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<li><a class="reference internal" href="#">Isotonic Regression</a></li>
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<p><a class="reference internal" href="#sphx-glr-download-auto-examples-miscellaneous-plot-isotonic-regression-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or 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="Link 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-Python 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">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</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.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.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.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-Python 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><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 fitted sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-211" type="checkbox" checked><label for="sk-estimator-id-211" class="sk-toggleable__label fitted sk-toggleable__label-arrow fitted"> LinearRegression<a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.4/modules/generated/sklearn.linear_model.LinearRegression.html">?<span>Documentation for LinearRegression</span></a><span class="sk-estimator-doc-link fitted">i<span>Fitted</span></span></label><div class="sk-toggleable__content fitted"><pre>LinearRegression()</pre></div> </div></div></div></div>
</div>
<br />
<br /><p>Plot results:</p>
<div class="highlight-Python 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>
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