Sliding Windows and Persistence: An Application of Topological Methods\n to Signal Analysis

2013/07/23 by José A. Perea, Perea, Jose, John Harer +2 · 12 citations
Computer Science · Medicine · Neuroscience · #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #FOS: Mathematics #Neuroinflammation and Neurodegeneration Mechanisms #Statistics Theory (math.ST) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1307.6188

openalex publication_date 2013/07/23 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We develop in this paper a theoretical framework for the topological study of\ntime series data. Broadly speaking, we describe geometrical and topological\nproperties of sliding window (or time-delay) embeddings, as seen through the\nlens of persistent homology. In particular, we show that maximum persistence at\nthe point-cloud level can be used to quantify periodicity at the signal level,\nprove structural and convergence theorems for the resulting persistence\ndiagrams, and derive estimates for their dependency on window size and\nembedding dimension. We apply this methodology to quantifying periodicity in\nsynthetic data sets, and compare the results with those obtained using\nstate-of-the-art methods in gene expression analysis. We call this new method\nSW1PerS which stands for Sliding Windows and 1-dimensional Persistence Scoring.\n

Citations

• Least-squares frequency analysis of unequally spaced data
• Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data
• Stability of Persistence Diagrams
• Probability measures on the space of persistence diagrams

Cited by

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• Flow-Aware Ellipsoidal Filtration for Persistent Homology of Recurrent Signals
• Seeing Beyond Sound: Visualization and Abstraction in Audio Data Representation
• Quantum topological data analysis algorithm for dynamical systems
• Tracking Temporal Evolution of Topological Features in Image Data
• Topology Structure Optimization of Reservoirs Using GLMY Homology
• Time Series Analysis of Spiking Neural Systems via Transfer Entropy and Directed Persistent Homology
• Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review

Related

• Exact multi-parameter persistent homology of time-series data: Fast and variable topological inferences
• Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data
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• Persistence Diagram Estimation : Beyond Plug-in Approaches
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