Department of Mathematical Sciences, Montana State University, Bozeman, MT, USA.
Biology Department, Duke University, Durham, NC, USA.
J Math Biol. 2020 Apr;80(5):1523-1557. doi: 10.1007/s00285-020-01471-4. Epub 2020 Feb 1.
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 [Formula: see text] by a sequence of intervals, each of which is guaranteed to contain an extremum for any function that [Formula: see text]-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 [Formula: see text]. 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.
实验时间序列为了解潜在动力系统提供了一个有价值的窗口，时间序列极值（或其导数）的时间包含了有关其结构的信息。然而，时间序列通常包含显著的测量误差。我们描述了一种方法，用于通过一系列区间来描述任何假设测量误差水平的时间序列[公式：见正文]，每个区间都保证包含任何[公式：见正文]逼近时间序列的函数的极值。基于连续函数的合并树，我们定义了一个称为归一化分支分解的新对象，它允许我们为任何级别[公式：见正文]计算区间。我们表明，对于单个时间序列，这些区间之间存在一个明确定义的全序，并且可以自然地将其扩展到包含数据集的一系列时间序列的偏序。我们在两个应用程序中使用提取区间的顺序。首先，可以使用描述单个数据集的偏序来与切换模型输出进行模式匹配（Cummins 等人在 SIAM J Appl Dyn Syst 17(2):1589-1616, 2018 中），这允许拒绝网络模型。其次，可以使用不同数据集的偏序图距离之间的比较来量化生物重复之间的相似性。
