Skip to main content
SpringerLink
Log in

Multivariate multiscale sample entropy of traffic time series

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

Abstract

Multivariate time series are common in the traffic system and are necessary for understanding the property of the traffic system. This paper introduces the multivariate multiscale sample entropy (MMSE) to evaluate the complexity in multiple data channels over different timescales. We illustrate the necessity and advantage of MMSE method by comparing MMSE results with the multiscale sample entropy results on original and shuffled traffic time series, respectively. MMSE is capable of revealing the long-range correlations and providing robust estimates for the complexity of traffic time series. Then, we utilize the MMSE to assess relative complexity of normalized multichannel temporal data in the traffic system and also reveal the weekday and weekend patterns containing in traffic signals by MMSE. MMSE can provide more accurate and helpful knowledge about the complexity of traffic time series for the dynamics and inner mechanism of traffic system from the view of multiple variables.

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

Similar content being viewed by others

References

  1. Belzoni, G.S., Colombo, R.M.: An n-populations model for traffic flow. Eur. J. Appl. Math. 14, 587–612 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Xu, N., Shang, P., Kamae, S.: Modeling traffic flow correlation using DFA and DCCA. Nonlinear Dyn. 61, 207–216 (2010)

    Article  MATH  Google Scholar 

  3. Yin, Y., Shang, P.: Multiscale multifractal detrended cross-correlation analysis of traffic flow. Nonlinear Dyn. 81, 1329–1347 (2015)

    Article  MathSciNet  Google Scholar 

  4. Yin, Y., Shang, P.: Multifractal cross-correlation analysis of traffic time series based on large deviation estimates. Nonlinear Dyn. 81, 1779–1794 (2015)

    Article  MathSciNet  Google Scholar 

  5. Colombo, R.M.: Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63, 708–721 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Klar, A.: Kinetic and Macroscopic Traffic Flow Models. Piano di Sorrento, Italy (2002)

    Google Scholar 

  7. Kerner, B.S.: Experimental feature of selforganization in traffic flow. Phys. Rev. Lett. 81, 3797–3800 (1998)

    Article  MATH  Google Scholar 

  8. Helbing, D., Huberman, B.A.: Coherent moving states in highway traffic. Nature 396, 738–740 (1998)

    Article  Google Scholar 

  9. Helbing, D.: Traffic and related self-driven manyparticle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)

    Article  Google Scholar 

  10. Vlahogianni, E.I., Karlaftis, M.G.: Comparing traffic flow time-series under fine and adverse weather conditions using recurrence-based complexity measures. Nonlinear Dyn. 69, 1949–1963 (2012)

    Article  Google Scholar 

  11. Choi, M.Y., Lee, H.Y.: Traffic flow and 1/f fluctuations. Phys. Rev. E 52, 5979–5984 (1995)

    Article  Google Scholar 

  12. Nagel, K., Rasmussen, S.: Traffic at the Edge of Chaos. MIT, Massachusetts (1994)

    Google Scholar 

  13. Leutzbach, W.: Introduction to the Theory of Traffic Flow. Springer, Berlin (1988)

    Book  Google Scholar 

  14. Kerner, B.S.: The Physics of Traffic. Springer, New York (2004)

    Book  Google Scholar 

  15. Shang, P., Li, X., Santi, K.: Chaotic analysis of traffic time series. Chaos Solitons Fractals 25, 121–128 (2005)

    Article  MATH  Google Scholar 

  16. Shang, P., Li, X., Santi, K.: Nonlinear analysis of traffic time series at different temporal scales. Phys. Lett. A 357, 314–318 (2006)

    Article  MATH  Google Scholar 

  17. Shang, P., Wan, M., Santi, K.: Fractal nature of highway traffic data. Comput. Math. Appl. 54, 107–116 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shang, P., Lu, Y., Santi, K.: Detecting longrange correlations of traffic time series with multifractal detrended fluctuation analysis. Chaos Solitons Fractals 36, 82–90 (2008)

    Article  Google Scholar 

  19. Shang, P., Lin, A., Liu, L.: Chaotic SVD method for minimizing the effect of exponential trends in detrended fluctuation analysis. Phys. A 388, 720–726 (2009)

