Advertisement

Nonlinear Dynamics

, Volume 98, Issue 3, pp 2147–2170 | Cite as

Complexity and uncertainty analysis of financial stock markets based on entropy of scale exponential spectrum

  • Boyi ZhangEmail author
  • Pengjian Shang
Original paper
  • 44 Downloads

Abstract

Research on complexity and uncertainty of nonlinear signals has great significance in dynamic system analysis. In order to further analyze the detailed information from the financial data to acquire deeper insight into the complex system and improve the ability of prediction, we propose the entropy of scale exponential spectrum (EOSES) as a new measure. Combined with multiscale theory, we get multiscale EOSES. The scale exponential spectrum (SES) is derived from the scale exponent of detrended fluctuation analysis. As for the entropy, we choose Rényi entropy and fractional cumulative residual entropy to compare and analyze the results. Simulated data and financial time series are used to obtain further in-depth information on the EOSES. Compared with traditional methods, we find that Rényi EOSES over moving window can provide more details of complexity which include fractal structure and scale properties. Also, it reduces the influence of degree of fitting polynomial and has higher noise immunity. In addition, through the SES and EOSES, we can better research the properties and stages of stock markets and distinguish stock markets with different characteristics.

Keywords

Complexity Rényi entropy Scale exponent Financial time series 

Notes

Acknowledgements

The financial support from the Fundamental Research Funds for the Central Universities (Grant No. 2018JBZ104) and the National Natural Science Foundation of China (Grant No. 61771035) are gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

