An Improved Weighted Total Least Squares Method with Applications in Linear Fitting and Coordinate Transformation

Abstract

This paper presents an improved weighted total least squares (IWTLS) method for the errors-in-variables (EIV) model with applications in linear fitting and coordinate transformation. In addition, an improved constrained weighted TLS (ICWTLS) method is further obtained based on the IWTLS algorithm. Following the weighted TLS solution (WTLSS) method in which the precisions of any two columns of the design matrix differ only by a scalar factor in linear orthogonal regression problems, the IWTLS method is derived for a more generic case in which there is no proportionality assumption for the cofactor matrix of the design matrix in the EIV model. Compared with existing research on the constrained TLS method under the assumption that both the constraining matrix and the right-hand-side (RHS) vector are error-free, or that only the RHS vector contains errors, the ICWTLS method is proposed for resolving the EIV model with constraints by integrating the observation equations and constraint equations under the assumption that the observation vector and design matrix in the observation equations, and the RHS vector and the constraining matrix in the constraint equations, contain errors. The applicability of the proposed methods is illustrated through empirical examples. The performances of our proposed methods are compared with those of existing methods in the applications of linear fitting and coordinate transformation. The analysis of the experimental results demonstrate that (1) the proposed IWTLS algorithm has not only the advantage of the WTLSS algorithm, which takes into account the errors in the design matrix in linear orthogonal regression applications, but also the capability of dealing with a more generic case in which the design matrix contains errors with different distributions; and (2) the proposed ICWTLS algorithm has the advantages of handling both the cases of equal and unequal weights in solving the EIV model with constraints and handling the case in which the constraints contain errors.