On the reliability of mixed LS adjustment models
Abstract
This paper examines the internal and external reliability criteria of the mixed LS adjustment model. We use the reliability concept to quantify the potential for detecting gross errors and to estimate their impact on the adjusted parameters. After a short introduction to the mixed adjustment model, the hat matrix and Baarda’s data snooping we describe the theoretical tools developed to define the internal and external reliability in the mixed adjustment model. The paper presents the results of an example of LS adjustment of transformation parameters between two coordinate systems, indicating that the reliability can be used effectively for this model.
Keyword : internal reliability, external reliability, mixed adjustment model, hat matrix
How to Cite
Even-Tzur, G. (2023). On the reliability of mixed LS adjustment models. Geodesy and Cartography, 49(1), 51–59. https://doi.org/10.3846/gac.2023.16893
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Cross, P. A. (1983). Advanced least squares applied to position fixing (Working Paper No. 6). North East London Polytechnic, Department of Land Surveying.
Ettlinger, A., & Neuner, H. (2020). Assessment of inner reliability in the Gauss-Helmert model. Journal of Applied Geodesy, 14(1), 13–28. https://doi.org/10.1515/jag-2019-0013
Even-Tzur, G. (1999). Reliability design and control of geodetic networks. Zeitschrift für Vermessungswesen, 124(4), 128–132.
Heck, B. (1981). Der Einfluss einzelner Beobachtungen auf das Ergebnis einer Ausleichung und die Suche nach Ausreissern in den Beobachtungen. Allgemeine Vermessungs-Nachrichten, 88(1), 17–34.
Hoaglin, D. C., & Welsch, R. E. (1978). The hat matrix in regression and ANOVA. The American Statistician, 32(1), 17–22. https://doi.org/10.1080/00031305.1978.10479237
Huber, P. J. (1981). Robust statistics. John Wiley & Sons. https://doi.org/10.1002/0471725250
Knight, N. L., Wang, J., & Rizos, C. (2010). Generalised measures of reliability for multiple outliers. Journal of Geodesy, 84(10), 625–635. https://doi.org/10.1007/s00190-010-0392-4
Koch, K. R. (1985). Test von Ausreissern in Beobachtungspaaren. Zeitschrift für Vermessungswesen, 110, 34–38.
Koch, K. R. (1999). Parameter estimation and hypothesis testing in linear models. Springer Science & Business Media. https://doi.org/10.1007/978-3-662-03976-2
Koch, K. R. (2014). Outlier detection for the nonlinear Gauss Helmert model with variance components by the expectation maximization algorithm. Journal of Applied Geodesy, 8(3), 185–194. https://doi.org/10.1515/jag-2014-0004
Krarup, T., Juhl, J., & Kubik, K. (1980). Götterdämmerung over least squares adjustment. In Proceedings of the14th Congress of the International Society for Photogrammetry (pp. 369–378), Hamburg, Germany.
Lehmann, R. (2012). Improved critical values for extreme normalized and studentized residuals in Gauss–Markov models. Journal of Geodesy, 86(12), 1137–1146. https://doi.org/10.1007/s00190-012-0569-0
Leick, A. (2004). GPS satellite surveying (3rd ed.). Wiley.
Mikhail, E. M., & Ackermann, F. E. (1976). Observations and least squares. IEP.
Neitzel, F. (2010). Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. Journal of Geodesy, 84(12), 751–762. https://doi.org/10.1007/s00190-010-0408-0
Pope, A. (1972). Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In Proceedings of the 38th Annual Meeting of the American Society of Photogrammetry (pp. 449–473), Washington.
Pope, A. J. (1976). The statistics of residuals and the outlier detection of outliers (NOAA Technical Report NOS 65, NGS 1). National Geodetic Survey, Rockville, MD.
Rofatto, V. F., Matsuoka, M. T., Klein, I., Veronez, M. R., Bonimani, M. L., & Lehmann, R. (2020). A half-century of Baarda’s concept of reliability: A review, new perspectives, and applications. Survey Review, 52(372), 261–277. https://doi.org/10.1080/00396265.2018.1548118
Teunissen, P. J. G. (1985). Quality control in geodetic networks. In Optimization and design of geodetic networks (pp. 526–547). Springer. https://doi.org/10.1007/978-3-642-70659-2_18
Teunissen, P. J. G. (1998). Minimal detectable biases of GPS data. Journal of Geodesy, 72(4), 236–244. https://doi.org/10.1007/s001900050163
Teunissen, P. J. G. (2006). Testing theory: An introduction (2nd ed.). Delft University Press.
