
.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/manifold/plot_mds.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_mds.py>`
        to download the full example code or to run this example in your browser via JupyterLite or Binder.

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

.. _sphx_glr_auto_examples_manifold_plot_mds.py:


=========================
Multi-dimensional scaling
=========================

An illustration of the metric and non-metric MDS on generated noisy data.

.. GENERATED FROM PYTHON SOURCE LINES 9-13

.. code-block:: Python


    # Authors: The scikit-learn developers
    # SPDX-License-Identifier: BSD-3-Clause








.. GENERATED FROM PYTHON SOURCE LINES 14-18

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

We start by uniformly generating 20 points in a 2D space.

.. GENERATED FROM PYTHON SOURCE LINES 18-37

.. code-block:: Python


    import numpy as np
    from matplotlib import pyplot as plt
    from matplotlib.collections import LineCollection

    from sklearn import manifold
    from sklearn.decomposition import PCA
    from sklearn.metrics import euclidean_distances

    # Generate the data
    EPSILON = np.finfo(np.float32).eps
    n_samples = 20
    rng = np.random.RandomState(seed=3)
    X_true = rng.randint(0, 20, 2 * n_samples).astype(float)
    X_true = X_true.reshape((n_samples, 2))

    # Center the data
    X_true -= X_true.mean()








.. GENERATED FROM PYTHON SOURCE LINES 38-41

Now we compute pairwise distances between all points and add
a small amount of noise to the distance matrix. We make sure
to keep the noisy distance matrix symmetric.

.. GENERATED FROM PYTHON SOURCE LINES 41-51

.. code-block:: Python


    # Compute pairwise Euclidean distances
    distances = euclidean_distances(X_true)

    # Add noise to the distances
    noise = rng.rand(n_samples, n_samples)
    noise = noise + noise.T
    np.fill_diagonal(noise, 0)
    distances += noise








.. GENERATED FROM PYTHON SOURCE LINES 52-53

Here we compute metric, non-metric, and classical MDS of the noisy distance matrix.

.. GENERATED FROM PYTHON SOURCE LINES 53-85

.. code-block:: Python


    mds = manifold.MDS(
        n_components=2,
        max_iter=3000,
        eps=1e-9,
        n_init=1,
        random_state=42,
        metric="precomputed",
        n_jobs=1,
        init="classical_mds",
    )
    X_mds = mds.fit(distances).embedding_

    nmds = manifold.MDS(
        n_components=2,
        metric_mds=False,
        max_iter=3000,
        eps=1e-12,
        metric="precomputed",
        random_state=42,
        n_jobs=1,
        n_init=1,
        init="classical_mds",
    )
    X_nmds = nmds.fit_transform(distances)

    cmds = manifold.ClassicalMDS(
        n_components=2,
        metric="precomputed",
    )
    X_cmds = cmds.fit_transform(distances)








.. GENERATED FROM PYTHON SOURCE LINES 86-87

Rescaling the non-metric MDS solution to match the spread of the original data.

.. GENERATED FROM PYTHON SOURCE LINES 87-90

.. code-block:: Python


    X_nmds *= np.sqrt((X_true**2).sum()) / np.sqrt((X_nmds**2).sum())








.. GENERATED FROM PYTHON SOURCE LINES 91-94

To make the visual comparisons easier, we rotate the original data and all MDS
solutions to their PCA axes. And flip horizontal and vertical MDS axes, if needed,
to match the original data orientation.

.. GENERATED FROM PYTHON SOURCE LINES 94-110

.. code-block:: Python


    # Rotate the data (CMDS does not need to be rotated, it is inherently PCA-aligned)
    pca = PCA(n_components=2)
    X_true = pca.fit_transform(X_true)
    X_mds = pca.fit_transform(X_mds)
    X_nmds = pca.fit_transform(X_nmds)

    # Align the sign of PCs
    for i in [0, 1]:
        if np.corrcoef(X_mds[:, i], X_true[:, i])[0, 1] < 0:
            X_mds[:, i] *= -1
        if np.corrcoef(X_nmds[:, i], X_true[:, i])[0, 1] < 0:
            X_nmds[:, i] *= -1
        if np.corrcoef(X_cmds[:, i], X_true[:, i])[0, 1] < 0:
            X_cmds[:, i] *= -1








.. GENERATED FROM PYTHON SOURCE LINES 111-112

Finally, we plot the original data and all MDS reconstructions.

.. GENERATED FROM PYTHON SOURCE LINES 112-145

.. code-block:: Python


    fig = plt.figure(1)
    ax = plt.axes([0.0, 0.0, 1.0, 1.0])

    s = 100
    plt.scatter(X_true[:, 0], X_true[:, 1], color="navy", s=s, lw=0, label="True Position")
    plt.scatter(X_mds[:, 0], X_mds[:, 1], color="turquoise", s=s, lw=0, label="MDS")
    plt.scatter(
        X_nmds[:, 0], X_nmds[:, 1], color="darkorange", s=s, lw=0, label="Non-metric MDS"
    )
    plt.scatter(
        X_cmds[:, 0], X_cmds[:, 1], color="lightcoral", s=s, lw=0, label="Classical MDS"
    )
    plt.legend(scatterpoints=1, loc="best", shadow=False)

    # Plot the edges
    start_idx, end_idx = X_mds.nonzero()
    # a sequence of (*line0*, *line1*, *line2*), where::
    #            linen = (x0, y0), (x1, y1), ... (xm, ym)
    segments = [
        [X_true[i, :], X_true[j, :]] for i in range(len(X_true)) for j in range(len(X_true))
    ]
    edges = distances.max() / (distances + EPSILON) * 100
    np.fill_diagonal(edges, 0)
    edges = np.abs(edges)
    lc = LineCollection(
        segments, zorder=0, cmap=plt.cm.Blues, norm=plt.Normalize(0, edges.max())
    )
    lc.set_array(edges.flatten())
    lc.set_linewidths(np.full(len(segments), 0.5))
    ax.add_collection(lc)

    plt.show()



.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_mds_001.png
   :alt: plot mds
   :srcset: /auto_examples/manifold/images/sphx_glr_plot_mds_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.182 seconds)


.. _sphx_glr_download_auto_examples_manifold_plot_mds.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/1.8.X?urlpath=lab/tree/notebooks/auto_examples/manifold/plot_mds.ipynb
        :alt: Launch binder
        :width: 150 px

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../lite/lab/index.html?path=auto_examples/manifold/plot_mds.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

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

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

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

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

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


.. include:: plot_mds.recommendations


.. only:: html

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

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