Some Results on Multifractal Spectral Analysis
DOI:
https://doi.org/10.32871/rmrj1301.02.07Keywords:
multifractal, monofractal, λ(s)- spectrum, Legendre spectrumAbstract
A multifractal spectrum, based on an earlier paper and different from the Legendre multifractal spectrum (Padua et al. 2013) was examined in this paper. The examination yielded interesting results which enhanced the utility of the developed λ(s)-multifractal spectrum in analyzing real data. One of the results show that a mixture of several monofractal observations can be represented as a single monofractal distribution but whose spectrum is different from the spectrum of the original data. Thus, high fractal dimensional distributions can be infinitely decomposed into component monofractal dimensions. Further, we also show that given a multifractal set of observations, observations that fall on smaller scales obey a normal distribution. The study ends by providing possible avenues for future research particularly in the area of analytic number theory in relation to the Riemann hypothesis about the distribution of primes.
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Schertzer D, Lovejoy S, Schmitt F, Chigirinsya Y, Marsan D. Multifractal cascade dynamics and turbulent intermittency. Fractals 1998;5:427-71.
Bittner HR, Tosi P. Braun C, Maeesman M, Kniffki KD. Counting statistics of f^(-b) fluctuations: a new method for analysis of earthquake data. Geol
Rundsch 1996;85:110-5.
Boschi E, Gasperini P, Mulargia F. Forecasting where larger crustal earthquakes arelikely 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 – organized critically in a crack – 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 – based Multifractal analysis of nonstationary and/or intermittent geophysical signals. In: Foufoula –
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 –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 – term capacity of reservoirs.Trans Am Soc Civ Eng 1951;116:770-808.
Ito K, Matsuzaki M. Earthquakes as a Self – 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 – 92.
Kagan YY, Jackson DD. Long – term earthquake clustering. Geophys J Int 1991;104:117-33.
Kagan YY, Knopoff L. Statistical short – term earthquakes prediction. Science 1987;236:1563.
Kantelhardt JW, Koscienly – 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^β 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) “Fractal Statistical Inference†( The Threshold, CHED-JAS Category A Journal, pp. 36- 44).
Padua, R. , Barabat, Edwin , Regalado, Dionesel Y. (2013) “ A Simple Test for Mono and Multifractality of Statistical Observations†(NORSU Prism Journal of Higher Education, CHED-JAS Category B, pp. 23- 24).
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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Published
2013-12-31
How to Cite
Padua, R. N., Barabat, E. O., Borres, M. S., & Salazar, R. K. (2013). Some Results on Multifractal Spectral Analysis. Recoletos Multidisciplinary Research Journal, 1(2). https://doi.org/10.32871/rmrj1301.02.07
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