改进的约束总体最小二乘法
|
摘要
针对传统的约束最小二乘模型和总体最小二乘模型的局限性,该文提出了一种改进的约束总体最小二乘法。假设约束总体最小二乘问题中约束方程系数矩阵也存在误差,然后构造函数模型的广义拉格朗日函数,采用最小二乘法迭代求解非线性的法方程,最终获得了改进的约束总体最小二乘法的牛顿-高斯迭代公式和平差模型精度的无偏估计。该算法采用了更接近实际的平差模型,能够获得更加接近真值的估计参数,同时平差模型的精度更加接近模拟数据加入的噪声水平。实验结果表明,本文算法可有效解决对参数进行约束时的数据处理问题。
In view of the limitations of the traditional constrained least-squares(CLS)model and the total least-squares(TLS)model,this paper presented an improved constrained total least-squares(ICTLS)method.we assumed that the coefficient matrix of constraint equations in the constrained total least-squares problem was also contaminated by error.So we seted up generalized Lagrange function of above adjustment model and then used iterative least-squares method for solving nonlinear normal equations to get GaussNewton iterative formula.The algorithm used a closer adjustment model to the reality,which could get the parameters evaluation closer to the true value,and the accuracy of the adjustment model was closer to the noise level of the simulated data.The experimental results showed that the ICTLS method was feasible for data processing which required parameter constraints.
In view of the limitations of the traditional constrained least-squares(CLS)model and the total least-squares(TLS)model,this paper presented an improved constrained total least-squares(ICTLS)method.we assumed that the coefficient matrix of constraint equations in the constrained total least-squares problem was also contaminated by error.So we seted up generalized Lagrange function of above adjustment model and then used iterative least-squares method for solving nonlinear normal equations to get GaussNewton iterative formula.The algorithm used a closer adjustment model to the reality,which could get the parameters evaluation closer to the true value,and the accuracy of the adjustment model was closer to the noise level of the simulated data.The experimental results showed that the ICTLS method was feasible for data processing which required parameter constraints.
引文
[1]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.
[2]CHANG G.On least-squares solution to 3Dsimilarity transformation problem under Gauss-Helmert model[J].Journal of Geodesy,2015,89(6):573-576.
[3]SCHAFFRIN B,WIESER A.On weighted total leastsquares adjustment for linear regression[J].Journal of Geodesy,2008,82(7):415-421.
[4]HUFFEL S V,VANDEWALLE J.The total least squares problem:computational aspects and analysis[M].Philadelphia:Society for Industrial and Applied Mathematics,1991.
[5]鲁铁定,周世健.总体最小二乘的迭代解法[J].武汉大学学报(信息科学版),2010,35(11):1351-1354.(LUTieding,ZHOU Shijian.An iterative algorithm for total least squares estimation[J].Geomatics and Information Science of Wuhan Univercity,2010,35(11):1351-1354.)
[6]SCHAFFRIN B.A note on constrained total leastsquares estimation[J].Linear Algebra&Its Applications,2006,417(1):245-258.
[7]GOLUB G H,VANLOAN C F.An analysis of the total least squares problem[J].Siam Journal on Numerical Analysis,1980,17(6):883-893.
[8]孔祥玉,魏瑞轩,韩崇昭,等.一种稳定的总体最小二乘自适应滤波算法[J].西安交通大学学报,2004,38(8):831-834.(KONG Xiangyu,WEI Ruixuan,HANChongzhao,et al.Stable total least mean square adaptive filter algorithm[J].Journal of Xi’an Jiaotong University,2004,38(8):831-834.)
[9]陆珏,陈义,郑波.总体最小二乘方法在三维坐标转换中的应用[J].大地测量与地球动力学.2008,28(5):77-81.(LU Jue,CHEN Yi,ZHENG Bo.Applying total least squares to three-dimensional datum transformation[J].Journal of Geodesy and Geodynamics,2008,28(5):77-81.)
[10]陈义,陆珏,郑波.总体最小二乘方法在空间后方交会中的应用[J].武汉大学学报(信息科学版).2008,33(12):1271-1274.(CHEN Yi,LU Jue,ZHENG Bo.Application of total least squares to space resection[J].Geomatics and Information Science of Wuhan University,2008,33(12):1271-1274.)
