Nonlinear Dynamics

, Volume 92, Issue 1, pp 41–58 | Cite as

Topological entropy and geometric entropy and their application to the horizontal visibility graph for financial time series

  • Lei Rong
  • Pengjian Shang
Original Paper


In this paper, we introduce topological entropy (TE) based on time series, which characterizes the total exponential complexity of a quantified system with a single number. Combined with multiscale theory, we propose geometric entropy (GE), aiming to examine the correlation among different time series. In order to detect the properties of TE and GE, we apply them to an original symbolic method utilized to measure time series irreversibility, namely horizontal visibility algorithm. On this basis, we propose a time series irreversibility measure, i.e., normalized index. Then, we employ TE and GE based on the horizontal visibility graph symbolic algorithm to simulated time series, which is generated by the logistic map with different parameters. Through the comparison of the results, we find out that different simulated data have the same variation tendency of TE, which means that TE is capable of reflecting the similarity among different time series. On the basic of these results, we further analyze the irreversibility of simulated data and also get some interesting findings. From the GE results comparison, we conclude that the GE method can distinguish different time series and expose their correlation efficiently. As a farther validation, we explore the effects of these methods on the analysis of different stock time series. Results show that they can reflect a large number of interrelationships, and successfully quantify the changes in the complexity of different stock market data.


Topological entropy (TE) Geometric entropy (GE) Horizontal visibility graph (HVg) Time series irreversibility Financial time series 



The financial supports from the funds of the China National Science (61371130) and the Beijing National Science (4162047) are gratefully acknowledged.


