Degree and noise power estimation from noisy polynomial data via AR modelling

Abstract

Copyright © 2021 The Author(s). An accurate estimation of the noise power from noisy data leads to better estimation of signal-to-noise ratio (SNR) and is useful in detection, estimation, and prediction. The major contributions of this paper are to estimate the polynomial degree and the noise power from data coming from an underlying polynomial with additive Gaussian noise, using an AR model. The two proposed methods have been inspired by the recent results that all finite degree polynomials have equivalent representation in finite order autoregressive (AR) models, with known AR coefficients and different constant terms. Preliminary experiments in a variety of scenarios provide estimations of the constant term and the standard deviation of these estimations, which are then used as a guide to developing theoretically the probability density functions. In the first stage, the degree of a polynomial is selected by minimizing the variance of the estimations of the constant term in the equivalent AR model. In the second stage, the noise variance is estimated using the estimated degree of a polynomial, a combination of the variance of the estimations of the constant term, and another known parameter. Further computer experiments have been carried out for evaluating the proposed methods for degree and noise power estimations. Four well-known and well-regarded maximum likelihood-based approaches have been used for comparisons.