Skip to main content
SpringerLink
Log in

Analysis of time series in the cumulative residual entropy plane based on oscillation roughness exponent

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this work, we propose the cumulative residual entropy (CRE) plane and CRE curve based on the weighted-multiscale cumulative residual Rényi/Tsallis permutation entropy and oscillation roughness exponent to analyze complex dynamic systems. The oscillation roughness exponent method and the cumulative residual distribution theorem adopted in our proposed methods are two core theories to depict more detailed information of complex dynamic systems in a more efficient way by reducing information loss and capture the statistics of the related time series’ roughness. Trials are operated on the logistic map model as numerical experiments, and we discover that our methods are capable of discriminating different types of complex data with high accuracy. Compared with the original methods, our methods are more superior in extracting more subtle details to distinguish different dynamic systems. In the experiments with the financial stocks, our methods are still found to be more reasonable in discriminating stock indices from different parts of the world by making comparisons with original methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman, San Francisco (1982)

    MATH  Google Scholar 

  2. Castellanos, M., Morató, M., Aguado, P., Monte, J., Tarquis, A.: Detrended fluctuation analysis for spatial characterisation of landscapes. Biosyst. Eng. 168, 14–25 (2018)

    Google Scholar 

  3. Tsuji, Y., Asakawa, T., Hitomi, Y., Todo, A., Yoshida, T., Mizuno, M.: Detrended fluctuation analysis of photoplethysmography in diabetic nephropathy patients on hemodialysis. In: Brain and Health Informatics, pp. 218–224 (2013)

    Google Scholar 

  4. Aytaç, A., Seçkin, K., Stelios, B.: Digital currency forecasting with chaotic meta-heuristic bio-inspired signal processing techniques. Chaos Solitons Fractals 126, 325–336 (2019)

    MathSciNet  Google Scholar 

  5. Ma, X., Tao, Z., Wang, Y., Yu, H.: Long short-term memory neural network for traffic speed prediction using remote microwave sensor data. Transp. Res. Part C Emerg. Technol. 54, 187–197 (2015)

    Google Scholar 

  6. Li, X., Peng, L., Yao, X., Cui, S., You, C., Chi, T.: Long short-term memory neural network for air pollutant concentration predictions: method development and evaluation. Environ. Pollut. 231, 997–1004 (2017)

    Google Scholar 

  7. Kolmogorov, A.: Three approaches to the quantitative definition of information. Probl. Inf. Transm. 2, 157–168 (1968)

    MathSciNet  MATH  Google Scholar 

  8. Shang, D., Shang, P., Liu, L.: Multidimensional scaling method for complex time series feature classification based on generalized complexity-invariant distance. Nonlinear Dyn. 95, 2875–2892 (2019)

    Google Scholar 

  9. Gustavo, E., Eamonn, J., Oben, M., Vińıcius, M.: CID: an efficient complexity invariant distance for time series. Data Min. Knowl. Disc. 28, 634–669 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Lamberti, P., Martin, M., Plastino, A., Rosso, O.: Intensive entropic non-triviality measure. Phys. A 334, 119–131 (2004)

    MathSciNet  Google Scholar 

  11. Lyapunov, A.: The general problem of the stability of motion. Int. J. Control 55, 531–534 (1992)

    MathSciNet  Google Scholar 

  12. Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Shannon, C.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 623–656 (1948)

    MathSciNet  MATH  Google Scholar 

  14. Chen, S., Shang, P., Wu, Y.: Multivariate multiscale fractional order weighted permutation entropy of nonlinear time series. Phys. A 515, 217–231 (2019)

    MathSciNet  Google Scholar 

  15. Dai, Y., He, J., Wu, Y., Chen, S., Shang, P.: Generalized entropy plane based on permutation entropy and distribution entropy analysis for complex time series. Phys. A 520, 217–231 (2019)

