Английская Википедия:Complex Wishart distribution

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Шаблон:Probability distribution_p</math> is the <math>p</math>-variate complex multivariate gamma function

• Шаблон:Math is the trace function

| cdf        =
| mean       =<math>\operatorname{E}[A]=n\Gamma</math>|
| median     =
| mode       =<math>(n-p) \mathbf{\Gamma}</math> for Шаблон:Math
| variance   =
| skewness   =
| kurtosis   =
| entropy    =
| mgf        =
| char       = <math>\det\left(I_p-i\mathbf{\Gamma}\mathbf{\Theta}\right)^{-n}</math>

}}

In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of <math>n</math> times the sample Hermitian covariance matrix of <math>n</math> zero-mean independent Gaussian random variables. It has support for <math>p\times p</math> Hermitian positive definite matrices.

The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let

<math> S_{p \times p} = \sum_{i=1}^n G_iG_i^H </math>

where each <math> G_i </math> is an independent column p-vector of random complex Gaussian zero-mean samples and <math> (.)^H </math> is an Hermitian (complex conjugate) transpose. If the covariance of G is <math> \mathbb{E}[GG^H] = M </math> then

<math> S \sim n\mathcal{CW}(M,n,p) </math>

where <math> \mathcal{CW}(M,n,p) </math> is the complex central Wishart distribution with n degrees of freedom and mean value, or scale matrix, M.

<math> f_S(\mathbf{S}) = \frac{

\left |\mathbf{S} \right|^{n-p} e^{-\operatorname{tr}(\mathbf M^{-1}\mathbf{S}) } } { \left|\mathbf{M}\right|^n\cdot \mathcal{C} \widetilde{\Gamma}_p(n) }, \;\;\; n\ge p, \;\;\; \left|\mathbf{M}\right| > 0</math> where

<math> \mathcal{C} \widetilde{\Gamma}_p^{} (n) = \pi^{p(p-1)/2} \prod_{j=1}^p \Gamma (n-j+1) </math>

is the complex multivariate Gamma function.

Using the trace rotation rule <math> \operatorname{tr}(ABC) = \operatorname{tr}(CAB) </math> we also get

<math> f_S(\mathbf{S}) = \frac{

\left |\mathbf{S} \right|^{n-p} } { \left|\mathbf{M}\right|^n\cdot \mathcal{C} \widetilde{\Gamma}_p(n) } \exp \left( -\sum_{i=1}^p G_i^H\mathbf M^{-1} G_i \right )

 </math>

which is quite close to the complex multivariate pdf of G itself. The elements of G conventionally have circular symmetry such that <math> \mathbb{E}[GG^T] = 0 </math>.

Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of <math> \mathbf{Y} = \mathbf{S^{-1}} </math> according to Goodman, Shaman is

<math> f_Y(\mathbf{Y}) = \frac{

\left |\mathbf{Y} \right|^{-(n+p)} e^{-\operatorname{tr}(\mathbf M\mathbf{Y^{-1}}) } } { \left|\mathbf{M}\right|^{-n}\cdot\mathcal{C}\widetilde{\Gamma}_p(n) }, \;\;\; n\ge p, \;\;\; \det \left(\mathbf{Y}\right) > 0</math> where <math> \mathbf{M} = \mathbf{\Gamma^{-1}}</math>.

If derived via a matrix inversion mapping, the result depends on the complex Jacobian determinant

<math> \mathcal{C}J_Y(Y^{-1}) = \left | Y \right |^{-2p-2} </math>

Goodman and others discuss such complex Jacobians.

Eigenvalues

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James and Edelman. For a <math> p \times p</math> matrix with <math>\nu \ge p </math> degrees of freedom we have

<math> f(\lambda_1\dots\lambda_p)=\tilde {K}_{\nu,p} \exp \left ( - \frac{1}{2} \sum_{i=1}^p \lambda_i \right )

 \prod_{i=1}^p \lambda_i^{\nu - p} \prod_{i<j} (\lambda_i - \lambda_j)^2 d\lambda_1 \dots d\lambda_p,

\;\;\; \lambda_i \in \mathbb{R} \ge 0</math> where

<math> \tilde {K}_{\nu,p}^{-1} = 2^{p\nu} \prod_{i=1}^p \Gamma (\nu - i+1) \Gamma (p-i+1) </math>

Note however that Edelman uses the "mathematical" definition of a complex normal variable <math> Z = X + iY </math> where iid X and Y each have unit variance and the variance of <math> Z = \mathbf{E} \left(X^2 + Y^2 \right ) = 2</math>. For the definition more common in engineering circles, with X and Y each having 0.5 variance, the eigenvalues are reduced by a factor of 2.

