一种Partial EIV半参数模型的系统误差处理方法
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摘要
在使用总体最小二乘求解参数时,若观测值中包含系统误差,此时得到的参数估值则会受到系统误差的影响,从而得到不可靠的解,因此必须削弱系统误差对参数估计的影响,以获得相对可靠的解。本文提出在partial errors-in-variables(Partial EIV)模型的基础上给观测值增加非参数部分(系统误差),从而构建Partial EIV半参数模型;基于补偿最小二乘准则进行公式推导,并分别通过选取适当的正则化矩阵及通过L曲线法确定平滑因子。通过算例结果分析表明,与传统方法相比,本文的方法在一定程度上能够削弱系统误差的影响,得到更为可靠的参数解,从而验证了该方法的有效性和可行性。
The estimated values are affected so that they are not reliable if the observations contain systematic errors under the solution of total least squares.Thus the bad effect on the estimated values should be weakened so the relatively reliable solution can be obtained.This paper adds non parametric part(systematic errors)in the partial errors-in-variables(Partial EIV)model,building Partial EIV semi-parametric model.The penalized least square criterion is introduced to derive formula,and choose the proper regularization matrix in the experiments.The smoothing factor is acquired by the method of L-curve.The results of experiments show that the algorithm of Partial EIV semi-parametric model can mitigate systematic errors to a certain extent and obtain the more reliable solution than the traditional method.So the validity of the algorithm is verified.
The estimated values are affected so that they are not reliable if the observations contain systematic errors under the solution of total least squares.Thus the bad effect on the estimated values should be weakened so the relatively reliable solution can be obtained.This paper adds non parametric part(systematic errors)in the partial errors-in-variables(Partial EIV)model,building Partial EIV semi-parametric model.The penalized least square criterion is introduced to derive formula,and choose the proper regularization matrix in the experiments.The smoothing factor is acquired by the method of L-curve.The results of experiments show that the algorithm of Partial EIV semi-parametric model can mitigate systematic errors to a certain extent and obtain the more reliable solution than the traditional method.So the validity of the algorithm is verified.
引文
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[16]SCHAFFRIN B,WIESER A.On Weighted Total LeastSquares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421.
[17]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.
[18]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.
[19]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.
[20]王乐洋,于冬冬.病态总体最小二乘问题的虚拟观测解法[J].测绘学报,2014,43(6):575-581.DOI:10.13485/j.cnki.11-2089.2014.0091.WANG Leyang,YU Dongdong.Virtual Observation Method to Ill-Posed Total Least Squares Problem[J].Acta Geodaetica et Cartographica Sinica,2014,43(6):575-581.DOI:10.13485/j.cnki.11-2089.2014.0091.
[21]王乐洋,吴飞,吴良才.GPS高程转换的总体最小二乘拟合推估模型[J].武汉大学学报(信息科学版),2016,41(9):1259-1264.WANG Leyang,WU Fei,WU Liangcai.Total Least Squares Fitting Estimation Model for GPS Height Transformation[J].Geomatics and Information Science of Wuhan University,2016,41(9):1259-1264.
[22]NEITZEL F.Generalization of Total Least-squares on Example of Unweighted and Weighted 2D Similarity Transformation[J].Journal of Geodesy,2010,84(12):751-762.
[23]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.
[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]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.
[26]赵俊,归庆明.部分变量误差模型的整体抗差最小二乘估计[J].测绘学报,2016,45(5):552-559.DOI:10.11947/j.AGCS.2016.20150374.ZHAO Jun,GUI Qingming.Total Robustified Least Squares Estimation in Partial Errors-in-variables Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(5):552-559.DOI:10.11947/j.AGCS.2016.20150374.
[27]SHI Yun,XU Peiliang,LIU Jingnan,et al.Alternative Formulae for Parameter Estimation in Partial Errors-invariables Models[J].Journal of Geodesy,2015,89(1):13-16.
[28]赵俊,归庆明,陈晨.TLS方法解算时间序列模型及其在卫星钟差预报中的应用[C]∥第六届中国卫星导航学术年会论文集.西安:中国卫星导航系统管理办公室学术交流中心,2015.ZHAO Jun,GUI Qingming,CHEN Chen.Times Series Model Solved by TLS Method and Applications in Prediction of Satellite Clock Error[C]∥Proceedings of the 6th China Satellite Navigation Conference.Xi'an:China Satellite Navigation Office-Academic Exchange Center,2015.
