Berry, EricCummins, BreeNerem, Robert R.Smith, Lauren M.Haase, Steven B.Gedeon, Tomas2021-05-212021-05-212020-02Berry, 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/s00285-020-01471-4.0303-6812https://scholarworks.montana.edu/handle/1/16356Experimental 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.en-USThis is a post-peer-review, pre-copyedit version of an article published in 'Journal of Mathematical Biology'. The final authenticated version is available online at: https://doi.org/10.1007/s00285-020-01471-4. The following terms of use apply: https://www.springer.com/gp/open- access/publication-policies/aam-terms-of-use.https://www.springer.com/gp/open- access/publication-policies/aam-terms-of-useArticle