A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, natural luminosity time series, and real-world scenes. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models, as well as the geometric Tweedie models. The first convergence effect yields monofractal sequences, and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences. Multifractal analysis is used to investigate datasets, often in conjunction with other methods of fractal and lacunarity analysis. The technique entails distorting datasets extracted from patterns to generate multifractal spectra that illustrate how scaling varies over the dataset. Multifractal analysis has been used to decipher the generating rules and functionalities of complex networks. Multifractal analysis techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images.
Definition In a multifractal system s {\displaystyle s} , the behavior around any point is described by a local power law:
s ( x → + a → ) − s ( x → ) ∼ a h ( x → ) . {\displaystyle s({\vec {x}}+{\vec {a}})-s({\vec {x}})\sim a^{h({\vec {x}})}.}
The exponent h ( x → ) {\displaystyle h({\vec {x}})} is called the singularity exponent, as it describes the local degree of singularity or regularity around the point x → {\displaystyle {\vec {x}}} . The ensemble formed by all the points that share the same singularity exponent is called the singularity manifold of exponent h, and is a fractal set of fractal dimension D ( h ) : {\displaystyle D(h):} the singularity spectrum. The curve D ( h ) {\displaystyle D(h)} versus h {\displaystyle h} is called the singularity spectrum and fully describes the statistical distribution of the variable s {\displaystyle s} . In practice, the multifractal behaviour of a physical system X {\displaystyle X} is not directly characterized by its singularity spectrum D ( h ) {\displaystyle D(h)} . Rather, data analysis gives access to the multiscaling exponents ζ ( q ) , q ∈ R {\displaystyle \zeta (q),\ q\in {\mathbb {R} }} . Indeed, multifractal signals generally obey a scale invariance property that yields power-law behaviours for multiresolution quantities, depending on their scale a {\displaystyle a} . Depending on the object under study, these multiresolution quantities, denoted by T X ( a ) {\displaystyle T_{X}(a)} , can be local averages in boxes of size a {\displaystyle a} , gradients over distance a {\displaystyle a} , wavelet coefficients at scale a {\displaystyle a} , etc. For multifractal objects, one usually observes a global power-law scaling of the form:
⟨ T X ( a ) q ⟩ ∼ a ζ ( q ) {\displaystyle \langle T_{X}(a)^{q}\rangle \sim a^{\zeta (q)}\ }
at least in some range of scales and for some range of orders q {\displaystyle q} . When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling.
Estimation
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