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.
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References
Akbaba-Altun, S.: Complexity of integrating computer technologies into education in Turkey. J. Educ. Technol. Soc. 9(1), 176–187 (2006)
Hidalgo, Hausmann, : The building blocks of quality. Health Serv. J. 119, 15 (2009). https://doi.org/10.2307/1312923
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
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
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
Tainter, J.A.: Social complexity and sustainability. Ecol. Complex. 3, 91–103 (2006). https://doi.org/10.1016/j.ecocom.2005.07.004
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)
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
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
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
Bialynicky-Birula, I., Mycielski, J.: Relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129–132 (1975)
Zyczkowski, K.: Rényi extrapolation of Shannon entropy. Open Syst. Inf. Dyn. 10, 297–310 (2003). https://doi.org/10.1023/A:1025128024427
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
Lin, J.: Divergence measures based on the shannon entropy. IEEE Trans. Inf. Theory 37, 145–151 (1991). https://doi.org/10.1109/18.61115
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
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
Szita, István, Lörincz, András: Learning Tetris using the noisy cross-entropy method. Neural Comput. 18(12), 2936–2941 (2006)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. arXiv preprint arXiv:0805.3823 (2008)
Machado, J.T.: Fractional order generalized information. Entropy 16, 2350–2361 (2014). https://doi.org/10.3390/e16042350
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
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
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
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
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
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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.
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Zhang, B., Shang, P. Complexity and uncertainty analysis of financial stock markets based on entropy of scale exponential spectrum. Nonlinear Dyn 98, 2147–2170 (2019). https://doi.org/10.1007/s11071-019-05314-w
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DOI: https://doi.org/10.1007/s11071-019-05314-w