New Insights into Largest Ellipsoid Estimation

The largest ellipsoid (LE) algorithm produces an unbiased estimate with the largest estimated covariance possible without being less confident than the a priori or measurement information. In this paper we further analyze the LE estimator and provide several insights. We derive the explicit formation of the LE estimator gain (prior work only performed fusion in the information domain) and use it to perform covariance-form analysis, which directly shows that LE fusion produces the correct covariance when the cross-correlations share a "component-wise aligned" structure, and that this structure is maintained after the fusion process. We compare it to the family of so-called "minimax" estimators and show that these estimators are also inconsistent and that the most easily implementable ones produce the same covariance estimate as LE estimation but at a higher computational cost. We also show that LE estimation with scalar measurements has a closed form solution, which eliminates the need for numerical singular value decompositions. Each insight is accompanied by a brief numerical or simulation example. The cumulative effect is a justification for using LE estimation over other inconsistent algorithms when covariance intersection performs poorly, and better system modeling isn’t an option.