Advertisement

Analysis of time series through complexity–entropy curves based on generalized fractional entropy

  • Yuanyuan WangEmail author
  • Pengjian Shang
  • Zhengli Liu
Original Paper
  • 9 Downloads

Abstract

In this paper, we propose the complexity–entropy causality plane based on the generalized fractional entropy. When applying the proposed method into artificial time series and empirical time series, we find that both results show that the stochastic and chaotic time series are clearly distinguished. On the one hand, we could distinguish them according to the trend of the normalized generalized fractional entropy H as the parameter \(\alpha \) increases. On the other hand, the stochastic and chaotic time series can be distinguished by the trend of their corresponding extreme values \(\alpha _C\) with the increase in embedding dimension m. However, compared with the q-complexity–entropy plane, the trend of their extreme value \(\alpha _C\) is irregular. Moreover, when applying the complexity–entropy causality plane into financial time series, we could obtain more accurate and clearer information on the classification of different regional financial markets.

Keywords

Complexity–entropy curves Generalized fractional entropy Permutation entropy Financial time series 

Notes

Acknowledgements

The financial supports from the funds of the Fundamental Research Funds for the Central Universities (2018YJS171, 2018JBZ104), the China National Science (61771035) and the Beijing National Science (4162047) are gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

We declare that we have no conflict of interest.

