.. 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. or to run this example in your browser via Binder

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

.. _sphx_glr_auto_examples_miscellaneous_plot_isotonic_regression.py:


===================
保序回归
===================

在生成的数据上展示保序回归（具有同方差均匀噪声的非线性单调趋势）。

保序回归算法在训练数据上找到函数的非递减近似，同时最小化均方误差。这种非参数模型的优点在于，它除了单调性之外，不对目标函数的形状做任何假设。为了比较，还展示了线性回归。

右侧的图显示了通过阈值点的线性插值得到的模型预测函数。阈值点是训练输入观测值的一个子集，其匹配的目标值由保序非参数拟合计算得出。

.. GENERATED FROM PYTHON SOURCE LINES 13-30

.. code-block:: Python


    # 作者：scikit-learn 开发者
    # 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 31-32

拟合等渗回归和线性回归模型：

.. GENERATED FROM PYTHON SOURCE LINES 32-40

.. 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






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <style>#sk-container-id-42 {
      /* Definition of color scheme common for light and dark mode */
      --sklearn-color-text: black;
      --sklearn-color-line: gray;
      /* Definition of color scheme for unfitted estimators */
      --sklearn-color-unfitted-level-0: #fff5e6;
      --sklearn-color-unfitted-level-1: #f6e4d2;
      --sklearn-color-unfitted-level-2: #ffe0b3;
      --sklearn-color-unfitted-level-3: chocolate;
      /* Definition of color scheme for fitted estimators */
      --sklearn-color-fitted-level-0: #f0f8ff;
      --sklearn-color-fitted-level-1: #d4ebff;
      --sklearn-color-fitted-level-2: #b3dbfd;
      --sklearn-color-fitted-level-3: cornflowerblue;

      /* Specific color for light theme */
      --sklearn-color-text-on-default-background: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, black)));
      --sklearn-color-background: var(--sg-background-color, var(--theme-background, var(--jp-layout-color0, white)));
      --sklearn-color-border-box: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, black)));
      --sklearn-color-icon: #696969;

      @media (prefers-color-scheme: dark) {
        /* Redefinition of color scheme for dark theme */
        --sklearn-color-text-on-default-background: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, white)));
        --sklearn-color-background: var(--sg-background-color, var(--theme-background, var(--jp-layout-color0, #111)));
        --sklearn-color-border-box: var(--sg-text-color, var(--theme-code-foreground, var(--jp-content-font-color1, white)));
        --sklearn-color-icon: #878787;
      }
    }

    #sk-container-id-42 {
      color: var(--sklearn-color-text);
    }

    #sk-container-id-42 pre {
      padding: 0;
    }

    #sk-container-id-42 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-42 div.sk-dashed-wrapped {
      border: 1px dashed var(--sklearn-color-line);
      margin: 0 0.4em 0.5em 0.4em;
      box-sizing: border-box;
      padding-bottom: 0.4em;
      background-color: var(--sklearn-color-background);
    }

    #sk-container-id-42 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-42 div.sk-text-repr-fallback {
      display: none;
    }

    div.sk-parallel-item,
    div.sk-serial,
    div.sk-item {
      /* draw centered vertical line to link estimators */
      background-image: linear-gradient(var(--sklearn-color-text-on-default-background), var(--sklearn-color-text-on-default-background));
      background-size: 2px 100%;
      background-repeat: no-repeat;
      background-position: center center;
    }

    /* Parallel-specific style estimator block */

    #sk-container-id-42 div.sk-parallel-item::after {
      content: "";
      width: 100%;
      border-bottom: 2px solid var(--sklearn-color-text-on-default-background);
      flex-grow: 1;
    }

    #sk-container-id-42 div.sk-parallel {
      display: flex;
      align-items: stretch;
      justify-content: center;
      background-color: var(--sklearn-color-background);
      position: relative;
    }

    #sk-container-id-42 div.sk-parallel-item {
      display: flex;
      flex-direction: column;
    }

