Generalised total least squares solution based on pseudo-observation method

In the generalised total least squares (GTLS) problem, observations can be perturbed by random errors that are dependently, inconsistently and normally distributed with a non-zero mean, and the coefficient matrix can hold any structure. In this contribution, a set of formulae for GTLS adjustment is derived using a pseudo-observation method. Based on the derived results, an iterative algorithm (algorithm 1) only for the estimation of parameters and a two-loop iterative algorithm (algorithm 2) for the estimation of parameters and variance factors are developed. Moreover, the derivative of a vector is introduced to deal with the structured TLS problem. A straight line fitting and a simulated 2D affine transformation experiment are performed to verify the applicability of the developed algorithms. The results show that algorithm1 can be used to simultaneously handle the structured coefficient matrix, correlated error and non-zero expectation problem, while algorithm 2 can be utilised to manage the variance component estimation problem with the non-zero expectation assumption. Under the identical statistical assumptions, the suggested algorithm can achieve the same results as the solutions of Schaffrin (2008), Shen (2011), Fang (2013) and Amiri-Simkooei (2013).