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<!DOCTYPE html>
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<title>gtda.homology: Persistent homology — giotto-tda 0.4.0 documentation</title>
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<li class="toctree-l2 current"><a class="current reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.homology</span></code>: Persistent homology</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#undirected-simplicial-homology">Undirected simplicial homology</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.VietorisRipsPersistence.html">VietorisRipsPersistence</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.WeightedRipsPersistence.html">WeightedRipsPersistence</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.SparseRipsPersistence.html">SparseRipsPersistence</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.WeakAlphaPersistence.html">WeakAlphaPersistence</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.EuclideanCechPersistence.html">EuclideanCechPersistence</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="#directed-simplicial-homology">Directed simplicial homology</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.FlagserPersistence.html">FlagserPersistence</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="#cubical-homology">Cubical homology</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/homology/gtda.homology.CubicalPersistence.html">CubicalPersistence</a></li>
</ul>
</li>
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</li>
<li class="toctree-l2"><a class="reference internal" href="diagrams.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.diagrams</span></code>: Persistence diagrams</a><ul>
<li class="toctree-l3"><a class="reference internal" href="diagrams.html#preprocessing">Preprocessing</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/preprocessing/gtda.diagrams.ForgetDimension.html">ForgetDimension</a></li>
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</ul>
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<li class="toctree-l3"><a class="reference internal" href="diagrams.html#distances">Distances</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/distances/gtda.diagrams.PairwiseDistance.html">PairwiseDistance</a></li>
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</li>
<li class="toctree-l3"><a class="reference internal" href="diagrams.html#representations">Representations</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/representations/gtda.diagrams.PersistenceLandscape.html">PersistenceLandscape</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/representations/gtda.diagrams.BettiCurve.html">BettiCurve</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/representations/gtda.diagrams.HeatKernel.html">HeatKernel</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/representations/gtda.diagrams.PersistenceImage.html">PersistenceImage</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/representations/gtda.diagrams.Silhouette.html">Silhouette</a></li>
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<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/features/gtda.diagrams.Amplitude.html">Amplitude</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/features/gtda.diagrams.PersistenceEntropy.html">PersistenceEntropy</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/features/gtda.diagrams.NumberOfPoints.html">NumberOfPoints</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/diagrams/features/gtda.diagrams.ComplexPolynomial.html">ComplexPolynomial</a></li>
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</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="curves.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.curves</span></code>: Curves</a><ul>
<li class="toctree-l3"><a class="reference internal" href="curves.html#preprocessing">Preprocessing</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/curves/preprocessing/gtda.curves.Derivative.html">Derivative</a></li>
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<li class="toctree-l3"><a class="reference internal" href="curves.html#feature-extraction">Feature extraction</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/curves/gtda.curves.StandardFeatures.html">StandardFeatures</a></li>
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<li class="toctree-l2"><a class="reference internal" href="point_clouds.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.point_clouds</span></code>: Point clouds</a><ul>
<li class="toctree-l3"><a class="reference internal" href="generated/gtda.point_clouds.ConsistentRescaling.html">ConsistentRescaling</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/gtda.point_clouds.ConsecutiveRescaling.html">ConsecutiveRescaling</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="time_series.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.time_series</span></code>: Time series</a><ul>
<li class="toctree-l3"><a class="reference internal" href="time_series.html#preprocessing">Preprocessing</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/preprocessing/gtda.time_series.SlidingWindow.html">SlidingWindow</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/preprocessing/gtda.time_series.Resampler.html">Resampler</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/preprocessing/gtda.time_series.Stationarizer.html">Stationarizer</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="time_series.html#time-delay-embedding">Time-delay embedding</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/embedding/gtda.time_series.TakensEmbedding.html">TakensEmbedding</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/embedding/gtda.time_series.SingleTakensEmbedding.html">SingleTakensEmbedding</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/embedding/gtda.time_series.takens_embedding_optimal_parameters.html">takens_embedding_optimal_parameters</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="time_series.html#target-preparation">Target preparation</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/target_preparation/gtda.time_series.Labeller.html">Labeller</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="time_series.html#dynamical-systems">Dynamical systems</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/dynamical_system/gtda.time_series.PermutationEntropy.html">PermutationEntropy</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="time_series.html#multivariate">Multivariate</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/time_series/multivariate/gtda.time_series.PearsonDissimilarity.html">PearsonDissimilarity</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="graphs.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.graphs</span></code>: Graphs</a><ul>