    #sk-container-id-42 div.sk-parallel-item:first-child::after {
      align-self: flex-end;
      width: 50%;
    }

    #sk-container-id-42 div.sk-parallel-item:last-child::after {
      align-self: flex-start;
      width: 50%;
    }

    #sk-container-id-42 div.sk-parallel-item:only-child::after {
      width: 0;
    }

    /* Serial-specific style estimator block */

    #sk-container-id-42 div.sk-serial {
      display: flex;
      flex-direction: column;
      align-items: center;
      background-color: var(--sklearn-color-background);
      padding-right: 1em;
      padding-left: 1em;
    }


    /* Toggleable style: style used for estimator/Pipeline/ColumnTransformer box that is
    clickable and can be expanded/collapsed.
    - Pipeline and ColumnTransformer use this feature and define the default style
    - Estimators will overwrite some part of the style using the `sk-estimator` class
    */

    /* Pipeline and ColumnTransformer style (default) */

    #sk-container-id-42 div.sk-toggleable {
      /* Default theme specific background. It is overwritten whether we have a
      specific estimator or a Pipeline/ColumnTransformer */
      background-color: var(--sklearn-color-background);
    }

    /* Toggleable label */
    #sk-container-id-42 label.sk-toggleable__label {
      cursor: pointer;
      display: block;
      width: 100%;
      margin-bottom: 0;
      padding: 0.5em;
      box-sizing: border-box;
      text-align: center;
    }

    #sk-container-id-42 label.sk-toggleable__label-arrow:before {
      /* Arrow on the left of the label */
      content: "▸";
      float: left;
      margin-right: 0.25em;
      color: var(--sklearn-color-icon);
    }

    #sk-container-id-42 label.sk-toggleable__label-arrow:hover:before {
      color: var(--sklearn-color-text);
    }

    /* Toggleable content - dropdown */

    #sk-container-id-42 div.sk-toggleable__content {
      max-height: 0;
      max-width: 0;
      overflow: hidden;
      text-align: left;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-42 div.sk-toggleable__content.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    #sk-container-id-42 div.sk-toggleable__content pre {
      margin: 0.2em;
      border-radius: 0.25em;
      color: var(--sklearn-color-text);
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-42 div.sk-toggleable__content.fitted pre {
      /* unfitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    #sk-container-id-42 input.sk-toggleable__control:checked~div.sk-toggleable__content {
      /* Expand drop-down */
      max-height: 200px;
      max-width: 100%;
      overflow: auto;
    }

    #sk-container-id-42 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {
      content: "▾";
    }

    /* Pipeline/ColumnTransformer-specific style */

    #sk-container-id-42 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-42 div.sk-label.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator-specific style */

    /* Colorize estimator box */
    #sk-container-id-42 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-42 div.sk-estimator.fitted input.sk-toggleable__control:checked~label.sk-toggleable__label {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    #sk-container-id-42 div.sk-label label.sk-toggleable__label,
    #sk-container-id-42 div.sk-label label {
      /* The background is the default theme color */
      color: var(--sklearn-color-text-on-default-background);
    }

    /* On hover, darken the color of the background */
    #sk-container-id-42 div.sk-label:hover label.sk-toggleable__label {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    /* Label box, darken color on hover, fitted */
    #sk-container-id-42 div.sk-label.fitted:hover label.sk-toggleable__label.fitted {
      color: var(--sklearn-color-text);
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Estimator label */

    #sk-container-id-42 div.sk-label label {
      font-family: monospace;
      font-weight: bold;
      display: inline-block;
      line-height: 1.2em;
    }

    #sk-container-id-42 div.sk-label-container {
      text-align: center;
    }

    /* Estimator-specific */
    #sk-container-id-42 div.sk-estimator {
      font-family: monospace;
      border: 1px dotted var(--sklearn-color-border-box);
      border-radius: 0.25em;
      box-sizing: border-box;
      margin-bottom: 0.5em;
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-0);
    }

    #sk-container-id-42 div.sk-estimator.fitted {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-0);
    }

