Complex Wishart distribution explained

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 x p Hermitian positive definite matrices.^[1]

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

S _{p} = \sum ^{n} _{i =1} G _{iG} ^{H} ^{i} 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 E [GG ^{H]} = M thenS \sim n l{CW}(M,n,p) where l{CW}(M,n,p) is the complex central Wishart distribution with n degrees of freedom and mean value, or scale matrix, M.f _{S(S)} = (\left | S \right| ^{n-p} e ^{-\operatorname{tr} ( M ^{-1} S ))/(} { \left| M \right|) ^{n ⋅} l{C} \widetilde{\Gamma} _{p(n) },} n\ge p, \left| M \right| > 0 wherel{C} \widetilde{\Gamma} _{p} (n) = \pi ^{p(p-1)/2} \prod ^{p} _{j =1} \Gamma (n - j +1 ) is the complex multivariate Gamma function.^[2]

Using the trace rotation rule

\operatorname{tr}(ABC) = \operatorname{tr}(CAB) we also getf _{S(S)} = (\left | S \right| ^{n-p})/(\left| M \right| ^{n ⋅} l{C) \widetilde{\Gamma} _{p(n)} } \exp \left( - \sum ^{p} _{i =1} G ^{H M} _{i} ^{-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 E [GG ^{T]} = 0 .

Inverse Complex WishartThe distribution of the inverse complex Wishart distribution of

Y = S ^{-1} according to Goodman,^[3] Shaman^[4] isf _{Y(Y)} = (\left | Y \right| ^{-(n+p)} e ^{-\operatorname{tr} ( M Y ^{-1} ))/(} { \left| M \right|) ^{-n} ⋅ l{C}\widetilde{\Gamma} _{p(n) },} n\ge p, \det \left( Y \right) > 0 where M = \Gamma ^{-1} .

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

l{C}J ^{-1} _{Y(Y} ) = \left | Y \right | ^{-2p-2}

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

Eigenvalues

The probability distribution of the eigenvalues of the complex Hermitian Wishart distribution are given by, for example, James^[6] and Edelman.^[7] For a

p x p matrix with \nu \ge p degrees of freedom we have f( λ _{1...λ} _{p)=\tilde} {K} _{\nu,p} \exp \left ( - (1)/(2) \sum ^{p} _{i =1} λ _{i} \right ) \prod ^{p} _{i =1} λ ^{\nu - p} _{i} \prod _{i<j} ( λ _{i} - λ ^{2} _{j)} d λ _{1} ... d λ _{p,} λ _{i} \in R \ge 0 where \tilde {K} ^{-1} _{\nu,p} = 2 ^{p\nu} \prod ^{p} _{i =1} \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 = 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 \le \kappa \le 1 such that S _{p} \sim l{CW}\left( 2 I , (p)/(\kappa) \right) then in the limit p → inf ty the distribution of eigenvalues converges in probability to the Marchenko–Pastur distribution functionp _{λ(λ)} = (\sqrt { [ λ / 2- ( \sqrt {\kappa)/(-1) ) ^{2} ][\sqrt {\kappa} +1 ) ^{2} - λ / 2 ] }} { 4 \pi \kappa ( λ / 2 )}, 2 ( \sqrt {\kappa} -1 ) ^{2} \le λ \le 2 (\sqrt {\kappa} +1 ) ^{2,} 0 \le \kappa \le 1 This distribution becomes identical to the real Wishart case, by replacing λ by 2λ , on account of the doubled sample variance, so in the case S _{p} \sim l{CW} \left( I , (p)/(\kappa) \right) , the pdf reduces to the real Wishart one:p _{λ(λ)} = (\sqrt {[ λ- ( \sqrt {\kappa)/(-1) ) ^{2} ][\sqrt {\kappa} +1 ) ^{2} - λ ] }} { 2 \pi \kappa λ }, (\sqrt {\kappa} -1 ) ^{2} \le λ \le (\sqrt {\kappa} +1 ) ^{2,} 0 \le \kappa \le 1

A special case is

\kappa = 1 p _{λ(λ)} = (1)/(4 \pi) \left ( (8-λ)/(λ) \right ) ^{(1)/(2)} , 0 \le λ \le 8 or, if a Var(Z) = 1 convention is used thenp _{λ(λ)} = (1)/(2 \pi) \left ( (4-λ)/(λ) \right ) ^{(1)/(2)} , 0 \le λ \le 4 .The Wigner semicircle distribution arises by making the change of variable y = \pm\sqrt{ λ } in the latter and selecting the sign of y randomly yielding pdfp _{y(y)} = (1)/(2 \pi) \left ( 4- y ^{2} \right ) ^{(1)/(2)} , -2 \le y \le 2

In place of the definition of the Wishart sample matrix above,

S _{p} = \sum ^{\nu} _{j =1} G _{jG} ^{H} ^{j} , we can define a Gaussian ensembleG _{i,j} = [G _{1} ... G _{\nu} ] \in C ^{p} such that S is the matrix product S = G G ^{H} . The real non-negative eigenvalues of S are then the modulus-squared singular values of the ensemble 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 G are invariant under transposition so, redefining \tilde{S} = G ^{H} G , then \tilde{S} _{\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

G are not linearly independent and \tilde{S} _{\nu} remains singular, a QR decomposition can be used to reduce G to a product likeG = Q \begin{bmatrix} R \ 0 \end{bmatrix} such that R _{q} , q \le \nu is upper triangular with full rank and \tilde\tilde{S} _{q} = R ^{H} 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 x p MIMO wireless channel which, to first approximation, is modeled as a zero-mean complex Gaussian ensemble.

Notes and References

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