<li class="toctree-l3"><a class="reference internal" href="graphs.html#graph-creation">Graph creation</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/graphs/creation/gtda.graphs.TransitionGraph.html">TransitionGraph</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/graphs/creation/gtda.graphs.KNeighborsGraph.html">KNeighborsGraph</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="graphs.html#graph-processing">Graph processing</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/graphs/processing/gtda.graphs.GraphGeodesicDistance.html">GraphGeodesicDistance</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="images.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.images</span></code>: Images</a><ul>
<li class="toctree-l3"><a class="reference internal" href="images.html#preprocessing">Preprocessing</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.Binarizer.html">Binarizer</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.Inverter.html">Inverter</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.Padder.html">Padder</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.ImageToPointCloud.html">ImageToPointCloud</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="images.html#filtrations">Filtrations</a><ul>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.HeightFiltration.html">HeightFiltration</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.RadialFiltration.html">RadialFiltration</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.DilationFiltration.html">DilationFiltration</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.ErosionFiltration.html">ErosionFiltration</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.SignedDistanceFiltration.html">SignedDistanceFiltration</a></li>
<li class="toctree-l4"><a class="reference internal" href="generated/images/gtda.images.DensityFiltration.html">DensityFiltration</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="plotting.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.plotting</span></code>: Plotting functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="generated/plotting/gtda.plotting.plot_point_cloud.html">plot_point_cloud</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/plotting/gtda.plotting.plot_heatmap.html">plot_heatmap</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/plotting/gtda.plotting.plot_diagram.html">plot_diagram</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/plotting/gtda.plotting.plot_betti_curves.html">plot_betti_curves</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/plotting/gtda.plotting.plot_betti_surfaces.html">plot_betti_surfaces</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="base.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.base</span></code>: Base</a><ul>
<li class="toctree-l3"><a class="reference internal" href="generated/base/gtda.base.TransformerResamplerMixin.html">TransformerResamplerMixin</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/base/gtda.base.PlotterMixin.html">PlotterMixin</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="pipeline.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.pipeline</span></code>: Pipeline</a><ul>
<li class="toctree-l3"><a class="reference internal" href="generated/base/gtda.pipeline.Pipeline.html">Pipeline</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/gtda.pipeline.make_pipeline.html">make_pipeline</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="metaestimators.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.metaestimators</span></code>: Meta-estimators</a><ul>
<li class="toctree-l3"><a class="reference internal" href="generated/base/gtda.metaestimators.CollectionTransformer.html">CollectionTransformer</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="utils.html"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.utils</span></code>: Utilities</a><ul>
<li class="toctree-l3"><a class="reference internal" href="generated/utils/gtda.utils.check_collection.html">check_collection</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/utils/gtda.utils.check_point_clouds.html">check_point_clouds</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/utils/gtda.utils.check_diagrams.html">check_diagrams</a></li>
<li class="toctree-l3"><a class="reference internal" href="generated/utils/gtda.utils.validate_params.html">validate_params</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../notebooks/index.html">Tutorials and examples</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../notebooks/tutorials.html">Tutorials</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/vietoris_rips_quickstart.html">Topological feature extraction using <code class="docutils literal notranslate"><span class="pre">VietorisRipsPersistence</span></code> and <code class="docutils literal notranslate"><span class="pre">PersistenceEntropy</span></code></a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/vietoris_rips_quickstart.html#generate-data">Generate data</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/vietoris_rips_quickstart.html#calculate-persistent-homology">Calculate persistent homology</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/vietoris_rips_quickstart.html#extract-features">Extract