Skip to main content
SpringerLink
Log in

Distance correlation entropy and ordinal distance complexity measure: efficient tools for complex systems

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

Abstract

Many distance measures are used to quantify the relationship between components in complex systems, such as the commonly used Euclidean distance and cos similarity. These distances have a strong connection with the Pearson coefficient. However, the Pearson coefficient sometimes ignores important nonlinear relations. Compared with the Pearson coefficient, the distance correlation coefficient can be a reliable measure of linear and nonlinear relationships. Inspired by this, we propose distance correlation entropy for uncertainty quantification and classifying different states. Unlike other entropy, the distribution of distance correlation is utilized to evaluate complexity. DCE retains the advantages of the previous methods, such as high consistency. The ordinal distance complexity measure is proposed as a supplement to DCE for quantifying information about pattern transitions in time series. Both DCE and ODCM are insensitive to the length of time series. Moreover, distance rank entropy derived from DCE and ODCM can be used to detect abnormal data. Experiments show that DCE and ODCM can distinguish periodic and chaotic behavior as well as different states in nonlinear dynamic systems, such as financial time series, while ODCM and distance rank entropy can be well combined with the uniform manifold approximation and projection method for data classification and visualization. The application in bearing data illustrates that they can be applied to fault diagnosis and get satisfactory results.

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
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Yoon, B.J., Qian, X., Dougherty, E.R.: Quantifying the objective cost of uncertainty in complex dynamical systems. IEEE Trans. Signal Process. 61(9), 2256–2266 (2013)

    Google Scholar 

  2. Schlitt, T., Brazma, A.: Current approaches to gene regulatory network modelling. BMC Bioinformatics 8(6), 1–22 (2007)

    Google Scholar 

  3. Zañudo, J.G.T., Yang, G., Albert, R.: Structure-based control of complex networks with nonlinear dynamics. Proc. Natl. Acad. Sci. 114(28), 7234–7239 (2017)

    Google Scholar 

  4. Özkaynak, F.: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. 92(2), 305–313 (2018)

    Google Scholar 

  5. Laurent, N., Meignen, S.: A novel ridge detector for nonstationary multicomponent signals: development and application to robust mode retrieval. IEEE Trans. Signal Process. 69, 3325–36 (2021)

    MathSciNet  Google Scholar 

  6. Jiang, B., Staroswiecki, M., Cocquempot, V.: Fault accommodation for nonlinear dynamic systems. IEEE Trans. Autom. Control 51(9), 1578–1583 (2006)

    MathSciNet  Google Scholar 

  7. Valenza, G., Lanata, A., Scilingo, E.P.: The role of nonlinear dynamics in affective valence and arousal recognition. IEEE Trans. Affect. Comput. 3(2), 237–249 (2011)

    Google Scholar 

  8. Guirao, J.L., Luo, A.C.: New trends in nonlinear dynamics and chaoticity. Nonlinear Dyn. 84, 1–2 (2016)

    MathSciNet  Google Scholar 

  9. Vannitsem, S.: Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics. Chaos Interdiscip. J. Nonlinear Sci. 27(3), 032101 (2017)

    MathSciNet  Google Scholar 

  10. Gao, M.C., Yeh, J.W., Liaw, P.K., Zhang, Y., et al.: High-Entropy Alloys. Springer International Publishing, Cham (2016)

    Google Scholar 

  11. Cao, Z., Lin, C.T.: Inherent fuzzy entropy for the improvement of eeg complexity evaluation. IEEE Trans. Fuzzy Syst. 26(2), 1032–1035 (2017)

    Google Scholar 

  12. Yilmaz, A., Unal, G.: Multiscale Higuchi’s fractal dimension method. Nonlinear Dyn. 101(2), 1441–1455 (2020)

    Google Scholar 

  13. Li, Y., Tang, B., Jiao, S.: So-slope entropy coupled with Svmd: A novel adaptive feature extraction method for ship-radiated noise. Ocean Eng. 280, 114677 (2023)

    Google Scholar 

  14. Li, Y., Geng, B., Tang, B.: Simplified coded dispersion entropy: A nonlinear metric for signal analysis. Nonlinear Dyn. 111(10), 9327–9344 (2023)

    Google Scholar 

  15. Mitchell, M.: Complexity: A Guided Tour. Oxford University Press, Oxford (2009)

    Google Scholar 

  16. Bar-Yam, Y.: General Features of Complex Systems Encyclopedia of Life Support Systems (EOLSS). UNESCO EOLSS Publishers, Oxford (2002)

    Google Scholar 

  17. Capra, F., Luisi, P.L.: The Systems View of Life: A Unifying Vision. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  18. Bar-Yam, Y.: About Complex Systems. Addison-Wesley, Reading (1997)

    Google Scholar 

  19. Carter, T.: An Introduction to Information Theory and Entropy. Complex systems summer school, Santa Fe (2007)

    Google Scholar 

  20. Fuchs, A.: Nonlinear Dynamics in Complex Systems. Springer, Cham (2014)

    Google Scholar 

  21. Buchman, T.G.: Nonlinear dynamics, complex systems, and the pathobiology of critical illness. Curr. Opin. Crit. Care 10(5), 378–382 (2004)

    Google Scholar 

  22. Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. 88(6), 2297–2301 (1991)

    MathSciNet  Google Scholar 

  23. Richman, J.S., Moorman, J.R.: Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol. 278(6), 2039 (2000)

    Google Scholar 

  24. Delgado-Bonal, A., Marshak, A.: Approximate entropy and sample entropy: A comprehensive tutorial. Entropy 21(6), 541 (2019)

