.. 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_prior_posterior.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_prior_posterior.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_prior_posterior.py:


==========================================================================
不同核函数下高斯过程的先验和后验示例
==========================================================================

本示例展示了使用不同核函数的
:class:`~sklearn.gaussian_process.GaussianProcessRegressor` 的先验和后验。
对于先验和后验分布，展示了均值、标准差和5个样本。

这里我们仅给出一些示例。要了解更多关于核函数的公式，请参考 :ref:`用户指南 <gp_kernels>` 。

.. GENERATED FROM PYTHON SOURCE LINES 13-17

.. code-block:: Python


    # 作者：scikit-learn 开发者
    # SPDX 许可证标识符：BSD-3-Clause








.. GENERATED FROM PYTHON SOURCE LINES 18-24

辅助函数
---------------

在介绍高斯过程的每个可用核之前，我们将定义一个辅助函数，允许我们绘制从高斯过程中抽取的样本。

此函数将接收一个 :class:`~sklearn.gaussian_process.GaussianProcessRegressor` 模型，并从高斯过程抽取样本。如果模型未经过拟合，则样本从先验分布中抽取；而在模型拟合后，样本从后验分布中抽取。

.. GENERATED FROM PYTHON SOURCE LINES 24-70

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np


    def plot_gpr_samples(gpr_model, n_samples, ax):
        """绘制从高斯过程模型中抽取的样本。

    如果高斯过程模型未经过训练，则抽取的样本来自先验分布。否则，样本来自后验分布。请注意，这里的样本对应于一个函数。

    Parameters
    ----------
    gpr_model :`GaussianProcessRegressor` 
        一个 :class:`~sklearn.gaussian_process.GaussianProcessRegressor` 模型。
    n_samples : int
        从高斯过程分布中抽取的样本数量。
    ax : matplotlib 轴
        用于绘制样本的 matplotlib 轴。
    """
        x = np.linspace(0, 5, 100)
        X = x.reshape(-1, 1)

        y_mean, y_std = gpr_model.predict(X, return_std=True)
        y_samples = gpr_model.sample_y(X, n_samples)

        for idx, single_prior in enumerate(y_samples.T):
            ax.plot(
                x,
                single_prior,
                linestyle="--",
                alpha=0.7,
                label=f"Sampled function #{idx + 1}",
            )
        ax.plot(x, y_mean, color="black", label="Mean")
        ax.fill_between(
            x,
            y_mean - y_std,
            y_mean + y_std,
            alpha=0.1,
            color="black",
            label=r"$\pm$ 1 std. dev.",
        )
        ax.set_xlabel("x")
        ax.set_ylabel("y")
        ax.set_ylim([-3, 3])









.. GENERATED FROM PYTHON SOURCE LINES 71-74

数据集和高斯过程生成
-----------------------
我们将创建一个训练数据集，并在不同部分中使用它。

.. GENERATED FROM PYTHON SOURCE LINES 74-79

.. code-block:: Python

    rng = np.random.RandomState(4)
    X_train = rng.uniform(0, 5, 10).reshape(-1, 1)
    y_train = np.sin((X_train[:, 0] - 2.5) ** 2)
    n_samples = 5








.. GENERATED FROM PYTHON SOURCE LINES 80-86

内核手册
---------------

在本节中，我们展示了从具有不同核函数的高斯过程的先验分布和后验分布中抽取的一些样本。

径向基函数核

.. GENERATED FROM PYTHON SOURCE LINES 86-108

.. code-block:: Python

    from sklearn.gaussian_process import GaussianProcessRegressor
    from sklearn.gaussian_process.kernels import RBF

    kernel = 1.0 * RBF(length_scale=1.0, length_scale_bounds=(1e-1, 10.0))
    gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

    fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

    # plot prior
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
    axs[0].set_title("Samples from prior distribution")

    # 绘制后验分布
    gpr.fit(X_train, y_train)
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
    axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
    axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
    axs[1].set_title("Samples from posterior distribution")

    fig.suptitle("Radial Basis Function kernel", fontsize=18)
    plt.tight_layout()




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_001.png
   :alt: Radial Basis Function kernel, Samples from prior distribution, Samples from posterior distribution
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 109-115

.. code-block:: Python

    print(f"Kernel parameters before fit:\n{kernel})")
    print(
        f"Kernel parameters after fit: \n{gpr.kernel_} \n"
        f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
    )





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

 .. code-block:: none

    Kernel parameters before fit:
    1**2 * RBF(length_scale=1))
    Kernel parameters after fit: 
    0.594**2 * RBF(length_scale=0.279) 
    Log-likelihood: -0.067




.. GENERATED FROM PYTHON SOURCE LINES 116-120

有理二次核
..........................