features</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/vietoris_rips_quickstart.html#use-the-new-features-in-a-standard-classifier">Use the new features in a standard classifier</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/vietoris_rips_quickstart.html#encapsulate-the-steps-above-in-a-pipeline">Encapsulate the steps above in a pipeline</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/plotting_api.html">Plotting in <code class="docutils literal notranslate"><span class="pre">giotto-tda</span></code></a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/plotting_api.html#basic-philosophy-and-plot-methods">1. Basic philosophy and <code class="docutils literal notranslate"><span class="pre">plot</span></code> methods</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/plotting_api.html#derived-convenience-methods-transform-plot-and-fit-transform-plot">2 Derived convenience methods: <code class="docutils literal notranslate"><span class="pre">transform_plot</span></code> and <code class="docutils literal notranslate"><span class="pre">fit_transform_plot</span></code></a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/mapper_quickstart.html">Getting started with Mapper</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#useful-references">Useful references</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#import-libraries">Import libraries</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#generate-and-visualise-data">Generate and visualise data</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#configure-the-mapper-pipeline">Configure the Mapper pipeline</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#visualise-the-mapper-graph">Visualise the Mapper graph</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#run-the-mapper-pipeline">Run the Mapper pipeline</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#creating-custom-filter-functions">Creating custom filter functions</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/mapper_quickstart.html#visualise-the-2d-mapper-graph-interactively-live-jupyter-session-needed">Visualise the 2D Mapper graph interactively (Live Jupyter session needed)</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/topology_time_series.html">Topology of time series</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#useful-references">Useful references</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#see-also">See also</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#from-time-series-to-time-delay-embeddings">From time series to time delay embeddings</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#a-periodic-example">A periodic example</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#a-non-periodic-example">A non-periodic example</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#from-time-delay-embeddings-to-persistence-diagrams">From time delay embeddings to persistence diagrams</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/topology_time_series.html#picking-the-embedding-dimension-and-time-delay">Picking the embedding dimension and time delay</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/time_series_forecasting.html">Topology in time series forecasting</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/time_series_forecasting.html#see-also">See also</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/time_series_forecasting.html#slidingwindow"><code class="docutils literal notranslate"><span class="pre">SlidingWindow</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/time_series_forecasting.html#multivariate-time-series-example-sliding-window-topology-pipeline">Multivariate time series example: Sliding window + topology <code class="docutils literal notranslate"><span class="pre">Pipeline</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/time_series_forecasting.html#univariate-time-series-takensembedding-and-singletakensembedding">Univariate time series – <code class="docutils literal notranslate"><span class="pre">TakensEmbedding</span></code> and <code class="docutils literal notranslate"><span class="pre">SingleTakensEmbedding</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/time_series_forecasting.html#endogeneous-target-preparation-with-labeller">Endogeneous target preparation with <code class="docutils literal notranslate"><span class="pre">Labeller</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/time_series_forecasting.html#where-to-next">Where to next?</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/persistent_homology_graphs.html">Topological feature extraction from graphs</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/persistent_homology_graphs.html#see-also">See also</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/persistent_homology_graphs.html#import-libraries">Import libraries</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/persistent_homology_graphs.html#undirected-graphs-vietorisripspersistence-and-sparseripspersistence">Undirected graphs – <code class="docutils literal notranslate"><span class="pre">VietorisRipsPersistence</span></code> and <code class="docutils literal notranslate"><span class="pre">SparseRipsPersistence</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/persistent_homology_graphs.html#directed-graphs-flagserpersistence">Directed graphs – <code class="docutils literal notranslate"><span class="pre">FlagserPersistence</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/persistent_homology_graphs.html#where-to-next">Where to next?