.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/dowker/plot_rbs_homology.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_dowker_plot_rbs_homology.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_dowker_plot_rbs_homology.py:


.. _dowker_rbs_homology_gallery:

Restricted Barycentric Subdivision (Vietoris-Rips)
===============================================================

The Restricted Barycentric Subdivision Homology of a hypergraph is the homology of the clique complex of the simple graph :math:`G` such that

- the vertices of :math:`G` are the hyperedges of the hypergraph
- the edges of :math:`G` are the pairs :math:`\{e,h\}` such that :math:`e \subseteq h` or :math:`h \subseteq e`

OAT doesn't offer a specialized function to compute this homology, but it does allow one to compute the *persistent homology* of any weighted graph. So, as a work around, we assign weight 0 to every edge of `G` and compute its persistent homology.

This notebook shows how to implement this strategy.

.. GENERATED FROM PYTHON SOURCE LINES 18-27

.. code-block:: Python

    import oat_python as oat

    import copy
    import plotly.graph_objects as go
    import numpy as np
    import networkx as nx
    import hypernetx as hnx
    import sklearn








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Define a hypergraph
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-33

The hypergraph has edges labeled "A", "B", "C", "D", with the following vertices:

.. GENERATED FROM PYTHON SOURCE LINES 33-36

.. code-block:: Python


    E = { "A": ["x"], "B": ["y"], "C": ["x","y","z",], "D": ["x","y","w",], }








.. GENERATED FROM PYTHON SOURCE LINES 37-39

Plot the hypergraph using HyperNetX
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-44

.. code-block:: Python

    hnxgraph = hnx.Hypergraph(E)
    hnx.drawing.draw(hnxgraph)




.. image-sg:: /auto_examples/dowker/images/sphx_glr_plot_rbs_homology_001.png
   :alt: plot rbs homology
   :srcset: /auto_examples/dowker/images/sphx_glr_plot_rbs_homology_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:150: DeprecationWarning:

    misc_props_col and misc_cell_props_col will be deprecated; all public references to these arguments will be removed in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/hypergraph.py:563: FutureWarning:

    This method or function will be deprecated in a future release. Public references to this method or function will be removed from the Hypergraph API in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:150: DeprecationWarning:

    misc_props_col and misc_cell_props_col will be deprecated; all public references to these arguments will be removed in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/hypergraph.py:563: FutureWarning:

    This method or function will be deprecated in a future release. Public references to this method or function will be removed from the Hypergraph API in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.





.. GENERATED FROM PYTHON SOURCE LINES 45-47

Compute homology
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-69

.. code-block:: Python


    # convert to a list of edges
    G                               =   list( E.values() )
    # graph representing the edge containment poset (forgetting direction)
    containment                     =   oat.hypergraph.edge_containment_graph_symmetrized( G )
    # graph whose edges form the set complement of the containment graph
    anticontainment                 =   nx.complement( containment )
    # adjacency matrix of the anticontainment graph
    anti_adjacency                  =   nx.adjacency_matrix( anticontainment ).todense()
    dissimilarity_matrix            =   oat.dissimilarity.sparse_matrix_for_dense(
                                            dissimilarity_matrix    =   anti_adjacency,
                                            max_dissimilarity       =   0.5
                                        )

    # decomposition boundary matrix
    decomposition                    =  oat.core.vietoris_rips.BoundaryMatrixDecomposition(
                                            dissimilarity_matrix    =   dissimilarity_matrix,
                                            max_homology_dimension  =   1, 
                                        )





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

 .. code-block:: none

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/checkouts/stable/examples/dowker/plot_rbs_homology.py:57: FutureWarning:

    adjacency_matrix will return a scipy.sparse array instead of a matrix in Networkx 3.0.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/networkx/linalg/graphmatrix.py:173: DeprecationWarning:



    The scipy.sparse array containers will be used instead of matrices
    in Networkx 3.0. Use `to_scipy_sparse_array` instead.





