偏差改正的Partial EIV模型方差分量估计

doi:10.11947/j.AGCS.2019.20170693

摘要/Abstract

摘要： 针对Partial EIV模型的方差分量估计中未考虑参数估值偏差所带来的影响，将Partial EIV模型视为非线性函数得到参数估值的偏差及二阶近似协方差表达式，计算得到偏差改正后的参数估值，结合方差分量估计方法，更新由参数估值影响的矩阵变量，给出了基于偏差改正的方差分量估计迭代方法。试验结果表明，参数估值及其协方差主要受参数估值偏差大小的影响，加入偏差改正能够得到更加合理的参数估值及方差分量估值，偏差改正后的方差分量估值可更加合理地评估参数估值的精度信息。

关键词: Partial EIV模型, 非线性, 偏差改正, 方差分量估计

Abstract: Considering the methods of variance components estimation (VCE) in Partial errors-in-variables (Partial EIV) model have not considered the effect of the bias of parameter estimates, the formulas of bias and second-order covariance matrix of parameter estimates are presented with the Partial EIV model regarded as a non-linear function and the parameter estimates after bias-correct are calculated. Combining the VCE method, the matrix variable influenced by the parameter estimates is updated, and an iterative method of variance components estimation based on bias-correct is given. The experiments show that the reasonable parameter estimates and its second-order approximate covariance results are affected by the bias of parameter estimates. The reasonable parameter estimates and variance components estimates can be obtained through the bias-correct and the second-order information obtained can reasonably evaluate precision of parameter estimates.

Key words: Partial EIV model, non-linear, bias correction, variance components estimation

中图分类号:

P207

王乐洋, 温贵森. 偏差改正的Partial EIV模型方差分量估计[J]. 测绘学报, 2019, 48(4): 412-421.

WANG Leyang, WEN Guisen. Bias-corrected variance components estimation of Partial EIV model[J]. Acta Geodaetica et Cartographica Sinica, 2019, 48(4): 412-421.