</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../notebooks/examples.html">Examples</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/gravitational_waves_detection.html">Gravitational wave detection</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/gravitational_waves_detection.html#useful-references">Useful references</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/gravitational_waves_detection.html#see-also">See also</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/gravitational_waves_detection.html#motivation">Motivation</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/gravitational_waves_detection.html#generate-the-data">Generate the data</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/gravitational_waves_detection.html#constant-signal-to-noise-ratio">Constant signal-to-noise ratio</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/classifying_shapes.html">Classifying 3D shapes</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/classifying_shapes.html#generate-simple-shapes">Generate simple shapes</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/classifying_shapes.html#from-data-to-persistence-diagrams">From data to persistence diagrams</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/classifying_shapes.html#train-a-classifier">Train a classifier</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/classifying_shapes.html#putting-it-all-together">Putting it all together</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/classifying_shapes.html#a-more-realistic-example">A more realistic example</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/classifying_shapes.html#improving-our-model">Improving our model</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/lorenz_attractor.html">The Lorenz attractor</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#see-also">See also</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#import-libraries">Import libraries</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#setting-up-the-lorenz-attractor-simulation">Setting up the Lorenz attractor simulation</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#resampling-the-time-series">Resampling the time series</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#takens-embedding">Takens Embedding</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#persistence-diagram">Persistence diagram</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#scikit-learnstyle-pipeline">Scikit-learn–style pipeline</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#rescaling-the-diagram">Rescaling the diagram</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#filtering-diagrams">Filtering diagrams</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#persistence-entropy">Persistence entropy</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#betti-curves">Betti Curves</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#distances-among-diagrams">Distances among diagrams</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/lorenz_attractor.html#new-distances-in-the-embedding-space-knn-graphs-and-geodesic-distances">New distances in the embedding space: kNN graphs and geodesic distances</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/MNIST_classification.html">Classifying handwritten digits</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/MNIST_classification.html#useful-references">Useful references</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/MNIST_classification.html#load-the-mnist-dataset">Load the MNIST dataset</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/MNIST_classification.html#from-pixels-to-topological-features">From pixels to topological features</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/MNIST_classification.html#building-a-full-blown-feature-extraction-pipeline">Building a full-blown feature extraction pipeline</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/MNIST_classification.html#training-a-classifier">Training a classifier</a></li>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/MNIST_classification.html#using-hyperparameter-search-with-topological-pipelines">Using hyperparameter search with topological pipelines</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../notebooks/voids_on_the_plane.html">Can two-dimensional topological voids exist in two dimensions?</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../notebooks/voids_on_the_plane.html#import-libraries">Import libraries</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../theory/glossary.html">Theory Glossary</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../theory/glossary.html#symbols">Symbols</a></li>
<li class="toctree-l2"><a class="reference internal" href="../theory/glossary.html#analysis">Analysis</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#metric-space">Metric space</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#normed-space">Normed space</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#inner-product-space">Inner product space</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#vectorization-amplitude-and-kernel">Vectorization, amplitude and kernel</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#euclidean-distance-and-l-p-norms">Euclidean distance and <span class="math notranslate nohighlight">\(l^p\)</span>-norms</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#distance-matrices-and-point-clouds">Distance matrices and point clouds</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#l-p-norms"><span class="math notranslate nohighlight">\(L^p\)</span>-norms</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../theory/glossary.html#homology">Homology</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#cubical-complex">Cubical complex</a><ul>
