Multifractal analysis of model fractal and multifractal signals
DOI:
https://doi.org/10.30837/rt.2022.4.211.05Keywords:
fractal, multifractal, signal, process, analysis, method, dimension, estimation, accuracy, correctionAbstract
One of the topical directions of modern fractal radio physics is the multifractal analysis of signals and processes of various origins. A set of deterministic and stochastic models of monofractal and multifractal signals and processes in the time domain is proposed. New multifractal signal characteristics, namely, the coefficient of asymmetry of the multifractal spectrum function, the relative width of the multifractal spectrum and the dimension of the multifractal support, are introduced, the necessity of their use is demonstrated on examples. Using Wavelet Transform Modulus Maxima Method and Multi-Fractal Detrended Fluctuation Analysis Method, a detailed multifractal analysis of model signals is performed. The features of multifractal analysis of monofractal, multifractal and non-fractal signals are established, the appropriate recommendations for practitioners are formulated. Convenient formats for presenting analysis results have been developed. It was found that during the transition of the multifractal signal to the monofractal regime, the function of the multifractal spectrum of the physical fractal does not collapse into a point, as it should happen in theory for a mathematical fractal. Threshold values of multifractal characteristics, which are indicators of the appearance of the monofractal, are proposed. It has been shown that multifractal analysis of non-fractal signals leads to the appearance of multifractal spectra with anomalous values of multifractal characteristics. The correction function method is modified for the methods of multifractal analysis of signals and processes. It is proved that its usage makes it possible to reduce the deviation of the obtained estimate of the generalized Hurst exponent from the true known value of the Hölder exponent of the analyzed signal from 5 – 90% to 3 – 8%.
References
Лазоренко О. В., Черногор Л. Ф. Фрактальная радиофизика. 1. Теоретические основы // Радиофизика и радиоастрономия. 2020. Т. 25, № 1. С. 3 – 77.
Kantelhardt J. W., Zschiegner S. A., Koscielny-Bunde E., Havlin S., Bunde A., Stanley H. E. Multifractal detrended fluctuation analysis of nonstationary time series // Physica A: Statistical Mechanics and Its Applications. 2002. Vol. 316, No. 1 – 4. P. 87 – 114.
Jaffard S. Multifractal Formalism for Functions. Part E Results Valid for All Functions // SIAM J. Math. Anal. 1997. Vol. 28, No. 4. P. 944-970.
Arneodo A., Audit B., Kestener P. and Roux S. Multifractal Formalism based on the Continuous Wavelet Transform // Scholarpedia. 2007, Vol. 3, P. 1-20.
Arneodo A., Grasseau G., and Holschneider M. Wavelet transform of multifractals // Phys. Rev. Lett. 1988. Vol. 61. P. 2281 – 2284.
Mallat S. A Wavelet Tour of Signal Processing. San Diego, CA: Academic Press, 1998.
Muzy J. F., Bacry E., Arneodo A. Wavelets and multifractal formalism for singular signals: Application to turbulence data // Physical Review Letters. 1991. Vol. 67, No. 25. P. 3515–3518.
Arneodo A., Bacry E., and Muzy J. F. The thermodynamics of fractals revisited with wavelets // Physica A. 1995. Vol. 213. P. 232–275.
Muzy J.-F., Bacry E., Arnéodo A. Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method // Physical Review E, American Physical Society (APS). 1993. Vol. 47, No. 2. P. 875 – 884.
Weiss B., Clemens Z., Bódizs R., Vágó Z., Halász P. Spatio-temporal analysis of mon-ofractal and multifractal properties of the human sleep EEG // Journal of Neuroscience Methods. 2009. Vol. 185. P. 116–124.
Ihlen E. A. F. Introduction to multifractal detrended fluctuation analysis in Matlab // Frontiers in Physiology. June 2012, Vol. 3, Article 141.
Telesca L., Lapenna V., Macchiato M. Mono- and multi-fractal investigation of scaling properties in temporal patterns of seismic sequences // Chaos, Solitons and Fractals. 2004. Vol. 19. P. 1–15.
Ge E., Leung Y. Detection of crossover time scales in multifractal detrended fluctuation analysis // Journal of Geographical Systems. 2012. Vol. 15, No. 2. P. 115 – 147.
Sarlis N. V., Skordas E. S., Mintzelas A., Papadopoulou K. A. Micro-scale, mid-scale, and macroscale in global seismicity identified by empirical mode decomposition and their multifractal characteristics // Scientific Reports. 2018. Vol. 8. P. 9206.
Astanin LY, Kostylev A A. Ultrawideband Radar Measurements: Analysis and Processing. London : The Institute of Electrical Engineers, 1997.
Feder J. Fractals. New York and London : Springer, 1988. 284 p.
Turcotte D. L. Fractals and Chaos in Geology and Geophysics. Cambridge: Cambridge University Press, 1997. 398 p.
Jaffard S. Multifractal Formalism for Functions. Part I: Results Valid for All Functions // SIAM J. Math. Anal. 1997. Vol. 28, No. 4. P. 944-970.
Лазоренко О. В., Онищенко А. А., Чорногор Л. Ф. Метод коригуючої функції для фрактального аналізу // Радіотехніка. 2022. Вип. 210. С. 177 – 187.
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