# 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…

## 219 Citations

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This paper will review recent developments and contributions where topological data analysis especially persistent homology has been applied to time series analysis, dynamical systems and signal processing and cover problem statements such as stability determination, risk analysis, systems behaviour, and predicting critical transitions in financial markets.

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A new distance is introduced on the space of persistence diagrams, and it is shown that it is useful in detecting changes in geometry and topology, which is essential for the supervised learning problem.

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