    /* on hover */
    #sk-container-id-42 div.sk-estimator:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-2);
    }

    #sk-container-id-42 div.sk-estimator.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-2);
    }

    /* Specification for estimator info (e.g. "i" and "?") */

    /* Common style for "i" and "?" */

    .sk-estimator-doc-link,
    a:link.sk-estimator-doc-link,
    a:visited.sk-estimator-doc-link {
      float: right;
      font-size: smaller;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1em;
      height: 1em;
      width: 1em;
      text-decoration: none !important;
      margin-left: 1ex;
      /* unfitted */
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
      color: var(--sklearn-color-unfitted-level-1);
    }

    .sk-estimator-doc-link.fitted,
    a:link.sk-estimator-doc-link.fitted,
    a:visited.sk-estimator-doc-link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    div.sk-estimator:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover,
    div.sk-label-container:hover .sk-estimator-doc-link:hover,
    .sk-estimator-doc-link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    div.sk-estimator.fitted:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover,
    div.sk-label-container:hover .sk-estimator-doc-link.fitted:hover,
    .sk-estimator-doc-link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    /* Span, style for the box shown on hovering the info icon */
    .sk-estimator-doc-link span {
      display: none;
      z-index: 9999;
      position: relative;
      font-weight: normal;
      right: .2ex;
      padding: .5ex;
      margin: .5ex;
      width: min-content;
      min-width: 20ex;
      max-width: 50ex;
      color: var(--sklearn-color-text);
      box-shadow: 2pt 2pt 4pt #999;
      /* unfitted */
      background: var(--sklearn-color-unfitted-level-0);
      border: .5pt solid var(--sklearn-color-unfitted-level-3);
    }

    .sk-estimator-doc-link.fitted span {
      /* fitted */
      background: var(--sklearn-color-fitted-level-0);
      border: var(--sklearn-color-fitted-level-3);
    }

    .sk-estimator-doc-link:hover span {
      display: block;
    }

    /* "?"-specific style due to the `<a>` HTML tag */

    #sk-container-id-42 a.estimator_doc_link {
      float: right;
      font-size: 1rem;
      line-height: 1em;
      font-family: monospace;
      background-color: var(--sklearn-color-background);
      border-radius: 1rem;
      height: 1rem;
      width: 1rem;
      text-decoration: none;
      /* unfitted */
      color: var(--sklearn-color-unfitted-level-1);
      border: var(--sklearn-color-unfitted-level-1) 1pt solid;
    }

    #sk-container-id-42 a.estimator_doc_link.fitted {
      /* fitted */
      border: var(--sklearn-color-fitted-level-1) 1pt solid;
      color: var(--sklearn-color-fitted-level-1);
    }

    /* On hover */
    #sk-container-id-42 a.estimator_doc_link:hover {
      /* unfitted */
      background-color: var(--sklearn-color-unfitted-level-3);
      color: var(--sklearn-color-background);
      text-decoration: none;
    }

    #sk-container-id-42 a.estimator_doc_link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
    }
    </style><div id="sk-container-id-42" 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-176" type="checkbox" checked><label for="sk-estimator-id-176" class="sk-toggleable__label fitted sk-toggleable__label-arrow fitted">&nbsp;&nbsp;LinearRegression<a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/dev/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 />

.. GENERATED FROM PYTHON SOURCE LINES 41-42

绘制结果：

.. GENERATED FROM PYTHON SOURCE LINES 42-64

.. 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 65-66

请注意，我们显式地将 `out_of_bounds="clip"` 传递给 `IsotonicRegression` 的构造函数，以控制模型在训练集中观察到的数据范围之外进行外推的方法。这种“剪辑”外推可以在右侧的决策函数图中看到。


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

   **Total running time of the script:** (0 minutes 0.066 seconds)


.. _sphx_glr_download_auto_examples_miscellaneous_plot_isotonic_regression.py:

.. only:: html

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

    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/miscellaneous/plot_isotonic_regression.ipynb
        :alt: Launch binder
        :width: 150 px

    .. 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>`_