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.
Multifractal systems are common in nature. They include the length of coastlines, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. 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 techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images.
Definition.
In a multifractal system formula_1, the behavior around any point is described by a local power law:
The exponent formula_3 is called the singularity exponent, as it describes the local degree of singularity or regularity around the point formula_4.
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 formula_5 the singularity spectrum. The curve formula_6 versus formula_7 is called the "singularity spectrum" and fully describes the statistical distribution of the variable formula_1.
In practice, the multifractal behaviour of a physical system formula_9 is not directly characterized by its singularity spectrum formula_6. Rather, data analysis gives access to the "multiscaling exponents" formula_11. Indeed, multifractal signals generally obey a "scale invariance" property that yields power-law behaviours for multiresolution quantities, depending on their scale formula_12. Depending on the object under study, these multiresolution quantities, denoted by formula_13, can be local averages in boxes of size formula_12, gradients over distance formula_12, wavelet coefficients at scale formula_12, etc. For multifractal objects, one usually observes a global power-law scaling of the form:
at least in some range of scales and for some range of orders formula_18. When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling.
Estimation.
Using so-called "multifractal formalism", it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum formula_6 and the multi-scaling exponents formula_20 through a Legendre transform. While the determination of formula_6 calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the formula_20 relies on the use of statistical averages and linear regressions in log-log diagrams. Once the formula_20 are known, one can deduce an estimate of formula_24 thanks to a simple Legendre transform.
Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The formula_20 are statistically interpreted, as they characterize the evolution of the distributions of the formula_13 as formula_12 goes from larger to smaller scales. This evolution is often called "statistical intermittency" and betrays a departure from Gaussian models.
Modelling as a multiplicative cascade also leads to estimation of multifractal properties. This methods works reasonably well, even for relatively small datasets. A maximum likely fit of a multiplicative cascade to the dataset not only estimates the complete spectrum but also gives reasonable estimates of the errors.
Estimating multifractal scaling from box counting.
Multifractal spectra can be determined from box counting on digital images. First, a box counting scan is done to determine how the pixels are distributed; then, this "mass distribution" becomes the basis for a series of calculations. The chief idea is that for multifractals, the probability formula_28 of a number of pixels formula_29, appearing in a box formula_30, varies as box size formula_31, to some exponent formula_32, which changes over the image, as in (NB: For monofractals, in contrast, the exponent does not change meaningfully over the set). formula_28 is calculated from the box-counting pixel distribution as in . 
formula_28 is used to observe how the pixel distribution behaves when distorted in certain ways as in and :
These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of formula_31-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:
Thus, a series of values for formula_50 can be found from the slopes of the regression line for the log of versus the log of formula_31 for each formula_44, based on :
In practice, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling.
Applications.
Multifractal analysis has been successfully used in many fields, including physical, information, and biological sciences. For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls.
Dataset distortion analysis.
Multifractal analysis has been used in several scientific fields to characterize various types of datasets. In essence, multifractal analysis applies a distorting factor to datasets extracted from patterns, to compare how the data behave at each distortion. This is done using graphs known as multifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in the illustration. Several types of multifractal spectra are used in practise.
DQ vs Q.
One practical multifractal spectrum is the graph of DQ vs Q, where DQ is the generalized dimension for a dataset and Q is an arbitrary set of exponents. The expression "generalized dimension" thus refers to a set of dimensions for a dataset (detailed calculations for determining the generalized dimension using box counting are described below).
Dimensional ordering.
The general pattern of the graph of DQ vs Q can be used to assess the scaling in a pattern. The graph is generally decreasing, sigmoidal around Q=0, where D(Q=0) ≥ D(Q=1) ≥ D(Q=2). As illustrated in the figure, variation in this graphical spectrum can help distinguish patterns. The image shows D(Q) spectra from a multifractal analysis of binary images of non-, mono-, and multi-fractal sets. As is the case in the sample images, non- and mono-fractals tend to have flatter D(Q) spectra than multifractals.
The generalized dimension also gives important specific information. D(Q=0) is equal to the capacity dimension, which—in the analysis shown in the figures here—is the box counting dimension. D(Q=1) is equal to the information dimension, and D(Q=2) to the correlation dimension. This relates to the "multi" in multifractal, where multifractals have multiple dimensions in the D(Q) versus Q spectra, but monofractals stay rather flat in that area.
formula_58 versus formula_32.
Another useful multifractal spectrum is the graph of formula_58 versus formula_32 (see calculations). These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall. Like DQ versus Q spectra, they also show typical patterns useful for comparing non-, mono-, and multi-fractal patterns. In particular, for these spectra, non- and mono-fractals converge on certain values, whereas the spectra from multifractal patterns typically form humps over a broader area.
Generalized dimensions of species abundance distributions in space.
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, 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.