Complex Wishart distribution

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

The complex Wishart distribution is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels.^[2]

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

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

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

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

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

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

${\displaystyle {\mathcal {C}}{\widetilde {\Gamma }}_{p}^{}(n)=\pi ^{p(p-1)/2}\prod _{j=1}^{p}\Gamma (n-j+1)}${\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.^[3]

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

${\displaystyle f_{S}(\mathbf {S} )={\frac {\left|\mathbf {S} \right|^{n-p}}{\left|\mathbf {M} \right|^{n}\cdot {\mathcal {C}}{\widetilde {\Gamma }}_{p}(n)}}\exp \left(-\sum _{i=1}^{p}G_{i}^{H}\mathbf {M} ^{-1}G_{i}\right)}${\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 ${\displaystyle \mathbb {E} [GG^{T}]=0}${\displaystyle \mathbb {E} [GG^{T}]=0}.

Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of ${\displaystyle \mathbf {Y} =\mathbf {S^{-1}} }${\displaystyle \mathbf {Y} =\mathbf {S^{-1}} } according to Goodman,^[3] Shaman^[4] is

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

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

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

Goodman and others^[5] discuss such complex Jacobians.

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James^[6] and Edelman.^[7] For a ${\displaystyle p\times p}${\displaystyle p\times p} matrix with ${\displaystyle \nu \geq p}${\displaystyle \nu \geq p} degrees of freedom we have

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

${\displaystyle {\tilde {K}}_{\nu ,p}^{-1}=2^{p\nu }\prod _{i=1}^{p}\Gamma (\nu -i+1)\Gamma (p-i+1)}${\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 ${\displaystyle Z=X+iY}${\displaystyle Z=X+iY} where iid X and Y each have unit variance and the variance of ${\displaystyle Z=\mathbf {E} \left(X^{2}+Y^{2}\right)=2}${\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.

This spectral density can be integrated to give the probability that all eigenvalues of a Wishart random matrix lie within an interval.^[8]

The spectral density can be also integrated to give the marginal distribution of eigenvalues.^{[9] [10]}

There are also approximations for marginal eigenvalue distributions. From Edelman we have that if S is a sample from the complex Wishart distribution with ${\displaystyle p=\kappa \nu ,\;\;0\leq \kappa \leq 1}${\displaystyle p=\kappa \nu ,\;\;0\leq \kappa \leq 1} such that ${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(2\mathbf {I} ,{\frac {p}{\kappa }}\right)}${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(2\mathbf {I} ,{\frac {p}{\kappa }}\right)} then in the limit ${\displaystyle p\rightarrow \infty }${\displaystyle p\rightarrow \infty } the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution function

${\displaystyle p_{\lambda }(\lambda )={\frac {\sqrt {[\lambda /2-({\sqrt {\kappa }}-1)^{2}][{\sqrt {\kappa }}+1)^{2}-\lambda /2]}}{4\pi \kappa (\lambda /2)}},\;\;\;2({\sqrt {\kappa }}-1)^{2}\leq \lambda \leq 2({\sqrt {\kappa }}+1)^{2},\;\;\;0\leq \kappa \leq 1}${\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 ${\displaystyle \lambda }${\displaystyle \lambda } by ${\displaystyle 2\lambda }${\displaystyle 2\lambda }, on account of the doubled sample variance, so in the case ${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(\mathbf {I} ,{\frac {p}{\kappa }}\right)}${\displaystyle S_{p\times p}\sim {\mathcal {CW}}\left(\mathbf {I} ,{\frac {p}{\kappa }}\right)}, the pdf reduces to the real Wishart one:

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

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

${\displaystyle p_{\lambda }(\lambda )={\frac {1}{2\pi }}\left({\frac {4-\lambda }{\lambda }}\right)^{\frac {1}{2}},\;0\leq \lambda \leq 4}${\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 ${\displaystyle y=\pm {\sqrt {\lambda }}}${\displaystyle y=\pm {\sqrt {\lambda }}} in the latter and selecting the sign of y randomly yielding pdf

${\displaystyle p_{y}(y)={\frac {1}{2\pi }}\left(4-y^{2}\right)^{\frac {1}{2}},\;-2\leq y\leq 2}${\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, ${\displaystyle S_{p\times p}=\sum _{j=1}^{\nu }G_{j}G_{j}^{H}}${\displaystyle S_{p\times p}=\sum _{j=1}^{\nu }G_{j}G_{j}^{H}}, we can define a Gaussian ensemble

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

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

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

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

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

such that ${\displaystyle \mathbf {R} _{q\times q},\;\;q\leq \nu }${\displaystyle \mathbf {R} _{q\times q},\;\;q\leq \nu } is upper triangular with full rank and ${\displaystyle {\tilde {\tilde {S}}}_{q\times q}=\mathbf {R^{H}} \mathbf {R} }${\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 ${\displaystyle \nu \times p}${\displaystyle \nu \times p} MIMO wireless channel which, to first approximation, is modeled as a zero-mean complex Gaussian ensemble.

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