Amultifractal system is a generalization of afractal system in which a single exponent (thefractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-calledsingularity spectrum) is needed.^[1]

Multifractal systems are common in nature. They include thelength 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 influid 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 thecentral limit theorem that have as foci of convergence the family of statistical distributions known as theTweedie exponential dispersion models,^[14] as well as the geometric Tweedie models.^[15] The first convergence effect yields monofractal sequences, and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences.^[16]

Multifractal analysis is used to investigate datasets, often in conjunction with other methods offractal andlacunarity 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.^[17]Multifractal analysis techniques have been applied in a variety of practical situations, such as predicting earthquakes and interpreting medical images.^[18][19][20]

In a multifractal system${\displaystyle s}$, the behavior around any point is described by a localpower law:

${\displaystyle s(\overrightarrow{x}+\overrightarrow{a})-s(\overrightarrow{x})\sim a^{h(\overrightarrow{x})}.}$

The exponent${\displaystyle h(\overrightarrow{x})}$ is called thesingularity exponent, as it describes the local degree ofsingularity or regularity around the point${\displaystyle \overrightarrow{x}}$.^[21]

The ensemble formed by all the points that share the same singularity exponent is called thesingularity manifold of exponent h, and is afractal set offractal dimension${\displaystyle D(h):}$ the singularity spectrum. The curve${\displaystyle D(h)}$ versus${\displaystyle h}$ is called thesingularity spectrum and fully describes the statistical distribution of the variable${\displaystyle s}$.^{[citation needed]}

In practice, the multifractal behaviour of a physical system${\displaystyle X}$ is not directly characterized by its singularity spectrum${\displaystyle D(h)}$. Rather, data analysis gives access to themultiscaling exponents${\displaystyle \zeta (q),q\in \mathbb{R}}$. Indeed, multifractal signals generally obey ascale invariance property that yields power-law behaviours for multiresolution quantities, depending on their scale${\displaystyle a}$. Depending on the object under study, these multiresolution quantities, denoted by${\displaystyle T_X(a)}$, can be local averages in boxes of size${\displaystyle a}$, gradients over distance${\displaystyle a}$, wavelet coefficients at scale${\displaystyle a}$, etc. For multifractal objects, one usually observes a global power-law scaling of the form:^{[citation needed]}

${\displaystyle ⟨T_X(a{)}^q⟩\sim a^{\zeta (q)}}$

at least in some range of scales and for some range of orders${\displaystyle q}$. When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling.^[22]

Using so-calledmultifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum${\displaystyle D(h)}$ and the multi-scaling exponents${\displaystyle \zeta (q)}$ through aLegendre transform. While the determination of${\displaystyle D(h)}$ calls for some exhaustive local analysis of the data, which would result in difficult and numerically unstable calculations, the estimation of the${\displaystyle \zeta (q)}$ relies on the use of statistical averages and linear regressions in log-log diagrams. Once the${\displaystyle \zeta (q)}$ are known, one can deduce an estimate of${\displaystyle D(h),}$ thanks to a simple Legendre transform.^{[citation needed]}

Multifractal systems are often modeled by stochastic processes such asmultiplicative cascades. The${\displaystyle \zeta (q)}$ are statistically interpreted, as they characterize the evolution of the distributions of the${\displaystyle T_X(a)}$ as${\displaystyle a}$ goes from larger to smaller scales. This evolution is often calledstatistical intermittency and betrays a departure fromGaussian models.^{[citation needed]}

Modelling as amultiplicative cascade also leads to estimation of multifractal properties.^[23] 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.^[24]

Multifractal spectra can be determined frombox 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.^[25][26][27] The chief idea is that for multifractals, the probability${\displaystyle P}$ of a number of pixels${\displaystyle m}$, appearing in a box${\displaystyle i}$, varies as box size${\displaystyle ϵ}$, to some exponent${\displaystyle \alpha }$, which changes over the image, as inEq.0.0 (NB: For monofractals, in contrast, the exponent does not change meaningfully over the set).${\displaystyle P}$ is calculated from the box-counting pixel distribution as inEq.2.0.

${\displaystyle P_{[i,ϵ]}\propto {ϵ}^{-{\alpha }_i}\therefore {\alpha }_i\propto \frac{\log P_{[i,ϵ]}}{\log {ϵ}^{-1}}}$

Eq.0.0

${\displaystyle ϵ}$ = an arbitrary scale (box size in box counting) at which the set is examined

${\displaystyle i}$ = the index for each box laid over the set for an${\displaystyle ϵ}$

${\displaystyle m_{[i,ϵ]}}$ = the number of pixels ormass in any box,${\displaystyle i}$, at size${\displaystyle ϵ}$

${\displaystyle N_{ϵ}}$ = the total boxes that contained more than 0 pixels, for each${\displaystyle ϵ}$

${\displaystyle M_{ϵ}=\sum _{i=1}^{N_{ϵ}}{m}_{[i,ϵ]}=}$ the total mass or sum of pixels in all boxes for this${\displaystyle ϵ}$

Eq.1.0

${\displaystyle P_{[i,ϵ]}=\frac{m_{[i,ϵ]}}{M_{ϵ}}=}$ the probability of this mass at${\displaystyle i}$ relative to the total mass for a box size

Eq.2.0

${\displaystyle P}$ is used to observe how the pixel distribution behaves when distorted in certain ways as inEq.3.0 andEq.3.1:

${\displaystyle Q}$ = an arbitrary range of values to use as exponents for distorting the data set

${\displaystyle I_{{(Q)}_{[ϵ]}}=\sum _{i=1}^{N_{ϵ}}{P}_{[i,ϵ]}^Q=}$ the sum of all mass probabilities distorted by being raised to this Q, for this box size

Eq.3.0

• When${\displaystyle Q=1}$,Eq.3.0 equals 1, the usual sum of all probabilities, and when${\displaystyle Q=0}$, every term is equal to 1, so the sum is equal to the number of boxes counted,${\displaystyle N_{ϵ}}$.

