Complex Wishart distribution

Complex Wishart

Notation	A ~ CWp([math]\displaystyle{ \Gamma }[/math], n)
Parameters	n > p − 1 degrees of freedom (real) [math]\displaystyle{ \Gamma }[/math] > 0 (p × p Hermitian pos. def)
Support	A (p × p) Hermitian positive definite matrix
PDF	[math]\displaystyle{ \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) } }[/math] [math]\displaystyle{ \mathcal{C}\widetilde{\mathbf{\Gamma}}_p }[/math] is the [math]\displaystyle{ p }[/math]-variate complex multivariate gamma function tr is the trace function
Mean	[math]\displaystyle{ \operatorname{E}[A]=n\Gamma }[/math]
Mode	[math]\displaystyle{ (n-p) \mathbf{\Gamma} }[/math] for n ≥ p + 1
CF	[math]\displaystyle{ \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]\displaystyle{ n }[/math] times the sample Hermitian covariance matrix of [math]\displaystyle{ n }[/math] zero-mean independent Gaussian random variables. It has support for [math]\displaystyle{ p\times p }[/math] Hermitian positive definite matrices.

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

[math]\displaystyle{ S_{p \times p} = \sum_{i=1}^n G_iG_i^H }[/math]

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

[math]\displaystyle{ S \sim n\mathcal{CW}(M,n,p) }[/math]

where [math]\displaystyle{ \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]\displaystyle{ 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| \gt 0 }[/math]

where

[math]\displaystyle{ \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]\displaystyle{ \operatorname{tr}(ABC) = \operatorname{tr}(CAB) }[/math] we also get

[math]\displaystyle{ 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]\displaystyle{ \mathbb{E}[GG^T] = 0 }[/math].

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

[math]\displaystyle{ 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) \gt 0 }[/math]

where [math]\displaystyle{ \mathbf{M} = \mathbf{\Gamma^{-1}} }[/math].

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

[math]\displaystyle{ \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]\displaystyle{ p \times p }[/math] matrix with [math]\displaystyle{ \nu \ge p }[/math] degrees of freedom we have

[math]\displaystyle{ 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\lt j} (\lambda_i - \lambda_j)^2 d\lambda_1 \dots d\lambda_p, \;\;\; \lambda_i \in \mathbb{R} \ge 0 }[/math]

where

[math]\displaystyle{ \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]\displaystyle{ Z = X + iY }[/math] where iid X and Y each have unit variance and the variance of [math]\displaystyle{ 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]\displaystyle{ p = \kappa \nu, \;\; 0 \le \kappa \le 1 }[/math] such that [math]\displaystyle{ S_{p \times p} \sim \mathcal{CW}\left( 2\mathbf{I}, \frac{p}{\kappa} \right) }[/math] then in the limit [math]\displaystyle{ p \rightarrow \infty }[/math] the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function

[math]\displaystyle{ 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]\displaystyle{ \lambda }[/math] by [math]\displaystyle{ 2\lambda }[/math], on account of the doubled sample variance, so in the case [math]\displaystyle{ S_{p \times p} \sim \mathcal{CW} \left( \mathbf{I}, \frac{p}{\kappa} \right) }[/math], the pdf reduces to the real Wishart one:

[math]\displaystyle{ 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]\displaystyle{ \kappa = 1 }[/math]

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ y = \pm\sqrt{\lambda} }[/math] in the latter and selecting the sign of y randomly yielding pdf

[math]\displaystyle{ 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]\displaystyle{ S_{p \times p} = \sum_{j=1}^\nu G_jG_j^H }[/math], we can define a Gaussian ensemble

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

such that S is the matrix product [math]\displaystyle{ 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]\displaystyle{ \mathbf{G} }[/math] and the moduli of the latter have a quarter-circle distribution.

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

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

[math]\displaystyle{ \mathbf{G} = Q \begin{bmatrix} \mathbf{R} \\ 0 \end{bmatrix} }[/math]

such that [math]\displaystyle{ \mathbf{R}_{q \times q}, \;\; q \le \nu }[/math] is upper triangular with full rank and [math]\displaystyle{ \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]\displaystyle{ \nu \times p }[/math] MIMO wireless channel which, to first approximation, is modeled as a zero-mean complex Gaussian ensemble.