Adjusting the Errors-In-Variables Model: Linearized Least-Squares vs. Nonlinear Total Least-Squares

Abstract

It has long been known that the Errors-In-Variables (EIV) Model is a special case of the nonlinear Gauss–Helmert Model (GHM) and can, therefore, be adjusted by standard least-squares techniques in iteratively linearized GH-Models, which is the approach by Helmert (Adjustment Computations Based on the Least-Squares Principle (in German), 1907) and – later – by Deming (Phil Mag 11:146–158, 1931; Phil Mag 17:804–829, 1934).

Apart from the fact that there are, at least, two other nonlinear models that are equivalent to the above GH-Model, thus allowing two more classical least-squares approaches based on iterative linearization, it was the seminal paper by Golub and van Loan (SIAM J Numer Anal 17:883–893, 1980) in which they proved that a purely nonlinear approach can be followed as well, thereby avoiding any model linearization. They called such an approach “Total Least-Squares adjustment” by which any normal equations may be replaced by a simple eigenvalue problem, as long as only diagonal dispersion matrices are involved.

Here, an attempt will be made to show the differences and parallels in various algorithms, even in the fully weighted case, which obviously all generate the same results, but without necessarily showing equal efficiency in doing so, as is well known since the publications by Schaffrin and Wieser (J Geodesy 82:415–421, 2008), Fang (Weighted Total Least-Squares solutions with applications in geodesy, 2011), and Mahboub (J Geodesy 86:359–367, 2012).