A simple iterative algorithm based on weighted least-squares for errors-in-variables models: Examples of coordinate transformations

Although weighted total least-squares (WTLS) adjustment within the errors-in-variables (EIV) model is a rigorous method developed for parameter estimation, its exact solution is complicated since the matrix operations are extremely time-consuming in the whole repeated iteration process, especially when dealing with large data sets. This paper rewrites the EIV model to a similar Gauss-Markov model by taking the random error of the design matrix and observations into account, and reformulates it as an iterative weighted least-squares (IWLS) method without complicated theoretical derivation. IWLS approximates the "exact solution" of the general WTLS and provides a good balance between computational efficiency and estimation accuracy. Because weighted LS (WLS) method has a natural advantage in solving the EIV model, we also investigate whether WLS can directly replace IWLS and WTLS to implement the EIV model when the parameters in the EIV model are small. The results of numerical experiments confirmed that IWLS can obtain almost the same solution as the general WTLS solution of Jazaeri [21] and WLS can achieve the same accuracy as the general WTLS when the parameters are small.