偏差改正的Partial EIV模型方差分量估计
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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。
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.
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.
引文
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[3] FANG Xing.Weighted total least squares solutions for applications in geodesy[D].Germany:Leibniz Universit?t 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.
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[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 ?.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.
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[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.
[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 Universit?t 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 ?.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.