[29]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.
[30]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.
[31]丁士俊.测量数据的建模与半参数估计[D].武汉:武汉大学,2005.DING Shijun.Survey Data Modeling and Semiparametric Estmating[D].Wuhan:Wuhan University,2005.
[32]王振杰,欧吉坤,曲国庆,等.用L-曲线法确定半参数模型中的平滑因子[J].武汉大学学报(信息科学版),2004,29(7):651-653.WANG Zhenjie,OU Jikun,QU Guoqing,et al.Determining the Smoothing Parameter in Semi-parametric Model Using L-Curve Method[J].Geomatics and Information Science of Wuhan University,2004,29(7):651-653.
[33]HANSEN P C,O'LEARY D P.The Use of the L-curve in the Regularization of Discrete Ill-posed Problems[J].SIAM Journal on Scientific Computing,1993,14(6):1487-1503.
[34]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.
[2]杨元喜,张双成.导航解算中的系统误差及其协方差矩阵拟合[J].测绘学报,2004,33(3):189-194.DOI:10.3321/j.issn:1001-1595.2004.03.001.YANG Yuanxi,ZHANG Shuangcheng.Fittings of Systematic Errors and Covariance Matrices in Navigation[J].Acta Geodaetica et Cartographica Sinica,2004,33(3):189-194.DOI:10.3321/j.issn:1001-1595.2004.03.001.
[3]康双双.半参数模型在航空重力测线网平差中的应用研究[D].武汉:中国地质大学,2012.KANG Shuangshuang.The Application of Semiparametric Model in Airline Network Adjustment of Airborne Gravimetry[D].Wuhan:China University of Geosciences,2012.
[4]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].2版.武汉:武汉大学出版社,2009.Surveying Adjustment Group of School of Geodesy and Geomatics,Wuhan University.Error Theory and Foundation of Surveying Adjustment[M].2ed ed.Wuhan:Wuhan University Press,2009.
[5]孙海燕,吴云.半参数回归与模型精化[J].武汉大学学报(信息科学版),2002,27(2):172-174,207.SUN Haiyan,WU Yun.Semiparametric Regression and Model Refining[J].Geomatics and Information Science of Wuhan University,2002,27(2):172-174,207.
[6]MORITZ H.Least-squares Collocation[J].Reviews of Geophysics,1978,16(3):421-430.
[7]王振杰,欧吉坤.半参数模型与拟合推估模型的比较[J].测绘通报,2004(9):4-6.DOI:10.3969/j.issn.0494-0911.2004.09.002.WANG Zhenjie,OU Jikun.Comparisons between Collocation Model and Semi-parametric Model[J].Bulletin of Surveying and Mapping,2004(9):4-6.DOI:10.3969/j.issn.0494-0911.2004.09.002.
[8]潘雄,孙海燕.半参数模型约束条件下估计量的统计诊断[J].湖北民族学院学报(自然科学版),2003,21(3):1-3.PAN Xiong,SUN Haiyan.Statistical Diagnosis of the Smoothing Parameter for Semi-parameter Model on Constrained Condition[J].Journal of Hubei Institute for Nationalities(Natural Science Edition),2003,21(3):1-3.
[9]潘雄.半参数模型的估计理论及其应用[D].武汉:武汉大学,2005.PAN Xiong.The Estimation Theory and Application Research in Semi-parametric Model[D].Wuhan:Wuhan University,2005.
[10]丁士俊,陶本藻.自然样条半参数模型与系统误差估计[J].武汉大学学报(信息科学版),2004,29(11):964-967.DING Shijun,TAO Benzao.Semiparametric Regression Model with Natural Spline and Systematic Error Estimation[J].Geomatics and Information Science of Wuhan University,2004,29(11):964-967.
[11]丁士俊,陶本藻.半参数模型核光滑估计与模拟分析[J].大地测量与地球动力学,2004,24(4):24-28.DING Shijun,TAO Benzao.Kernel Smoothing Estimate of Semiparametric Model and Simulation Analysis[J].Journal of Geodesy and Geodynamics,2004,24(4):24-28.
