Weighted total least squares formulated by standard least squares theory

This contribution presents a simple, attractive, and exible formulation for the weighted total least squares (WTLS) problem. It is simple because it is based on the well-known standard least squares theory; it is attractive because it allows one to directly use the existing body of knowledge of the least squares theory; and it is exible because it can be used to a broad eld of applications in the error-invariable (EIV) models. Two empirical examples using real and simulated data are presented. The rst example, a linear regression model, takes the covariance matrix of the coefficient matrix as QA = Qn ⊗ Qm , while the second example, a 2-D affine transformation, takes a general structure of the covariance matrix QA . The estimates for the unknown parameters along with their standard deviations of the estimates are obtained for the two examples. The results are shown to be identical to those obtained based on the nonlinear GaussHelmert model (GHM). We aim to have an impartial evaluation of WTLS and GHM. We further explore the high potential capability of the presented formulation.One can simply obtain the covariancematrix of theWTLS estimates. In addition, one cangeneralize theorthogonal projectors of the standard least squares fromwhich estimates for the residuals and observations (alongwith their covariancematrix), and the variance of the unit weight can directly be derived. Also, the constrainedWTLS, variance component estimation for an EIVmodel, and the theory of reliability and data snooping can easily be established, which are in progress for future publications.