On the Weighted Total Least Squares Solutions

Abstract

AbstractNowadays the terminology Total Least Squares (TLS) is frequently used as a standard name of the estimation method for the errors-in-variables (EIV) model. Although a significant number of contribution have been published to adjust the EIV model, the computational advantages of the TLS problem are still largely unknown. In this contribution various approaches are applied for solving the weighted TLS problem, where the covariance matrix of the observation vector can be fully populated: 1. The auxiliary Lagrange multipliers are applied to give some implementations for solving the problem. 2. In contrast to the nonlinear Gauss–Helmert model (GHM) proposed by other authors, the model matrices and the inconsistency vector are analytically formulated within the GHM. 3. The gradient of the objective function is given when the weighted TLS problem is expressed as an unconstrained optimization problem. If the gradient equals to zero, the necessary conditions for the optimality are identical with the normal equation which is derived by Lagrange multipliers. Furthermore, a numerical example demonstrates the identical solution by the proposed algorithms.KeywordsErrors-in-variablesGauss–Helmert modelGradientLagrange multipliersTotal least squares