.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/gaussian_process/plot_gpr_co2.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_gaussian_process_plot_gpr_co2.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_gaussian_process_plot_gpr_co2.py:


====================================================================================
使用高斯过程回归（GPR）对莫纳罗亚数据集的CO2水平进行预测
====================================================================================

本示例基于《高斯过程机器学习》第5.4.3节[1]。它展示了复杂核工程和使用梯度上升法对对数边际似然进行超参数优化的示例。数据包括1958年至2001年在夏威夷莫纳罗亚天文台收集的每月平均大气CO2浓度（以体积百万分之一（ppm）为单位）。目标是将CO2浓度建模为时间 :math:`t` 的函数，并对2001年以后的年份进行外推。

.. rubric:: 参考文献

.. [1] `Rasmussen, Carl Edward. "Gaussian processes in machine learning."
    Summer school on machine learning. Springer, Berlin, Heidelberg, 2003
    <http://www.gaussianprocess.org/gpml/chapters/RW.pdf>`_  .

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python


    print(__doc__)

    # 作者：scikit-learn 开发者
    # SPDX-License-Identifier：BSD-3-Clause








.. GENERATED FROM PYTHON SOURCE LINES 22-26

构建数据集
---------

我们将从毛纳罗亚天文台获取一个收集空气样本的数据集。我们感兴趣的是估算二氧化碳的浓度并将其外推到未来几年。首先，我们加载在OpenML上可用的原始数据集，并将其作为pandas数据框。这将在 `fetch_openml` 添加对Polars的原生支持后被替换。

.. GENERATED FROM PYTHON SOURCE LINES 26-31

.. code-block:: Python

    from sklearn.datasets import fetch_openml

    co2 = fetch_openml(data_id=41187, as_frame=True)
    co2.frame.head()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>year</th>
          <th>month</th>
          <th>day</th>
          <th>weight</th>
          <th>flag</th>
          <th>station</th>
          <th>co2</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>1958</td>
          <td>3</td>
          <td>29</td>
          <td>4</td>
          <td>0</td>
          <td>MLO</td>
          <td>316.1</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1958</td>
          <td>4</td>
          <td>5</td>
          <td>6</td>
          <td>0</td>
          <td>MLO</td>
          <td>317.3</td>
        </tr>
        <tr>
          <th>2</th>
          <td>1958</td>
          <td>4</td>
          <td>12</td>
          <td>4</td>
          <td>0</td>
          <td>MLO</td>
          <td>317.6</td>
        </tr>
        <tr>
          <th>3</th>
          <td>1958</td>
          <td>4</td>
          <td>19</td>
          <td>6</td>
          <td>0</td>
          <td>MLO</td>
          <td>317.5</td>
        </tr>
        <tr>
          <th>4</th>
          <td>1958</td>
          <td>4</td>
          <td>26</td>
          <td>2</td>
          <td>0</td>
          <td>MLO</td>
          <td>316.4</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 32-33

首先，我们处理原始数据框以创建一个日期列，并选择它和CO2列。

.. GENERATED FROM PYTHON SOURCE LINES 33-41

.. code-block:: Python


    import polars as pl

    co2_data = pl.DataFrame(co2.frame[["year", "month", "day", "co2"]]).select(
        pl.date("year", "month", "day"), "co2"
    )
    co2_data.head()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div><style>
    .dataframe > thead > tr,
    .dataframe > tbody > tr {
      text-align: right;
      white-space: pre-wrap;
    }
    </style>
    <small>shape: (5, 2)</small><table border="1" class="dataframe"><thead><tr><th>date</th><th>co2</th></tr><tr><td>date</td><td>f64</td></tr></thead><tbody><tr><td>1958-03-29</td><td>316.1</td></tr><tr><td>1958-04-05</td><td>317.3</td></tr><tr><td>1958-04-12</td><td>317.6</td></tr><tr><td>1958-04-19</td><td>317.5</td></tr><tr><td>1958-04-26</td><td>316.4</td></tr></tbody></table></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 42-44

.. code-block:: Python

    co2_data["date"].min(), co2_data["date"].max()





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    (datetime.date(1958, 3, 29), datetime.date(2001, 12, 29))



.. GENERATED FROM PYTHON SOURCE LINES 45-46

我们可以看到，从1958年3月到2001年12月的某些天，我们得到了CO2浓度数据。我们可以绘制这些原始信息，以便更好地理解。

.. GENERATED FROM PYTHON SOURCE LINES 46-54

.. code-block:: Python


    import matplotlib.pyplot as plt

    plt.plot(co2_data["date"], co2_data["co2"])
    plt.xlabel("date")
    plt.ylabel("CO$_2$ concentration (ppm)")
    _ = plt.title("Raw air samples measurements from the Mauna Loa Observatory")




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_co2_001.png
   :alt: Raw air samples measurements from the Mauna Loa Observatory
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_co2_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 55-56

