Construction of a Simple Multifractal Spectrum as an Alternative to the Legendre Spectra in Multifractal Formalisms
Keywords:monofractal, multifractal statistical observations, power-law
It is important to realize at the beginning of a statistical analysis whether the data are from a monofractal or multifractal distribution because the methods of analysis are different for each. In seismic sequence analysis, for instance, the monofractal method uses the R/S and DFA (range-scale and detrended fluctuation analysis, respectively) techniques while the multifractal formalism uses the partition function technique (PFT) and the Legendre spectra outputting three parameters: maximum Â of the spectrum, asymmetry B and width W of the curve (Lapenna et al. (2003)). In this paper, we introduce a simple test of mono or multifractality of data sets. The test is based on fitting a power-law distribution to a random sample obtained from some unknown distribution G(.). For each quantile, a fractal dimension Â is obtained. This corresponds to the Legendre spectra or multifractal spectra. A regression function is fitted to the points (tk, Â ) and the slope b of this line is tested. If b = 0, then the observations are deduced to have come from a monofractal distribution f(x). The paper proposed a test for monofractality which, in effect, also tests for multifractality or non-fractality of a set of observations. For monofractal observations, the proposed new multifractal spectral analysis revealed a single point (singularity at a point) while for multifractal observations, a single-humped continuous quadratic function is observed. The parameters of the quadratic function are interpreted as the measure of asymmetry (B), ruggedness (C) and width (W). The new proposed multifractal spectrum function is easier to calculate and is consistent with the more complicated Legendre spectrum proposed in the literature.
Bittner HR, Tosi P. Braun C, Maeesman M, Kniffki KD. Counting statistics of fluctuations: a new method for analysis of earthquake data. Geol
Boschi E, Gasperini P, Mulargia F. Forecasting where larger crustal earthquakes are likely to occur in Italy in the near future. Bull Seism Soc Am 1995;85:1475-82.
Bruno R, BAvassano B, Pietropaolo E, Carbone V, Veltri P. Effects of intermittency on interplanetary velocity and magnetic field fluctuations anisptropy. Geophys Res Lett 1999;26:3185-8.
Carlson JM, Langer JS. Properties of earthquakes generated by fault dynamics. Phys Rev Lett 1989;62:2632-6.
Chen K, Bak P, Obukhov SP. Self â€“ organized critically in a crack â€“ propagation model of earthquakes. Phys Rev A 1991;43:625-30.
Corrieg AM, Urquizu M, Vila J, Manrubia S. Analysis of the temporal occurrence of seismicity at Deception Island (Antartica). A nonlinear approach. Pageoph 1997;149:553-74.
Cox DR. Isham V. Point Processes. London: Chapman and Hall; 1980.
Davis A, Marshak A, Wiscombe W. Wavelet â€“ based Multifractal analysis of nonstationary and/or intermittent geophysical signals. In: Foufoula â€“
Geogiou E, Kumar P, editors. Wavelet in geophysics. New York: Academic Press; 1994. P. 249-98.
Davis A, Marshack A, Wiscombe W, Cahalan R. Multifractal characterization of nonstationary and intermittency in geophysical fields:
observed, retrieved or simulated. J Geophys Res 1994;99:8055-72.
Diego JM, Martinez â€“Gonazales E, Sanz JL, Mollerach S, Mart VJ. Partition function based analysis of cosmic microwave background maps. Mon Not R Astron Soc 1999;306:427-36.
Gutenberg B, Richter CF. Frequency of earthquakes in California. Bull Seism Soc Am 1944;34:185-8.
Hurst HE. Long â€“ term capacity of reservoirs.Trans Am Soc Civ Eng 1951;116:770-808.
Ito K, Matsuzaki M. Earthquakes as a Self â€“ organized critical phenomena. J Geophys Res 1990;95:6853-60.
Johnson, R. and W. Wichern (2000) Applied Multivariate Statistics (Wiley Series: New York)
Kagan YY. Observation evidence for earthquakes as a nonlinear dynamics process.Physica D 1994;77:160 â€“ 92.
Kagan YY, Jackson DD. Long â€“ term earthquake clustering. Geophys J Int 1991;104:117-33.
Kagan YY, Knopoff L. Statistical short â€“ term earthquakes prediction. Science 1987;236:1563.
Kantelhardt JW, Koscienly â€“ Bunde E, Rego HNA, Havlin S, Bunde A. Detecting long-range correlations with detrended fluactuation analysis. Physica A 2001;295:441-54.
Lapenna V, Macchiato M, Telesca L. 1/f^(-Î²) Fluctuations and self-similarity in earthquakes dynamics: observational evidences in southern Italy. Phys Earth Planet INt 1998;106:115-27.
Mandelbrot BB. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J Fluid Mech 1974;62:331-58.
Meneveau C, Screenivasan KR. The Multifractal nature of turbulent energy dissipation. J Fluid Mech 1991;224:429-84.
Padua, R., Adanza, Joel G., Mirasol, Joy M. (2013) â€œFractal Statistical Inferenceâ€ ( The Threshold, CHED-JAS Category A Journal, pp. 36- 44).
Peng C-K, Halvin S, Stanley HE, Goldberger Al. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series CHAOS 1995;5:82-7.
Schertzer D, Lovejoy S, Schmitt F, Chigirinsya Y, Marsan D. Multifractal cascade dynamics and turbulent intermittency. Fractals 1998;5:427-71.
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