Euler Characteristic Surfaces: A Stable Multiscale Topological Summary of Time Series Data

We present Euler Characteristic Surfaces as a multiscale spatiotemporal topological summary of time series data encapsulating the topology of the system at different time instants and length scales. Euler Characteristic Surfaces with an appropriate metric is used to quantify stability and locate critical changes in a dynamical system with respect to variations in a parameter, while being substantially computationally cheaper than available alternate methods such as persistent homology. The stability of the construction is demonstrated by a quantitative comparison bound with persistent homology, and a quantitative stability bound under small changes in time is established. The proposed construction is used to analyze two different kinds of simulated disordered flow situations.