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A multifractal system is a generalization of a
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
system in which a single exponent (the
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called
singularity spectrum The singularity spectrum is a function used in Multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same Hölder exponent. Intuitively, the singularity spectrum ...
) is needed. Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, real-world scenes,
heartbeat A heartbeat is one cardiac cycle of the heart. Heartbeat, heart beat, heartbeats, and heart beats may refer to: Computing *Heartbeat (computing), a periodic signal to indicate normal operation or to synchronize parts of a system *Heartbeat, clus ...
dynamics,
human gait A gait is a pattern of limb movements made during locomotion. Human gaits are the various ways in which humans can move, either naturally or as a result of specialized training. Human gait is defined as bipedal, biphasic forward propulsion of ...
and activity,
human brain The human brain is the central organ of the human nervous system, and with the spinal cord makes up the central nervous system. The brain consists of the cerebrum, the brainstem and the cerebellum. It controls most of the activities of the ...
activity, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
to internet traffic, finance, image modeling, texture synthesis, meteorology,
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
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 In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
and
lacunarity Lacunarity, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. B ...
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, the behavior around any point is described by a local
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
: :s(\vec+\vec)-s(\vec) \sim a^. The exponent h(\vec) is called the
singularity exponent Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
, as it describes the local degree of singularity or regularity around the point \vec. 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 In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
of
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
D(h): the singularity spectrum. The curve D(h) versus h is called the ''singularity spectrum'' and fully describes the statistical distribution of the variable s. In practice, the multifractal behaviour of a physical system X is not directly characterized by its singularity spectrum D(h). Rather, data analysis gives access to the ''multiscaling exponents'' \zeta(q),\ q\in. Indeed, multifractal signals generally obey a ''scale invariance'' property that yields power-law behaviours for multiresolution quantities, depending on their scale a. Depending on the object under study, these multiresolution quantities, denoted by T_X(a), can be local averages in boxes of size a, gradients over distance a, wavelet coefficients at scale a, etc. For multifractal objects, one usually observes a global power-law scaling of the form: :\langle T_X(a)^q \rangle \sim a^\ at least in some range of scales and for some range of orders q. 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 D(h) and the multi-scaling exponents \zeta(q) through a
Legendre transform In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
. While the determination of D(h) calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the \zeta(q) relies on the use of statistical averages and linear regressions in log-log diagrams. Once the \zeta(q) are known, one can deduce an estimate of D(h), thanks to a simple Legendre transform. Multifractal systems are often modeled by stochastic processes such as
multiplicative cascade In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. Definition The plots above are examples of multiplicative cascade multifractals. To create thes ...
s. The \zeta(q) are statistically interpreted, as they characterize the evolution of the distributions of the T_X(a) as a goes from larger to smaller scales. This evolution is often called ''statistical intermittency'' and betrays a departure from
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
models. Modelling as a
multiplicative cascade In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. Definition The plots above are examples of multiplicative cascade multifractals. To create thes ...
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 Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the pro ...
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 P of a number of pixels m, appearing in a box i, varies as box size \epsilon, to some exponent \alpha, which changes over the image, as in ( NB: For monofractals, in contrast, the exponent does not change meaningfully over the set). P is calculated from the box-counting pixel distribution as in . :\epsilon = an arbitrary scale ( box size in box counting) at which the set is examined :i = the index for each box laid over the set for an \epsilon :m_ = the number of pixels or ''mass'' in any box, i, at size \epsilon :N_\epsilon = the total boxes that contained more than 0 pixels, for each \epsilon P is used to observe how the pixel distribution behaves when distorted in certain ways as in and : :Q = an arbitrary range of values to use as exponents for distorting the data set :*When Q=1, equals 1, the usual sum of all probabilities, and when Q=0, every term is equal to 1, so the sum is equal to the number of boxes counted, N_\epsilon. These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of \epsilon-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following: :*An important feature of is that it can also be seen to vary according to scale raised to the exponent \tau in : Thus, a series of values for \tau_ can be found from the slopes of the regression line for the log of versus the log of \epsilon for each Q, based on : :*For the generalized dimension: :*\alpha_ is estimated as the slope of the regression line for versus where: :*Then f_ is found from . :*The mean \tau_ is estimated as the slope of the log-log regression line for \tau_ versus \epsilon, where: 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 An illustration is a decoration, interpretation or visual explanation of a text, concept or process, designed for integration in print and digital published media, such as posters, flyers, magazines, books, teaching materials, animations, vid ...
. 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 Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the pro ...
are described
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
).


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 Figure may refer to: General *A shape, drawing, depiction, or geometric configuration *Figure (wood), wood appearance *Figure (music), distinguished from musical motif *Noise figure, in telecommunication *Dance figure, an elementary dance pattern ...
, 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 A box (plural: boxes) is a container used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides. Boxes can be very small (like a matchbox) or very large (like a shipping box for furniture), and can ...
. D(Q=1) is equal to the
information dimension In information theory, information dimension is an information measure for random vectors in Euclidean space, based on the normalized entropy of finely quantized versions of the random vectors. This concept was first introduced by Alfréd Rényi ...
, and D(Q=2) to the
correlation dimension In chaos theory, the correlation dimension (denoted by ''ν'') is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For example, if we have a set of random points on ...
. 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.


f(α) versus α

Another useful multifractal spectrum is the graph of f(\alpha) versus \alpha (see
calculations A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to th ...
). These graphs generally rise to a maximum that approximates the
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
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 abundance Relative species abundance is a component of biodiversity and is a measure of how common or rare a species is relative to other species in a defined location or community.Hubbell, S. P. 2001. ''The unified neutral theory of biodiversity and biogeogr ...
s 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.


See also

*
Fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gauss ...
*
Detrended fluctuation analysis In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory process ...
*
Tweedie distributions In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the cla ...
*
Markov switching multifractal In financial econometrics (the application of Statistics, statistical methods to economic data), the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent-Emmanuel Calvet, Laurent E. Calvet and Adlai J. Fisher that in ...
*
Weighted planar stochastic lattice (WPSL) Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next ...


References


Further reading

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External links

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Movies of visualizations of multifractals
{{DEFAULTSORT:Multifractal System Fractals Dimension theory