PECULARITIES OF HIRST EXPONENT ESTIMATION FOR NATURAL PHYSICAL PROCESSES
Keywords:
nonlinear paradigm, natural physical process, fractal paradigm, fractal analysis, fractal dimension, corrective function, rescaled range analysis method, Hurst exponent.
Abstract
In bounds of the non-linear and system paradigms, been formulated by L. F. Chernogor in the last 1980th, all processes in open, non-linear, dynamical systems are very complex, non-linear, ultra-wideband or fractal ones.
According to the fractal paradigm put forward in the early 2000s by V. V. Yanovsky, fractality is one of the fundamental properties of the surrounding world. Therefore, the study of fractal characteristics, in particular, of natural physical processes is actual, interesting and useful.
The fractal dimension based on the Hurst exponent is one of the oldest and most famous ones. Based on the study of model fractal signals, it is demonstrated that the dependence between the estimate of the Hurst fractal dimension, obtained by the normalized range method, and its true value is significantly non-linear. To decrease of influence of the errors arising as a result of this, it is proposed to use the method of the corrective function.
The practical effectiveness of the proposed method is demonstrated on the example of the analysis of experimental results obtained in the middle 1960s by H. E. Hurst, which discovered the presence of a somewhat strange grouping of the values of the Hurst fractal dimension around the value of 1.27 for various natural physical processes. A hypothesis about the possibility of explaining this fact precisely by the nonlinearity of the mentioned dependence for R/S-method was proposed.
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References
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37. T. Gneiting, H. Sevcikova and D. B. Percival. Statist. Sci., 27, 247 (2012). https://doi.org/10.1214/11-STS370
38. M. S. Taqqu. Stochastic Processes and Their Applications, 7(1), 55 (1978). https://doi.org/10.1016/0304-4149(78)90037-6
39. H. E. Hurst, R. P. Black, Y. M. Simaika. Long-term storage: an experimental study, London, Constable (1965), 145 p. https://doi.org/10.2307/2982267
40. O. V. Lazorenko, A. A. Onishchenko, L. F. Chernogor. Radiotekhnika: All-Ukr. Sci. Inter- dep. Mag., 210. 177 (2022). (in Ukrainian). https://doi.org/10.30837/rt.2022.3.210.15.
41. C. Bandt, M. Barnsley, R. Devaney, K. J. Falconer, V. Kannan, P. B. Vinod Kumar, editors. Fractals, Wavelets, and their Applications: Contributions from the Int. Conference and Workshop on Fractals and Wavelets (Springer Proceedings in Mathematics & Statistics), Switzerland, Springer Int. Publ. (2014), 508 p. https://www.amazon.com/Fractals-Wavelets-their-Applications-Contributions-ebook/dp/B00PUM0AQ2
2. K. J. Falconer. Fractal Geometry. Mathematical Foundations and Applications, Chichester, Wiley & Sons (1990), 288 p. https://doi.org/10.1002/0470013850
3. J. Feder. Fractals, New York, Plenum Press (1988), 305 p. https://link.springer.com/book/10.1007/978-1-4899-2124-6
4. E. Chandrasekhar, V. P. Dimri, V. M. Gadre, editors. Wavelets and Fractals in Earth System Sciences, CRC Press (2014), 294 p. https://doi.org/10.1201/b16046
5. D. P. Feldman. Chaos and Fractals. An Elementary Introduction, Oxford, University Press (2012), 408 p. https://doi.org/10.1093/acprof:oso/9780199566433.001.0001
6. B. B. Mandelbrot. Fractals and Chaos: The Mandelbrot Set and Beyond, Springer (2005), 400 p. https://doi.org/10.1186/1475-925X-4-30
7. R. M. Crownover. Introduction to Fractals and Chaos, Boston, Jones and Barlett Publishers (1995), 320 p. https://www.amazon.com/Introduction-Fractals-Chaos-Bartlett-Mathematics/dp/0867204648
8. B. B. Mandelbrot. Multifractals and 1/f Noise, Springer (1999), 442 p. https://doi.org/10.1007/978-1-4612-2150-0
9. D. Harte. Multifractals. Theory and Applications, Boca Raton, Chapman and Hall/CRC Press (2001), 264 p. https://doi.org/10.1201/9781420036008
10. M. Schroeder. Fractals, Chaos, Power Laws. Minutes from Infinite Paradise, New York, W. H. Freeman and Company (1991), 528 p. https://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/0486472043
11. F. C. Moon. Chaotic Vibrations. An Introduction for Applied Scientists and Engineers, New York, Wiley and Sons (2004), 309 p. https://doi.org/10.1002/3527602844
12. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 25 (1), 3 (2020) (in Russian). https://doi.org/10.15407/rpra25.01.003
13. V. V. Yanovsky. Universitates, 3, 32 (2003) (In Russian).
14. V. V. Yanovsky. Lectures on Nonlinear Phenomena. Volume 1, Kharkiv, Institut monokristallov Publ. (2006), 456 p. (in Russian).
15. L. F. Chernogor. On the Nonlinearity in Nature and Science, Kharkiv, V. N. Karazin Kharkiv National University (2008), 528 p. (In Russian).