.. GENERATED FROM PYTHON SOURCE LINES 120-142

.. code-block:: Python


    from sklearn.gaussian_process.kernels import RationalQuadratic

    kernel = 1.0 * RationalQuadratic(length_scale=1.0, alpha=0.1, alpha_bounds=(1e-5, 1e15))
    gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

    fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

    # plot prior
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
    axs[0].set_title("Samples from prior distribution")

    # 绘制后验分布
    gpr.fit(X_train, y_train)
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
    axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
    axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
    axs[1].set_title("Samples from posterior distribution")

    fig.suptitle("Rational Quadratic kernel", fontsize=18)
    plt.tight_layout()




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_002.png
   :alt: Rational Quadratic kernel, Samples from prior distribution, Samples from posterior distribution
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 143-149

.. code-block:: Python

    print(f"Kernel parameters before fit:\n{kernel})")
    print(
        f"Kernel parameters after fit: \n{gpr.kernel_} \n"
        f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
    )





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

 .. code-block:: none

    Kernel parameters before fit:
    1**2 * RationalQuadratic(alpha=0.1, length_scale=1))
    Kernel parameters after fit: 
    0.594**2 * RationalQuadratic(alpha=6.22e+06, length_scale=0.279) 
    Log-likelihood: -0.067




.. GENERATED FROM PYTHON SOURCE LINES 150-151

指数正弦平方核

.. GENERATED FROM PYTHON SOURCE LINES 151-178

.. code-block:: Python


    from sklearn.gaussian_process.kernels import ExpSineSquared

    kernel = 1.0 * ExpSineSquared(
        length_scale=1.0,
        periodicity=3.0,
        length_scale_bounds=(0.1, 10.0),
        periodicity_bounds=(1.0, 10.0),
    )
    gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

    fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

    # plot prior
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
    axs[0].set_title("Samples from prior distribution")

    # 绘制后验分布
    gpr.fit(X_train, y_train)
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
    axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
    axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
    axs[1].set_title("Samples from posterior distribution")

    fig.suptitle("Exp-Sine-Squared kernel", fontsize=18)
    plt.tight_layout()




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_003.png
   :alt: Exp-Sine-Squared kernel, Samples from prior distribution, Samples from posterior distribution
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 179-185

.. code-block:: Python

    print(f"Kernel parameters before fit:\n{kernel})")
    print(
        f"Kernel parameters after fit: \n{gpr.kernel_} \n"
        f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
    )





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

 .. code-block:: none

    Kernel parameters before fit:
    1**2 * ExpSineSquared(length_scale=1, periodicity=3))
    Kernel parameters after fit: 
    0.799**2 * ExpSineSquared(length_scale=0.791, periodicity=2.87) 
    Log-likelihood: 3.394




.. GENERATED FROM PYTHON SOURCE LINES 186-188

点积核函数


.. GENERATED FROM PYTHON SOURCE LINES 188-212

.. code-block:: Python


    from sklearn.gaussian_process.kernels import ConstantKernel, DotProduct

    kernel = ConstantKernel(0.1, (0.01, 10.0)) * (
        DotProduct(sigma_0=1.0, sigma_0_bounds=(0.1, 10.0)) ** 2
    )
    gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

    fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

    # plot prior
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
    axs[0].set_title("Samples from prior distribution")

    # 绘制后验分布
    gpr.fit(X_train, y_train)
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
    axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
    axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
    axs[1].set_title("Samples from posterior distribution")

    fig.suptitle("Dot-product kernel", fontsize=18)
    plt.tight_layout()




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_004.png
   :alt: Dot-product kernel, Samples from prior distribution, Samples from posterior distribution
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_004.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    /app/scikit-learn-main-origin/sklearn/gaussian_process/_gpr.py:594: ConvergenceWarning:

    lbfgs failed to converge (status=2):
    ABNORMAL_TERMINATION_IN_LNSRCH.

    Increase the number of iterations (max_iter) or scale the data as shown in:
        https://scikit-learn.org/stable/modules/preprocessing.html





.. GENERATED FROM PYTHON SOURCE LINES 213-219

.. code-block:: Python

    print(f"Kernel parameters before fit:\n{kernel})")
    print(
        f"Kernel parameters after fit: \n{gpr.kernel_} \n"
        f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
    )





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

 .. code-block:: none

    Kernel parameters before fit:
    0.316**2 * DotProduct(sigma_0=1) ** 2)
    Kernel parameters after fit: 
    0.664**2 * DotProduct(sigma_0=2.1) ** 2 
    Log-likelihood: -7959532552.057




.. GENERATED FROM PYTHON SOURCE LINES 220-222

Matérn 核函数
..............

.. GENERATED FROM PYTHON SOURCE LINES 222-244

.. code-block:: Python


    from sklearn.gaussian_process.kernels import Matern

    kernel = 1.0 * Matern(length_scale=1.0, length_scale_bounds=(1e-1, 10.0), nu=1.5)
    gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

    fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

    # plot prior
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
    axs[0].set_title("Samples from prior distribution")

    # 绘制后验分布
    gpr.fit(X_train, y_train)
    plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
    axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
    axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
    axs[1].set_title("Samples from posterior distribution")

    fig.suptitle("Matérn kernel", fontsize=18)
    plt.tight_layout()




.. image-sg:: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_005.png
   :alt: Matérn kernel, Samples from prior distribution, Samples from posterior distribution
   :srcset: /auto_examples/gaussian_process/images/sphx_glr_plot_gpr_prior_posterior_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 245-250

.. code-block:: Python

    print(f"Kernel parameters before fit:\n{kernel})")
    print(
        f"Kernel parameters after fit: \n{gpr.kernel_} \n"
        f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
    )




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

 .. code-block:: none

    Kernel parameters before fit:
    1**2 * Matern(length_scale=1, nu=1.5))
    Kernel parameters after fit: 
    0.609**2 * Matern(length_scale=0.484, nu=1.5) 
    Log-likelihood: -1.185





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

   **Total running time of the script:** (0 minutes 1.171 seconds)


.. _sphx_glr_download_auto_examples_gaussian_process_plot_gpr_prior_posterior.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_prior_posterior.ipynb
        :alt: Launch binder
        :width: 150 px

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

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

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

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

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

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


.. include:: plot_gpr_prior_posterior.recommendations


.. only:: html

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

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