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Wishart distribution

Generalization of gamma distribution to multiple dimensions From Wikipedia, the free encyclopedia

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In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first formulated the distribution in 1928.[1] Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).[2]

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It is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random vector.[3]

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Definition

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Suppose G is a p × n matrix, each column of which is independently drawn from a p-variate normal distribution with zero mean:

It means :

Then the Wishart distribution is the probability distribution of the p × p random matrix [4]

known as the scatter matrix. One indicates that S has that probability distribution by writing

The positive integer n is the number of degrees of freedom. Sometimes this is written W(V, p, n). For np the matrix S is invertible with probability 1 if V is invertible.

If p = V = 1 then this distribution is a chi-squared distribution with n degrees of freedom.

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Occurrence

The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a multivariate normal distribution. It occurs frequently in likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of random matrices[citation needed] and in multidimensional Bayesian analysis.[5] It is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels .[6]

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Probability density function

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Thumb
Spectral density of Wishart-Laguerre ensemble with dimensions (8, 15). A reconstruction of Figure 1 of [7].

The Wishart distribution can be characterized by its probability density function as follows:

Let X be a p × p symmetric matrix of random variables that is positive semi-definite. Let V be a (fixed) symmetric positive definite matrix of size p × p.

Then, if np, X has a Wishart distribution with n degrees of freedom if it has the probability density function

where is the determinant of and Γp is the multivariate gamma function defined as

The density above is not the joint density of all the elements of the random matrix X (such -dimensional density does not exist because of the symmetry constrains ), it is rather the joint density of elements for (,[1] page 38). Also, the density formula above applies only to positive definite matrices for other matrices the density is equal to zero.

Spectral density

The joint-eigenvalue density for the eigenvalues of a random matrix is,[8][9]

where is a constant. The spectral density can be marginalized to yield the density of a single eigenvalue, by evaluating a Selberg integral.

In fact the above definition can be extended to any real n > p − 1. If np − 1, then the Wishart no longer has a densityinstead it represents a singular distribution that takes values in a lower-dimension subspace of the space of p × p matrices.[10]

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Use in Bayesian statistics

In Bayesian statistics, in the context of the multivariate normal distribution, the Wishart distribution is the conjugate prior to the precision matrix Ω = Σ−1, where Σ is the covariance matrix.[11]:135[12]

Choice of parameters

The least informative, proper Wishart prior is obtained by setting n = p.[citation needed]

A common choice for V leverages the fact that the mean of X ~Wp(V, n) is nV. Then V is chosen so that nV equals an initial guess for X. For instance, when estimating a precision matrix Σ−1 ~ Wp(V, n) a reasonable choice for V would be n−1Σ0−1, where Σ0 is some prior estimate for the covariance matrix Σ.

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Properties

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Log-expectation

The following formula plays a role in variational Bayes derivations for Bayes networks involving the Wishart distribution. From equation (2.63),[13]

where is the multivariate digamma function (the derivative of the log of the multivariate gamma function).

Log-variance

The following variance computation could be of help in Bayesian statistics:

where is the trigamma function. This comes up when computing the Fisher information of the Wishart random variable.

Entropy

The information entropy of the distribution has the following formula:[11]:693

where B(V, n) is the normalizing constant of the distribution:

This can be expanded as follows:

Cross-entropy

The cross-entropy of two Wishart distributions with parameters and with parameters is

Note that when and we recover the entropy.

KL-divergence

The Kullback–Leibler divergence of from is

Characteristic function

The characteristic function of the Wishart distribution is

where E[⋅] denotes expectation. (Here Θ is any matrix with the same dimensions as V, 1 indicates the identity matrix, and i is a square root of −1).[9] Properly interpreting this formula requires a little care, because noninteger complex powers are multivalued; when n is noninteger, the correct branch must be determined via analytic continuation.[14]

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Theorem

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If a p × p random matrix X has a Wishart distribution with m degrees of freedom and variance matrix V — write — and C is a q × p matrix of rank q, then [15]

Corollary 1

If z is a nonzero p × 1 constant vector, then:[15]

In this case, is the chi-squared distribution and (note that is a constant; it is positive because V is positive definite).

Corollary 2

Consider the case where zT = (0, ..., 0, 1, 0, ..., 0) (that is, the j-th element is one and all others zero). Then corollary 1 above shows that

gives the marginal distribution of each of the elements on the matrix's diagonal.

George Seber points out that the Wishart distribution is not called the “multivariate chi-squared distribution” because the marginal distribution of the off-diagonal elements is not chi-squared. Seber prefers to reserve the term multivariate for the case when all univariate marginals belong to the same family.[16]

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Estimator of the multivariate normal distribution

The Wishart distribution is the sampling distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution.[17] A derivation of the MLE uses the spectral theorem.

Bartlett decomposition

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The Bartlett decomposition of a matrix X from a p-variate Wishart distribution with scale matrix V and n degrees of freedom is the factorization:

where L is the Cholesky factor of V, and:

where and nij ~ N(0, 1) independently.[18] This provides a useful method for obtaining random samples from a Wishart distribution.[19]

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Marginal distribution of matrix elements

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Let V be a 2 × 2 variance matrix characterized by correlation coefficient −1 < ρ < 1 and L its lower Cholesky factor:

Multiplying through the Bartlett decomposition above, we find that a random sample from the 2 × 2 Wishart distribution is

The diagonal elements, most evidently in the first element, follow the χ2 distribution with n degrees of freedom (scaled by σ2) as expected. The off-diagonal element is less familiar but can be identified as a normal variance-mean mixture where the mixing density is a χ2 distribution. The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution

where Kν(z) is the modified Bessel function of the second kind.[20] Similar results may be found for higher dimensions. In general, if follows a Wishart distribution with parameters, , then for , the off-diagonal elements

. [21]

It is also possible to write down the moment-generating function even in the noncentral case (essentially the nth power of Craig (1936)[22] equation 10) although the probability density becomes an infinite sum of Bessel functions.

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The range of the shape parameter

It can be shown [23] that the Wishart distribution can be defined if and only if the shape parameter n belongs to the set

This set is named after Simon Gindikin, who introduced it[24] in the 1970s in the context of gamma distributions on homogeneous cones. However, for the new parameters in the discrete spectrum of the Gindikin ensemble, namely,

the corresponding Wishart distribution has no Lebesgue density.

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Relationships to other distributions

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See also

References

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