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Alternative formulae for robust Weighted Total Least-Squares solutions for Errors-In-Variables models

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Abstract

Weighted Total Least-Squares (WTLS) can optimally solve the issue of parameter estimation in the Errors-In-Variables (EIV) model; however, this method is relatively sensitive to outliers that may exist in the observation vector and/or the coefficient matrix. Hence, an attempt to identify/suppress those outliers is in progress and will ultimately lead to a novel robust estimation procedure similar to the one used in the Gauss-Markov model. The method can be considered as a follow-up to the WTLS solution formulated with the standard Least-Squares framework. We utilize the standardized total residuals to construct the equivalent weights, and apply the median method to obtain a robust estimator of the variance to provide good robustness in the observation and structure spaces. Moreover, a preliminary analysis for the robustness of related estimators within the EIV model is conducted, which shows that the redescending M-estimates are more robust than the monotonic ones. Finally, the efficacy of the proposed algorithm is demonstrated through two applications, i.e. 2D affine transformation and linear regression on simulated data and on real data with some assumptions. Unfortunately, the proposed algorithm may not be reliable for detecting multiple outliers. Therefore, MM-estimates within the EIV model need to be investigated in further research.

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Acknowledgements

We would like to acknowledge useful comments of the Associate Editor, P.J.G. Teunissen and of the anonymous reviewers, that improved the paper. This research is supported by the National Natural Science Foundation of PR China (Grant Nos 41674016 and 41904039) and the Natural Science Foundation of Jiangxi Province (Grant No. 20202BABL214055).

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  1. School of Civil Engineering and Architecture, East China Jiaotong University, Nanchang, 330013, China

    Zhipeng Lv

  2. Institute of Surveying and Mapping, Information Engineering University, Zhengzhou, 450001, China

    Lifen Sui

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  1. Zhipeng Lv
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Lv, Z., Sui, L. Alternative formulae for robust Weighted Total Least-Squares solutions for Errors-In-Variables models. Stud Geophys Geod 65, 105–127 (2021). https://doi.org/10.1007/s11200-019-0282-z

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  • DOI: https://doi.org/10.1007/s11200-019-0282-z

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