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<li><a class="reference internal" href="#">1.15. Isotonic regression</a></li>
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  <section id="isotonic-regression">
<span id="isotonic"></span><h1><span class="section-number">1.15. </span>Isotonic regression<a class="headerlink" href="#isotonic-regression" title="Permalink to this heading">¶</a></h1>
<p>The class <a class="reference internal" href="generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">IsotonicRegression</span></code></a> fits a non-decreasing real function to
1-dimensional data. It solves the following problem:</p>
<blockquote>
<div><p>minimize <span class="math notranslate nohighlight">\(\sum_i w_i (y_i - \hat{y}_i)^2\)</span></p>
<p>subject to <span class="math notranslate nohighlight">\(\hat{y}_i \le \hat{y}_j\)</span> whenever <span class="math notranslate nohighlight">\(X_i \le X_j\)</span>,</p>
</div></blockquote>
<p>where the weights <span class="math notranslate nohighlight">\(w_i\)</span> are strictly positive, and both <code class="docutils literal notranslate"><span class="pre">X</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code> are
arbitrary real quantities.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">increasing</span></code> parameter changes the constraint to
<span class="math notranslate nohighlight">\(\hat{y}_i \ge \hat{y}_j\)</span> whenever <span class="math notranslate nohighlight">\(X_i \le X_j\)</span>. Setting it to
‘auto’ will automatically choose the constraint based on <a class="reference external" href="https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient">Spearman’s rank
correlation coefficient</a>.</p>
<p><a class="reference internal" href="generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">IsotonicRegression</span></code></a> produces a series of predictions
<span class="math notranslate nohighlight">\(\hat{y}_i\)</span> for the training data which are the closest to the targets
<span class="math notranslate nohighlight">\(y\)</span> in terms of mean squared error. These predictions are interpolated
for predicting to unseen data. The predictions of <a class="reference internal" href="generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">IsotonicRegression</span></code></a>
thus form a function that is piecewise linear:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/miscellaneous/plot_isotonic_regression.html"><img alt="../_images/sphx_glr_plot_isotonic_regression_001.png" src="../_images/sphx_glr_plot_isotonic_regression_001.png" /></a>
</figure>
</section>


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