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
A functional model named EIO(Errors-In-Observations) is proposed for general TLS(total least-squares)adjustment. The EIO model only considers the correction of the observation vector, but doesn't consider to corre...A functional model named EIO(Errors-In-Observations) is proposed for general TLS(total least-squares)adjustment. The EIO model only considers the correction of the observation vector, but doesn't consider to correct all elements in the design matrix as the EIV(Errors-In-Variables) model does, furthermore, the dimension of cofactor matrix is much smaller. Iterative algorithms for the parameter estimation and their precise covariance matrix are derived rigorously, and the computation steps are also presented. The proposed approach considers the correction of the observations in the coefficient matrix, and ensures their agreements in every matrix elements. Parameters and corrections can be solved at the same time.An approximate solution and a precise solution of the covariance matrix can be achieved by corresponding algorithms. Applications of EIO model and the proposed algorithms are demonstrated with several examples. The results and comparative studies show that the proposed EIO model and algorithms are feasible and reliable for general adjustment problems.展开更多
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.
基金supported by the Open Fund of Engineering laboratory of Spatial Information Technology of Highway Geological Disaster Early Warning in Hunan Province(Changsha University of Science & Technology, Grant No:KFJ150602)Hunan Province Science and Technology Program Funded Projects, China (Grant No:2015NK3035)
文摘A functional model named EIO(Errors-In-Observations) is proposed for general TLS(total least-squares)adjustment. The EIO model only considers the correction of the observation vector, but doesn't consider to correct all elements in the design matrix as the EIV(Errors-In-Variables) model does, furthermore, the dimension of cofactor matrix is much smaller. Iterative algorithms for the parameter estimation and their precise covariance matrix are derived rigorously, and the computation steps are also presented. The proposed approach considers the correction of the observations in the coefficient matrix, and ensures their agreements in every matrix elements. Parameters and corrections can be solved at the same time.An approximate solution and a precise solution of the covariance matrix can be achieved by corresponding algorithms. Applications of EIO model and the proposed algorithms are demonstrated with several examples. The results and comparative studies show that the proposed EIO model and algorithms are feasible and reliable for general adjustment problems.
