加权EIV模型的经典最小二乘算法
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摘要
采用GHM方法将EIV模型在最优解处线性化,得到解的近似方差。然后,将EIV模型表达成与Gauss-Markov模型相似的形式,利用标准最小二乘理论推导EIV模型的解及近似方差矩阵,得到与已有算法等价的结论。最后,推导观测值估值和残差的统计性质,建立起一整套EIV模型参数估计和精度评定的体系。
First, the EIV model is linearized at the optimal solution through the GHM method and the approximate variance matrix is derived. Then, the EIV model is reformulated in the form of the Gauss-Markov model. The solution to EIV model and its approximate dispersion matrix are derived using the standard least squares theory, which is equivalent to the existing results. Finally, the statistical properties of the estimation of observations and residuals are derived and the system of parameter estimation and accuracy assessment of EIV model are established.
First, the EIV model is linearized at the optimal solution through the GHM method and the approximate variance matrix is derived. