Skip to main content
SpringerLink
Log in

Multiscale Higuchi’s fractal dimension method

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

Abstract

The aim of this study is to introduce a new method for the evaluation of complexity properties of time series by extending Higuchi’s fractal dimension (HFD) over multiple scales. Multiscale Higuchi’s fractal dimension (MSHG) is presented and demonstrated on a number of stochastic time series and chaotic time series, starting with the examination of the selection of the effective scaling filter among several widely used filtering methods and then diving into the application of HFD through the scales obtained by coarse-graining procedure. Moreover, on the basis of MSHG, fractal dimension and Hurst exponent relationship are studied by employing MSHG method with computation of Hurst value in multiple scales, simultaneously. Consequently, it is found that the proposed method, MSHG produces remarkable results by exposing unique complexity features of time series in multiple scales. It is also discovered that MSHG with multiscale Hurst exponent calculation leads to revelation of distinguishing patterns between verifying stochastic time series and diverging chaotic time series. In light of these findings, it can be inferred that the proposed methods can be utilized for the characterization and classification of time series in terms of complexity.

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

Similar content being viewed by others

References

  1. Mandelbrot, B.B.: The Fractal Geometry of Nature, vol. 173. WH Freeman, New York (1983)

    Google Scholar 

  2. Reishofer, G., Koschutnig, K., Enzinger, C., Ebner, F., Ahammer, H.: Fractal dimension and vessel complexity in patients with cerebral arteriovenous malformations. PLoS ONE 7(7), e41148 (2012)

    Google Scholar 

  3. Mustafa, N., Ahearn, T.S., Waiter, G.D., Murray, A.D., Whalley, L.J., Staff, R.T.: Brain structural complexity and life course cognitive change. Neuroimage 61(3), 694–701 (2012)

    Google Scholar 

  4. Acharya, R., Bhat, P.S., Kannathal, N., Rao, A., Lim, C.M.: Analysis of cardiac health using fractal dimension and wavelet transformation. ITBM-RBM 26(2), 133–139 (2005)

    Google Scholar 

  5. Watari, S.: Fractal dimensions of solar activity. Solar Phys. 158(2), 365–377 (1995)

    Google Scholar 

  6. Kalauzi, A., Cukic, M., Millán, H., Bonafoni, S., Biondi, R.: Comparison of fractal dimension oscillations and trends of rainfall data from Pastaza Province, Ecuador and Veneto, Italy. Atmos. Res. 93(4), 673–679 (2009)

    Google Scholar 

  7. Lee, E.-T., Eun, H.-C.: Damage detection of damaged beam by constrained displacement curvature. J. Mech. Sci. Technol. 22(6), 1111–1120 (2008)

    Google Scholar 

  8. Zhou, Z.-M.: Measurement of time-dependent fractal dimension for time series of silicon content in pig iron. Phys. A 376, 133–138 (2007)

    Google Scholar 

  9. Klonowski, W., Olejarczyk, E., Stepien, R.: A new simple fractal method for nanomaterials science and nanosensors. Mater. Sci.-Poland 23(3), 607–612 (2005)

    Google Scholar 

  10. Samadder, S., Ghosh, K., Basu, T.: Fractal analysis of prime indian stock market indices. Fractals 21(01), 1350003 (2013)

    MathSciNet  Google Scholar 

  11. Higuchi, T.: Approach to an irregular time series on the basis of the fractal theory. Phys. D 31(2), 277–283 (1988)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  13. Li, D., Li, X., Liang, Z., Voss, L.J., Sleigh, J.W.: Multiscale permutation entropy analysis of eeg recordings during sevoflurane anesthesia. J. Neural Eng. 7(4), 046010 (2010)

    Google Scholar 

  14. Ouyang, G., Dang, C., Li, X.: Complexity analysis of eeg data with multiscale permutation entropy. In: Advances in Cognitive Neurodynamics (II), pp. 741–745, Springer, Berlin (2011)

  15. Morabito, F.C., Labate, D., La Foresta, F., Bramanti, A., Morabito, G., Palamara, I.: Multivariate multi-scale permutation entropy for complexity analysis of alzheimer’s disease eeg. Entropy 14(7), 1186–1202 (2012)

    MATH  Google Scholar 

  16. Shaobo, H., Kehui, S., Huihai, W.: Modified multiscale permutation entropy algorithm and its application for multiscroll chaotic systems. Complexity 21(5), 52–58 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Wu, S.-D., Wu, C.-W., Lee, K.-Y., Lin, S.-G.: Modified multiscale entropy for short-term time series analysis. Phys. A 392(23), 5865–5873 (2013)

    Google Scholar 

  18. Azami, H., Escudero, J.: Improved multiscale permutation entropy for biomedical signal analysis: interpretation and application to electroencephalogram recordings. Biomed. Signal Process. Control 23, 28–41 (2016)

    Google Scholar 

  19. Zhao, X., Sun, Y., Li, X., Shang, P.: Multiscale transfer entropy: measuring information transfer on multiple time scales. Commun. Nonlinear Sci. Numer. Simul. 62, 202–212 (2018)

    MathSciNet  Google Scholar 

  20. Nixon, M., Aguado, A.S.: Feature Extraction and Image Processing for Computer Vision. Academic Press, New York (2012)

    Google Scholar 

  21. Vaseghi, S.V.: Advanced Digital Signal Processing and Noise Reduction. Wiley, New York (2008)

    Google Scholar 

  22. Fukunaga, K., Hostetler, L.: The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans. Inf. Theory 21(1), 32–40 (1975)

    MathSciNet  MATH  Google Scholar 

  23. Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Mach. Intell. 5, 603–619 (2002)