[11]鲁铁定,陶本藻,周世健.基于整体最小二乘法的线性回归建模和解法[J].武汉大学学报(信息科学版),2008,33(5):504-507.(LU Tieding,TAO Benzao,ZHOU Shijian.Modeling and algorithm of linear regression based on total least squares[J].Geomatics and Information Science of Wuhan University,2008,33(5):504-507.)
[12]钱传俊,舒国栋,江宝锋.改进总体最小二乘迭代解法在病态平差模型中的应用[J].现代测绘,2016,39(4):1-2.(QIAN Chuanjun,SHU Guodong,JIANGBaofeng.The application of total least squares iterative method in Ill posed adjustment model[J].Modern Surveying and Mapping,2016,39(4):1-2.)
[13]SILLARD P,BOUCHER C.A review of algebraic constraints in terrestrial reference frame datum definition[J].Journal of Geodesy,2001,75(2/3):63-73.
[14]GRAFAREND E,SCHAFFRIN B.Unbiased free net adjustment[J].Survey Review,2013,22(171):200-218.
[15]HWANG C,WANG C G,LEE L H.Adjustment of relative gravity measurements using weighted and datum-free constraints[J].Computers&Geosciences,2002,28(9):1005-1015.
[16]MIKHAIL E M.Observations and least squares[M].New York:IEP,1976.
[17]陈义,沈云中,刘大杰.适用于大旋转角的三维基准转换的一种简便模型[J].武汉大学学报(信息科学版),2004,29(12):1101-1105.(CHEN Yi,SHENYunzhong,LIU Dajie.A simplified model of three dimensional-datum transformation adapted to big rotation angle[J].Geomatics and Information Science of Wuhan Univercity,2004,29(12):1101-1105.)
[18]SIMA D M,HUFFEL S V,GOLUB G H.Regularized total least squares based on quadratic eigenvalue problem solvers[J].BIT Numerical Mathematics,2004,44(4):793-812.
[19]DOWLING E M,DEGROAT R D,LINEBARGER D A.Total least squares with linear constraints[C]//IEEEInternational Conference on Acoustics.San Francisco:IEEE,1992:341-344.
[20]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.
[21]SCHAFFRIN B,FELUS Y A.On total least-squares adjustment with constraints[M]//International Association of Geodesy Symposia.Berlin:Springer Berlin Heidelberg,2005:417-421.
[22]FANG X.A structured and constrained total leastsquares solution with cross-covariances[J].Studia Geophysica et Geodaetica,2014,58(1):1-16.
[23]MAHBOUB V,SHARIFI M A.On weighted total least-squares with linear and quadratic constraints[J].Journal of Geodesy,2012,87(3):279-286.
[24]MAHBOUB V,SHARIFI M A.Erratum to:on weighted total least-squares with linear and quadratic constraints[J].Journal of Geodesy,2013,87(6):607-608.
[25]FANG X.Weighted total least squares:necessary and sufficient conditions,fixed and random parameters[J].Journal of Geodesy,2013,87(8):733-749.
[2]CHANG G.On least-squares solution to 3Dsimilarity transformation problem under Gauss-Helmert model[J].Journal of Geodesy,2015,89(6):573-576.
[3]SCHAFFRIN B,WIESER A.On weighted total leastsquares adjustment for linear regression[J].Journal of Geodesy,2008,82(7):415-421.
[4]HUFFEL S V,VANDEWALLE J.The total least squares problem:computational aspects and analysis[M].Philadelphia:Society for Industrial and Applied Mathematics,1991.
[5]鲁铁定,周世健.总体最小二乘的迭代解法[J].武汉大学学报(信息科学版),2010,35(11):1351-1354.(LUTieding,ZHOU Shijian.An iterative algorithm for total least squares estimation[J].Geomatics and Information Science of Wuhan Univercity,2010,35(11):1351-1354.)
[6]SCHAFFRIN B.A note on constrained total leastsquares estimation[J].Linear Algebra&Its Applications,2006,417(1):245-258.
[7]GOLUB G H,VANLOAN C F.An analysis of the total least squares problem[J].Siam Journal on Numerical Analysis,1980,17(6):883-893.
