Jordan Algebras, the EM Algorithm, and Covariance Matrices

Abstract

Using Jordan algebras, Galois field theory and the EM algorithm we show how to obtain either essentially closed-form or simplified solutions to the ML equations for estimation of the covariance matrix for multivariate normal data in the following situations: (a) Data having a patterned covariance matrix; equivalently, data with a linear covariance structure. This case includes the problems of variance and variance-covariance components estimation; unbalanced repeated measures designs; and some time series models; (b) Data vectors with values missing completely at random; in particular, retrospective analyses of long term, clinical trials, incomplete repeated measures, and time series; (c) The intersection of (a) and (b): multivariate normal data assumed to have a linear covariance structure but with some values missing (completely at random); KeywordsCovariance MatriceStatistical ApplicationJordan AlgebraNuisance ParameterToeplitz MatriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.