The main goal of the work described in this proposal is to introduce and perform a thorough theoretical and numerical analysis of the class of beta random matrix ensembles. The investigators will generalize and extend to any postive beta the classical real, complex, and quaternion random matrix ensembles which correspond to the cases 1, 2, 4, respectively. Already many results in infinite random matrix theory and a few results in finite random matrix theory suggest that the use of beta as a continuous parameter is reasonable. The key new concept in this proposal is the notion of a beta-random variable, an object which, for all practical purposes behaves as a beta-dimensional algebra over the reals. To develop the theory the PIs use the notions of "ghosts" and "shadows". A "ghost" is a beta-dimensional random variable and a "shadow" is a derived real or complex quantity that can be sampled. Along with the derivation of theoretical results, a major goal of this project is to provide algorithms for computation with these random matrix ensembles.
A vast number of practical application ranging from bioinformatics, and genomics (population classification) to wireless communications (network capacity optimization) and military applications (automatic target classification) rely on the methods of multivariate statistics and in turn on random matrix theory. The proposed research will provide new algorithmic and theoretical tools for these applications as well as enable new applications and research directions in these fields.
The research under the above award resulted in the following outcomes: The random matrix technique of ``ghosts’’ and ``shadows’’ was developed. In particular, we introduced the notion of a beta-random Gaussian variable for any positive beta, thus generalizing the notion of real, complex, and quaternion Gaussian random variables to any beta greater than zero. This technique was been instrumental in the development of the other results in our research. We introduced two beta random matrix ensembles, the beta-Wishart ensemble and the beta-MANOVA random matrix ensembles with covariance. These ensembles generalize the classical real, complex, and quaternion counterparts (beta=1,2,4, respectively) to any positive beta. These results generalize earlier results on beta ensembles with identity covariance by Dumitriu and Edelman (2002) to arbitrary covariance. 3. We developed empirical models for each of these ensembles as well as comprehensive eigenvalue theory for each of the above two beta ensembles. We derived explicit expressions for the joint eigenvalue density, the extreme eigenvalues, and trace in terms of the hypergeometric function of a matrix argument. These expressions are well suited for practical computation, which we demonstrate. The software to perform these computations is available to the general public. We established many new identities for the hypergeometric function of a matrix argument, which are of independent theoretical interest, but were also instrumental in the development and establishment of the theoretical properties of the above beta random matrix ensembles. The research supported under this grant has thus far resulted in 9 papers (7 published in refereed journals and 2 submitted as of this writing) and two student theses (one Ph.D. thesis, defended in 2014 and one M.Sc thesis. to be defended in 2015). We presented our findings at a number of national and international conferences. We made our software publicly available. We collaborated with researchers from other fields of science and engineering in our research. As a service to the general mathematical community we transcribed two highly cited, but unpublished manuscripts of Prof. Ian Macdonald on the topic of hypergeometric functions. With Prof. Macdonald’s permission we published these on arXiv.
