Weighted Total Least Squares with Singular Covariance Matrices Subject to Weighted and Hard Constraints
Publication: Journal of Surveying Engineering
Volume 143, Issue 4
Abstract
Weighted total least squares (WTLS) has been widely used as a standard method to optimally adjust an errors-in-variables (EIV) model containing random errors both in the observation vector and in the coefficient matrix. An earlier work provided a simple and flexible formulation for WTLS based on the standard least-squares (SLS) theory. The formulation allows one to directly apply the available SLS theory to the EIV models. Among such applications, this contribution formulates the WTLS problem subject to weighted or hard linear(ized) equality constraints on unknown parameters. The constraints are to be properly incorporated into the system of equations in an EIV model of which a general structure for the (singular) covariance matrix of the coefficient matrix is used. The formulation can easily take into consideration any number of weighted linear and nonlinear constraints. Hard constraints turn out to be a special case of the general formulation of the weighted constraints. Because the formulation is based on the SLS theory, the method automatically approximates the covariance matrix of the estimates from which the precision of the constrained estimates can be obtained. Three numerical examples with different scenarios are used to demonstrate the efficacy of the proposed algorithm for geodetic applications.
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Acknowledgments
The author acknowledges Professor Athanasios Dermanis at the Aristotle University of Thessaloniki for his unpublished paper on singular covariance matrices of the EIV models, which improved the quality of this paper.
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Journal of Surveying Engineering
Volume 143 • Issue 4 • November 2017
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© 2017 American Society of Civil Engineers.
History
Received: Dec 7, 2016
Accepted: Apr 10, 2017
Published online: Jul 5, 2017
Published in print: Nov 1, 2017
Discussion open until: Dec 5, 2017
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