Complex Wishart distribution

Complex Wishart
Notation	A ~ CWp(, n)
Parameters	n > p − 1 degrees of freedom (real); > 0 (p × p Hermitian pos. def)
Support	A (p × p) Hermitian positive definite matrix
PDF	is the -variate complex multivariate gamma function; tr is the trace function;
Mean
Mode	for n ≥ p + 1
CF

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.^[1]

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

S_{p\times p}=\sum _{i=1}^{n}G_{i}G_{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\geq 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.^[2]

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,^[2] Shaman^[3] 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\geq 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-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 $p\times p$ matrix with $\nu \geq p$ 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<j}(\lambda _{i}-\lambda _{j})^{2}d\lambda _{1}\dots d\lambda _{p},\;\;\;\lambda _{i}\in \mathbb {R} \geq 0

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 $p=\kappa \nu ,\;\;0\leq \kappa \leq 1$ 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}\leq \lambda \leq 2({\sqrt {\kappa }}+1)^{2},\;\;\;0\leq \kappa \leq 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}\leq \lambda \leq ({\sqrt {\kappa }}+1)^{2},\;\;\;0\leq \kappa \leq 1

A special case is $\kappa =1$

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

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 $y=\pm {\sqrt {\lambda }}$ 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\leq y\leq 2

In place of the definition of the Wishart sample matrix above, $S_{p\times p}=\sum _{j=1}^{\nu }G_{j}G_{j}^{H}$ , 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 $\mathbf {R} _{q\times q},\;\;q\leq \nu$ 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

^ N. R. Goodman (1963). "The distribution of the determinant of a complex Wishart distributed matrix". The Annals of Mathematical Statistics. 34 (1): 178–180. doi:10.1214/aoms/1177704251.
^ ^a ^b Goodman, N R (1963). "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)". Ann. Math. Statist. 34: 152–177. doi:10.1214/aoms/1177704250.
^ Shaman, Paul (1980). "The Inverted Complex Wishart Distribution and Its Application to Spectral Estimation". Journal of Multivariate Analysis. 10: 51–59. doi:10.1016/0047-259X(80)90081-0.
^ Cross, D J (May 2008). "On the Relation between Real and Complex Jacobian Determinants" (PDF). drexel.edu.
^ James, A. T. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Ann. Math. Statist. 35 (2): 475–501. doi:10.1214/aoms/1177703550.
^ Edelman, Alan (October 1988). "Eigenvalues and Condition Numbers of Random Matrices" (PDF). SIAM J. Matrix Anal. Appl. 9 (4): 543–560. doi:10.1137/0609045. hdl:1721.1/14322.

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[1] N. R. Goodman (1963). "The distribution of the determinant of a complex Wishart distributed matrix". The Annals of Mathematical Statistics. 34 (1): 178–180. doi:10.1214/aoms/1177704251.

[Goodman1963-2] Goodman, N R (1963). "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)". Ann. Math. Statist. 34: 152–177. doi:10.1214/aoms/1177704250.

[3] Shaman, Paul (1980). "The Inverted Complex Wishart Distribution and Its Application to Spectral Estimation". Journal of Multivariate Analysis. 10: 51–59. doi:10.1016/0047-259X(80)90081-0.

[4] Cross, D J (May 2008). "On the Relation between Real and Complex Jacobian Determinants" (PDF). drexel.edu.

[5] James, A. T. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Ann. Math. Statist. 35 (2): 475–501. doi:10.1214/aoms/1177703550.

[6] Edelman, Alan (October 1988). "Eigenvalues and Condition Numbers of Random Matrices" (PDF). SIAM J. Matrix Anal. Appl. 9 (4): 543–560. doi:10.1137/0609045. hdl:1721.1/14322.

[1]

Notation	$\Gamma$
Parameters	$n > p - 1$ degrees of freedom (real) $\Gamma$ ( $p \times p$ Hermitian pos. def)
Support	$A (p \times p)$ 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 $tr$ is the trace function
Mean	$\operatorname {E} [A]=n\Gamma$
Mode	$(n-p)\mathbf {\Gamma }$ for $n \geq p + 1$
CF	$\det \left(I_{p}-i\mathbf {\Gamma } \mathbf {\Theta } \right)^{-n}$