A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

Especially in metrology and geodesy, but also in many other disciplines, the solution of overdetermined linear systems of the form Ax≈b with individual uncertainties not only in b but also in A is an important task. The problem is known in literature as weighted total least squares. In the most general case, correlations between the elements of [A,b] exist as well. The problem becomes more complicated and can—except for special cases—only be solved numerically. While the formulation of this problem and even its solution is straightforward, its implementation—when the focus is on reliability and computational costs—is not. In this paper, a robust, fast, and universal method for computing the solution of such linear systems as well as their covariance matrix is presented. The results were confirmed by applying the method to several special cases for which an analytical or numerical solution is available. If individual coefficients can be considered to be free of errors, this can be taken into account in a simple way. An implementation of the code in MATLAB is provided.

Weighted Total Least Squares (WTLS) Solutions for Straight Line Fitting to 3D Point Data

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.

Weighted total least squares problems with inequality constraints solved by standard least squares theory

<p>The errors-in-variables (EIV) model is applied to surveying and mapping fields such as empirical coordinate transformation, line/plane fitting and rigorous modelling of point clouds and so on as it takes the errors both in coefficient matrix and observation vector into account. In many cases, not all of the elements in coefficient matrix are random or some of the elements are functionally dependent. The partial EIV (PEIV) model is more suitable in dealing with such structured coefficient matrix. Furthermore, when some reliable prior information expressed by inequality constraints is considered, the adjustment result of inequality constrained PEIV (ICPEIV) model is expected to be improved. There are two kinds of algorithms to solve the ICPEIV model under the weighted total least squares (WTLS) criterion currently. On the one hand, one can linearize the PEIV model and transform it into a sequence of quadratic programming (QP) sub-problems. On the other hand, one can directly solve the nonlinear target function by common used programming algorithms.All the QP algorithms and nonlinear programming methods are complicated and not familiar to the geodesists, so the ICPEIV model is not widely used in geodesy.   </p><p>In this contribution, an algorithm based on standard least squares is proposed. First, the estimation of model parameters and random variables in coefficient matrix are separated according to the Karush-Kuhn-Tucker (KKT) conditions of the minimization problem. The model parameters are obtained by solving the QP sub-problems while the variables are determined by the functional relationship between them. Then the QP problem is transformed to a system of linear equations with nonnegative Lagrange multipliers which is solved by an improved Jacobi iterative algorithm. It is similar to the equality-constrained least squares problem. The algorithm is simple because the linearization process is not required and it has the same form of classical least squares adjustment. Finally, two empirical examples are presented. The linear approximation algorithm, the sequential quadratic programming algorithm and the standard least squares algorithm are used. The examples show that the new method is efficient in computation and easy to implement, so it is a beneficial extension of classical least squares theory.</p>