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Complex Wishart Distribution
In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of n times the sample Hermitian covariance matrix of n zero-mean independent Gaussian random variables. It has support for p\times p Hermitian positive definite matrices. The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let : S_ = \sum_^n G_iG_i^H where each G_i is an independent column ''p''-vector of random complex Gaussian zero-mean samples and (.)^H is an Hermitian (complex conjugate) transpose. If the covariance of ''G'' is \mathbb G^H= M then : S \sim n\mathcal(M,n,p) where \mathcal(M,n,p) is the complex central Wishart distribution with ''n'' degrees of freedom and mean value, or scale matrix, ''M''. : f_S(\mathbf) = \frac , \;\;\; n\ge p, \;\;\; \left, \mathbf\ > 0 where : \mathcal \widetilde_p^ (n) = \pi^ \prod_^p \Gamma (n-j+1) is the complex multivariate Gamma function. Using the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Degrees Of Freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself. For example, if the variance is to be estimated from a random sample of N independent scores, then the degrees of freedom is equal to the number of independent scores (''N'') minus the number of parameters estimated as intermediate steps (one, namely, the sample mean) and is therefore equal to N-1. Mathematically, degrees of freedom is the number of dimensions of the domain of a random vector, or e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conjugate Prior Distributions
Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change of sign of the imaginary part of a complex number *Conjugate (square roots), the change of sign of a square root in an expression *Conjugate element (field theory), a generalization of the preceding conjugations to roots of a polynomial of any degree *Conjugate transpose, the complex conjugate of the transpose of a matrix * Harmonic conjugate in complex analysis * Conjugate (graph theory), an alternative term for a line graph, i.e. a graph representing the edge adjacencies of another graph *In group theory, various notions are called conjugation: **Inner automorphism, a type of conjugation homomorphism **Conjugacy class in group theory, related to matrix similarity in linear algebra **Conjugation (group theory), the image of an element und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Random Matrices
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the study of properties of random matrices, often as they become large. RMT provides techniques like mean-field theory, diagrammatic methods, the cavity method, or the replica method to compute quantities like traces, spectral densities, or scalar products between eigenvectors. Many physical phenomena, such as the spectrum of nuclei of heavy atoms, the thermal conductivity of a lattice, or the emergence of quantum chaos, can be modeled mathematically as problems concerning large, random matrices. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Covariance And Correlation
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways. If ''X'' and ''Y'' are two random variables, with means (expected values) ''μX'' and ''μY'' and standard deviations ''σX'' and ''σY'', respectively, then their covariance and correlation are as follows: ; covariance :\text_ = \sigma_ = E X-\mu_X)\,(Y-\mu_Y)/math> ; correlation :\text_ = \rho_ = E X-\mu_X)\,(Y-\mu_Y)(\sigma_X \sigma_Y)\,, so that \rho_ = \sigma_ / (\sigma_X \sigma_Y) where ''E'' is the expected value operator. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. If ''Y'' always takes on the same values as ''X'', we have the covariance of a variable with itself (i.e. \sigma_), which is called the variance and is more commonly denoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Multivariate Continuous Distributions
Multivariate is the quality of having multiple variables. It may also refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial * Multivariate interpolation * Multivariate optimization In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate optical computing In statistics * Multivariate analysis * Multivariate random variable * Multivariate regression * Multivariate statistics See also * Univariate In mathematics, a univariate object is an expression (mathematics), expression, equation, function (mathematics), function or polynomial involving only one Variable (mathematics), variable. Objects involving more than one variable are ''wikt:multi ... * Bivariate (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Continuous Distributions
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic section ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
MIMO
In radio, multiple-input and multiple-output (MIMO) () is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wireless communication standards including IEEE 802.11n (Wi-Fi 4), IEEE 802.11ac (Wi-Fi 5), HSPA+ (3G), WiMAX, and Long Term Evolution (LTE). More recently, MIMO has been applied to power-line communication for three-wire installations as part of the ITU G.hn standard and of the HomePlug AV2 specification. At one time, in wireless the term "MIMO" referred to the use of multiple antennas at the transmitter and the receiver. In modern usage, "MIMO" specifically refers to a class of techniques for sending and receiving more than one data signal simultaneously over the same radio channel by exploiting the difference in signal propagation between different antennas (e.g. due to multipath propagation). Additionally, modern MIMO usage often refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
QR Decomposition
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthonormal matrix ''Q'' and an upper triangular matrix ''R''. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Cases and definitions Square matrix Any real square matrix ''A'' may be decomposed as : A = QR, where ''Q'' is an orthogonal matrix (its columns are orthogonal unit vectors meaning and ''R'' is an upper triangular matrix (also called right triangular matrix). If ''A'' is invertible, then the factorization is unique if we require the diagonal elements of ''R'' to be positive. If instead ''A'' is a complex square matrix, then there is a decomposition ''A'' = ''QR'' where ''Q'' is a unitary matrix (so the conjugate transpose If ''A'' has ''n'' linearly independent columns, then the first ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Modulus Squared
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 (caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is '' quadratic''. The square of an integer may also be called a ''square number'' or a ''perfect square''. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Wigner Semicircle Distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution defined on the domain minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle, i.e. a semi-ellipse, centered at (0, 0): :f(x)=\sqrt\, for −''R'' ≤ ''x'' ≤ ''R'', and ''f''(''x'') = 0 if '', x, '' > ''R''. The parameter R is commonly referred to as the "radius" parameter of the distribution. The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise. General properties Because of symmetry, all of the odd-order moments of the Wigner distribution are zero. For positive integers , the -th moment of this distribution is :\frac\left(\right)^ \, In the typical special case that , this sequence coincides with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Support (measure Theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure \mu on a measurable topological space (X, \operatorname(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest ( closed) subset of X for which every open neighbourhood of every point of the set has positive measure. Motivation A (non-negative) measure \mu on a measurable space (X, \Sigma) is really a function \mu : \Sigma \to , +\infty Therefore, in terms of the usual definition of support, the support of \mu is a subset of the σ-algebra \Sigma: \operatorname (\mu) := \overline, where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on \Sigma. What we really want to know is where in the space X the measure \mu is non-zero. Consider two examples: # Lebesgue measure \lambda on the real line \Reals. It seems clear that \lambda "lives on" the wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
