About the Effectiveness of Different Methods for the Estimation of the Multifractal Spectrum of Natural Series

Abstract

Complex natural systems present characteristics of scalar invariance. This behavior has been experimentally verified and a large related bibliography has been reported. Multifractal Formalism is a way to evaluate this kind of behavior. In the past years, different numerical methods to estimate the multifractal spectrum have been proposed. These methods could be classified into those that originated from the wavelet analysis and others from numerical approximations like the Multifractal Detrended Fluctuation Analysis (MFDFA), proposed by Kantelhardt and Stanley. Recently, S. Jaffard and co-workers proposed the Wavelet Leaders (WL) method that exploits the potential of wavelet analysis and the efficiency of the Multiresolution Wavelet Schema. In a previous work, we checked that both methods are equivalent for estimating fractal properties in a series from singular measures. Now, we apply MFDFA and WL to natural signals with self-similar structures, but unknown multifractal spectrum. We observe that some differences appear in their respective estimations, particularly when the signals are corrupted with fractional Gaussian noise.

Wavelet Leaders Based Vibration Signals Multifractal Features of Plunger Pump in Truck Crane

“…This is attributed to the two spectra having a similar shape. Furthermore, the three of four parameters used to characterize the spectrum [15], which are the minimum and the maximum of the Holder index corresponding to the two spectra, and the range of the Holder index, are similar; as for the fourth parameter, which is the ℎ * of the wavelet leaders with Chhabra regression method that corresponds to the Holder index of the maximum Hausdorff dimension, it is about 0.65 and agrees with the analytical method.…”

“…Furthermore, all the multifractal spectra of the three conditions are unimodal functions, which also verifies that the signals have multifractal characters. But the precise singularities of these three signals are different as to the multifractal parameters [15], though all of the maximum of max (ℎ) are close to 1. From the curve of multifractal spectrum, for the hoisting condition, the Δℎ = ℎ max − ℎ min is about 1.65, which is larger than 0.55 of rotating condition and 0.73 of luffing.…”

Pitfalls in fractal time series analysis: fMRI BOLD as an exemplary case

“…Monofractal analysis methods are quite robust with respect of noise, thus in case of monofractals do not require preprocessing (Bassingthwaighte and Raymond, 1995). When uncorrelated noise is added to a multifractal process, the shape of its singularity spectrum will also be preserved (Figliola et al, 2010). However with correlated noise present, – known to impact fMRI BOLD time series – preprocessing should be considered (Friston et al, 2000), and if carried out, it should be done with an appropriate adaptive filter (Gao et al, 2010, 2011; Tung et al, 2011).…”

Application of multifractal analysis to segmentation of water bodies in optical satellite images

“…Multifractal theory can be seen as an extension of fractal formalism, centered in subsets with different scaling behavior coexisting simultaneously. Since its introduction, multifractal analysis has been developed and applied in a wide range of disciplines, like medicine [6], neuroscience [7], economy [8][9][10], ecology [11] and theoretical physics [12][13][14]. Many methods have been proposed for approximating multifractal spectra.…”