multifractal system

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

:$s(\backslash vec\{x\}+\backslash vec\{a\})-s(\backslash vec\{x\})\; \backslash sim\; a^\{h(\backslash vec\{x\})\}.$

The exponent $h(\backslash vec\{x\})$ is called the singularity exponent, as it describes the local degree of singularity or regularity around the point $\backslash vec\{x\}$.{{Cite book |last=Falconer |first=Kenneth J. |title=Fractal geometry: mathematical foundations and applications |date=2014 |publisher=Wiley |isbn=978-1-119-94239-9 |edition=3. ed., 1. publ |location=Chichester |chapter=17. Multifractal measures}}

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

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 $\backslash zeta(q),\backslash \; q\backslash in\{\backslash mathbb\; R\}$. 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:{{Citation needed|date=November 2018}}

:$\backslash langle\; T\_X(a)^q\; \backslash rangle\; \backslash sim\; a^\{\backslash zeta(q)\}\backslash$

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.{{cite journal |author1-first=A.J. |author1-last=Roberts |author2-first= A.|author2-last =Cronin |title=Unbiased estimation of multi-fractal dimensions of finite data sets |journal=Physica A |volume=233 |issue=3 |year=1996 |pages=867–878 |doi=10.1016/S0378-4371(96)00165-3 |arxiv=chao-dyn/9601019 |bibcode=1996PhyA..233..867R |s2cid=14388392 }}

{{anchor|techniques}}

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 $\backslash zeta(q)$ through a Legendre transform. 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 $\backslash zeta(q)$ relies on the use of statistical averages and linear regressions in log-log diagrams. Once the $\backslash zeta(q)$ are known, one can deduce an estimate of $D(h),$ thanks to a simple Legendre transform.{{Citation needed|date=November 2018}}

Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The $\backslash zeta(q)$ 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 models.{{Citation needed|date=November 2018}}

Modelling as a multiplicative cascade also leads to estimation of multifractal properties.{{sfn|Roberts|Cronin|1996}} 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.{{cite web |last=Roberts |first=A. J. |title=Multifractal estimation—maximum likelihood |url=http://www.maths.adelaide.edu.au/anthony.roberts/multifractal.php |date=7 August 2014 |website=University of Adelaide |access-date=4 June 2019}}

{{anchor|calculations}}

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.{{citation |author=Karperien, A |title=What are Multifractals? |publisher=ImageJ |access-date=2012-02-10 |url=http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Multifractals.htm |year=2002 |archive-url=https://web.archive.org/web/20111020190126/http://rsbweb.nih.gov/ij/plugins/fraclac/FLHelp/Multifractals.htm |archive-date=2011-10-20 |url-status=live }}{{Cite journal | last1 = Chhabra | first1 = A. | last2 = Jensen | first2 = R. | doi = 10.1103/PhysRevLett.62.1327 | title = Direct determination of the f(α) singularity spectrum | journal = Physical Review Letters | volume = 62 | issue = 12 | pages = 1327–1330 | year = 1989 | pmid = 10039645|bibcode = 1989PhRvL..62.1327C }}{{Cite journal | last1 = Posadas | first1 = A. N. D. | last2 = Giménez | first2 = D. | last3 = Bittelli | first3 = M. | last4 = Vaz | first4 = C. M. P. | last5 = Flury | first5 = M. | title = Multifractal Characterization of Soil Particle-Size Distributions | doi = 10.2136/sssaj2001.6551361x | journal = Soil Science Society of America Journal | volume = 65 | issue = 5 | pages = 1361 | year = 2001 | bibcode = 2001SSASJ..65.1361P }} 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 $\backslash epsilon$, to some exponent $\backslash alpha$, 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). $P$ is calculated from the box-counting pixel distribution as in {{EquationNote|Eq.2.0}}.

