Abstract
The multivariate multiscale complexity-entropy causality plane (MMCECP) is introduced for evaluating the dynamical complexity and long-range correlations of multivariate nonlinear systems. Numerical simulations from different classes of systems are applied to confirm the effectiveness of the proposed measure. We observe that the MMCECP not only can characterize the deterministic properties of the systems, but also can distinguish Gaussian and non-Gaussian processes. Moreover, it is immune to varying degrees of noises at large scales. Then we apply it to financial time series analysis, mainly investigating the classification of stock market dynamics. Empirical results illustrate that the MMCECP is robust and valid to detect the physical structures of stock markets.
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Okaly, J.B., Mvogo, A., Woulache, R.L., Kofane, T.C.: Nonlinear dynamics of damped dna systems with long-range interactions. Commun. Nonlinear Sci. Numer. Simul. 55, 183–193 (2017)
Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Probl. Inf. Transm. 1(1), 1–7 (1965)
Mandelbrot, B.B., Wheeler, J.A.: The fractal geometry of nature. J. R. Stat. Soc. 147(4), 468 (1983)
Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 31(3), 353–354 (1994)
Shannon, C.E.: A mathematical theory of communication. ACM Sigmob. Mob. Comput. Commun. Rev. 5(3), 379–423 (1948)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)
Pincus, S.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA 88(6), 2297–2301 (1991)
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 (2000)
Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. 88(17), 174102 (2002)
Zhang, Y., Shang, P.: Permutation entropy analysis of financial time series based on hills diversity number. Commun. Nonlinear Sci. Numer. Simul. 6(1), 1659–1671 (2017)
Zhao, X., Shang, P., Huang, J.: Permutation complexity and dependence measures of time series. EPL 102(4), 40005 (2013)
Yin, Y., Shang, P.: Weighted multiscale permutation entropy of financial time series. Nonlinear Dyn. 78(4), 2921–2939 (2014)
Lopezruiz, R., Mancini, H.L., Calbet, X.: A statistical measure of complexity. Phys. Lett. A 209(5), 321–326 (2010)
Martin, M.T., Plastino, A.R., Rosso, O.A.: Statistical complexity and disequilibrium. Phys. Lett. A 311(2), 126–132 (2003)
Lamberti, P.W., Martin, M.T., Plastino, A., Rosso, O.A.: Intensive entropic non-triviality measure. Phys. A Stat. Mech. Appl. 334(1), 119–131 (2004)
Rosso, O.A., Larrondo, H.A., Martin, M.T., Plastino, A., Fuentes, M.A.: Distinguishing noise from chaos. Phys. Rev. Lett. 99(15), 154102 (2007)
Rosso, O.A., Zunino, L., Perez, D.G., Figliola, A., Larrondo, H.A., Garavaglia, M., Martin, M.T., Plastino, A.: Extracting features of gaussian self-similar stochastic processes via the Bandt–Pompe approach. Phys. Rev. E 76(6), 061114 (2007)
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(2), 023101 (2015)
Zunino, L., Zanin, M., Tabak, B.M., Perez, D.G., Rosso, O.A.: Complexity-entropy causality plane: a useful approach to quantify the stock market inefficiency. Phys. A Stat. Mech. Appl. 389(9), 1891–1901 (2010)
Rosso, O.A., Olivares, F., Zunino, L., De Micco, L., Andre, L.L., Plastino, A., Larrondo, H.A.: Characterization of chaotic maps using the permutation Bandt–Pompe probability distribution. Eur. Phys. J. B 86(4), 116 (2013)
Siddagangaiah, S., Li, Y., Guo, X., Chen, X., Zhang, Q., Yang, K., Yang, Y.: A complexity-based approach for the detection of weak signals in ocean ambient noise. Entropy 18(3), 101 (2016)
Ribeiro, H.V., Jauregui, M., Zunino, L., Lenzi, E.K.: Characterizing time series via complexity-entropy curves. Phys. Rev. E 95(6–1), 062106 (2017)
