• J. Wanliss Department of Physics, Presbyterian College, Clinton, SC 29325 USA https://orcid.org/0000-0002-3291-6529
  • R. Hernandez Arriaza Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 29208 USA
  • G. Wanliss Department of Physics, Presbyterian College, Clinton, SC 29325 USA
  • S. Gordon Department of Biology, Presbyterian College, Clinton, SC 29325 USA




Dynamics, Fractals, Numerical Analysis, Signal Processing Algorithms

Abstract [English]

Background and Objective: Higuchi’s method of determining fractal dimension (HFD) occupies a valuable place in the study of a wide variety of physical signals. In comparison to other methods, it provides more rapid, accurate estimations for the entire range of possible fractal dimensions. However, a major difficulty in using the method is the correct choice of tuning parameter (kmax) to compute the most accurate results. In the past researchers have used various ad hoc methods to determine the appropriate kmax choice for their particular data. We provide a more objective method of determining, a priori, the best value for the tuning parameter, given a particular length data set. Methods: We create numerous simulations of fractional Brownian motion to perform Monte Carlo simulations of the distribution of the calculated HFD. Results: Experimental results show that HFD depends not only on kmax but also on the length of the time series, which enable derivation of an expression to find the appropriate kmax for an input time series of unknown fractal dimension. Conclusion: The Higuchi method should not be used indiscriminately without reference to the type of data whose fractal dimension is examined. Monte Carlo simulations with different fractional Brownian motions increases the confidence of evaluation results.


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Abry, P. and Sellan, F., (1996). The Wavelet-Based Synthesis for Fractional Brownian Motion Proposed by F. Sellan and Y. Meyer: Remarks and Fast Implementation. Applied and Computational Harmonic Analysis, 3(4), pp.377-383. Retrieved from https://doi.org/10.1006/acha.1996.0030 DOI: https://doi.org/10.1006/acha.1996.0030

Affinito, M., Carrozzi, M., Accardo, A. and Bouquet, F., (1997). Use of the fractal dimension for the analysis of electroencephalographic time series. Biological Cybernetics, 77(5), pp.339-350. Retrieved from https://doi.org/10.1007/s004220050394 DOI: https://doi.org/10.1007/s004220050394

Bardet, J.-M.; G. Lang, G. Oppenheim, A. Philippe, S. Stoev, M.S. Taqqu (2003), "Generators of long-range dependence processes: a survey," Theory and appli-cations of long-range dependence, Birkhäuser, 579-623.

Cersosimo, D. O., and J. A. Wanliss (2007), Initial studies of high latitude magnetic field data during different magnetospheric conditions, Earth Planets Space, 59(1), 39- 43. Retrieved from https://doi.org/10.1186/BF03352020 DOI: https://doi.org/10.1186/BF03352020

Doyle, T., Dugan, E., Humphries, B. and Newton, R., (2004). Discriminating between elderly and young using a fractal dimension analysis of centre of pressure. International Journal of Medical Sciences, pp.11-20. Retrieved from https://doi.org/10.7150/ijms.1.11 DOI: https://doi.org/10.7150/ijms.1.11

Esteller, R., Vachtsevanos, G., Echauz, J. and Litt, B., (2001). A comparison of waveform fractal dimension algorithms. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48(2), pp.177-183. Retrieved from https://doi.org/10.1109/81.904882 DOI: https://doi.org/10.1109/81.904882

Gomolka, R., Kampusch, S., Kaniusas, E., Thürk, F., Széles, J. and Klonowski, W., (2018). Higuchi Fractal Dimension of Heart Rate Variability During Percutaneous Auricular Vagus Nerve Stimulation in Healthy and Diabetic Subjects. Frontiers in Physiology, 9. Retrieved from https://doi.org/10.3389/fphys.2018.01162 DOI: https://doi.org/10.3389/fphys.2018.01162

Gómez, C., Mediavilla, Á., Hornero, R., Abásolo, D. and Fernández, A., (2009). Use of the Higuchi's fractal dimension for the analysis of MEG recordings from Alzheimer's disease patients. Medical Engineering & Physics, 31(3), pp.306-313. Retrieved from https://doi.org/10.1016/j.medengphy.2008.06.010 DOI: https://doi.org/10.1016/j.medengphy.2008.06.010

Higuchi, T., (1988). Approach to an irregular time series on the basis of the fractal theory. Physica D: Nonlinear Phenomena, 31(2), pp.277-283. Retrieved from https://doi.org/10.1016/0167-2789(88)90081-4 DOI: https://doi.org/10.1016/0167-2789(88)90081-4

Kawe, T., Shadli, S. and McNaughton, N., (2019). Higuchi's fractal dimension, but not frontal or posterior alpha asymmetry, predicts PID-5 anxiousness more than depressivity. Scientific Reports, 9(1). Retrieved from https://doi.org/10.1038/s41598-019-56229-w

