.. _sphx_glr_api_gallery_topology_data:

Topological data analysis
-------------------------

Complexes built from point clouds, persistent homology, and Mapper.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="To define homology for any compact metric space, Leopold Vietoris joined every set of points that are pairwise close. Today this is the Vietoris-Rips complex: at scale r (balls of radius r), a simplex for every set of points at most 2r apart. Sampled from a figure eight, the complex at small scales is dust. At the right scale it has \beta = (1, 2), the figure eight&#x27;s two loops. At large scales everything fills in. Triangles are enough to count loops, and mod-2 coefficients keep the rank computation fast.">

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  .. image:: /api/gallery/topology/data/images/thumb/sphx_glr_plot_01_vietoris_rips_complex_thumb.png
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  :doc:`/api/gallery/topology/data/plot_01_vietoris_rips_complex`

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      <div class="sphx-glr-thumbnail-title">Vietoris's complex of a point cloud (1927)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Eduard Čech built homology from the nerve of a cover: a simplex for every set of cover sets that share a point. Borsuk&#x27;s nerve theorem (1948) says that when every intersection is contractible, as for round discs, the nerve has the homotopy type of the union. Below, the Čech complex of discs around noisy samples of two touching rings has the same Betti numbers as the union of the discs, counted independently from a pixel image. The Vietoris-Rips complex at the same radius also fills in triangles whose three discs meet pairwise but share no common point. At r = 0.6 that closes the small ring&#x27;s hole while the discs still leave it open.">

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  .. image:: /api/gallery/topology/data/images/thumb/sphx_glr_plot_02_cech_nerve_theorem_thumb.png
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  :doc:`/api/gallery/topology/data/plot_02_cech_nerve_theorem`

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      <div class="sphx-glr-thumbnail-title">Čech complexes and the nerve theorem (1932-1948)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="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&#x27;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.">

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  .. image:: /api/gallery/topology/data/images/thumb/sphx_glr_plot_03_persistent_homology_thumb.png
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  :doc:`/api/gallery/topology/data/plot_03_persistent_homology`

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      <div class="sphx-glr-thumbnail-title">Persistent homology (Edelsbrunner, Letscher and Zomorodian 2002)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Persistence is useful because it is stable. Change a function by at most \varepsilon everywhere, and its persistence diagram moves by at most \varepsilon in the bottleneck distance: d_B(\mathrm{Dgm} f, \mathrm{Dgm}\, g) \le \|f - g\|_\infty. Noise creates new points, but only near the diagonal. Below, a signal&#x27;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.">

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  .. image:: /api/gallery/topology/data/images/thumb/sphx_glr_plot_04_persistence_stability_thumb.png
    :alt:

  :doc:`/api/gallery/topology/data/plot_04_persistence_stability`

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      <div class="sphx-glr-thumbnail-title">Stability of persistence diagrams (Cohen-Steiner, Edelsbrunner and Harer 2007)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Mapper summarizes a data set as a graph. Slice the data by a lens function, cluster each slice, and join clusters that share points. It is a discrete Reeb graph. Here 3-dimensional data sampled near a circle with a dangling branch are seen through their height. Mapper recovers one loop and one flare, and the graph&#x27;s Betti numbers (1, 1) match the loop.">

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  .. image:: /api/gallery/topology/data/images/thumb/sphx_glr_plot_05_mapper_thumb.png
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  :doc:`/api/gallery/topology/data/plot_05_mapper`

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      <div class="sphx-glr-thumbnail-title">Mapper (Singh, Mémoli and Carlsson 2007)</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/topology/data/plot_01_vietoris_rips_complex
   /api/gallery/topology/data/plot_02_cech_nerve_theorem
   /api/gallery/topology/data/plot_03_persistent_homology
   /api/gallery/topology/data/plot_04_persistence_stability
   /api/gallery/topology/data/plot_05_mapper