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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] |
Tsallis Entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. It is proportional to the expectation of the q-logarithm of a distribution. History The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identical in form to Havrda–Charvát structural α-entropy, introduced in 1967 within information theory. Definition Given a discrete set of probabilities \ with the condition \sum_i p_i=1, and q any real number, the Tsallis entropy is defined as :S_q() = k \cdot \frac \left( 1 - \sum_i p_i^q \right), where q is a real parameter sometimes called ''entropic-index'' and k a positive constant. In the limit as q \to 1, the usual Boltzmann–Gibbs entropy is recovered, namely :S_\text = S_1(p) = -k \sum_i p_i \ln p_i , where one identifies k with the Boltzmann constant k_B. For continuous probability distributions, we define the entropy as :S_q = \left( 1 - \int (p(x)) ... [...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] |
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] |
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] |
Function Of Inverse
Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented programming * Function (computer programming), a callable sequence of instructions Music * Function (music), a relationship of a chord to a tonal centre * Function (musician) (born 1973), David Charles Sumner, American techno DJ and producer * "Function" (song), a 2012 song by American rapper E-40 featuring YG, Iamsu! & Problem * "Function", song by Dana Kletter from '' Boneyard Beach'' 1995 Other uses * Function (biology), the effect of an activity or process * Function (engineering), a specific action that a system can perform * Function (language), a way of achieving an aim using language * Function (mathematics), a relation that associates an input to a single output * Function (sociology), an activity's role in society * Fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , then is the logarithm of to base , written , so . As a single-variable function, the logarithm to base is the inverse of exponentiation with base . The logarithm base is called the ''decimal'' or ''common'' logarithm and is commonly used in science and engineering. The ''natural'' logarithm has the number as its base; its use is widespread in mathematics and physics because of its very simple derivative. The ''binary'' logarithm uses base and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Q-exponential
The term ''q''-exponential occurs in two contexts. The q-exponential distribution, based on the Tsallis q-exponential is discussed in elsewhere. In combinatorial mathematics, a ''q''-exponential is a ''q''-analog of the exponential function, namely the eigenfunction of a ''q''-derivative. There are many ''q''-derivatives, for example, the classical ''q''-derivative, the Askey–Wilson operator, etc. Therefore, unlike the classical exponentials, ''q''-exponentials are not unique. For example, e_q(z) is the ''q''-exponential corresponding to the classical ''q''-derivative while \mathcal_q(z) are eigenfunctions of the Askey–Wilson operators. The ''q''-exponential is also known as the quantum dilogarithm. Definition The ''q''-exponential e_q(z) is defined as :e_q(z)= \sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty z^n\frac where _q is the ''q''-factorial and :(q;q)_n=(1-q^n)(1-q^)\cdots (1-q) is the ''q''-Pochhammer symbol. That this is the ''q''-analog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
George Box
George Edward Pelham Box (18 October 1919 – 28 March 2013) was a British statistician, who worked in the areas of quality control, time-series analysis, design of experiments, and Bayesian inference. He has been called "one of the great statistical minds of the 20th century". He is famous for the quote "All models are wrong but some are useful". Education and early life He was born in Gravesend, Kent, England. Upon entering university he began to study chemistry, but was called up for service before finishing. During World War II, he performed experiments for the British Army exposing small animals to poison gas. To analyze the results of his experiments, he taught himself statistics from available texts. After the war, he enrolled at University College London and obtained a bachelor's degree in mathematics and statistics. He received a PhD from the University of London in 1953, under the supervision of Egon Pearson and Herman Otto Hartley, HO Hartley. Career and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
David Cox (statistician)
Sir David Roxbee Cox (15 July 1924 – 18 January 2022) was a British statistician and educator. His wide-ranging contributions to the field of statistics included introducing logistic regression, the proportional hazards model and the Cox process, a point process named after him. He was a professor of statistics at Birkbeck, University of London, Birkbeck College, London, Imperial College London and the University of Oxford, and served as Warden (college), Warden of Nuffield College, Oxford. The first recipient of the International Prize in Statistics, he also received the Guy Medal, Guy, George Box Medal, George Box and Copley Medal, Copley medals, as well as a Knight bachelor, knighthood. Early life Cox was born in Birmingham on 15 July 1924. His father was a die (manufacturing), die sinker and part-owner of a jewellery shop, and they lived near the Jewellery Quarter. The aeronautical engineer Harold Roxbee Cox was a distant cousin. He attended Handsworth Grammar School, B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tsallis Distribution
In statistics, a Tsallis distribution is a probability distribution derived from the maximization of the Tsallis entropy under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the 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. Similarly, if the domain of the variable is constrained to be positive in the maximum entropy procedure, the q-exponential distribution is derived. The Tsallis distributions have been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distributions are often used for their heavy tails. Note that Tsallis distributions are obtained as Box–Cox transformation over usual distributions, with deformation parameter \lambda=1-q. This deformatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Box–Cox Transformation
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 between variables), and for other data stabilization procedures. Power transforms are used in multiple fields, including multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas. Definition The power transformation is defined as a continuous function of power parameter ''λ'', typically given in piece-wise form that makes it continuous at the point of singularity (''λ'' = 0). For data vectors (''y''1,..., ''y''''n'') in which each ''y''''i'' > 0, the power transform is : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