<li class="toctree-l4"><a class="reference internal" href="../theory/glossary.html#reference">Reference:</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#simplicial-complex">Simplicial complex</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#abstract-simplicial-complex">Abstract simplicial complex</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#ordered-simplicial-complex">Ordered simplicial complex</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#directed-simplicial-complex">Directed simplicial complex</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#chain-complex">Chain complex</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#homology-and-cohomology">Homology and cohomology</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#simplicial-chains-and-simplicial-homology">Simplicial chains and simplicial homology</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#cubical-chains-and-cubical-homology">Cubical chains and cubical homology</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="../theory/glossary.html#persistence">Persistence</a><ul>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#filtered-complex">Filtered complex</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#cellwise-filtration">Cellwise filtration</a></li>
<li class="toctree-l3"><a class="reference internal" href="../theory/glossary.html#clique-and-flag-complexes">Clique and flag complexes</a></li>
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<li><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.homology</span></code>: Persistent homology</li>
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<div class="section" id="module-gtda.homology">
<span id="gtda-homology-persistent-homology"></span><h1><a class="reference internal" href="#module-gtda.homology" title="gtda.homology"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.homology</span></code></a>: Persistent homology<a class="headerlink" href="#module-gtda.homology" title="Permalink to this headline">¶</a></h1>
<p>The module <a class="reference internal" href="#module-gtda.homology" title="gtda.homology"><code class="xref py py-mod docutils literal notranslate"><span class="pre">gtda.homology</span></code></a> implements transformers to generate
persistence diagrams.</p>
<div class="section" id="undirected-simplicial-homology">
<h2>Undirected simplicial homology<a class="headerlink" href="#undirected-simplicial-homology" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
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<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/homology/gtda.homology.VietorisRipsPersistence.html#gtda.homology.VietorisRipsPersistence" title="gtda.homology.VietorisRipsPersistence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homology.VietorisRipsPersistence</span></code></a>([metric, …])</p></td>
<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <a class="reference internal" href="../theory/glossary.html#vietoris-rips-complex-and-vietoris-rips-persistence"><span class="std std-ref">Vietoris–Rips filtrations</span></a>.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/homology/gtda.homology.WeightedRipsPersistence.html#gtda.homology.WeightedRipsPersistence" title="gtda.homology.WeightedRipsPersistence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homology.WeightedRipsPersistence</span></code></a>([metric, …])</p></td>
<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <span class="xref std std-ref">weighted Vietoris–Rips filtrations</span> as in <a href="#id3"><span class="problematic" id="id1">[3]_</span></a>.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/homology/gtda.homology.SparseRipsPersistence.html#gtda.homology.SparseRipsPersistence" title="gtda.homology.SparseRipsPersistence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homology.SparseRipsPersistence</span></code></a>([metric, …])</p></td>
<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <a class="reference internal" href="../theory/glossary.html#vietoris-rips-complex-and-vietoris-rips-persistence"><span class="std std-ref">Sparse Vietoris–Rips filtrations</span></a>.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/homology/gtda.homology.WeakAlphaPersistence.html#gtda.homology.WeakAlphaPersistence" title="gtda.homology.WeakAlphaPersistence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homology.WeakAlphaPersistence</span></code></a>([…])</p></td>
<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <span class="xref std std-ref">weak alpha filtrations</span>.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/homology/gtda.homology.EuclideanCechPersistence.html#gtda.homology.EuclideanCechPersistence" title="gtda.homology.EuclideanCechPersistence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homology.EuclideanCechPersistence</span></code></a>([…])</p></td>
<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <a class="reference external" href="cech_complex_and_cech_persistence">Cech filtrations</a>.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="directed-simplicial-homology">
<h2>Directed simplicial homology<a class="headerlink" href="#directed-simplicial-homology" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
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<tr class="row-odd"><td><p><a class="reference internal" href="generated/homology/gtda.homology.FlagserPersistence.html#gtda.homology.FlagserPersistence" title="gtda.homology.FlagserPersistence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">homology.FlagserPersistence</span></code></a>([…])</p></td>
<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <a class="reference internal" href="../theory/glossary.html#filtered-complex"><span class="std std-ref">filtrations</span></a> of <a class="reference internal" href="../theory/glossary.html#clique-and-flag-complexes"><span class="std std-ref">directed or undirected flag complexes</span></a> <a href="#id4"><span class="problematic" id="id2">[1]_</span></a>.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="cubical-homology">
<h2>Cubical homology<a class="headerlink" href="#cubical-homology" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
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<td><p><a class="reference internal" href="../theory/glossary.html#persistence-diagram"><span class="std std-ref">Persistence diagrams</span></a> resulting from <a class="reference internal" href="../theory/glossary.html#cubical-complex"><span class="std std-ref">filtered cubical complexes</span></a>.</p></td>
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