Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise $L_{\infty}$ distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.

翻译：实验时间序列中的本地最大值和微型值,或极端值事件,可以用作粗略的概括性数据特征。然而,在记录实验测量中的离散抽样表明实验期间的极限值真实时间的不确定性。这反过来又给时间序列中的极限值的时间顺序带来不确定性。我们利用基因时间序列和生物网络分析的应用,建造了一个加权定向循环图(DAG),它使用从持久性同系物学到测量噪音的可靠技术,称为极端值事件DAG。此外,我们根据字符串之间的编辑距离界定了极端值事件DAG之间的距离。我们证明了若干特性,包括极端事件DAG距离对时间序列数据中函数之间距离的本地稳定性。最后,我们提供了算法、公开免费软件和对极端值事件DAG的构建和比较实施。