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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>
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<li class="toctree-l2 current active"><a class="current reference internal" href="#">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>
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<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-l2"><a class="reference internal" href="classification_threshold.html">3.3. Tuning the decision threshold for class prediction</a></li>
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<li class="toctree-l2"><a class="reference internal" href="learning_curve.html">3.5. Validation curves: plotting scores to evaluate models</a></li>
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<li class="toctree-l2"><a class="reference internal" href="compose.html">6.1. Pipelines and composite estimators</a></li>
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<li class="toctree-l2"><a class="reference internal" href="unsupervised_reduction.html">6.5. Unsupervised dimensionality reduction</a></li>
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<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">6.7. Kernel Approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="metrics.html">6.8. Pairwise metrics, Affinities and Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">6.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../datasets.html">7. 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">7.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">7.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">7.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">7.4. Loading other datasets</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">8.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">8.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">8.3. Parallelism, resource management, and configuration</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">9. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">10. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">11. 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">11.1. Array API support (experimental)</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">12. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">13. External Resources, Videos and Talks</a></li>
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<section id="decision-trees">
<span id="tree"></span><h1><span class="section-number">1.10. </span>Decision Trees<a class="headerlink" href="#decision-trees" title="Link to this heading">#</a></h1>
<p><strong>Decision Trees (DTs)</strong> are a non-parametric supervised learning method used
for <a class="reference internal" href="#tree-classification"><span class="std std-ref">classification</span></a> and <a class="reference internal" href="#tree-regression"><span class="std std-ref">regression</span></a>. The goal is to create a model that predicts the value of a
target variable by learning simple decision rules inferred from the data
features. A tree can be seen as a piecewise constant approximation.</p>
<p>For instance, in the example below, decision trees learn from data to
approximate a sine curve with a set of if-then-else decision rules. The deeper
the tree, the more complex the decision rules and the fitter the model.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/tree/plot_tree_regression.html"><img alt="../_images/sphx_glr_plot_tree_regression_001.png" src="../_images/sphx_glr_plot_tree_regression_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p>Some advantages of decision trees are:</p>
<ul class="simple">
<li><p>Simple to understand and to interpret. Trees can be visualized.</p></li>
<li><p>Requires little data preparation. Other techniques often require data
normalization, dummy variables need to be created and blank values to
be removed. Some tree and algorithm combinations support
<a class="reference internal" href="#tree-missing-value-support"><span class="std std-ref">missing values</span></a>.</p></li>
<li><p>The cost of using the tree (i.e., predicting data) is logarithmic in the
number of data points used to train the tree.</p></li>
<li><p>Able to handle both numerical and categorical data. However, the scikit-learn
implementation does not support categorical variables for now. Other
techniques are usually specialized in analyzing datasets that have only one type
of variable. See <a class="reference internal" href="#tree-algorithms"><span class="std std-ref">algorithms</span></a> for more
information.</p></li>
<li><p>Able to handle multi-output problems.</p></li>
<li><p>Uses a white box model. If a given situation is observable in a model,
the explanation for the condition is easily explained by boolean logic.
By contrast, in a black box model (e.g., in an artificial neural
network), results may be more difficult to interpret.</p></li>
<li><p>Possible to validate a model using statistical tests. That makes it
possible to account for the reliability of the model.</p></li>
<li><p>Performs well even if its assumptions are somewhat violated by
the true model from which the data were generated.</p></li>
</ul>
<p>The disadvantages of decision trees include:</p>
<ul class="simple">
<li><p>Decision-tree learners can create over-complex trees that do not
generalize the data well. This is called overfitting. Mechanisms
such as pruning, setting the minimum number of samples required
at a leaf node or setting the maximum depth of the tree are
necessary to avoid this problem.</p></li>
<li><p>Decision trees can be unstable because small variations in the
data might result in a completely different tree being generated.
This problem is mitigated by using decision trees within an
ensemble.</p></li>
<li><p>Predictions of decision trees are neither smooth nor continuous, but
piecewise constant approximations as seen in the above figure. Therefore,
they are not good at extrapolation.</p></li>
<li><p>The problem of learning an optimal decision tree is known to be
NP-complete under several aspects of optimality and even for simple
concepts. Consequently, practical decision-tree learning algorithms
are based on heuristic algorithms such as the greedy algorithm where
locally optimal decisions are made at each node. Such algorithms
cannot guarantee to return the globally optimal decision tree. This
can be mitigated by training multiple trees in an ensemble learner,
where the features and samples are randomly sampled with replacement.</p></li>
<li><p>There are concepts that are hard to learn because decision trees
do not express them easily, such as XOR, parity or multiplexer problems.</p></li>
<li><p>Decision tree learners create biased trees if some classes dominate.
