Efficient Maximum Likelihood Identification of a Positive Semi-Definite Covariance of Initial Population Statistics

A method is presented for constrained maximum likelihood identification of the (positive semi-definite) initial covariance of an otherwise known linear discrete time dynamical system, with guaranteed positive semi-definite estimate at each step. The technique is a modification of Newton-Raphson or Scoring procedures transformed linearly to the space of the Cholesky square root matrix. The required algorithm is specified completely, and numerical and analytic difficulties and their solutions are discussed. It is shown that in cases of interest this procedure can result in order of magnitude reduction in computational costs compared to other iterative ML schemes which guarantee a semi-definite covariance estimate at each step. Formal extension to the maximum likelihood identification of time constants and power spectral densities is presented.