[12]潘雄,孙海燕.半参数测量平差模型参数的二阶段估计[J].测绘科学,2004,29(3):19-21.PAN Xiong,SUN Haiyan.Two Stage Estimation of Parameter for Semiparametric Survey Adjustment Models[J].Science of Surveying and Mapping,2004,29(3):19-21.
[13]王振杰.大地测量中不适定问题的正则化解法研究[D].武汉:中国科学院测量与地球物理研究所,2003.WANG Zhenjie.Research on the Regularization Solutions of Ill-Posed Problems in Geodesy[D].Wuhan:Institute of Geodesy and Geophysics Chinese Academy of Sciences,2003.
[14]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.
[15]SCHAFFRIN B.A Note on Constrained Total Least-squares Estimation[J].Linear Algebra and Its Applications,2006,417(1):245-258.
[16]SCHAFFRIN B,WIESER A.On Weighted Total LeastSquares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421.
[17]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.
[18]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.
[19]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.
[20]王乐洋,于冬冬.病态总体最小二乘问题的虚拟观测解法[J].测绘学报,2014,43(6):575-581.DOI:10.13485/j.cnki.11-2089.2014.0091.WANG Leyang,YU Dongdong.Virtual Observation Method to Ill-Posed Total Least Squares Problem[J].Acta Geodaetica et Cartographica Sinica,2014,43(6):575-581.DOI:10.13485/j.cnki.11-2089.2014.0091.
[21]王乐洋,吴飞,吴良才.GPS高程转换的总体最小二乘拟合推估模型[J].武汉大学学报(信息科学版),2016,41(9):1259-1264.WANG Leyang,WU Fei,WU Liangcai.Total Least Squares Fitting Estimation Model for GPS Height Transformation[J].Geomatics and Information Science of Wuhan University,2016,41(9):1259-1264.
[22]NEITZEL F.Generalization of Total Least-squares on Example of Unweighted and Weighted 2D Similarity Transformation[J].Journal of Geodesy,2010,84(12):751-762.
[23]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.
[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]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.
[26]赵俊,归庆明.部分变量误差模型的整体抗差最小二乘估计[J].测绘学报,2016,45(5):552-559.DOI:10.11947/j.AGCS.2016.20150374.ZHAO Jun,GUI Qingming.Total Robustified Least Squares Estimation in Partial Errors-in-variables Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(5):552-559.DOI:10.11947/j.AGCS.2016.20150374.
[27]SHI Yun,XU Peiliang,LIU Jingnan,et al.Alternative Formulae for Parameter Estimation in Partial Errors-invariables Models[J].Journal of Geodesy,2015,89(1):13-16.
[28]赵俊,归庆明,陈晨.TLS方法解算时间序列模型及其在卫星钟差预报中的应用[C]∥第六届中国卫星导航学术年会论文集.西安:中国卫星导航系统管理办公室学术交流中心,2015.ZHAO Jun,GUI Qingming,CHEN Chen.Times Series Model Solved by TLS Method and Applications in Prediction of Satellite Clock Error[C]∥Proceedings of the 6th China Satellite Navigation Conference.Xi'an:China Satellite Navigation Office-Academic Exchange Center,2015.
[29]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.
[30]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.
[31]丁士俊.测量数据的建模与半参数估计[D].武汉:武汉大学,2005.DING Shijun.Survey Data Modeling and Semiparametric Estmating[D].Wuhan:Wuhan University,2005.
[32]王振杰,欧吉坤,曲国庆,等.用L-曲线法确定半参数模型中的平滑因子[J].武汉大学学报(信息科学版),2004,29(7):651-653.WANG Zhenjie,OU Jikun,QU Guoqing,et al.Determining the Smoothing Parameter in Semi-parametric Model Using L-Curve Method[J].Geomatics and Information Science of Wuhan University,2004,29(7):651-653.
[33]HANSEN P C,O'LEARY D P.The Use of the L-curve in the Regularization of Discrete Ill-posed Problems[J].SIAM Journal on Scientific Computing,1993,14(6):1487-1503.
[34]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.