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-:orphan:
-
-.. To get rid of WARNING: document isn't included in any toctree
-
-Rips complex user manual
-=========================
-Definition
-----------
-
-==================================================================== ================================ ======================
-:Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3
-==================================================================== ================================ ======================
-
-+-------------------------------------------+----------------------------------------------------------------------+
-| :doc:`rips_complex_user` | :doc:`rips_complex_ref` |
-+-------------------------------------------+----------------------------------------------------------------------+
-
-The `Rips complex <https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex>`_ is a simplicial complex that
-generalizes proximity (:math:`\varepsilon`-ball) graphs to higher dimensions. The vertices correspond to the input
-points, and a simplex is present if and only if its diameter is smaller than some parameter α. Considering all
-parameters α defines a filtered simplicial complex, where the filtration value of a simplex is its diameter.
-The filtration can be restricted to values α smaller than some threshold, to reduce its size.
-
-The input discrete metric space can be provided as a point cloud plus a distance function, or as a distance matrix.
-
-When creating a simplicial complex from the graph, :doc:`RipsComplex <rips_complex_ref>` first builds the graph and
-inserts it into the data structure. It then expands the simplicial complex (adds the simplices corresponding to cliques)
-when required. The expansion can be stopped at dimension `max_dimension`, by default 1.
-
-A vertex name corresponds to the index of the point in the given range (aka. the point cloud).
-
-.. figure::
- ../../doc/Rips_complex/rips_complex_representation.png
- :align: center
-
- Rips-complex one skeleton graph representation
-
-On this example, as edges (4,5), (4,6) and (5,6) are in the complex, simplex (4,5,6) is added with the filtration value
-set with :math:`max(filtration(4,5), filtration(4,6), filtration(5,6))`. And so on for simplex (0,1,2,3).
-
-If the `RipsComplex` interfaces are not detailed enough for your need, please refer to rips_persistence_step_by_step.cpp
-C++ example, where the graph construction over the Simplex_tree is more detailed.
-
-A Rips complex can easily become huge, even if we limit the length of the edges
-and the dimension of the simplices. One easy trick, before building a Rips
-complex on a point cloud, is to call `sparsify_point_set` which removes points
-that are too close to each other. This does not change its persistence diagram
-by more than the length used to define "too close".
-
-A more general technique is to use a sparse approximation of the Rips
-introduced by Don Sheehy :cite:`sheehy13linear`. We are using the version
-described in :cite:`buchet16efficient` (except that we multiply all filtration
-values by 2, to match the usual Rips complex), which proves a
-:math:`\frac{1+\varepsilon}{1-\varepsilon}`-interleaving, although in practice the
-error is usually smaller. A more intuitive presentation of the idea is
-available in :cite:`cavanna15geometric`, and in a video
-:cite:`cavanna15visualizing`. Passing an extra argument `sparse=0.3` at the
-construction of a `RipsComplex` object asks it to build a sparse Rips with
-parameter :math:`\varepsilon=0.3`, while the default `sparse=None` builds the
-regular Rips complex.
-
-
-Point cloud
------------
-
-Example from a point cloud
-^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-This example builds the neighborhood graph from the given points, up to max_edge_length.
-Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it.
-
-Finally, it is asked to display information about the simplicial complex.
-
-.. testcode::
-
- import gudhi
- rips_complex = gudhi.RipsComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]],
- max_edge_length=12.0)
-
- simplex_tree = rips_complex.create_simplex_tree(max_dimension=1)
- result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
- repr(simplex_tree.num_simplices()) + ' simplices - ' + \
- repr(simplex_tree.num_vertices()) + ' vertices.'
- print(result_str)
- fmt = '%s -> %.2f'
- for filtered_value in simplex_tree.get_filtration():
- print(fmt % tuple(filtered_value))
-
-When launching (Rips maximal distance between 2 points is 12.0, is expanded
-until dimension 1 - one skeleton graph in other words), the output is:
-
-.. testoutput::
-
- Rips complex is of dimension 1 - 18 simplices - 7 vertices.