    Google Scholar 

  16. Gao, J., Shang, P.: Multiscale weighted Rényi entropy causality plane for financial time series. Int. J. Mod. Phys. C 30, 1950037 (2019)

    Google Scholar 

  17. He, J., Shang, P., Zhang, Y.: PID: a PDF-induced distance based on permutation cross-distribution entropy. Nonlinear Dyn. 97, 1329–1342 (2019)

    MATH  Google Scholar 

  18. Li, C., Shang, P.: Multiscale Tsallis permutation entropy analysis for complex physiological time series. Phys. A 523, 10–20 (2019)

    Google Scholar 

  19. Li, J., Shang, P., Zhang, X.: Financial time series analysis based on fractional and multiscale permutation entropy. Commun. Nonlinear Sci. Numer. Simul. 78, 104880 (2019)

    MathSciNet  Google Scholar 

  20. Isohata, Y., Hayashi, M.: Power spectrum and mutual information analyses of DNA base (nucleotide) sequences. J. Phys. Soc. Jpn. 72, 735–742 (2003)

    Google Scholar 

  21. Akselrod, S., Gordon, D., Ubel, F., Shannon, D., Berger, A., Cohen, R.: Power spectrum analysis of heart rate fluctuation: a quantitative probe of beat-to-beat cardiovascular control. Science 213, 220–222 (1981)

    Google Scholar 

  22. Mccraty, R., Atkinson, M., Tiller, W., Rein, G., Watkins, A.: The effects of emotions on short-term power spectrum analysis of heart rate variability. Am. J. Cardiol. 76, 1089 (1995)

    Google Scholar 

  23. Li, Z., Zhu, M., Chu, F., Xiao, Y.: Mechanical fault diagnosis method based on empirical wavelet transform. Chin. J. Sci. Instrum. 35, 2423–2432 (2014)

    Google Scholar 

  24. Amezquita-Sanchez, J., Adeli, H.: A new music-empirical wavelet transform methodology for time-frequency analysis of noisy nonlinear and non-stationary signals. Digit. Signal Proc. 45, 55–68 (2015)

    Google Scholar 

  25. Richman, J., Randall, M.: Physiological time-series analysis, using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol. 278, H2039–H2049 (2000)

    Google Scholar 

  26. Pincus, S.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. 88, 2297–2301 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88, 174102 (2002)

    Google Scholar 

  28. Kosko, B.: Fuzzy entropy and conditioning. Inf. Sci. 40, 165–174 (1986)

    MathSciNet  MATH  Google Scholar 

  29. Pincus, S., Singer, B.: Randomness and degrees of irregularity. Proc. Natl. Acad. Sci. 93, 2083–2088 (1996)

    MathSciNet  MATH  Google Scholar 

  30. Richman, J., Lake, D., Moorman, J.: Sample entropy. Methods Enzymol. 384, 172–184 (2004)

    Google Scholar 

  31. Barnett, L., Bossomaier, T.: Transfer entropy as a log-likelihood ratio. Phys. Rev. Lett. 109, 138105 (2012)

    Google Scholar 

  32. Rényi, A.: Probability Theory. North Holland, Amsterdam (1970)

    MATH  Google Scholar 

  33. Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 5, 479–487 (1988)

    MathSciNet  MATH  Google Scholar 

  34. Fadlallah, B., Chen, B., Keil, A., Principe, J.: Weighted-permutation entropy: a complexity measure for time series incorporating amplitude information. Phys. Rev. E 87, 022911 (2013)

    Google Scholar 

  35. Rao, M., Chen, Y., Vemuri, B., Wang, F.: Cumulative residual entropy: a new measure of information. IEEE Trans. Inf. Theory 50, 1220–1228 (2004)

    MathSciNet  MATH  Google Scholar 

  36. https://en.wikipedia.org/wiki/R%C3%A9nyientropy

  37. Madan, M., Nitin, G.: Some characterization results on dynamic cumulative residual Tsallis entropy. J. Probab. Stat. 2015, 1–8 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Albuquerque, M., Esquef, I., Mello, A.: Image thresholding using Tsallis entropy. Pattern Recognit. Lett. 25, 1059–1065 (2004)

