Complex Wishart distribution

Complex Wishart

Notation	A ~ CWp(${\displaystyle \Gamma }$, n)
Parameters	n > p − 1 degrees of freedom (real) ${\displaystyle \Gamma }$ > 0 (p × p Hermitian pos. def)
Support	A (p × p) Hermitian positive definite matrix
PDF	${\displaystyle {\frac {\det \left(\mathbf {A} \right)^{(n-p)}e^{-\operatorname {tr} (\mathbf {\Gamma } ^{-1}\mathbf {A} )}}{\det \left(\mathbf {\Gamma } \right)^{n}\cdot {\mathcal {C}}{\widetilde {\Gamma }}_{p}(n)}}}$ ${\displaystyle {\mathcal {C}}{\widetilde {\mathbf {\Gamma } }}_{p}}$ is the ${\displaystyle p}$-variate complex multivariate gamma function tr is the trace function
Mean	${\displaystyle \operatorname {E} [A]=n\Gamma }$
Mode	${\displaystyle (n-p)\mathbf {\Gamma } }$ for n ≥ p + 1
CF	${\displaystyle \det \left(I_{p}-i\mathbf {\Gamma } \mathbf {\Theta } \right)^{-n}}$

In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of ${\displaystyle n}$ times the sample Hermitian covariance matrix of ${\displaystyle n}$ zero-mean independent Gaussian random variables. It has support for ${\displaystyle p\times p}$ Hermitian positive definite matrices.^[1]

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

${\displaystyle S_{p\times p}=\sum _{i=1}^{n}G_{i}G_{i}^{H}}$

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

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

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

${\displaystyle f_{S}(\mathbf {S} )={\frac {\left|\mathbf {S} \right|^{n-p}e^{-\operatorname {tr} (\mathbf {M} ^{-1}\mathbf {S} )}}{\left|\mathbf {M} \right|^{n}\cdot {\mathcal {C}}{\widetilde {\Gamma }}_{p}(n)}},\;\;\;n\geq p,\;\;\;\left|\mathbf {M} \right|>0}$

where

${\displaystyle {\mathcal {C}}{\widetilde {\Gamma }}_{p}^{}(n)=\pi ^{p(p-1)/2}\prod _{j=1}^{p}\Gamma (n-j+1)}$

is the complex multivariate Gamma function.^[2]

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

${\displaystyle f_{S}(\mathbf {S} )={\frac {\left|\mathbf {S} \right|^{n-p}}{\left|\mathbf {M} \right|^{n}\cdot {\mathcal {C}}{\widetilde {\Gamma }}_{p}(n)}}\exp \left(-\sum _{i=1}^{p}G_{i}^{H}\mathbf {M} ^{-1}G_{i}\right)}$

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

Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of ${\displaystyle \mathbf {Y} =\mathbf {S^{-1}} }$ according to Goodman,^[2] Shaman^[3] is

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

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

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

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

Goodman and others^[4] discuss such complex Jacobians.

Eigenvalues

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James^[5] and Edelman.^[6] For a ${\displaystyle p\times p}$ matrix with ${\displaystyle \nu \geq p}$ degrees of freedom we have

${\displaystyle f(\lambda _{1}\dots \lambda _{p})={\tilde {K}}_{\nu ,p}\exp \left(-{\frac {1}{2}}\sum _{i=1}^{p}\lambda _{i}\right)\prod _{i=1}^{p}\lambda _{i}^{\nu -p}\prod _{i<j}(\lambda _{i}-\lambda _{j})^{2}d\lambda _{1}\dots d\lambda _{p},\;\;\;\lambda _{i}\in \mathbb {R} \geq 0}$

where

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

Note however that Edelman uses the "mathematical" definition of a complex normal variable ${\displaystyle Z=X+iY}$ where iid X and Y each have unit variance and the variance of ${\displaystyle Z=\mathbf {E} \left(X^{2}+Y^{2}\right)=2}$. 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 ${\displaystyle p=\kappa \nu ,\;\;0\leq \kappa \leq 1}$ such that ${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(2\mathbf {I} ,{\frac {p}{\kappa }}\right)}$ then in the limit ${\displaystyle p\rightarrow \infty }$ the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function

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

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

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

A special case is ${\displaystyle \kappa =1}$

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

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

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

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

${\displaystyle p_{y}(y)={\frac {1}{2\pi }}\left(4-y^{2}\right)^{\frac {1}{2}},\;-2\leq y\leq 2}$

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

${\displaystyle \mathbf {G} _{i,j}=[G_{1}\dots G_{\nu }]\in \mathbb {C} ^{\,p\times \nu }}$

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

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

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

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

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

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

References