Application of the microcanonical multifractal formalism to monofractal systems

Abstract

The design of appropriate multifractal analysis algorithms, able to correctly characterize the scaling properties of multifractal systems from experimental, discretized data, is a major challenge in the study of such scale invariant systems. In the recent years, a growing interest for the application of the microcanonical formalism has taken place, as it allows a precise localization of the fractal components as well as a statistical characterization of the system. In this paper, we deal with the specific problems arising when systems that are strictly monofractal are analyzed using some standard microcanonical multifractal methods. We discuss the adaptations of these methods needed to give an appropriate treatment of monofractal systems.

Empirical evidences of a common multifractal signature in economic, biological and physical systems

“…Let us formalize the requirements on a real signal in order to accept that MMF holds for it: These conditions and what they imply are discussed in detail in Ref. [42,27,43]. For the context of this paper, we will just verify the extent of validity of conditions (i), (ii) and (iii) for the datasets we have analyzed.…”

“…Gradients are dimensional and scale-dependent: they are estimated at a given resolution and depend on the local amplitude of the signal. We need a different approach to invariantly analyze the multifractal signals and the key tool to carry out this analysis is MMF [41,9,42,27,43].…”

“…Although the roots of MMF can be tracked in the works by Chhabra et al almost twenty years ago [41], they have not been set to efficiently process real signals up to recently [9,42,27,43]. In MMF, signal processing is geometrical instead of statistical, and the goal is to obtain the local scaling exponents.…”

An Optimized Algorithm for the Evaluation of Local Singularity Exponents in Digital Signals

“…But as we are mainly interested in the most singular structures, it is hence harmful to our interests to project across multiple scales. Instead, we will make use of point estimates [29,15] of the singularity exponents, namely:…”

“…This new approach to singularity analysis is in the basis of the so-called Microcanonical Multiscale Formalism (MMF) [32], which has been shown to be useful to assess physical properties of turbulent flows and other multiscale systems [24,6,31,32,28,15,17,16]. It has been hence demonstrated that obtaining the singularity exponents at each point of a signal can reveal many useful information not only about the image, but also about the physical processes giving raise to it.…”