Mutual Information for Quaternion Time Series

  • Michał PiórekEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9842)


Mutual Information method is a widely used method for estimation of time delay value in the process of time delay embedding. It’s designed for a univariate scalar time series. In the real systems often many outputs of investigated system are available. In this case a multivariate time delay estimation method is necessary if one may require to perform the uniform time delay embedding. The special case of multivariate data is a kinematic time series(e.g. quaternion time series or Euler angles time series). The main goal of this paper is to provide a method for this case: Mutual Information method’s extension for quaternion time series. The results are also compared with previosly presented quaternion’s angle method. The method was tested on the real kinematic data - the recordings of human gait.


Mutual information Quaternions Deterministic chaos Nonlinear time series analysis Human motion analysis Human gait data 



This work has been supported by the “Młoda Kadra” project from the Wrocław University of Science and Technology.

The author also would like to acknowledge the Polish-Japanese Institute of Information Technology for the gait sequences recorded in the Human Motion Laboratory (HML).


  1. 1.
    Abarbanel, H.: Analysis of Observed Chaotic Data. Springer, New York (1996)zbMATHCrossRefGoogle Scholar
  2. 2.
    Albano, A.M., et al.: Using high-order correlations to define an embedding window. Phys. D 54, 85–97 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brown, R., Kennel, M., Abarbanel, H.: Determinig embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45(6), 3403 (1992)CrossRefGoogle Scholar
  4. 4.
    Buzug, T., Pfister, G.: Comparison of algorithms calculating optimal embedding parameters for delay time coordinates. Phys. D 58, 127–137 (1992)CrossRefGoogle Scholar
  5. 5.
    Buzug, T., Pfister, G.: Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static, local dynamical behaviour of strange attractors. Phys. Rev. A 45, 7073–7084 (1992)CrossRefGoogle Scholar
  6. 6.
    Cao, L., Mees, A., Judd, K.: Dynamics from multivariate time series. Phys. D 121, 75–88 (1998)zbMATHCrossRefGoogle Scholar
  7. 7.
    Cao, L.: Practical method for determining the minimum embed-ding dimension of a scalar time series. Phys. D: Nonlinear Phenom. 110(1), 43–50 (1997)CrossRefGoogle Scholar
  8. 8.
    Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chong-zhao, H., Hong-guang, M.: Selection of embedding dimension and delay time in phase space reconstruction. Front. Electr. Electron. Eng. 1, 111–114 (2006)CrossRefGoogle Scholar
  10. 10.
    Jablonski, B.: Quaternion dynamic time warping. IEEE Trans. Sig. Process. 60(3), 1174–1183 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Huang, Z., Lin, J., Wang, Y., Shen, Z.: Selection of proper time-delay in phase space reconstruction of speech signals. Sig. Process. 15, 220–225 (1999)Google Scholar
  12. 12.
    James MacQueen et al.: Some methods for classification and analysis of multivariate observations, vol. 1(14), pp. 281–297 (1967)Google Scholar
  13. 13.
    Maus, A., Sprott, J.C.: Neural network method for determining embedding dimension of a time series. Commun. Nonlinear Sci. Numer. Simul. 16, 3294–3302 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Montalto, A., Faes, L., Marinazzo, D.: Mute: a matlab toolbox to compare established and novel estimators of the multivariate transfer entropy. PloS one 9(10), e109462 (2014)CrossRefGoogle Scholar
  15. 15.
    Piorek, M.: Computer Information Systems and Industrial Management. Chaotic Properties of Gait Kinematic Data, pp. 111–119. Springer International Publishing, Cham (2015)CrossRefGoogle Scholar
  16. 16.
    Rokach, L., Maimon, O.: Clustering methods, (2005)Google Scholar
  17. 17.
    Colins, J.J., Rossenstein, M.T., de Luca, C.J.: Reconstruction expansion as a geometry-based framework for choosing proper delay times. Phys. D 73, 82–98 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Takens, F.: Detecting strange attractors in turbulence. Springer-Verlag, Berlin (1981)zbMATHCrossRefGoogle Scholar
  19. 19.
    Vlachos, I., Kugiumtzis, D.: Nonuniform state-space recon- struction and coupling detection. Phys. Rev. E 82(1), 016207 (2010)CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (, which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  1. 1.Department of Computer EngineeringWrocław University of Science and TechnologyWrocławPoland

Personalised recommendations