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Q-Weibull Distribution
In statistics, the ''q''-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution. Characterization Probability density function The probability density function of a ''q''-Weibull random variable is: : f(x;q,\lambda,\kappa) = \begin (2-q)\frac\left(\frac\right)^ e_q(-(x/\lambda)^)& x\geq0 ,\\ 0 & x 0 is the ''scale parameter'' of the distribution and :e_q(x) = \begin \exp(x) & \textq=1, \\ pt +(1-q)x & \textq \ne 1 \text 1+(1-q)x >0, \\ pt0^ & \textq \ne 1\text1+(1-q)x \le 0, \\ pt\end is the ''q''-exponential Cumulative distribution function The cumulative distribution function of a ''q''-Weibull random variable is: :\begin1- e_^ & x\geq0\\ 0 & x<0\end where : : Mean The mean of the ''q''-Weibull distribution is : |
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''-deformation 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] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. 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 microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Tsallis Entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. Overview 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. In scientific literature, the physical relevance of the Tsallis entropy has been debated. However, from the years 2000 on, an increasingly wide spectrum of natural, artificial and social complex systems have been identified which confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive statistical mechanics, which generalizes the Boltzmann–Gibbs theory. Among the various experimental verifications and applications presently available in the literature, the following ones deserve a special mention: # The distribution characterizing the motion of cold atoms in dissipative optical lattices pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constantino Tsallis
Constantino Tsallis (; el, Κωνσταντίνος Τσάλλης ; 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. In 2011 he gave a talk ''From Nonlinear Statistical Mechanics to Nonlinear Quantum Mechanics — Concepts and Applications'' at the international symposium on subnuclear physics held in Vatican City. 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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, 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 () of log45 ≈ 1.16 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heavy-tailed Distribution
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy. There are three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions and the subexponential distributions. In practice, all commonly used heavy-tailed distributions belong to the subexponential class. There is still some discrepancy over the use of the term heavy-tailed. There are two other definitions in use. Some authors use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a finite variance. The definition given in this article is the most general in use, and includes all dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family of probability distributions is such that there is a parameter ''s'' (and other parameters ''θ'') for which the cumulative distribution function satisfies :F(x;s,\theta) = F(x/s;1,\theta), \! then ''s'' is called a scale parameter, since its value determines the " scale" or statistical dispersion of the probability distribution. If ''s'' is large, then the distribution will be more spread out; if ''s'' is small then it will be more concentrated. If the probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies :f_s(x) = f(x/s)/s, \! where ''f'' is the density of a standardized version of the density, i.e. f(x) \equiv f_(x). An estimator of a scale p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
