A Polynomial Approximation Algorithm for Real-Time Maximum-Likelihood Estimation

• Published in: IEEE transactions on signal processing Vol. 57; no. 6; pp. 2085 - 2095
• Main Authors: Villien, C., Ostertag, E.P.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Accuracy; Algorithms; Applied sciences; Approximation; Approximation algorithms; Detection, estimation, filtering, equalization, prediction; Estimators; Exact sciences and technology; Filters; Information, signal and communications theory; Loss measurement; Mathematical analysis; Mathematical models; Maximum likelihood estimation; Measurement standards; Miscellaneous; Parameter estimation; Performance evaluation; polynomial approximation; Polynomials; Raw; Real time; real-time estimation; Signal and communications theory; Signal processing; Signal, noise; Smoothing methods; Studies; Telecommunications and information theory; Time measurement; Performance evaluation; Infinite impulse response filter; Parameter estimation; Polynomial approximation; Non linear function; Processing time; Maximum-likelihood estimation; Polynomial method; Algorithm; Energy savings; Optimization; Distributed processing; Accuracy; Algorithm performance; Signal processing; Maximum likelihood; Real time processing; real-time estimation; Deterministic approach
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2009.2016875

Maximum-likelihood estimation subject to nonlinear measurement functions is generally performed through optimization algorithms when accuracy is required and enough processing time is available, or with recursive filters for real-time applications but at the expense of a loss of accuracy. In this paper, we propose a new estimator for parameter estimation based on a polynomial approximation of the measurement signal. The raw dataset is replaced by n + 1 independent polynomial samples (PS) for a smoothing polynomial of order n, resulting in a reduction of the computational burden. It is shown that the PSs must be sampled at some deterministic instants and an approximate formula for the variance of the PSs is also provided. Moreover, it is also proved and illustrated on three examples that the new estimator which processes the PSs is equivalent to the standard maximum-likelihood estimator based on the raw dataset, provided that the measurement function and its first derivatives can be approximated with a polynomial of order n. Since this algorithm proceeds from a compact representation of a measurement signal, it can find applications in real-time processing, power saving processing, or estimation based on compressed data, even if this latter field has not been investigated from a theoretical perspective. Its structure which is made up of several separate tasks is also adapted to distributed processing problems. Because the performance of the method is related to the polynomial approximation quality, the algorithm is well suited for smooth measurement functions like in trajectory estimation applications.