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Logit-normal Distribution
In probability theory, a logit-normal distribution is a probability distribution of a random variable whose logit has a normal distribution. If ''Y'' is a random variable with a normal distribution, and ''t'' is the standard logistic function, then ''X'' = ''t''(''Y'') has a logit-normal distribution; likewise, if ''X'' is logit-normally distributed, then ''Y'' = logit(''X'')= log (''X''/(1-''X'')) is normally distributed. It is also known as the logistic normal distribution, which often refers to a multinomial logit version (e.g.). A variable might be modeled as logit-normal if it is a proportion, which is bounded by zero and one, and where values of zero and one never occur. Characterization Probability density function The probability density function (PDF) of a logit-normal distribution, for 0 Using moment properties of the Dirichlet distribution, the solution can be written in terms of the digamma \psi and trigamma \psi' functions: : \mu_i^* = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kumaraswamy Distribution
In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous probability distributions defined on the interval (0,1). It is similar to the Beta distribution, but much simpler to use especially in simulation studies since its probability density function, cumulative distribution function and quantile functions can be expressed in closed-form solution, closed form. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded with a zero-inflation. This was extended to inflations at both extremes [0,1] in later work with S. G . Fletcher. Characterization Probability density function The probability density function of the Kumaraswamy distribution without considering any inflation is : f(x; a,b) = a b x^^, \ \ \mbox \ \ x \in (0,1), and where ''a'' and ''b'' are non-negative shape parameters. Cumulative distribution function The cumulative distribution function is :F(x; a,b)=\int_^f(\xi; a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas ''beta distribution of the second kind'' is an alternative name for the beta prime distribution. The generalization to mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigamma Function
In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by : \psi_1(z) = \frac \ln\Gamma(z). It follows from this definition that : \psi_1(z) = \frac \psi(z) where is the digamma function. It may also be defined as the sum of the series : \psi_1(z) = \sum_^\frac, making it a special case of the Hurwitz zeta function : \psi_1(z) = \zeta(2,z). Note that the last two formulas are valid when is not a natural number. Calculation A double integral representation, as an alternative to the ones given above, may be derived from the series representation: : \psi_1(z) = \int_0^1\!\!\int_0^x\frac\,dy\,dx using the formula for the sum of a geometric series. Integration over yields: : \psi_1(z) = -\int_0^1\frac\,dx An asymptotic expansion as a Laurent series is : \psi_1(z) = \frac + \frac + \sum_^\frac = \sum_^\frac if we have chosen , i.e. the Bernoulli numbers of the second kind. Recurrence and reflection formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictly concave on (0,\infty). The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning double-gamma). Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the derivative with respect to gives: :\Gamma'(z+1)=z\Gamma'(z)+\Gamma(z) \, Dividing by or the equivalent gives: :\frac=\frac+\frac or: :\psi(z+1)=\psi(z)+\frac Since the harmonic numbers are defined for positive integers as :H_n=\sum_^n \frac 1 k, the digamma function is related to them by :\psi(n)=H_-\gamma, where and is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values : \psi \left(n+\tfrac12\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kullback–Leibler Divergence
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different from a second, reference probability distribution ''Q''. A simple interpretation of the KL divergence of ''P'' from ''Q'' is the expected excess surprise from using ''Q'' as a model when the actual distribution is ''P''. While it is a distance, it is not a metric, the most familiar type of distance: it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances). In the simple case, a relative entropy of 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Normal Approximation To Dirichlet Distribution
Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, the inverse of the logistic function ** Logistic distribution, the derivative of the logistic function, a continuous probability distribution, used in probability theory and statistics * Mathematical logic, subfield of mathematics exploring the applications of formal logic to mathematics Other uses * Logistics, the management of resources and their distributions ** Logistic engineering, the scientific study of logistics ** Military logistics, the study of logistics at the service of military units and operations See also *Logic (other) Logic is the study of the principles and criteria of valid inference and demonstration Logic may also refer to: *Mathematical logic, a branch of mathematics that grew out of symbolic logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Aitchison
John Aitchison (22 July 1926 – 23 December 2016) was a Scottish statistician. Career John Aitchison studied at the Universitiy of Edinburgh after being uncomfortable explaining to his headmaster that he didn’t plan to attend university. He graduated in 1947 with an MA in mathematics. After two years wherein he did actuarial work, he also attended Trinity College, Cambridge. He had a scholarship to do so, and graduated in 1951 with a BA focused on statistics. The year after he graduated, he joined the Department of Applied Economics at Cambridge as a statistician. He continued his work at Cambridge until 1956, when he was offered the position of Lecturer of Statistics at the University of Glasgow. During his time at Glasgow, he wrote ''The lognormal distribution, with special reference to its uses in economics (1957)'' with J A C Brown (who he met at Cambridge). However, he left Glasgow in 1962, when the University of Liverpool offered him the positions of Senior Lect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compositional Data
In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, percentages, and ppm can all be thought of as compositional data. Ternary plot Compositional data in three variables can be plotted via ternary plots. The use of a barycentric plot on three variables graphically depicts the ratios of the three variables as positions in an equilateral triangle. Simplicial sample space In general, John Aitchison defined compositional data to be proportions of some whole in 1982. In particular, a compositional data point (or ''composition'' for short) can be represented by a real vector with positive components. The sample space of compositional data is a simplex: :: \mathcal^D=\left\. \ The only information is given by the ratios between components, so the information of a composition is preserved un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \boldsymbol\alpha of positive reals. It is a multivariate generalization of the beta distribution, (Chapter 49: Dirichlet and Inverted Dirichlet Distributions) hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet distribution is the ''Dirichlet process''. Definitions Probability density function The Dirichlet distribution of order ''K'' ≥ 2 with parameters ''α''1, ..., ''α''''K'' > 0 has a probability density function with respect to Lebesgue m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian And Logistic Normal Pdfs
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective ''Gaussian'' is pronounced . Mathematics Algebra and linear algebra Geometry and differential geometry Number theory Cyclotomic fields * Gaussian period * Gaussian rational *Gauss sum, an exponential sum over Dirichlet characters **Elliptic Gauss sum, an analog of a Gauss sum **Quadratic Gauss sum Analysis, numerical analysis, vector calculus and calculus of variations Complex analysis and convex analysis * Gauss–Lucas theorem * Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions * Gauss's criterion – described oEncyclopedia of Mathematics* Gauss's hypergeometric theorem, an identity on hypergeometric series *Gauss plane Statistics *Gauss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
