病态变量含误差模型的分步正则化算法
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摘要
针对数值逼近理论的病态变量含误差模型正则化算法无法顾及模型的随机性质,以及获得的参数估值不具有统计意义的问题。该文在对现有算法进行拓展的基础上,提出了分步的正则化算法:首先通过构造约束矩阵改善模型的病态性,获得稳定的参数初值;然后应用参数的最小二乘正则化解作为初值,建立附有不等式约束的总体最小二乘参数估计模型;最后,通过实例对已有算法与本文所建立的算法进行比较。结果表明,该算法弥补了现有的算法单一通过正则化参数实现模型正则化存在的不足,避免了总体最小二乘算法具有的降正则化性质导致的参数估计发散,具有稳定的收敛性质。
Current solutions for ill-posed errors-in-variables model regularization are based on theory of numerical approximation,however,these kinds of solutions can not take into account stochastic properties of model,and there is no statistic meaning of parameters estimation.To overcome the shortages of current solutions,a new solution which has two steps for regularization was established by modifying current algorithms:firstly,regularization of ill-posed model is based on constraint matrix to obtain stable value of parameters;secondly,parameters computed by least squares estimation were taken as initial value,and inequality constraints was established under total least squares estimation;at last,an instance was applied to compare current solution with the proposed solution.The numerical results of the instance demonstrated that the proposed algorithm could overcome the shortages of solutions which utilized unique parameter for regularization,besides,it could guarantee that parameters estimation convergences on the process of computation while the algorithm had ill-posed property,and it had stable convergence character.
Current solutions for ill-posed errors-in-variables model regularization are based on theory of numerical approximation,however,these kinds of solutions can not take into account stochastic properties of model,and there is no statistic meaning of parameters estimation.To overcome the shortages of current solutions,a new solution which has two steps for regularization was established by modifying current algorithms:firstly,regularization of ill-posed model is based on constraint matrix to obtain stable value of parameters;secondly,parameters computed by least squares estimation were taken as initial value,and inequality constraints was established under total least squares estimation;at last,an instance was applied to compare current solution with the proposed solution.The numerical results of the instance demonstrated that the proposed algorithm could overcome the shortages of solutions which utilized unique parameter for regularization,besides,it could guarantee that parameters estimation convergences on the process of computation while the algorithm had ill-posed property,and it had stable convergence character.
引文
[1]GOLUB G H,VAN LOAN C F.An analysis of thetotal least squares problem[J].SIAM J Number,Anal,1980,17(5):883-893.
[2]HUFFEL S V,VANDEWALLE J.The total least-squares problem:computational aspects and analysis[M].Philadelphia:Society for Industrial and AppliedMathematics,1991.
[3]SCHAFFRIN B,WIESER A.On weighted total least-squares adjustment for linear regression[J].Journal ofGeodesy,2008,82(7):415-421.
[4]SHEN Y Z,LI B F,CHEN Y.An iterative solution ofweighted total least-squares adjustment[J].Journal ofGeodesy,2011,85(4):229-238.
[5]XING F.Weighted total least squares:necessary andsufficient conditions,fixed and random parameters[J].Journal of Geodesy,2013,87(8):733-749.
[6]AMIRI-SIMKOOEI A R.Application of least squaresvariance component estimation to errors-in-variablesmodels[J].Journal of Geodesy,2013,87(10/12):935-944.
[7]XING F.Weighted total least-squares with constraints:auniversal formula for geodetic symmetrical transfor-mations[J].Journal of Geodesy,2015,89(5):459-469.
[8]NEITZEL F.Generalization of total least-squares onexample of unweighted and weighted 2D similaritytransformation[J].Journal of Geodesy,2010,84(12):751-762.
[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 distinguishabilityof gross error in total least squares[J].Science ofSurveying and Mapping,2017,42(7):46-51.)
[10]陈义,陆珏.以三维坐标转换为例解算稳健总体最小二乘方法[J].测绘学报,2012,41(5):715-722.(CHENYi,LU Jue.Performing 3Dsimilarity transformationby robust total least squares[J].Acta Geodaetica etCartographica Sinica,2012,41(5):715-722)
[11]MAHBOUB V.On weighted total least-squares forgeodetic transformations[J].Journal of Geodesy,2012,86(5):359-367.
[12]方兴,曾文宪,刘经南,等.三维坐标转换的通用整体最小二乘算法[J].测绘学报,2014,43(11):1139-1143.(FANG Xing,ZENG Wenxian,LIU Jiangnan,et al.Ageneral total least squares algorithm for three-dimen-sional coordinate transformations[J].Acta Geodaeticaet Cartographica Sinica,2014,43(11):1139-1143.)
[13]TAO Y Q,GAO J X,YAO Y F.TLS algorithm forGPS height fitting based on robust estimation[J].Survey Review,2014,46(336):184-188.