It is therefore recommended to balance the dataset prior to fitting
with the decision tree.</p></li>
</ul>
<section id="classification">
<span id="tree-classification"></span><h2><span class="section-number">1.10.1. </span>Classification<a class="headerlink" href="#classification" title="Link to this heading">#</a></h2>
<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">DecisionTreeClassifier</span></code></a> is a class capable of performing multi-class
classification on a dataset.</p>
<p>As with other classifiers, <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">DecisionTreeClassifier</span></code></a> takes as input two arrays:
an array X, sparse or dense, of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_features)</span></code> holding the
training samples, and an array Y of integer values, shape <code class="docutils literal notranslate"><span class="pre">(n_samples,)</span></code>,
holding the class labels for the training samples:</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</span> <span class="kn">import</span> <span class="n">tree</span>
<span class="gp">>>> </span><span class="n">X</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="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">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="n">clf</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeClassifier</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>
</pre></div>
</div>
<p>After being fitted, the model can then be used to predict the class of samples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([1])</span>
</pre></div>
</div>
<p>In case that there are multiple classes with the same and highest
probability, the classifier will predict the class with the lowest index
amongst those classes.</p>
<p>As an alternative to outputting a specific class, the probability of each class
can be predicted, which is the fraction of training samples of the class in a
leaf:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([[0., 1.]])</span>
</pre></div>
</div>
<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">DecisionTreeClassifier</span></code></a> is capable of both binary (where the
labels are [-1, 1]) classification and multiclass (where the labels are
[0, …, K-1]) classification.</p>
<p>Using the Iris dataset, we can construct a tree as follows:</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.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">tree</span>
<span class="gp">>>> </span><span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</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">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeClassifier</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>
</pre></div>
</div>
<p>Once trained, you can plot the tree with the <a class="reference internal" href="generated/sklearn.tree.plot_tree.html#sklearn.tree.plot_tree" title="sklearn.tree.plot_tree"><code class="xref py py-func docutils literal notranslate"><span class="pre">plot_tree</span></code></a> function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">tree</span><span class="o">.</span><span class="n">plot_tree</span><span class="p">(</span><span class="n">clf</span><span class="p">)</span>
<span class="go">[...]</span>
</pre></div>
</div>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/tree/plot_iris_dtc.html"><img alt="../_images/sphx_glr_plot_iris_dtc_002.png" src="../_images/sphx_glr_plot_iris_dtc_002.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="alternative-ways-to-export-trees">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Alternative ways to export trees<a class="headerlink" href="#alternative-ways-to-export-trees" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">We can also export the tree in <a class="reference external" href="https://www.graphviz.org/">Graphviz</a> format using the <a class="reference internal" href="generated/sklearn.tree.export_graphviz.html#sklearn.tree.export_graphviz" title="sklearn.tree.export_graphviz"><code class="xref py py-func docutils literal notranslate"><span class="pre">export_graphviz</span></code></a>
exporter. If you use the <a class="reference external" href="https://conda.io">conda</a> package manager, the graphviz binaries
and the python package can be installed with <code class="docutils literal notranslate"><span class="pre">conda</span> <span class="pre">install</span> <span class="pre">python-graphviz</span></code>.</p>
<p class="sd-card-text">Alternatively binaries for graphviz can be downloaded from the graphviz project homepage,
and the Python wrapper installed from pypi with <code class="docutils literal notranslate"><span class="pre">pip</span> <span class="pre">install</span> <span class="pre">graphviz</span></code>.</p>
<p class="sd-card-text">Below is an example graphviz export of the above tree trained on the entire
iris dataset; the results are saved in an output file <code class="docutils literal notranslate"><span class="pre">iris.pdf</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">graphviz</span>
<span class="gp">>>> </span><span class="n">dot_data</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">export_graphviz</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">out_file</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">graph</span> <span class="o">=</span> <span class="n">graphviz</span><span class="o">.</span><span class="n">Source</span><span class="p">(</span><span class="n">dot_data</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">graph</span><span class="o">.</span><span class="n">render</span><span class="p">(</span><span class="s2">"iris"</span><span class="p">)</span>
</pre></div>
</div>
<p class="sd-card-text">The <a class="reference internal" href="generated/sklearn.tree.export_graphviz.html#sklearn.tree.export_graphviz" title="sklearn.tree.export_graphviz"><code class="xref py py-func docutils literal notranslate"><span class="pre">export_graphviz</span></code></a> exporter also supports a variety of aesthetic
