An Improved Weighted Total Least Squares Method with Applications in Linear Fitting and Coordinate Transformation
Publication: Journal of Surveying Engineering
Volume 137, Issue 4
Abstract
This paper presents an improved weighted total least squares (IWTLS) method for the errors-in-variables (EIV) model with applications in linear fitting and coordinate transformation. In addition, an improved constrained weighted TLS (ICWTLS) method is further obtained based on the IWTLS algorithm. Following the weighted TLS solution (WTLSS) method in which the precisions of any two columns of the design matrix differ only by a scalar factor in linear orthogonal regression problems, the IWTLS method is derived for a more generic case in which there is no proportionality assumption for the cofactor matrix of the design matrix in the EIV model. Compared with existing research on the constrained TLS method under the assumption that both the constraining matrix and the right-hand-side (RHS) vector are error-free, or that only the RHS vector contains errors, the ICWTLS method is proposed for resolving the EIV model with constraints by integrating the observation equations and constraint equations under the assumption that the observation vector and design matrix in the observation equations, and the RHS vector and the constraining matrix in the constraint equations, contain errors. The applicability of the proposed methods is illustrated through empirical examples. The performances of our proposed methods are compared with those of existing methods in the applications of linear fitting and coordinate transformation. The analysis of the experimental results demonstrate that (1) the proposed IWTLS algorithm has not only the advantage of the WTLSS algorithm, which takes into account the errors in the design matrix in linear orthogonal regression applications, but also the capability of dealing with a more generic case in which the design matrix contains errors with different distributions; and (2) the proposed ICWTLS algorithm has the advantages of handling both the cases of equal and unequal weights in solving the EIV model with constraints and handling the case in which the constraints contain errors.
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Acknowledgments
The writers thank the anonymous reviewers and the editor for their valuable comments on the manuscript. The work described in this paper was substantially supported by the National Natural Science Foundation of China (Project No. NSFC40771174 and NSFC40771175), High-Tech Research and Development Program of China (Project No. UNSPECIFIED2009AA12Z214 and UNSPECIFIED2009AA12Z131), Foundation of Shanghai Dawn Scholarship and National Science and Technology Support Program (Project No. UNSPECIFIED07SG24 and UNSPECIFIED2008BAJ10B01).
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Published In
Journal of Surveying Engineering
Volume 137 • Issue 4 • November 2011
Pages: 120 - 128
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Feb 9, 2010
Accepted: Jan 11, 2011
Published online: Jan 13, 2011
Published in print: Nov 1, 2011
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