References

  1. 1.
    Akbaba-Altun, S.: Complexity of integrating computer technologies into education in Turkey. J. Educ. Technol. Soc. 9(1), 176–187 (2006)Google Scholar
  2. 2.
    Hidalgo, Hausmann, : The building blocks of quality. Health Serv. J. 119, 15 (2009).  https://doi.org/10.2307/1312923 Google Scholar
  3. 3.
    Ma, J., Bangura, H.I.: Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances. Nonlinear Dyn. 70, 2313–2326 (2012).  https://doi.org/10.1007/s11071-012-0336-z Google Scholar
  4. 4.
    Holzinger, A., Kickmeier-Rust, M., Albert, D.: Dynamic media in computer science education; content complexity and learning performance: is less more? Educ. Technol. Soc. 11, 279–290 (2008).  https://doi.org/10.1037/a0014320 Google Scholar
  5. 5.
    Bond, A.B., Kamil, A.C., Balda, R.P.: Social complexity and transitive inference in corvids. Anim. Behav. 65, 479–487 (2003).  https://doi.org/10.1006/anbe.2003.2101 Google Scholar
  6. 6.
    Tainter, J.A.: Social complexity and sustainability. Ecol. Complex. 3, 91–103 (2006).  https://doi.org/10.1016/j.ecocom.2005.07.004 Google Scholar
  7. 7.
    Kim, Dai-Jin, et al.: An estimation of the first positive Lyapunov exponent of the EEG in patients with schizophrenia. Psychiatry Res. Neuroimaging 98, 177–189 (2000)Google Scholar
  8. 8.
    Rosenstein, M.T., Collins, J.J., Luca, C.J.De: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65(65), 117–134 (1992).  https://doi.org/10.1016/0167-2789(93)90009-P Google Scholar
  9. 9.
    Sun, K., Mou, S., Qiu, J., Wang, T., Gao, H.: Adaptive fuzzy control for non-triangular structural stochastic switched nonlinear systems with full state constraints. IEEE Trans. Fuzzy Syst. (2018).  https://doi.org/10.1109/TFUZZ.2018.2883374 Google Scholar
  10. 10.
    Qiu, J., Sun, K., Wang, T., Gao, H.: Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems with prescribed performance. IEEE Trans. Fuzzy Syst. (2019).  https://doi.org/10.1109/tfuzz.2019.2895560 Google Scholar
  11. 11.
    Bialynicky-Birula, I., Mycielski, J.: Relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129–132 (1975)Google Scholar
  12. 12.
    Zyczkowski, K.: Rényi extrapolation of Shannon entropy. Open Syst. Inf. Dyn. 10, 297–310 (2003).  https://doi.org/10.1023/A:1025128024427 Google Scholar
  13. 13.
    Hammer, D., Romashchenko, A., Shen, A., Vereshchagin, N.: Inequalities for Shannon entropy and Kolmogorov complexity. J. Comput. Syst. Sci. 60, 442–464 (2000).  https://doi.org/10.1006/jcss.1999.1677 Google Scholar
  14. 14.
    Lin, J.: Divergence measures based on the shannon entropy. IEEE Trans. Inf. Theory 37, 145–151 (1991).  https://doi.org/10.1109/18.61115 Google Scholar
  15. 15.
    Wu, Y., Zhou, Y., Saveriades, G., Agaian, S., Noonan, J.P., Natarajan, P.: Local Shannon entropy measure with statistical tests for image randomness. Inf. Sci. (Ny) 222, 323–342 (2013).  https://doi.org/10.1016/j.ins.2012.07.049 Google Scholar
  16. 16.
    Gao, J., Hu, J., Tung, W.W.: Entropy measures for biological signal analyses. Nonlinear Dyn. 68, 431–444 (2012).  https://doi.org/10.1007/s11071-011-0281-2 Google Scholar
  17. 17.
    Szita, István, Lörincz, András: Learning Tetris using the noisy cross-entropy method. Neural Comput. 18(12), 2936–2941 (2006)Google Scholar
  18. 18.
    Zhai, J.hai, Xu, H.yu, Wang, X.zhao: Dynamic ensemble extreme learning machine based on sample entropy. Soft Comput. 16, 1493–1502 (2012).  https://doi.org/10.1007/s00500-012-0824-6 Google Scholar
  19. 19.
    Jia, Y., Gu, H.: Identifying nonlinear dynamics of brain functional networks of patients with schizophrenia by sample entropy. Nonlinear Dyn. (2019).  https://doi.org/10.1007/s11071-019-04924-8 Google Scholar
  20. 20.
    McGrath, D., Yentes, J.M., Kaipust, J.P., Stergiou, N., Schmid, K.K., Hunt, N.: The appropriate use of approximate entropy and sample entropy with short data sets. Ann. Biomed. Eng. 41, 349–365 (2012).  https://doi.org/10.1007/s10439-012-0668-3 Google Scholar
  21. 21.
    Robinson, S., Cattaneo, A., El-Said, M.: Updating and estimating a social accounting matrix using cross entropy methods. Econ. Syst. Res. 13, 47–64 (2001).  https://doi.org/10.1080/09535310120026247 Google Scholar
  22. 22.
    Lenzi, E.K., Mendes, R.S., Da Silva, L.R.: Statistical mechanics based on Renyi entropy. Phys. A Stat. Mech. Appl. 280, 337–345 (2000).  https://doi.org/10.1016/S0378-4371(00)00007-8 Google Scholar
  23. 23.
    Wang, Vemuri, Chen, Rao: Cumulative residual entropy, a new measure of information. IEEE Trans. Inf. Theory 50 1, 548–553 (2005).  https://doi.org/10.1109/iccv.2003.1238395 Google Scholar
  24. 24.
    Drissi, N., Chonavel, T., Boucher, J.M.: Generalized cumulative residual entropy for distributions with unrestricted supports. Res. Lett. Signal Process. 2008, 1–5 (2008).  https://doi.org/10.1155/2008/790607 Google Scholar
  25. 25.
    Kristoufek, L.: Fractal markets hypothesis and the global financial crisis: scaling, investment horizons and liquidity. Adv. Complex Syst. (2012).  https://doi.org/10.1142/S0219525912500658 Google Scholar
