杜寒露,陶叶青,蔡安宁,周露

• 1:淮阴师范学院城市与环境学院

摘要(Abstract)：

针对解决变量中含有误差(EIV)模型参数估计算法的降正则化性导致即使模型参数初值可靠,参数估值也可能在迭代过程中发散的问题,该文分析现有EIV模型参数估计算法具有的降正则化性质,讨论EIV模型参数估计算法具有的降正则化性对模型正则化的影响,建立一种病态EIV模型的实时修正算法。通过算例验证该文所建立的算法,算例结果表明,该文建立的算法能够有效解决EIV模型参数估计存在的上述问题。该文所建立的病态EIV模型正则化算法更具普适性。

关键词(KeyWords)： 病态EIV模型;正则化;迭代算法

Abstract:

Keywords:

基金项目(Foundation): 国家自然科学基金项目(41601501);; 江苏省社会科学基金项目(17EYB006);; 江苏省高校自然科学基金项目(16KJD420001);; 江苏省住房和城乡建设厅科技计划项目(2017ZD259)

作者(Author): 杜寒露,陶叶青,蔡安宁,周露

DOI: 10.16251/j.cnki.1009-2307.2019.08.005

参考文献(References)：

• [1]汪奇生,叶险峰.顾及系数矩阵结构性的加权总体最小二乘解算[J].测绘科学,2017,42(4):133-136.(WANG Qisheng,YE Xianfeng.A weighted total least squares algorithm considering the coefficient matrix structure[J].Science of Surveying and Mapping,2017,42(4):133-136.)
• [2]FANG Xing.Weighted total least-squares with constraints:a universal formula for geodetic symmetrical transformations[J].Journal of Geodesy,2015,89(5):459-469.
• [3]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,WULiangcai.Total least squares fitting estimation model for GPS height transformation[J].Geomatics and Information Science of Wuhan University,2016,41(9):1259-1264.)
• [4]余岸竹,姜挺,郭文月,等.总体最小二乘用于线阵卫星遥感影像光束法平差解算[J].测绘学报,2016,45(4):442-449,457.(YU Anzhu,JIANG Ting,GUO Wenyue,et al.Bundle adjustment for satellite linear array images based on total least squares[J].Acta Geodaetica et Cartographica Sinica,2016,45(4):442-449,457.)
• [5]李思达,刘志平.多维直线概括模型的总体最小二乘估计[J].测绘科学,2015,40(12):55-58.(LI Sida,LIUZhiping.Total least squares method for general model of multi-dimensional straight line fitting[J].Science of Surveying and Mapping,2015,40(12):55-58.)
• [6]AMIRI-SIMKOOEI A R.Application of least squares variance component estimation to errors-in-variables models[J].Journal of Geodesy,2013,87(10/11/12):935-944.
• [7]FANG Xing,WANG Jin,LI Bofeng,et al.On total least squares for quadratic form estimation[J].Studio Geophysics et Geodaetica,2015,59(3):366-379.
• [8]XU Peiliang.The effect of errors-in-variables on variance component estimation[J].Journal of Geodesy,2016,90(8):681-701.
• [9]陶武勇,鲁铁定,李香莲.总体最小二乘平差中粗差的可区分性[J].测绘科学,2017,42(7):46-51.(TAOWuyong,LU Tieding,LI Xianglian.The distinguishability of gross error in total least square[J].Science of Surveying and Mapping,2017,42(7):46-51.)
• [10]陶叶青,高井祥,姚一飞.基于中位数法的抗差总体最小二乘估计[J].测绘学报,2016,45(3):297-301.(TAO Yeqing,GAO Jingxiang,YAO Yifei.Solution for robust total least squares estimation based on median method[J].Acta Geodaetica et Cartographica Sinica,2016,45(3):297-301.)
• [11]SCHAFFRIN B,SNOW K.Total least-squares regularization of Tykhonov type and an ancient racetrack in corinth[J].Linear Algebra and Its Applications,2010,432(8):2061-2076.
• [12]王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报(信息科学版),2010,35(11):1347-1350.(WANG Leyang,XU Caijun,LUTieding.Ridge estimation method in ill-posed weighted total least squares adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1347-1350.)
• [13]ZHANG Songlin,TONG Xiaohua,ZHANG Kun,et al.A solution to EIV model with inequality constraints and its geodetic applications[J].Journal of Geodesy,2013,87(1):23-28.
• [14]FANG Xing.On non-combinatorial weighted total least squares with inequality constraints[J].Journal of Geodesy,2014,88(8):805-816.
• [15]TIKHONOV A N.Solution of incorrectly formulated problems and the regularization method[J].Soviet Mathematics Doklady,1963,5(4):1035-1038.
• [16]HANSEN P C.Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank[J].SIAM Journal on Scientific and Statistical Computing,1990,11(3):503-518.
• [17]HOERL A E,KENNARD R W.Ridge regression:biased estimation for nonorthogonal problems[J].Technometrics,1970,12(1):55-67.
• [18]XU Peiliang.Truncated SVD methods for discrete linear ill-posed problems[J].Geophysical Journal International,1998,135(2):505-514.
• [19]葛旭明,伍吉仓.病态总体最小二乘问题的广义正则化[J].测绘学报,2012,41(3):372-377.(GE Xuming,WU Jicang.Generalized regularization to ill-posed total least squares problem[J].Acta Geodaetica et Cartographica Sinica,2012,41(3):372-377.)
• [20]葛旭明,伍吉仓.误差限的病态总体最小二乘解算[J].测绘学报,2013,42(2):196-202.(GE Xuming,WU Jicang.A regularization method to ill-posed total least squares with error limits[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):196-202.)
• [21]BECK A,BEN-TAL A.On the solution of the Tikhonov regularization of the total least squares problem[J].SIAM Journal on Optimization,2006,17(1):98-118.
• [22]FIERRO R D,GOLUB G H,HANSEN P C,et al.Regularization by truncated total least squares[J].SIAM Journal on Scientific Computing,1997,18(4):1223-1241.
• [23]GOLUB G H,HANSEN P C,O’LEARY D P.Tikhonov regularization and total least squares[J].SIAMJournal on Matrix Analysis and Applications,1999,21(1):185-194.
• [24]CHANG Guobin.On least-squares solution to 3Dsimilarity transformation problem under Gauss-Helmert model[J].Journal of Geodesy,2015,89(6):573-576.
• [25]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.
• [26]鲁铁定.总体最小二乘平差理论及其在测绘数据处理中的应用[D].武汉:武汉大学,2010.(LU Tieding.Research on the total least squares and its application in surveying data processing[D].Wuhan:Wuhan University,2010.)
• [27]林东方,朱建军,宋迎春,等.正则化的奇异值分解参数构造法[J].测绘学报,2016,45(8):883-889.(LINDongfang,ZHU Jianjun,SONG Yingchun,et al.Construction method of regularization by singular value decomposition of design marix[J].Acta Geodaetica et Cartographica Sinica,2016,45(8):883-889.)
• [28]顾勇为,归庆明,张璇,等.大地测量与地球物理中病态性问题的正则化迭代解法[J].测绘学报,2014,43(4):331-336.(GU Yongwei,GUI Qingming,ZHANGXuan,et al.Iterative solution of regularization to illconditioned problems in geodesy and geophysics[J].Acta Geodaetica et Cartographica Sinica,2014,43(4):331-336.)