参考文献

[1] 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.
[2] 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.
[3] FANG Xing. Weighted total least squares solutions for applications in geodesy[D]. Germany:Leibniz Universität Hannover, 2011.
[4] 王乐洋, 许才军. 总体最小二乘研究进展[J]. 武汉大学学报(信息科学版), 2013, 38(7):850-856. WANG Leyang, XU Caijun. Progress in total least squares[J]. Geomatics and Information Science of Wuhan University, 2013, 38(7):850-856.
[5] AMIRI-SIMKOOEI A, JAZAERI S. Weighted total least squares formulated by standard least squares theory[J]. Journal of Geodetic Science, 2012, 2(2):113-124.
[6] 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.
[7] FANG Xing. Weighted total least squares:necessary and sufficient conditions, fixed and random parameters[J]. Journal of Geodesy, 2013, 87(8):733-749.
[8] FANG Xing. A structured and constrained total least-squares solution with cross-covariances[J]. Studia Geophysica et Geodaetica, 2014, 58(1):1-16.
[9] SCHAFFRIN B, FELUS Y A. On total least-squares adjustment with constraints[M]//SANSÒ F. A Window on the Future of Geodesy. Berlin, Heidelberg:Springer, 2005:417-421.
[10] SCHAFFRIN B, FELUS Y A. An algorithmic approach to the total least-squares problem with linear and quadratic constraints[J]. Studia Geophysica et Geodaetica, 2009, 53(1):1-16.
[11] MAHBOUB V, SHARIFI M A. On weighted total least-squares with linear and quadratic constraints[J]. Journal of Geodesy, 2013, 87(3):279-286.
[12] WANG Ding, ZHANG Li, WU Ying. Constrained total least squares algorithm for passive location based on bearing-only measurements[J]. Science in China Series F:Information Sciences, 2007, 50(4):576-586.
[13] FANG Xing. Weighted total least-squares with constraints:a universal formula for geodetic symmetrical transformations[J]. Journal of Geodesy, 2015, 89(5):459-469.
[14] FANG Xing. On non-combinatorial weighted total least squares with inequality constraints[J]. Journal of Geodesy, 2014, 88(8):805-816.
[15] AYDIN C, MERCAN H, UYGUR S Ö. Increasing numerical efficiency of iterative solution for total least-squares in datum transformations[J]. Studia Geophysica et Geodaetica, 2018, 62(2):223-242.
[16] 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.
[17] 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.
[18] 王乐洋, 余航, 陈晓勇. Partial EIV模型的解法[J]. 测绘学报, 2016, 45(1):22-29. DOI:10.11947/j.AGCS.2016.20140560. WANG Leyang, YU Hang, CHEN Xiaoyong. An algorithm for partial EIV model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1):22-29. DOI:10.11947/j.AGCS.2016.20140560.
[19] 王乐洋, 许光煜, 温贵森. 一种相关观测的Partial EIV模型求解方法[J]. 测绘学报, 2017, 46(8):978-987. DOI:10.11947/j.AGCS.2017.20160430. WANG Leyang, XU Guangyu, WEN Guisen. A method for partial EIV model with correlated observations[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(8):978-987. DOI:10.11947/j.AGCS.2017.20160430.
[20] 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.
[21] 赵俊, 归庆明, 郭飞宵. 基于改进目标函数的partial EIV模型WTLS估计的新算法[J]. 武汉大学学报(信息科学版), 2017, 42(8):1179-1184. ZHAO Jun, GUI Qingming, GUO Feixiao. A new algorithm of weighted total least squares estimate of partial EIV model based on an improved objective function[J]. Geomatics and Information Science of Wuhan University, 2017, 42(8):1179-1184.
[22] 王乐洋, 许光煜, 陈晓勇. 附有相对权比的PEIV模型总体最小二乘平差[J]. 武汉大学学报(信息科学版), 2017, 42(6):857-863. WANG Leyang, XU Guangyu, CHEN Xiaoyong. Total least squares adjustment of partial errors-in-variables model with weight scaling factor[J]. Geomatics and Information Science of Wuhan University, 2017, 42(6):857-863.
[23] 许光煜. Partial EIV模型的总体最小二乘方法及应用研究[D]. 南昌:东华理工大学, 2015. XU Guangyu. The total least squares method and its application of partial errors-in-variables model[D]. Nanchang:East China University of Technology, 2015.
[24] 王乐洋, 温贵森. 一种基于Partial EIV模型的多项式拟合解法[J]. 大地测量与地球动力学, 2017, 37(7):737-742. WANG Leyang, WEN Guisen. A kind of polynomial fitting method based on partial EIV model[J]. Journal of Geodesy and Geodynamics, 2017, 37(7):737-742.
[25] 王乐洋, 熊露云. 水准测量中尺度比参数的附加系统参数的Partial EIV模型解法[J]. 大地测量与地球动力学, 2017, 37(8):856-859, 875. WANG Leyang, XIONG Luyun. The partial EIV model solution for additional systematic parameters of scale parameters in leveling[J]. Journal of Geodesy and Geodynamics, 2017, 37(8):856-859, 875.
[26] SCHAFFRIN B, WIESER A. On weighted total least-squares adjustment for linear regression[J]. Journal of Geodesy, 2008, 82(7):415-421.
[27] SHEN Yunzhong, LI Bofeng, CHEN Yi. An iterative solution of weighted total least-squares adjustment[J]. Journal of Geodesy, 2011, 85(4):229-238.
[28] 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.
[29] BOX M J. Bias in nonlinear estimation[J]. Journal of the Royal Statistical Society. Series B (Methodological), 1971, 33(2):171-201.
[30] GERHOLD G A. Least-squares adjustment of weighted data to a general linear equation[J]. American Journal of Physics, 1969, 37(2):156-161.
[31] AMIRI-SIMKOOEI A R, ZANGENEH-NEJAD F, ASGARI J. On the covariance matrix of weighted total least-squares estimates[J]. Journal of Surveying Engineering, 2016, 142(3):04015014.
[32] 王乐洋, 赵英文, 陈晓勇, 等. 多元总体最小二乘问题的牛顿解法[J]. 测绘学报, 2016, 45(4):411-417, 424. DOI:10.11947/j.AGCS.2016.20150246. WANG Leyang, ZHAO Yingwen, CHEN Xiaoyong, et al. A newton algorithm for multivariate total least squares problems[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(4):411-417, 424. DOI:10.11947/j.AGCS.2016.20150246.
[33] WANG Leyang, ZHAO Yingwen. Unscented transformation with scaled symmetric sampling strategy for precision estimation of total least squares[J]. Studia Geophysica et Geodaetica, 2017, 61(3):385-411.
[34] 赵英文. 总体最小二乘精度评定方法研究[D]. 南昌:东华理工大学, 2017. ZHAO Yingwen. Research on precision estimation method for total least squares[D]. Nanchang:East China University of Technology, 2017.
[35] 於宗俦. 方差-协方差分量估计的统一理论[J]. 测绘学报, 1991, 20(3):161-171. YU Zongchou. The uniformity theory of estimation of variance-covariance components[J]. Acta Geodaetica et Cartographica Sinica, 1991, 20(3):161-171.
[36] TEUNISSEN P J G, AMIRI-SIMKOOEI A R. Least-squares variance component estimation[J]. Journal of Geodesy, 2008, 82(2):65-82.
[37] 崔希璋, 於宗俦, 陶本藻, 等. 广义测量平差(新版)[M]. 武汉:武汉大学出版社, 2005. CUI Xizhang, YU Zongchou, TAO Benzao, et al. Generalized surveying adjustment (New Edition)[M]. Wuhan:Wuhan University Press, 2005.
[38] 李博峰. 无缝仿射基准转换模型的方差分量估计[J]. 测绘学报, 2016, 45(1):30-35. DOI:10.11947/j.AGCS.2016.20140676. LI Bofeng. Variance component estimation in the seamless affine transformation model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1):30-35. DOI:10.11947/j.AGCS.2016.20140676.
[39] 余航. 总体最小二乘联合平差方法及其应用研究[D]. 南昌:东华理工大学, 2016. YU Hang. Research on the total least squares joint adjustment and its application[D]. Nanchang:East China University of Technology, 2016.
[40] AMIRI-SIMKOOEI A R. Application of least squares variance component estimation to errors-in-variables models[J]. Journal of Geodesy, 2013, 87(10-12):935-944.
[41] WANG Leyang, XU Guangyu. Variance component estimation for partial errors-in-variables models[J]. Studia Geophysica et Geodaetica, 2016, 60(1):35-55.
[42] 王乐洋, 温贵森. Partial EIV模型的非负最小二乘方差分量估计[J]. 测绘学报, 2017, 46(7):857-865. DOI:10.11947/j.AGCS.2017.20160501. WANG Leyang, WEN Guisen. Non-negative least squares variance component estimation of Partial EIV model[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(7):857-865. DOI:10.11947/j.AGCS.2017.20160501.
[43] XU Peiliang, LIU Jingnan. Variance components in errors-in-variables models:estimability, stability and bias analysis[J]. Journal of Geodesy, 2014, 88(8):719-734.
[44] RATKOWSKY D A. Nonlinear regression modeling:a unified practical approach[M]. New York:Marcel Dekker Inc, 1983.