
.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/miscellaneous/plot_isotonic_regression.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_miscellaneous_plot_isotonic_regression.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_miscellaneous_plot_isotonic_regression.py:


===================
Isotonic Regression
===================

An illustration of the isotonic regression on generated data (non-linear
monotonic trend with homoscedastic uniform noise).

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.

The plot on the right-hand side shows the model prediction function that
results from the linear interpolation of threshold points. The threshold
points are a subset of the training input observations and their matching
target values are computed by the isotonic non-parametric fit.

.. GENERATED FROM PYTHON SOURCE LINES 21-38

.. code-block:: Python


    # Authors: The scikit-learn developers
    # SPDX-License-Identifier: BSD-3-Clause

    import matplotlib.pyplot as plt
    import numpy as np
    from matplotlib.collections import LineCollection

    from sklearn.isotonic import IsotonicRegression
    from sklearn.linear_model import LinearRegression
    from sklearn.utils import check_random_state

    n = 100
    x = np.arange(n)
    rs = check_random_state(0)
    y = rs.randint(-50, 50, size=(n,)) + 50.0 * np.log1p(np.arange(n))








.. GENERATED FROM PYTHON SOURCE LINES 39-40

Fit IsotonicRegression and LinearRegression models:

.. GENERATED FROM PYTHON SOURCE LINES 40-47

.. code-block:: Python


    ir = IsotonicRegression(out_of_bounds="clip")
    y_ = ir.fit_transform(x, y)

    lr = LinearRegression()
    lr.fit(x[:, np.newaxis], y)  # x needs to be 2d for LinearRegression






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    .total_features {
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      justify-content: center;
      margin-top: 0.5em;
    }
    </style><body><div id="sk-container-id-37" tabindex="0" class="sk-top-container sk-global"><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 sk-global" id="sk-estimator-id-132" type="checkbox" checked><label for="sk-estimator-id-132" class="sk-toggleable__label fitted sk-toggleable__label-arrow"><div><div>LinearRegression</div></div><div><a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/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></div></label><div class="sk-toggleable__content fitted" data-param-prefix="">
            <div class="estimator-table">
                <details>
                    <summary>Parameters</summary>
                    <table class="parameters-table">
                      <tbody>
                    
            <tr class="default">
                <td><i class="copy-paste-icon"
                     onclick="copyToClipboard('fit_intercept',
                              this.parentElement.nextElementSibling)"
                ></i></td>
                <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-fit_intercept;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=fit_intercept,-bool%2C%20default%3DTrue">
                fit_intercept
                <span class="param-doc-description"
                style="position-anchor: --doc-link-fit_intercept;">
                fit_intercept: bool, default=True<br><br>Whether to calculate the intercept for this model. If set<br>to False, no intercept will be used in calculations<br>(i.e. data is expected to be centered).</span>
            </a>
        </td>
                <td class="value">True</td>
            </tr>
    

            <tr class="default">
                <td><i class="copy-paste-icon"
                     onclick="copyToClipboard('copy_X',
                              this.parentElement.nextElementSibling)"
                ></i></td>
                <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-copy_X;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=copy_X,-bool%2C%20default%3DTrue">
                copy_X
                <span class="param-doc-description"
                style="position-anchor: --doc-link-copy_X;">
                copy_X: bool, default=True<br><br>If True, X will be copied; else, it may be overwritten.</span>
            </a>
        </td>
                <td class="value">True</td>
            </tr>
    

            <tr class="default">
                <td><i class="copy-paste-icon"
                     onclick="copyToClipboard('tol',
                              this.parentElement.nextElementSibling)"
                ></i></td>
                <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-tol;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=tol,-float%2C%20default%3D1e-6">
                tol
                <span class="param-doc-description"
                style="position-anchor: --doc-link-tol;">
                tol: float, default=1e-6<br><br>The precision of the solution (`coef_`) is determined by `tol` which<br>specifies the convergence criterion of the underlying solver. `tol` is<br>set as `atol` and `btol` of :func:`scipy.sparse.linalg.lsqr` when<br>fitting on sparse training data. `tol` is set as `cond` of<br>:func:`scipy.linalg.lstsq` when fitting on dense training data.<br><br>.. versionadded:: 1.7<br>.. versionchanged:: 1.9<br>    Now supported on dense data, interpreted as the `cond` parameter.</span>
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                <td class="value">1e-06</td>
            </tr>
    

            <tr class="default">
                <td><i class="copy-paste-icon"
                     onclick="copyToClipboard('n_jobs',
                              this.parentElement.nextElementSibling)"
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                <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-n_jobs;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=n_jobs,-int%2C%20default%3DNone">
                n_jobs
                <span class="param-doc-description"
                style="position-anchor: --doc-link-n_jobs;">
                n_jobs: int, default=None<br><br>The number of jobs to use for the computation. This will only provide<br>speedup in case of sufficiently large problems, that is if firstly<br>`n_targets &gt; 1` and secondly `X` is sparse or if `positive` is set<br>to `True`. ``None`` means 1 unless in a<br>:obj:`joblib.parallel_backend` context. ``-1`` means using all<br>processors. See :term:`Glossary &lt;n_jobs&gt;` for more details.</span>
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        </td>
                <td class="value">None</td>
            </tr>
    

