Witness Complexes for Time Series Analysis, Nikki Sanderson (University of Colorado) Seminar

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Time series analysis relying upon statistics and frequency analyses makes restrictive assumptions about the underlying data - i.e. nonlinearity, non-stationarity. We believe topological data analysis (TDA) can be of benefit in these situations. Yet time series do not necessarily have interesting topology in their own right. The process of delay coordinate reconstruction ``unfolds” a scalar time-series into a point cloud in Rm. We can then compute the persistent homology of the reconstructed data to obtain a topological signature. With the ultimate goal of online regime shift detection in mind, we choose to use the witness complex - a sparse simplicial complex - for these computations. Topologically accurate delay reconstruction requires appropriate choices for the dimension m and time delay. We introduce novel witness relations that incorporate time and improve the robustness of the resulting homology with respect to choice of delay. These new relations seek to inhibit data points from witnessing landmarks traveling in dissimilar directions, as these can create false connections. We explore how these relations simultaneously address challenges that arise when dealing with non-uniform samples of strange attractors from chaotic dynamical systems.