References

  1. 1.
    Bercher, J.F., Vignat, C.: On minimum Fisher information distributions with restricted support and fixed variance. Inf. Sci. 179, 3832–3842 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berger, A.L., Della-Pietra, V.J., Della-Pietra, S.A.: A maximum entropy approach to natural language processing. Comput. Linguist. 22, 39–71 (1996)Google Scholar
  3. 3.
    Della-Pietra, S.A., Della-Pietra, V.J., Lafferty, J.: Inducing features of random fields. IEEE Trans. Pattern Anal. Mach. Int. 19, 380–393 (1997)CrossRefGoogle Scholar
  4. 4.
    Frieden, B.R.: Physics from Fisher Information, vol. 33, pp. 327–343. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  5. 5.
    Wang, Y.Y., Shang, P.J.: Analysis of financial stock markets through multidimensional scaling based on information measures. Nonlinear Dyn. 89, 1827–1844 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Xiong, H., Shang, P.J.: Weighted multifractal cross-correlation analysis based on Shannon entropy. Commun. Nonlinear Sci. Numer. Simul. 30, 268–283 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fouda, J.S.A.E., Koepf, W.: Detecting regular dynamics from time series using permutations slopes. Commun. Nonlinear Sci. Numer. Simul. 27, 216–227 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lopes, A.M., Machado, J.A.T.: Analysis of temperature time-series: embedding dynamics into the MDS method. Commun. Nonlinear Sci. Numer. Simul. 19, 851–871 (2014)CrossRefGoogle Scholar
  9. 9.
    Xia, J.N., Shang, P.J.: Multiscale entropy analysis of financial time series. Fluct. Noise Lett. 11, 1250033 (2012)CrossRefGoogle Scholar
  10. 10.
    Yin, Y., Shang, P.J.: Comparison of multiscale methods in the stock markets for detrended cross-correlation analysis and cross-sample entropy. Fluct. Noise Lett. 13, 1450023 (2014)CrossRefGoogle Scholar
  11. 11.
    Tian, Q., Shang, P.J., Feng, G.C.: Financial time series analysis based on information categorization method. Physica A 416, 183–191 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Shannon, C.E.: A mathematical theory of communication. Bell. Syst. Tech. J. 27, 379–423 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Probl. Peredachi Inf. 2, 157–168 (1965)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mandelbrot, B.B.: The Fractal Geometry of Nature, vol. 147, pp. 286–287. Freeman, San Francisco (1983)Google Scholar
  16. 16.
    Lyapunov, A.M.: The General Problem of the Stability of Motion, vol. 11. Taylor & Francis, London (1992)zbMATHGoogle Scholar
  17. 17.
    Perc, M.: Nonlinear time series analysis of the human electrocardiogram. Eur. J. Phys. 26, 757–768 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88, 174102 (2002)CrossRefGoogle Scholar
  19. 19.
    Aragoneses, A., Carpi, L., Tarasov, N., Churkin, D.V., Torrent, M.C., Masoller, C., Turitsyn, S.K.: Unveiling temporal correlations characteristic of a phase transition in the output intensity of a fiber laser. Phys. Rev. Lett. 116, 033902 (2016)CrossRefGoogle Scholar
  20. 20.
    Lin, H., Khurram, A., Hong, Y.: Time-delay signatures in multi-transverse mode VCSELs subject to double-cavity polarization-rotated optical feedback. Opt. Commun. 377, 128–138 (2016)CrossRefGoogle Scholar
  21. 21.
    Weck, P.J., Schaffner, D.A., Brown, M.R., Wicks, R.T.: Permutation entropy and statistical complexity analysis of turbulence in laboratory plasmas and the solar wind. Phys. Rev. E 91, 023101 (2015)CrossRefGoogle Scholar
  22. 22.
    Li, Q., Zuntao, F.: Permutation entropy and statistical complexity quantifier of non-stationarity effect in the vertical velocity records. Phys. Rev. E 89, 012905 (2014)CrossRefGoogle Scholar
  23. 23.
    Bian, C., Qin, C., Ma, Q.D.Y., Shen, Q.: Modified permutation-entropy analysis of heartbeat dynamics. Phys. Rev. E 85, 021906 (2015)CrossRefGoogle Scholar
  24. 24.
    Yang, Y.G., Pan, Q.X., Sun, S.J., Xu, P.: Novel image encryption based on quantum walks. Sci. Rep. 5, 7784 (2015)CrossRefGoogle Scholar
  25. 25.
    Aragoneses, A., Rubido, N., Tiana-Alsina, J., Torrent, M.C., Masoller, C.: Distinguishing signatures of determinism and stochasticity in spiking complex systems. Sci. Rep. 3, 1778 (2013)CrossRefGoogle Scholar
  26. 26.
    Rosso, O.A., Larrondo, H.A., Martin, M.T., Plastino, A., Fuentes, M.A.: Distinguishing noise from chaos. Phys. Rev. Lett. 99, 154102 (2017)CrossRefGoogle Scholar
  27. 27.
    Jovanovic, T., García, S., Gall, H., Mejía, A.: Complexity as a streamflow metric of hydrologic alteration. Environ. Res. Risk. Assess. 31, 2107–2119 (2017)CrossRefGoogle Scholar
  28. 28.
    Stosic, T., Telesca, L., 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
  29. 29.
    Ribeiro, H.V., Zunino, L., Mendes, R.S., Lenzi, E.K.: Complexity–entropy causality plane: a useful approach for distinguishing songs. Physica A 391, 2421–2428 (2012)CrossRefGoogle Scholar
  30. 30.
    Ribeiro, H.V., Jauregui, M., Zunino, L., Lenzi, E.K.: Characterizing time series via complexity–entropy curves. Phys. Rev. E 95, 062106 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1998)zbMATHGoogle Scholar
  32. 32.
    Hilfer, R.: Application of Fractional Calculus in Physics, vol. 21, pp. 1021–1032. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  33. 33.