            <tr class="default">
                <td><i class="copy-paste-icon"
                     onclick="copyToClipboard('positive',
                              this.parentElement.nextElementSibling)"
                ></i></td>
                <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-positive;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=positive,-bool%2C%20default%3DFalse">
                positive
                <span class="param-doc-description"
                style="position-anchor: --doc-link-positive;">
                positive: bool, default=False<br><br>When set to ``True``, forces the coefficients to be positive. This<br>option is only supported for dense arrays.<br><br>For a comparison between a linear regression model with positive constraints<br>on the regression coefficients and a linear regression without such constraints,<br>see :ref:`sphx_glr_auto_examples_linear_model_plot_nnls.py`.<br><br>.. versionadded:: 0.24</span>
            </a>
        </td>
                <td class="value">False</td>
            </tr>
    
                      </tbody>
                    </table>
                </details>
            </div>
    
            <div class="estimator-table">
                <details>
                    <summary>Fitted attributes</summary>
                    <table class="parameters-table">
                        <tbody>
                            <tr>
                            <th>Name</th>
                            <th>Type</th>
                            <th>Value</th>
                            </tr>
                        
           <tr class="default">
               <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-coef_;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=coef_,-array%20of%20shape%20%28n_features%2C%20%29%20or%20%28n_targets%2C%20n_features%29">
                coef_
                <span class="param-doc-description"
                style="position-anchor: --doc-link-coef_;">
                coef_: array of shape (n_features, ) or (n_targets, n_features)<br><br>Estimated coefficients for the linear regression problem.<br>If multiple targets are passed during the fit (y 2D), this<br>is a 2D array of shape (n_targets, n_features), while if only<br>one target is passed, this is a 1D array of length n_features.</span>
            </a>
        </td>
               <td class="fitted-att-type">ndarray[float64](1,)</td>
               <td>[1.31]</td>


           </tr>
    

           <tr class="default">
               <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-intercept_;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=intercept_,-float%20or%20array%20of%20shape%20%28n_targets%2C%29">
                intercept_
                <span class="param-doc-description"
                style="position-anchor: --doc-link-intercept_;">
                intercept_: float or array of shape (n_targets,)<br><br>Independent term in the linear model. Set to 0.0 if<br>`fit_intercept = False`.</span>
            </a>
        </td>
               <td class="fitted-att-type">float64</td>
               <td>115.1</td>


           </tr>
    

           <tr class="default">
               <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-n_features_in_;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=n_features_in_,-int">
                n_features_in_
                <span class="param-doc-description"
                style="position-anchor: --doc-link-n_features_in_;">
                n_features_in_: int<br><br>Number of features seen during :term:`fit`.<br><br>.. versionadded:: 0.24</span>
            </a>
        </td>
               <td class="fitted-att-type">int</td>
               <td>1</td>


           </tr>
    

           <tr class="default">
               <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-rank_;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=rank_,-int">
                rank_
                <span class="param-doc-description"
                style="position-anchor: --doc-link-rank_;">
                rank_: int<br><br>Rank of matrix `X`. Only available when `X` is dense.</span>
            </a>
        </td>
               <td class="fitted-att-type">int</td>
               <td>1</td>


           </tr>
    

           <tr class="default">
               <td class="param">
            <a class="param-doc-link"
                style="anchor-name: --doc-link-singular_;"
                rel="noreferrer" target="_blank" href="https://scikit-learn.org/1.9/modules/generated/sklearn.linear_model.LinearRegression.html#:~:text=singular_,-array%20of%20shape%20%28min%28X%2C%20y%29%2C%29">
                singular_
                <span class="param-doc-description"
                style="position-anchor: --doc-link-singular_;">
                singular_: array of shape (min(X, y),)<br><br>Singular values of `X`. Only available when `X` is dense.</span>
            </a>
        </td>
               <td class="fitted-att-type">ndarray[float64](1,)</td>
               <td>[288.66]</td>


           </tr>
    
                        </tbody>
                    </table>
                </details>
            </div>
        </div></div></div></div></div><script>/*  Authors: The scikit-learn developers
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.. GENERATED FROM PYTHON SOURCE LINES 48-49

Plot results:

.. GENERATED FROM PYTHON SOURCE LINES 49-71

.. code-block:: Python


    segments = [[[i, y[i]], [i, y_[i]]] for i in range(n)]
    lc = LineCollection(segments, zorder=0)
    lc.set_array(np.ones(len(y)))
    lc.set_linewidths(np.full(n, 0.5))

    fig, (ax0, ax1) = plt.subplots(ncols=2, figsize=(12, 6))

    ax0.plot(x, y, "C0.", markersize=12)
    ax0.plot(x, y_, "C1.-", markersize=12)
    ax0.plot(x, lr.predict(x[:, np.newaxis]), "C2-")
    ax0.add_collection(lc)
    ax0.legend(("Training data", "Isotonic fit", "Linear fit"), loc="lower right")
    ax0.set_title("Isotonic regression fit on noisy data (n=%d)" % n)

    x_test = np.linspace(-10, 110, 1000)
    ax1.plot(x_test, ir.predict(x_test), "C1-")
    ax1.plot(ir.X_thresholds_, ir.y_thresholds_, "C1.", markersize=12)
    ax1.set_title("Prediction function (%d thresholds)" % len(ir.X_thresholds_))

    plt.show()




.. image-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_isotonic_regression_001.png
   :alt: Isotonic regression fit on noisy data (n=100), Prediction function (36 thresholds)
   :srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_isotonic_regression_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 72-76

Note that we explicitly passed `out_of_bounds="clip"` to the constructor of
`IsotonicRegression` 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.


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.129 seconds)


.. _sphx_glr_download_auto_examples_miscellaneous_plot_isotonic_regression.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_isotonic_regression.ipynb <plot_isotonic_regression.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_isotonic_regression.py <plot_isotonic_regression.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_isotonic_regression.zip <plot_isotonic_regression.zip>`


.. include:: plot_isotonic_regression.recommendations


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
