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Multiscale fractional-order approximate entropy analysis of financial time series based on the cumulative distribution matrix

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Abstract

In this paper, a generalized method, fractional-order approximate entropy (FOApEn), is proposed with the objective of distinguishing the complexity of time processes from statistical perspectives and characterizing differences and changes in dynamical systems. Moreover, we generalized approximate entropy (ApEn) to multiscales, which can detect complexity of time series in more scales and probe the multiscale properties containing in time series. This fractional-order approximate entropy, which provides an assessment on the multiscale complexity between measurements, is defined in terms of the FOApEn method and the multiscale method. The implementation of multiscale FOApEn is illustrated with simulated time series and financial time series. Examples taken from simulated and financial data demonstrate that tuning the fractional order allows a high sensitivity to the signal evolution and how the FOApEn for complex systems behaves on different scales and determination, which is helpful in describing the dynamics of complex systems.

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References

  1. Kolmogorov, A.N.: A new metric invariant of transient dynamical systems and automorphisms in lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S) 951(5), 861–864 (1958)

    MathSciNet  MATH  Google Scholar 

  2. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(4), 1115–1115 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Goldberger, A.L., West, B.J.: Applications of nonlinear dynamics to clinical cardiology. Ann. N. Y. Acad. Sci. 504(1), 195213 (1987)

    Google Scholar 

  4. Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica D 9(1), 189–208 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl Acad. Sci. USA 88(6), 2297–301 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Pincus, S.M., Gladstone, I.M., Ehrenkranz, R.A.: A regularity statistic for medical data analysis. J. Clin. Monit. 7(4), 335–345 (1991)

    Article  Google Scholar 

  7. Pincus, S.: Approximate entropy as a complexity measure. Chaos 5(1), 110 (1995)

    Article  MathSciNet  Google Scholar 

  8. Yeragani, V.K., Pohl, R., Mallavarapu, M., Balon, R.: Approximate entropy of symptoms of mood: an effective technique to quantify regularity of mood. Bipolar Disorders 5(4), 279–286 (2015)

    Article  Google Scholar 

  9. Shen, C.P., Chen, C.C., Hsieh, S.L., Chen, W.H., Chen, J.M., Chen, C.M., Lai, F., Chiu, M.J.: High-performance seizure detection system using a wavelet-approximate entropy-fsvm cascade with clinical validation. Clin. EEG Neurosci. 44(4), 247 (2013)

    Article  Google Scholar 

  10. Saumitra, B.: Applying approximate entropy to speculative bubble in the stock market. J. Emerg. Market Finance 13(1), 43–68 (2012)

    Google Scholar 

  11. Udhayakumar, R.K., Karmakar, C., Palaniswami, M.: Approximate entropy profile: a novel approach to comprehend irregularity of short-term hrv signal. Nonlinear Dyn. 88(2), 1–15 (2016)

    MATH  Google Scholar 

  12. Anastasiadis, A.: Special issue: Tsallis entropy. Entropy 14(2), 174–176 (2012)

    Article  MATH  Google Scholar 

  13. Li, X., Essex, C., Davison, M., Hoffmann, K.H., Schulzky, C.: Fractional diffusion, irreversibility and entropy. J. Non-Equilibrium Thermodyn. 28(3), 279–291 (2003)

    Article  Google Scholar 

  14. Mathai, A.M., Haubold, H.J.: Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy. Physica A 375(1), 110–122 (2007)

    Article  MathSciNet  Google Scholar 

  15. Prkopa, A., Ganczer, S., Dek, I., Patyi, K.: Tsallis entropy and Jaynes’ information theory formalism. Braz. J. Phys. 29(1), 50–60 (1999)

    Article  Google Scholar 

  16. Hentenryck, P., Bent, R., Upfal, E.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    Google Scholar 

  17. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  18. Kvitsinski, A.A.: Fractional integrals and derivatives: theory and applications. Theor. Math. Phys 3, 397–414 (1987)

    Google Scholar 

  19. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  20. Ben-Naim, A.: Farewell to Entropy: Statistical Thermodynamics Based on Information. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  21. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Boltzmann–Gibbs entropy versus Tsallis entropy: recent contributions to resolving the argument of Einstein concerning “neither Herr Boltzmann nor Herr Planck has given a definition of w”? Astrophys. Space Sci. 290(3–4), 241–245 (2004)

