Linear observation based total least squares

Abstract This paper presents a total least squares (TLS) method in an iterative way when the observations are linear with applications in two-dimensional linear regression and three-dimensional coordinate transformation. The second order smaller terms are preserved and the unbiased solution and the variance component estimate are both obtained rigorously from traditional non-linear least squares theory. Compared with the traditional TLS algorithm dealing with the so called errors-in-variables (EIV) model, this algorithm can be used to analyse all the observations involved in the observation vector and the design matrix coequally; the non-linear adjustment with constraints or partial EIV model can also be solved using the same method. In addition, all the errors of observations can be considered in a heteroscedastic or correlated case, and the calculation of the solution and the variance component estimate are much simpler than the traditional TLS and its related improved algorithms. Experiments using statistical methods show the deviations between the designed true value of the variables and the estimated ones using this algorithm and the traditional least squares algorithm respectively, and the mean value of the posteriori variance in 1000 simulations of coordinate transformation is computed as well to test and verify the efficiency and unbiased estimation of this algorithm.