Weighted total least squares: necessary and sufficient conditions, fixed and random parameters

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 computa...

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Bibliographic Details
Published in:Journal of geodesy 2013-08, Vol.87 (8), p.733-749
Main Author: Fang, Xing
Format: Article
Language:English
Subjects:
Earth and Environmental Science
Earth Sciences
Error analysis
Geodetics
Geophysics/Geodesy
Markov analysis
Markov chains
Matrix
Numerical analysis
Original Article
Parameter estimation
Randomized algorithms
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Summary: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.
ISSN:0949-7714
1432-1394
DOI:10.1007/s00190-013-0643-2