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<ul>
<li><a class="reference internal" href="#">Precision-Recall</a></li>
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<div class="section" id="precision-recall">
<span id="example-model-selection-plot-precision-recall-py"></span><h1>Precision-Recall<a class="headerlink" href="#precision-recall" title="Permalink to this headline">¶</a></h1>
<p>Example of Precision-Recall metric to evaluate classifier output quality.</p>
<p>In information retrieval, precision is a measure of result relevancy, while
recall is a measure of how many truly relevant results are returned. A high
area under the curve represents both high recall and high precision, where high
precision relates to a low false positive rate, and high recall relates to a
low false negative rate. High scores for both show that the classifier is
returning accurate results (high precision), as well as returning a majority of
all positive results (high recall).</p>
<p>A system with high recall but low precision returns many results, but most of
its predicted labels are incorrect when compared to the training labels. A
system with high precision but low recall is just the opposite, returning very
few results, but most of its predicted labels are correct when compared to the
training labels. An ideal system with high precision and high recall will
return many results, with all results labeled correctly.</p>
<p>Precision (<img class="math" src="../../_images/math/f48b617185733d8dd6712643f1ab17c736661a06.png" alt="P"/>) is defined as the number of true positives (<img class="math" src="../../_images/math/d3a93ce5fdd86a1a6cea538e4b3ad7616156ff25.png" alt="T_p"/>)
over the number of true positives plus the number of false positives
(<img class="math" src="../../_images/math/c65c2d023af7f064e3391aa7678358f3ed1ff117.png" alt="F_p"/>).</p>
<p><img class="math" src="../../_images/math/cfe9bd9df7580025210fdf53add52cb820e6872f.png" alt="P = \frac{T_p}{T_p+F_p}"/></p>
<p>Recall (<img class="math" src="../../_images/math/9d86170e7de539c0ff999de09621ee0c7b6c8ed0.png" alt="R"/>) is defined as the number of true positives (<img class="math" src="../../_images/math/d3a93ce5fdd86a1a6cea538e4b3ad7616156ff25.png" alt="T_p"/>)
over the number of true positives plus the number of false negatives
(<img class="math" src="../../_images/math/bacdaf2f979238c05ec1690cd4777be7056b07d7.png" alt="F_n"/>).</p>
<p><img class="math" src="../../_images/math/f8d5d8d5e7d9ea881f1b65e56a38ae02156a773f.png" alt="R = \frac{T_p}{T_p + F_n}"/></p>
<p>These quantities are also related to the (<img class="math" src="../../_images/math/86b53a99470bfe410dd9108afdb7a4464f2461fb.png" alt="F_1"/>) score, which is defined
as the harmonic mean of precision and recall.</p>
<p><img class="math" src="../../_images/math/51b9aeb9655b757d685a814093e49924caee8e33.png" alt="F1 = 2\frac{P \times R}{P+R}"/></p>
<p>It is important to note that the precision may not decrease with recall. The
definition of precision (<img class="math" src="../../_images/math/d07adc1a4155b273cdcd51dabb24c19b31c436a4.png" alt="\frac{T_p}{T_p + F_p}"/>) shows that lowering
the threshold of a classifier may increase the denominator, by increasing the
number of results returned. If the threshold was previously set too high, the
new results may all be true positives, which will increase precision. If the
previous threshold was about right or too low, further lowering the threshold
will introduce false positives, decreasing precision.</p>
<p>Recall is defined as <img class="math" src="../../_images/math/43632e5e903214a015edb525608b1996530563d9.png" alt="\frac{T_p}{T_p+F_n}"/>, where <img class="math" src="../../_images/math/8cd6937b04645af63fcf95a6a147ff833b150ace.png" alt="T_p+F_n"/> does
not depend on the classifier threshold. This means that lowering the classifier
threshold may increase recall, by increasing the number of true positive
results. It is also possible that lowering the threshold may leave recall
unchanged, while the precision fluctuates.</p>
<p>The relationship between recall and precision can be observed in the
stairstep area of the plot - at the edges of these steps a small change
in the threshold considerably reduces precision, with only a minor gain in
recall. See the corner at recall = .59, precision = .8 for an example of this
phenomenon.</p>
<p>Precision-recall curves are typically used in binary classification to study
the output of a classifier. In order to extend Precision-recall curve and
average precision to multi-class or multi-label classification, it is necessary
to binarize the output. One curve can be drawn per label, but one can also draw
a precision-recall curve by considering each element of the label indicator
matrix as a binary prediction (micro-averaging).</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<dl class="last docutils">
<dt>See also <a class="reference internal" href="../../modules/generated/sklearn.metrics.average_precision_score.html#sklearn.metrics.average_precision_score" title="sklearn.metrics.average_precision_score"><tt class="xref py py-func docutils literal"><span class="pre">sklearn.metrics.average_precision_score</span></tt></a>,</dt>
<dd><a class="reference internal" href="../../modules/generated/sklearn.metrics.recall_score.html#sklearn.metrics.recall_score" title="sklearn.metrics.recall_score"><tt class="xref py py-func docutils literal"><span class="pre">sklearn.metrics.recall_score</span></tt></a>,
<a class="reference internal" href="../../modules/generated/sklearn.metrics.precision_score.html#sklearn.metrics.precision_score" title="sklearn.metrics.precision_score"><tt class="xref py py-func docutils literal"><span class="pre">sklearn.metrics.precision_score</span></tt></a>,
<a class="reference internal" href="../../modules/generated/sklearn.metrics.f1_score.html#sklearn.metrics.f1_score" title="sklearn.metrics.f1_score"><tt class="xref py py-func docutils literal"><span class="pre">sklearn.metrics.f1_score</span></tt></a></dd>
</dl>
</div>
<img alt="../../_images/plot_precision_recall_001.png" class="align-center" src="../../_images/plot_precision_recall_001.png" />
<p><strong>Python source code:</strong> <a class="reference download internal" href="../../_downloads/plot_precision_recall.py"><tt class="xref download docutils literal"><span class="pre">plot_precision_recall.py</span></tt></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="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">svm</span><span class="p">,</span> <span class="n">datasets</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.metrics.precision_recall_curve.html#sklearn.metrics.precision_recall_curve"><span class="n">precision_recall_curve</span></a>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.metrics.average_precision_score.html#sklearn.metrics.average_precision_score"><span class="n">average_precision_score</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="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.preprocessing.label_binarize.html#sklearn.preprocessing.label_binarize"><span class="n">label_binarize</span></a>
<span class="kn">from</span> <span class="nn">sklearn.multiclass</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.multiclass.OneVsRestClassifier.html#sklearn.multiclass.OneVsRestClassifier"><span class="n">OneVsRestClassifier</span></a>
<span class="c"># import some data to play with</span>
<span class="n">iris</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">X</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">data</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span>
