N. Sanderson, E. Shugerman, S. Molnar, J. Meiss E. Bradley, "Computational Topology Techniques for Characterizing Time-Series Data", Advances in Intelligent Data Analysis XVI 16th International Symposium, IDA 2017, London, UK, October 26–28, 2017, Proceedings, October 2017, pp.284-296, doi: 10.1007/978-3-319-68765-0_24
Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure—counting pieces and holes—could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems—e.g., the same note played on different musical instruments.