Kawe, T.N.J., Shadli, S.M. & McNaughton, N. (2019) Higuchi's fractal dimension, but not frontal or posterior alpha asymmetry, predicts PID-5 anxiousness more than depressivity. Sci Rep 9, 19666. https://doi.org/10.1038/s41598-019-56229-w Retrieved from https://doi.org/10.1038/s41598-019-56229-w DOI: https://doi.org/10.1038/s41598-019-56229-w

Kesić, S. and Spasić, S., (2016). Application of Higuchi's fractal dimension from basic to clinical neurophysiology: A review. Computer Methods and Programs in Biomedicine, 133, pp.55-70. Retrieved from https://doi.org/10.1016/j.cmpb.2016.05.014 DOI: https://doi.org/10.1016/j.cmpb.2016.05.014

Khoa, T., Ha, V. and Toi, V., (2012). Higuchi Fractal Properties of Onset Epilepsy Electroencephalogram. Computational and Mathematical Methods in Medicine, 2012, pp.1-6. Retrieved from https://doi.org/10.1155/2012/461426 DOI: https://doi.org/10.1155/2012/461426

Klonowski, W., Olejarczyk, E. and Stepien, R., (2004). 'Epileptic seizures' in economic organism. Physica A: Statistical Mechanics and its Applications, 342(3-4), pp.701-707. Retrieved from https://doi.org/10.1016/j.physa.2004.05.045 DOI: https://doi.org/10.1016/j.physa.2004.05.045

Mandelbrot, Benoit B., and John W. Van Ness. (1968) "Fractional Brownian Motions, Fractional Noises and Applications." SIAM Review, vol. 10, no. 4, Society for Industrial and Applied Mathematics, pp. 422-37. Retrieved from https://doi.org/10.1137/1010093 DOI: https://doi.org/10.1137/1010093

Mitsutake G, Otsuka K, Oinuma S, Ferguson I, Cornélissen G, Wanliss J, Halberg F (2004) Does exposure to an artificial ULF magnetic field affect blood pressure, heart rate variability and mood? Biomed Pharmacother 58:S20-S27. Retrieved from https://doi.org/10.1016/S0753-3322(04)80004-0 DOI: https://doi.org/10.1016/S0753-3322(04)80004-0

Paramanathan, P. and Uthayakumar, R., (2008). Application of fractal theory in analysis of human electroencephalographic signals. Computers in Biology and Medicine, 38(3), pp.372-378. Retrieved from https://doi.org/10.1016/j.compbiomed.2007.12.004 DOI: https://doi.org/10.1016/j.compbiomed.2007.12.004

Peng, C., Buldyrev, S., Goldberger, A., Havlin, S., Sciortino, F., Simons, M. and Stanley, H., (1992). Long-range correlations in nucleotide sequences. Nature, 356(6365), pp.168-170. Retrieved from https://doi.org/10.1038/356168a0 DOI: https://doi.org/10.1038/356168a0

Shamsi, Elham, Mohammad Ali Ahmadi-Pajouh, Tirdad Seifi Ala, (2021) Higuchi fractal dimension: An efficient approach to detection of brain entrainment to theta binaural beats, Biomedical Signal Processing and Control, Volume 68, 102580. Retrieved from https://doi.org/10.1016/j.bspc.2021.102580 DOI: https://doi.org/10.1016/j.bspc.2021.102580

Wajnsztejn, R., Carvalho, T., Garner, D., Raimundo, R., Vanderlei, L., Godoy, M., Ferreira, C., Valenti, V. and Abreu, L., (2016). Higuchi fractal dimension applied to RR intervals in children with Attention Defi cit Hyperactivity Disorder. Journal of Human Growth and Development, 26(2), p.147. Retrieved from https://doi.org/10.7322/jhgd.119256 DOI: https://doi.org/10.7322/jhgd.119256

Wanliss, J. A., and M. A. Reynolds (2003), Measurement of the stochasticity of low-latitude geomagnetic temporal variations, Ann. Geophys., 21, 2025. Retrieved from https://doi.org/10.5194/angeo-21-2025-2003 DOI: https://doi.org/10.5194/angeo-21-2025-2003

Zappasodi F, Olejarczyk E, Marzetti L, Assenza G, Pizzella V, Tecchio F (2014) Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke. PLoS ONE 9(6): e100199. Retrieved from https://doi.org/10.1371/journal.pone.0100199 DOI: https://doi.org/10.1371/journal.pone.0100199




How to Cite

Wanliss, J., Arriaza, R. H., Wanliss, G., & Gordon, S. (2021). OPTIMIZATION OF THE HIGUCHI METHOD . International Journal of Research -GRANTHAALAYAH, 9(11), 202–213. https://doi.org/10.29121/granthaalayah.v9.i11.2021.4393