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<li><a class="reference internal" href="#">Comparison of Calibration of Classifiers</a><ul>
<li><a class="reference internal" href="#dataset">Dataset</a></li>
<li><a class="reference internal" href="#calibration-curves">Calibration curves</a></li>
<li><a class="reference internal" href="#references">References</a></li>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-calibration-plot-compare-calibration-py"><span class="std std-ref">here</span></a>
to download the full example code or to run this example in your browser via Binder</p>
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<section class="sphx-glr-example-title" id="comparison-of-calibration-of-classifiers">
<span id="sphx-glr-auto-examples-calibration-plot-compare-calibration-py"></span><h1>Comparison of Calibration of Classifiers<a class="headerlink" href="#comparison-of-calibration-of-classifiers" title="Permalink to this heading">¶</a></h1>
<p>Well calibrated classifiers are probabilistic classifiers for which the output
of <a class="reference internal" href="../../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> can be directly interpreted as a confidence level.
For instance, a well calibrated (binary) classifier should classify the samples
such that for the samples to which it gave a <a class="reference internal" href="../../glossary.html#term-predict_proba"><span class="xref std std-term">predict_proba</span></a> value close
to 0.8, approximately 80% actually belong to the positive class.</p>
<p>In this example we will compare the calibration of four different
models: <a class="reference internal" href="../../modules/linear_model.html#logistic-regression"><span class="std std-ref">Logistic regression</span></a>, <a class="reference internal" href="../../modules/naive_bayes.html#gaussian-naive-bayes"><span class="std std-ref">Gaussian Naive Bayes</span></a>,
<a class="reference internal" href="../../modules/ensemble.html#forest"><span class="std std-ref">Random Forest Classifier</span></a> and <a class="reference internal" href="../../modules/svm.html#svm-classification"><span class="std std-ref">Linear SVM</span></a>.</p>
<p>Author: Jan Hendrik Metzen <<a class="reference external" href="mailto:jhm%40informatik.uni-bremen.de">jhm<span>@</span>informatik<span>.</span>uni-bremen<span>.</span>de</a>>
License: BSD 3 clause.</p>
<section id="dataset">
<h2>Dataset<a class="headerlink" href="#dataset" title="Permalink to this heading">¶</a></h2>
<p>We will use a synthetic binary classification dataset with 100,000 samples
and 20 features. Of the 20 features, only 2 are informative, 2 are
redundant (random combinations of the informative features) and the
remaining 16 are uninformative (random numbers). Of the 100,000 samples,
100 will be used for model fitting and the remaining for testing.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.make_classification.html#sklearn.datasets.make_classification" title="sklearn.datasets.make_classification" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_classification</span></a>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">train_test_split</span></a>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_classification.html#sklearn.datasets.make_classification" title="sklearn.datasets.make_classification" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_classification</span></a><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="mi">100_000</span><span class="p">,</span> <span class="n">n_features</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">n_informative</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_redundant</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">42</span>
<span class="p">)</span>
<span class="n">train_samples</span> <span class="o">=</span> <span class="mi">100</span> <span class="c1"># Samples used for training the models</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">train_test_split</span></a><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">shuffle</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">test_size</span><span class="o">=</span><span class="mi">100_000</span> <span class="o">-</span> <span class="n">train_samples</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
</section>
<section id="calibration-curves">
<h2>Calibration curves<a class="headerlink" href="#calibration-curves" title="Permalink to this heading">¶</a></h2>
<p>Below, we train each of the four models with the small training dataset, then
plot calibration curves (also known as reliability diagrams) using
predicted probabilities of the test dataset. Calibration curves are created
by binning predicted probabilities, then plotting the mean predicted
probability in each bin against the observed frequency (‘fraction of
positives’). Below the calibration curve, we plot a histogram showing
the distribution of the predicted probabilities or more specifically,
the number of samples in each predicted probability bin.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearSVC</span></a>
<span class="k">class</span> <span class="nc">NaivelyCalibratedLinearSVC</span><span class="p">(</span><a href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearSVC</span></a><span class="p">):</span>
<span class="w"> </span><span class="sd">"""LinearSVC with `predict_proba` method that naively scales</span>
<span class="sd"> `decision_function` output."""</span>
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</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="nb">super</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="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">df_min_</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">min</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">df_max_</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">predict_proba</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Min-max scale output of `decision_function` to [0,1]."""</span>
<span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">calibrated_df</span> <span class="o">=</span> <span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">df_min_</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">df_max_</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">df_min_</span><span class="p">)</span>
<span class="n">proba_pos_class</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.clip.html#numpy.clip" title="numpy.clip" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">clip</span></a><span class="p">(</span><span class="n">calibrated_df</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="n">proba_neg_class</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">proba_pos_class</span>
<span class="n">proba</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.c_.html#numpy.c_" title="numpy.c_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">c_</span></a><span class="p">[</span><span class="n">proba_neg_class</span><span class="p">,</span> <span class="n">proba_pos_class</span><span class="p">]</span>
