Time delay estimation for inhomogeneous Poisson processes in the presence of Gaussian noise.

Abstract

This thesis develops Maximum Likelihood based approaches to the estimation of Poisson intensity parameters for filtered Poisson processes observed in additive Gaussian measurement noise. While the methods developed are generally applicable to arbitrary intensity parameterizations, the thesis focuses on the case of time shift parameters. The general estimation problem arises in applications where one is interested in the intensity of a sequence of partially observed discrete events occurring at random points in time or space. Examples include: photon detection for optical communications, positron emission tomography and high energy physics; processing of seismic reflections for geological remote sensing and oil exploration; and neurological activity evaluation based on evoked compound action potentials. For the observation model considered here, the exact likelihood function is analytically intractable due to imperfect observation of the Poisson point process. This intractability is attributable to the presence of additive noise and to the distortion due to filtering the points of the process. Two classes of approximations to the Maximum Likelihood estimator are developed. The first class consists of asymptotic forms of the exact likelihood function under various limiting regimes such as: low measurement noise power, high filter bandwidth, high and low average intensity levels. These asymptotic expressions are of closed form and can be maximized directly to find the maximum likelihood estimate under the respective limiting regimes. The second class of approximations is based on an iterative method founded on the Expectation-Maximization algorithm. This iterative Maximum Likelihood method alternates between two successive operations: approximation of the log-likelihood function for the perfectly observed Poisson process; and maximization of this approximate log-likelihood over unknown parameter values. A linearization of the EM algorithm is developed for the filtered Poisson process model and its convergence properties are analyzed. Results of simulations are given which indicate that significant improvements in estimator performance can be obtained in only a few iterations for time delay estimation. In particular, after less than five iterations, the algorithm yields estimates which are unbiased and come very close to achieving a theoretical lower bound on mean-square estimation error of any unbiased estimator.