${\displaystyle {\mu }_{{(Q)}_{[i,ϵ]}}=\frac{P_{[i,ϵ]}^Q}{I_{{(Q)}_{[ϵ]}}}=}$ how the distorted mass probability at a box compares to the distorted sum over all boxes at this box size

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${\displaystyle ϵ}$-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:

• An important feature ofEq.3.0 is that it can also be seen to vary according to scale raised to the exponent${\displaystyle \tau }$ inEq.4.0:

${\displaystyle I_{{(Q)}_{[ϵ]}}\propto {ϵ}^{{\tau }_{(Q)}}}$

Eq.4.0

Thus, a series of values for${\displaystyle {\tau }_{(Q)}}$ can be found from the slopes of the regression line for the log ofEq.3.0 versus the log of${\displaystyle ϵ}$ for each${\displaystyle Q}$, based onEq.4.1:

${\displaystyle {\tau }_{(Q)}=\underset{ϵ\to 0}{lim}\left[\frac{\log I_{{(Q)}_{[ϵ]}}}{\log ϵ}\right]}$

Eq.4.1

• For the generalized dimension:

${\displaystyle D_{(Q)}=\underset{ϵ\to 0}{lim}\left[\frac{\log I_{{(Q)}_{[ϵ]}}}{\log {ϵ}^{-1}}\right](1-Q{)}^{-1}}$

Eq.5.0

${\displaystyle D_{(Q)}=\frac{{\tau }_{(Q)}}{Q-1}}$

Eq.5.1

${\displaystyle {\tau }_{{(Q)}_{}}=D_{(Q)}\left(Q-1\right)}$

Eq.5.2

${\displaystyle {\tau }_{(Q)}={\alpha }_{(Q)}Q-f_{\left({\alpha }_{(Q)}\right)}}$

Eq.5.3

• ${\displaystyle {\alpha }_{(Q)}}$ is estimated as the slope of the regression line forlog A_{${\displaystyle ϵ}$,Q} versuslog${\displaystyle ϵ}$ where:

${\displaystyle A_{ϵ,Q}=\sum _{i=1}^{N_{ϵ}}{\mu }_{{i,ϵ}_Q}{P}_{{i,ϵ}_Q}}$

Eq.6.0

• Then${\displaystyle f_{\left({\alpha }_{(Q)}\right)}}$ is found fromEq.5.3.

• The mean${\displaystyle {\tau }_{(Q)}}$ is estimated as the slope of the log-log regression line for${\displaystyle {\tau }_{{(Q)}_{[ϵ]}}}$ versus${\displaystyle ϵ}$, where:

${\displaystyle {\tau }_{(Q{)}_{[ϵ]}}=\frac{\sum _{i=1}^{N_{ϵ}}{P}_{[i,ϵ]}^{Q-1}}{N_{ϵ}}}$

Eq.6.1

In practice, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling.^[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.^[31]

Multifractal analysis has been used in several scientific fields to characterize various types of datasets.^[32][5][8] 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 asmultifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in theillustration.^[25] Several types of multifractal spectra are used in practise.

One practical multifractal spectrum is the graph of D_Q vs Q, where D_Q is thegeneralized dimension for a dataset and Q is an arbitrary set of exponents. The expressiongeneralized dimension thus refers to a set of dimensions for a dataset (detailed calculations for determining the generalized dimension usingbox counting are describedbelow).

The general pattern of the graph of D_Q 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 thefigure, 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 thecapacity dimension, which—in the analysis shown in the figures here—is thebox counting dimension. D_(Q=1) is equal to theinformation dimension, and D_(Q=2) to thecorrelation 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.^[25][26]

Another useful multifractal spectrum is the graph of${\displaystyle f(\alpha )}$ versus${\displaystyle \alpha }$ (seecalculations). These graphs generally rise to a maximum that approximates thefractal dimension at Q=0, and then fall. Like D_Q 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.

One application of D_q versus Q in ecology is characterizing the distribution of species. Traditionally therelative 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 theneutral theory of biodiversity,metacommunity dynamics, orniche theory.^[33][34]

• de Rham curve – Continuous fractal curve obtained as the image of Cantor space
• Fractional Brownian motion – Probability theory concept
• Detrended fluctuation analysis – Method to detect power-law scaling in time series
• Tweedie distributions – Family of probability distributionsPages displaying short descriptions of redirect targets
• Markov switching multifractal – model of asset returnsPages displaying wikidata descriptions as a fallback
• Weighted planar stochastic lattice – mathematical structure sharing some of the properties both of lattices and of graphsPages displaying wikidata descriptions as a fallback

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• Veneziano, Daniele; Essiam, Albert K. (June 1, 2003)."Flow through porous media with multifractal hydraulic conductivity".Water Resources Research.39 (6): 1166.Bibcode:2003WRR....39.1166V.doi:10.1029/2001WR001018.ISSN 1944-7973.
• Movies of visualizations of multifractals

• Fractals
• Dimension theory

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