    MathSciNet  Google Scholar 

  25. Zhou, R., Wang, X., Wan, J., Xiong, N.: Edm-fuzzy: An Euclidean distance based multiscale fuzzy entropy technology for diagnosing faults of industrial systems. IEEE Trans. Indus. Inform. 17, 4046 (2020)

    Google Scholar 

  26. Wang, X., Si, S., Li, Y.: Multiscale diversity entropy: A novel dynamical measure for fault diagnosis of rotating machinery. IEEE Trans. Indus. Inform. 17, 5419 (2020)

    Google Scholar 

  27. Székely, G.J., Rizzo, M.L., Bakirov, N.K., et al.: Measuring and testing dependence by correlation of distances. Ann. Stat. 35(6), 2769–2794 (2007)

    MathSciNet  Google Scholar 

  28. Székely, G.J., Rizzo, M.L., et al.: Brownian distance covariance. Annal. Appl. Stat. 3(4), 1236–1265 (2009)

    MathSciNet  Google Scholar 

  29. Borges, J.B., Ramos, H.S., Mini, R.A., Rosso, O.A., Frery, A.C., Loureiro, A.A.: Learning and distinguishing time series dynamics via ordinal patterns transition graphs. Appl. Math. Comput. 362, 124554 (2019)

    MathSciNet  Google Scholar 

  30. Ruan, Y., Donner, R.V., Guan, S., Zou, Y.: Ordinal partition transition network based complexity measures for inferring coupling direction and delay from time series. Chaos Interdiscip. J. Nonlinear Sci. 29(4), 043111 (2019)

    MathSciNet  Google Scholar 

  31. Wang, Y., Shi, W., Yeh, C.H.: A novel measure of cardiopulmonary coupling during sleep based on the synchrosqueezing transform algorithm. IEEE J. Biomed. Health Inform. 27(4), 1790–1800 (2023)

    Google Scholar 

  32. Shi, W., Yeh, C.H., Hong, Y.: Cross-frequency transfer entropy characterize coupling of interacting nonlinear oscillators in complex systems. IEEE Trans. Biomed. Eng. 66(2), 521–529 (2018)

  33. McInnes, L., Healy, J., Melville, J.: Umap: Uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426 (2018)

  34. Berthold, M.R., Höppner, F.: On clustering time series using euclidean distance and pearson correlation. arXiv preprint arXiv:1601.02213 (2016)

  35. Székely, G.J., Rizzo, M.L.: The energy of data. Annual Rev. Stat. Appl. 4, 447–479 (2017)

    Google Scholar 

  36. Edelmann, D., Fokianos, K., Pitsillou, M.: An updated literature review of distance correlation and its applications to time series. Int. Stat. Rev. 87(2), 237–262 (2019)

    MathSciNet  Google Scholar 

  37. Daw, C.S., Finney, C.E.A., Tracy, E.R.: A review of symbolic analysis of experimental data. Rev. Sci. Instrum. 74(2), 915–930 (2003)

    Google Scholar 

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

    Google Scholar 

  39. Yin, Y., Shang, P.: Multivariate weighted multiscale permutation entropy for complex time series. Nonlinear Dyn. 88, 1707–1722 (2017)

    MathSciNet  Google Scholar 

  40. Tarasova, V.V., Tarasov, V.E.: Logistic map with memory from economic model. Chaos, Solitons Fractals 95, 84–91 (2017)

    MathSciNet  Google Scholar 

  41. Azami, H., Fernández, A., Escudero, J.: Refined multiscale fuzzy entropy based on standard deviation for biomedical signal analysis. Med. Biol. Eng. Comput. 55, 2037–2052 (2017)

    Google Scholar 

  42. Chen, W., Wang, Z., Xie, H., Yu, W.: Characterization of surface emg signal based on fuzzy entropy. IEEE Trans. Neural Syst. Rehabil. Eng. 15(2), 266–272 (2007)

    Google Scholar 

  43. Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of biological signals. Phys. Rev. E 71(2), 021906 (2005)

    MathSciNet  Google Scholar 

  44. Becht, E., McInnes, L., Healy, J., Dutertre, C.A., Kwok, I.W., Ng, L.G., Ginhoux, F., Newell, E.W.: Dimensionality reduction for visualizing single-cell data using umap. Nat. Biotechnol. 37(1), 38–44 (2019)

    Google Scholar 

  45. Dorrity, M.W., Saunders, L.M., Queitsch, C., Fields, S., Trapnell, C.: Dimensionality reduction by umap to visualize physical and genetic interactions. Nat. Commun. 11(1), 1537 (2020)

    Google Scholar 

  46. Diaz-Papkovich, A., Anderson-Trocmé, L., Gravel, S.: A review of umap in population genetics. J. Hum. Genet. 66(1), 85–91 (2021)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

    Boyi Zhang & Pengjian Shang

Authors
  1. Boyi Zhang
    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 Boyi Zhang.

Ethics declarations

Funding

This study is supported by the funds of the Fundamental Research Funds for the Central Universities (2022YJS097), China Academy of Railway Science Cooperation Limited (2019YJ153) and the National Natural Science Foundation of China (62171018).

Data Availability

The stock datasets analyzed during the current study are available, [https://cn.investing.com/] The bearing datasets analyzed during the current study are available, [https://engineering.case.edu/]

Conflict of interest

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

Additional information

Publisher's Note

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

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

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

Zhang, B., Shang, P. Distance correlation entropy and ordinal distance complexity measure: efficient tools for complex systems. Nonlinear Dyn 112, 1153–1172 (2024). https://doi.org/10.1007/s11071-023-09080-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09080-8

Keywords

Search

Navigation