Nonlinear Dynamics

, Volume 97, Issue 4, pp 2813–2828 | Cite as

A novel method of visualizing q-complexity-entropy curve in the multiscale fashion

  • Chien-Hung Yeh
  • Yu Fang
  • Wenbin ShiEmail author
  • Yang Hong
Original paper


Increasing reports indicated that multiscale fashion is a general and efficient approach to investigate various natural and physiological states. To this end, a generalization of the q-complexity-entropy curve in the multiscale fashion (GMCE curve) is introduced. In addition, the concept of the minimum permutation entropy and the maximum statistical complexity are proposed to simplify the visualization of entropy/complexity in the multiscale fashion, so-called generalized multiscale permutation entropy (GMPE), which is proposed to yield a spectrum of entropy and complexity. We demonstrate that the proposed GMCE curve or the GMPE method is capable of identifying an irregular oscillation is either stochastic or chaotic, highly predictable or long-term correlated. Simulations include Gaussian white noise and 1 / f noise, stationary and fractionally integrated autoregressive processes, logistic map and Hénon map. In the application of the GMCE curve and the GMPE method in real datasets, the GMCE loop of the daily streamflow of hydrologic alteration is found to be able of serving as an indicator to basin imperviousness or the level of urbanization. High basin imperviousness corresponds to narrower q-complexity-entropy loop. In the sleep analyses, both the minimum permutation entropy and the maximum statistical complexity show significant differences across sleep stages \((p < 0.0001^*)\), whereas all five sleep stages are differentiable in the multiple comparisons. Both the real datasets present clearer visualization and discrimination among basin imperviousness or sleep stages than the standard multiscale entropy. Our approach enables us to investigate irregular oscillations with generalization in timescales and Tsallis q-entropy.


Permutation entropy Complexity-entropy curve Multiscale Sleep Hydrologic alteration 



This study is supported by the National Key Research and Development Program of China (Grant No. 2016YFE01024 00) and Beijing Institute of Technology Research Fund Program for Young Scholars (Grant No. 3050011181811).