16. O. V. Lazorenko, L. F. Chernogor. Radio Phys. Radio Astron., 28(1), 5 (2023) (in Ukrainian). https://doi.org/10.15407/rpra28.01.005
17. H. E. Hurst. Trans. Amer. Soc. Civ. Eng., 116, 770 (1951).
18. H. E. Hurst, R. P. Black, Y. M. Simaika. Long-term storage: an experimental study, London, Constable (1965), 145 p.
19. L. F. Chernogor, O. V. Lazorenko, A. A. Onishchenko. Low Temperature Physics, 49(4), 459 (2023). https://doi.org/10.1063/10.0017581
20. H. H. Hardy, R. A. Beier. Fractals in Reservoir Engineering. Singapore, New Jersey, London, Hong Kong, World Scientific (1994), 359 p. https://doi.org/10.1142/2574
21. L. Seuront. Fractals and Multifractals in Ecology and Aquatic Science, Boca Raton, London, New York, CRC Press (2010), 344 p. https://doi.org/10.1201/9781420004243
22. B. Mandelbrot, J. R. Wallis. Water Resources Res., 5(1) 228 (1969). https://doi.org/10.1029/WR005I001P00228
23. J. B. Bassingthwaighte. News Physiol. Sci., 3, 5 (1988). https://doi.org/10.1152/physiologyonline.1988.3.1.5
24. N. Scafetta, P. Grigolini. Phys. Rev. E, 66(3), 036130 (2002). https://doi.org/10.1103/physreve.66.036130
25. J. H. Van Beek, S. A. Roger, J. B. Bassingthwaighte. Am. J. Physiol., 257(5), H1670 (1989). https://doi.org/10.1152/ajpheart.1989.257.5.h1670
26. H. M. Hastings, G. Sugihara. Fractals: A User’s Guide for the Natural Science, Oxford, Oxford University Press (1993), 248 p. https://www.amazon.com/Fractals-Natural-Sciences-Science-Publications/dp/0198545975
27. R. F. Peltier, J. Lévy-Véhel. Research report, INRIA Rocqencourt (1994). https://hal.inria.fr/inria-00074279
28. J. Beran. Statistics for Long-Memory Processes, Chapman and Hall (1994), 328 p. https://doi.org/10.2307/2983481
29. C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, A. L. Goldberger. Phys. Rev. E., 49, 1685 (1994). https://doi.org/10.1103/PhysRevE.49.1685
30. M. S. Taqqu, V. Teverovsky, W. Willinger. Fractals, 03(04), 785 (1995). https://doi.org/10.1142/s0218348x95000692
31. M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, J. B. Bassingthwaighte. Physica A: Statistical Mechanics and Its Applications, 241(3-4), 606 (1997). https://doi.org/10.1016/s0378-4371(97)00252-5
32. N. Vandewalle, M. Ausloos. Phys. Rev. E, 58(5), 6832 (1998). https://doi.org/10.1103/physreve.58.6832
33. A. Eke, P. Hermán, J. Bassingthwaighte, G. Raymond, D. Percival, M. Cannon, … C. Ikrényi. Pflügers Archiv - European Journal of Physiology, 439(4), 403 (2000). https://doi.org/10.1007/s004249900135
34. S. M. Prigarin, K. Hahn, G. Winkler. Numerical Analysis and Applications, 2(4), 352 (2009). https://doi.org/10.1134/s1995423909040077
35. M. A. Riley, S. Bonnette, N. Kuznetsov, S. Wallot, J. Gao. Frontiers in Physiology, 3 (2012). https://doi.org/10.3389/fphys.2012.00371
36. M. J. Sánchez-Granero, M. Fernández-Martínez, J. Trinidad-Segovia. The European Physical Journal B, 85(3), 86 (2012). https://doi.org/10.1140/epjb/e2012-20803-2
37. T. Gneiting, H. Sevcikova and D. B. Percival. Statist. Sci., 27, 247 (2012). https://doi.org/10.1214/11-STS370
38. M. S. Taqqu. Stochastic Processes and Their Applications, 7(1), 55 (1978). https://doi.org/10.1016/0304-4149(78)90037-6
39. H. E. Hurst, R. P. Black, Y. M. Simaika. Long-term storage: an experimental study, London, Constable (1965), 145 p. https://doi.org/10.2307/2982267
40. O. V. Lazorenko, A. A. Onishchenko, L. F. Chernogor. Radiotekhnika: All-Ukr. Sci. Inter- dep. Mag., 210. 177 (2022). (in Ukrainian). https://doi.org/10.30837/rt.2022.3.210.15.
41. C. Bandt, M. Barnsley, R. Devaney, K. J. Falconer, V. Kannan, P. B. Vinod Kumar, editors. Fractals, Wavelets, and their Applications: Contributions from the Int. Conference and Workshop on Fractals and Wavelets (Springer Proceedings in Mathematics & Statistics), Switzerland, Springer Int. Publ. (2014), 508 p. https://www.amazon.com/Fractals-Wavelets-their-Applications-Contributions-ebook/dp/B00PUM0AQ2
Published
2024-05-30
How to Cite
Lazorenko, O. V., Onishchenko, A. A., Taranova, I. A., & Udovenko, M. A. (2024). PECULARITIES OF HIRST EXPONENT ESTIMATION FOR NATURAL PHYSICAL PROCESSES. Journal of V. N. Karazin Kharkiv National University. Series Physics, (40), 25-34. https://doi.org/10.26565/2222-5617-2023-40-02
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