    Google Scholar 

  24. Paris, S., Kornprobst, P., Tumblin, J., Durand, F.: A gentle introduction to bilateral filtering and its applications. In: ACM SIGGRAPH 2007 Courses, p. 1. ACM (2007)

  25. Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: ICCV, vol. 98, p. 2 (1998)

  26. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)

    MathSciNet  MATH  Google Scholar 

  27. Selesnick, I.W., Bayram, I.: Total Variation Filtering. White paper (2010)

  28. Standard deviation filter. https://reference.wolfram.com/language/ref/StandardDeviationFilter.html. Accessed 11 Sept 2019

  29. Burger, W., Burge, M.J.: Digital Image Processing: An Algorithmic Introduction Using Java. Springer, Berlin (2016)

    Google Scholar 

  30. Marques, O.: Practical Image and Video Processing Using MATLAB. Wiley, New York (2011)

    Google Scholar 

  31. Harmonic mean filter. https://reference.wolfram.com/language/ref/HarmonicMeanFilter.html. Accessed 11 Sept 2019

  32. Gao, J., Cao, Y., Tung, W.-W., Hu, J.: Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond. Wiley, New York (2007)

    MATH  Google Scholar 

  33. Mandelbrot, B.B.: Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E. Springer, Berlin (2013)

    Google Scholar 

  34. Fernández-Martínez, M., Sánchez-Granero, M.: Fractal Dimension for Fractal Structures with Applications to Finance. Springer, Switzerland (2019)

    MATH  Google Scholar 

  35. Coyt, G.G., Diosdado, A.M., Brown, F.A., et al.: Higuchi’s method applied to the detection of periodic components in time series and its application to seismograms. Rev. Mex. de Física S 59(1), 1–6 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Arqub, O.A., Shawagfeh, N.: Application of reproducing kernel algorithm for solving dirichlet time-fractional diffusion-gordon types equations in porous media. J. Porous Media 22(4), 411–434 (2019)

    Google Scholar 

  37. Accardo, A., Affinito, M., Carrozzi, M., Bouquet, F.: Use of the fractal dimension for the analysis of electroencephalographic time series. Biol. Cybern. 77(5), 339–350 (1997)

    MATH  Google Scholar 

  38. Mandelbrot, B.B., Van Ness, J.W.: Fractional brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

    MathSciNet  MATH  Google Scholar 

  39. Brooks, C.: Introductory Econometrics for Finance. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  40. Box, G.E., Jenkins, G.M., Reinsel, G.C., Ljung, G.M.: Time Series Analysis: Forecasting and Control. Wiley, New York (2015)

    MATH  Google Scholar 

  41. Sheng, H., Chen, Y., Qiu, T.: Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Springer, Berlin (2011)

    MATH  Google Scholar 

  42. Granger, C.W., Joyeux, R.: An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1(1), 15–29 (1980)

    MathSciNet  MATH  Google Scholar 

  43. Baillie, R.T., Bollerslev, T., Mikkelsen, H.O.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econom. 74(1), 3–30 (1996)

    MathSciNet  MATH  Google Scholar 

  44. Yilmaz, A., Unal, G.: Chaoticity properties of fractionally integrated generalized autoregressive conditional heteroskedastic processes. Bull. Math. Sci. Appl. 15, 69–82 (2016)

    Google Scholar 

  45. Hénon, M.: A two-dimensional mapping with a strange attractor. In: The Theory of Chaotic Attractors, pp. 94–102. Springer, Berlin (1976)

  46. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, Berlin (2003)

    MATH  Google Scholar 

  47. Ikeda, K.: Multiple-valued stationary state and its instability of the trasmitted light by a ring cavity system. Opt. Commun. 30(2), 257–261 (1979)

    Google Scholar 

  48. Tong, S., Zhang, J., Bao, Y., Lai, Q., Lian, X., Li, N., Bao, Y.: Analyzing vegetation dynamic trend on the mongolian plateau based on the hurst exponent and influencing factors from 1982–2013. J. Geograph. Sci. 28(5), 595–610 (2018)

    Google Scholar 

  49. Ohu, I.P., Carlson, J.N., Piovesan, D.: The hurst exponent: a novel approach for assessing focus during trauma resuscitation. In: Signal Processing in Medicine and Biology, pp. 139–160. Springer, Berlin (2020)

  50. Singh, A., Bhargawa, A.: An early prediction of 25th solar cycle using hurst exponent. Astrophys. Space Sci. 362(11), 199 (2017)

    Google Scholar 

  51. Ramos-Requena, J.P., Trinidad-Segovia, J., Sánchez-Granero, M.: Introducing hurst exponent in pair trading. Phys. A 488, 39–45 (2017)

    Google Scholar 

  52. Hurst, H.E.: Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770–799 (1951)

    Google Scholar 

  53. Pipiras, V., Taqqu, M.S.: Long-Range Dependence and Self-similarity, vol. 45. Cambridge University Press, Cambridge (2017)

    MATH  Google Scholar 

  54. Mandelbrot, B.: Long-run linearity, locally gaussian process, h-spectra and infinite variances. Int. Econ. Rev. 10(1), 82–111 (1969)

    MATH  Google Scholar 

  55. Raimundo, M.S., Okamoto Jr., J.: Application of hurst exponent (h) and the r/s analysis in the classification of forex securities. Int. J. Model. Optim 8, 116–124 (2018)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yeditepe University, 34755, Istanbul, Turkey

    A. Yilmaz

  2. Bahcesehir University, 34349, Istanbul, Turkey

    G. Unal

Authors
  1. A. Yilmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Unal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Yilmaz.

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

Yilmaz, A., Unal, G. Multiscale Higuchi’s fractal dimension method. Nonlinear Dyn 101, 1441–1455 (2020). https://doi.org/10.1007/s11071-020-05826-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05826-w

Keywords

Search

Navigation