Bayesian Estimation and Tracking: A Practical Guide

14.1 Error Ellipses

In applications of track estimation filters when the true state is unknown, such as for a fielded system tracking a real ship, or when only a small number of field tests are conducted and the truth state is only approximately known, the best way to gauge the performance of a tracking filter is to use the estimated covariance matrix output by the filter to generate and plot error ellipses around the filtered track state. For clarity of presentation only the two-dimensional Cartesian coordinate system positions are used due to the inherent two-dimensional nature of displays.

At any discrete time t_n, the probability density function of the track state x_n is given by the Gaussian

(14.1)

where () are the output of the Kalman filter update equations in Cartesian coordinates. Now, dropping the time subscript, examine only the two-dimensional position components given by . Examination of Figure 2.9 shows that the contour ellipses of constant probability (before the affine transformation) are generally elongated along the major axis and may also be skewed so that the axes of the ellipse are not aligned with the axes associated with the state position vector. The question ...