  1. 1.
    Machado, J.A.T.: Entropy analysis of integer and fractional dynamical systems. Nonlinear Dyn. 62(1–2), 371–378 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Yin, Y., Shang, P.: Weighted multiscale permutation entropy of financial time series. Nonlinear Dyn. 78(4), 2921–2939 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fontaine, S., Dia, S., Renner, M.: Nonlinear friction dynamics on fibrous materials, application to the characterization of surface quality. Part I: global characterization of phase spaces. Nonlinear Dyn. 66(4), 647–665 (2011)CrossRefGoogle Scholar
  4. 4.
    Xiong, H., Shang, P.: Weighted multifractal cross-correlation analysis based on Shannon entropy. Commun. Nonlinear Sci. Numer. Simul. 30(1C3), 268–283 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Faranda, D., Pons, F.M.E., Giachino, E., Vaienti, S.: Early warnings indicators of financial crises via auto regressive moving average models. Commun. Nonlinear Sci. Numer. Simul. 29(1C3), 233–239 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H.E.: A theory of power-law distributions in financial market fluctuations. Nature 423(6937), 267–270 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Mantegna, R.N., Stanley, H.E., Chriss, N.A.: An introduction to econophysics: correlations and complexity in finance. Phys. Today 53(12), 570–571 (2000)CrossRefGoogle Scholar
  8. 8.
    Piqueira, J.R.C., Mortoza, L.P.D.: Brazilian exchange rate complexity: financial crisis effects. Commun. Nonlinear Sci. Numer. Simul. 17(4), 1690–1695 (2012)CrossRefGoogle Scholar
  9. 9.
    Machado, J.T., Duarte, F.B., Duarte, G.M.: Analysis of stock market indices through multidimensional scaling. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4610–4618 (2011)CrossRefGoogle Scholar
  10. 10.
    Xu, M., Shang, P., Huang, J.: Modified generalized sample entropy and surrogate data analysis for stock markets. Commun. Nonlinear Sci. Numer. Simul. 35, 17–24 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Flanagan, R., Lacasa, L.: Irreversibility of financial time series: a graph-theoretical approach. Phys. Lett. A 380(20), 1689–1697 (2016)CrossRefGoogle Scholar
  12. 12.
    Forbes, K., Rigobon, R.: No contagion, only interdependence: measuring stock market comovements. J. Financ. 57(5), 2223–2261 (2002)CrossRefGoogle Scholar
  13. 13.
    Peter, F.J., Dimpfl, T., Huergo, L.: Using transfer entropy to measure information flows between financial markets. Stud. Nonlinear Dyn. Econom. 17(1), 85–102 (2015)MathSciNetGoogle Scholar
  14. 14.
    Yin, Y., Shang, P.: Weighted permutation entropy based on different symbolic approaches for financial time series. Physica A 443, 137–148 (2016)CrossRefGoogle Scholar
  15. 15.
    Schreiber, T.: Measuring information transfer. Phys. Rev. Lett. 85(2), 461–464 (2000)CrossRefGoogle Scholar
  16. 16.
    Richman, J.S., Moorman, J.R.: Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol. 278(6), H2039–H2049 (2000)CrossRefGoogle Scholar
  17. 17.
    Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)CrossRefGoogle Scholar
  18. 18.
    Marschinski, R., Kantz, H.: Analysing the information flow between financial time series. Eur. Phys. J. B 30(2), 275–281 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhao, X., Shang, P., Pang, Y.: Power law and stretched exponential effects of extreme events in chinese stock markets. Fluct. Noise Lett. 09(2), 203–217 (2012)CrossRefGoogle Scholar
  20. 20.
    Adler, R.L., Marcus, B.: Topological Entropy and Equivalence of Dynamical Systems. American Mathematical Society, Providence (1979)zbMATHGoogle Scholar
  21. 21.
    Nilsson, J.: On the entropy of a family of random substitutions. Monatshefte Fr Mathematik 168(3–4), 563–577 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Denker, M., Grillenberger, C., Sigmund, K.: Topological entropy. In: Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol. 527, pp. 82–91. Springer, Berlin (1976)Google Scholar
  23. 23.
    Zheleznyak, A.L.: An Approach to the Computation of the Topological Entropy. Springer, Berlin (1990)CrossRefGoogle Scholar
  24. 24.
    Weiss, H.: Some variational formulas for Hausdorff dimension, topological entropy, and SRB entropy for hyperbolic dynamical systems. J. Stat. Phys. 69(3), 879–886 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Savkin, A.V.: Analysis and synthesis of networked control systems: topological entropy, observability, robustness and optimal control. Automatica 42, 51–62 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Queffélec, M.: Dynamical Systems Associated with Sequences. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  27. 27.
    Ghys, E., Langevin, R., Walczak, P.: Entropie geometrique des feuilletages. Acta Math. 160, 105–142 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang, S., Zhou, L., Zhou, Y.: Geometric entropy of group actions on regular curves. Adv. Math. 39(4), 467–471 (2010)MathSciNetGoogle Scholar
  29. 29.
    Fujita, M.: Geometric entropy and hagedorn/deconfinement transition. J. High Energy Phys. 2008(9), 1–17 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Cysarz, D., Bettermann, H., Van, L.P.: Entropies of short binary sequences in heart period dynamics. Am. J. Physiol. Heart Circ. Physiol. 278(6), 183–202 (2000)CrossRefGoogle Scholar
  31. 31.
    Cysarz, D., Porta, A., Montano, N., Leeuwen, P.V., Kurths, J., Wessel, N.: Quantifying heart rate dynamics using different approaches of symbolic dynamics. Eur. Phys. J. Spec. Top. 222(2), 487–500 (2013)CrossRefGoogle Scholar
  32. 32.
    Lacasa, L., Nuñez, A., Roldán, É., Parrondo, J.M.R., Luque, B.: Time series irreversibility: a visibility graph approach. Eur. Phys. J. B 85(6), 217 (2012)Google Scholar
  33. 33.
    Lacasa, L., Luque, B., Ballesteros, F., Luque, J., Nuño, J.C.: From time series to complex networks: the visibility graph. Proc. Natl. Acad. Sci. U. S. A. 105(13), 4972–4975 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 71(2 Pt 1), 021906 (2005)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xia, J., Shang, P.: Multiscale entropy analysis of financial time series. Fluct. Noise Lett. 11(11), 333–342 (2012)Google Scholar
  36. 36.
    Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 92(8), 705–708 (2002)Google Scholar
  37. 37.
    Luque, B., Lacasa, L., Ballesteros, F., Luque, J.: Horizontal visibility graphs: exact results for random time series. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(2), 593–598 (2009)Google Scholar
  38. 38.
    Hsieh, D.A.: Chaos and nonlinear dynamics: application to financial markets. J. Financ. 46(5), 1839–1877 (1991)CrossRefGoogle Scholar
  39. 39.
    Guegan, D.: Chaos in economics and finance. Annu. Rev. Control 33(1), 89–93 (2009)CrossRefGoogle Scholar
  40. 40.
    Sauer, T., Yorke, J.A., Casdagli, M.: Embedology. J. Stat. Phys. 65(3), 579–616 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kazem, A., Sharifi, E., Hussain, F.K., Saberi, M., Hussain, O.K.: Support vector regression with chaos-based firefly algorithm for stock market price forecasting. Appl. Soft Comput. 13(2), 947–958 (2013)CrossRefGoogle Scholar
  42. 42.
    Masoller, C., Hong, Y., Ayad, S., Gustave, F., Barland, S., Pons, A.J., Gómez, S., Arenas, A.: Quantifying sudden changes in dynamical systems using symbolic networks. New J. Phys. 17(2), 023068 (2015)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Diks, C., Van Houwelingen, J., Takens, F., DeGoede, J.: Reversibility as a criterion for discriminating time series. Phys. Lett. A 201(2–3), 221–228 (1995)CrossRefGoogle Scholar
  44. 44.
    Cao, Y., Tung, W., Gao, J., Protopopescu, V.A., Hively, L.M.: Detecting dynamical changes in time series using the permutation entropy. Phys. Rev. E 70(4), 046217 (2004)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Jayawardena, A., Li, W., Xu, P.: Neighbourhood selection for local modelling and prediction of hydrological time series. J. Hydrol. 258(1), 40–57 (2002)CrossRefGoogle Scholar
  46. 46.
    Schittenkopf, C., Dorffner, G., Dockner, E.J.: On nonlinear, stochastic dynamics in economic and financial time series. Stud. Nonlinear Dyn. Econom. 4(3), 101–121 (2000)CrossRefzbMATHGoogle Scholar
  47. 47.
    Hong, W.C., Dong, Y., Chen, L.Y., Wei, S.Y.: SVR with hybrid chaotic genetic algorithms for tourism demand forecasting. Appl. Soft Comput. 11(2), 1881–1890 (2011)CrossRefGoogle Scholar
  48. 48.
    Hommes, C.H., Manzan, S.: Comments on testing for nonlinear structure and chaos in economic time series. J. Macroecon. 28(1), 169–174 (2006)CrossRefGoogle Scholar
  49. 49.
    Jin, L., Xin-Bao, N., Wei, W., Xiao-Fei, M.: Detecting dynamical complexity changes in time series using the base-scale entropy. Chin. Phys. 14(12), 2428 (2005)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceBeijing Jiaotong UniversityBeijingChina

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