  1. 1.
    Costa, M., Goldberger, A.L., Peng, C.-K.: Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 89(6), 068102 (2002)CrossRefGoogle Scholar
  2. 2.
    Shi, W., Shang, P., Ma, Y., Sun, S., Yeh, C.-H.: A comparison study on stages of sleep: quantifying multiscale complexity using higher moments on coarse-graining. Commun. Nonlinear Sci. Numer. Simul. 44, 292–303 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ma, Y., Shi, W., Peng, C.-K., Yang, A.C.: Nonlinear dynamical analysis of sleep electroencephalography using fractal and entropy approaches. Sleep Med. Rev. 37, 85–93 (2018)CrossRefGoogle Scholar
  4. 4.
    Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50(5), 346 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531–534 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)CrossRefGoogle Scholar
  7. 7.
    Humeau-Heurtier, A., Wu, C.-W., Wu, S.-D.: Refined composite multiscale permutation entropy to overcome multiscale permutation entropy length dependence. IEEE Signal Process. Lett. 22(12), 2364–2367 (2015)CrossRefGoogle Scholar
  8. 8.
    Frank, B., Pompe, B., Schneider, U., Hoyer, D.: Permutation entropy improves fetal behavioural state classification based on heart rate analysis from biomagnetic recordings in near term fetuses. Med. Biol. Eng. Comput. 44(3), 179 (2006)CrossRefGoogle Scholar
  9. 9.
    Stosic, T., Telesca, L., de Souza Ferreira, D.V., Stosic, B.: Investigating anthropically induced effects in streamflow dynamics by using permutation entropy and statistical complexity analysis: a case study. J. Hydrol. 540, 1136–1145 (2016)CrossRefGoogle Scholar
  10. 10.
    Zunino, L., Zanin, M., Tabak, B.M., Pérez, D.G., Rosso, O.A.: Forbidden patterns, permutation entropy and stock market inefficiency. Physica A Stat. Mech. Appl. 388(14), 2854–2864 (2009)CrossRefGoogle Scholar
  11. 11.
    Aziz, W., Arif, M.: Multiscale permutation entropy of physiological time series. In: 9th International Multitopic Conference, IEEE INMIC 2005, pp. 1–6. IEEE (2005)Google Scholar
  12. 12.
    Fadlallah, B., Chen, B., Keil, A., Príncipe, J.: Weighted-permutation entropy: a complexity measure for time series incorporating amplitude information. Phys. Rev. E 87(2), 022911 (2013)CrossRefGoogle Scholar
  13. 13.
    Shi, W., Shang, P., Lin, A.: The coupling analysis of stock market indices based on cross-permutation entropy. Nonlinear Dyn. 79(4), 2439–2447 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rosso, O.A., Larrondo, H.A., Martin, M.T., Plastino, A., Fuentes, M.A.: Distinguishing noise from chaos. Phys. Rev. Lett. 99(15), 154102 (2007)CrossRefGoogle Scholar
  15. 15.
    Ribeiro, H.V., Zunino, L., Mendes, R.S., Lenzi, E.K.: Complexity-entropy causality plane: a useful approach for distinguishing songs. Physica A Stat. Mech. Appl. 391(7), 2421–2428 (2012)CrossRefGoogle Scholar
  16. 16.
    Ribeiro, H.V., Zunino, L., Lenzi, E.K., Santoro, P.A., Mendes, R.S.: Complexity-entropy causality plane as a complexity measure for two-dimensional patterns. PLoS ONE 7(8), e40689 (2012)CrossRefGoogle Scholar
  17. 17.
    Xu, M., Shang, P.: Generalized permutation entropy analysis based on the two-index entropic form s q, \(\delta \). Chaos Interdiscip. J. Nonlinear Sci. 25(5), 053114 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ribeiro, H.V., Jauregui, M., Zunino, L., Lenzi, E.K.: Characterizing time series via complexity-entropy curves. Phys. Rev. E 95(6), 062106 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shi, W., Shang, P., Wang, J.: Large deviations estimates for the multiscale analysis of traffic speed time series. Physica A Stat. Mech. Appl. 421, 562–570 (2015)CrossRefGoogle Scholar
  20. 20.
    Ibáñez-Molina, A.J., Iglesias-Parro, S., Soriano, M.F., Aznarte, J.I.: Multiscale lempel-ziv complexity for eeg measures. Clin. Neurophysiol. 126(3), 541–548 (2015)CrossRefGoogle Scholar
  21. 21.
    Costa, M.D., Peng, C.-K., Goldberger, A.L.: Multiscale analysis of heart rate dynamics: entropy and time irreversibility measures. Cardiovasc. Eng. 8(2), 88–93 (2008)CrossRefGoogle Scholar
  22. 22.
    Costa, M.D., Henriques, T., Munshi, M.N., Segal, A.R., Goldberger, A.L.: Dynamical glucometry: use of multiscale entropy analysis in diabetes. Chaos Interdiscip. J. Nonlinear Sci. 24(3), 033139 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Li, Q., Zuntao, F.: Permutation entropy and statistical complexity quantifier of nonstationarity effect in the vertical velocity records. Phys. Rev. E 89(1), 012905 (2014)CrossRefGoogle Scholar
  24. 24.
    Martin, M.T., Plastino, A., Rosso, O.A.: Generalized statistical complexity measures: geometrical and analytical properties. Physica A Stat. Mech. Appl. 369(2), 439–462 (2006)CrossRefGoogle Scholar
  25. 25.
    Granger, C.W.J., Joyeux, R.: An introduction to long-memory time series models and fractional differencing. J. Time Series Anal. 1(1), 15–29 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Podobnik, B., Fu, D.F., Stanley, H.E., Ivanov, P.Ch.: Power-law autocorrelated stochastic processes with long-range cross-correlations. Eur. Phys. J. B 56(1), 47–52 (2007)Google Scholar
  27. 27.
    Xiong, W., Faes, L., Ivanov, P.C.: Entropy measures, entropy estimators, and their performance in quantifying complex dynamics: effects of artifacts, nonstationarity, and long-range correlations. Phys. Rev. E 95(6), 062114 (2017)CrossRefGoogle Scholar
  28. 28.
    Shuster, W.D., Bonta, J., Thurston, H., Warnemuende, E., Smith, D.R.: Impacts of impervious surface on watershed hydrology: a review. Urban Water J. 2(4), 263–275 (2005)CrossRefGoogle Scholar
  29. 29.
    Jovanovic, T., García, S., Gall, H., Mejía, A.: Complexity as a streamflow metric of hydrologic alteration. Stoch. Environ. Res. Risk Assess. 31(8), 2107–2119 (2017)CrossRefGoogle Scholar
  30. 30.
    Rosso, O.A., Zunino, L., Pérez, D.G., Figliola, A., Larrondo, H.A., Garavaglia, M., Martín, M.T., Plastino, A.: Extracting features of Gaussian self-similar stochastic processes via the bandt-pompe approach. Phys. Rev. E 76(6), 061114 (2007)CrossRefGoogle Scholar
  31. 31.
    Brandes, D., Cavallo, G.J., Nilson, M.L.: Base flow trends in urbanizing watersheds of the delaware river basin 1. JAWRA J. Am. Water Resour. Assoc. 41(6), 1377–1391 (2005)CrossRefGoogle Scholar
  32. 32.
    Yeh, C.-H., Shi, W.: Generalized multiscale lempel-ziv complexity of cyclic alternating pattern during sleep. Nonlinear Dyn. 93(4), 1899–1910 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Nuffield Department of Clinical NeurosciencesUniversity of OxfordOxfordUK
  2. 2.State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic EngineeringTsinghua UniversityBeijingChina
  3. 3.School of Information and ElectronicsBeijing Institute of TechnologyBeijingChina
  4. 4.Institute of Remote Sensing and Geographic Information SystemPeking UniversityBeijingChina
  5. 5.Department of Civil Engineering and Environmental ScienceUniversity of OklahomaNormanUSA

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