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

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.