<span class="k">return</span> <span class="n">proba</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.calibration</span> <span class="kn">import</span> <span class="n">CalibrationDisplay</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier" class="sphx-glr-backref-module-sklearn-ensemble sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RandomForestClassifier</span></a>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LogisticRegression</span></a>
<span class="kn">from</span> <span class="nn">sklearn.naive_bayes</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB" class="sphx-glr-backref-module-sklearn-naive_bayes sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianNB</span></a>
<span class="c1"># Create classifiers</span>
<span class="n">lr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LogisticRegression</span></a><span class="p">()</span>
<span class="n">gnb</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB" class="sphx-glr-backref-module-sklearn-naive_bayes sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GaussianNB</span></a><span class="p">()</span>
<span class="n">svc</span> <span class="o">=</span> <span class="n">NaivelyCalibratedLinearSVC</span><span class="p">(</span><span class="n">C</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
<span class="n">rfc</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier" class="sphx-glr-backref-module-sklearn-ensemble sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">RandomForestClassifier</span></a><span class="p">()</span>
<span class="n">clf_list</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">(</span><span class="n">lr</span><span class="p">,</span> <span class="s2">"Logistic"</span><span class="p">),</span>
<span class="p">(</span><span class="n">gnb</span><span class="p">,</span> <span class="s2">"Naive Bayes"</span><span class="p">),</span>
<span class="p">(</span><span class="n">svc</span><span class="p">,</span> <span class="s2">"SVC"</span><span class="p">),</span>
<span class="p">(</span><span class="n">rfc</span><span class="p">,</span> <span class="s2">"Random forest"</span><span class="p">),</span>
<span class="p">]</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">matplotlib.gridspec</span> <span class="kn">import</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.gridspec.GridSpec.html#matplotlib.gridspec.GridSpec" title="matplotlib.gridspec.GridSpec" class="sphx-glr-backref-module-matplotlib-gridspec sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSpec</span></a>
<span class="n">fig</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">gs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.gridspec.GridSpec.html#matplotlib.gridspec.GridSpec" title="matplotlib.gridspec.GridSpec" class="sphx-glr-backref-module-matplotlib-gridspec sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSpec</span></a><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">get_cmap</span><span class="p">(</span><span class="s2">"Dark2"</span><span class="p">)</span>
<span class="n">ax_calibration_curve</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">gs</span><span class="p">[:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">])</span>
<span class="n">calibration_displays</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">markers</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"^"</span><span class="p">,</span> <span class="s2">"v"</span><span class="p">,</span> <span class="s2">"s"</span><span class="p">,</span> <span class="s2">"o"</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">clf_list</span><span class="p">):</span>
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="n">display</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.calibration.CalibrationDisplay.html#sklearn.calibration.CalibrationDisplay.from_estimator" title="sklearn.calibration.CalibrationDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-calibration-CalibrationDisplay sphx-glr-backref-type-py-method"><span class="n">CalibrationDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="n">clf</span><span class="p">,</span>
<span class="n">X_test</span><span class="p">,</span>
<span class="n">y_test</span><span class="p">,</span>
<span class="n">n_bins</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="n">name</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">ax_calibration_curve</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">(</span><span class="n">i</span><span class="p">),</span>
<span class="n">marker</span><span class="o">=</span><span class="n">markers</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
<span class="p">)</span>
<span class="n">calibration_displays</span><span class="p">[</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">display</span>
<span class="n">ax_calibration_curve</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax_calibration_curve</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Calibration plots"</span><span class="p">)</span>
<span class="c1"># Add histogram</span>
<span class="n">grid_positions</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="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">clf_list</span><span class="p">):</span>
<span class="n">row</span><span class="p">,</span> <span class="n">col</span> <span class="o">=</span> <span class="n">grid_positions</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">gs</span><span class="p">[</span><span class="n">row</span><span class="p">,</span> <span class="n">col</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span>
<span class="n">calibration_displays</span><span class="p">[</span><span class="n">name</span><span class="p">]</span><span class="o">.</span><span class="n">y_prob</span><span class="p">,</span>
<span class="nb">range</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="n">bins</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="n">name</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">(</span><span class="n">i</span><span class="p">),</span>
<span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">title</span><span class="o">=</span><span class="n">name</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="s2">"Mean predicted probability"</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">"Count"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_compare_calibration_001.png" srcset="../../_images/sphx_glr_plot_compare_calibration_001.png" alt="Calibration plots, Logistic, Naive Bayes, SVC, Random forest" class = "sphx-glr-single-img"/><div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>/home/circleci/project/sklearn/calibration.py:1176: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "s-" (-> marker='s'). The keyword argument will take precedence.
