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


=========================================
流形学习方法的比较
=========================================

使用各种流形学习方法对S曲线数据集进行降维的示例。

有关这些算法的讨论和比较，请参见
:ref:`流形模块页面 <manifold>` 

有关类似的示例，其中这些方法应用于
球面数据集，请参见 :ref:`sphx_glr_auto_examples_manifold_plot_manifold_sphere.py` 

请注意，MDS的目的是找到数据的低维表示（此处为2D），其中距离很好地反映了原始高维空间中的距离，与其他流形学习算法不同，它不寻求数据在低维空间中的各向同性表示。

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.. code-block:: Python


    # Author: Jake Vanderplas -- <vanderplas@astro.washington.edu>








.. GENERATED FROM PYTHON SOURCE LINES 21-25

数据集准备
-------------------

我们首先生成S形曲线数据集。

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.. code-block:: Python


    import matplotlib.pyplot as plt

    # 未使用但需要导入以使用低于3.2版本的matplotlib进行3D投影
    import mpl_toolkits.mplot3d  # noqa: F401
    from matplotlib import ticker

    from sklearn import datasets, manifold

    n_samples = 1500
    S_points, S_color = datasets.make_s_curve(n_samples, random_state=0)








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让我们看看原始数据。同时定义一些辅助函数，我们将在后面使用它们。

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.. code-block:: Python




    def plot_3d(points, points_color, title):
        x, y, z = points.T

        fig, ax = plt.subplots(
            figsize=(6, 6),
            facecolor="white",
            tight_layout=True,
            subplot_kw={"projection": "3d"},
        )
        fig.suptitle(title, size=16)
        col = ax.scatter(x, y, z, c=points_color, s=50, alpha=0.8)
        ax.view_init(azim=-60, elev=9)
        ax.xaxis.set_major_locator(ticker.MultipleLocator(1))
        ax.yaxis.set_major_locator(ticker.MultipleLocator(1))
        ax.zaxis.set_major_locator(ticker.MultipleLocator(1))

        fig.colorbar(col, ax=ax, orientation="horizontal", shrink=0.6, aspect=60, pad=0.01)
        plt.show()


    def plot_2d(points, points_color, title):
        fig, ax = plt.subplots(figsize=(3, 3), facecolor="white", constrained_layout=True)
        fig.suptitle(title, size=16)
        add_2d_scatter(ax, points, points_color)
        plt.show()


    def add_2d_scatter(ax, points, points_color, title=None):
        x, y = points.T
        ax.scatter(x, y, c=points_color, s=50, alpha=0.8)
        ax.set_title(title)
        ax.xaxis.set_major_formatter(ticker.NullFormatter())
        ax.yaxis.set_major_formatter(ticker.NullFormatter())


    plot_3d(S_points, S_color, "Original S-curve samples")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_001.png
   :alt: Original S-curve samples
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_001.png
   :class: sphx-glr-single-img





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定义流形学习的算法
-------------------

流形学习是一种非线性降维方法。用于此任务的算法基于这样一个想法：许多数据集的维度只是人为地高。

请参阅 :ref:`用户指南 <manifold>` 了解更多信息。

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.. code-block:: Python


    n_neighbors = 12  # neighborhood which is used to recover the locally linear structure
    n_components = 2  # number of coordinates for the manifold








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局部线性嵌入
^^^^^^^^^^^^^^

局部线性嵌入（LLE）可以看作是一系列局部主成分分析，通过全局比较来找到最佳的非线性嵌入。
在 :ref:`用户指南 <locally_linear_embedding>` 中阅读更多内容。

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.. code-block:: Python


    params = {
        "n_neighbors": n_neighbors,
        "n_components": n_components,
        "eigen_solver": "auto",
        "random_state": 0,
    }

    lle_standard = manifold.LocallyLinearEmbedding(method="standard", **params)
    S_standard = lle_standard.fit_transform(S_points)

    lle_ltsa = manifold.LocallyLinearEmbedding(method="ltsa", **params)
    S_ltsa = lle_ltsa.fit_transform(S_points)

    lle_hessian = manifold.LocallyLinearEmbedding(method="hessian", **params)
    S_hessian = lle_hessian.fit_transform(S_points)

    lle_mod = manifold.LocallyLinearEmbedding(method="modified", **params)
    S_mod = lle_mod.fit_transform(S_points)








