complex Wishart distribution

{{Probability distribution

| name = Complex Wishart

| type =density

| pdf_image =

| cdf_image =

| notation ={{math|A ~ CW_p( $\Gamma$ , n)}}

| parameters ={{math|n > p − 1}} degrees of freedom (real)
{{math| $\Gamma$ > 0}} ({{math|p × p}} Hermitian pos. def)

| support ={{math|A (p × p)}} Hermitian positive definite matrix

| pdf = $\frac{
\det\left(\mathbf{A}\right)^{(n-p)} e^{-\operatorname{tr}(\mathbf{\Gamma}^{-1}\mathbf{A})}
}{
\det\left(\mathbf{\Gamma}\right)^{n}\cdot \mathcal{C} \widetilde{\Gamma}_p(n)
}$

$\mathcal{C}\widetilde{\mathbf{\Gamma}}_p$ is the $p$ -variate complex multivariate gamma function
{{math|tr}} is the trace function

| cdf =

| mean = $\operatorname{E}[A]=n\Gamma$ |

| median =

| mode = $(n-p) \mathbf{\Gamma}$ for {{math|n ≥ p + 1}}

| variance =

| skewness =

| kurtosis =

| entropy =

| mgf =

| char = $\det\left(I_p-i\mathbf{\Gamma}\mathbf{\Theta}\right)^{-n}$

}}

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.{{cite journal |author= N. R. Goodman |year= 1963 |title= The distribution of the determinant of a complex Wishart distributed matrix | journal =The Annals of Mathematical Statistics |volume= 34 |number= 1 |pages= 178–180|doi= 10.1214/aoms/1177704251 |doi-access= free }}

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

: $S_{p \times p} = \sum_{i=1}^n G_iG_i^H$

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 $\mathbb{E}[GG^H] = M$ then

: $S \sim n\mathcal{CW}(M,n,p)$

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

: $f_S(\mathbf{S}) = \frac{
\left |\mathbf{S} \right|^{n-p} e^{-\operatorname{tr}(\mathbf M^{-1}\mathbf{S}) }
}
{
\left|\mathbf{M}\right|^n\cdot \mathcal{C} \widetilde{\Gamma}_p(n)
}, \;\;\; n\ge p, \;\;\; \left|\mathbf{M}\right| > 0$

where

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

is the complex multivariate Gamma function.{{Cite journal|last=Goodman|first=N R|date=1963|title=Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)|journal=Ann. Math. Statist.|volume=34|pages=152–177|doi=10.1214/aoms/1177704250|doi-access=free}}

Using the trace rotation rule $\operatorname{tr}(ABC) = \operatorname{tr}(CAB)$ we also get

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

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

Inverse Complex Wishart

The distribution of the inverse complex Wishart distribution of $\mathbf{Y} = \mathbf{S^{-1}}$ according to Goodman,{{Cite journal|last=Goodman|first=N R|date=1963|title=Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)|journal=Ann. Math. Statist.|volume=34|pages=152–177|doi=10.1214/aoms/1177704250|doi-access=free}} Shaman{{Cite journal|last=Shaman|first=Paul|date=1980|title=The Inverted Complex Wishart Distribution and Its Application to Spectral Estimation|journal=Journal of Multivariate Analysis|volume=10|pages=51–59|doi=10.1016/0047-259X(80)90081-0|doi-access=free}} is

: $f_Y(\mathbf{Y}) = \frac{
\left |\mathbf{Y} \right|^{-(n+p)} e^{-\operatorname{tr}(\mathbf M\mathbf{Y^{-1}}) }
}
{
\left|\mathbf{M}\right|^{-n}\cdot\mathcal{C}\widetilde{\Gamma}_p(n)
}, \;\;\; n\ge p, \;\;\; \det \left(\mathbf{Y}\right) > 0$

where $\mathbf{M} = \mathbf{\Gamma^{-1}}$ .

