.. 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

        Click :ref:`here <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:


=========================================
Comparison of Manifold Learning methods
=========================================

An illustration of dimensionality reduction on the S-curve dataset
with various manifold learning methods.

For a discussion and comparison of these algorithms, see the
:ref:`manifold module page <manifold>`

For a similar example, where the methods are applied to a
sphere dataset, see :ref:`sphx_glr_auto_examples_manifold_plot_manifold_sphere.py`

Note that the purpose of the MDS is to find a low-dimensional
representation of the data (here 2D) in which the distances respect well
the distances in the original high-dimensional space, unlike other
manifold-learning algorithms, it does not seeks an isotropic
representation of the data in the low-dimensional space.

.. GENERATED FROM PYTHON SOURCE LINES 22-25

.. code-block:: default


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








.. GENERATED FROM PYTHON SOURCE LINES 26-30

Dataset preparation
-------------------

We start by generating the S-curve dataset.

.. GENERATED FROM PYTHON SOURCE LINES 30-45

.. code-block:: default


    from numpy.random import RandomState
    import matplotlib.pyplot as plt
    from matplotlib import ticker

    # unused but required import for doing 3d projections with matplotlib < 3.2
    import mpl_toolkits.mplot3d  # noqa: F401

    from sklearn import manifold, datasets

    rng = RandomState(0)

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








.. GENERATED FROM PYTHON SOURCE LINES 46-48

Let's look at the original data. Also define some helping
functions, which we will use further on.

.. GENERATED FROM PYTHON SOURCE LINES 48-87

.. code-block:: default



    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





.. GENERATED FROM PYTHON SOURCE LINES 88-96

Define algorithms for the manifold learning
-------------------------------------------

Manifold learning is an approach to non-linear dimensionality reduction.
Algorithms for this task are based on the idea that the dimensionality of
many data sets is only artificially high.

Read more in the :ref:`User Guide <manifold>`.

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.. code-block:: default


    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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Locally Linear Embeddings
^^^^^^^^^^^^^^^^^^^^^^^^^

Locally linear embedding (LLE) can be thought of as a series of local
Principal Component Analyses which are globally compared to find the
best non-linear embedding.
Read more in the :ref:`User Guide <locally_linear_embedding>`.

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.. code-block:: default


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

    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", modified_tol=0.8, **params)
    S_mod = lle_mod.fit_transform(S_points)








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.. code-block:: default

    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 Embedding
^^^^^^^^^^^^^^^^

Non-linear dimensionality reduction through Isometric Mapping.
Isomap seeks a lower-dimensional embedding which maintains geodesic
distances between all points. Read more in the :ref:`User Guide <isomap>`.

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.. code-block:: default


    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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Multidimensional scaling
^^^^^^^^^^^^^^^^^^^^^^^^

Multidimensional scaling (MDS) seeks a low-dimensional representation
of the data in which the distances respect well the distances in the
original high-dimensional space.
Read more in the :ref:`User Guide <multidimensional_scaling>`.

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.. code-block:: default


    md_scaling = manifold.MDS(
        n_components=n_components, max_iter=50, n_init=4, random_state=rng
    )
    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





.. GENERATED FROM PYTHON SOURCE LINES 176-182

Spectral embedding for non-linear dimensionality reduction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

This implementation uses Laplacian Eigenmaps, which finds a low dimensional
representation of the data using a spectral decomposition of the graph Laplacian.
Read more in the :ref:`User Guide <spectral_embedding>`.

.. GENERATED FROM PYTHON SOURCE LINES 182-190

.. code-block:: default


    spectral = manifold.SpectralEmbedding(
        n_components=n_components, n_neighbors=n_neighbors
    )
    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





.. GENERATED FROM PYTHON SOURCE LINES 191-199

T-distributed Stochastic Neighbor Embedding
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

It converts similarities between data points to joint probabilities and
tries to minimize the Kullback-Leibler divergence between the joint probabilities
of the low-dimensional embedding and the high-dimensional data. t-SNE has a cost
function that is not convex, i.e. with different initializations we can get
different results. Read more in the :ref:`User Guide <t_sne>`.

.. GENERATED FROM PYTHON SOURCE LINES 199-211

.. code-block:: default


    t_sne = manifold.TSNE(
        n_components=n_components,
        learning_rate="auto",
        perplexity=30,
        n_iter=250,
        init="random",
        random_state=rng,
    )
    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  11.397 seconds)


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