病态问题导致参数估值方差较大,在参数真值未知的情况下,难以对正则化与TSVD(truncated singular value decomposition)方法进行精度比较。针对此问题,提出一种均方误差相对比较分析方法。首先,基于正则化估值相对变化量与TSVD估值相对...病态问题导致参数估值方差较大,在参数真值未知的情况下,难以对正则化与TSVD(truncated singular value decomposition)方法进行精度比较。针对此问题,提出一种均方误差相对比较分析方法。首先,基于正则化估值相对变化量与TSVD估值相对变化量确定二者相对于最小二乘估值的相对偏差,避免偏差计算对真值的依赖;然后,利用相对偏差以及相对标准差确定均方根误差相对下降量,通过比较相对下降量大小确定最优的解算方法;最后,通过两组实验验证均方误差相对比较分析方法的可行性与有效性。展开更多
Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with rand...Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.展开更多
三维基准转换广泛应用于大地测量、摄影测量、点云配准等领域,求解大角度、任意比例尺的三维基准转换参数的研究有很多。然而,当观测值中含有粗差时,得到的转换参数估值会受到不利影响甚至被严重扭曲。为处理含有粗差的大角度三维基准...三维基准转换广泛应用于大地测量、摄影测量、点云配准等领域,求解大角度、任意比例尺的三维基准转换参数的研究有很多。然而,当观测值中含有粗差时,得到的转换参数估值会受到不利影响甚至被严重扭曲。为处理含有粗差的大角度三维基准转换问题,本文首先将大角度三维基准转换问题抽象为具有等式约束的最小二乘问题(Constrained least squares, CLS),推导参数在正交约束条件下的最小二乘解。然后,将灵敏度分析方法应用到CLS问题中,研究残差加权平方和对观测值扰动的局部敏感性,并基于这些敏感度指标构造局部检验统计量,进而推导出一个适用于CLS问题的粗差探测算法。最后,为核实该算法的有效性进行了仿真与实测数据实验。实验结果表明:本文提出的基于灵敏度检验统计量的数据探测算法可以降低粗差的负面影响,得到可靠的参数估值,从而有效解决大角度三维基准转换中的粗差处理问题。展开更多
文摘病态问题导致参数估值方差较大,在参数真值未知的情况下,难以对正则化与TSVD(truncated singular value decomposition)方法进行精度比较。针对此问题,提出一种均方误差相对比较分析方法。首先,基于正则化估值相对变化量与TSVD估值相对变化量确定二者相对于最小二乘估值的相对偏差,避免偏差计算对真值的依赖;然后,利用相对偏差以及相对标准差确定均方根误差相对下降量,通过比较相对下降量大小确定最优的解算方法;最后,通过两组实验验证均方误差相对比较分析方法的可行性与有效性。
基金the financial support of the National Natural Science Foundation of China(Grant No.42074016,42104025,42274057and 41704007)Hunan Provincial Natural Science Foundation of China(Grant No.2021JJ30244)Scientific Research Fund of Hunan Provincial Education Department(Grant No.22B0496)。
文摘Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.
文摘三维基准转换广泛应用于大地测量、摄影测量、点云配准等领域,求解大角度、任意比例尺的三维基准转换参数的研究有很多。然而,当观测值中含有粗差时,得到的转换参数估值会受到不利影响甚至被严重扭曲。为处理含有粗差的大角度三维基准转换问题,本文首先将大角度三维基准转换问题抽象为具有等式约束的最小二乘问题(Constrained least squares, CLS),推导参数在正交约束条件下的最小二乘解。然后,将灵敏度分析方法应用到CLS问题中,研究残差加权平方和对观测值扰动的局部敏感性,并基于这些敏感度指标构造局部检验统计量,进而推导出一个适用于CLS问题的粗差探测算法。最后,为核实该算法的有效性进行了仿真与实测数据实验。实验结果表明:本文提出的基于灵敏度检验统计量的数据探测算法可以降低粗差的负面影响,得到可靠的参数估值,从而有效解决大角度三维基准转换中的粗差处理问题。
