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<li><a class="reference internal" href="#">Isotonic Regression</a></li>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-miscellaneous-plot-isotonic-regression-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="isotonic-regression">
<span id="sphx-glr-auto-examples-miscellaneous-plot-isotonic-regression-py"></span><h1>Isotonic Regression<a class="headerlink" href="#isotonic-regression" title="Permalink to this heading">¶</a></h1>
<p>An illustration of the isotonic regression on generated data (non-linear
monotonic trend with homoscedastic uniform noise).</p>
<p>The isotonic regression algorithm finds a non-decreasing approximation of a
function while minimizing the mean squared error on the training data. The
benefit of such a non-parametric model is that it does not assume any shape for
the target function besides monotonicity. For comparison a linear regression is
also presented.</p>
<p>The plot on the right-hand side shows the model prediction function that
results from the linear interpolation of thresholds points. The thresholds
points are a subset of the training input observations and their matching
target values are computed by the isotonic non-parametric fit.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Author: Nelle Varoquaux <[email protected]></span>
<span class="c1"># Alexandre Gramfort <[email protected]></span>
<span class="c1"># License: BSD</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</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.collections</span> <span class="kn">import</span> <a href="https://matplotlib.org/stable/api/collections_api.html#matplotlib.collections.LineCollection" title="matplotlib.collections.LineCollection" class="sphx-glr-backref-module-matplotlib-collections sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LineCollection</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.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearRegression</span></a>
<span class="kn">from</span> <span class="nn">sklearn.isotonic</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression" class="sphx-glr-backref-module-sklearn-isotonic sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">IsotonicRegression</span></a>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.utils.check_random_state.html#sklearn.utils.check_random_state" title="sklearn.utils.check_random_state" class="sphx-glr-backref-module-sklearn-utils sphx-glr-backref-type-py-function"><span class="n">check_random_state</span></a>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">rs</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.utils.check_random_state.html#sklearn.utils.check_random_state" title="sklearn.utils.check_random_state" class="sphx-glr-backref-module-sklearn-utils sphx-glr-backref-type-py-function"><span class="n">check_random_state</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">rs</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="mi">50</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,))</span> <span class="o">+</span> <span class="mf">50.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.log1p.html#numpy.log1p" title="numpy.log1p" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">log1p</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="n">n</span><span class="p">))</span>
</pre></div>
</div>
<p>Fit IsotonicRegression and LinearRegression models:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">ir</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression" class="sphx-glr-backref-module-sklearn-isotonic sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">IsotonicRegression</span></a><span class="p">(</span><span class="n">out_of_bounds</span><span class="o">=</span><span class="s2">"clip"</span><span class="p">)</span>
<span class="n">y_</span> <span class="o">=</span> <span class="n">ir</span><span class="o">.</span><span class="n">fit_transform</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">lr</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression" class="sphx-glr-backref-module-sklearn-linear_model sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LinearRegression</span></a><span class="p">()</span>
<span class="n">lr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">],</span> <span class="n">y</span><span class="p">)</span> <span class="c1"># x needs to be 2d for LinearRegression</span>
</pre></div>
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<style>#sk-container-id-47 {color: black;background-color: white;}#sk-container-id-47 pre{padding: 0;}#sk-container-id-47 div.sk-toggleable {background-color: white;}#sk-container-id-47 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-47 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-47 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-47 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-47 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-47 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-47 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-47 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-47 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-47 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-47 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-47 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-47 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-47 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-47 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-47 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-47 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-47 div.sk-item {position: relative;z-index: 1;}#sk-container-id-47 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-47 div.sk-item::before, #sk-container-id-47 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-47 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-47 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-47 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-47 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-47 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-47 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-47 div.sk-label-container {text-align: center;}#sk-container-id-47 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-47 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-47" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>LinearRegression()</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-215" type="checkbox" checked><label for="sk-estimator-id-215" class="sk-toggleable__label sk-toggleable__label-arrow">LinearRegression</label><div class="sk-toggleable__content"><pre>LinearRegression()</pre></div></div></div></div></div>
</div>
<br />
<br /><p>Plot results:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">segments</span> <span class="o">=</span> <span class="p">[[[</span><span class="n">i</span><span class="p">,</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]],</span> <span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">y_</span><span class="p">[</span><span class="n">i</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</span><span class="p">)]</span>
<span class="n">lc</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/collections_api.html#matplotlib.collections.LineCollection" title="matplotlib.collections.LineCollection" class="sphx-glr-backref-module-matplotlib-collections sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LineCollection</span></a><span class="p">(</span><span class="n">segments</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">lc</span><span class="o">.</span><span class="n">set_array</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.ones.html#numpy.ones" title="numpy.ones" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">)))</span>
<span class="n">lc</span><span class="o">.</span><span class="n">set_linewidths</span><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.full.html#numpy.full" title="numpy.full" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">full</span></a><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">))</span>
<span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax0</span><span class="p">,</span> <span class="n">ax1</span><span class="p">)</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" 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">subplots</span></a><span class="p">(</span><span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</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="s2">"C0."</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</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="s2">"C1.-"</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">lr</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.newaxis" title="numpy.newaxis" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]),</span> <span class="s2">"C2-"</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">add_collection</span><span class="p">(</span><span class="n">lc</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s2">"Training data"</span><span class="p">,</span> <span class="s2">"Isotonic fit"</span><span class="p">,</span> <span class="s2">"Linear fit"</span><span class="p">),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Isotonic regression fit on noisy data (n=</span><span class="si">%d</span><span class="s2">)"</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
<span class="n">x_test</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">110</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">ir</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x_test</span><span class="p">),</span> <span class="s2">"C1-"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ir</span><span class="o">.</span><span class="n">X_thresholds_</span><span class="p">,</span> <span class="n">ir</span><span class="o">.</span><span class="n">y_thresholds_</span><span class="p">,</span> <span class="s2">"C1."</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Prediction function (</span><span class="si">%d</span><span class="s2"> thresholds)"</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">ir</span><span class="o">.</span><span class="n">X_thresholds_</span><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_isotonic_regression_001.png" srcset="../../_images/sphx_glr_plot_isotonic_regression_001.png" alt="Isotonic regression fit on noisy data (n=100), Prediction function (36 thresholds)" class = "sphx-glr-single-img"/><p>Note that we explicitly passed <code class="docutils literal notranslate"><span class="pre">out_of_bounds="clip"</span></code> to the constructor of
<code class="docutils literal notranslate"><span class="pre">IsotonicRegression</span></code> to control the way the model extrapolates outside of the
range of data observed in the training set. This “clipping” extrapolation can
be seen on the plot of the decision function on the right-hand.</p>
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