.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/data/plot_04_persistence_stability.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_04_persistence_stability.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_04_persistence_stability.py:


Stability of persistence diagrams (Cohen-Steiner, Edelsbrunner and Harer 2007)
==============================================================================

Persistence is useful because it is stable. Change a function by at
most :math:`\varepsilon` everywhere, and its persistence diagram moves by
at most :math:`\varepsilon` in the bottleneck distance:
:math:`d_B(\mathrm{Dgm} f, \mathrm{Dgm}\, g) \le \|f - g\|_\infty`. Noise
creates new points, but only near the diagonal. Below, a signal's
sublevel-set diagram (its minima paired with the maxima that merge
them) is compared with noisy copies at growing noise levels. The
bottleneck distance always stays below the noise amplitude.

.. GENERATED FROM PYTHON SOURCE LINES 16-54



.. image-sg:: /api/gallery/topology/data/images/sphx_glr_plot_04_persistence_stability_001.png
   :alt: a signal and a noisy copy, $H_0$ diagrams: $d_B$ = 0.144, stability: $d_B \leq \|f - g\|_\infty$
   :srcset: /api/gallery/topology/data/images/sphx_glr_plot_04_persistence_stability_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    ||f - g|| <= 0.00: d_B = 0.000
    ||f - g|| <= 0.05: d_B = 0.044
    ||f - g|| <= 0.10: d_B = 0.088
    ||f - g|| <= 0.15: d_B = 0.145
    ||f - g|| <= 0.20: d_B = 0.197
    ||f - g|| <= 0.25: d_B = 0.243
    ||f - g|| <= 0.30: d_B = 0.281
    ||f - g|| <= 0.35: d_B = 0.338
    ||f - g|| <= 0.40: d_B = 0.382
    ||f - g|| <= 0.45: d_B = 0.438
    ||f - g|| <= 0.50: d_B = 0.478

    Text(0.5, 1.0, 'stability: $d_B \\leq \\|f - g\\|_\\infty$')





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import SimplicialComplex, bottleneck_distance, lower_star_filtration, persistent_homology
    from mathematicskit.topology.visualizers import plot_persistence_diagram

    x = np.linspace(0, 1, 400)
    f = np.sin(6 * np.pi * x) * (1 - x) + 0.4 * np.sin(17 * x)
    path = SimplicialComplex([[i, i + 1] for i in range(len(x) - 1)])
    diagram_f = persistent_homology(lower_star_filtration(path, f))

    rng = np.random.default_rng(0)
    amplitudes = np.linspace(0.0, 0.5, 11)
    distances = []
    for eps in amplitudes:
        g = f + rng.uniform(-eps, eps, len(x))
        d = bottleneck_distance(diagram_f.diagram(0), persistent_homology(lower_star_filtration(path, g)).diagram(0))
        distances.append(d)
        print(f"||f - g|| <= {eps:.2f}: d_B = {d:.3f}")

    g = f + rng.uniform(-0.15, 0.15, len(x))
    diagram_g = persistent_homology(lower_star_filtration(path, g))
    fig, axes = plt.subplots(1, 3, figsize=(15, 4.4))
    axes[0].plot(x, g, color="0.6", lw=0.8, label="g = f + noise (±0.15)")
    axes[0].plot(x, f, lw=2, label="f")
    axes[0].legend()
    axes[0].set_title("a signal and a noisy copy")
    plot_persistence_diagram(diagram_g, ax=axes[1], dims=[0])
    d = diagram_f.diagram(0)
    deaths = np.where(np.isinf(d[:, 1]), axes[1].get_ylim()[1] / 1.08, d[:, 1])
    axes[1].scatter(d[:, 0], deaths, s=90, facecolors="none", edgecolors="k", linewidths=1.5, zorder=4, label="Dgm f")
    axes[1].legend(loc="lower right")
    axes[1].set_title(f"$H_0$ diagrams: $d_B$ = {bottleneck_distance(d, diagram_g.diagram(0)):.3f}")
    axes[2].plot(amplitudes, amplitudes, "k--", label=r"$\|f - g\|_\infty$ bound")
    axes[2].plot(amplitudes, distances, "o-", label="bottleneck distance")
    axes[2].set_xlabel("noise amplitude")
    axes[2].legend()
    axes[2].set_title("stability: $d_B \\leq \\|f - g\\|_\\infty$")


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

   **Total running time of the script:** (0 minutes 0.105 seconds)


.. _sphx_glr_download_api_gallery_topology_data_plot_04_persistence_stability.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_04_persistence_stability.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_persistence_stability.ipynb <plot_04_persistence_stability.ipynb>`

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

      :download:`Download Python source code: plot_04_persistence_stability.py <plot_04_persistence_stability.py>`

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

      :download:`Download zipped: plot_04_persistence_stability.zip <plot_04_persistence_stability.zip>`


.. only:: html

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

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