Spatiotemporal Persistence Landscapes

TL;DR

This paper introduces spatiotemporal persistence landscapes, a novel method combining zigzag and multiparameter persistent homology to analyze and visualize persistent features in time series across space and time.

Contribution

It develops a new invariant called spatiotemporal persistence landscapes using extended zigzag modules, enabling statistical and machine learning applications.

Findings

Defines spatiotemporal persistence landscapes for time series.

Proves stability of the invariant under an interleaving distance.

Provides a framework for statistical analysis of persistent features.

Abstract

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.