Converted Total Least Squares Method and Gauss-Helmert Model with Applications to Coordinate Transformations
In this paper, the three kind of solutions of TLS problem, the common solution by singular value decomposition (SVD), the iteration solution and Partial-EIV model are firstly reviewed with respect to their advantages and disadvantages. Then a newly develo
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Abstract
In this paper, the three kind of solutions of TLS problem, the common solution by singular value decomposition (SVD), the iteration solution and Partial-EIV model are firstly reviewed with respect to their advantages and disadvantages. Then a newly developed Converted Total Least Squares (CTLS) dealing with the errors-in-variables (EIV) model is introduced. The basic idea of CTLS has been proposed by the authors in 2010, which is to take the stochastic design matrix elements as virtual observations, and to transform the TLS problem into a traditional Least Squares problem. This new method has the advantages that it cannot only easily consider the weight of observations and the weight of stochastic design matrix, but also deal with TLS problem without complicated iteration processing, if the suitable approximates of parameters are available, which enriches the TLS algorithm and solves the bottleneck restricting the application of TLS solutions. CTLS method, together with all the three TLS models reviewed here has been successfully integrated in our coordinate transformation programs and verified with the real case study of 6parameters Affine coordinate transformation. Furthermore, the comparison and connection of this notable CLTS method and estimation of Gauss-Helmert model are also discussed in detail with applications of coordinate transformations. Keywords
Converted TLS Errors-In-Variables (EIV) Gauss-Helmert model Total Least Squares (TLS) Virtual observation
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Introduction
Total Least Squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector and in the design matrix in computational mathematics and engineering, which is also referred as ErrorsIn-Variables (EIV) modelling or orthogonal regression in the statistical community. The TLS/EIV principle studied by Adcock (1878) and Pearson (1901) already more than J. Cai () · D. Dong · N. Sneeuw Institute of Geodesy, University of Stuttgart, Stuttgart, Germany e-mail: [email protected] Y. Yao School of Geodesy and Geomatics, Wuhan University, Wuhan, China
one century ago. Kendall and Stuart (1969) described this problem as structural relationship model models. In geodetic application this method was discussed by Koch (2002) and studied recently by Schaffrin (2005). How to obtain the best parameter estimation values and give the statistical information of parameters in the EIV model is not ‘perfectly’ solved. Nevertheless, the EIV model is still becoming increasingly widespread in remote sensing (Felus and Schaffrin 2005) and geodetic datum transformation (Schaffrin and Felus 2005, 2008; Schaffrin and Wieser 2008; Akyilmaz 2007; Cai and Grafarend 2009; Shen et al. 2011;Amiri-Simkooeil and Jazaeri 2012). In 1980, the mathematical structure of TLS was completed by Golub and Van Loan (1980), who gave the first numerically stable algorithm based on matrix singular value decomposition. With the rapid development of the
International Association of Geodesy Symposia, https://doi.org/10.1007/1
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