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Multifractal
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. They include the length of coastlines, mountain topography, 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 (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, as well as the geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Analysis
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 generaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Set
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singularity Spectrum
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. More formally, the singularity spectrum D(\alpha) of a function, f(x), is defined as: :D(\alpha) = D_F\ Where \alpha(x) is the function describing the Hölder exponent, \alpha(x) of f(x) at the point x. D_F\ is the Hausdorff dimension of a point set. See also * Multifractal analysis * Holder exponent * Hausdorff dimension * Fractal * Fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gauss ... References * . Fractals {{fractal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Cascade
In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. Definition The plots above are examples of multiplicative cascade multifractals. To create these distributions there are a few steps to take. Firstly, we must create a lattice of cells which will be our underlying probability density field. Secondly, an iterative process is followed to create multiple levels of the lattice: at each iteration the cells are split into four equal parts (cells). Each new cell is then assigned a probability randomly from the set \lbrace p_1,p_2,p_3,p_4 \rbrace without replacement, where p_i \in ,1/math>. This process is continued to the ''N''th level. For example, in constructing such a model down to level 8 we produce a 48 array of cells. Thirdly, the cells are filled as follows: We take the probability of a cell being occupied as the product of the cell's own ''p''''i'' and those of all its parents ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Counting
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 (see Figure 1). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Counting
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 (see Figure 1). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tweedie Distribution
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models. The Tweedie distributions were named by Bent Jørgensen after Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984. Definitions The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship. A random variable ''Y'' is Tweedie distributed ''Twp(μ, σ2)'', if Y \sim \mathrm(\mu, \sigma^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lacunarity
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 particu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nota Bene
(, or ; plural form ) is a Latin phrase meaning "note well". It is often abbreviated as NB, n.b., or with the ligature and first appeared in English writing . In Modern English, it is used, particularly in legal papers, to draw the attention of the reader to a certain (side) aspect or detail of the subject being addressed. While ''NB'' is also often used in academic writing, ''note'' is a common substitute. The markings used to draw readers' attention in medieval manuscripts are also called marks. The common medieval markings do not, however, include the abbreviation ''NB''. The usual medieval equivalents are anagrams from the four letters in the word , the abbreviation DM from ("worth remembering"), or a symbol of a little hand (☞), called a manicule or index, with the index finger pointing towards the beginning of the significant passage.Raymond Clemens and Timothy Graham, Introduction to Manuscript Studies (Ithaca: Cornell University Press, 2007), p. 44. Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
