Estimating Tracking Variance From Target Returns: No More Noise Variance Assumptions

Abstract

A weighting function formulated from polynomial least squares (LS) is combined with statistical estimation theory to produce the statistical estimate, expected value, and variance of arbitrary points on the LS estimated trajectory polynomial from samples of the polynomial corrupted with statistical observation noise. The statistical noise variance is estimated as the sample average of the deviations of noisy samples from the estimated polynomial. This approach obviates the need to assume observation and/or state noises. Although state noise in the Kalman filter (KF) is actually fictitious and not really a noise at all but merely a threshold, such noise can actually exist but is not taken into accounted by Monte Carlo simulations of the KF. It along with all other unknown and unspecified noises (atmospheric, interference, etc.) that do exist are accounted for in the sample average. Perhaps most important is the estimated tracking variance at arbitrary points on the estimated polynomial. It is minimum at the centroid of horizontal axis samples and increases as the distance squared of a point from that centroid along the horizontal axis. Moreover, the weighting function solves the problem that existed in LS invented by Gauss two and a quarter centuries ago: The absence of the statistical tracking variance at arbitrary points on estimated polynomials. An example demonstrates the tracking variance at four points on an estimated polynomial: the centroid, last sample location, next sample location, and the point where the tracking variance exceeds the noise sample variance.