.. GENERATED FROM PYTHON SOURCE LINES 70-74

The persistent homology dataframe contains the persistent homology of the Vietoris-Rips complex.
Because every edge in the underlying graph has weight 0, this is the same as the RBS homology of the hypergraph.
The dataframe contains a cycle basis for the RBS homology, in which each cycle technically
represets a homology class that is born at filtration parameter 0 and never dies.

.. GENERATED FROM PYTHON SOURCE LINES 74-80

.. code-block:: Python

    persistent_homology             =   decomposition.persistent_homology_dataframe(
                                            return_cycle_representatives    =   True,
                                            return_bounding_chains          =   True,
                                        )
    persistent_homology






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>dimension</th>
          <th>interval_length</th>
          <th>birth_filtration</th>
          <th>birth_simplex</th>
          <th>death_filtration</th>
          <th>death_simplex</th>
          <th>cycle_representative</th>
          <th>num_cycle_simplices</th>
          <th>bounding_chain</th>
          <th>num_bounding_simplices</th>
        </tr>
        <tr>
          <th>id</th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>0</td>
          <td>inf</td>
          <td>0.0</td>
          <td>(0,)</td>
          <td>inf</td>
          <td>None</td>
          <td>simplex  filtration coefficient
    0    (0,)   ...</td>
          <td>1</td>
          <td>None</td>
          <td>None</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1</td>
          <td>inf</td>
          <td>0.0</td>
          <td>(1, 3)</td>
          <td>inf</td>
          <td>None</td>
          <td>simplex  filtration coefficient
    0  (0, 2)   ...</td>
          <td>4</td>
          <td>None</td>
          <td>None</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 81-83

Inspect a cycle representative
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 85-88

.. code-block:: Python

    cycle           =   persistent_homology["cycle_representative"][1]
    cycle






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
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        .dataframe thead th {
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    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>simplex</th>
          <th>filtration</th>
          <th>coefficient</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>(0, 2)</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
        <tr>
          <th>1</th>
          <td>(0, 3)</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>2</th>
          <td>(1, 2)</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>3</th>
          <td>(1, 3)</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 89-94

Relabel each vertex 
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Recall that a vertex in the Vietoris-Rips complex corresponds to a hyperedge in the initial hypergraph.
These hyperedges have labels "A", "B", "C", "D", so lets relabel the vertices in the cycle representative accordingly.

.. GENERATED FROM PYTHON SOURCE LINES 97-98

Recall that each vertex in RBS homology represents an edge in the reduced hypergraph (which corresponds to a set of hyperedges in the initial hypergraph).  Here we relabel each vertex with **one** of the edges that maps to it.

.. GENERATED FROM PYTHON SOURCE LINES 100-108

.. code-block:: Python


    original_edge_labels=   { p: k for p,k in enumerate(E.keys()) }

    cycle               =   copy.deepcopy(persistent_homology["cycle_representative"][1])
    cycle["simplex"]    =   [ [original_edge_labels[x] for x in simplex] for simplex in cycle["simplex"] ]

    cycle






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
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        </tr>
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      <tbody>
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          <td>[A, D]</td>
          <td>0.0</td>
          <td>-1</td>
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          <th>2</th>
          <td>[B, C]</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>3</th>
          <td>[B, D]</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 109-112

Plot a cycle representative
-----------------------------------------


.. GENERATED FROM PYTHON SOURCE LINES 114-134

.. code-block:: Python

    edges                   =   persistent_homology["cycle_representative"][1]["simplex"] # the edges in the cycle
    points                  =   oat.plot.vertex_embedding_for_simplices(
                                    simplices           =   edges.tolist(),
                                    dimension           =   3,
                                ) 

    fig                     =   oat.plot.fig_3d_for_simplices( 
                                    edges, 
                                    points=points,
                                    kwargs_points       =   dict(
                                                                mode="markers+text", 
                                                                text= [ val for val in original_edge_labels.values() ],
                                                                marker_size=8,
                                                                textfont_size=18,
                                                            ),                                
                                )
    fig.update_layout(template="plotly_dark")








