Weighted total least squares applied to mixed observation model

Abstract

This contribution presents the weighted total least squares (WTLS) formulation for a mixed errors-in-variables (EIV) model, generally consisting of two erroneous coefficient matrices and two erroneous observation vectors. The formulation is conceptually simple because it is formulated based on the standard least squares theory. It is also flexible because the existing body of knowledge of the least squares theory can directly be generalised to the mixed EIV model. For example, without any derivation, estimate for the variance factor of unit weight and a first approximation for the covariance matrix of the unknown parameters can directly be provided. Further, the constrained WTLS, variance component estimation and the theory of reliability and data snooping can easily be established to the mixed EIV model. The mixed WTLS formulation is also attractive because it can simply handle the two special cases of EIV models: the conditioned EIV model and the parametric EIV model. The WTLS formulation has been applied to three examples. The first two examples are simulated ones, the results of which are shown to be identical to those obtained by the non-linear Gauss-Helmert method. Further, the covariance matrix of the WTLS estimates is shown to closely approximate that obtained through a large number of simulations. The third is a real example of which two object points are photographed by three terrestrial cameras. Three scenarios are employed to show the efficiency of the proposed formulation on this last example.

Keywords:

• Weighted total least squares
• Errors-in-variables model
• Mixed observation model
• Non-linear Gauss-Helmert model

Acknowledgement

Authors would like to acknowledge useful comments of two anonymous reviewers, which improved the presentation and quality of the paper.