Algorithms and statistical analysis for linear structured weighted total least squares problem

Abstract

Weighted total least squares (WTLS) have been regarded as the standard tool for the errors-in-variables (EIV) model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors. However, in many geodetic applications, some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix. It is called the linear structured EIV (LSEIV) model. Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications. On the one hand, the functional part of the LSEIV model is modified into the errors-in-observations (EIO) model. On the other hand, the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix. The algorithms are derived through the Lagrange multipliers method and linear approximation. The estimation principles and iterative formula of the parameters are proven to be consistent. The first-order approximate variance-covariance matrix (VCM) of the parameters is also derived. A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach. Afterwards, the least squares (LS), total least squares (TLS) and linear structured weighted total least squares (LSWTLS) solutions are compared and the accuracy evaluation formula is proven to be feasible and effective. Finally, the LSWTLS is applied to the field of deformation analysis, which yields a better result than the traditional LS and TLS estimations.