options, including coloring nodes by their class (or value for regression) and
using explicit variable and class names if desired. Jupyter notebooks also
render these plots inline automatically:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">dot_data</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">export_graphviz</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">out_file</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">feature_names</span><span class="o">=</span><span class="n">iris</span><span class="o">.</span><span class="n">feature_names</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">class_names</span><span class="o">=</span><span class="n">iris</span><span class="o">.</span><span class="n">target_names</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">filled</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">special_characters</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">graph</span> <span class="o">=</span> <span class="n">graphviz</span><span class="o">.</span><span class="n">Source</span><span class="p">(</span><span class="n">dot_data</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">graph</span>
</pre></div>
</div>
<figure class="align-center">
<img alt="../_images/iris.svg" src="../_images/iris.svg" />
</figure>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/tree/plot_iris_dtc.html"><img alt="../_images/sphx_glr_plot_iris_dtc_001.png" src="../_images/sphx_glr_plot_iris_dtc_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p class="sd-card-text">Alternatively, the tree can also be exported in textual format with the
function <a class="reference internal" href="generated/sklearn.tree.export_text.html#sklearn.tree.export_text" title="sklearn.tree.export_text"><code class="xref py py-func docutils literal notranslate"><span class="pre">export_text</span></code></a>. This method doesn’t require the installation
of external libraries and is more compact:</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.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_text</span>
<span class="gp">>>> </span><span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">decision_tree</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</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">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">decision_tree</span> <span class="o">=</span> <span class="n">decision_tree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">r</span> <span class="o">=</span> <span class="n">export_text</span><span class="p">(</span><span class="n">decision_tree</span><span class="p">,</span> <span class="n">feature_names</span><span class="o">=</span><span class="n">iris</span><span class="p">[</span><span class="s1">'feature_names'</span><span class="p">])</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
<span class="go">|--- petal width (cm) <= 0.80</span>
<span class="go">| |--- class: 0</span>
<span class="go">|--- petal width (cm) > 0.80</span>
<span class="go">| |--- petal width (cm) <= 1.75</span>
<span class="go">| | |--- class: 1</span>
<span class="go">| |--- petal width (cm) > 1.75</span>
<span class="go">| | |--- class: 2</span>
</pre></div>
</div>
</div>
</details><p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/tree/plot_iris_dtc.html#sphx-glr-auto-examples-tree-plot-iris-dtc-py"><span class="std std-ref">Plot the decision surface of decision trees trained on the iris dataset</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/tree/plot_unveil_tree_structure.html#sphx-glr-auto-examples-tree-plot-unveil-tree-structure-py"><span class="std std-ref">Understanding the decision tree structure</span></a></p></li>
</ul>
</section>
<section id="regression">
<span id="tree-regression"></span><h2><span class="section-number">1.10.2. </span>Regression<a class="headerlink" href="#regression" title="Link to this heading">#</a></h2>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/tree/plot_tree_regression.html"><img alt="../_images/sphx_glr_plot_tree_regression_001.png" src="../_images/sphx_glr_plot_tree_regression_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p>Decision trees can also be applied to regression problems, using the
<a class="reference internal" href="generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="sklearn.tree.DecisionTreeRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">DecisionTreeRegressor</span></code></a> class.</p>
<p>As in the classification setting, the fit method will take as argument arrays X
and y, only that in this case y is expected to have floating point values
instead of integer values:</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</span> <span class="kn">import</span> <span class="n">tree</span>
<span class="gp">>>> </span><span class="n">X</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="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</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeRegressor</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">clf</span><span class="o">.</span><span class="n">predict</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="go">array([0.5])</span>
</pre></div>
</div>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/tree/plot_tree_regression.html#sphx-glr-auto-examples-tree-plot-tree-regression-py"><span class="std std-ref">Decision Tree Regression</span></a></p></li>
</ul>
</section>
<section id="multi-output-problems">
<span id="tree-multioutput"></span><h2><span class="section-number">1.10.3. </span>Multi-output problems<a class="headerlink" href="#multi-output-problems" title="Link to this heading">#</a></h2>
<p>A multi-output problem is a supervised learning problem with several outputs
