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Q-exponential Distribution
The ''q''-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The ''q''-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as q \rightarrow 1. Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for q=1-\lambda, a particular case of power transform in statistics. Characterization Probability density function The ''q''-exponential distribution has the probability density function :(2-q) \lambda e_q(-\lambda x) where :e_q(x) = +(1-q)x is the ''q''-exponential if . When , ''e''''q''(x) is just exp(''x''). Derivation In a similar procedure to how the exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
The Probability Density Function Of Q-exponential Distribution
''The'' is a grammatical article in English, denoting nouns that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pronoun ''thee'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Power Transform
In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association (such as the Pearson correlation In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviation ... between variables), and for other data stabilization procedures. Power transforms are used in multiple fields, including Multiresolution analysis, multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, Geochemical modeling, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas. Definition The power tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Q-Gaussian
The ''q''-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The ''q''-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The normal distribution is recovered as ''q'' → 1. The ''q''-Gaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < ''q'' < 3. For the ''q''-Gaussian distribution is the PDF of a bounded . This makes in biology and other domains the ''q''-Gaussian distribution more suitable than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Copula (probability Theory)
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval , 1 Copulas are used to describe / model the dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but only metaphoricly related to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Constantino Tsallis
Constantino Tsallis (; ; born 4 November 1943) is a naturalized Brazilian physicist of Greek descent, working in Rio de Janeiro at Centro Brasileiro de Pesquisas Físicas (CBPF), Brazil. Biography Tsallis was born in Greece, and grew up in Argentina, where he studied physics at Instituto Balseiro, in Bariloche. In 1974, he received a ''Doctorat d'État ès Sciences Physiques'' degree from the University of Paris-Sud. He moved to Brazil in 1975 with his wife and daughter. Tsallis is an External Professor of the Santa Fe Institute. Research Tsallis is credited with introducing the notion of what is known as Tsallis entropy and Tsallis statistics in his 1988 paper "Possible generalization of Boltzmann–Gibbs statistics" published in the ''Journal of Statistical Physics''. The generalization is considered to be a good candidate for formulating a theory of non- extensive thermodynamics. The resulting theory is not intended to replace Boltzmann–Gibbs statistics, but rather suppleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Phys
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Inverse Transform Sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u between 0 and 1, interpreted as a probability, and then returns the smallest number x\in\mathbb R such that F(x)\ge u for the cumulative distribution function F of a random variable. For example, imagine that F is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution. We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pareto Distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial science, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population. The ''Pareto principle'' or "80:20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value () precisely reflect it. Empirical observation has shown that this 80:20 distribution fits a wide range of cases, including natural phenomena and human activities. Definitions If ''X'' is a random variable with a Pareto (Type I) distribution, then the probability that ''X'' is greater than some nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lomax Distribution
The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero. Characterization Probability density function The probability density function (pdf) for the Lomax distribution is given by :p(x) = \frac\alpha\lambda \left(1 + \frac x\lambda \right)^, \qquad x \geq 0, with shape parameter \alpha > 0 and scale parameter \lambda > 0. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is: :p(x) = \frac. Non-central moments The \nuth non-central moment E\left ^\nu\right/math> exists only if the shape parameter \alpha strictly exceeds \nu, when the moment has the value :E\left(X^\nu\right) = \frac. Related distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Generalized Pareto Distribution
In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location \mu, scale \sigma, and shape \xi. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as \kappa = - \xi \,. With shape \xi > 0 and location \mu = \sigma/\xi, the GPD is equivalent to the Pareto distribution with scale x_m=\sigma/\xi and shape \alpha=1/\xi. Definition The cumulative distribution function of X \sim GPD(\mu, \sigma, \xi) (\mu\in\mathbb R, \sigma>0, and \xi\in\mathbb R) is : F_(x) = \begin 1 - \left(1+ \frac\right)^ & \text\xi \neq 0, \\ 1 - \exp \left(-\frac\right) & \text\xi = 0, \end where the support of X is x \geqslant \mu when \xi \geqslant 0 \,, and \mu \leqslant x \leqslant \mu - \sigma /\xi when \xi < 0. The probability density function (pdf) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tsallis Statistics
The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using that collection, it is possible to derive Tsallis distributions from the optimization of the Tsallis entropic form. A continuous real parameter ''q'' can be used to adjust the distributions, so that distributions which have properties intermediate to that of Gaussian and Lévy distributions can be created. The parameter ''q'' represents the degree of non- extensivity of the distribution. Tsallis statistics are useful for characterising complex, anomalous diffusion. Tsallis functions The ''q''-deformed exponential and logarithmic functions were first introduced in Tsallis statistics in 1994. However, the ''q''-logarithm is the Box–Cox transformation for q=1-\lambda, proposed by George Box and David Cox in 1964. ''q''-exponential The ''q''-exponential is a deformation of the exponential function using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
