Complex inverse Wishart distribution

Complex inverse Wishart Distribution

Notation	${\displaystyle {\mathcal {CW}}^{-1}({\mathbf {\Psi } },\nu ,p)}$
Parameters	${\displaystyle \nu >p-1}$ degrees of freedom (real) ${\displaystyle \mathbf {\Psi } >0}$, ${\displaystyle p\times p}$ scale matrix (pos. def.)
Support	${\displaystyle \mathbf {X} }$ is p × p positive definite Hermitian
PDF	${\displaystyle {\frac {\left|\mathbf {\Psi } \right|^{\nu }}{{\mathcal {C}}\Gamma _{p}(\nu )}}\left|\mathbf {x} \right|^{-(\nu +p)}e^{-\operatorname {tr} (\mathbf {\Psi } \mathbf {x} ^{-1})}}$ ${\displaystyle {\mathcal {C}}\Gamma _{p}}$ is the complex multivariate gamma function ${\displaystyle {\mathcal {C}}\Gamma _{p}(\nu )=\pi ^{{\tfrac {1}{2}}p(p-1)}\prod _{j=1}^{p}\Gamma (\nu -j+1)}$ ${\displaystyle \operatorname {tr} }$ is the trace function
Mean	${\displaystyle {\frac {\mathbf {\Psi } }{\nu -p}}}$ for ${\displaystyle \nu >p+1}$
Variance	see below

The complex inverse Wishart distribution is a matrix probability distribution defined on complex-valued positive-definite matrices and is the complex analog of the real inverse Wishart distribution. The complex Wishart distribution was extensively investigated by Goodman ^[1] while the derivation of the inverse is shown by Shaman ^[2] and others. It has greatest application in least squares optimization theory applied to complex valued data samples in digital radio communications systems, often related to Fourier Domain complex filtering.

Letting ${\displaystyle \mathbf {S} _{p\times p}=\sum _{j=1}^{\nu }G_{j}G_{j}^{H}}$ be the sample covariance of independent complex p-vectors ${\displaystyle G_{j}}$ whose Hermitian covariance has complex Wishart distribution ${\displaystyle \mathbf {S} \sim {\mathcal {CW}}(\mathbf {\Sigma } ,\nu ,p)}$ with mean value ${\displaystyle \mathbf {\Sigma } {\text{ and }}\nu }$ degrees of freedom, then the pdf of ${\displaystyle \mathbf {X} =\mathbf {S^{-1}} }$ follows the complex inverse Wishart distribution.

Density

If ${\displaystyle \mathbf {S} _{p\times p}}$ is a sample from the complex Wishart distribution ${\displaystyle {\mathcal {CW}}({\mathbf {\Sigma } },\nu ,p)}$ such that, in the simplest case, ${\displaystyle \nu \geq p{\text{ and }}\left|\mathbf {S} \right|>0}$ then ${\displaystyle \mathbf {X} =\mathbf {S} ^{-1}}$ is sampled from the inverse complex Wishart distribution ${\displaystyle {\mathcal {CW}}^{-1}({\mathbf {\Psi } },\nu ,p){\text{ where }}\mathbf {\Psi } =\mathbf {\Sigma } ^{-1}}$.

The density function of ${\displaystyle \mathbf {X} }$ is

${\displaystyle f_{\mathbf {x} }(\mathbf {x} )={\frac {\left|\mathbf {\Psi } \right|^{\nu }}{{\mathcal {C}}\Gamma _{p}(\nu )}}\left|\mathbf {x} \right|^{-(\nu +p)}e^{-\operatorname {tr} (\mathbf {\Psi } \mathbf {x} ^{-1})}}$

where ${\displaystyle {\mathcal {C}}\Gamma _{p}(\nu )}$ is the complex multivariate Gamma function

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

Moments

The variances and covariances of the elements of the inverse complex Wishart distribution are shown in Shaman's paper above while Maiwald and Kraus ^[3] determine the 1-st through 4-th moments.

Shaman finds the first moment to be

${\displaystyle \mathbf {E} [{\mathcal {C}}\mathbf {W^{-1}} ]={\frac {1}{n-p}}\mathbf {\Psi ^{-1}} ,\;n>p}$

and, in the simplest case ${\displaystyle \mathbf {\Psi } =\mathbf {I} _{p\times p}}$, given ${\displaystyle d={\frac {1}{n-p}}}$, then

${\displaystyle \mathbf {\mathbf {E} \left[vec({\mathcal {C}}W_{3}^{-1})\right]} ={\begin{bmatrix}d&0&0&0&d&0&0&0&d\\\end{bmatrix}}}$

The vectorised covariance is

${\displaystyle \mathbf {Cov\left[vec({\mathcal {C}}W_{p}^{-1})\right]} =b\left(\mathbf {I} _{p}\otimes I_{p}\right)+c\,\mathbf {vecI_{p}} \left(\mathbf {vecI_{p}} \right)^{T}+(a-b-c)\mathbf {J} }$

where ${\displaystyle \mathbf {J} }$ is a ${\displaystyle p^{2}\times p^{2}}$ identity matrix with ones in diagonal positions ${\displaystyle 1+(p+1)j,\;j=0,1,\dots p-1}$ and ${\displaystyle a,b,c}$ are real constants such that for ${\displaystyle n>p+1}$

${\displaystyle a={\frac {1}{(n-p)^{2}(n-p-1)}}}$, marginal diagonal variances

${\displaystyle b={\frac {1}{(n-p+1)(n-p)(n-p-1)}}}$, off-diagonal variances.

${\displaystyle c={\frac {1}{(n-p+1)(n-p)^{2}(n-p-1)}}}$, intra-diagonal covariances

For ${\displaystyle \mathbf {\Psi } =\mathbf {I} _{3}}$, we get the sparse matrix:

${\displaystyle \mathbf {Cov\left[vec({\mathcal {C}}W_{3}^{-1})\right]} ={\begin{bmatrix}a&\cdot &\cdot &\cdot &c&\cdot &\cdot &\cdot &c\\\cdot &b&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &b&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &b&\cdot &\cdot &\cdot &\cdot &\cdot \\c&\cdot &\cdot &\cdot &a&\cdot &\cdot &\cdot &c\\\cdot &\cdot &\cdot &\cdot &\cdot &b&\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &b&\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &b&\cdot \\c&\cdot &\cdot &\cdot &c&\cdot &\cdot &\cdot &a\\\end{bmatrix}}}$

Eigenvalue distributions

The joint distribution of the real eigenvalues of the inverse complex (and real) Wishart are found in Edelman's paper ^[4] who refers back to an earlier paper by James. ^[5] In the non-singular case, the eigenvalues of the inverse Wishart are simply the inverted values for the Wishart. Edelman also characterises the marginal distributions of the smallest and largest eigenvalues of complex and real Wishart matrices.

References

1. Goodman, N R (1963). "Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution: an Introduction". Ann. Math. Statist. 34 (1): 152–177. doi:10.1214/aoms/1177704250.
2. 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.
3. Maiwald, Dirk; Kraus, Dieter (1997). "On Moments of Complex Wishart and Complex Inverse Wishart Distributed Matrices". 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing. IEEE Icassp 1997. Vol. 5. pp. 3817–3820. doi:10.1109/ICASSP.1997.604712. ISBN   0-8186-7919-0. S2CID   14918978.
4. Edelman, Alan (October 1998). "Eigenvalues and Condition Numbers of Random Matrices". SIAM J. Matrix Anal. Appl. 9 (4): 543–560. doi:10.1137/0609045. hdl:1721.1/14322.
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.