<span class="c"># Binarize the output</span>
<span class="n">y</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.preprocessing.label_binarize.html#sklearn.preprocessing.label_binarize"><span class="n">label_binarize</span></a><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">classes</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="n">n_classes</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="c"># Add noisy features</span>
<span class="n">random_state</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">0</span><span class="p">)</span>
<span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="n">X</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.6.0/reference/generated/numpy.c_.html#numpy.c_"><span class="n">np</span><span class="o">.</span><span class="n">c_</span></a><span class="p">[</span><span class="n">X</span><span class="p">,</span> <span class="n">random_state</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">200</span> <span class="o">*</span> <span class="n">n_features</span><span class="p">)]</span>
<span class="c"># Split into training and test</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.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">test_size</span><span class="o">=.</span><span class="mi">5</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">)</span>
<span class="c"># Run classifier</span>
<span class="n">classifier</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.multiclass.OneVsRestClassifier.html#sklearn.multiclass.OneVsRestClassifier"><span class="n">OneVsRestClassifier</span></a><span class="p">(</span><a href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC"><span class="n">svm</span><span class="o">.</span><span class="n">SVC</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s">'linear'</span><span class="p">,</span> <span class="n">probability</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">))</span>
<span class="n">y_score</span> <span class="o">=</span> <span class="n">classifier</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="o">.</span><span class="n">decision_function</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c"># Compute Precision-Recall and plot curve</span>
<span class="n">precision</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="n">recall</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="n">average_precision</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_classes</span><span class="p">):</span>
<span class="n">precision</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">recall</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">_</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.precision_recall_curve.html#sklearn.metrics.precision_recall_curve"><span class="n">precision_recall_curve</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span>
<span class="n">y_score</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span>
<span class="n">average_precision</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.average_precision_score.html#sklearn.metrics.average_precision_score"><span class="n">average_precision_score</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">y_score</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span>
<span class="c"># Compute micro-average ROC curve and ROC area</span>
<span class="n">precision</span><span class="p">[</span><span class="s">"micro"</span><span class="p">],</span> <span class="n">recall</span><span class="p">[</span><span class="s">"micro"</span><span class="p">],</span> <span class="n">_</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.precision_recall_curve.html#sklearn.metrics.precision_recall_curve"><span class="n">precision_recall_curve</span></a><span class="p">(</span><span class="n">y_test</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span>
<span class="n">y_score</span><span class="o">.</span><span class="n">ravel</span><span class="p">())</span>
<span class="n">average_precision</span><span class="p">[</span><span class="s">"micro"</span><span class="p">]</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.metrics.average_precision_score.html#sklearn.metrics.average_precision_score"><span class="n">average_precision_score</span></a><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_score</span><span class="p">,</span>
<span class="n">average</span><span class="o">=</span><span class="s">"micro"</span><span class="p">)</span>
<span class="c"># Plot Precision-Recall curve</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.clf"><span class="n">plt</span><span class="o">.</span><span class="n">clf</span></a><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">recall</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">precision</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s">'Precision-Recall curve'</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="s">'Recall'</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="s">'Precision'</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="mf">0.0</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.xlim"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</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>
<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="s">'Precision-Recall example: AUC={0:0.2f}'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">average_precision</span><span class="p">[</span><span class="mi">0</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="s">"lower left"</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>
<span class="c"># Plot Precision-Recall curve for each class</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.clf"><span class="n">plt</span><span class="o">.</span><span class="n">clf</span></a><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">recall</span><span class="p">[</span><span class="s">"micro"</span><span class="p">],</span> <span class="n">precision</span><span class="p">[</span><span class="s">"micro"</span><span class="p">],</span>
<span class="n">label</span><span class="o">=</span><span class="s">'micro-average Precision-recall curve (area = {0:0.2f})'</span>
<span class="s">''</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">average_precision</span><span class="p">[</span><span class="s">"micro"</span><span class="p">]))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_classes</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">recall</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">precision</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
<span class="n">label</span><span class="o">=</span><span class="s">'Precision-recall curve of class {0} (area = {1:0.2f})'</span>
<span class="s">''</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">average_precision</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
<a href="http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.xlim"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</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>
<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="mf">0.0</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="s">'Recall'</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="s">'Precision'</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="s">'Extension of Precision-Recall curve to multi-class'</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="s">"lower right"</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> 0.30 seconds
( 0 minutes 0.30 seconds)</p>
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