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

Abstract

We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring.

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Acknowledgments

Both authors were supported in part by DARPA under Grants D12AP00001 and D12AP00025-002 and by the AFOSR under Grant FA9550-10-1-0436.

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Correspondence to Jose A. Perea.

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Communicated by Herbert Hedelsbrunner.

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Perea, J.A., Harer, J. Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis. Found Comput Math 15, 799–838 (2015). https://doi.org/10.1007/s10208-014-9206-z

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Keywords

  • Persistent homology
  • Time-delay embeddings
  • Periodicity

Mathematics Subject Classification

  • Primary 55U99
  • 37M10
  • 68W05
  • Secondary 57M99