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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. Multifractal systems are common in nature, especially geophysics. They include fully developed turbulence, stock market time series, real world scenes, the Sun’s magnetic field time series, heartbeat dynamics, human gait, 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 (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〔Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, ''1/f'' noise and multifractality. ''Phys. Rev E'' 84, 066120〕 as well as the geometric Tweedie models.〔Jørgensen B, Kokonendji CC (2011) Dispersion models for geometric sums. ''Braz J Probab Stat'' 25, 263-293〕 The first convergence effect yields monofractal sequences and the second convergence effect is responsible for variation in the fractal dimension of the monofractal sequences.〔Kendal WS (2014) Multifractality attributed to dual central limit-like convergence effects. ''Physica A'' 401, 22-33〕 From a practical perspective, multifractal analysis uses the mathematical basis of multifractal theory 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. The techniques of multifractal analysis 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 D(h). The curve D(h) versus h 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 D(h). 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: : 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「multifractal system」の詳細全文を読む スポンサード リンク
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