to predict, that is when Y is a 2d array of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_outputs)</span></code>.</p>
<p>When there is no correlation between the outputs, a very simple way to solve
this kind of problem is to build n independent models, i.e. one for each
output, and then to use those models to independently predict each one of the n
outputs. However, because it is likely that the output values related to the
same input are themselves correlated, an often better way is to build a single
model capable of predicting simultaneously all n outputs. First, it requires
lower training time since only a single estimator is built. Second, the
generalization accuracy of the resulting estimator may often be increased.</p>
<p>With regard to decision trees, this strategy can readily be used to support
multi-output problems. This requires the following changes:</p>
<ul class="simple">
<li><p>Store n output values in leaves, instead of 1;</p></li>
<li><p>Use splitting criteria that compute the average reduction across all
n outputs.</p></li>
</ul>
<p>This module offers support for multi-output problems by implementing this
strategy in both <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">DecisionTreeClassifier</span></code></a> and
<a class="reference internal" href="generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="sklearn.tree.DecisionTreeRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">DecisionTreeRegressor</span></code></a>. If a decision tree is fit on an output array Y
of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_outputs)</span></code> then the resulting estimator will:</p>
<ul class="simple">
<li><p>Output n_output values upon <code class="docutils literal notranslate"><span class="pre">predict</span></code>;</p></li>
<li><p>Output a list of n_output arrays of class probabilities upon
<code class="docutils literal notranslate"><span class="pre">predict_proba</span></code>.</p></li>
</ul>
<p>The use of multi-output trees for regression is demonstrated in
<a class="reference internal" href="../auto_examples/tree/plot_tree_regression_multioutput.html#sphx-glr-auto-examples-tree-plot-tree-regression-multioutput-py"><span class="std std-ref">Multi-output Decision Tree Regression</span></a>. In this example, the input
X is a single real value and the outputs Y are the sine and cosine of X.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/tree/plot_tree_regression_multioutput.html"><img alt="../_images/sphx_glr_plot_tree_regression_multioutput_001.png" src="../_images/sphx_glr_plot_tree_regression_multioutput_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p>The use of multi-output trees for classification is demonstrated in
<a class="reference internal" href="../auto_examples/miscellaneous/plot_multioutput_face_completion.html#sphx-glr-auto-examples-miscellaneous-plot-multioutput-face-completion-py"><span class="std std-ref">Face completion with a multi-output estimators</span></a>. In this example, the inputs
X are the pixels of the upper half of faces and the outputs Y are the pixels of
the lower half of those faces.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/miscellaneous/plot_multioutput_face_completion.html"><img alt="../_images/sphx_glr_plot_multioutput_face_completion_001.png" src="../_images/sphx_glr_plot_multioutput_face_completion_001.png" style="width: 750.0px; height: 847.5px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/tree/plot_tree_regression_multioutput.html#sphx-glr-auto-examples-tree-plot-tree-regression-multioutput-py"><span class="std std-ref">Multi-output Decision Tree Regression</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/miscellaneous/plot_multioutput_face_completion.html#sphx-glr-auto-examples-miscellaneous-plot-multioutput-face-completion-py"><span class="std std-ref">Face completion with a multi-output estimators</span></a></p></li>
</ul>
<p class="rubric">References</p>
<ul class="simple">
<li><p>M. Dumont et al, <a class="reference external" href="http://www.montefiore.ulg.ac.be/services/stochastic/pubs/2009/DMWG09/dumont-visapp09-shortpaper.pdf">Fast multi-class image annotation with random subwindows
and multiple output randomized trees</a>,
International Conference on Computer Vision Theory and Applications 2009</p></li>
</ul>
</section>
<section id="complexity">
<span id="tree-complexity"></span><h2><span class="section-number">1.10.4. </span>Complexity<a class="headerlink" href="#complexity" title="Link to this heading">#</a></h2>
<p>In general, the run time cost to construct a balanced binary tree is
<span class="math notranslate nohighlight">\(O(n_{samples}n_{features}\log(n_{samples}))\)</span> and query time
<span class="math notranslate nohighlight">\(O(\log(n_{samples}))\)</span>. Although the tree construction algorithm attempts
to generate balanced trees, they will not always be balanced. Assuming that the
subtrees remain approximately balanced, the cost at each node consists of
searching through <span class="math notranslate nohighlight">\(O(n_{features})\)</span> to find the feature that offers the
largest reduction in the impurity criterion, e.g. log loss (which is equivalent to an
information gain). This has a cost of
<span class="math notranslate nohighlight">\(O(n_{features}n_{samples}\log(n_{samples}))\)</span> at each node, leading to a
total cost over the entire trees (by summing the cost at each node) of
<span class="math notranslate nohighlight">\(O(n_{features}n_{samples}^{2}\log(n_{samples}))\)</span>.</p>
</section>
<section id="tips-on-practical-use">
<h2><span class="section-number">1.10.5. </span>Tips on practical use<a class="headerlink" href="#tips-on-practical-use" title="Link to this heading">#</a></h2>
<ul>
<li><p>Decision trees tend to overfit on data with a large number of features.
Getting the right ratio of samples to number of features is important, since