Fitting polynomials to data in the presence of noise

Procedures are examined for fitting a polynomial to data consisting of uniformly spaced samples of either the polynomial directly or the derivative of the polynomial plus measurement noise. For the time invariant polynomial, the maximum likelihood (ML) and maximum a posteriori (MAP) estimators are presented. For the time varying case, the dynamic system and measurement equations required for a recursive estimator are specified. The estimator variance for a polynomial fit metric of distortion is discussed. The ML and MAP estimator variances reach a lower bound when the measurement noise is zero mean. For measurement noise with unknown means, expressions for the resultant biased estimator excess mean squared error are presented. The important case of a second order polynomial is considered as an example.