A Structured Total Least Squares Method Based on Nonlinear Gauss-Helmert Model

The coefficient matrix of errors-in-variables(EIV) model may contain structural features, and this situation can be extended to the observation vector. Here, we define the the augmented matrix of the coefficient matrix that consists of the coefficient matrix and the observation vector. Firstly, we reformulate the EIV model as the nonlinear Gauss-Helmert model. The augmented matrix is expanded as an affine function form, and then the random elements in it are extracted by variable projection method. Finally, we derive a novel algorithm for the structural total least squares (STLS) problem and its first-order approximation precision estimator based on the nonlinear least squares (NLS) theory. The proposed algorithm unifies the general STLS method, the structural data least squares (SDLS) method and the least squares (LS) method. Furthermore, this algorithm is applied to a real example and a simulation example, e. g. straight line fitting and plane fitting. The results of the two examples show that the proposed algorithm is consistent with the existing structural or weighted total least squares methods which can solve the structure problem of the EIV model, thus the results verifies the effectiveness of the proposed algorithm. The method of extracting structural features in this paper is simple in the concept and easy in the implementation. And the STLS problem is transformed into a condition adjustment problem with parameters, which is incorporated into the least squares adjustment theory system to facilitate its extended application. Additionally, the characteristic of the estimated errors of plane fitting is analyzed qualitatively. It can be seen from the analysis that the relative size of the parameters has a direct impact on the consistency of the estimated errors, which indicates that the consistency between the estimated errors and the true errors is relatively complex in the EIV model.