Pareto optimality solution of the Gauss-Helmert model
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The Pareto optimality method is applied to the parameter estimation of the Gauss-Helmert weighted 2D similarity transformation assuming that there are measurement errors and/or modeling inconsistencies.
In some cases of parametric modeling, the residuals to be minimized can be expressed in different forms resulting in different values for the estimated parameters. Sometimes these objectives may compete in the Pareto sense, namely a small change in the parameters can result in an increase in one of the objectives on the one hand, and a decrease of the other objective on the other hand. In this study, the Pareto optimality approach was employed to find the optimal trade-off solution between the conflicting objectives and the results compared to those from ordinary least squares (OLS), total least squares (TLS) techniques and the least geometric mean deviation (LGMD) approach.
The results indicate that the Pareto optimality can be considered as their generalization since the Pareto optimal solution produces a set of optimal parameters represented by the Pareto-set containing the solutions of these techniques (error models). From the Pareto-set, a single optimal solution can be selected on the basis of the decision maker’s criteria. The application of Pareto optimality needs nonlinear multi-objective optimization, which can be easily achieved in parallel via hybrid genetic algorithms built-in engineering software systems such as Matlab. A real-word problem is investigated to illustrate the effectiveness of this approach.
KeywordsPareto optimality Gauss-Helmert transformation Parameter estimation Measurement and modeling errors Least squares approach Genetic algorithm
The authors thank Prof. Bernhard Heck and Dr. Kevin Fleming for proof reading the manuscript, but accept all responsibility for any errors. J.L. Awange acknowledges the financial support of the Alexander von Humbold Foundation (Ludwig Leichhardt Memorial Fellowship) and a Curtin Research Fellowship. He is grateful for the warm welcome and conducive working atmosphere provided by his host, Prof. Bernhard Heck and his team, at the Geodetic Institute, Karlsruhe Institute of Technology (KIT). This work was partially funded by OTKA project No. 76231. The authors are indebted to the reviewer whose remarks, advices and suggestions helped to improve the paper considerably.
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