[14]FANG X,WANG J,Li B F,et al.On total least squaresfor quadratic form estimation[J].Studio Geophysics etGeodaetica,2015,59(3):366-379.
[15]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.
[16]GOLUB G H,HANSEN P C,O′LEARY D P.Tikhonov regularization and total least squares[J].SIAM Journal on Matrix Analysis and Applications,1999,21(1):185-194.
[17]BECK A,BENT A.On the solution of the Tikhonov regu-larization of the total least squares problem[J].SIAMJournal on Optimization,2006,17(1):98-118.
[18]SCHAFFRIN B,SNOW K.Total least-squares regu-larization of Tykhonov type and an ancient racetrackin Corinth[J].Linear Algebra and its Applications,2010,432(8):2061-2076.
[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-posedtotal least squares problem[J].Acta Geodaetica etCartographica 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,WUJicang.A regularization method to ill-posed total leastsquares with error limits[J].Acta Geodaetica etCartographica Sinica,2013,42(2):196-202.)
[21]袁振超,沈云中,周泽波.病态总体最小二乘模型的正则化算法[J].大地测量与地球动力学,2009,29(2):131-134.(YUAN Zhenchao,SHENG Yunzhong,ZHOU Zebo.Regularized total least-squares solutionto ill-posed error-in-variable MODEL[J].Journal ofGeodesy and Geodynamics,2009,29(2):131-134)
[22]于冬冬,王乐洋.病态总体最小二乘问题的共轭梯度解法[J].测绘科学,2018,43(2):95-100.(YU Dong-dong,WANG Leyang.Conjunction gradient method toill-posed total least squares problem[J].Science ofSurveying and Mapping,2018,43(2)95-100.)
[23]顾勇为,归庆明,赵俊.病态加权总体最小二乘靶向病灶的正则化方法[J].大地测量与地球动力学,2016,36(3):253-256.(GU Yongwei,GUI Qingming,ZHAOJun.Target focus regularization as compared withill-posed weighted total least square[J].Journal ofGeodesy and Geodynamics,2016,36(3):253-256.)
[24]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 applicationin surveying data processing[D].Wuhan:WuhanUniversity,2010.)
[25]欧吉坤.测量平差中不适定问题解的统一表达与选权拟合法[J].测绘学报,2004,33(4):283-288.(OUJikun.Uniform expression of solutions of ill-posedproblems in surveying adjustment and the fittingmethod by selection of the parameter weights[J].ActaGeodaetica et Cartographica Sinica,2004,33(4):283-288.)
[26]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):883-886.(LINDongfang,ZHU Jianjun,SONG Yingchun,et al.Construction method of regularization by singular valuedecomposition of design marix[J].Acta Geodaetica etCartographica Sinica,2016,45(8):883-889.)
[27]ZHANG S L,TONG X H,ZHANG K L,et al.Asolution to EIV model with inequality constraints andits geodetic applications[J].Journal of Geodesy,2013,87(1):23-28.
[28]MAHBOUB V,SHARIFI M A.On weighted totalleast-squares with linear and quadratic constraints[J].Journal of Geodesy,2013,87(3):279-286.
[29]宋迎春,左廷英,朱建军.带有线性不等式约束平差模型的算法研究[J].测绘学报,2008,37(4):433-437.(SONG Yingchun,ZUO Tingying,ZHU Jiangjun.Research on algorithm of adjustment model with linearinequality constrained parameters[J].Acta Geodaeticaet Cartographica Sinica,2008,37(4):433-437.)
[30]FANG X.On non-combinatorial weighted total leastsquares with inequality constraints[J].Journal ofGeodesy,2014,88(8):805-816.
[2]HUFFEL S V,VANDEWALLE J.The total least-squares problem:computational aspects and analysis[M].Philadelphia:Society for Industrial and AppliedMathematics,1991.
[3]SCHAFFRIN B,WIESER A.On weighted total least-squares adjustment for linear regression[J].Journal ofGeodesy,2008,82(7):415-421.
[4]SHEN Y Z,LI B F,CHEN Y.An iterative solution ofweighted total least-squares adjustment[J].Journal ofGeodesy,2011,85(4):229-238.
[5]XING F.Weighted total least squares:necessary andsufficient conditions,fixed and random parameters[J].Journal of Geodesy,2013,87(8):733-749.
[6]AMIRI-SIMKOOEI A R.Application of least squaresvariance component estimation to errors-in-variablesmodels[J].Journal of Geodesy,2013,87(10/12):935-944.
[7]XING F.Weighted total least-squares with constraints:auniversal formula for geodetic symmetrical transfor-mations[J].Journal of Geodesy,2015,89(5):459-469.
[8]NEITZEL F.Generalization of total least-squares onexample of unweighted and weighted 2D similaritytransformation[J].Journal of Geodesy,2010,84(12):751-762.