{{NumBlk|:|$P\_\{[i,\backslash epsilon]\}\; \backslash varpropto\; \backslash epsilon^\{-\backslash alpha\_i\}\; \backslash therefore\backslash alpha\_i\; \backslash varpropto\; \backslash frac\{\backslash log\{P\_\{[i,\backslash epsilon]\}\}\}\{\backslash log\{\backslash epsilon^\{-1\}\}\}$|{{EquationRef|Eq.0.0}}}}

:$\backslash epsilon$ = 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 $\backslash epsilon$

:$m\_\{[i,\backslash epsilon]\}$ = the number of pixels or mass in any box, $i$, at size $\backslash epsilon$

:$N\_\backslash epsilon$ = the total boxes that contained more than 0 pixels, for each $\backslash epsilon$

{{NumBlk|:|$M\_\backslash epsilon\; =\; \backslash sum\_\{i=1\}^\{N\_\backslash epsilon\}m\_\{[i,\backslash epsilon]\}\; =$ the total mass or sum of pixels in all boxes for this $\backslash epsilon$|{{EquationRef|Eq.1.0}}}}

{{NumBlk|:|$P\_\{[i,\backslash epsilon]\}\; =\; \backslash frac\{m\_\{[i,\backslash epsilon]\}\}\{M\_\backslash epsilon\}\; =$ the probability of this mass at $i$ relative to the total mass for a box size|{{EquationRef|Eq.2.0}}}}

$P$ 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}}:

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

{{NumBlk|:|$I\_\{\{(Q)\}\_\{[\backslash epsilon]\}\}\; =\; \backslash sum\_\{i=1\}^\{N\_\backslash epsilon\}\; \{P\_\{[i,\backslash epsilon]\}^Q\}\; =$ the sum of all mass probabilities distorted by being raised to this Q, for this box size|{{EquationRef |Eq.3.0}}}}

:*When $Q=1$, {{EquationNote|Eq.3.0}} 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, $N\_\backslash epsilon$.

{{NumBlk|:|$\backslash mu\_\{\{(Q)\}\_\{[i,\backslash epsilon]\}\}\; =\; \backslash frac\{P\_\{[i,\backslash epsilon]\}^Q\}\{I\_\{\{(Q)\}\_\{[\backslash epsilon]\}\}\}\; =$ how the distorted mass probability at a box compares to the distorted sum over all boxes at this box size|{{EquationRef|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 $\backslash epsilon$-sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:

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

{{NumBlk|:|$I\_\{\{(Q)\}\_\{[\backslash epsilon]\}\}\; \backslash varpropto\; \backslash epsilon^\{\backslash tau\_\{(Q)\}\}$|{{EquationRef|Eq.4.0}}}}

Thus, a series of values for $\backslash tau\_\{(Q)\}$ can be found from the slopes of the regression line for the log of {{EquationNote|Eq.3.0}} versus the log of $\backslash epsilon$ for each $Q$, based on {{EquationNote|Eq.4.1}}:

{{NumBlk|:|$\backslash tau\_\{(Q)\}\; =\; \{\backslash lim\_\{\backslash epsilon\backslash to0\}\{\backslash left[\; \backslash frac\; \{\backslash log\{I\_\{\{(Q)\}\_\{[\backslash epsilon]\}\}\}\}\; \{\backslash log\{\backslash epsilon\}\}\; \backslash right\; ]\}\}$|{{EquationRef|Eq.4.1}}}}

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

{{NumBlk|:|$D\_\{(Q)\}\; =\; \{\backslash lim\_\{\backslash epsilon\backslash to0\}\; \{\; \backslash left\; [\; \backslash frac\{\backslash log\{I\_\{\{(Q)\}\_\{[\backslash epsilon]\}\}\}\}\{\backslash log\{\backslash epsilon^\{-1\}\}\}\; \backslash right\; ]\}\}\; \{(1-Q)^\{-1\}\}$|{{EquationRef|Eq.5.0}}}}

{{NumBlk|:|$D\_\{(Q)\}\; =\; \backslash frac\{\backslash tau\_\{(Q)\}\}\{Q-1\}$|{{EquationRef|Eq.5.1}}}}

{{NumBlk|:|$\backslash tau\_\{\{(Q)\}\_\{\}\}\; =\; D\_\{(Q)\}\backslash left(Q-1\backslash right)$|{{EquationRef|Eq.5.2}}}}

