In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of
times the sample Hermitian covariance matrix of
zero-mean independent Gaussian random variables. It has support for
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae43cd3ecc984afc2106d75776bedb0a4a40ca0a)
where each
is an independent column p-vector of random complex Gaussian zero-mean samples and
is an Hermitian (complex conjugate) transpose. If the covariance of G is
then
![{\displaystyle S\sim n{\mathcal {CW}}(M,n,p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37d99aa87a6a853a3b45ffdc8b81f2ff791fa20)
where
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47afe4390115510221927ae1c431281660e22871)
where
![{\displaystyle {\mathcal {C}}{\widetilde {\Gamma }}_{p}^{}(n)=\pi ^{p(p-1)/2}\prod _{j=1}^{p}\Gamma (n-j+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2000850314555772a4e1717976ed2fae30e14f5c)
is the complex multivariate Gamma function.[2]
Using the trace rotation rule
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c36d4a41a8c69b8acfccacfd4d36c7fd5969e48)
which is quite close to the complex multivariate pdf of G itself. The elements of G conventionally have circular symmetry such that
.
Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65ef835eea398e218cbb148bf47ddb9ce4362181)
where
.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71b5acc33259a2c6dd2319ca883a1e80fd56773e)
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
matrix with
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eeba92426ace6d72758511ae235f46306e36c5c)
where
![{\displaystyle {\tilde {K}}_{\nu ,p}^{-1}=2^{p\nu }\prod _{i=1}^{p}\Gamma (\nu -i+1)\Gamma (p-i+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc45ca26cbbf582e3624be70175d7ad308fd4e9)
Note however that Edelman uses the "mathematical" definition of a complex normal variable
where iid X and Y each have unit variance and the variance of
. 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
such that
then in the limit
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaaee4158d7953e7e3cbca03fb2fa270cda943ea)
This distribution becomes identical to the real Wishart case, by replacing
by
, on account of the doubled sample variance, so in the case
, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a57700bfbfce4f11915b621a09a40998ececbf9)
A special case is
![{\displaystyle p_{\lambda }(\lambda )={\frac {1}{4\pi }}\left({\frac {8-\lambda }{\lambda }}\right)^{\frac {1}{2}},\;0\leq \lambda \leq 8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f56229fe1da497a59b0031098e3c80cf4f3b2f)
or, if a Var(Z) = 1 convention is used then
.
The Wigner semicircle distribution arises by making the change of variable
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da85af33475139bea5f75a8207c474b1687dd1a6)
In place of the definition of the Wishart sample matrix above,
, we can define a Gaussian ensemble
![{\displaystyle \mathbf {G} _{i,j}=[G_{1}\dots G_{\nu }]\in \mathbb {C} ^{\,p\times \nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93416a61d716e3bf3ae66c0f9dd62cda961a9910)
such that S is the matrix product
. The real non-negative eigenvalues of S are then the modulus-squared singular values of the ensemble
and the moduli of the latter have a quarter-circle distribution.
In the case
such that
then
is rank deficient with at least
null eigenvalues. However the singular values of
are invariant under transposition so, redefining
, then
has a complex Wishart distribution, has full rank almost certainly, and eigenvalue distributions can be obtained from
in lieu, using all the previous equations.
In cases where the columns of
are not linearly independent and
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e59d1f8edba7d63e3921d3ad670da5ef0750dbc6)
such that
is upper triangular with full rank and
has further reduced dimensionality.
The eigenvalues are of practical significance in radio communications theory since they define the Shannon channel capacity of a
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.