Optimization Method Based on Minimization M-Order Central Moments Used In Surveying Engineering Problems

Sławomir Cellmer

University of Warmia and Mazury in Olsztyn

Andrzej Bobojć




Abstract

A new optimization method presented in this work – the Least m-Order Central Moments method, is a generalization of the Least Squares method. It allows fitting a geometric object into a set of points in such a way that the maximum shift between the object and the points after fitting is smaller than in the Least Squares method. This property can be very useful in some engineering tasks, e.g. in the realignment of a railway track or gantry rails. The theoretical properties of the proposed optimization method are analyzed. The computational problems are discussed. The appropriate computational techniques are proposed to overcome these problems. The detailed computational algorithm and formulas of iterative processes have been derived. The numerical tests are presented, in order to illustrate the operation of proposed techniques. The results have been analyzed, and the conclusions were then formulated.


Keywords:

the Least Squares method, the Newton method, objective function, m-estimation, surveying engineering


AVRIEL M. 2003. Nonlinear Programming: Analysis and Methods. Dover Publishing.   Google Scholar

CASPARY W., HAEN W. 1990. Simultaneous estimation of location and scale parameters in the context of robust M-estimation. Manuscripta Geodaetica, 15: 273–282.   Google Scholar

CELLMER S. 2014. Least fourth powers: optimisation method favouring outliers. Survey Review, 47(345): 417. DOI: https://doi.org/10.1179/1752270614Y.0000000142.   Google Scholar

CHANG X.W., GUO Y. 2005. Huber’s M-estimation in relative GPS positioning: computational aspects. Journal of Geodesy, 79: 351–362. DOI: https://doi.org/10.1007/s00190-005-0473-y.   Google Scholar

DUCHNOWSKI R., WIŚNIEWSKI Z. 2012. Estimation of the shift between parameters of functional models of geodetic observations by applying M split estimation. Journal of Surveying Engineering, 138: 1–8. DOI: 10.1061/(ASCE)SU.1943-5428.0000062.   Google Scholar

FLETCHER R. 1987. Practical methods of optimization. 2nd ed. John Wiley & Sons, New York.   Google Scholar

HAMPEL F.R., RONCHETTI E., ROUSSEEUW P.J., STAHEL W.A. 1986. Robust statistics: the approach based on influence function. Wiley, New York.   Google Scholar

HUBER P.J. 1981. Robust statistics. Wiley, New York.   Google Scholar

KADAJ R. 1988. Eine verallgemeinerte Klasse von Schätzverfahren mit praktischen Anwendungen. Z Vermessungs-wesen, 113(4): 157–166.   Google Scholar

KAMIŃSKI W., WIŚNIEWSKI Z. 1992. Analysis of some, robust, adjustment methods. Geodezja i Kartografia, 41(3-4): 173-182.   Google Scholar

KOCH K.R. 1996. Robuste Parameterschätzung. Allgemeine Vermessungs-Nachrichten, 103(11): 1–18.   Google Scholar

LIEW C.K. 1976. Inequality constrained least squares estimation. Journal of the American Statistical Association, 71: 746–751.   Google Scholar

MARTINS T.C., TSUZUKI M.S.G. 2009. Placement over containers with fixed dimensions solved with adaptive neighborhood simulated annealing. Bulletin of the Polish Academy of Sciences. Technical Sciences, 57(3): 273-280.   Google Scholar

MEAD J.L., RENAUT R.A. 2010. Least squares problems with inequality constraints as quadratic constraints. Linear Algebra and its Applications, 432(8): 1936–1949.   Google Scholar

NEUMANN J. von, MORGENSTERN O. 1947. Theory of games and economic behavior. Princeton Univ. Press. Princeton, New Jersey.   Google Scholar

NOCEDAL J., WRIGHT S.J. 1999. Numerical Optimization. Springer-Verlag, Berlin.   Google Scholar

SKAŁA-SZYMAŃSKA M., CELLMER S., RAPIŃSKI J. 2014. Use of Nelder-Mead simplex method to arc fitting for railway track realignment. The 9th International Conference Environmental Engineering, selected papers. DOI: 10.3846/enviro.2014.244.   Google Scholar

WERNER H.J. 1990. On inequality constrained generalized least-squares estimation. Linear Algebra and its Applications, 27: 379–392. DOI: http://dx.doi.org/10.1179/1752270614Y.0000000142.   Google Scholar

WIŚNIEWSKI Z. 2009. Estimation of parameters in a split functional model of geodetic observations (M split estimation). Journal of Geodesy, 83: 105–120. DOI: https://doi.org/10.1007/s00190-008-0241-x.   Google Scholar

WIŚNIEWSKI Z. 2010. M split(q) estimation: estimation of parameters in a multi split functional model of geodetic observations. Journal of Geodesy, 84: 355–372. DOI: https://doi.org/10.1007/s00190-010-0373-7.   Google Scholar

XU P. 1989. On robust estimation with correlated observations. Bulletin Géodésique, 63: 237–252.   Google Scholar

YANG Y.1999. Robust estimation of geodetic datum transformation. Journal of Geodesy, 73: 268–274. DOI: https://doi.org/10.1007/s001900050243.   Google Scholar

YANG Y., SONG I., XU T. 2002. Robust estimation for correlated observations based on bifactor equivalent weights. Journal of Geodesy, 76: 353–358. DOI: https://doi.org/10.1007/s00190-002-0256-7.   Google Scholar

ZHONG D. 1997. Robust estimation and optimal selection of polynomial parameters for the interpolation of GPS geoid heights. Journal of Geodesy, 71: 552–561. DOI: https://doi.org/10.1007/s001900050123.   Google Scholar

ZHU J. 1996. Robustness and the robust estimate. Journal of Geodesy, 70: 586–590. DOI: https://doi.org/10.1007/BF00867867.   Google Scholar

Download


Published
2021-05-31 — Updated on 2021-06-15

Cited by

Cellmer, S., & Bobojć, A. (2021). Optimization Method Based on Minimization M-Order Central Moments Used In Surveying Engineering Problems. Technical Sciences, 24(1), 39–49. https://doi.org/10.31648/ts.6555

Sławomir Cellmer 
University of Warmia and Mazury in Olsztyn
Andrzej Bobojć 




License

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.





-->