.. _sphx_glr_auto_examples_plot_OT_2D_samples.py:


====================================================
2D Optimal transport between empirical distributions
====================================================

Illustration of 2D optimal transport between discributions that are weighted
sum of diracs. The OT matrix is plotted with the samples.




.. code-block:: python


    # Author: Remi Flamary <remi.flamary@unice.fr>
    #         Kilian Fatras <kilian.fatras@irisa.fr>
    #
    # License: MIT License

    import numpy as np
    import matplotlib.pylab as pl
    import ot
    import ot.plot







Generate data
-------------



.. code-block:: python


    #%% parameters and data generation

    n = 50  # nb samples

    mu_s = np.array([0, 0])
    cov_s = np.array([[1, 0], [0, 1]])

    mu_t = np.array([4, 4])
    cov_t = np.array([[1, -.8], [-.8, 1]])

    xs = ot.datasets.make_2D_samples_gauss(n, mu_s, cov_s)
    xt = ot.datasets.make_2D_samples_gauss(n, mu_t, cov_t)

    a, b = np.ones((n,)) / n, np.ones((n,)) / n  # uniform distribution on samples

    # loss matrix
    M = ot.dist(xs, xt)
    M /= M.max()







Plot data
---------



.. code-block:: python


    #%% plot samples

    pl.figure(1)
    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
    pl.legend(loc=0)
    pl.title('Source and target distributions')

    pl.figure(2)
    pl.imshow(M, interpolation='nearest')
    pl.title('Cost matrix M')




.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_001.png
            :scale: 47

    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_002.png
            :scale: 47




Compute EMD
-----------



.. code-block:: python


    #%% EMD

    G0 = ot.emd(a, b, M)

    pl.figure(3)
    pl.imshow(G0, interpolation='nearest')
    pl.title('OT matrix G0')

    pl.figure(4)
    ot.plot.plot2D_samples_mat(xs, xt, G0, c=[.5, .5, 1])
    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
    pl.legend(loc=0)
    pl.title('OT matrix with samples')





.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_005.png
            :scale: 47

    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_006.png
            :scale: 47




Compute Sinkhorn
----------------



.. code-block:: python


    #%% sinkhorn

    # reg term
    lambd = 1e-3

    Gs = ot.sinkhorn(a, b, M, lambd)

    pl.figure(5)
    pl.imshow(Gs, interpolation='nearest')
    pl.title('OT matrix sinkhorn')

    pl.figure(6)
    ot.plot.plot2D_samples_mat(xs, xt, Gs, color=[.5, .5, 1])
    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
    pl.legend(loc=0)
    pl.title('OT matrix Sinkhorn with samples')

    pl.show()





.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_009.png
            :scale: 47

    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_010.png
            :scale: 47




Emprirical Sinkhorn
----------------



.. code-block:: python


    #%% sinkhorn

    # reg term
    lambd = 1e-3

    Ges = ot.bregman.empirical_sinkhorn(xs, xt, lambd)

    pl.figure(7)
    pl.imshow(Ges, interpolation='nearest')
    pl.title('OT matrix empirical sinkhorn')

    pl.figure(8)
    ot.plot.plot2D_samples_mat(xs, xt, Ges, color=[.5, .5, 1])
    pl.plot(xs[:, 0], xs[:, 1], '+b', label='Source samples')
    pl.plot(xt[:, 0], xt[:, 1], 'xr', label='Target samples')
    pl.legend(loc=0)
    pl.title('OT matrix Sinkhorn from samples')

    pl.show()



.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_013.png
            :scale: 47

    *

      .. image:: /auto_examples/images/sphx_glr_plot_OT_2D_samples_014.png
            :scale: 47


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

 Out::

    Warning: numerical errors at iteration 0


**Total running time of the script:** ( 0 minutes  2.616 seconds)



.. only :: html

 .. container:: sphx-glr-footer


  .. container:: sphx-glr-download

     :download:`Download Python source code: plot_OT_2D_samples.py <plot_OT_2D_samples.py>`



  .. container:: sphx-glr-download

     :download:`Download Jupyter notebook: plot_OT_2D_samples.ipynb <plot_OT_2D_samples.ipynb>`


.. only:: html

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

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