- [0] -> 0.00
- [1] -> 0.00
- [2] -> 0.00
- [3] -> 0.00
- [4] -> 0.00
- [5] -> 0.00
- [6] -> 0.00
- [2, 3] -> 5.00
- [4, 5] -> 5.39
- [0, 2] -> 5.83
- [0, 1] -> 6.08
- [1, 3] -> 6.32
- [1, 2] -> 6.71
- [5, 6] -> 7.28
- [2, 4] -> 8.94
- [0, 3] -> 9.43
- [4, 6] -> 9.49
- [3, 6] -> 11.00
-
-Notice that if we use
-
-.. code-block:: python
-
- rips_complex = gudhi.RipsComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]],
- max_edge_length=12.0, sparse=2)
-
-asking for a very sparse version (theory only gives some guarantee on the meaning of the output if `sparse<1`),
-2 to 5 edges disappear, depending on the random vertex used to start the sparsification.
-
-Example from OFF file
-^^^^^^^^^^^^^^^^^^^^^
-
-This example builds the :doc:`RipsComplex <rips_complex_ref>` from the given
-points in an OFF file, and max_edge_length value.
-Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it.
-
-Finally, it is asked to display information about the Rips complex.
-
-
-.. testcode::
-
- import gudhi
- point_cloud = gudhi.read_off(off_file=gudhi.__root_source_dir__ + '/data/points/alphacomplexdoc.off')
- rips_complex = gudhi.RipsComplex(points=point_cloud, max_edge_length=12.0)
- simplex_tree = rips_complex.create_simplex_tree(max_dimension=1)
- result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
- repr(simplex_tree.num_simplices()) + ' simplices - ' + \
- repr(simplex_tree.num_vertices()) + ' vertices.'
- print(result_str)
- fmt = '%s -> %.2f'
- for filtered_value in simplex_tree.get_filtration():
- print(fmt % tuple(filtered_value))
-
-the program output is:
-
-.. testoutput::
-
- Rips complex is of dimension 1 - 18 simplices - 7 vertices.
- [0] -> 0.00
- [1] -> 0.00
- [2] -> 0.00
- [3] -> 0.00
- [4] -> 0.00
- [5] -> 0.00
- [6] -> 0.00
- [2, 3] -> 5.00
- [4, 5] -> 5.39
- [0, 2] -> 5.83
- [0, 1] -> 6.08
- [1, 3] -> 6.32
- [1, 2] -> 6.71
- [5, 6] -> 7.28
- [2, 4] -> 8.94
- [0, 3] -> 9.43
- [4, 6] -> 9.49
- [3, 6] -> 11.00
-
-Distance matrix
----------------
-
-Example from a distance matrix
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-This example builds the one skeleton graph from the given distance matrix, and max_edge_length value.
-Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it.
-
-Finally, it is asked to display information about the simplicial complex.
-
-.. testcode::
-
- import gudhi
- rips_complex = gudhi.RipsComplex(distance_matrix=[[],
- [6.0827625303],
- [5.8309518948, 6.7082039325],
- [9.4339811321, 6.3245553203, 5],
- [13.0384048104, 15.6524758425, 8.94427191, 12.0415945788],
- [18.0277563773, 19.6468827044, 13.152946438, 14.7648230602, 5.3851648071],
- [17.88854382, 17.1172427686, 12.0830459736, 11, 9.4868329805, 7.2801098893]],
- max_edge_length=12.0)
-
- simplex_tree = rips_complex.create_simplex_tree(max_dimension=1)
- result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
- repr(simplex_tree.num_simplices()) + ' simplices - ' + \
- repr(simplex_tree.num_vertices()) + ' vertices.'
- print(result_str)
- fmt = '%s -> %.2f'
- for filtered_value in simplex_tree.get_filtration():
- print(fmt % tuple(filtered_value))
-
-When launching (Rips maximal distance between 2 points is 12.0, is expanded
-until dimension 1 - one skeleton graph in other words), the output is:
-
-.. testoutput::
-
- Rips complex is of dimension 1 - 18 simplices - 7 vertices.
- [0] -> 0.00
- [1] -> 0.00
- [2] -> 0.00
- [3] -> 0.00
- [4] -> 0.00
- [5] -> 0.00
- [6] -> 0.00
- [2, 3] -> 5.00
- [4, 5] -> 5.39
- [0, 2] -> 5.83
- [0, 1] -> 6.08
- [1, 3] -> 6.32
- [1, 2] -> 6.71
- [5, 6] -> 7.28
- [2, 4] -> 8.94
- [0, 3] -> 9.43
- [4, 6] -> 9.49
- [3, 6] -> 11.00
-
-Example from csv file
-^^^^^^^^^^^^^^^^^^^^^
-
-This example builds the :doc:`RipsComplex <rips_complex_ref>` from the given
-distance matrix in a csv file, and max_edge_length value.
-Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it.
-
-Finally, it is asked to display information about the Rips complex.
-
-
-.. testcode::
-
- import gudhi
- distance_matrix = gudhi.read_lower_triangular_matrix_from_csv_file(csv_file=gudhi.__root_source_dir__ + \
- '/data/distance_matrix/full_square_distance_matrix.csv')
- rips_complex = gudhi.RipsComplex(distance_matrix=distance_matrix, max_edge_length=12.0)
- simplex_tree = rips_complex.create_simplex_tree(max_dimension=1)
- result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
- repr(simplex_tree.num_simplices()) + ' simplices - ' + \
- repr(simplex_tree.num_vertices()) + ' vertices.'
- print(result_str)
- fmt = '%s -> %.2f'
- for filtered_value in simplex_tree.get_filtration():
- print(fmt % tuple(filtered_value))
-
-the program output is:
-
-.. testoutput::
-
- Rips complex is of dimension 1 - 18 simplices - 7 vertices.
- [0] -> 0.00
- [1] -> 0.00
- [2] -> 0.00
- [3] -> 0.00
- [4] -> 0.00
- [5] -> 0.00
- [6] -> 0.00
- [2, 3] -> 5.00
- [4, 5] -> 5.39
- [0, 2] -> 5.83
- [0, 1] -> 6.08
- [1, 3] -> 6.32
- [1, 2] -> 6.71
- [5, 6] -> 7.28
- [2, 4] -> 8.94
- [0, 3] -> 9.43
- [4, 6] -> 9.49
- [3, 6] -> 11.00
-
-Correlation matrix
-------------------
-
-Example from a correlation matrix
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-Analogously to the case of distance matrix, Rips complexes can be also constructed based on correlation matrix.
-Given a correlation matrix M, comportment-wise 1-M is a distance matrix.
-This example builds the one skeleton graph from the given corelation matrix and threshold value.
-Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it.
-
-Finally, it is asked to display information about the simplicial complex.
-
-.. testcode::
-
- import gudhi
- import numpy as np
-
- # User defined correlation matrix is:
- # |1 0.06 0.23 0.01 0.89|
- # |0.06 1 0.74 0.01 0.61|
- # |0.23 0.74 1 0.72 0.03|
- # |0.01 0.01 0.72 1 0.7 |
- # |0.89 0.61 0.03 0.7 1 |
- correlation_matrix=np.array([[1., 0.06, 0.23, 0.01, 0.89],
- [0.06, 1., 0.74, 0.01, 0.61],
- [0.23, 0.74, 1., 0.72, 0.03],
- [0.01, 0.01, 0.72, 1., 0.7],
- [0.89, 0.61, 0.03, 0.7, 1.]], float)
-
- distance_matrix = np.ones((correlation_matrix.shape),float) - correlation_matrix
- rips_complex = gudhi.RipsComplex(distance_matrix=distance_matrix, max_edge_length=1.0)
-
- simplex_tree = rips_complex.create_simplex_tree(max_dimension=1)
- result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
- repr(simplex_tree.num_simplices()) + ' simplices - ' + \
- repr(simplex_tree.num_vertices()) + ' vertices.'
- print(result_str)
- fmt = '%s -> %.2f'
- for filtered_value in simplex_tree.get_filtration():
- print(fmt % tuple(filtered_value))
-
-When launching (Rips maximal distance between 2 points is 12.0, is expanded
-until dimension 1 - one skeleton graph in other words), the output is:
-
-.. testoutput::
-
- Rips complex is of dimension 1 - 15 simplices - 5 vertices.
- [0] -> 0.00
- [1] -> 0.00
- [2] -> 0.00
- [3] -> 0.00
- [4] -> 0.00
- [0, 4] -> 0.11
- [1, 2] -> 0.26
- [2, 3] -> 0.28
- [3, 4] -> 0.30
- [1, 4] -> 0.39
- [0, 2] -> 0.77
- [0, 1] -> 0.94
- [2, 4] -> 0.97
- [0, 3] -> 0.99
- [1, 3] -> 0.99
-
-.. note::
- As persistence diagrams points will be under the diagonal,
- bottleneck distance and persistence graphical tool will not work properly,
- this is a known issue.
generated by cgit v1.2.3 (git 2.39.1) at 2024-07-03 11:06:43 +0000