Topological Data Analyses of Time Series Using Witness Complexes

Graduate Thesis Or Dissertation

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/mc87pq327

Abstract

Real-time regime shift detection between chaotic dynamical systems via time series analysis demands quick and correct, theoretically guaranteed methods. Often the best implemented techniques in the field are well-motivated heuristics and even interpretable statistics are scarce. Topological data analysis can contribute to the canon of traditional methods for analyzing nonlinear time series but is not computationally cheap. We introduce a topological membership test for sliding windows of time series data that uses a sparse simplicial complex - the witness complex - to model the data and assess its performance across a range of model parameters affecting computational efficiency. We then explore how the topology of witness complexes changes across this range of model parameters. We next define a simplicial complex whose construction incorporates the temporal information available with time series data. We experimentally show that this construction results in filtrations with fewer simplices and improved topological signature. We apply our techniques to synthetic time series data including numerical solutions of classical low dimensional chaotic systems Lorenz and Rössler systems of ODEs as well as regimes of the higher dimensional Brunel neuronal network model and experimental live voltage recordings of musical instruments.

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