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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] |
Multifractal Analysis
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 geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Exponent
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Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holder Exponent
Holder may refer to: Law * Holder (law), a person that has in their custody a promissory note, bill of exchange or cheque * '' Holder v Holder'', an English trusts law case * '' Holder v. Humanitarian Law Project'' (2010), a U.S. Supreme Court decision People * Holder (surname) * Holder da Silva (born 1988), Guinea-Bissauan sprinter Places * Holder, Australian Capital Territory, a suburb of Canberra * Holder, South Australia, a locality * Holder, Florida, United States, an unincorporated community * Holder, Illinois, United States, a town * Holder Plantation, Jackson County, Georgia, United States, on the National Register of Historic Places * Holder Peak, Princess Elizabeth Land, Antarctica Ships * , a US destroyer escort * , a US destroyer Other uses * Holder (American football) * Holder baronets, a title in the Baronetage of the United Kingdom See also * Hölder (other) * Holder Formation, a geologic formation in the Sacramento Mountains of New Mexico * Ho ... [...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] |
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 Gaussian process ''BH''(''t'') on , ''T'' that starts at zero, has expectation zero for all ''t'' in , ''T'' and has the following covariance function: :E _H(t) B_H (s)\tfrac (, t, ^+, s, ^-, t-s, ^), where ''H'' is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by . The value of ''H'' determines what kind of process the ''fBm'' is: * if ''H'' = 1/2 then the process is in fact a Brownian motion or Wiener process; * if ''H'' > 1/2 then the increments of the process are positively correlated; * if ''H'' < 1/2 then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
