Using extremal events to characterize noisy time series
Nerem, Robert R.
Smith, Lauren M.
Haase, Steven B.
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Experimental time series provide an informative window into the underlying dynamical system, and the timing of the extrema of a time series (or its derivative) contains information about its structure. However, the time series often contain significant measurement errors. We describe a method for characterizing a time series for any assumed level of measurement error 𝜀 by a sequence of intervals, each of which is guaranteed to contain an extremum for any function that 𝜀-approximates the time series. Based on the merge tree of a continuous function, we define a new object called the normalized branch decomposition, which allows us to compute intervals for any level 𝜀. We show that there is a well-defined total order on these intervals for a single time series, and that it is naturally extended to a partial order across a collection of time series comprising a dataset. We use the order of the extracted intervals in two applications. First, the partial order describing a single dataset can be used to pattern match against switching model output (Cummins et al. in SIAM J Appl Dyn Syst 17(2):1589–1616, 2018), which allows the rejection of a network model. Second, the comparison between graph distances of the partial orders of different datasets can be used to quantify similarity between biological replicates.
Berry, Eric, Bree Cummins, Robert R. Nerem, Lauren M. Smith, Steven B. Haase, and Tomas Gedeon. “Using Extremal Events to Characterize Noisy Time Series.” Journal of Mathematical Biology 80, no. 5 (February 1, 2020): 1523–1557. doi:10.1007/s00285-020-01471-4.