Multifractal system
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.
They include the length of coastlines, mountain topography,[2] fully developed turbulence, real-world scenes, heartbeat dynamics,[3] human gait[4] and activity,[5] human brain activity,[6][7][8][9][10][11][12] and natural luminosity time series.
[13] 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.
[citation needed] 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,[14] as well as the geometric Tweedie models.
Multifractal analysis has been used to decipher the generating rules and functionalities of complex networks.
[17] Multifractal analysis techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images.
, the behavior around any point is described by a local power law: The exponent
[21] 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
is called the singularity spectrum and fully describes the statistical distribution of the variable
[citation needed] In practice, the multifractal behaviour of a physical system
Depending on the object under study, these multiresolution quantities, denoted by
When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling.
Using so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum
calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the
relies on the use of statistical averages and linear regressions in log-log diagrams.
[citation needed] Multifractal systems are often modeled by stochastic processes such as multiplicative cascades.
This evolution is often called statistical intermittency and betrays a departure from Gaussian models.
[citation needed] Modelling as a multiplicative cascade also leads to estimation of multifractal properties.
Multifractal spectra can be determined from box counting on digital images.
is used to observe how the pixel distribution behaves when distorted in certain ways as in Eq.3.0 and Eq.3.1: These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of
[25] Multifractal analysis has been successfully used in many fields, including physical,[28][29] information, and biological sciences.
[30] For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls.
Multifractal analysis has been used in several scientific fields to characterize various types of datasets.
This is done using graphs known as multifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in the illustration.
As illustrated in the figure, variation in this graphical spectrum can help distinguish patterns.
As is the case in the sample images, non- and mono-fractals tend to have flatter D(Q) spectra than multifractals.
These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall.
One application of Dq versus Q in ecology is characterizing the distribution of species.
Traditionally the relative species abundances is calculated for an area without taking into account the locations of the individuals.
An equivalent representation of relative species abundances are species ranks, used to generate a surface called the species-rank surface,[33] which can be analyzed using generalized dimensions to detect different ecological mechanisms like the ones observed in the neutral theory of biodiversity, metacommunity dynamics, or niche theory.