    Article  MATH  Google Scholar 

  22. Rényi, A.: On measures of entropy and information. Maximum Entropy Bayesian Methods 1(2), 547–561 (1961)

    MathSciNet  MATH  Google Scholar 

  23. Baleanu, D.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012)

    Book  MATH  Google Scholar 

  24. Combescure, M.: Hamiltonian chaos and fractional dynamics. J. Phys. Gen. Phys. 38(23), 5380 (2005)

    Article  Google Scholar 

  25. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  26. Ionescu, C.M.: The Human Respiratory System: An Analysis of the Interplay Between Anatomy, Structure, Breathing and Fractal Dynamics. Series in BioEngineering (2013)

  27. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  28. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  29. Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Dordrecht (2010)

    Book  MATH  Google Scholar 

  30. Machado, J.A.T.: Entropy analysis of integer and fractional dynamical systems. Nonlinear Dyn. 62(1), 371–378 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Machado, J.A.T.: Fractional dynamics of a system with particles subjected to impacts. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4596–4601 (2011)

    Article  MATH  Google Scholar 

  32. Machado, J.A.T.: Entropy analysis of fractional derivatives and their approximation. J. Appl. Nonlinear Dyn. 1(1), 109–112 (2012)

    Article  Google Scholar 

  33. Xu, X., Qiao, Z., Lei, Y.: Repetitive transient extraction for machinery fault diagnosis using multiscale fractional order entropy infogram. Mech. Syst. Signal Process. 103, 312–326 (2018)

    Article  Google Scholar 

  34. Lopes, A.M., Machado, J.A.T.: Integer and fractional-order entropy analysis of earthquake data series. Nonlinear Dyn. 84(1), 79–90 (2016)

    Article  MathSciNet  Google Scholar 

  35. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(4), 379–423 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  36. Beck, C.: Generalised information and entropy measures in physics. Contemp. Phys. 50(4), 495–510 (2009)

    Article  Google Scholar 

  37. Gray, R.M.: Entropy and Information Theory, pp. 319–320. Springer, New York (1990)

    Book  Google Scholar 

  38. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  39. Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover, New York (1957)

    MATH  Google Scholar 

  40. Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373(30), 2516–2519 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Machado, J.A.T., Galhano, A.M., Oliveira, A.M., Tar, J.K.: Approximating fractional derivatives through the generalized mean. Commun. Nonlinear Sci. Numer. Simul. 14(11), 3723–3730 (2009)

    Article  MATH  Google Scholar 

  42. Valrio, 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(8), 1827–1846 (2013)

    Article  Google Scholar 

  43. Machado, J.: Fractional order generalized information. Entropy 16(4), 2350–2361 (2014)

    Article  Google Scholar 

  44. Teng, Y., Shang, P.: Transfer entropy coefficient: quantifying level of information flow between financial time series. Physica A 469, 60–70 (2017)

    Article  MATH  Google Scholar 

  45. May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261(5560), 459–467 (1976)

    Article  MATH  Google Scholar 

  46. Dirac, P.A.M.: The physical interpretation of quantum mechanics. Proc. Roy. Soc. Lond. A 26(4), 1–40 (1942)

    MathSciNet  MATH  Google Scholar 

  47. Hiley, B.J., Peat, F.D., Zeilinger, A.: Quantum implications: essays in honour of David Bohm. Phys. Today 39(3), 120 (1988)

    Google Scholar 

  48. Machado, J.A.T.: Fractional coins and fractional derivatives. Abstract and Applied Analysis (2013)

  49. Wellington, S.L., Vinegar, H.J., Rouffignac, E.P.D., Berchenko, I.E., Stegemeier, G.L., Zhang, E., Shahin Jr., G.T., Fowler, T.D., Ryan, R.C.: A taxonomy of robot deception and its benefits in hri. Georgia Inst. Technol. 8215(2), 2328–2335 (2013)

    Google Scholar 

  50. Tarasova, V.V.: Economic interpretation of fractional derivatives. Papers 3(1), 1–7 (2017)

    Google Scholar 

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Acknowledgements

The financial supports from the funds of the Fundamental Research Funds for the Central Universities (2018JBZ104, 2019YJS193), the National Natural Science Foundation of China (61771035) and the Beijing National Science (4162047) are gratefully acknowledged.

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  1. School of Science, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Yue Teng, Pengjian Shang & Jiayi He

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  1. Yue Teng
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  2. Pengjian Shang
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Correspondence to Yue Teng.

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Teng, Y., Shang, P. & He, J. Multiscale fractional-order approximate entropy analysis of financial time series based on the cumulative distribution matrix. Nonlinear Dyn 97, 1067–1085 (2019). https://doi.org/10.1007/s11071-019-05033-2

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