    Article  Google Scholar 

  20. Safonov, L.A., Tomer, E., Strygin, V.V., Ashkenazy, Y., Havlin, S.: Delay-induce chaos with multifractal attractor in a traffic flow model. Europhys. Lett. 57, 151–158 (2002)

    Article  Google Scholar 

  21. Daoudi, K., Lévy Véhel, J.: Signal representation and segmentation based on multifractal stationarity. Signal Process. 82, 2015–2024 (2002)

    Article  MATH  Google Scholar 

  22. Gasser, I., Sirito, G., Werner, B.: Bifurcation analysis of a class of ‘car following’ traffic models. Phys. D 197, 222–241 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wilson, R.E.: Mechanisms for spatio-temporal pattern formation in highway traffic models. Philos. Trans. R. Soc. A 366, 2017–2032 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, J., Shang, P., Zhao, X., Xia, J.: Multiscale entropy analysis of traffic time series. Int. J. Mod. Phys. C 24, 1350006 (2013)

    Article  MathSciNet  Google Scholar 

  25. Kumar, M., Rawat, T.K.: Fractional order digital differentiator design based on power function and least-squares. Int. J. Electron. (2016). doi:10.1080/00207217.2016.1138520

    Google Scholar 

  26. Aggarwal, A., Kumar, M., Rawat, T.K., Upadhyay, D.K.: Optimal design of 2-D FIR filters with quadrantally symmetric properties using fractional derivative constraints. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-016-0283-x

    Google Scholar 

  27. Kumar, M., Rawat, T.K.: Design of a variable fractional delay filter using comprehensive least square method encompassing all delay values. J. Circuits Syst. Comput. 24, 1550116 (2015)

    Article  Google Scholar 

  28. Goldberger, A.L., Peng, C.-K., Lipsitz, L.A.: What is physiologic complexity and how does it change with aging and disease. Neurobiol. Aging 23, 23–26 (2002)

  29. Costa, M., Goldberger, A.L., Peng, C.-K.: Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 89, 068102 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Thuraisingham, R.A., Gottwald, G.A.: On multiscale entropy analysis for physiological data. Phys. A 366, 323–332 (2006)

    Article  Google Scholar 

  32. Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy: a tool for complexity analysis of multichannel data. Phys. Rev. E 84, 3067–3076 (2011)

    Article  Google Scholar 

  33. Ahmed, M.U., Mandic, D.P.: Multivariate multiscale entropy analysis. IEEE Signal Process. Lett. 19, 91–94 (2012)

    Article  Google Scholar 

  34. Richman, J.S., Moorman, J.R.: Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. Heart Circ. Physiol. 278, 2039–2049 (2000)

    Google Scholar 

  35. Grassberger, P., Schreiber, T., Schaffrath, C.: Nonlinear time sequence analysis. Int. J. Bifurc. Chaos 1, 521–547 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134–1140 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Albano, A.M., Mees, A.I., deGuzman, G.C., Rapp, P.E.: Data requirements for reliable estimation of correlation dimensions. In: Degn, H., Holden, A.V., Olsen, L.F. (eds.) Chaos Biol. Syst., pp. 207–220. Plenum, New York (1987)

    Chapter  Google Scholar 

  38. Liu, L.Z., Qian, X.Y., Lu, H.Y.: Cross-sample entropy of foreign exchange time series. Phys. A 389, 4785–4792 (2010)

    Article  Google Scholar 

  39. Zhao, X., Shang, P., Lin, A., Chen, G.: Multifractal Fourier detrended cross-correlation analysis of traffic signals. Phys. A 390, 3670–3678 (2011)

    Article  Google Scholar 

  40. Wang, J., Shang, P., Dong, K.: Effect of linear and nonlinear filters on multifractal analysis. Appl. Math. Comput. 224, 337–345 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Chianca, C.V., Tinoca, A., Penna, T.J.P.: Fourier-detrended fluctuation analysis. Phys. A 357, 447–454 (2005)

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by “the Fundamental Research Funds for the Central Universities” (2016YJS160).

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Yi Yin & Pengjian Shang

Authors
  1. Yi Yin
    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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yin, Y., Shang, P. Multivariate multiscale sample entropy of traffic time series. Nonlinear Dyn 86, 479–488 (2016). https://doi.org/10.1007/s11071-016-2901-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2901-3

Keywords

Search

Navigation