Witness complexes for time series analysis

Time series analysis traditionally relies upon statistics and frequency analyses that make restrictive assumptions about the data - i.e. nonlinearity, non-stationarity. We believe topological data analysis (TDA) can be of benefit in these situations. The process of delay coordinate reconstruction ``unfolds” a scalar time-series into a point cloud in R^m. We can then compute the persistent homology of the reconstructed data to obtain a topological signature. With the ultimate goal of 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. The 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 can ameliorate additional challenges that arise when dealing with non-uniform samples of strange attractors.