我们将通过取每月平均值来预处理数据集，并删除未收集测量值的月份。这样的处理将对数据产生平滑效果。

.. GENERATED FROM PYTHON SOURCE LINES 56-71

.. code-block:: Python



    co2_data = (
        co2_data.sort(by="date")
        .group_by_dynamic("date", every="1mo")
        .agg(pl.col("co2").mean())
        .drop_nulls()
    )
    plt.plot(co2_data["date"], co2_data["co2"])
    plt.xlabel("date")
    plt.ylabel("Monthly average of CO$_2$ concentration (ppm)")
    _ = plt.title(
        "Monthly average of air samples measurements\nfrom the Mauna Loa Observatory"
    )




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_co2_002.png
   :alt: Monthly average of air samples measurements from the Mauna Loa Observatory
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_co2_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 72-75

本示例的目的是根据日期预测二氧化碳浓度。我们还对2001年之后的未来年份进行外推感兴趣。

作为第一步，我们将划分数据和目标进行估计。由于数据是日期，我们会将其转换为数值。

.. GENERATED FROM PYTHON SOURCE LINES 75-80

.. code-block:: Python

    X = co2_data.select(
        pl.col("date").dt.year() + pl.col("date").dt.month() / 12
    ).to_numpy()
    y = co2_data["co2"].to_numpy()








.. GENERATED FROM PYTHON SOURCE LINES 81-87

设计合适的内核
------------------------

为了设计用于高斯过程的核函数，我们可以对手头的数据做出一些假设。我们观察到它们具有以下几个特征：长期上升趋势、明显的季节性变化和一些较小的不规则性。我们可以使用不同的合适核函数来捕捉这些特征。

首先，可以使用具有大尺度参数的径向基函数（RBF）核来拟合长期上升趋势。具有大尺度的RBF核会使这一部分变得平滑。没有强制要求趋势性增加，以便为我们的模型提供一定的自由度。具体的尺度和幅度是自由超参数。

.. GENERATED FROM PYTHON SOURCE LINES 87-91

.. code-block:: Python

    from sklearn.gaussian_process.kernels import RBF

    long_term_trend_kernel = 50.0**2 * RBF(length_scale=50.0)








.. GENERATED FROM PYTHON SOURCE LINES 92-93

季节性变化由具有固定周期为1年的周期指数正弦平方核解释。该周期成分的长度尺度控制其平滑度，是一个自由参数。为了允许偏离精确周期性的衰减，采用了与RBF核的乘积。该RBF成分的长度尺度控制衰减时间，也是另一个自由参数。这种类型的核也被称为局部周期核。

.. GENERATED FROM PYTHON SOURCE LINES 93-102

.. code-block:: Python


    from sklearn.gaussian_process.kernels import ExpSineSquared

    seasonal_kernel = (
        2.0**2
        * RBF(length_scale=100.0)
        * ExpSineSquared(length_scale=1.0, periodicity=1.0, periodicity_bounds="fixed")
    )








.. GENERATED FROM PYTHON SOURCE LINES 103-104

小的不规则性可以通过有理二次核成分来解释，其长度尺度和量化长度尺度扩散性的 alpha 参数需要确定。有理二次核等价于具有多个长度尺度的 RBF 核，并且能够更好地适应不同的不规则性。

.. GENERATED FROM PYTHON SOURCE LINES 104-109

.. code-block:: Python


    from sklearn.gaussian_process.kernels import RationalQuadratic

    irregularities_kernel = 0.5**2 * RationalQuadratic(length_scale=1.0, alpha=1.0)








.. GENERATED FROM PYTHON SOURCE LINES 110-111

最终，数据集中的噪声可以通过一个包含RBF核贡献的核来解释，该核可以解释局部天气现象等相关噪声成分，以及一个用于白噪声的白核贡献。相对幅度和RBF的长度尺度是进一步的自由参数。

.. GENERATED FROM PYTHON SOURCE LINES 111-118

.. code-block:: Python


    from sklearn.gaussian_process.kernels import WhiteKernel

    noise_kernel = 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(
        noise_level=0.1**2, noise_level_bounds=(1e-5, 1e5)
    )








.. GENERATED FROM PYTHON SOURCE LINES 119-120

因此，我们的最终核是所有先前核的总和。

.. GENERATED FROM PYTHON SOURCE LINES 120-126

.. code-block:: Python


    co2_kernel = (
        long_term_trend_kernel + seasonal_kernel + irregularities_kernel + noise_kernel
    )
    co2_kernel





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    50**2 * RBF(length_scale=50) + 2**2 * RBF(length_scale=100) * ExpSineSquared(length_scale=1, periodicity=1) + 0.5**2 * RationalQuadratic(alpha=1, length_scale=1) + 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01)



.. GENERATED FROM PYTHON SOURCE LINES 127-131

模型拟合与外推
-------------------------------

现在，我们准备使用高斯过程回归器并拟合可用数据。为了遵循文献中的示例，我们将从目标值中减去均值。我们本可以使用 `normalize_y=True` 。然而，这样做还会缩放目标值（将 `y` 除以其标准差）。因此，不同核的超参数将具有不同的含义，因为它们将不再以百万分率（ppm）表示。

