有序点云的距离矩阵及其持续同调
Distance Matrices of Ordered Point Clouds and Their Persistent Homology
中文总结 AI 辅助
该研究建立了时间序列的距离矩阵滤过与其状态空间嵌入的Čech/Vietoris–Rips滤过的一次链映射关系，简化了时间序列分析的循环特征计算，可用于分析循环运动的转变。
中文摘要 AI 辅助
有限点云的距离矩阵可可视化为热图。当数据源自时间序列时，该图像的子水平集被称为递归图，广泛用于时间序列分析。受此视角启发，我们以一次链映射的形式建立了时间序列的距离矩阵滤过与其状态空间嵌入的Čech（或Vietoris–Rips）滤过之间的关系。我们研究了诱导的同调映射，表明从H₀到H₁的映射是本质满射，并给出了从H₁到H₂的映射非平凡的例子。这些链映射可用于简化循环特征计算中产生的图像持续同调计算，循环特征是一种用于时间序列分析的拓扑工具，此外，这些计算能产生更精细的信息，可用于分析不同类型循环运动之间的转变。
英文摘要
The distance matrix of a finite point cloud can be visualized as a heatmap. When the data arise from a time series, the sublevel sets of this image are known as recurrence plots and are widely used in time series analysis. Motivated by this perspective, we establish a relationship between the distance-matrix filtration of the time series and the Čech (or Vietoris--Rips) filtration of its state-space embedding in the form of a degree-one chain map. We study the induced maps in homology, showing that the map from $H_0$ into $H_1$ is essentially surjective and providing an example where the map from $H_1$ into $H_2$ is nontrivial. These chain maps can be applied to simplify image persistence computations arising in the computation of cycling signatures, a topological tool for time series analysis. Moreover, these computations yield finer information that allows the analysis of transitions between different types of cycling motion.
