Skip to main content
SpringerLink
Log in

Simplified coded dispersion entropy: a nonlinear metric for signal analysis

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

Abstract

Recently, coded permutation entropy has been proposed, which enhances the noise immunity by quadratic partitioning on the basis of permutation entropy. However, coded permutation entropy and permutation entropy only consider the order of amplitude values and ignore some information related to amplitude. To overcome these defects, this paper applies the concept of quadratic partitioning to dispersion entropy (DE), takes advantage of the fact that DE can effectively measure amplitude information, and proposes coded DE (CDE), which increases the number of patterns and improves the divisibility by further coding the dispersion patterns in DE. Moreover, to reduce the computational consumption of CDE, we simplify the division criterion in quadratic partitioning while guaranteeing that no effective information is lost and propose simplified CDE (SCDE). Several simulation experiments demonstrate the advantages of SCDE and CDE over DE, permutation entropy, and coded permutation entropy in detecting the nonlinear dynamic changes within chaotic and synthetic signals. In addition, real-world experiments on electroencephalogram signals, bearing signals, and ship signals show that SCDE has better performance in medical diagnosis, fault diagnosis and signal classification, and the accuracy of SCDE-based classification methods is higher than that of other entropy-based 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

Similar content being viewed by others

Data availability

The datasets analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

DE:

Dispersion entropy

PE:

Permutation entropy

CPE:

Coded permutation entropy

CDE:

Coded dispersion entropy

SCDE:

Simplified coded permutation entropy

LZC:

Lempel–Ziv complexity

RPE:

Reverse permutation entropy

RDE:

Reverse dispersion entropy

NCDF:

Normal cumulative distribution function

FRDE:

Fluctuation-based reverse dispersion entropy

FuzzDE:

Fuzzy dispersion entropy

STD:

Standard deviation

CV:

Coefficient of variation

KNN:

K-nearest neighbor

EEG:

Electroencephalogram

References

  1. Bai, Y., Liang, Z., Li, X.: A permutation Lempel-Ziv complexity measure for EEG analysis. Biomed. Signal Process. Control 19, 102–114 (2015)

    Article  Google Scholar 

  2. Li, Y., Geng, B., Jiao, S.: Dispersion entropy-based Lempel-Ziv complexity: A new metric for signal analysis. Chaos, Solitons Fractals 161, 112400 (2022)

    Article  MathSciNet  Google Scholar 

  3. Lin, H., Scs, Y.: Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors. J. Appl. Mech. 63(2), 509–516 (1996)

    Article  MathSciNet  Google Scholar 

  4. Simiu E. Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of the Melnikov Method in Engineering, Physics, and Neuroscience. Princeton University Press, 2002.

  5. Yao, W., Dai, J., Perc, M.: Permutation-based time irreversibility in epileptic electroencephalograms. Nonlinear Dyn 100, 907–919 (2020)

    Article  Google Scholar 

  6. Gao, Z., Dang, W., Wang, X.: Complex networks and deep learning for EEG signal analysis. Cogn Neurodyn 15, 369–388 (2021)

    Article  Google Scholar 

  7. Enders, W.: Applied Econometric Time Series. Wiley, New York (1994)

    Google Scholar 

  8. Azami, H., Rostaghi, M., Abásolo, D., Escudero, J.: Refined composite multiscale dispersion entropy and its application to biomedical signals. IEEE Trans. Biomed. Eng. 64(12), 2872–2879 (2017)

    Article  Google Scholar 

  9. Tenreiro, M., Lopes, A.: Entropy analysis of human death uncertainty. Nonlinear Dyn 104, 3897–3911 (2021)

    Article  Google Scholar 

  10. Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inf. Theory 22(1), 75–81 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yeh, C., Shi, W.: Generalized multiscale Lempel-Ziv complexity of cyclic alternating pattern during sleep. Nonlinear Dyn 93, 1899–1910 (2018)

    Article  Google Scholar 

  12. Babloyantz, A., Destexhe, A.: Is the normal heart a periodic oscillator? Biol. Cybern. 58, 203–211 (1988)

    Article  MathSciNet  Google Scholar 

  13. Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50, 346 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, Y., Li, Y., Chen, X., Yu, J.: A novel feature extraction method for ship-radiated noise based on variational mode decomposition and multi-scale permutation entropy. Entropy 19, 342 (2017)

    Article  Google Scholar 

  15. Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88(17), 174102 (2002)

    Article  Google Scholar 

  16. Zanin, M., Zunino, L., Rosso, O.: Permutation entropy and its main biomedical and econophysics applications: a review. Entropy, 14(8) (2012)

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

    Article  Google Scholar 

  18. Xia, J., Shang, P., Wang, J., Shi, W.: Permutation and weighted-permutation entropy analysis for the complexity of nonlinear time series. Commun. Nonlinear Sci. Numer. Simul. 31, 60–68 (2016)

    Article  MATH  Google Scholar 

  19. Bandt, C.: A new kind of permutation entropy used to classify sleep stages from invisible EEG microstructure. Entropy 19, 197 (2017)

    Article  Google Scholar 

  20. Kang, H., Zhang, X., Zhang, G.: Coded permutation entropy: a measure for dynamical changes based on the secondary partitioning of amplitude information. Entropy 22(2), 187 (2020)

    Article  MathSciNet  Google Scholar 

  21. Mao, X., Shang, P., Xu, M.: Measuring time series based on multiscale dispersion Lempel-Ziv complexity and dispersion entropy plane. Chaos, Solitons Fractals 137, 109868 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rostaghi, M., Khatibi, M., Ashory, M., Azami, H.: Fuzzy dispersion entropy: a nonlinear measure for signal analysis. IEEE Trans. Fuzzy Syst. 30(9), 3785–3796 (2022)

    Article  Google Scholar 

  23. Zheng, J., Pan, H.: Use of generalized refined composite multiscale fractional dispersion entropy to diagnose the faults of rolling bearing. Nonlinear Dyn. 101, 1417–1440 (2021)

    Article  Google Scholar 

  24. Rostaghi, M,, Azami, H.: Dispersion entropy: a measure for time-series analysis. IEEE Signal Process. Lett., 610–614 (2016).

  25. Azami, H., Fernandez, A., Escudero, J.: Multivariate multiscale dispersion entropy of biomedical times series. Entropy 21(9), 913 (2017)

    Article  MathSciNet  Google Scholar 

  26. Li, Y., Gao, X., Wang, L.: Reverse dispersion entropy: a new complexity measure for sensor signal. Sensors 19(23), 5203 (2019)

    Article  Google Scholar 

  27. Jiao, S., Geng, B., Li, Y.: Fluctuation-based reverse dispersion entropy and its applications to signal classification. Appl. Acoust. 175(4), 107857 (2021)

    Article  Google Scholar 

  28. Azami, H., Sanei, S., Rajji, T.: Ensemble entropy: a low bias approach for data analysis. Knowl.-Based Syst. 256, 109876 (2022)

    Article  Google Scholar 

  29. Ocampo, S.: A new special function and its application in probability. Int. J. Math. Math. Sci. 2018(12), 5146794 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Dobson, A.: Introduction to Statistical Modelling. Chapman and Hall, New York (1983)

    Book  MATH  Google Scholar 

  31. Wu, G., Baleanu, D.: Chaos synchronization of the discrete fractional logistic map. Signal Process. 102(9), 96–99 (2014)

    Article  Google Scholar 

  32. Li, Y., Liu, F., Wang, S.: Multi-scale symbolic lempel-ziv: an effective feature extraction approach for fault diagnosis of railway vehicle systems. IEEE Trans. Industr. Inf. 17(1), 199–208 (2021)

    Article  Google Scholar 

  33. Chen, F., Baleanu, D.: Chaos synchronization of fractional chaotic maps based on the stability condition. Physica Stat. Mech. Appl. 460, 374–383 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. [Online]. https://blog.csdn.net/weixin_40521823/article/details/103862255. Accessed 30 Oct 2022

  35. [Online]. http://csegroups.case.edu/bearingdatacenter/pages/ download-data-file. Accessed 30 Oct 2022

Download references

Funding

This work was supported by the Natural Science Foundation of Shaanxi Province (No. 2022JM-337).

Author information

Authors and Affiliations

  1. School of Automation and Information Engineering, Xi’an University of Technology, Xi’an, 710048, China

    Yuxing Li, Bo Geng & Bingzhao Tang

  2. Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi’an University of Technology, Xi’an, 710048, China

    Yuxing Li

Authors
  1. Yuxing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Geng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bingzhao Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuxing Li.

Ethics declarations

Conflict of interest

The authors have no financial or proprietary interests in any material discussed in this article.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Geng, B. & Tang, B. Simplified coded dispersion entropy: a nonlinear metric for signal analysis. Nonlinear Dyn 111, 9327–9344 (2023). https://doi.org/10.1007/s11071-023-08339-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08339-4

Keywords

Search

Navigation