标题： 时空持久性景观 显示英文标题

标题： Spatiotemporal Persistence Landscapes

摘要： 提出了一种应用和可视化时间序列持久同调的方法。 该方法在空间和时间上捕捉持久特征，与现有方法不同，现有方法通常选择其中一个而固定另一个。 定义了一个从时间序列构建的扩展Zigzag模。 该模结合了Zigzag持久同调和多参数持久同调的思想。 使用对秩不变量的最新推广（Kim, Mémoli, 2021），为扩展Zigzag模的情况定义了持久景观。 这种新的不变量称为时空持久景观。 在某些有限性假设下，时空持久景观是一族取值于Lebesgue空间的函数，为持久景观空间赋予了一个距离。 展示了该不变量相对于扩展Zigzag模的适应交错距离的稳定性。 作为一种取值于Banach空间的不变量，时空持久景观可以用于统计分析以及作为机器学习算法的输入。 显示英文摘要

摘要： 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.