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“Parametric Spectrum Estimation for Contaminated Random Fields”
by Richard R. Hansen, Jr.
December 1988
In this dissertation we propose and investigate new methods of parametric spectrum estimation for two-dimensional random fields which are adequately represented with spatial interaction models. The maximum likelihood (ML) estimator is considered optimal for strictly Gaussian fields and, because of its invariance property, yields the ML estimate of the spectrum. However, many observed fields do not conform to strict distributional assumptions because in inherent contamination (noise) or other isolated imperfections (outliers) resulting from the data measurement and recording processes.
When the observed signal is Gaussian we show that a noncausal autoagressive signal plus noise model that accounts for possible contamination yields better spectrum estimates than conventional signal only models. The theoretical properties of the ML parameter estimates for noncausal autoagressive plus noise model are stated and proved, and the numerical properties are experimentally evaluated and compared with the conventional signal only model for direction of arrival estimation in planar array signal processing.
When the Gaussian assumption is suspect we propose robust techniques for estimating the parameters of spatial interaction model spectra. First, we extend the time series generalized M-estimator to two-dimensional nonsymmetric half-plane and Gaussian-Markov random field models, and then we analyze this robust estimator's performance in series of parameter and spectrum estimation experiments.
The generalized M-estimator will not perform well for noncausal autoagressive models. Additionally, the covariance and conditional probability structure of noncausal models preclude a solution to the optimal robust problem. Thus, we work directly from the observed data and propose empirical estimators for the noncausal autoagressive and Gaussian-Markov random field models. The robust empirical estimators performance in the contaminated situation is shown through a series of experiments to be better than the performance of optimal methods based on a strict Gaussian assumption.