Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Twisty Takens: a geometric characterization of good observations on dense trajectories

Twisty Takens: a geometric characterization of good observations on dense trajectories In nonlinear time series analysis and dynamical systems theory, Takens’ embedding theorem states that the sliding window embedding of a generic observation along trajectories in a state space, recovers the region traversed by the dynamics. This can be used, for instance, to show that sliding window embeddings of periodic signals recover topological loops, and that sliding window embeddings of quasiperiodic signals recover high-dimensional torii. However, in spite of these motivating examples, Takens’ theorem does not in general prescribe how to choose such an observation function given particular dynamics in a state space. In this work, we state conditions on observation functions defined on compact Riemannian manifolds, that lead to successful reconstructions for particular dynamics. We apply our theory and construct families of time series whose sliding window embeddings trace tori, Klein bottles, spheres, and projective planes. This greatly enriches the set of examples of time series known to concentrate on various shapes via sliding window embeddings, and will hopefully help other researchers in identifying them in naturally occurring phenomena. We also present numerical experiments showing how to recover low dimensional representations of the underlying dynamics on state space, by using the persistent cohomology of sliding window embeddings and Eilenberg–MacLane (i.e. circular and real projective) coordinates. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

Twisty Takens: a geometric characterization of good observations on dense trajectories

Download PDF
29 pages

Loading next page...
 
/lp/springer-journals/twisty-takens-a-geometric-characterization-of-good-observations-on-05YPWI2zX4
Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Algebraic Topology; Computational Science and Engineering; Mathematical and Computational Biology
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-019-00036-9
Publisher site
See Article on Publisher Site

Abstract

In nonlinear time series analysis and dynamical systems theory, Takens’ embedding theorem states that the sliding window embedding of a generic observation along trajectories in a state space, recovers the region traversed by the dynamics. This can be used, for instance, to show that sliding window embeddings of periodic signals recover topological loops, and that sliding window embeddings of quasiperiodic signals recover high-dimensional torii. However, in spite of these motivating examples, Takens’ theorem does not in general prescribe how to choose such an observation function given particular dynamics in a state space. In this work, we state conditions on observation functions defined on compact Riemannian manifolds, that lead to successful reconstructions for particular dynamics. We apply our theory and construct families of time series whose sliding window embeddings trace tori, Klein bottles, spheres, and projective planes. This greatly enriches the set of examples of time series known to concentrate on various shapes via sliding window embeddings, and will hopefully help other researchers in identifying them in naturally occurring phenomena. We also present numerical experiments showing how to recover low dimensional representations of the underlying dynamics on state space, by using the persistent cohomology of sliding window embeddings and Eilenberg–MacLane (i.e. circular and real projective) coordinates.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Aug 16, 2019

References

  • Measuring structural similarity in music
    Bello, JP
  • Segregation of information about emotional arousal and valence in horse whinnies
    Briefer, EF; Maigrot, A-L; Mandel, R; Freymond, SB; Bachmann, I; Hillmann, E
  • Topology and data
    Carlsson, G
  • Decomposition of pointwise finite-dimensional persistence modules
    Crawley-Boevey, W
  • Persistent cohomology and circular coordinates
    Silva, V; Morozov, D; Vejdemo-Johansson, M
  • Persistent homology—a survey
    Edelsbrunner, H; Harer, J
  • Computational Topology: An Introduction
    Edelsbrunner, H; Harer, J
  • Three-dimensional alpha shapes
    Edelsbrunner, H; Mücke, EP
  • Elementary Applied Topology
    Ghrist, RW
  • Quasi-periodic intermittency in oscillating cylinder flow
    Glaz, B; Mezić, I; Fonoberova, M; Loire, S
  • Algebraic Topology
    Hatcher, A
  • Analysis of vocal disorders with methods from nonlinear dynamics
    Herzel, H; Berry, D; Titze, IR; Saleh, M
  • Nonlinear Time Series Analysis
    Kantz, H; Schreiber, T
  • Chatter detection in turning using persistent homology
    Khasawneh, FA; Munch, E
  • Hamiltonian systems and transformation in hilbert space
    Koopman, BO
  • Analysis of fluid flows via spectral properties of the koopman operator
    Mezić, I
  • The tangled tale of phase space
    Nolte, DD
  • Multiscale projective coordinates via persistent cohomology of sparse filtrations
    Perea, JA
  • Topological time series analysis
    Perea, JA
  • Sliding windows and persistence: an application of topological methods to signal analysis
    Perea, JA; Harer, J
  • Sw1pers: Sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data
    Perea, JA; Deckard, A; Haase, SB; Harer, J
  • Detection of the electrocardiogramfiducial points in the phase space using area calculation
    Plesnik, E; Malgina, O; Tasic, JF; Tomazic, S; Zajc, M
  • Cross recurrence quantification for cover song identification
    Serra, J; Serra, X; Andrzejak, RG
  • Nonlinear dynamical analysis of eeg and meg: review of an emerging field
    Stam, CJ
  • Detecting strange attractors in turbulence
    Takens, F
  • Ripser.py: A lean persistent homology library for python
    Tralie, C; Saul, N; Barr-On, R
  • (Quasi) Periodicity quantification in video data, using topology
    Tralie, CJ; Perea, JA
  • Flat surfaces
    Zorich, A