1. Definition

2. Estimation

3. Practical application of multifractal spectra
     D_Q vs Q    Dimensional ordering    vs    Generalized dimensions of species abundance distributions in space  

4. Estimating multifractal scaling from box counting

5. Applications

6. See also

7. References

8. External links

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.^[1]

Multifractal systems are common in nature. They include the length of coastlines, fully developed turbulence, real world scenes, the Sun's magnetic field time series{{Citation needed|date=November 2018}}, heartbeat dynamics,^[2] human gait{{Citation needed|date=November 2018}} and activity,^[3] human brain activity,^{[4][5][6][7][8][9][10]} and natural luminosity time series{{Citation needed|date=November 2018}}. 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|date=November 2018}}. 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^[11] as well as the geometric Tweedie models.^[12] The first convergence effect yields monofractal sequences and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences.^[13]

Multifractal analysis uses mathematical basis to investigate datasets, often in conjunction with other methods of fractal analysis 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.^[14][15][16]

Definition

In a multifractal system , the behavior around any point is described by a local power law:

The exponent is called the singularity exponent, as it describes the local degree of singularity or regularity around the point .{{Citation needed|date=November 2018}}

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 : the singularity spectrum. The curve versus h is called the singularity spectrum and fully describes the (statistical) distribution of the variable .{{Citation needed|date=November 2018}}

In practice, the multifractal behaviour of a physical system is not directly characterized by its singularity spectrum . Data analysis rather gives access to the multiscaling exponents . Indeed, multifractal signals generally obey a scale invariance property which yields power law behaviours for multiresolution quantities depending on their scale . Depending on the object under study, these multiresolution quantities, denoted by in the following, can be local averages in boxes of size , gradients over distance , wavelet coefficients at scale ... For multifractal objects, one usually observes a global power law scaling of the form:{{Citation needed|date=November 2018}}

at least in some range of scales and for some range of orders . When such a behaviour is observed, one talks of scale invariance, self-similarity or multiscaling.^[17]

Estimation

Using the so-called multifractal formalism, it can be shown that, under some well-suited assumptions, there exists a correspondence between the singularity spectrum and the multi scaling exponents through a Legendre transform. While the determination of 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. Once the are known, one can deduce an estimate of thanks to a simple Legendre transform.{{Citation needed|date=November 2018}}

Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The receives some statistical interpretation as they characterize the evolution of the distributions of the as goes from larger to smaller scales. This evolution is often called statistical intermittency and betrays a departure from Gaussian models.{{Citation needed|date=November 2018}}

Modelling as a multiplicative cascade also leads to estimation of multifractal properties ({{harvnb|Roberts|Cronin|1996}}). This methods works reasonably well even for relatively small datasets A maximum likelihood fit of a multiplicative cascade to the dataset not only estimates the complete spectrum, but also gives reasonable estimates of the errors (see the web service  ).

Practical application of multifractal spectra

D_Q vs Q

Dimensional ordering

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 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 offers some 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 whereby multifractals have multiple dimensions in the D_(Q) vs Q spectra but monofractals stay rather flat in that area.^[21][23]

vs

Another useful multifractal spectrum is the graph of vs (see calculations). These graphs generally rise to a maximum that approximates the fractal dimension at Q=0, and then fall. Like D_Q vs 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 are typically humped over a broader extent.

Generalized dimensions of species abundance distributions in space

One application of D_q vs q in ecology is the characterization of the abundance distribution of species. Traditionally the relative species abundances is calculated for an area without taking into account the positions of the individuals. An equivalent representation of relative species abundances are species ranks, used to generate a surface called the species-rank surface,^[19] which can be analyzed using generalized dimensions to detect different ecological mechanisms like the ones observed in neutral theory of biodiversity, metacommunity dynamics or niche theory.^[19][20]

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.^[21][22][23] The chief idea is that for multifractals, the probability of a number of pixels , appearing in a box , varies as box size , to some exponent , which changes over the image, as in {{EquationNote|Eq.0.0}}. NB: For monofractals, in contrast, the exponent does not change meaningfully over the set. is calculated from the box counting pixel distribution as in {{EquationNote|Eq.2.0}}.

= an arbitrary scale (box size in box counting) at which the set is examined

= the index for each box laid over the set for an

= the number of pixels or mass in any box, , at size

= the total boxes that contained more than 0 pixels, for each

is used to observe how the pixel distribution behaves when distorted in certain ways as in {{EquationNote|Eq.3.0}} and {{EquationNote|Eq.3.1}}:

= an arbitrary range of values to use as exponents for distorting the data set

• When , {{EquationNote|Eq.3.0}} equals 1, the usual sum of all probabilities, and when , every term is equal to 1, so the sum is equal to the number of boxes counted, .

These distorting equations are further used to address how the set behaves when scaled or resolved or cut up into a series of -sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:

• An important feature of {{EquationNote|Eq.3.0}} is that it can also be seen to vary according to scale raised to the exponent in {{EquationNote|Eq.4.0}}:

Thus, a series of values for can be found from the slopes of the regression line for the log of {{EquationNote|Eq.3.0}} versus the log of for each , based on {{EquationNote|Eq.4.1}}:

• {{anchor|generalized dimension}}For the generalized dimension:

• is estimated as the slope of the regression line for {{math|log A_,Q}} versus {{math|log }} where:

• Then is found from {{EquationNote|Eq.5.3}}.

• The mean is estimated as the slope of the log-log regression line for versus , where:

In practice, the probability distribution depends on how the dataset is sampled, so optimizing algorithms have been developed to ensure adequate sampling.^[21]

Applications

Multifractal analysis has been successfully used in many fields including biology and physical sciences.^[24] For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls.^[25]

See also

• Multifractal Model of Asset Returns (MMAR)
• Multifractal Random Walk model (MRW)
• Fractional Brownian motion
• Mandelbrot cascade, continuous cascade and lognormal cascade
• Detrended fluctuation analysis
• Tweedie distributions
• Markov switching multifractal
• Weighted planar stochastic lattice (WPSL) ^[26]

References

{{cite journal |title=Flow through porous media with multifractal hydraulic conductivity |date=June 1, 2003 |issn=1944-7973 |doi=10.1029/2001WR001018 |issue=6 |pages = 1166|volume=39 |journal=Water Resources Research |first1=Daniele |last1=Veneziano |first2=Albert K. |last2=Essiam |bibcode=2003WRR....39.1166V}}

External links

• {{cite journal |author=Stanley H.E., Meakin P. |title=Multifractal phenomena in physics and chemistry |journal=Nature |volume=335 |year=1988 |pages=405–9 |url=http://polymer.bu.edu/hes/articles/sm88.pdf |format=Review |doi=10.1038/335405a0 |issue=6189|bibcode=1988Natur.335..405S }}
• {{cite journal |first1=Alain |last1=Arneodo |first2=Benjamin |last2=Audit |first3=Pierre |last3=Kestener |first4=Stephane |last4=Roux |title=Wavelet-based multifractal analysis |journal=Scholarpedia |volume=3 |issue=3 |pages=4103 |year=2008 |url=http://www.scholarpedia.org/article/Wavelet-based_multifractal_analysis |doi=10.4249/scholarpedia.4103|issn=1941-6016|bibcode=2008SchpJ...3.4103A }}
• Movies of visualizations of multifractals

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