Weighted Total Least Squares of Universal EIV Adjustment Model

doi:10.11947/j.AGCS.2016.20150156

Abstract

Abstract: This paper proposes the universal errors-in-variables (EIV) adjustment model based on the fundamental adjustment theory, which covers the parametric adjustment model, conditional adjustment model,conditional adjustment model with parameters and parametric adjustment model with constrains. Applying total least squares (TLS) principle, we deduce the weighted TLS (WTLS) algorithm and the approximated precision of the EIV model. The universal EIV adjustment model and its estimator of WTLS contribute to the integrity of theory of EIV model estimation. The proposed uniform WTLS algorithm is appropriate for programming in software, which can contribute to the geodetic application of the theory of the EIV model estimation.

Key words: universal EIV adjustment model, weighted total least squares, precision, nonlinear optimization

CLC Number:

P207

ZENG Wenxian, FANG Xing, LIU Jingnan, YAO Yibin. Weighted Total Least Squares of Universal EIV Adjustment Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(8): 890-894.

References

[1] 武汉大学测绘学院测量平差学科组.误差理论与测量平差基础[M].武汉:武汉大学出版社,2003.The Group of Surveying Adjustment in the School of Geodesy and Geomatics,Wuhan University.Error Theory and Foundation of Surveying Adjustment[M].Wuhan:Wuhan University Press,2003.
[2] VAN HUFFEL S,VANDEWALLE J.The Total Least Squares Problem:Computational Aspects and Analysis[M].Philadelphia:Society for Industrial and Applied Mathematics,1991.
[3] ADCOCK R J.Note on the Method of Least Squares[J].The Analyst,1877,4(6):183-184.
[4] DEMING W E.The Application of Least Squares[J].The London,Edinburgh,and Dublin Philosophical Magazine and Journal of Science:Series 7,1931,11(68):146-158.
[5] GERHOLD G A.Least-squares Adjustment of Weighted Data to a General Linear Equation[J].American Journal of Physics,1969,37(2):156-161.
[6] SCHAFFRIN B,WIESER A.On Weighted Total Least-squares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421.
[7] 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.
[8] LI Bofeng,SHEN Yunzhong,ZHANG Xingfu et al.Seamless Multivariate Affine Error-in-variables Transformation and Its Application to Map Rectification[J].International Journal of Geographical Information Science,2013,27(8):1572-1592.
[9] 王乐洋.地壳应变参数反演的总体最小二乘方法[J].大地测量与地球动力学,2013,33(3):106-110.WANG Leyang.Inversion of Crustal Strain Parameters Based on Total Least Squares[J].Journal of Geodesy and Geodynamics,2013,33(3):106-110.
[10] XU Peiliang,LIU Jingnan,SHI Chuang.Total Least Squares Adjustment in Partial Errors-in-variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675.
[11] SHEN Yunzhong,LI Bofeng,CHEN Yi.An Iterative Solution of Weighted Total Least-squares Adjustment[J].Journal of Geodesy,2010,85(4):229-238.
[12] 姚宜斌,孔建.顾及设计矩阵随机误差的最小二乘组合新解法[J].武汉大学学报(信息科学版),2014,39(9):1028-1032.YAO Yibin,KONG Jian.A New Combined LS Method Considering Random Errors of Design Matrix[J].Geomatics and Information Science of Wuhan University,2014,39(9):1028-1032.
[13] FANG Xing.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749.
[14] 方兴,曾文宪,刘经南,等.三维坐标转换的通用整体最小二乘算法[J].测绘学报,2014,43(11):1139-1143.DOI:10.13485/j.cnki.11-2089.2014.0193.FANG Xing,ZENG Wenxian,LIU Jingnan,et al.A General Total Least Squares Algorithm for Three-dimensional Coordinate Transformations[J].Acta Geodaetica et Cartographica Sinica,2014,43(11):1139-1143.DOI:10.13485/j.cnki.11-2089.2014.0193.
[15] FANG Xing.A Total Least Squares Solution for Geodetic Datum Transformations[J].Acta Geodaetica et Geophysica,2014,49(2):189-207.
[16] SCHAFFRIN B,FELUS Y A.On Total Least-squares Adjustment with Constraints[C]//Proceedings of the International Association of Geodesy IAG General Assembly Sapporo.Berlin:Springer,2005,128:417-421.
[17] MAHBOUB V,SHARIFI M A.On Weighted Total Least-squares with Linear and Quadratic Constraints[J].Journal of Geodesy,2013,87(3):279-286.
[18] ZENG Wenxian,LIU Jingnan,Yao Yibin.On Partial Errors-in-variables Models with Inequality Constraints of Parameters and Variables[J].Journal of Geodesy,2015,89(2):111-119.
[19] 曾文宪,方兴,刘经南,等.附有不等式约束的加权整体最小二乘算法[J].测绘学报,2014,43(10):1013-1018.DOI:10.13485/j.cnki.11-2089.2014.0173.ZENG Wenxian,FANG Xing,LIU Jingnan,et al.Weighted Total Least Squares Algorithm with Inequality Constraints[J].Acta Geodaetica et Cartographica Sinica,2014,43(10):1013-1018.DOI:10.13485/j.cnki.11-2089.2014.0173.
[20] FANG Xing.A Structured and Constrained Total Least-squares Solution with Cross-covariances[J].Studia Geophysica et Geodaetica,2014,58(1):1-16.
[21] FANG Xing.On Non-combinatorial Weighted Total Least Squares with Inequality Constraints[J].Journal of Geodesy,2014,88(8):805-816.
[22] FANG Xing.Weighted Total Least-squares with Constraints:A Universal Formula for Geodetic Symmetrical Transformations[J].Journal of Geodesy,2015,89(5):459-469.
[23] TONG Xiaohua,JIN Yanmin,ZHANG Songlin,et al.Bias-corrected Weighted Total Least-squares Adjustment of Condition Equations[J].Journal of Surveying Engineering,2014,141(2).DOI:10.1061/(ASCE) SU.1943-5428.0000140.
[24] 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.
[25] GLESER L J.Estimation in a Multivariate "Errors in Variables" Regression Model:Large Sample Results[J].The Annals of Statistics,1981,9(1):24-44.
[26] 许超钤,姚宜斌,张豹,等.基于整体最小二乘的参数估计新方法及精度评定[J].测绘通报,2011(10):1-4.XU Chaoqian,YAO Yibin,ZHANG Bao,et al.New Method of Parameters Estimation and Accuracy Evaluation Based on TLS[J].Bulletin of Surveying and Mapping,2011(10):1-4.
[27] 孔建,姚宜斌,黄承猛.非线性模型的一阶偏导数确定方法及其在TLS精度评定中的应用[J].大地测量与地球动力学,2011,31(3):110-114.KONG Jian,YAO Yibin,HUANG Chengmeng.Method for Determining First-order Partial Derivative of NonlinearModel and Its Application in TLS Accuracy Assessment[J].Journal of Geodesy and Geodynamics,2011,31(3):110-114.