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+
+
+.. _sphx_glr_auto_examples_plot_barycenter_1D.py:
+
+
+==============================
+1D Wasserstein barycenter demo
+==============================
+
+This example illustrates the computation of regularized Wassersyein Barycenter
+as proposed in [3].
+
+
+[3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015).
+Iterative Bregman projections for regularized transportation problems
+SIAM Journal on Scientific Computing, 37(2), A1111-A1138.
+
+
+
+
+.. code-block:: python
+
+
+    # Author: Remi Flamary <remi.flamary@unice.fr>
+    #
+    # License: MIT License
+
+    import numpy as np
+    import matplotlib.pylab as pl
+    import ot
+    # necessary for 3d plot even if not used
+    from mpl_toolkits.mplot3d import Axes3D  # noqa
+    from matplotlib.collections import PolyCollection
+
+
+
+
+
+
+
+Generate data
+-------------
+
+
+
+.. code-block:: python
+
+
+    #%% parameters
+
+    n = 100  # nb bins
+
+    # bin positions
+    x = np.arange(n, dtype=np.float64)
+
+    # Gaussian distributions
+    a1 = ot.datasets.make_1D_gauss(n, m=20, s=5)  # m= mean, s= std
+    a2 = ot.datasets.make_1D_gauss(n, m=60, s=8)
+
+    # creating matrix A containing all distributions
+    A = np.vstack((a1, a2)).T
+    n_distributions = A.shape[1]
+
+    # loss matrix + normalization
+    M = ot.utils.dist0(n)
+    M /= M.max()
+
+
+
+
+
+
+
+Plot data
+---------
+
+
+
+.. code-block:: python
+
+
+    #%% plot the distributions
+
+    pl.figure(1, figsize=(6.4, 3))
+    for i in range(n_distributions):
+        pl.plot(x, A[:, i])
+    pl.title('Distributions')
+    pl.tight_layout()
+
+
+
+
+.. image:: /auto_examples/images/sphx_glr_plot_barycenter_1D_001.png
+    :align: center
+
+
+
+
+Barycenter computation
+----------------------
+
+
+
+.. code-block:: python
+
+
+    #%% barycenter computation
+
+    alpha = 0.2  # 0<=alpha<=1
+    weights = np.array([1 - alpha, alpha])
+
+    # l2bary
+    bary_l2 = A.dot(weights)
+
+    # wasserstein
+    reg = 1e-3
+    bary_wass = ot.bregman.barycenter(A, M, reg, weights)
+
+    pl.figure(2)
+    pl.clf()
+    pl.subplot(2, 1, 1)
+    for i in range(n_distributions):
+        pl.plot(x, A[:, i])
+    pl.title('Distributions')
+
+    pl.subplot(2, 1, 2)
+    pl.plot(x, bary_l2, 'r', label='l2')
+    pl.plot(x, bary_wass, 'g', label='Wasserstein')
+    pl.legend()
+    pl.title('Barycenters')
+    pl.tight_layout()
+
+
+
+
+.. image:: /auto_examples/images/sphx_glr_plot_barycenter_1D_003.png
+    :align: center
+
+
+
+
+Barycentric interpolation
+-------------------------
+
+
+
+.. code-block:: python
+
+
+    #%% barycenter interpolation
+
+    n_alpha = 11
+    alpha_list = np.linspace(0, 1, n_alpha)
+
+
+    B_l2 = np.zeros((n, n_alpha))
+
+    B_wass = np.copy(B_l2)
+
+    for i in range(0, n_alpha):
+        alpha = alpha_list[i]
+        weights = np.array([1 - alpha, alpha])
+        B_l2[:, i] = A.dot(weights)
+        B_wass[:, i] = ot.bregman.barycenter(A, M, reg, weights)
+
+    #%% plot interpolation
+
+    pl.figure(3)
+
+    cmap = pl.cm.get_cmap('viridis')
+    verts = []
+    zs = alpha_list
+    for i, z in enumerate(zs):
+        ys = B_l2[:, i]
+        verts.append(list(zip(x, ys)))
+
+    ax = pl.gcf().gca(projection='3d')
+
+    poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
+    poly.set_alpha(0.7)
+    ax.add_collection3d(poly, zs=zs, zdir='y')
+    ax.set_xlabel('x')
+    ax.set_xlim3d(0, n)
+    ax.set_ylabel('$\\alpha$')
+    ax.set_ylim3d(0, 1)
+    ax.set_zlabel('')
+    ax.set_zlim3d(0, B_l2.max() * 1.01)
+    pl.title('Barycenter interpolation with l2')
+    pl.tight_layout()
+
+    pl.figure(4)
+    cmap = pl.cm.get_cmap('viridis')
+    verts = []
+    zs = alpha_list
+    for i, z in enumerate(zs):
+        ys = B_wass[:, i]
+        verts.append(list(zip(x, ys)))
+
+    ax = pl.gcf().gca(projection='3d')
+
+    poly = PolyCollection(verts, facecolors=[cmap(a) for a in alpha_list])
+    poly.set_alpha(0.7)
+    ax.add_collection3d(poly, zs=zs, zdir='y')
+    ax.set_xlabel('x')
+    ax.set_xlim3d(0, n)
+    ax.set_ylabel('$\\alpha$')
+    ax.set_ylim3d(0, 1)
+    ax.set_zlabel('')
+    ax.set_zlim3d(0, B_l2.max() * 1.01)
+    pl.title('Barycenter interpolation with Wasserstein')
+    pl.tight_layout()
+
+    pl.show()
+
+
+
+.. rst-class:: sphx-glr-horizontal
+
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_barycenter_1D_005.png
+            :scale: 47
+
+    *
+
+      .. image:: /auto_examples/images/sphx_glr_plot_barycenter_1D_006.png
+            :scale: 47
+
+
+
+
+**Total running time of the script:** ( 0 minutes  0.413 seconds)
+
+
+
+.. only :: html
+
+ .. container:: sphx-glr-footer
+
+
+  .. container:: sphx-glr-download
+
+     :download:`Download Python source code: plot_barycenter_1D.py <plot_barycenter_1D.py>`
+
+
+
+  .. container:: sphx-glr-download
+
+     :download:`Download Jupyter notebook: plot_barycenter_1D.ipynb <plot_barycenter_1D.ipynb>`
+
+
+.. only:: html
+
+ .. rst-class:: sphx-glr-signature
+
+    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_