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.. GENERATED FROM PYTHON SOURCE LINES 135-137

Suspend and repeat
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 139-143

.. code-block:: Python

    F = { "A": ["x"], "B": ["y"], "C": ["x","y","z",], "D": ["x","y","w",], "E":["x","y","z","w","a"], "F":["x","y","z","w","b"] }
    hnxgraph = hnx.Hypergraph(F)
    hnx.drawing.draw(hnxgraph)




.. image-sg:: /auto_examples/dowker/images/sphx_glr_plot_rbs_homology_002.png
   :alt: plot rbs homology
   :srcset: /auto_examples/dowker/images/sphx_glr_plot_rbs_homology_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:150: DeprecationWarning:

    misc_props_col and misc_cell_props_col will be deprecated; all public references to these arguments will be removed in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/hypergraph.py:563: FutureWarning:

    This method or function will be deprecated in a future release. Public references to this method or function will be removed from the Hypergraph API in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:150: DeprecationWarning:

    misc_props_col and misc_cell_props_col will be deprecated; all public references to these arguments will be removed in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/hypergraph.py:563: FutureWarning:

    This method or function will be deprecated in a future release. Public references to this method or function will be removed from the Hypergraph API in a future release.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/hypernetx/classes/entityset.py:618: FutureWarning:

    The default of observed=False is deprecated and will be changed to True in a future version of pandas. Pass observed=False to retain current behavior or observed=True to adopt the future default and silence this warning.





.. GENERATED FROM PYTHON SOURCE LINES 144-145

convert to a list of edges

.. GENERATED FROM PYTHON SOURCE LINES 145-163

.. code-block:: Python

    G                               =   list( F.values() )
    # graph representing the edge containment poset (forgetting direction)
    containment                     =   oat.hypergraph.edge_containment_graph_symmetrized( G )
    # graph whose edges form the set complement of the containment graph
    anticontainment                 =   nx.complement( containment )
    # adjacency matrix of the anticontainment graph
    anti_adjacency                  =   nx.adjacency_matrix( anticontainment ).todense()
    dissimilarity_matrix            =   oat.dissimilarity.sparse_matrix_for_dense(
                                            dissimilarity_matrix    =   anti_adjacency,
                                            max_dissimilarity       =   0.5
                                        )

    # decomposition boundary matrix
    decomposition                   =   oat.core.vietoris_rips.BoundaryMatrixDecomposition(
                                            dissimilarity_matrix    =   dissimilarity_matrix,
                                            max_homology_dimension  =   2, 
                                        )





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

 .. code-block:: none

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/checkouts/stable/examples/dowker/plot_rbs_homology.py:151: FutureWarning:

    adjacency_matrix will return a scipy.sparse array instead of a matrix in Networkx 3.0.

    /home/docs/checkouts/readthedocs.org/user_builds/oat-python/envs/stable/lib/python3.9/site-packages/networkx/linalg/graphmatrix.py:173: DeprecationWarning:



    The scipy.sparse array containers will be used instead of matrices
    in Networkx 3.0. Use `to_scipy_sparse_array` instead.





.. GENERATED FROM PYTHON SOURCE LINES 164-165

The persistent homology dataframe:

.. GENERATED FROM PYTHON SOURCE LINES 165-171

.. code-block:: Python

    persistent_homology             =   decomposition.persistent_homology_dataframe(
                                            return_cycle_representatives    =   True,
                                            return_bounding_chains          =   True,
                                        )
    persistent_homology






