Nonlinear estimation with a known covariance structure over time

Abstract

This dissertation provides a means of parameter estimation for nonlinear models in which the observations are known to be correlated over time. The proposed estimation procedure, by incorporating the known (as a function of the parameters to be estimated) covariance structure of the observations, provides RBAN estimators for the parameters. In addition, identification of the asymptotic covariance matrix of the estimators makes possible the construction of approximate confidence intervals and regions for the parameters. Since the need for such an estimation procedure has been apparent in the area of compartments analysis, a large portion of this paper is devoted to describing the application of the recommended procedure to certain compartmental models for which the covariance structure of the observations over time is known. For each particular model considered, certain secondary results which enhance the general procedure as applied to that model, are presented. In particular, the explicit form of the inverse of the covariance matrix for a two-compartment model with reversible flows is identified. For purposes of illustration, the proposed estimation procedure is applied to simulated data, and comparisons are made to the widely used nonlinear least squares procedure, which does not allow for correlations over time. The dissertation concludes with a brief discussion of some more-general models, as well as an outline of several possible areas of future research.