Multifractal formalism by enforcing the universal behavior of scaling functions

Abstract

Despite its solid foundations, multifractal analysis is still a challenging task. The 'inversed' singularity spectrum is a major pitfall in standard multifractal analyses especially for empirical signals. To resolve this issue, we identified the fan-like convergent geometry of scaling functions yielding a limit value (termed focus) for all moments at the largest scale. Building on this behavior of scaling, we introduce the novel concept of focus-based multifractal formalism. It relies on enforcing this universal behavior when the moment-wise scaling exponents are assessed for the scaling functions. Besides developing focus-based variants of the established multifractal detrended fluctuation analysis and the wavelet leader method, we present a novel analytical tool of multifractal signal summation conversion. All methods are extensively tested on exact multifractal signals synthesized by the generalized binomial multifractal model in terms of precision and incidence of 'inversed' singularity spectra. Our focus-based variants never yielded 'inversed' spectra and their precision was found similar to that of standard methods. Our approach allowed computing a moment-wise and a global error parameter describing the impact of finite size effect and degree of multifractality as compared to that of the fitted exact multifractal model. We demonstrate that the standard approach to multifractal analyses contains a central element that is essentially monofractal due to its regression scheme assessing the scaling exponents for each and every moment, separately. Hence these methods can yield reliable estimates only for ideally behaving multifractal signals. In contrast, our focus-based variants due to their genuine multifractal model fitting always yield reliable estimates accompanied by goodness-of-fit statistics. The presented novel multifractal tools offer means of dealing with the consequences of endogenous impurities of potentially multifractal empirical signals.