[8]孔祥玉,魏瑞轩,韩崇昭,等.一种稳定的总体最小二乘自适应滤波算法[J].西安交通大学学报,2004,38(8):831-834.(KONG Xiangyu,WEI Ruixuan,HANChongzhao,et al.Stable total least mean square adaptive filter algorithm[J].Journal of Xi’an Jiaotong University,2004,38(8):831-834.)
[9]陆珏,陈义,郑波.总体最小二乘方法在三维坐标转换中的应用[J].大地测量与地球动力学.2008,28(5):77-81.(LU Jue,CHEN Yi,ZHENG Bo.Applying total least squares to three-dimensional datum transformation[J].Journal of Geodesy and Geodynamics,2008,28(5):77-81.)
[10]陈义,陆珏,郑波.总体最小二乘方法在空间后方交会中的应用[J].武汉大学学报(信息科学版).2008,33(12):1271-1274.(CHEN Yi,LU Jue,ZHENG Bo.Application of total least squares to space resection[J].Geomatics and Information Science of Wuhan University,2008,33(12):1271-1274.)
[11]鲁铁定,陶本藻,周世健.基于整体最小二乘法的线性回归建模和解法[J].武汉大学学报(信息科学版),2008,33(5):504-507.(LU Tieding,TAO Benzao,ZHOU Shijian.Modeling and algorithm of linear regression based on total least squares[J].Geomatics and Information Science of Wuhan University,2008,33(5):504-507.)
[12]钱传俊,舒国栋,江宝锋.改进总体最小二乘迭代解法在病态平差模型中的应用[J].现代测绘,2016,39(4):1-2.(QIAN Chuanjun,SHU Guodong,JIANGBaofeng.The application of total least squares iterative method in Ill posed adjustment model[J].Modern Surveying and Mapping,2016,39(4):1-2.)
[13]SILLARD P,BOUCHER C.A review of algebraic constraints in terrestrial reference frame datum definition[J].Journal of Geodesy,2001,75(2/3):63-73.
[14]GRAFAREND E,SCHAFFRIN B.Unbiased free net adjustment[J].Survey Review,2013,22(171):200-218.
[15]HWANG C,WANG C G,LEE L H.Adjustment of relative gravity measurements using weighted and datum-free constraints[J].Computers&Geosciences,2002,28(9):1005-1015.
[16]MIKHAIL E M.Observations and least squares[M].New York:IEP,1976.
[17]陈义,沈云中,刘大杰.适用于大旋转角的三维基准转换的一种简便模型[J].武汉大学学报(信息科学版),2004,29(12):1101-1105.(CHEN Yi,SHENYunzhong,LIU Dajie.A simplified model of three dimensional-datum transformation adapted to big rotation angle[J].Geomatics and Information Science of Wuhan Univercity,2004,29(12):1101-1105.)
[18]SIMA D M,HUFFEL S V,GOLUB G H.Regularized total least squares based on quadratic eigenvalue problem solvers[J].BIT Numerical Mathematics,2004,44(4):793-812.
[19]DOWLING E M,DEGROAT R D,LINEBARGER D A.Total least squares with linear constraints[C]//IEEEInternational Conference on Acoustics.San Francisco:IEEE,1992:341-344.
[20]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.
[21]SCHAFFRIN B,FELUS Y A.On total least-squares adjustment with constraints[M]//International Association of Geodesy Symposia.Berlin:Springer Berlin Heidelberg,2005:417-421.
[22]FANG X.A structured and constrained total leastsquares solution with cross-covariances[J].Studia Geophysica et Geodaetica,2014,58(1):1-16.
[23]MAHBOUB V,SHARIFI M A.On weighted total least-squares with linear and quadratic constraints[J].Journal of Geodesy,2012,87(3):279-286.
[24]MAHBOUB V,SHARIFI M A.Erratum to:on weighted total least-squares with linear and quadratic constraints[J].Journal of Geodesy,2013,87(6):607-608.
[25]FANG X.Weighted total least squares:necessary and sufficient conditions,fixed and random parameters[J].Journal of Geodesy,2013,87(8):733-749.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |