Expectations of useful complex Wishart forms

Tague, J.A.; Caldwell, Curtis I.

doi:10.1007/bf00980709

Cited by 89 publications

(42 citation statements)

References 8 publications

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“…For complex Gaussian as above, the use of the Jeffreys prior results in a posterior that is complex Wishart distributed according to [28]. Hence, the posterior mean [29] will equal the standard sample covariance estimate in (7). This supports the use of the Jeffreys prior as the resulting estimate is commonly used.…”

Section: A Jeffreys Priormentioning

confidence: 99%

“…Compared to the Jeffreys prior, this prior is reinforced in a region where all eigenvalues are close to each other. For such a choice, the posterior distribution of is complex Wishart according to , and the usage of this prior gives [29] …”

Section: B Family Of Jeffreys-like Priorsmentioning

confidence: 99%

“…Finding the posterior distribution in (35) requires both and . The first is given by (29), recalling that here, and , whereas the second can be expressed as (37) Considering the previous discussions, choosing a prior is not a trivial task. For convenience, we use a uniform prior , hoping that the choice of prior is of minor importance.…”

Section: Treating the Regularization Parametermentioning

confidence: 99%

“…As the mean is independent of , we have . Further, since , then ; see, e.g., [29]. The covariance matrix of is thereby given as Cov (57) Moreover, writing (50) in vectorized form gives (58)…”

Section: Appendixmentioning

confidence: 99%

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On posterior distributions for signals in Gaussian noise with unknown covariance matrix

Svensson

Lundberg

2005

IEEE Trans. Signal Process.

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Abstract-A Bayesian approach to estimate parameters of signals embedded in complex Gaussian noise with unknown color is presented. The study specifically focuses on a Bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems. By integrating out uncertainties regarding the noise color, an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited. Several noninformative priors for the covariance matrix, such as the reference prior, the Jeffreys prior, and modifications to this, are considered. Some of the priors result in analytical solutions, whereas others demand numerical approximations. In the linear signal model, connections are made between the standard Adaptive Maximum Likelihood (AML) estimate and a Bayesian solution using the Jeffreys prior. With adjustments to the Jeffreys prior, correspondence to the regularized solution is also established. This in turn enables a formal treatment of the regularization parameter. Simulations indicate that significant improvements, compared to the AML estimator, can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior. The simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged.

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Section: A Jeffreys Priormentioning

confidence: 99%

Section: B Family Of Jeffreys-like Priorsmentioning

confidence: 99%

Section: Treating the Regularization Parametermentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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On posterior distributions for signals in Gaussian noise with unknown covariance matrix

Svensson

Lundberg

2005

IEEE Trans. Signal Process.

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“…Shaman [25] gives the first and second moments of the inverted complex Wishart distribution. Tague and Caldwell [29] derive expressions for E(WAW ) and E(W −1 AW −1 ) when W is a complex Wishart matrix and A a constant matrix. Sultan and Tracy [28] find the first four moments of the central and noncentral complex Wishart distributions by differentiating their characteristic functions using a matrix differential operator.…”

Section: Introductionmentioning

confidence: 99%

Moments and cumulants for the complex Wishart

Withers¹,

Nadarajah

2012

Journal of Multivariate Analysis

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a b s t r a c tWe summarize the main results known for the complex normal and complex Wishart, then give the cumulants of the central and noncentral complex Wishart. Their moments are expressed explicitly in terms of multivariate Bell polynomials, believed to be used here for the first time. Multivariate Bell polynomials are easily written down from their univariate forms, which are widely accessible in most computer algebra packages. This is shown to be the natural way of obtaining the moments for any sum of independent and identically distributed (i.i.d.) random variables. An extension is given to the weighted complex Wishart.

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References

2010

Cooperative Communications

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Expectations of useful complex Wishart forms

Cited by 89 publications

References 8 publications

On posterior distributions for signals in Gaussian noise with unknown covariance matrix

On posterior distributions for signals in Gaussian noise with unknown covariance matrix

Moments and cumulants for the complex Wishart

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