Nonlinear Dynamics

, Volume 81, Issue 4, pp 1779–1794 | Cite as

Multifractal cross-correlation analysis of traffic time series based on large deviation estimates

Original Paper


Multiscale analysis of cross-correlation between traffic signals is a developing research area, which helps to better understand the characteristics of traffic systems. Multifractal analysis is a multiscale analysis and almost exclusively develops around the concept of Legendre singularity spectrum, but which is structurally blind to subtle features like non-concavity or nonscaling of the distributions in some degree. Large deviations theory overcomes these limitations, and recently, performing estimators were proposed to reliably compute the corresponding large deviations singularity spectrum. In this paper, we apply the large deviations spectrum based on detrend cross-correlation analysis roughness exponent to both artificial and traffic time series and verify that this kind of approach is able to reveal significant information that remains hidden with Legendre spectrum. Meanwhile, we illustrate the presence of non-concavities in the spectra of cross-correlation between traffic series by the presence or absence of accidents and quantify the presence or absence of scale invariance and the generating mechanism of multifractality for cross-correlation between traffic signals.


Legendre spectrum Large deviations spectrum Multifractal analysis Cross-correlation Traffic signals 



Financial support by the Fundamental Research Funds for the Central Universities (2015YJS168) is gratefully acknowledged.


