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<meta property="og:description" content="The class IsotonicRegression fits a non-decreasing real function to 1-dimensional data. It solves the following problem:\min \sum_i w_i (y_i - \hat{y}_i)^2 subject to\hat{y}_i \le \hat{y}_j wheneve..." />
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<meta name="description" content="The class IsotonicRegression fits a non-decreasing real function to 1-dimensional data. It solves the following problem:\min \sum_i w_i (y_i - \hat{y}_i)^2 subject to\hat{y}_i \le \hat{y}_j wheneve..." />
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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="Link 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>
<div class="math notranslate nohighlight">
\[\min \sum_i w_i (y_i - \hat{y}_i)^2\]</div>
<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>,
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>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/miscellaneous/plot_isotonic_regression.html#sphx-glr-auto-examples-miscellaneous-plot-isotonic-regression-py"><span class="std std-ref">Isotonic Regression</span></a></p></li>
</ul>
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