Distance Matrices of Ordered Point Clouds and Their Persistent Homology

Abstract

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 \v{C}ech (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.