Solution for Robust Total Least Squares Estimation Based on Median Method

doi:10.11947/j.AGCS.2016.20150234

Abstract

Abstract: Because the present algorithms of total least squares for robust estimation have disadvantages, solution for computation primary model parameters based on median method is proposed. And to get over the influence that estimation error of random model and error of mean square has, computation weight matrix of observation vector and coefficient matrix separately are proposed. Iterative algorithm of robust total least squares is established based on median method, and to prove the proposed solution to be feasible, an instance is cited. The numerical results of the instance clearly demonstrate that the presented solution is more accurate than the traditional method for line fitting.

Key words: total least squares, robust estimation, median method

CLC Number:

P207

TAO Yeqing, GAO Jingxiang, YAO Yifei. Solution for Robust Total Least Squares Estimation Based on Median Method[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(3): 297-301.

References

[1] PEARSON K. On Lines and Planes of Closest Fit to Systems of Points in Space[J]. Philosophical Magazine Series 6, 1901, 2(11):559-572.
[2] GOLUB G H, VAN LOAN C F. An Analysis of the Total Least Squares Problem[J]. SIAM Journal on Numerical Analysis, 1980, 17(6):883-893.
[3] VAN HUFFEL S, VANDEWALLE J. Analysis and Solution of the Nongeneric Total Least Squares Problem[J]. SIAM Journal on Matrix Analysis and Applications, 1988, 9(3):360-372.
[4] SCHAFFRIN B, FELUS Y A. On Total Least-squares Adjustment with Constraints[J]. International Association of Geodesy Symposia, 2005, 128:417-421.
[5] SCHAFFRIN B, LEE I, CHOI Y, et al. Total Least Squares(TLS) for Geodetic Straight-line and Plane Adjustment[J]. Bollettino di Geodesia e Scienze Affini, 2006, 65(3):141-168.
[6] MAHBOUB V. On Weighted Total Least-squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86(5):359-367.
[7] SCHAFFRIN B, FELUS Y A. On the Multivariate Total Least-squares Approach to Empirical Coordinate Transformations:Three Algorithms[J]. Journal of Geodesy, 2008, 82(6):373-383.
[8] KUPFERER S. Application of the Total Least-squares Technology with Geodetic Problem Definitions[D]. Karlsruhe:University of Karlsruhe, 2005.
[9] SCHAFFRIN B, WIESER A. On Weighted Total Least-squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7):415-421.
[10] FULLER W A. Properties of Some Estimators for the Errors-in-Variables Model[J]. The Annals of Statistics, 1980, 8(2):407-422.
[11] TONG Xiaohua, JIN Yanmin, LI Lingyun. An Improved Weighted Total Least Squares Method with Applications in Linear Fitting and Coordinate Transformation[J]. Journal of Surveying Engineering, 2011, 137(4):120-128.
[12] SHEN Yunzhong, LI Bofeng, CHEN Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy, 2011, 85(4):229-238.
[13] 葛旭明, 伍吉仓. 病态总体最小二乘问题的广义正则化[J]. 测绘学报, 2012, 41(3):372-377. GE Xuming, WU Jicang. Generalized Regularization to Ⅲ-posed Total Least Squares Problem[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(3):372-377.
[14] SCHAFFRIN B, SNOW K. Total Least-squares Regularization of Tykhonov Type and an Ancient Racetrack in Corinth[J]. Linear Algebra and Its Applications, 2010, 432(8):2061-2076.
[15] 陈义, 陆珏. 以三维坐标转换为例解算稳健总体最小二乘方法[J]. 测绘学报, 2012, 41(5):715-722. CHEN Yi,LU Jue.Performing 3D Similarity Transformation by Robust Total Least Squares[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(5):715-722.
[16] SCHAFFRIN B. A Note on Constrained Total Least Squares Estimation[J]. Linear Algebra and Its Applications, 2006, 417(1):245-258.
[17] ZHANG Songlin, TONG Xiaohua, ZHANG Kun. A Solution to EIV Model with Inequality Constraints and Its Geodetic Applications[J]. Journal of Geodesy, 2013, 87(1):23-28.
[18] TAO Yeqing, GAO Jinxiang, YAO Yifei. TLS Algorithm for GPS Height Fitting Based on Robust Estimation[J] Survey Review, 2014, 46(336):184-188.
[19] 周江文. 经典误差理论与抗差估计[J]. 测绘学报, 1989, 18(2):115-120. ZHOU Jiangwen. Classic Theory of Errors and Robust Estimation[J]. Acta Geodaetica et Cartographica Sinica, 1989, 18(2):115-120.
[20] ROUSSEEUW P J, WAGNER J. Robust Regression with a Distributed Intercept Using Least Median of Squares[J]. Computational Statistics & Data Analysis, 1994, 17(1):65-76.
[21] YANG Yuanxi. Robust Estimation of Geodetic Datum Transformation[J]. Journal of Geodesy, 1999, 73(5):268-274.
[22] 刘经南, 曾文宪, 徐培亮. 整体最小二乘估计的研究进展[J]. 武汉大学学报(信息科学版), 2013, 38(5):505-512. LIU Jingnan, ZENG Wenxian, XU Peiliang. Overview of Total Least Squares Methods[J]. Geomatics and Information Science of Wuhan University, 2013, 38(5):505-512.