Abstract
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.
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Notes
- 1.
The work we describe in this paper calls upon areas of mathematics—including dynamical systems, topology and persistent homology—that may not be commonly used in the data-analysis community. As a full explanation of these would require several textbook length treatments, we content ourselves with discussing how these ideas can be applied, leaving the details of the theory to references.
- 2.
Landmark choice is another issue. There are a number of ways to do this; here, we evenly space the landmarks across the data.
- 3.
Choosing the range and increment for \(\epsilon \) in such a plot requires some experimentation; in this paper, we use \(\epsilon _{\text {step}} = 20\) and \(\epsilon _{\text {max}}\) set for each instrument when the first 20-dimensional simplex is witnessed. This is a good compromise between effectiveness and efficiency for the data sets that we studied.
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Sanderson, N., Shugerman, E., Molnar, S., Meiss, J.D., Bradley, E. (2017). Computational Topology Techniques for Characterizing Time-Series Data. In: Adams, N., Tucker, A., Weston, D. (eds) Advances in Intelligent Data Analysis XVI. IDA 2017. Lecture Notes in Computer Science(), vol 10584. Springer, Cham. https://doi.org/10.1007/978-3-319-68765-0_24
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