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How Does State Space Definition Influence the Measure of Chaotic Behavior?

  • Henryk JosińskiEmail author
  • Adam Świtoński
  • Agnieszka Michalczuk
  • Marzena Wojciechowska
  • Konrad Wojciechowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11432)

Abstract

In the case of experimental data the largest Lyapunov exponent is a measure which is used to quantify the amount of chaos in a time series on the basis of a trajectory reconstructed in a phase (state) space. The authors’ goal was to analyze the influence of a state space definition on the measure of chaos. The time series which represent the joint angles of hip, knee and ankle joints were recorded using the motion capture technique in the CAREN Extended environment. Fourteen elderly subjects (‘65+’) participated in the experiments. Six state spaces based on univariate or multivariate time series describing a movement at individual joints were taken into consideration. The authors proposed a modified version of the False Nearest Neighbors algorithm adjusted for determining the embedding dimension in the case of a multivariate time series representing gait data (MultiFNN). The largest short-term Lyapunov exponent was computed in two variants for six scenarios of trials based on different assumptions regarding walking speed, platform inclination, and optional external perturbation. A statistical analysis confirmed a significant difference between values of the Lyapunov exponent for different state spaces. In addition, computation time was measured and averaged across the spaces.

Keywords

Nonlinear time series analysis State space Largest Lyapunov exponent Human motion analysis CAREN Extended system 

Notes

Acknowledgments

Data used in this project were obtained from the Centre for Research and Development of the Polish-Japanese Academy of Information Technology (PJAIT) (http://bytom.pja.edu.pl/). This work was supported by Statutory Research funds of Institute of Informatics, Silesian University of Technology, Gliwice, Poland (grant No BK-213/RAU2/2018).

References

  1. 1.
    Baker, G.B., Gollub, J.P.: Chaotic Dynamics: An Introduction. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  2. 2.
    Bruijn, S.M., Meijer, O.G., Beek, P.J., van Dieёn, J.H.: Assessing the stability of human locomotion: a review of current measures. J. Royal Soc. Interface 10, 20120999 (2013)CrossRefGoogle Scholar
  3. 3.
    Buzzi, U.H., Stergiou, N., Kurz, M.J., Hageman, P.A., Heidel, J.: Nonlinear dynamics indicates aging affects variability during gait. Clin. Biomech. (Bristol, Avon) 18(5), 435–443 (2003)CrossRefGoogle Scholar
  4. 4.
    Dingwell, J.B., Cusumano, J.P.: Nonlinear time series analysis of normal and pathological human walking. Chaos 10(4), 848–863 (2000)CrossRefGoogle Scholar
  5. 5.
    Gates, D.H., Dingwell, J.B.: Comparison of different state space definitions for local dynamic stability analyses. J. Biomech. 42(9), 1345–1349 (2009)CrossRefGoogle Scholar
  6. 6.
    Henry, B., Lovell, N., Camacho, F.: Nonlinear dynamics time series analysis. In: Akay, M. (ed.) Nonlinear Biomedical Signal Processing, Volume 2: Dynamic Analysis and Modeling, pp. 1–39. Wiley/IEEE Press (2000)Google Scholar
  7. 7.
    Kennel, M.B., Brown, R., Abarbanel, H.D.I.: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45(6), 3403–3411 (1992)CrossRefGoogle Scholar
  8. 8.
    Liu, K., Wang, H., Xiao, J.: The multivariate largest lyapunov exponent as an age-related metric of quiet standing balance. Comput. Math. Methods Med. 2, 1–11 (2015)zbMATHGoogle Scholar
  9. 9.
    Piórek, M., Josiński, H., Michalczuk, A., Świtoński, A., Szczęsna, A.: Quaternions and joint angles in an analysis of local stability of gait for different variants of walking speed and treadmill slope. Inf. Sci. 384, 263–280 (2017)CrossRefGoogle Scholar
  10. 10.
    Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65, 117–134 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stergiou, N., Decker, L.M.: Human movement variability, nonlinear dynamics, and pathology: is there a connection? Hum. Mov. Sci. 30(5), 869–888 (2011)CrossRefGoogle Scholar
  12. 12.
    Takens, F.: Detecting strange attractors in turbulence. In: Rand, D., Young, L.-S. (eds.) Dynamical Systems and Turbulence, Warwick 1980. LNM, vol. 898, pp. 366–381. Springer, Heidelberg (1981).  https://doi.org/10.1007/BFb0091924CrossRefGoogle Scholar
  13. 13.
    Vlachos, I., Kugiumtzis, D.: State space reconstruction for multivariate time series prediction. Nonlinear Phenom. Complex Syst. 11(2), 241–249 (2008)MathSciNetGoogle Scholar
  14. 14.
    Zhang, Ch.-T., Guo, J., Ma, Q.-L., Peng, H., Zhang, X.-D.: Phase space reconstruction and prediction of multivariate chaotic time series. In: Proceedings of the 9th International Conference on Machine Learning and Cybernetics, pp. 2428–2433. IEEE (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Henryk Josiński
    • 1
    Email author
  • Adam Świtoński
    • 1
  • Agnieszka Michalczuk
    • 1
  • Marzena Wojciechowska
    • 2
  • Konrad Wojciechowski
    • 2
  1. 1.Silesian University of TechnologyGliwicePoland
  2. 2.Polish-Japanese Academy of Information TechnologyWarsawPoland

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