Departmental Seminar

Witness Complex for Time Series Analysis

Date & time

Thu 22 Jun 2017, 10–10.30am



Le Couteur Seminar Room (3.17)


Members of RSPE welcome


Ms Nikki Anderson
University of Colorado

A scalar time-series can be ``unfolded" into $\mathbb{R}^d$ by delay coordinate reconstruction. In the best case, this gives an attractor that is topologically equivalent to that of the underlying dynamical system.  We can then compute the persistent homology of the reconstructed data using a variety of complexes, e.g., \v{C}ech, Vietoris-Rips, or alpha. To be more computationally efficient we use a witness complex: it can provide a sparser representation and yet be faithful to the homology. Topologically accurate delay reconstruction requires choice of appropriate values for a $d$ and a time delay. In practice, these must be estimated from the data, and the estimation procedures are heuristic and often problematic. Recent work of Garland et al. demonstrates that accurate persistent homology computations are possible from witness complexes built from delay reconstructions with $d$ below that demanded by the theory. Following this, we introduce novel witness relations that incorporate time and explore 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 disparate directions and that are on distinct branches of an attractor, as these can suggest a false connection due to the particular reconstruction.

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