  26. 26.
    Weron, A., Weron, R.: Fractal market hypothesis and two power-laws. Chaos Solitons Fractals 11, 289–296 (2000).  https://doi.org/10.1016/S0960-0779(98)00295-1 Google Scholar
  27. 27.
    Peng, C.K., Havlin, S., Stanley, H.E., Goldberger, A.L.: Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 5, 82–87 (1995).  https://doi.org/10.1063/1.166141 Google Scholar
  28. 28.
    Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Stanley, H.E.: Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64, 19 (2001).  https://doi.org/10.1103/PhysRevE.64.011114 Google Scholar
  29. 29.
    Talkner, P., Weber, R.O.: Power spectrum and detrended fluctuation analysis: application to daily temperatures. Phys. Rev. E 62, 150–160 (2000).  https://doi.org/10.1103/PhysRevE.62.150 Google Scholar
  30. 30.
    He, L.Y., Chen, S.P.: Nonlinear bivariate dependency of pricevolume relationships in agricultural commodity futures markets: a perspective from multifractal detrended cross-correlation analysis. Phys. A Stat. Mech. Appl. 390, 297–308 (2010).  https://doi.org/10.1016/j.physa.2010.09.018 Google Scholar
  31. 31.
    Kumar, S., Deo, N.: Multifractal properties of the Indian financial market. Phys. A 388, 1593–1602 (2009).  https://doi.org/10.1016/j.physa.2008.12.017 Google Scholar
  32. 32.
    Norouzzadeh, P., Jafari, G.R.: Application of multifractal measures to Tehran price index. Phys. A Stat. Mech. Appl. 356, 609–627 (2005).  https://doi.org/10.1016/j.physa.2005.02.046 Google Scholar
  33. 33.
    Serinaldi, F.: Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series. Phys. A Stat. Mech. Appl. 389, 2770–2781 (2010).  https://doi.org/10.1016/j.physa.2010.02.044 Google Scholar
  34. 34.
    Qian, B., Rasheed, K.: Hurst exponent and financial market predictability. In: Proceedings of the Second IASTED International Conference on Financial Engineering and Applications, pp. 203–209 (2004)Google Scholar
  35. 35.
    Eom, C., Choi, S., Oh, G., Jung, W.S.: Hurst exponent and prediction based on weak-form efficient market hypothesis of stock markets. Phys. A Stat. Mech. Appl. 387, 4630–4636 (2008).  https://doi.org/10.1016/j.physa.2008.03.035 Google Scholar
  36. 36.
    Morales, R., Di Matteo, T., Gramatica, R., Aste, T.: Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series. Phys. A Stat. Mech. Appl. 391, 3180–3189 (2012).  https://doi.org/10.1016/j.physa.2012.01.004 Google Scholar
  37. 37.
    Grech, D., Pamuła, G.: The local Hurst exponent of the financial time series in the vicinity of crashes on the Polish stock exchange market. Phys. A Stat. Mech. Appl. 387, 4299–4308 (2008).  https://doi.org/10.1016/j.physa.2008.02.007 Google Scholar
  38. 38.
    Movahed, M.S., Jafari, G.R., Ghasemi, F., Rahvar, S., Tabar, M.R.R.: Multifractal detrended fluctuation analysis of sunspot time series. J. Stat. Mech. Theory Exp. (2006).  https://doi.org/10.1088/1742-5468/2006/02/P02003 Google Scholar
  39. 39.
    Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.E.: Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A Stat. Mech. Appl. 316, 87–114 (2002).  https://doi.org/10.1016/S0378-4371(02)01383-3 Google Scholar
  40. 40.
    Machado, J.A.T., Galhano, A.M.S.F., Trujillo, J.J.: On development of fractional calculus during the last fifty years. Scientometrics 98, 577–582 (2014).  https://doi.org/10.1007/s11192-013-1032-6 Google Scholar
  41. 41.
    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. arXiv preprint arXiv:0805.3823 (2008)
  42. 42.
    Machado, J.T.: Fractional order generalized information. Entropy 16, 2350–2361 (2014).  https://doi.org/10.3390/e16042350 Google Scholar
  43. 43.
    Costa, M., Peng, C.K., Goldberger, A.L., Hausdorff, J.M.: Multiscale entropy analysis of human gait dynamics. Phys. A Stat. Mech. Appl. 330, 53–60 (2003).  https://doi.org/10.1016/j.physa.2003.08.022 Google Scholar
  44. 44.
    Alvarez-Ramirez, J., Rodriguez, E., Echeverría, J.C.: Detrending fluctuation analysis based on moving average filtering. Phys. A Stat. Mech. Appl. 354, 199–219 (2005).  https://doi.org/10.1016/j.physa.2005.02.020 Google Scholar
  45. 45.
    Xiong, H., Shang, P.: Weighted multifractal cross-correlation analysis based on Shannon entropy. Commun. Nonlinear Sci. Numer. Simul. 30, 268–283 (2016).  https://doi.org/10.1016/j.cnsns.2015.06.029 Google Scholar
  46. 46.
    Panas, E.: Estimating fractal dimension using stable distributions and exploring long memory through ARFIMA models in Athens stock exchange. Appl. Financ. Econ. 11, 395–402 (2001).  https://doi.org/10.1080/096031001300313956 Google Scholar
  47. 47.
    Leite, A., Rocha, A.P., Silva, M.E., Gouveia, S., Carvalho, J., Costa, O.: Long-range dependence in heart rate variability data: ARFIMA modelling vs detrended fluctuation analysis. Comput. Cardiol. 34, 21–24 (2007).  https://doi.org/10.1109/CIC.2007.4745411 Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

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

Personalised recommendations