While this expression gives little insight, there are approximations for marginal eigenvalue distributions. From Edelman we have that if S is a sample from the complex Wishart distribution with <math> p = \kappa \nu, \;\; 0 \le \kappa \le 1 </math> such that <math> S_{p \times p} \sim \mathcal{CW}\left( 2\mathbf{I}, \frac{p}{\kappa} \right) </math> then in the limit <math> p \rightarrow \infty </math> the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function

<math> p_\lambda(\lambda) = \frac

{\sqrt { [\lambda/2 - ( \sqrt {\kappa} -1 )^2 ][\sqrt {\kappa} +1 )^2 - \lambda /2 ] }} { 4\pi \kappa (\lambda /2)}, \;\;\; 2( \sqrt {\kappa} -1)^2 \le \lambda \le 2(\sqrt {\kappa} +1 )^2, \;\;\; 0 \le \kappa \le 1 </math> This distribution becomes identical to the real Wishart case, by replacing <math> \lambda</math> by <math>2\lambda </math>, on account of the doubled sample variance, so in the case <math> S_{p \times p} \sim \mathcal{CW} \left( \mathbf{I}, \frac{p}{\kappa} \right) </math>, the pdf reduces to the real Wishart one:

<math> p_\lambda(\lambda) = \frac

{\sqrt {[\lambda - ( \sqrt {\kappa} -1 )^2 ][\sqrt {\kappa} +1 )^2 - \lambda ] }} { 2\pi \kappa \lambda}, \;\;\; (\sqrt {\kappa} -1)^2 \le \lambda \le (\sqrt {\kappa} +1 )^2, \;\;\; 0 \le \kappa \le 1 </math>

A special case is <math> \kappa = 1 </math>

<math> p_\lambda(\lambda) = \frac {1}{4\pi} \left (\frac {8-\lambda}{\lambda} \right )^{\frac{1}{2}}, \; 0 \le \lambda \le 8 </math>

or, if a Var(Z) = 1 convention is used then

<math> p_\lambda(\lambda) = \frac {1}{2\pi} \left (\frac {4-\lambda}{\lambda} \right )^{\frac{1}{2}}, \; 0 \le \lambda \le 4 </math>.

The Wigner semicircle distribution arises by making the change of variable <math> y = \pm\sqrt{\lambda} </math> in the latter and selecting the sign of y randomly yielding pdf

<math> p_y(y) = \frac {1}{2\pi} \left ( 4-y^2 \right )^{\frac{1}{2}}, \; -2 \le y \le 2 </math>

In place of the definition of the Wishart sample matrix above, <math> S_{p \times p} = \sum_{j=1}^\nu G_jG_j^H </math>, we can define a Gaussian ensemble

<math> \mathbf{G}_{i,j} = [G_1 \dots G_\nu ] \in \mathbb{C}^{\,p \times \nu } </math>

such that S is the matrix product <math> S = \mathbf{G}\mathbf{G^H} </math>. The real non-negative eigenvalues of S are then the modulus-squared singular values of the ensemble <math> \mathbf{G}</math> and the moduli of the latter have a quarter-circle distribution.

In the case <math> \kappa > 1</math> such that <math>\nu < p</math> then <math>S </math> is rank deficient with at least <math> p - \nu </math> null eigenvalues. However the singular values of <math> \mathbf{G} </math> are invariant under transposition so, redefining <math> \tilde{S} = \mathbf{G^H}\mathbf{G} </math>, then <math> \tilde{S}_{\nu \times \nu} </math> has a complex Wishart distribution, has full rank almost certainly, and eigenvalue distributions can be obtained from <math> \tilde{S} </math> in lieu, using all the previous equations.

In cases where the columns of <math> \mathbf{G} </math> are not linearly independent and <math> \tilde{S}_{\nu \times \nu} </math> remains singular, a QR decomposition can be used to reduce G to a product like

<math>

 \mathbf{G}  =  Q \begin{bmatrix} \mathbf{R} \\ 0 \end{bmatrix}

</math> such that <math> \mathbf{R}_{q \times q}, \;\; q \le \nu </math> is upper triangular with full rank and <math> \tilde\tilde{S}_{q \times q} = \mathbf{R^H}\mathbf{R} </math> has further reduced dimensionality.

The eigenvalues are of practical significance in radio communications theory since they define the Shannon channel capacity of a <math> \nu \times p </math> MIMO wireless channel which, to first approximation, is modeled as a zero-mean complex Gaussian ensemble.