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Spatiotemporal Persistence Landscapes

Martina Flammer, Knut Hüper

TL;DR

The paper develops spatiotemporal persistence landscapes, a Banach-space-valued invariant for time-series data built on extended zigzag modules that fuse zigzag and multiparameter persistence ideas. It defines landscapes via a generalized rank invariant on a spatiotemporal filtration, proves stability under an adapted interleaving distance, and shows these landscapes admit statistical treatment as random variables in $L^p$ spaces. An algorithmic pipeline leverages rank computations along boundary zigzag paths and FastZigzag to produce landscapes from time-series data, with practical use demonstrated on sinusoidal signals and Hopf-bifurcation-like dynamics. The method enables robust, scalable analysis of features persistent in both space and time, with potential for integration into statistical and machine-learning workflows.

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, 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.

Spatiotemporal Persistence Landscapes

TL;DR

The paper develops spatiotemporal persistence landscapes, a Banach-space-valued invariant for time-series data built on extended zigzag modules that fuse zigzag and multiparameter persistence ideas. It defines landscapes via a generalized rank invariant on a spatiotemporal filtration, proves stability under an adapted interleaving distance, and shows these landscapes admit statistical treatment as random variables in spaces. An algorithmic pipeline leverages rank computations along boundary zigzag paths and FastZigzag to produce landscapes from time-series data, with practical use demonstrated on sinusoidal signals and Hopf-bifurcation-like dynamics. The method enables robust, scalable analysis of features persistent in both space and time, with potential for integration into statistical and machine-learning workflows.

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, 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.

Paper Structure

This paper contains 26 sections, 18 theorems, 47 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistent homology
  4. Persistence landscapes
  5. Oneparameter persistence landscapes
  6. Multiparameter persistence landscape
  7. Persistent homology for time series
  8. Time delay embedding
  9. Zigzag persistent homology for time series
  10. Generalized Rank
  11. Spatiotemporal persistence landscapes
  12. Definition
  13. Statistics
  14. Stability of Spatiotemporal Persistence Landscapes
  15. Block extension functor for zigzag modules
  16. ...and 11 more sections

Key Result

Lemma 1

Let $M$ be a multiparameter persistence module with rank function $\beta_0(\cdot,\cdot)$. Let $1\in\mathbb{R}^n$ be the vector where every entry is $1$. For all $h\geq 0$ we have $\beta_0(x-\|h\|_\infty 1,x+\|h\|_\infty 1)\leq \beta_0(x-h,x+h)$.

Figures (7)

  • Figure 1: The left hand side shows a persistence diagram (black dots) and the right hand side the corresponding landscapes $\lambda_1$ and $\lambda_2$.
  • Figure 2: Inclusion of the zigzag poset into $\mathop{\mathrm{\mathbb{Z}}}\nolimits^\mathrm{op}\times\mathop{\mathrm{\mathbb{Z}}}\nolimits$.
  • Figure 3: Extension of zigzag intervals to block intervals for the four different types $(\cdot,\cdot)$, $[\cdot,\cdot)$, $(\cdot,\cdot]$ and $[\cdot,\cdot]$ (in that order). Cf. Figure 3 in botnan2018.
  • Figure 4: (a) A visualization of the block extension functor and Lemma \ref{['lemma_extension_functor_explicit']}. The unique map is given by the universal property of colimits. Cf. Figure 2 in botnan2018. (b) The set $U_L$.
  • Figure 5: First row: noisy data; second row: delay embedding of the first and last windows; third row: first persistence landscape.
  • ...and 2 more figures

Theorems & Definitions (60)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Example 1
  • Definition 5
  • Remark 3
  • ...and 50 more