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<div class="section" id="probability-calibration-of-classifiers">
<span id="example-calibration-plot-calibration-py"></span><h1>Probability calibration of classifiers<a class="headerlink" href="#probability-calibration-of-classifiers" title="Permalink to this headline">¶</a></h1>
<p>When performing classification you often want to predict not only
the class label, but also the associated probability. This probability
gives you some kind of confidence on the prediction. However, not all
classifiers provide well-calibrated probabilities, some being over-confident
while others being under-confident. Thus, a separate calibration of predicted
probabilities is often desirable as a postprocessing. This example illustrates
two different methods for this calibration and evaluates the quality of the
returned probabilities using Brier’s score
(see <a class="reference external" href="http://en.wikipedia.org/wiki/Brier_score">http://en.wikipedia.org/wiki/Brier_score</a>).</p>
<p>Compared are the estimated probability using a Gaussian naive Bayes classifier
without calibration, with a sigmoid calibration, and with a non-parametric
isotonic calibration. One can observe that only the non-parametric model is able
to provide a probability calibration that returns probabilities close to the
expected 0.5 for most of the samples belonging to the middle cluster with
heterogeneous labels. This results in a significantly improved Brier score.</p>
<ul class="horizontal">
<li><a class="first reference internal image-reference" href="../../_images/plot_calibration_001.png"><img alt="../../_images/plot_calibration_001.png" src="../../_images/plot_calibration_001.png" style="width: 376.0px; height: 282.0px;" /></a>
</li>
<li><a class="first reference internal image-reference" href="../../_images/plot_calibration_002.png"><img alt="../../_images/plot_calibration_002.png" src="../../_images/plot_calibration_002.png" style="width: 376.0px; height: 282.0px;" /></a>
</li>
</ul>
<p><strong>Script output</strong>:</p>
<div class="highlight-python"><div class="highlight"><pre>Brier scores: (the smaller the better)
No calibration: 0.104
With isotonic calibration: 0.084
With sigmoid calibration: 0.109
</pre></div>
</div>
<p><strong>Python source code:</strong> <a class="reference download internal" href="../../_downloads/plot_calibration.py"><code class="xref download docutils literal"><span class="pre">plot_calibration.py</span></code></a></p>
<div class="highlight-python"><div class="highlight"><pre><span class="k">print</span><span class="p">(</span><span class="n">__doc__</span><span class="p">)</span>
<span class="c1"># Author: Mathieu Blondel <[email protected]></span>
<span class="c1"># Alexandre Gramfort <[email protected]></span>
<span class="c1"># Balazs Kegl <[email protected]></span>
<span class="c1"># Jan Hendrik Metzen <[email protected]></span>
<span class="c1"># License: BSD Style.</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs"><span class="n">make_blobs</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"><span class="n">GaussianNB</span></a>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.metrics.brier_score_loss.html#sklearn.metrics.brier_score_loss"><span class="n">brier_score_loss</span></a>
<span class="kn">from</span> <span class="nn">sklearn.calibration</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV"><span class="n">CalibratedClassifierCV</span></a>
<span class="kn">from</span> <span class="nn">sklearn.cross_validation</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.cross_validation.train_test_split.html#sklearn.cross_validation.train_test_split"><span class="n">train_test_split</span></a>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">50000</span>
<span class="n">n_bins</span> <span class="o">=</span> <span class="mi">3</span> <span class="c1"># use 3 bins for calibration_curve as we have 3 clusters here</span>
<span class="c1"># Generate 3 blobs with 2 classes where the second blob contains</span>
<span class="c1"># half positive samples and half negative samples. Probability in this</span>
<span class="c1"># blob is therefore 0.5.</span>
<span class="n">centers</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">),</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">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)]</span>
<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_blobs.html#sklearn.datasets.make_blobs"><span class="n">make_blobs</span></a><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">cluster_std</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
<span class="n">centers</span><span class="o">=</span><span class="n">centers</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="bp">False</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">y</span><span class="p">[:</span><span class="n">n_samples</span> <span class="o">//</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">y</span><span class="p">[</span><span class="n">n_samples</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">sample_weight</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="c1"># split train, test for calibration</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="p">,</span> <span class="n">sw_train</span><span class="p">,</span> <span class="n">sw_test</span> <span class="o">=</span> \
<a href="../../modules/generated/sklearn.cross_validation.train_test_split.html#sklearn.cross_validation.train_test_split"><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">sample_weight</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.9</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="c1"># Gaussian Naive-Bayes with no calibration</span>
<span class="n">clf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB"><span class="n">GaussianNB</span></a><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="c1"># GaussianNB itself does not support sample-weights</span>
<span class="n">prob_pos_clf</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test</span><span class="p">)[:,</span> <span class="mi">1</span><span class="p">]</span>
<span class="c1"># Gaussian Naive-Bayes with isotonic calibration</span>
<span class="n">clf_isotonic</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV"><span class="n">CalibratedClassifierCV</span></a><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s1">'isotonic'</span><span class="p">)</span>
<span class="n">clf_isotonic</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">sw_train</span><span class="p">)</span>
<span class="n">prob_pos_isotonic</span> <span class="o">=</span> <span class="n">clf_isotonic</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test</span><span class="p">)[:,</span> <span class="mi">1</span><span class="p">]</span>
<span class="c1"># Gaussian Naive-Bayes with sigmoid calibration</span>
<span class="n">clf_sigmoid</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.calibration.CalibratedClassifierCV.html#sklearn.calibration.CalibratedClassifierCV"><span class="n">CalibratedClassifierCV</span></a><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">cv</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s1">'sigmoid'</span><span class="p">)</span>
