A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed. An extended zigzag module that is built from a time series is defined. This module combines ideas from zigzag persistent homology and multiparameter persistent homology. Persistence landscapes are defined for the case of extended zigzag modules using a recent generalization of the rank invariant (Kim, M\'emoli, 2021). This new invariant is called spatiotemporal persistence landscapes. Under certain finiteness assumptions, spatiotemporal persistence landscapes are a family of functions that take values in Lebesgue spaces, endowing the space of persistence landscapes with a distance. Stability of this invariant is shown with respect to an adapted interleaving distance for extended zigzag modules. Being an invariant that takes values in a Banach space, spatiotemporal persistence landscapes can be used for statistical analysis as well as for input to machine learning algorithms.

翻译：本文提出了一种应用于时间序列持续性同调的方法及其可视化技术。该方法能够同时捕捉空间和时间维度上的持续性特征，与现有通常固定一个维度而分析另一个维度的流程形成对比。我们定义了一种基于时间序列构建的扩展之字形模。该模融合了之字形持续性同调与多参数持续性同调的思想。利用秩不变量的最新推广成果（Kim, Mémoli, 2021），我们为扩展之字形模定义了持续性景观。这种新不变量称为时空持续性景观。在一定的有限性假设下，时空持续性景观是一族取值于勒贝格空间的函数，从而赋予持续性景观空间以距离度量。我们证明了该不变量相对于扩展之字形模的适配交错距离具有稳定性。作为取值于巴拿赫空间的不变量，时空持续性景观既可用于统计分析，也可作为机器学习算法的输入数据。