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

: $\mathcal{C}J_Y(Y^{-1}) = \left | Y \right |^{-2p}$

Goodman and others{{Cite web|url=http://www.physics.drexel.edu/~dcross/academics/papers/jacobian.pdf|title=On the Relation between Real and Complex Jacobian Determinants|last=Cross|first=D J|date=May 2008|website=drexel.edu}} discuss such complex Jacobians.

Eigenvalues

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James{{Cite journal|last=James|first=A. T.|date=1964|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples|journal=Ann. Math. Statist.|volume=35|issue=2|pages=475–501|doi=10.1214/aoms/1177703550|doi-access=free}} and Edelman.{{Cite journal|last=Edelman|first=Alan|date=October 1988|title=Eigenvalues and Condition Numbers of Random Matrices|url=https://dspace.mit.edu/bitstream/1721.1/14322/2/21864285-MIT.pdf|journal=SIAM J. Matrix Anal. Appl.|volume=9 |issue=4|pages=543–560|doi=10.1137/0609045|hdl=1721.1/14322|hdl-access=free}} For a $p \times p$ matrix with $\nu \ge p$ degrees of freedom we have

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

where

: $\tilde {K}_{\nu,p}^{-1} = 2^{p\nu} \prod_{i=1}^p \Gamma (\nu - i+1) \Gamma (p-i+1)$

Note however that Edelman uses the "mathematical" definition of a complex normal variable $Z = X + iY$ where iid X and Y each have unit variance and the variance of $Z = \mathbf{E} \left(X^2 + Y^2 \right ) = 2$ . 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 $p = \kappa \nu, \;\; 0 \le \kappa \le 1$ such that $S_{p \times p} \sim \mathcal{CW}\left( 2\mathbf{I}, \frac{p}{\kappa} \right)$

then in the limit $p \rightarrow \infty$ the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function

: $p_\lambda(\lambda) = \frac
{\sqrt { [\lambda/2 - ( \sqrt {\kappa} -1 )^2 ][\sqrt {\kappa} +1 )^2 - \lambda /2 ] }}
{ 4\pi \kappa (\lambda /2)}, \;\;\; 2( \sqrt {\kappa} -1)^2 \le \lambda \le 2(\sqrt {\kappa} +1 )^2,
\;\;\; 0 \le \kappa \le 1$

This distribution becomes identical to the real Wishart case, by replacing $\lambda$ by $2\lambda$ , on account of the doubled sample variance, so in the case $S_{p \times p} \sim \mathcal{CW} \left( \mathbf{I}, \frac{p}{\kappa} \right)$ , the pdf reduces to the real Wishart one:

: $p_\lambda(\lambda) = \frac
{\sqrt {[\lambda - ( \sqrt {\kappa} -1 )^2 ][\sqrt {\kappa} +1 )^2 - \lambda ] }}
{ 2\pi \kappa \lambda}, \;\;\; (\sqrt {\kappa} -1)^2 \le \lambda \le (\sqrt {\kappa} +1 )^2,
\;\;\; 0 \le \kappa \le 1$

A special case is $\kappa = 1$

: $p_\lambda(\lambda) = \frac {1}{4\pi} \left (\frac {8-\lambda}{\lambda} \right )^{\frac{1}{2}}, \; 0 \le \lambda \le 8$

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

: $p_\lambda(\lambda) = \frac {1}{2\pi} \left (\frac {4-\lambda}{\lambda} \right )^{\frac{1}{2}}, \; 0 \le \lambda \le 4$ .

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

: $p_y(y) = \frac {1}{2\pi} \left ( 4-y^2 \right )^{\frac{1}{2}}, \; -2 \le y \le 2$

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

: $\mathbf{G}_{i,j} = [G_1 \dots G_\nu ] \in \mathbb{C}^{\,p \times \nu }$

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

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

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

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

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

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

References

Category:Continuous distributions

Category:Multivariate continuous distributions

Category:Covariance and correlation

Category:Random matrices

Category:Conjugate prior distributions

Category:Exponential family distributions

Category:Complex distributions