Teunissen, P. J. G. (2018). Distributional theory for the DIA method. Journal of Geodesy, 92(1), 59–80. https://doi.org/10.1007/s00190-017-1045-7
Wang, B., Fang, X., Liu, C., & Zhu, B. (2020). Data Snooping for the equality constrained nonlinear Gauss–Helmert model using sensitivity analysis. Journal of Surveying Engineering, 146(4), 04020015. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000326
Yang, L., Wang, J., Knight, N. L., & Shen, Y. (2013). Outlier separability analysis with a multiple alternative hypotheses test. Journal of Geodesy, 87(6), 591–604. https://doi.org/10.1007/s00190-013-0629-0
Yang, L., Li, B., Shen, Y., & Rizos, C. (2017). Extension of internal reliability analysis regarding separability analysis. Journal of Surveying Engineering, 143(3), 04017002. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000220
Cross, P. A. (1983). Advanced least squares applied to position fixing (Working Paper No. 6). North East London Polytechnic, Department of Land Surveying.
Ettlinger, A., & Neuner, H. (2020). Assessment of inner reliability in the Gauss-Helmert model. Journal of Applied Geodesy, 14(1), 13–28. https://doi.org/10.1515/jag-2019-0013
Even-Tzur, G. (1999). Reliability design and control of geodetic networks. Zeitschrift für Vermessungswesen, 124(4), 128–132.
Heck, B. (1981). Der Einfluss einzelner Beobachtungen auf das Ergebnis einer Ausleichung und die Suche nach Ausreissern in den Beobachtungen. Allgemeine Vermessungs-Nachrichten, 88(1), 17–34.
Hoaglin, D. C., & Welsch, R. E. (1978). The hat matrix in regression and ANOVA. The American Statistician, 32(1), 17–22. https://doi.org/10.1080/00031305.1978.10479237
Huber, P. J. (1981). Robust statistics. John Wiley & Sons. https://doi.org/10.1002/0471725250
Knight, N. L., Wang, J., & Rizos, C. (2010). Generalised measures of reliability for multiple outliers. Journal of Geodesy, 84(10), 625–635. https://doi.org/10.1007/s00190-010-0392-4
Koch, K. R. (1985). Test von Ausreissern in Beobachtungspaaren. Zeitschrift für Vermessungswesen, 110, 34–38.
Koch, K. R. (1999). Parameter estimation and hypothesis testing in linear models. Springer Science & Business Media. https://doi.org/10.1007/978-3-662-03976-2
Koch, K. R. (2014). Outlier detection for the nonlinear Gauss Helmert model with variance components by the expectation maximization algorithm. Journal of Applied Geodesy, 8(3), 185–194. https://doi.org/10.1515/jag-2014-0004
Krarup, T., Juhl, J., & Kubik, K. (1980). Götterdämmerung over least squares adjustment. In Proceedings of the14th Congress of the International Society for Photogrammetry (pp. 369–378), Hamburg, Germany.
Lehmann, R. (2012). Improved critical values for extreme normalized and studentized residuals in Gauss–Markov models. Journal of Geodesy, 86(12), 1137–1146. https://doi.org/10.1007/s00190-012-0569-0
Leick, A. (2004). GPS satellite surveying (3rd ed.). Wiley.
Mikhail, E. M., & Ackermann, F. E. (1976). Observations and least squares. IEP.
Neitzel, F. (2010). Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. Journal of Geodesy, 84(12), 751–762. https://doi.org/10.1007/s00190-010-0408-0
Pope, A. (1972). Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In Proceedings of the 38th Annual Meeting of the American Society of Photogrammetry (pp. 449–473), Washington.
Pope, A. J. (1976). The statistics of residuals and the outlier detection of outliers (NOAA Technical Report NOS 65, NGS 1). National Geodetic Survey, Rockville, MD.
Rofatto, V. F., Matsuoka, M. T., Klein, I., Veronez, M. R., Bonimani, M. L., & Lehmann, R. (2020). A half-century of Baarda’s concept of reliability: A review, new perspectives, and applications. Survey Review, 52(372), 261–277. https://doi.org/10.1080/00396265.2018.1548118
Teunissen, P. J. G. (1985). Quality control in geodetic networks. In Optimization and design of geodetic networks (pp. 526–547). Springer. https://doi.org/10.1007/978-3-642-70659-2_18
Teunissen, P. J. G. (1998). Minimal detectable biases of GPS data. Journal of Geodesy, 72(4), 236–244. https://doi.org/10.1007/s001900050163
Teunissen, P. J. G. (2006). Testing theory: An introduction (2nd ed.). Delft University Press.
Teunissen, P. J. G. (2018). Distributional theory for the DIA method. Journal of Geodesy, 92(1), 59–80. https://doi.org/10.1007/s00190-017-1045-7
Wang, B., Fang, X., Liu, C., & Zhu, B. (2020). Data Snooping for the equality constrained nonlinear Gauss–Helmert model using sensitivity analysis. Journal of Surveying Engineering, 146(4), 04020015. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000326
Yang, L., Wang, J., Knight, N. L., & Shen, Y. (2013). Outlier separability analysis with a multiple alternative hypotheses test. Journal of Geodesy, 87(6), 591–604. https://doi.org/10.1007/s00190-013-0629-0
Yang, L., Li, B., Shen, Y., & Rizos, C. (2017). Extension of internal reliability analysis regarding separability analysis. Journal of Surveying Engineering, 143(3), 04017002. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000220