    Google Scholar 

  39. Jalab, H., Ibrahim, R., Ahmed, A.: Image denoising algorithm based on the convolution of fractional Tsallis entropy with the Riesz fractional derivative. Neural Comput. Appl. 28, 217–2232 (2016)

    Google Scholar 

  40. Stariolo, D., Tsallis, C.: Optimization by Simulated Annealing: Recent Progress. Annual Reviews of Computational Physics II. World Scientific, Singapore (1995)

    Google Scholar 

  41. Tsallis, C., Stariolo, D.: Generalized simulated annealing. Phys. A 233, 395–406 (1996)

    Google Scholar 

  42. Andricioaei, I., Straub, J.: On Monte Carlo and molecular dynamics methods inspired by Tsallis statistics: methodology, optimization, and application to atomic clusters. J. Chem. Phys. 107, 9117–9124 (1997)

    Google Scholar 

  43. Zunino, L., Pérez, D., Kowalski, A., Martin, M., Garavaglia, M., Plastino, A., Rosso, O.: Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy. Phys. A 387, 6057–6068 (2008)

    MathSciNet  MATH  Google Scholar 

  44. Mao, X., Shang, P., Li, Q.: Multivariate multiscale complexity-entropy causality plane analysis for complex time series. Nonlinear Dyn. 96, 2449–2462 (2019)

    Google Scholar 

  45. Zunino, L., Zanin, M., Tabak, B., Pérez, D., Rosso, O.: Complexity-entropy causality plane: a useful approach to quantify the stock market inefficiency. Phys. A 389, 1891–1901 (2010)

    Google Scholar 

  46. Wiedermann, M., Donges, J., Kurths, J., Donner, R.: Mapping and discrimination of networks in the complexity-entropy plane. Phys. Rev. E 96, 042304 (2017)

    Google Scholar 

  47. Ribeiro, H., Zunino, L., Lenzi, E., Santoro, P., Mendes, R.: Complexity-entropy causality plane as a complexity measure for two-dimensional patterns. PLoS ONE 7(e40689), 22 (2012)

    Google Scholar 

  48. Jauregui, M., Zunino, L., Lenzi, E., Mendes, R., Ribeiro, H.: Characterization of time series via Renyi complexity-entropy curves. Phys. A 498, 74–85 (2018)

    MathSciNet  Google Scholar 

  49. Loiseau, P., Médigue, C., Gonçalves, P., Attia, N.: Large deviations estimates for the multiscale analysis of heart rate variability. Phys. A 391, 5658–5671 (2012)

    Google Scholar 

  50. Barral, J., Gonçalves, P.: On the estimation of the large deviations spectrum. J. Stat. Phys. 144, 1256–1283 (2011)

    MathSciNet  MATH  Google Scholar 

  51. Staniek, M., Lehnertz, K.: Parameter selection for permutation entropy measurements. Int. J. Bifurc. Chaos 17, 3729–3733 (2007)

    MATH  Google Scholar 

  52. Sunoj, S., Linu, M.: Dynamic cumulative residual Renyi’s entropy. Statistics 46, 41–56 (2012)

    MathSciNet  MATH  Google Scholar 

  53. http://finance.yahoo.com

Download references

Acknowledgements

The financial support from the funds of the Fundamental Research Funds for the Central Universities (2019YJS203) is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, China

    Du Shang & Pengjian Shang

Authors
  1. Du Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pengjian Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pengjian Shang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shang, D., Shang, P. Analysis of time series in the cumulative residual entropy plane based on oscillation roughness exponent. Nonlinear Dyn 100, 2167–2186 (2020). https://doi.org/10.1007/s11071-020-05646-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05646-y

Keywords

Search

Navigation