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An introduction to persistent homology for time series

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Abstract Topological data analysis (TDA) uses information from topological structures in complex data for statistical analysis and learning. This paper discusses persistent homology, a part of computational (algorithmic) topology that converts data into simplicial complexes and elicits information about the persistence of homology classes in the data. It computes and outputs the birth and death of such topologies via a persistence diagram. Data inputs for persistent homology are usually represented as point clouds or as functions, while the outputs depend on the nature of the analysis and commonly consist of either a persistence diagram, or persistence landscapes. This paper gives an introductory level tutorial on computing these summaries for time series using R, followed by an overview on using these approaches for time series classification and clustering. This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Applications of Computational Statistics > Computational Mathematics
Persistence diagram corresponding to a point cloud. (a) Shows the raw point cloud and (b) shows the persistence diagram
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Persistence diagrams using Walsh‐Fourier transforms
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Persistence landscapes of the persistence diagram pers.diag.2
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Persistence diagrams using Walsh‐Fourier transforms
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Persistence diagrams using second‐order spectrum
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Distance‐to‐measure function and the persistence diagram
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Construction of a persistence diagram corresponding to a one‐dimensional continuous real function. (a) Is the function and (h) is the persistence diagram. (c), (e), and (g) show the sublevel set filtration procedure, while (b), (d), and (f) are the intermediate steps for constructing the persistence diagram
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Persistence diagrams of periodic time series with different shapes
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Pure periodic signals to persistence diagrams
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(a) Rotation of the persistence diagram and (b) barcode representation of persistent homology
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Persistence diagram corresponding to a point cloud. (a) Shows the raw point cloud and (h) shows the persistence diagram. (c), (e), and (g) are intermediate steps for the filtration by varying r, while (b), (d), and (f) are intermediate steps for constructing the persistence diagram
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Applications of Computational Statistics > Computational Mathematics
Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data
Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification

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