  1. 1.
    Leutzbach, W.: Introduction to the Theory of Traffic Flow. Springer, Berlin (1988)CrossRefMATHGoogle Scholar
  2. 2.
    Kerner, B.S.: The Physics of Traffic. Springer, New York (2004)CrossRefGoogle Scholar
  3. 3.
    Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329, 199–329 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Helbing, D.: Traffic and related self-driven manyparticle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)CrossRefGoogle Scholar
  5. 5.
    Safonov, L.A., Tomer, E., Strygin, V.V., Ashkenazy, Y., Havlin, S.: Delay-induced chaos with multifractal attractor in a traffic flow model. Europhys. Lett. 57, 151–158 (2002)CrossRefGoogle Scholar
  6. 6.
    Daoudi, K., Lévy Véhel, J.: Signal representation and segmentation based on multifractal stationarity. Signal Process 82, 2015–2024 (2002)CrossRefMATHGoogle Scholar
  7. 7.
    Gasser, I., Sirito, G., Werner, B.: Bifurcation analysis of a class of ‘car following’ traffic models. Phys. D 197, 222–241 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Wilson, R.E.: Mechanisms for spatio-temporal pattern formation in highway traffic models. Philos. Trans. R. Soc. A 366, 2017–2032 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Bai, M.Y., Zhu, H.B.: Power law and multiscaling properties of the Chinese stock market. Phys. A 389, 1883–1890 (2010)CrossRefGoogle Scholar
  10. 10.
    Wang, J., Shang, P., Zhao, X., Xia, J.: Multiscale entropy analysis of traffic time series. Int. J. Mod. Phys. C 24, 1350006 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shang, P., Lu, Y., Kamae, S.: Detecting long-range correlations of traffic time series with multifractal detrended fluctuation analysis. Chaos Solitons Fractals 36, 82–90 (2008)CrossRefGoogle Scholar
  12. 12.
    Nicholson, H., SwannThe, C.D.: Prediction of traffic flow volumes based on spectral analysis. Transp. Res. 8, 533C538 (1974)CrossRefGoogle Scholar
  13. 13.
    Stathopoulos, A., Karlaftis, M.G.: Spectral and cross-spectral analysis of urban traffic flows. In: 2001 IEEE intelligent transportation systems conference proceedings, pp. 820–825 (2001)Google Scholar
  14. 14.
    Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.E.: Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A 316, 87–114 (2002)CrossRefGoogle Scholar
  15. 15.
    Ivanov, P.C., Amaral, L.A.N., Goldberger, A.L., Havlin, S., Rosenblum, M.G., Struzik, Z., Stanley, H.E.: Multifractality in human heartbeat dynamics. Nature 399, 461–465 (1999)CrossRefGoogle Scholar
  16. 16.
    Sassi, R., Signorini, M.G., Cerutti, S.: Multifractality and heart rate variability. Chaos 19, 028507 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mandelbrot, B.: Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C.R. Acad. Sci. Paris 278(289–292), 355–358 (1974)MathSciNetMATHGoogle Scholar
  18. 18.
    Meyer, M., Stiedl, O.: Self-affine fractal variability of human heartbeat interval dynamics in health and disease. Eur. J. Appl. Physiol. 90, 305–316 (2003)CrossRefGoogle Scholar
  19. 19.
    Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685–1689 (1994)CrossRefGoogle Scholar
  20. 20.
    Hu, K., Chen, Z., Ivanov, P.C., Carpena, P., Stanley, H.E.: Effect of trends on detrended fluctuation analysis. Phys. Rev. E 64, 011114 (2001)CrossRefGoogle Scholar
  21. 21.
    Chen, Z., Ivanov, P.C., Hu, K., Stanley, H.E.: Effect of nonstationarities on detrended fluctuation analysis. Phys. Rev. E 65, 041107 (2002)CrossRefGoogle Scholar
  22. 22.
    Loiseau, P., Médigue, C., Gonçalves, P., Attia, N., Seuret, S., Cottin, F., Chemla, D., Sorine, M., Barral, J.: Large deviations estimates for the multiscale analysis of heart variability. Phys. A 391, 5658–5671 (2012)CrossRefGoogle Scholar
  23. 23.
    Podobnik, B., Stanley, H.E.: Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. Phys. Rev. Lett. 100, 084102 (2008)CrossRefGoogle Scholar
  24. 24.
    Lin, A., Shang, P., Zhao, X.: The cross-correlations of stock markets based on DCCA and time-delay DCCA. Nonlinear Dyn. 67, 425–435 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Siqueira, E.L., Stosic, T., Bejan, L., Stosic, B.: Statistical mechanics and its applications. Phys. A 389, 2739–2743 (2010)CrossRefGoogle Scholar
  26. 26.
    Podobnik, B., Horvatic, D., Petersen, A., Stanley, H.E.: Cross-correlations between volume change and price change. Proc. Natl. Acad. Sci. USA 106, 22079–22084 (2009)CrossRefMATHGoogle Scholar
  27. 27.
    Podobnik, B., Jiang, Z.Q., Zhou, W.X., Stanley, H.E.: Statistical tests for power-law cross-correlated processes. Phys. Rev. E 84, 066118 (2011)CrossRefGoogle Scholar
  28. 28.
    Horvatic, D., Stanley, H.E., Podobnik, B.: Detrended cross-correlation analysis for non-stationary time series with periodic trends. EPL 94, 18007–18012 (2011)CrossRefGoogle Scholar
  29. 29.
    Podobnik, B., Grosse, I., Horvatic, D., Ilic, S., Ivanov, P.C., Stanley, H.E.: Quantifying cross-correlations using local and global detrending approaches. Eur. Phys. J. B 71, 243–250 (2009)CrossRefGoogle Scholar
  30. 30.
    Yin, Y., Shang, P.: Modified DFA and DCCA approach for quantifying the multiscale correlation structure of financial markets. Phys. A 392, 6442–6457 (2013)CrossRefGoogle Scholar
  31. 31.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  32. 32.
    Riedi, R., Lévy Véhel, J.: Multifractal properties of TCP traffic: a numerical study, Technical Report No. 3129, INRIA Rocquencourt, France (1997).
  33. 33.
    Canus, C., Lévy-Véhel, J., Tricot, C.: Continuous large deviation multifractal spectrum: definition and estimation. In: Novak, M. (ed.), Proceedings of fractal 98 conference: “fractals and beyond: complexities in the sciences”, pp. 117–128Google Scholar
  34. 34.
    Lévy Véhel, J., Tricot, C.: On various multifractal spectra. Prog. Probab. 57, 23–42 (2004)Google Scholar
  35. 35.
    Barral, J., Gonçalves, P.: On the estimation of the large deviations spectrum. J. Stat. Phys. 144, 1256–1283 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Makse, H.A., Havlin, S., Schwartz, M., Stanley, H.E.: Method for generating long-range correlations for large systems. Phys. Rev. E 53, 5445–5449 (1996)CrossRefGoogle Scholar
  37. 37.
    Hosking, J.: Fractional differencing. Biometrika 68, 165–176 (1981)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Podobnik, B., Ivanov, PCh., Jazbinsek, V., Trontelj, Z., Stanley, H.E., Grosse, I.: Power-law correlated processes with asymmetric distributions. Phys. Rev. E 71, 025104 (2005)CrossRefMATHGoogle Scholar
  39. 39.
    Podobnik, B., Ivanov, P.C., Biljakovic, K., Horvatic, D., Stanley, H.E., Grosse, I.: Fractionally integrated process with power-law correlations in variables and magnitudes. Phys. Rev. E 72, 026121 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China

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