Consistent covariance estimation for PMHT
Wayne R. Blanding, Peter Willett, Roy L. Streit, Darin T. Dunham · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2007
The Probabilistic Multi-Hypothesis Tracker (PMHT) has been demonstrated to be an effective multi-target tracker while retaining linear computational complexity in the number of measurements and targets. However PMHT only provides a point estimate for target tracks. The returned by the PMHT is a byproduct of applying the Expectation-Maximization algorithm to maximize the PMHT likelihood function and is not intended to be the track estimate covariance. In this paper we derive a consistent covariance estimator for PMHT. By re-introducing the constraint that the sum of the PMHT weights (posterior probabilities that a measurement is target-originated) across measurements sum to unity, a covariance based on Probabilistic Data Association (PDA) principles is derived. We show through simulations that the resulting covariance provides a consistent covariance for the PMHT track estimates. There has been some work both in the statistics and engineering literature that gives the posterior covariance for ML Gaussian-mixture estimation, and the PMHT can be viewed as a tracker whose genesis is of MAP Gaussian-mixture estimation with a Gaussian prior. The expressions and calculations are, unfortunately, complicated. Consequently we also report on a novel and intuitive way to derive these via calculus.