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Matrix T-distribution
In statistics, the matrix ''t''-distribution (or matrix variate ''t''-distribution) is the generalization of the multivariate ''t''-distribution from vectors to matrices.Zhu, Shenghuo and Kai Yu and Yihong Gong (2007)"Predictive Matrix-Variate ''t'' Models."In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, ''NIPS '07: Advances in Neural Information Processing Systems'' 20, pages 1721–1728. MIT Press, Cambridge, MA, 2008. The notation is changed a bit in this article for consistency with the matrix normal distribution article. The matrix ''t''-distribution shares the same relationship with the multivariate ''t''-distribution that the matrix normal distribution shares with the multivariate normal distribution: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrix ''t''-distribution is the compound distribution that results from an infinite mixture of a matrix norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Location Parameter
In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x_0, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways: * either as having a probability density function or probability mass function f(x - x_0); or * having a cumulative distribution function F(x - x_0); or * being defined as resulting from the random variable transformation x_0 + X, where X is a random variable with a certain, possibly unknown, distribution. See also . A direct example of a location parameter is the parameter \mu of the normal distribution. To see this, note that the probability density function f(x , \mu, \sigma) of a normal distribution \mathcal(\mu,\sigma^2) can have the parameter \mu factored out and be written as: : g(x' = x - \mu , \sigma) = \frac \exp\left(-\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Multivariate Gamma Function
In mathematics, the multivariate gamma function Γ''p'' is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution. It has two equivalent definitions. One is given as the following integral over the p \times p positive-definite real matrices: : \Gamma_p(a)= \int_ \exp\left( -(S)\right)\, \left, S\^ dS, where , S, denotes the determinant of S. The other one, more useful to obtain a numerical result is: : \Gamma_p(a)= \pi^\prod_^p \Gamma(a+(1-j)/2). In both definitions, a is a complex number whose real part satisfies \Re(a) > (p-1)/2. Note that \Gamma_1(a) reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for p\ge 2: : \Gamma_p(a) = \pi^ \Gamma(a) \Gamma_(a-\tfrac) = \pi^ \Gamma_(a) \Gamma(a+(1-p)/2). Thus * \Gamma_2(a)=\pi^\Gamma(a)\G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Matrix Normal Distribution
In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. Definition The probability density function for the random matrix X (''n'' × ''p'') that follows the matrix normal distribution \mathcal_(\mathbf, \mathbf, \mathbf) has the form: : p(\mathbf\mid\mathbf, \mathbf, \mathbf) = \frac where \mathrm denotes Trace (linear algebra), trace and M is ''n'' × ''p'', U is ''n'' × ''n'' and V is ''p'' × ''p'', and the density is understood as the probability density function with respect to the standard Lebesgue measure in \mathbb^, i.e.: the measure corresponding to integration with respect to dx_ dx_\dots dx_ dx_\dots dx_\dots dx_. The matrix normal is related to the multivariate normal distribution in the following way: :\mathbf \sim \mathcal_(\mathbf, \mathbf, \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Multivariate T-distribution
In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix ''t''-distribution is distinct and makes particular use of the matrix structure. Definition One common method of construction of a multivariate ''t''-distribution, for the case of p dimensions, is based on the observation that if \mathbf y and u are independent and distributed as N(,) and \chi^2_\nu (i.e. multivariate normal and chi-squared distributions) respectively, the matrix \mathbf\, is a ''p'' × ''p'' matrix, and is a constant vector then the random variable =/\sqrt + has the density : \frac\left +\frac(-)^T^(-)\right and is said to be distributed as a multivariate ''t''-distribution with para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nonsingular Matrices
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse. Definition An -by- square matrix is called invertible if there exists an -by- square matrix such that\mathbf = \mathbf = \mathbf_n ,where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field, a square matrix that is ''not'' invertible is called singular or degenerate. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Inverse Multivariate Gamma Distribution
Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the original number, yields zero * Compositional inverse, a function that "reverses" another function * Inverse element * Inverse function, a function that "reverses" another function **Generalized inverse, a matrix that has some properties of the inverse matrix but not necessarily all of them * Multiplicative inverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1 ** Inverse matrix of an Invertible matrix Other uses * Invert level, the base interior level of a pipe, trench or tunnel * Inverse (website), ''Inverse'' (website), an online magazine * An outdated term for an LGBT person; see Sexual inversion (sexology) See also * Inversion (other) * Inverter (other) * Opp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shape Parameter
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter). Such a parameter must affect the ''shape (geometry), shape'' of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does). For example, "peakedness" refers to how round the main peak is. Estimation Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moment (mathematics), moments, using the Method of moments (statistics), method of moments, as in the ''skewness'' (3rd moment) or ''kurtosis'' (4th moment), if the higher moments are defined and finite. Estimato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