.. GENERATED FROM PYTHON SOURCE LINES 131-137

.. code-block:: Python

    from sklearn.gaussian_process import GaussianProcessRegressor

    y_mean = y.mean()
    gaussian_process = GaussianProcessRegressor(kernel=co2_kernel, normalize_y=False)
    gaussian_process.fit(X, y - y_mean)






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      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-28 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-28 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-28 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-28 a.estimator_doc_link.fitted:hover {
      /* fitted */
      background-color: var(--sklearn-color-fitted-level-3);
    }
    </style><div id="sk-container-id-28" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>GaussianProcessRegressor(kernel=50**2 * RBF(length_scale=50) + 2**2 * RBF(length_scale=100) * ExpSineSquared(length_scale=1, periodicity=1) + 0.5**2 * RationalQuadratic(alpha=1, length_scale=1) + 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01))</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-111" type="checkbox" checked><label for="sk-estimator-id-111" class="sk-toggleable__label fitted sk-toggleable__label-arrow fitted">&nbsp;&nbsp;GaussianProcessRegressor<a class="sk-estimator-doc-link fitted" rel="noreferrer" target="_blank" href="https://scikit-learn.org/dev/modules/generated/sklearn.gaussian_process.GaussianProcessRegressor.html">?<span>Documentation for GaussianProcessRegressor</span></a><span class="sk-estimator-doc-link fitted">i<span>Fitted</span></span></label><div class="sk-toggleable__content fitted"><pre>GaussianProcessRegressor(kernel=50**2 * RBF(length_scale=50) + 2**2 * RBF(length_scale=100) * ExpSineSquared(length_scale=1, periodicity=1) + 0.5**2 * RationalQuadratic(alpha=1, length_scale=1) + 0.1**2 * RBF(length_scale=0.1) + WhiteKernel(noise_level=0.01))</pre></div> </div></div></div></div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 138-144

现在，我们将使用高斯过程进行预测：

- 训练数据用于检查拟合优度；
- 未来数据用于查看模型的外推结果。

因此，我们从1958年到当前月份创建合成数据。此外，我们还需要添加在训练期间计算的减去的均值。

.. GENERATED FROM PYTHON SOURCE LINES 144-154

.. code-block:: Python

    import datetime

    import numpy as np

    today = datetime.datetime.now()
    current_month = today.year + today.month / 12
    X_test = np.linspace(start=1958, stop=current_month, num=1_000).reshape(-1, 1)
    mean_y_pred, std_y_pred = gaussian_process.predict(X_test, return_std=True)
    mean_y_pred += y_mean








.. GENERATED FROM PYTHON SOURCE LINES 155-171

.. code-block:: Python

    plt.plot(X, y, color="black", linestyle="dashed", label="Measurements")
    plt.plot(X_test, mean_y_pred, color="tab:blue", alpha=0.4, label="Gaussian process")
    plt.fill_between(
        X_test.ravel(),
        mean_y_pred - std_y_pred,
        mean_y_pred + std_y_pred,
        color="tab:blue",
        alpha=0.2,
    )
    plt.legend()
    plt.xlabel("Year")
    plt.ylabel("Monthly average of CO$_2$ concentration (ppm)")
    _ = plt.title(
        "Monthly average of air samples measurements\nfrom the Mauna Loa Observatory"
    )




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_co2_003.png
   :alt: Monthly average of air samples measurements from the Mauna Loa Observatory
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_co2_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 172-178

我们的拟合模型能够很好地拟合先前的数据，并且有信心外推到未来的年份。

核超参数的解释
----------------------------------------

现在，我们可以看看核函数的超参数。

.. GENERATED FROM PYTHON SOURCE LINES 178-180

.. code-block:: Python

    gaussian_process.kernel_





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    44.8**2 * RBF(length_scale=51.6) + 2.64**2 * RBF(length_scale=91.5) * ExpSineSquared(length_scale=1.48, periodicity=1) + 0.536**2 * RationalQuadratic(alpha=2.89, length_scale=0.968) + 0.188**2 * RBF(length_scale=0.122) + WhiteKernel(noise_level=0.0367)



.. GENERATED FROM PYTHON SOURCE LINES 181-182

因此，大部分目标信号（减去均值后）可以通过一个约45 ppm的长期上升趋势和约52年的长度尺度来解释。周期性成分的振幅约为2.6 ppm，衰减时间约为90年，长度尺度约为1.5。长衰减时间表明我们有一个非常接近季节性周期的成分。相关噪声的振幅约为0.2 ppm，长度尺度约为0.12年，白噪声贡献约为0.04 ppm。因此，总体噪声水平非常小，表明数据可以很好地用模型解释。


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

   **Total running time of the script:** (0 minutes 2.635 seconds)


.. _sphx_glr_download_auto_examples_gaussian_process_plot_gpr_co2.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/gaussian_process/plot_gpr_co2.ipynb
        :alt: Launch binder
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_gpr_co2.ipynb <plot_gpr_co2.ipynb>`

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

      :download:`Download Python source code: plot_gpr_co2.py <plot_gpr_co2.py>`

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

      :download:`Download zipped: plot_gpr_co2.zip <plot_gpr_co2.zip>`


.. include:: plot_gpr_co2.recommendations


.. only:: html

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

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