Polynomial least squares multiple-model estimation: simple, optimal, adaptive, practical

This paper presents a polynomial least squares (LS) method of overcoming numerous weaknesses and deficiencies in the interacting multiple model (IMM), which comprises multiple Kalman filter models as the state-of-the-art algorithm for tracking maneuvering targets. The paper also addresses several polynomial LS misunderstandings and flaws applied in econometrics. These aims are achieved by first uniquely deriving a very simple version of conventional LS, which fits discrete deterministic data by minimizing the sum of squared deviations of data from an assumed polynomial (often called data-fitting). The contrasting contemporary polynomial LS method filters out corrupting statistical noise from already existing known polynomials, as in target tracking. Contemporary polynomial LS is developed from the conventional LS method derived by Gauss. Polynomial LS conventionally applied to data-fitting and contemporarily applied to noise filtering are shown to be different problems that are not to be conflated as done in econometrics, where data-fitting is described as regression using a 1st degree polynomial called ordinary LS. Most significantly, analogous to the IMM, the polynomial LS multiple model (LSMM) is derived. Contrary to the IMM, which is not optimized; the LSMM is optimized with a trade-off between the statistical variance and the deterministic bias-squared to minimize the mean-square-error (variance plus bias-squared). A sequence of optimal LSMMs matched to accelerations covering the spectrum of acceleration between zero and the assumed maximum are derived and applied in designing an adaptive algorithm. Results demonstrate improved target tracking and accuracy by the LSMM of maneuvering targets compared with published IMM performances.