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

@article{Perea2015SlidingWA,
  title={Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis},
  author={Jose A. Perea and John Harer},
  journal={Foundations of Computational Mathematics},
  year={2015},
  volume={15},
  pages={799-838}
}
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… 
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