Error-Covariance Analysis of the Total Least-Squares Problem

Abstract

This paper derives and analyzes the estimate error covariances associated with both the nonstationary and stationary noise process cases with uncorrelated elementwise components for the total least-squares problem. The nonstationary case is derived directly from the associated unconstrained total least-squares loss function. The stationary case is derived by using a linear expansion of the total least-squares estimate equation, which involves a first-order expansion of the associated singular value decomposition matrices. The actual solution for the error covariance is evaluated at the true variables, which are unknown in practice. Two common approaches to overcome this difficulty are used; the first involves using the measurements directly and the second involves using the estimates, which are more accurate than the measurements. This paper shows that using the latter greatly simplifies the error-covariance solution for the stationary case. Simulation results using bearings-only point estimation are shown to quantify the theoretical derivations.