[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 distinguishabilityof gross error in total least squares[J].Science ofSurveying and Mapping,2017,42(7):46-51.)
[10]陈义,陆珏.以三维坐标转换为例解算稳健总体最小二乘方法[J].测绘学报,2012,41(5):715-722.(CHENYi,LU Jue.Performing 3Dsimilarity transformationby robust total least squares[J].Acta Geodaetica etCartographica Sinica,2012,41(5):715-722)
[11]MAHBOUB V.On weighted total least-squares forgeodetic transformations[J].Journal of Geodesy,2012,86(5):359-367.
[12]方兴,曾文宪,刘经南,等.三维坐标转换的通用整体最小二乘算法[J].测绘学报,2014,43(11):1139-1143.(FANG Xing,ZENG Wenxian,LIU Jiangnan,et al.Ageneral total least squares algorithm for three-dimen-sional coordinate transformations[J].Acta Geodaeticaet Cartographica Sinica,2014,43(11):1139-1143.)
[13]TAO Y Q,GAO J X,YAO Y F.TLS algorithm forGPS height fitting based on robust estimation[J].Survey Review,2014,46(336):184-188.
[14]FANG X,WANG J,Li B F,et al.On total least squaresfor quadratic form estimation[J].Studio Geophysics etGeodaetica,2015,59(3):366-379.
[15]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.
[16]GOLUB G H,HANSEN P C,O′LEARY D P.Tikhonov regularization and total least squares[J].SIAM Journal on Matrix Analysis and Applications,1999,21(1):185-194.
[17]BECK A,BENT A.On the solution of the Tikhonov regu-larization of the total least squares problem[J].SIAMJournal on Optimization,2006,17(1):98-118.
[18]SCHAFFRIN B,SNOW K.Total least-squares regu-larization of Tykhonov type and an ancient racetrackin Corinth[J].Linear Algebra and its Applications,2010,432(8):2061-2076.
[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-posedtotal least squares problem[J].Acta Geodaetica etCartographica 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,WUJicang.A regularization method to ill-posed total leastsquares with error limits[J].Acta Geodaetica etCartographica Sinica,2013,42(2):196-202.)
[21]袁振超,沈云中,周泽波.病态总体最小二乘模型的正则化算法[J].大地测量与地球动力学,2009,29(2):131-134.(YUAN Zhenchao,SHENG Yunzhong,ZHOU Zebo.Regularized total least-squares solutionto ill-posed error-in-variable MODEL[J].Journal ofGeodesy and Geodynamics,2009,29(2):131-134)
[22]于冬冬,王乐洋.病态总体最小二乘问题的共轭梯度解法[J].测绘科学,2018,43(2):95-100.(YU Dong-dong,WANG Leyang.Conjunction gradient method toill-posed total least squares problem[J].Science ofSurveying and Mapping,2018,43(2)95-100.)
[23]顾勇为,归庆明,赵俊.病态加权总体最小二乘靶向病灶的正则化方法[J].大地测量与地球动力学,2016,36(3):253-256.(GU Yongwei,GUI Qingming,ZHAOJun.Target focus regularization as compared withill-posed weighted total least square[J].Journal ofGeodesy and Geodynamics,2016,36(3):253-256.)
[24]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 applicationin surveying data processing[D].Wuhan:WuhanUniversity,2010.)
[25]欧吉坤.测量平差中不适定问题解的统一表达与选权拟合法[J].测绘学报,2004,33(4):283-288.(OUJikun.Uniform expression of solutions of ill-posedproblems in surveying adjustment and the fittingmethod by selection of the parameter weights[J].ActaGeodaetica et Cartographica Sinica,2004,33(4):283-288.)
[26]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):883-886.(LINDongfang,ZHU Jianjun,SONG Yingchun,et al.Construction method of regularization by singular valuedecomposition of design marix[J].Acta Geodaetica etCartographica Sinica,2016,45(8):883-889.)
[27]ZHANG S L,TONG X H,ZHANG K L,et al.Asolution to EIV model with inequality constraints andits geodetic applications[J].Journal of Geodesy,2013,87(1):23-28.
[28]MAHBOUB V,SHARIFI M A.On weighted totalleast-squares with linear and quadratic constraints[J].Journal of Geodesy,2013,87(3):279-286.
[29]宋迎春,左廷英,朱建军.带有线性不等式约束平差模型的算法研究[J].测绘学报,2008,37(4):433-437.(SONG Yingchun,ZUO Tingying,ZHU Jiangjun.Research on algorithm of adjustment model with linearinequality constrained parameters[J].Acta Geodaeticaet Cartographica Sinica,2008,37(4):433-437.)
[30]FANG X.On non-combinatorial weighted total leastsquares with inequality constraints[J].Journal ofGeodesy,2014,88(8):805-816.
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