    Zaslavsky, G.: Hamiltonian Chaos and Fractional Dynamics, vol. 23, p. 5380. Oxford University Press, Oxford (2008)Google Scholar
  34. 34.
    Tarasov, V.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  35. 35.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, vol. 368. Imperial College Press, London (2010)CrossRefzbMATHGoogle Scholar
  36. 36.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods; Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012)CrossRefzbMATHGoogle Scholar
  37. 37.
    Ionescu, C.: The Human Respiratory System: An Analysis of the Interplay Between Anatomy, Structure, Breathing and Fractal Dynamics. Series in BioEngineering. Springer, London (2013)CrossRefzbMATHGoogle Scholar
  38. 38.
    Machado, J.A.T.: Entropy analysis of integer and fractional dynamical systems. J. Appl. Nonlinear Dyn. 62, 371–378 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Machado, J.A.T.: Fractional dynamics of a system with particles subjected to impacts. Commun. Nonlinear Sci. Numer. Simul. 16, 4596–4601 (2011)CrossRefzbMATHGoogle Scholar
  40. 40.
    Machado, J.A.T.: Entropy analysis of fractional derivatives and their approximation. J. Appl. Nonlinear Dyn. 1, 109–112 (2012)CrossRefGoogle Scholar
  41. 41.
    Machado, J.A.T.: Fractional order generalized information. Entropy 16, 2350–2361 (2014)CrossRefGoogle Scholar
  42. 42.
    Podobnik, B., Horvatic, D., Ng, A.L., Stanley, H.E., Ivanov, P.C.: Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes. Physica A 387, 3954–3959 (2008)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Dean, M.F., Muir, H., Benson, P.F., Button, L.R., Boylston, A., Mowbray, J.: Enzyme replacement therapy by fibroblast transplantation in a case of Hunter syndrome. Nature 261, 323–325 (1976)CrossRefGoogle Scholar
  44. 44.
    Gray, R.: Entropy and Information Theory. Springer, New York (1990)CrossRefzbMATHGoogle Scholar
  45. 45.
    Beck, C.: Generalised information and entropy measures in physics. Contemp. Phys. 50, 495–510 (2009)CrossRefGoogle Scholar
  46. 46.
    Ubriaco, M.: Entropies based on fractional calculus. Phys. Lett. A 373, 2516–2519 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Machado, J.A.T., Galhano, A.M.S.: Approximating fractional derivatives in the perspective of system control. Nonlinear Dyn. 56, 401–407 (2009)CrossRefzbMATHGoogle Scholar
  48. 48.
    Machado, J.A.T., Galhano, A.M.S., Oliveira, A.A., Tar, J.K.: Approximating fractional derivatives through the generalized mean. Commun. Nonlinear Sci. Numer. Simul. 14, 3723–3730 (2009)CrossRefzbMATHGoogle Scholar
  49. 49.
    Valério, D., Trujillo, J.J., Rivero, M., Machado, J.A.T., Baleanu, D.: Fractional calculus: a survey of useful formulas. Eur. Phys. J. Spec. Top. 222, 1827–1846 (2013)CrossRefGoogle Scholar
  50. 50.
    López-Ruiz, R., Mancini, H.L., Calbet, X.: A statistical measure of complexity. Phys. Lett. A 209, 321–326 (1995)CrossRefGoogle Scholar
  51. 51.
    Taneja, I., Pardo, L., Morales, D., Ménandez, L.: Generalized information measures and their applications: a brief survey. Qüestiió 13, 47–73 (1989)MathSciNetGoogle Scholar
  52. 52.
    Lin, J.: Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory 37, 145–151 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Cha, S.H.: Measures between probability density functions. Int. J. Math. Models Methods Appl. Sci. 1, 300–307 (2007)Google Scholar
  54. 54.
    Deza, M.M., Deza, E.: Encyclopedia of Distances. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  55. 55.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Political Econ. 81, 637–654 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Hübner, U., Abraham, N., Weiss, C.O.: Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser. Phys. Rev. A 40, 6354–6365 (1989)CrossRefGoogle Scholar
  57. 57.
  58. 58.
    Yang, X.J., Gao, F., Srivastava, H.M.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. 339, 285–296 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yang, X.J., Machado, J.A.T., Baleanu, D.: Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25, 1740006 (2017)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Yang, X.J., Srivastava, H.M., Machado, J.A.T.: A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm. Sci. 20, 753–756 (2016)CrossRefGoogle Scholar
  61. 61.
    Yang, X.J., Gao, F., Machado, J.A.T., Baleanu, D.: A new fractional derivative involving the normalized sinc function without singular kernel. Eur. Phys. J. Spec. Top. 226, 3567–3575 (2017)CrossRefGoogle Scholar
  62. 62.
    Yang, X.J.: New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 19, 45–52 (2018)MathSciNetGoogle Scholar
  63. 63.
    Yang, X.J.: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm. Sci. 21, 1161–1171 (2017)CrossRefGoogle Scholar
  64. 64.
    Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transforms and Their Applications. Academic Press, New York (2015)zbMATHGoogle Scholar
  65. 65.
    Time series A of the Santa Fe time series competition. https://rdrr.io/cran/TSPred/man/SantaFe.A.html

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China

Personalised recommendations