.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/data/plot_03_persistent_homology.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_topology_data_plot_03_persistent_homology.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_topology_data_plot_03_persistent_homology.py:


Persistent homology (Edelsbrunner, Letscher and Zomorodian 2002)
================================================================

No single scale is right for a noisy sample, so persistent homology
tracks all scales at once. As the radius grows, each homology class is
born at some radius and dies at a larger one. One reduction of the
filtration's boundary matrix pairs every birth with its death. Short
bars are noise. The long bars are the shape: two loops for this sample
of two circles of different sizes, each dying when its disc fills in.

.. GENERATED FROM PYTHON SOURCE LINES 14-42



.. image-sg:: /api/gallery/topology/data/images/sphx_glr_plot_03_persistent_homology_001.png
   :alt: Two long $H_1$ bars: two loops, dying near their radii $\times \sqrt{3}/2$, two noisy circles, radii 1 and 0.5, barcode, persistence diagram
   :srcset: /api/gallery/topology/data/images/sphx_glr_plot_03_persistent_homology_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    47360 simplices in the filtration
    H_1 class born at r = 0.218, dies at r = 0.839, persistence 0.621
    H_1 class born at r = 0.132, dies at r = 0.402, persistence 0.270
    the other 2 H_1 classes persist at most 0.014

    Text(0.5, 0.98, 'Two long $H_1$ bars: two loops, dying near their radii $\\times \\sqrt{3}/2$')





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import persistent_homology, vietoris_rips_filtration
    from mathematicskit.topology.visualizers import plot_barcode, plot_persistence_diagram

    rng = np.random.default_rng(2)
    t = rng.uniform(0, 2 * np.pi, 90)
    points = np.vstack([np.column_stack([np.cos(t[:55]), np.sin(t[:55])]), np.column_stack([2.6 + 0.5 * np.cos(t[55:]), 0.5 * np.sin(t[55:])])])
    points += rng.normal(0, 0.05, points.shape)

    filtration = vietoris_rips_filtration(points, max_radius=1.2, max_dim=2)
    diagram = persistent_homology(filtration, max_dim=1)
    print(f"{len(filtration)} simplices in the filtration")
    order = np.argsort(-diagram.persistence(1))
    for birth, death in diagram.diagram(1)[order[:2]]:
        print(f"H_1 class born at r = {birth:.3f}, dies at r = {death:.3f}, persistence {death - birth:.3f}")
    print(f"the other {len(order) - 2} H_1 classes persist at most {diagram.persistence(1)[order[2:]].max():.3f}")

    fig, axes = plt.subplots(1, 3, figsize=(15, 4.6))
    axes[0].scatter(*points.T, s=10)
    axes[0].set_aspect("equal")
    axes[0].set_title("two noisy circles, radii 1 and 0.5")
    plot_barcode(diagram, ax=axes[1])
    axes[1].set_title("barcode")
    plot_persistence_diagram(diagram, ax=axes[2])
    axes[2].set_title("persistence diagram")
    fig.suptitle("Two long $H_1$ bars: two loops, dying near their radii $\\times \\sqrt{3}/2$")


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.305 seconds)


.. _sphx_glr_download_api_gallery_topology_data_plot_03_persistent_homology.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/topology/data/plot_03_persistent_homology.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_persistent_homology.ipynb <plot_03_persistent_homology.ipynb>`

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

      :download:`Download Python source code: plot_03_persistent_homology.py <plot_03_persistent_homology.py>`

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

      :download:`Download zipped: plot_03_persistent_homology.zip <plot_03_persistent_homology.zip>`


.. only:: html

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

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