Total Least-Squares Adjustment of Condition Equations
ABSTRACT The usual least-squares adjustment within an Errors-in-Variables (EIV) model is often described as Total Least-Squares Solution (TLSS), just as the usual least-squares adjustment within a Random Effects Model (REM) has become popular under the name of Least-Squares Collocation (without trend). In comparison to the standard Gauss-Markov Model (GMM), the EIV-Model is less informative whereas the REM is more informative. It is known under which conditions exactly the GMM or the REM can be equivalently replaced by a model of Condition Equations or, more generally, by a Gauss-Helmert-Model (GHM). Such equivalency conditions are, however, still unknown for the EIV-Model once it is transformed into such a model of Condition Equations. In a first step, it is shown in this contribution how the respective residual vector and residual matrix would look like if the Total Least-Squares Solution is applied to condition equations with a random coefficient matrix to describe the transformation of the random error vector. The results are demonstrated using numeric examples which show that this approach may be valuable in its own right.
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ABSTRACT: In an earlier work, a simple and flexible formulation for the weighted total least squares (WTLS) problem was presented. The formulation allows one to directly apply the existing body of knowledge of the least squares theory to the errors-in-variables (EIV) models of which the complete description of the covariance matrices of the observation vector and of the design matrix can be employed. This contribution presents one of the well-known theories—least squares variance component estimation (LS-VCE)—to the total least squares problem. LS-VCE is adopted to cope with the estimation of different variance components in an EIV model having a general covariance matrix obtained from the (fully populated) covariance matrices of the functionally independent variables and a proper application of the error propagation law. Two empirical examples using real and simulated data are presented to illustrate the theory. The first example is a linear regression model and the second example is a 2-D affine transformation. For each application, two variance components—one for the observation vector and one for the coefficient matrix—are simultaneously estimated. Because the formulation is based on the standard least squares theory, the covariance matrix of the estimates in general and the precision of the estimates in particular can also be presented.Journal of Geodesy 11/2013; 87(10-12). · 3.92 Impact Factor
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ABSTRACT: A new proof is presented of the desirable property of the weighted total least-squares (WTLS) approach in preserving the structure of the coefficient matrix in terms of the functional independent elements. The WTLS considers the full covariance matrix of observed quantities in the observation vector and in the coefficient matrix; possible correlation between entries in the observation vector and the coefficient matrix are also considered. The WTLS approach is then equipped with constraints in order to produce the constrained structured TLS (CSTLS) solution. The proposed approach considers the correlation between the observation vector and the coefficient matrix of an Error-In-Variables model, which is not considered in other, recently proposed approaches. A rigid transformation problem is done by preservation of the structure and satisfying the constraints simultaneously.Studia Geophysica et Geodaetica 01/2014; · 0.75 Impact Factor
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ABSTRACT: A standard errors-in-variables (EIV) model refers to a Gauss–Markov model with an uncertain model matrix from a geodetic perspective. Least squares within the EIV model is usually called the total least squares (TLS) technique because of its symmetrical adjustment. However, the solutions and computational advantages of the weighted TLS problem with a general weight matrix (WTLS) are mostly unknown. In this study, the WTLS problem was solved using three different approaches: iterative methods based on the normal equation, the iteratively linearized Gauss–Helmert model with algebraic Jacobian matrices, and numerical analysis. Furthermore, sufficient conditions for WTLS optimization were investigated systematically as proposed solutions yield only necessary conditions for optimality. A WTLS solution was considered to treat random parameters within the EIV model. Last, applications to test these novel algorithms are presented.Journal of Geodesy 08/2013; 87(8). · 3.92 Impact Factor