/home/circleci/project/sklearn/calibration.py:1176: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "s-" (-> marker='s'). The keyword argument will take precedence.
/home/circleci/project/sklearn/calibration.py:1176: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "s-" (-> marker='s'). The keyword argument will take precedence.
/home/circleci/project/sklearn/calibration.py:1176: UserWarning:
marker is redundantly defined by the 'marker' keyword argument and the fmt string "s-" (-> marker='s'). The keyword argument will take precedence.
</pre></div>
</div>
<p><a class="reference internal" href="../../modules/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">LogisticRegression</span></code></a> returns well calibrated
predictions as it directly optimizes log-loss. In contrast, the other methods
return biased probabilities, with different biases for each method:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../../modules/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">GaussianNB</span></code></a> tends to push
probabilities to 0 or 1 (see histogram). This is mainly
because the naive Bayes equation only provides correct estimate of
probabilities when the assumption that features are conditionally
independent holds <a class="footnote-reference brackets" href="#id6" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a>. However, features tend to be positively correlated
and is the case with this dataset, which contains 2 features
generated as random linear combinations of the informative features. These
correlated features are effectively being ‘counted twice’, resulting in
pushing the predicted probabilities towards 0 and 1 <a class="footnote-reference brackets" href="#id7" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>.</p></li>
<li><p><a class="reference internal" href="../../modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RandomForestClassifier</span></code></a> shows the opposite
behavior: the histograms show peaks at approx. 0.2 and 0.9 probability,
while probabilities close to 0 or 1 are very rare. An explanation for this
is given by Niculescu-Mizil and Caruana <a class="footnote-reference brackets" href="#id5" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>: “Methods such as bagging and
random forests that average predictions from a base set of models can have
difficulty making predictions near 0 and 1 because variance in the
underlying base models will bias predictions that should be near zero or
one away from these values. Because predictions are restricted to the
interval [0,1], errors caused by variance tend to be one- sided near zero
and one. For example, if a model should predict p = 0 for a case, the only
way bagging can achieve this is if all bagged trees predict zero. If we add
noise to the trees that bagging is averaging over, this noise will cause
some trees to predict values larger than 0 for this case, thus moving the
average prediction of the bagged ensemble away from 0. We observe this
effect most strongly with random forests because the base-level trees
trained with random forests have relatively high variance due to feature
subsetting.” As a result, the calibration curve shows a characteristic
sigmoid shape, indicating that the classifier is under-confident
and could return probabilities closer to 0 or 1.</p></li>
<li><p>To show the performance of <a class="reference internal" href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a>, we naively
scale the output of the <a class="reference internal" href="../../glossary.html#term-decision_function"><span class="xref std std-term">decision_function</span></a> into [0, 1] by applying
min-max scaling, since SVC does not output probabilities by default.
<a class="reference internal" href="../../modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a> shows an
even more sigmoid curve than the
<a class="reference internal" href="../../modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RandomForestClassifier</span></code></a>, which is typical for
maximum-margin methods <a class="footnote-reference brackets" href="#id5" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> as they focus on difficult to classify samples
that are close to the decision boundary (the support vectors).</p></li>
</ul>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this heading">¶</a></h2>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id5" role="note">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id3">1</a>,<a role="doc-backlink" href="#id4">2</a>)</span>
<p><a class="reference external" href="https://dl.acm.org/doi/pdf/10.1145/1102351.1102430">Predicting Good Probabilities with Supervised Learning</a>,
A. Niculescu-Mizil & R. Caruana, ICML 2005</p>
</aside>
<aside class="footnote brackets" id="id6" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">2</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.ics.uci.edu/~pazzani/Publications/mlc96-pedro.pdf">Beyond independence: Conditions for the optimality of the simple
bayesian classifier</a>
Domingos, P., & Pazzani, M., Proc. 13th Intl. Conf. Machine Learning.
1996.</p>
</aside>
<aside class="footnote brackets" id="id7" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">3</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://citeseerx.ist.psu.edu/doc_view/pid/4f67a122ec3723f08ad5cbefecad119b432b3304">Obtaining calibrated probability estimates from decision trees and
naive Bayesian classifiers</a>
Zadrozny, Bianca, and Charles Elkan. Icml. Vol. 1. 2001.</p>
</aside>
</aside>
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