<span class="n">clf_sigmoid</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">sw_train</span><span class="p">)</span>
<span class="n">prob_pos_sigmoid</span> <span class="o">=</span> <span class="n">clf_sigmoid</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test</span><span class="p">)[:,</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"Brier scores: (the smaller the better)"</span><span class="p">)</span>
<span class="n">clf_score</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.brier_score_loss.html#sklearn.metrics.brier_score_loss"><span class="n">brier_score_loss</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">prob_pos_clf</span><span class="p">,</span> <span class="n">sw_test</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"No calibration: </span><span class="si">%1.3f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">clf_score</span><span class="p">)</span>
<span class="n">clf_isotonic_score</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.brier_score_loss.html#sklearn.metrics.brier_score_loss"><span class="n">brier_score_loss</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">prob_pos_isotonic</span><span class="p">,</span> <span class="n">sw_test</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"With isotonic calibration: </span><span class="si">%1.3f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">clf_isotonic_score</span><span class="p">)</span>
<span class="n">clf_sigmoid_score</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.brier_score_loss.html#sklearn.metrics.brier_score_loss"><span class="n">brier_score_loss</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">prob_pos_sigmoid</span><span class="p">,</span> <span class="n">sw_test</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">"With sigmoid calibration: </span><span class="si">%1.3f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">clf_sigmoid_score</span><span class="p">)</span>
<span class="c1">###############################################################################</span>
<span class="c1"># Plot the data and the predicted probabilities</span>
<a href="http://matplotlib.org/api/figure_api.html#matplotlib.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<span class="n">y_unique</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.unique.html#numpy.unique"><span class="n">np</span><span class="o">.</span><span class="n">unique</span></a><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="n">cm</span><span class="o">.</span><span class="n">rainbow</span><span class="p">(</span><a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.linspace.html#numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="n">y_unique</span><span class="o">.</span><span class="n">size</span><span class="p">))</span>
<span class="k">for</span> <span class="n">this_y</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">y_unique</span><span class="p">,</span> <span class="n">colors</span><span class="p">):</span>
<span class="n">this_X</span> <span class="o">=</span> <span class="n">X_train</span><span class="p">[</span><span class="n">y_train</span> <span class="o">==</span> <span class="n">this_y</span><span class="p">]</span>
<span class="n">this_sw</span> <span class="o">=</span> <span class="n">sw_train</span><span class="p">[</span><span class="n">y_train</span> <span class="o">==</span> <span class="n">this_y</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">this_X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">this_X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="n">this_sw</span> <span class="o">*</span> <span class="mi">50</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"Class </span><span class="si">%s</span><span class="s2">"</span> <span class="o">%</span> <span class="n">this_y</span><span class="p">)</span>
<a href="http://matplotlib.org/api/legend_api.html#matplotlib.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Data"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/figure_api.html#matplotlib.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<span class="n">order</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.lexsort.html#numpy.lexsort"><span class="n">np</span><span class="o">.</span><span class="n">lexsort</span></a><span class="p">((</span><span class="n">prob_pos_clf</span><span class="p">,</span> <span class="p">))</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">prob_pos_clf</span><span class="p">[</span><span class="n">order</span><span class="p">],</span> <span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'No calibration (</span><span class="si">%1.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="n">clf_score</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">prob_pos_isotonic</span><span class="p">[</span><span class="n">order</span><span class="p">],</span> <span class="s1">'g'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'Isotonic calibration (</span><span class="si">%1.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="n">clf_isotonic_score</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">prob_pos_sigmoid</span><span class="p">[</span><span class="n">order</span><span class="p">],</span> <span class="s1">'b'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'Sigmoid calibration (</span><span class="si">%1.3f</span><span class="s1">)'</span> <span class="o">%</span> <span class="n">clf_sigmoid_score</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.linspace.html#numpy.linspace"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">y_test</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="mi">51</span><span class="p">)[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">],</span>
<span class="n">y_test</span><span class="p">[</span><span class="n">order</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">25</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span>
<span class="s1">'k'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">r'Empirical'</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.ylim"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">([</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span> <span class="mf">1.05</span><span class="p">])</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"Instances sorted according to predicted probability "</span>
<span class="s2">"(uncalibrated GNB)"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"P(y=1)"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/legend_api.html#matplotlib.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper left"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Gaussian naive Bayes probabilities"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
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<p><strong>Total running time of the example:</strong> 0.45 seconds
( 0 minutes 0.45 seconds)</p>
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