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, 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 in fluid 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 the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie 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 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 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 , 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 . [21]
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 is called the singularity spectrum and fully describes the statistical distribution of the variable .[ citation needed ]
In practice, the multifractal behaviour of a physical system is not directly characterized by its singularity spectrum . Rather, data analysis gives access to the multiscaling exponents. Indeed, multifractal signals generally obey a scale invariance property that yields power-law behaviours for multiresolution quantities, depending on their scale . Depending on the object under study, these multiresolution quantities, denoted by , can be local averages in boxes of size , gradients over distance , wavelet coefficients at scale , etc. For multifractal objects, one usually observes a global power-law scaling of the form:[ citation needed ]
at least in some range of scales and for some range of orders . When such behaviour is observed, one talks of scale invariance, self-similarity, or multiscaling. [22]
Using 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 ]
Multifractal systems are often modeled by stochastic processes such as multiplicative cascades. The are statistically interpreted, 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 ]
Modelling as a multiplicative cascade also leads to estimation of multifractal properties. 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. [23]
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. [24] [25] [26] 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 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 Eq.2.0 .
(Eq.0.0) |
the total mass or sum of pixels in all boxes for this | (Eq.1.0) |
the probability of this mass at relative to the total mass for a box size | (Eq.2.0) |
is used to observe how the pixel distribution behaves when distorted in certain ways as in Eq.3.0 and Eq.3.1 :
the sum of all mass probabilities distorted by being raised to this Q, for this box size | (Eq.3.0) |
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 -sized pieces and distorted by Q, to find different values for the dimension of the set, as in the following:
(Eq.4.0) |
Thus, a series of values for can be found from the slopes of the regression line for the log of Eq.3.0 versus the log of for each , based on Eq.4.1 :
(Eq.4.1) |
(Eq.5.0) |
(Eq.5.1) |
(Eq.5.2) |
(Eq.5.3) |
(Eq.6.0) |
(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. [24]
Multifractal analysis has been successfully used in many fields, including physical, [27] [28] information, and biological sciences. [29] For example, the quantification of residual crack patterns on the surface of reinforced concrete shear walls. [30]
Multifractal analysis has been used in several scientific fields to characterize various types of datasets. [31] [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 as multifractal spectra, analogous to viewing the dataset through a "distorting lens", as shown in the illustration. [24] Several types of multifractal spectra are used in practise.
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).
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. [24] [25]
Another useful multifractal spectrum is the graph of versus (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.
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, [32] 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. [32] [33]
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. The rate of change exhibited in these relationships is said to be multiplicative.
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by
In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.
In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities described by critical exponents, universality, fractal behaviour, and ergodicity breaking. Critical phenomena take place in second order phase transitions, although not exclusively.
In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as Hofstadter wrote, "the large gaps [in the graph] form a very striking pattern somewhat resembling a butterfly."
A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe near the initial gravitational singularity, described by an anisotropic, chaotic solution of the Einstein field equation of gravitation. According to this model, the universe is chaotically oscillating around a gravitational singularity in which time and space become equal to zero or, equivalently, the spacetime curvature becomes infinitely big. This singularity is physically real in the sense that it is a necessary property of the solution, and will appear also in the exact solution of those equations. The singularity is not artificially created by the assumptions and simplifications made by the other special solutions such as the Friedmann–Lemaître–Robertson–Walker, quasi-isotropic, and Kasner solutions.
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (1880–1978), who was the lead researcher in these studies; the use of the standard notation H for the coefficient also relates to his name.
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 processes or 1/f noise.
The Binder parameter or Binder cumulant in statistical physics, also known as the fourth-order cumulant is defined as the kurtosis of the order parameter, s, introduced by Austrian theoretical physicist Kurt Binder. It is frequently used to determine accurately phase transition points in numerical simulations of various models.
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including topography, natural geometric objects, ecology and aquatic sciences, sound, market fluctuations, heart rates, frequency domain in electroencephalography signals, digital images, molecular motion, and data science. Fractal analysis is now widely used in all areas of science. An important limitation of fractal analysis is that arriving at an empirically determined fractal dimension does not necessarily prove that a pattern is fractal; rather, other essential characteristics have to be considered. Fractal analysis is valuable in expanding our knowledge of the structure and function of various systems, and as a potential tool to mathematically assess novel areas of study. Fractal calculus was formulated which is a generalization of ordinary calculus.
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. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular.
A coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear systems. They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases.
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.
The wavelet transform modulus maxima (WTMM) is a method for detecting the fractal dimension of a signal.
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 gives a value for how "fractal" a set of points are in a function.
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 process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern. Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and multifractal analysis.
In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies and the potential . here a is a classical turning point so , the quantum energies of the model are the roots of the Riemann Xi function and . In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by ; with N(x) being the eigenvalue staircase and H(x) is the Heaviside step function.
JCMsuite is a finite element analysis software package for the simulation and analysis of electromagnetic waves, elasticity and heat conduction. It also allows a mutual coupling between its optical, heat conduction and continuum mechanics solvers. The software is mainly applied for the analysis and optimization of nanooptical and microoptical systems. Its applications in research and development projects include dimensional metrology systems, photolithographic systems, photonic crystal fibers, VCSELs, Quantum-Dot emitters, light trapping in solar cells, and plasmonic systems. The design tasks can be embedded into the high-level scripting languages MATLAB and Python, enabling a scripting of design setups in order to define parameter dependent problems or to run parameter scans.