Zunino, L., Tabak, B.M., Serinaldi, F., Zanin, M., Perez, D.G., Rosso, O.A.: Commodity predictability analysis with a permutation information theory approach. Phys. A Stat. Mech. Appl. 390(5), 876–890 (2011)
Ribeiro, H.V., Zunino, L., Mendes, R.S., Lenzi, E.K.: Complexity-entropy causality plane: a useful approach for distinguishing songs. Phys. A Stat. Mech. Appl. 391(7), 2421–2428 (2012)
Morabito, F.C., Labate, D., La Foresta, F., Bramanti, A., Morabito, G., Palamara, I.: Multivariate multi-scale permutation entropy for complexity analysis of alzheimers disease eeg. Entropy 14(7), 1186–1202 (2012)
He, S., Sun, K., Wang, H.: Multivariate permutation entropy and its application for complexity analysis of chaotic systems. Phys. A Stat. Mech. Appl. 461, 812–823 (2016)
Yin, Y., Shang, P.: Multivariate weighted multiscale permutation entropy for complex time series. Nonlinear Dyn. 88(3), 1707–1722 (2017)
Grassberger, P.: Toward a quantitative theory of self-generated complexity. Int. J. Theor. Phys. 25(9), 907–938 (1986)
Feldman, D.P., Mctague, C.S., Crutchfield, J.P.: The organization of intrinsic computation: complexity-entropy diagrams and the diversity of natural information processing. Chaos 18(4), 148–201 (2008)
Staniek, M., Lehnertz, K.: Parameter selection for permutation entropy measurements. Int. J. Bifurc. Chaos 17(10), 3729–3733 (2011)
Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 89(6), 068102 (2002)
Plastino, A.R., Plastino, A.: Symmetries of the Fokker–Planck equation and the Fisher–Frieden arrow of time. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 54(4), 4423 (1996)
Martin, M.T., Plastino, A., Rosso, O.A.: Generalized statistical complexity measures: geometrical and analytical properties. Phys. A Stat. Mech. Appl. 369(2), 439–462 (2012)
Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 94–102 (1976)
Whitehead, R.R., Macdonald, N.: A chaotic mapping that displays its own homoclinic structure. Phys. D Nonlinear Phenom. 13(3), 401–407 (1984)
Devaney, R.L.: A piecewise linear model for the zones of instability of an area-preserving map. Phys. D Nonlinear Phenom. 10(3), 387–393 (1984)
Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)
Rössler, O.E.: An equation for continuous chaos. Phys. Lett. B 57(5), 397–398 (1976)
Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 09(07), 1465–1466 (2011)
Tang, Y., Zhao, A., Ren, Y.Y., Dou, F.X., De Jin, N.: Gasliquid two-phase flow structure in the multi-scale weighted complexity entropy causality plane. Phys. A Stat. Mech. Appl. 449, 324–335 (2016)
Stratimirovic, D., Sarvan, D., Miljkovic, V., Blesic, S.: Analysis of cyclical behavior in time series of stock market returns. Commun. Nonlinear Sci. Numer. Simul. 54, 21–33 (2018)
Schmitt, T.A., Chetalova, D., Schafer, R., Guhr, T.: Non-stationarity in financial time series: generic features and tail behavior. Europhys. Lett. 103(103), 58003 (2013)
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The financial supports from the funds of the Fundamental Research Funds for the Central Universities (2019YJS205, 2018JBZ104), the China National Science (61771035) and the Beijing National Science (4162047) are gratefully acknowledged.
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The authors declare that they have no conflict of interest concerning the publication of this manuscript. Name: Xuegeng Mao
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Mao, X., Shang, P. & Li, Q. Multivariate multiscale complexity-entropy causality plane analysis for complex time series. Nonlinear Dyn 96, 2449–2462 (2019). https://doi.org/10.1007/s11071-019-04933-7
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DOI: https://doi.org/10.1007/s11071-019-04933-7