Spatiotemporal persistence landscapes

A method to apply and visualize persistent homology of time series is proposed. The method captures features that are persistent with respect to 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 and Mémoli 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. This invariant can be applied whenever data is given that can be transformed into a finite sequence of point clouds, including uni- and multivariate time series as well as a time-varying point cloud.