An Algorithm for Partial EIV Model

doi:10.11947/j.AGCS.2016.20140560

Abstract

Abstract: A new thinking for solving partial errors-in-variables (partial EIV) model was proposed. Through the transposition processing in partial EIV model, a new functional model was reconstructed. Adjustment of indirect observations has been used two times to calculate the model parameters and the stochastic elements in coefficient matrix, translating total least squares problem to least squares problem. It also achieves high convergence rate through some simple variables transformation. Finally, real and simulation data were implemented to compare with the existing algorithms and to analysis the applicability of the proposed algorithms. The results show that the new algorithms are feasible and it can achieve the same values with the existing algorithms.

Key words: total least squares, partial errors-in-variables model, indirect observations adjustment, coordinate transformation, straight line fit

CLC Number:

P207

WANG Leyang, YU Hang, CHEN Xiaoyong. An Algorithm for Partial EIV Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1): 22-29.

References

[1] 王乐洋. 基于总体最小二乘的大地测量反演理论及应用研究[J]. 测绘学报, 2012, 41(4): 629. WANG Leyang. Research on Theory and Application of Total Least Squares in Geodetic Inversion[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(4): 629.
[2] ADCOCK R J. Note on the Method of Least Squares[J]. The Analysis, 1877, 4: 183-184.
[3] PEARSON K. On Line and Planes of Closest Fit to Systems of Points in Space[J]. Philosophical Magazine Series 6, 1901, 2(11): 559-572.
[4] MADANSKY A. The Fitting of Straight Lines When Both Variables are Subject to Error[J]. Journal of the American Statistical Association, 1959, 54(285): 173-205.
[5] YORK D. Least-Squares Fitting of a Straight Line[J]. Canadian Journal of Physics, 1966, 44(5): 1079-1086.
[6] 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.
[7] SCHAFFRIN B, WIESER A. On Weighted Total Least-Squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7): 415-421.
[8] 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.
[9] 鲁铁定, 宁津生. 总体最小二乘平差理论及其应用[M]. 北京: 中国科学技术出版社, 2011. LU Tieding, NING Jinsheng. Total Least Squares Adjustment Theory and Its Applications[M]. Beijing: China Science and Technology Press, 2011.
[10] MAHBOUB V. On Weighted Total Least Squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86(5): 359-367.
[11] FANG Xing. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749.
[12] AMIRI-SIMKOOEI A R, JAZAERI S. Weighted Total Least Squares Formulated by Standard Least Squares Theory[J]. Journal of Geodetic Science, 2012, 2(2): 113-124.
[13] 周拥军, 朱建军, 邓才华. 附参数的条件平差与按行独立的加权总体最小二乘法估计的一致性研究[J]. 测绘学报, 2012, 41(1): 48-53. ZHOU Yongjun, ZHU Jianjun, DENG Caihua. The Consistency between Row-wised Weighted Total Least Squares and Condition Adjustment with Parameters[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(1): 48-53.
[14] 胡川, 陈义. 非线性整体最小二乘平差迭代算法[J]. 测绘学报, 2014, 43(7): 668-674. HU Chuan, CHEN Yi. An Iterative Algorithm for Nonlinear Total Least Squares Adjustment[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(7): 668-674.
[15] SHEN Yunzhong, LI Bofeng, CHEN Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238.
[16] 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.
[17] 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.
[18] 王乐洋, 许才军, 鲁铁定. 边长变化反演应变参数的总体最小二乘方法[J]. 武汉大学学报(信息科学版), 2010, 35(2): 181-184. WANG Leyang, XU Caijun, LU Tieding. Inversion of Strain Parameter Using Distance Changes Based on Total Least Squares[J]. Geomatics and Information Science of Wuhan University, 2010, 35(2): 181-184.
[19] JAZAERI S, AMIRI-SIMKOOEI A R, Sharifi M A. Iterative Algorithm for Weighted Total Least Squares Adjustment[J]. Survey Review, 2014, 46(334): 19-27.
[20] 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.
[21] 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.
[22] SHI Yun, XU Peiliang, LIU Jingnan, et al. Alternative Formulae for Parameter Estimation in Partial Errors-in-variables Models[J]. Journal of Geodesy, 2015, 89(1): 13-16.
[23] VAN HUFFEL S, VANDEWALLE J. The Total Least Squares Problem: Computational Aspects and Analysis[M]. Philadelphia: SIAM, 1991.
[24] NERI F, SAITTA G, CHIOFALO S. An Accurate and Straightforward Approach to Line Regression Analysis of Error-affected Experimental Data[J]. Journal of Physics E: Scientific Instruments, 1989, 22(4): 215-217.
[25] 曾文宪. 系数矩阵误差对EIV模型平差结果的影响研究[D]. 武汉: 武汉大学, 2013. ZENG Wenxian. Effect of the Random Design Matrix on Adjustment of an EIV Model and Its Reliability Theory[D]. Wuhan: Wuhan University, 2013.
[26] 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. 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.