{{NumBlk|:|$\backslash tau\_\{(Q)\}\; =\; \backslash alpha\_\{(Q)\}Q\; -\; f\_\{\backslash left(\backslash alpha\_\{(Q)\}\backslash right)\}$|{{EquationRef|Eq.5.3}}}}

:*$\backslash alpha\_\{(Q)\}$ is estimated as the slope of the regression line for {{math|log A_{$\backslash epsilon$,Q}}} versus {{math|log $\backslash epsilon$}} where:

{{NumBlk|:|$A\_\{\backslash epsilon,Q\}\; =\; \backslash sum\_\{i=1\}^\{N\_\backslash epsilon\}\{\backslash mu\_\{\{i,\backslash epsilon\}\_\{Q\}\}\{P\_\{\{i,\backslash epsilon\}\_\{Q\}\}\}\}$|{{EquationRef|Eq.6.0}}}}

:*Then $f\_\{\backslash left(\backslash alpha\_\{\{(Q)\}\}\backslash right)\}$ is found from {{EquationNote|Eq.5.3}}.

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

{{NumBlk|:|$\backslash tau\_\{(Q)\_\{[\backslash epsilon]\}\}\; =\; \backslash frac\{\backslash sum\_\{i=1\}^\{N\_\backslash epsilon\}\; \{P\_\{[i,\backslash epsilon]\}^\{Q-1\}\}\}\; \{N\_\backslash epsilon\}$|{{EquationRef|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.

Multifractal analysis has been successfully used in many fields, including physical,{{Cite journal |last1=Amin |first1=Kazi Rafsanjani |last2=Nagarajan |first2=Ramya |last3=Pandit |first3=Rahul |last4=Bid |first4=Aveek |date=2022-10-26 |title=Multifractal Conductance Fluctuations in High-Mobility Graphene in the Integer Quantum Hall Regime |url=https://link.aps.org/doi/10.1103/PhysRevLett.129.186802 |journal=Physical Review Letters |volume=129 |issue=18 |pages=186802 |doi=10.1103/PhysRevLett.129.186802|pmid=36374690 |arxiv=2112.14018 |bibcode=2022PhRvL.129r6802A |s2cid=245537293 }}{{Cite journal |last1=Amin |first1=Kazi Rafsanjani |last2=Ray |first2=Samriddhi Sankar |last3=Pal |first3=Nairita |last4=Pandit |first4=Rahul |last5=Bid |first5=Aveek |date=2018-02-22 |title=Exotic multifractal conductance fluctuations in graphene |journal=Communications Physics |language=en |volume=1 |issue=1 |pages=1–7 |doi=10.1038/s42005-017-0001-4 |bibcode=2018CmPhy...1....1A |s2cid=55555526 |issn=2399-3650|doi-access=free |arxiv=1804.04454 }} information, and biological sciences.{{Cite journal|date=2009|title=Fractal and multifractal analysis: A review|journal=Medical Image Analysis|volume=13|issue=4|pages=634–649|doi=10.1016/j.media.2009.05.003|pmid=19535282|last1=Lopes|first1=R.|last2=Betrouni|first2=N.}} For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls.{{Cite journal|last1=Ebrahimkhanlou|first1=Arvin|last2=Farhidzadeh|first2=Alireza|last3=Salamone|first3=Salvatore|date=2016-01-01|title=Multifractal analysis of crack patterns in reinforced concrete shear walls|journal=Structural Health Monitoring|language=en|volume=15|issue=1|pages=81–92|doi=10.1177/1475921715624502|s2cid=111619405|issn=1475-9217|doi-access=free}}

{{anchor|distort}}

File:Distort.png.]]

Multifractal analysis has been used in several scientific fields to characterize various types of datasets.{{Cite journal | last1 = Trevino | first1 = J. | last2 = Liew | first2 = S. F. | last3 = Noh | first3 = H. | last4 = Cao | first4 = H. | last5 = Dal Negro | first5 = L. | title = Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals | doi = 10.1364/OE.20.003015 | journal = Optics Express | volume = 20 | issue = 3 | pages = 3015–33 | year = 2012 | pmid = 22330539| bibcode = 2012OExpr..20.3015T | doi-access = free }} 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.