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
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        .dataframe tbody tr th {
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        .dataframe thead th {
            text-align: right;
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    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>dimension</th>
          <th>interval_length</th>
          <th>birth_filtration</th>
          <th>birth_simplex</th>
          <th>death_filtration</th>
          <th>death_simplex</th>
          <th>cycle_representative</th>
          <th>num_cycle_simplices</th>
          <th>bounding_chain</th>
          <th>num_bounding_simplices</th>
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        <tr>
          <th>id</th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
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        <tr>
          <th>0</th>
          <td>0</td>
          <td>inf</td>
          <td>0.0</td>
          <td>(0,)</td>
          <td>inf</td>
          <td>None</td>
          <td>simplex  filtration coefficient
    0    (0,)   ...</td>
          <td>1</td>
          <td>None</td>
          <td>None</td>
        </tr>
        <tr>
          <th>1</th>
          <td>2</td>
          <td>inf</td>
          <td>0.0</td>
          <td>(1, 3, 5)</td>
          <td>inf</td>
          <td>None</td>
          <td>simplex  filtration coefficient
    0  (0, 2,...</td>
          <td>8</td>
          <td>None</td>
          <td>None</td>
        </tr>
      </tbody>
    </table>
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    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 172-173

A cycle representative:

.. GENERATED FROM PYTHON SOURCE LINES 173-176

.. code-block:: Python

    cycle                       =   persistent_homology["cycle_representative"][1]
    cycle






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          <th>simplex</th>
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          <th>coefficient</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>(0, 2, 4)</td>
          <td>0.0</td>
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          <th>1</th>
          <td>(0, 2, 5)</td>
          <td>0.0</td>
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          <th>2</th>
          <td>(0, 3, 4)</td>
          <td>0.0</td>
          <td>1</td>
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          <th>3</th>
          <td>(0, 3, 5)</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
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          <th>4</th>
          <td>(1, 2, 4)</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
        <tr>
          <th>5</th>
          <td>(1, 2, 5)</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>6</th>
          <td>(1, 3, 4)</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>7</th>
          <td>(1, 3, 5)</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 177-178

A cycle with reverted vertex labels

.. GENERATED FROM PYTHON SOURCE LINES 178-184

.. code-block:: Python

    original_edge_labels                       =   { p: k for p,k in enumerate(F.keys()) }
    cycle                       =   copy.deepcopy(persistent_homology["cycle_representative"][1])
    cycle["simplex"]            =   [ [original_edge_labels[x] for x in simplex] for simplex in cycle["simplex"] ]
    cycle







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    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
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        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>[A, C, E]</td>
          <td>0.0</td>
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        </tr>
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          <th>1</th>
          <td>[A, C, F]</td>
          <td>0.0</td>
          <td>1</td>
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          <th>2</th>
          <td>[A, D, E]</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
        <tr>
          <th>3</th>
          <td>[A, D, F]</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>4</th>
          <td>[B, C, E]</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
        <tr>
          <th>5</th>
          <td>[B, C, F]</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>6</th>
          <td>[B, D, E]</td>
          <td>0.0</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>7</th>
          <td>[B, D, F]</td>
          <td>0.0</td>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 185-186

Plot the cycle representative

.. GENERATED FROM PYTHON SOURCE LINES 186-206

.. code-block:: Python

    triangles               =   persistent_homology["cycle_representative"][1]["simplex"] # the edges in the cycle
    points                  =   oat.plot.vertex_embedding_for_simplices(
                                    simplices           =   triangles.tolist(),
                                    dimension           =   3,
                                ) 

    fig                     =   oat.plot.fig_3d_for_simplices( 
                                    triangles, 
                                    points=points,
                                    kwargs_points       =   dict(
                                                                mode="markers+text", 
                                                                text= [ val for val in original_edge_labels.values() ],
                                                                marker_size=8,
                                                                textfont_size=18,
                                                            ),                                
                                )
    fig.update_layout(template="plotly_dark")








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.. GENERATED FROM PYTHON SOURCE LINES 207-208

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.. GENERATED FROM PYTHON SOURCE LINES 210-211

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.. GENERATED FROM PYTHON SOURCE LINES 211-215

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.. _sphx_glr_download_auto_examples_dowker_plot_rbs_homology.py:

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

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

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

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

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


.. only:: html

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