Article
Total LeastSquares Adjustment of Condition Equations
Studia Geophysica et Geodaetica (Impact Factor: 0.81). 07/2011; 55(3). DOI: 10.1007/s1120001100323
ABSTRACT
The usual leastsquares adjustment within an ErrorsinVariables (EIV) model is often described as Total LeastSquares Solution (TLSS), just as the usual leastsquares adjustment within a Random Effects Model (REM) has become popular under the name of LeastSquares Collocation (without trend). In comparison to the standard GaussMarkov Model (GMM), the EIVModel 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 GaussHelmertModel (GHM). Such equivalency conditions are, however, still unknown for the EIVModel 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 LeastSquares 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.

 "In geodetic literature Teunissen (1988) was the first who solved an EIV model in an exact form. The equivalent form of the EIV model appears as the condition equation with a random coefficient matrix (Schaffrin and Wieser 2011). From a geodetic point of view, the EIV model is a special case of the nonlinear Gauss Helmert Model (GHM), which generates the standard LS solution after iterative linearization (Neitzel 2010; Schaffrin and Snow 2010; Fang 2011, 2013a; Bányai 2012; Snow 2012). "
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ABSTRACT: It is well known that the errorsinvariables (EIV) model has been treated as a special case of the traditional geodetic model, the nonlinear Gauss–Helmert model (GHM), for more than a century. In this contribution, an adjustment of the EIV model with equality and inequality constraints is investigated based on the nonlinear GHM. In each iteration, the constrained EIV model is linearized to form a quadratic program. Furthermore, the precision description is investigated for the mixed constrained problem. The demonstrated results from the numerical examples show that this approach avoids the large computational expenses of the existing combinatorial solution that normally accompany the number of inequality constraints. Keywords Total LeastSquares (TLS) Á Errorsinvariables model Á Equality and inequality constraints Á Gauss–Helmert model Á Convex quadratic program 
 "Wieser (2008), Schaffrin and Felus (2009), Schaffrin and Wieser (2009), Schaffrin and Wieser (2011), Tong et al. (2011) and Shen et al. (2011) "
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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 errorsinvariables (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 wellknown theories—least squares variance component estimation (LSVCE)—to the total least squares problem. LSVCE 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 2D 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. 
Article: Weighted total least squares: Necessary and sufficient conditions, fixed and random parameters
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ABSTRACT: A standard errorsinvariables (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.
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