Construction of a Simple Multifractal Spectrum as an Alternative to the Legendre Spectra in Multifractal Formalisms

Abstract

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  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  is obtained. This corresponds to the Legendre spectra or multifractal spectra. A regression function is fitted to the points (tk,  ) 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.