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.. code-block:: Python

    fig, axs = plt.subplots(
        nrows=2, ncols=2, figsize=(7, 7), facecolor="white", constrained_layout=True
    )
    fig.suptitle("Locally Linear Embeddings", size=16)

    lle_methods = [
        ("Standard locally linear embedding", S_standard),
        ("Local tangent space alignment", S_ltsa),
        ("Hessian eigenmap", S_hessian),
        ("Modified locally linear embedding", S_mod),
    ]
    for ax, method in zip(axs.flat, lle_methods):
        name, points = method
        add_2d_scatter(ax, points, S_color, name)

    plt.show()




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_002.png
   :alt: Locally Linear Embeddings, Standard locally linear embedding, Local tangent space alignment, Hessian eigenmap, Modified locally linear embedding
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_002.png
   :class: sphx-glr-single-img





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Isomap 嵌入
^^^^^^^^^^^^

通过等距映射进行非线性降维。Isomap 寻求一种低维嵌入，以保持所有点之间的测地距离。更多内容请参阅：:ref:`用户指南 <isomap>` 。

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.. code-block:: Python


    isomap = manifold.Isomap(n_neighbors=n_neighbors, n_components=n_components, p=1)
    S_isomap = isomap.fit_transform(S_points)

    plot_2d(S_isomap, S_color, "Isomap Embedding")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_003.png
   :alt: Isomap Embedding
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_003.png
   :class: sphx-glr-single-img





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多维尺度分析
^^^^^^^^^^^^^^^^

多维尺度分析（MDS）旨在寻找数据的低维表示，其中的距离能够很好地反映原始高维空间中的距离。
更多内容请参阅：:ref:`用户指南 <multidimensional_scaling>` 。

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.. code-block:: Python


    md_scaling = manifold.MDS(
        n_components=n_components,
        max_iter=50,
        n_init=4,
        random_state=0,
        normalized_stress=False,
    )
    S_scaling = md_scaling.fit_transform(S_points)

    plot_2d(S_scaling, S_color, "Multidimensional scaling")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_004.png
   :alt: Multidimensional scaling
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_004.png
   :class: sphx-glr-single-img





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非线性降维的谱嵌入

该实现使用拉普拉斯特征映射，通过图拉普拉斯算子的谱分解找到数据的低维表示。更多内容请参阅：:ref:`用户指南 <spectral_embedding>` 。

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.. code-block:: Python


    spectral = manifold.SpectralEmbedding(
        n_components=n_components, n_neighbors=n_neighbors, random_state=42
    )
    S_spectral = spectral.fit_transform(S_points)

    plot_2d(S_spectral, S_color, "Spectral Embedding")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_005.png
   :alt: Spectral Embedding
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_005.png
   :class: sphx-glr-single-img





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T分布随机邻嵌入

它将数据点之间的相似度转换为联合概率，并尝试最小化低维嵌入和高维数据的联合概率之间的Kullback-Leibler散度。t-SNE的代价函数不是凸的，即不同的初始化可能会得到不同的结果。更多内容请参阅：:ref:`用户指南 <t_sne>` 。

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.. code-block:: Python


    t_sne = manifold.TSNE(
        n_components=n_components,
        perplexity=30,
        init="random",
        max_iter=250,
        random_state=0,
    )
    S_t_sne = t_sne.fit_transform(S_points)

    plot_2d(S_t_sne, S_color, "T-distributed Stochastic  \n Neighbor Embedding")




.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_006.png
   :alt: T-distributed Stochastic    Neighbor Embedding
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_006.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 9.337 seconds)


.. _sphx_glr_download_auto_examples_manifold_plot_compare_methods.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/manifold/plot_compare_methods.ipynb
        :alt: Launch binder
        :width: 150 px

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

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

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

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

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

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


.. include:: plot_compare_methods.recommendations


.. only:: html

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

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