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: It is shown here how – similarly to the unconstrained case – the Constrained Total Least Squares Estimate (CTLSE) can be generated by solving a certain sequence of eigenvalue problems iteratively. For this, the normal matrix from the constrained (standard) least-squares approach has to be suitably augmented by one row and one column. Further modification of the augmented row and column allows the treatment of “fiducial constraints” for which the RHS vector is affected by random errors, but not the constraining matrix itself.Linear Algebra and Its Applications - LINEAR ALGEBRA APPL. 01/2006; 417(1):245-258.
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ABSTRACT: Error-contaminated systems Ax ≈ b, for which A is ill-conditioned, are considered. Such systems may,be solved using Tikhonov-like regularized total least squares (RTLS) methods. Golub, Hansen, and O’Leary [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185–194] presented a parameter-dependent direct algorithm for the solution of the augmented,Lagrange formulation for the RTLS problem, and Sima, Van Huffel, and Golub [Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers, Tech. Report SCCM-03-03, SCCM, Stanford University, Stanford, CA, 2003] have introduced a technique for solution based on a quadratic eigenvalue prob- lem, RTLSQEP. Guo and Renaut [A regularized total least squares algorithm, in Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications, S. Van Huffel and P. Lem- merling, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002, pp. 57–66] derived an eigenproblem for the RTLS which can be solved using the iterative inverse power method. Here we present an alternative derivation of the eigenproblem for constrained TLS through the augmented Lagrangian for the constrained normalized residual. This extends the analysis of the eigenproblem and leads to derivation of more efficient algorithms compared,to the original formulation. Additional algorithms based on bisection search and a standard L-curve approach are presented. These algo- rithms vary with respect to the parameters that need to be prescribed. Numerical and convergence results supporting the different versions and contrasting with RTLSQEP are presented. Key words. total least squares, regularization, ill-posedness, Rayleigh quotient iteration AMS subject classifications. 65F22, 65F30 DOI. 10.1137/S0895479802419889SIAM J. Matrix Analysis Applications. 01/2004; 26:457-476.
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ABSTRACT: The weighted total least-squares solution (WTLSS) is presented for an errors-in-variables model with fairly general variance–covariance matrices. In particular, the observations can be heteroscedastic and correlated, but the variance–covariance matrix of the dependent variables needs to have a certain block structure. An algorithm for the computation of the WTLSS is presented and applied to a straight-line fit problem where the data have been observed with different precision, and to a multiple regression problem from recently published climate change research.Journal of Geodesy 01/2008; 82(7):415-421. · 2.81 Impact Factor