code class="docutils literal notranslate"><span class="pre">log2(n_classes)</span> <span class="pre">/</span> <span class="pre">n_classes</span></code> is sufficient to
represent each class unambiguously. However, in practice, it may not lead to
good accuracy since <code class="docutils literal notranslate"><span class="pre">log2(n_classes)</span></code> is much smaller than <code class="docutils literal notranslate"><span class="pre">n_classes</span></code>.</p>
<p>A number greater than 1 will require more classifiers than
one-vs-the-rest. In this case, some classifiers will in theory correct for
the mistakes made by other classifiers, hence the name “error-correcting”.
In practice, however, this may not happen as classifier mistakes will
typically be correlated. The error-correcting output codes have a similar
effect to bagging.</p>
<p>Below is an example of multiclass learning using Output-Codes:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">datasets</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.multiclass</span><span class="w"> </span><span class="kn">import</span> <span class="n">OutputCodeClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.svm</span><span class="w"> </span><span class="kn">import</span> <span class="n">LinearSVC</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">OutputCodeClassifier</span><span class="p">(</span><span class="n">LinearSVC</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="n">code_size</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">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="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,</span>
<span class="go"> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,</span>
<span class="go"> 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1,</span>
<span class="go"> 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1,</span>
<span class="go"> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,</span>