.. note::
    :class: sphx-glr-download-link-note

    Click :ref:`here <sphx_glr_download_auto_examples_manifold_plot_mds.py>` to download the full example code
.. 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.

The reconstructed points using the metric MDS and non metric MDS are slightly
shifted to avoid overlapping.



.. code-block:: python


    # Author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
    # License: BSD

    print(__doc__)
    import numpy as np

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

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

    n_samples = 20
    seed = np.random.RandomState(seed=3)
    X_true = seed.randint(0, 20, 2 * n_samples).astype(np.float)
    X_true = X_true.reshape((n_samples, 2))
    # Center the data
    X_true -= X_true.mean()

    similarities = euclidean_distances(X_true)

    # Add noise to the similarities
    noise = np.random.rand(n_samples, n_samples)
    noise = noise + noise.T
    noise[np.arange(noise.shape[0]), np.arange(noise.shape[0])] = 0
    similarities += noise

    mds = manifold.MDS(n_components=2, max_iter=3000, eps=1e-9, random_state=seed,
                       dissimilarity="precomputed", n_jobs=1)
    pos = mds.fit(similarities).embedding_

    nmds = manifold.MDS(n_components=2, metric=False, max_iter=3000, eps=1e-12,
                        dissimilarity="precomputed", random_state=seed, n_jobs=1,
                        n_init=1)
    npos = nmds.fit_transform(similarities, init=pos)

    # Rescale the data
    pos *= np.sqrt((X_true ** 2).sum()) / np.sqrt((pos ** 2).sum())
    npos *= np.sqrt((X_true ** 2).sum()) / np.sqrt((npos ** 2).sum())

    # Rotate the data
    clf = PCA(n_components=2)
    X_true = clf.fit_transform(X_true)

    pos = clf.fit_transform(pos)

    npos = clf.fit_transform(npos)

    fig = plt.figure(1)
    ax = plt.axes([0., 0., 1., 1.])

    s = 100
    plt.scatter(X_true[:, 0], X_true[:, 1], color='navy', s=s, lw=0,
                label='True Position')
    plt.scatter(pos[:, 0], pos[:, 1], color='turquoise', s=s, lw=0, label='MDS')
    plt.scatter(npos[:, 0], npos[:, 1], color='darkorange', s=s, lw=0, label='NMDS')
    plt.legend(scatterpoints=1, loc='best', shadow=False)

    similarities = similarities.max() / similarities * 100
    similarities[np.isinf(similarities)] = 0

    # Plot the edges
    start_idx, end_idx = np.where(pos)
    # 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(pos)) for j in range(len(pos))]
    values = np.abs(similarities)
    lc = LineCollection(segments,
                        zorder=0, cmap=plt.cm.Blues,
                        norm=plt.Normalize(0, values.max()))
    lc.set_array(similarities.flatten())
    lc.set_linewidths(0.5 * np.ones(len(segments)))
    ax.add_collection(lc)

    plt.show()

**Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_manifold_plot_mds.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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


.. only:: html

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

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