Multifractal system
A multifractal system is a generalization of a fractal system in which a single exponent is not enough to describe its dynamics; instead, a continuous spectrum of exponents is needed.
Multifractal systems are common in nature. They include the length of coastlines, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. 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. The origin of multifractality in sequential 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, 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 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.
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.
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.
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:
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.
Estimation
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.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.
Modelling as a multiplicative cascade 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 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 of a number of pixels, appearing in a box, varies as box size, to some exponent, which changes over the image, as in . is calculated from the box-counting pixel distribution as in.is used to observe how the pixel distribution behaves when distorted in certain ways as in and :
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:
Thus, a series of values for can be found from the slopes of the regression line for the log of versus the log of for each, based on :
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]. 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.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 ≥ D ≥ D. As illustrated in the [|figure], variation in this graphical spectrum can help distinguish patterns. The image shows D 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 spectra than multifractals.The generalized dimension also gives important specific information. D is equal to the capacity dimension, which—in the analysis shown in the figures here—is the box counting dimension. D is equal to the information dimension, and D to the correlation dimension. This relates to the "multi" in multifractal, where multifractals have multiple dimensions in the D versus Q spectra, but monofractals stay rather flat in that area.