{{anchor|dqvsq}}

{{anchor|dimensional ordering}}

File:Dqvsq.png (empirical box counting dimension = 1.49), and multifractal Hénon map (empirical box counting dimension = 1.29).]]

One practical multifractal spectrum is the graph of D_Q vs Q, where D_Q 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).

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 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.

Another useful multifractal spectrum is the graph of $f(\backslash alpha)$ versus $\backslash alpha$ (see calculations). These graphs generally rise to a maximum that approximates the fractal 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 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,{{Cite journal|last=Saravia|first=Leonardo A.|date=2015-08-01|title=A new method to analyse species abundances in space using generalized dimensions|url=https://peerj.com/preprints/745v5/|journal=Methods in Ecology and Evolution|volume=6|issue=11|pages=1298–1310|doi=10.1111/2041-210X.12417|issn=2041-210X|doi-access=free|bibcode=2015MEcEv...6.1298S }} 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.{{Cite journal|title = mfSBA: Multifractal analysis of spatial patterns in ecological communities|journal = F1000Research|date = 2014-01-01|pmc = 4197745|pmid = 25324962|doi = 10.12688/f1000research.3-14.v2|first = Leonardo A.|last = Saravia|volume=3|pages=14 | doi-access=free }}

• {{annotated link|de Rham curve}}
• {{annotated link|Fractional Brownian motion}}
• {{annotated link|Detrended fluctuation analysis}}
• {{annotated link|Tweedie distributions}}
• {{annotated link|Markov switching multifractal}}

{{Reflist}}

• {{Cite book |last=Falconer |first=Kenneth J. |title=Fractal geometry: mathematical foundations and applications |date=2014 |publisher=Wiley |isbn=978-1-119-94239-9 |edition=3. ed., 1. publ |location=Chichester |chapter=17. Multifractal measures}}
• {{Citation |title=Multi-affine surfaces |date=1995 |work=Fractal Concepts in Surface Growth |pages=262–268 |editor-last=Barabási |editor-first=A.- L. |url=https://www.cambridge.org/core/books/fractal-concepts-in-surface-growth/multiaffine-surfaces/369764B5F6B50994C42E7A558472CFC7 |access-date=2024-06-05 |place=Cambridge |publisher=Cambridge University Press |doi=10.1017/CBO9780511599798.026 |isbn=978-0-521-48318-6 |editor2-last=Stanley |editor2-first=H. E.|url-access=subscription }}
• {{Cite journal |last1=G |first1=Evertsz C. J.|last2=Mandelbrot |first2=Benoît B. |date=1992 |title=Multifractal measures |url=https://users.math.yale.edu/~bbm3/web_pdfs/136multifractal.pdf |journal=Chaos and Fractals New Frontiers of Science |pages=922–953|archive-url=https://web.archive.org/web/20230713074018/https://users.math.yale.edu/~bbm3/web_pdfs/136multifractal.pdf |archive-date=2023-07-13 }}
• {{Cite book |last=Mandelbrot |first=Benoît B. |title=Fractals and scaling in finance: discontinuity, concentration, risk|date=1997 |publisher=Springer |isbn=978-0-387-98363-9 |series=Selecta |location=New York, NY Berlin Heidelberg}}
• {{Cite book |last=Harte |first=David |url=http://dx.doi.org/10.1201/9781420036008 |title=Multifractals |date=2001-06-26 |publisher=Chapman and Hall/CRC |doi=10.1201/9781420036008 |isbn=978-0-429-12366-5}}
• {{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 |s2cid=4318433}}
• {{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 |doi=10.4249/scholarpedia.4103|issn=1941-6016|bibcode=2008SchpJ...3.4103A |doi-access=free }}

• {{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 |doi-access=free}}
• [http://www2.warwick.ac.uk/fac/sci/physics/research/theory/research/disqs/media Movies of visualizations of multifractals]

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{{DEFAULTSORT:Multifractal System}}

Category:Fractals

Category:Dimension theory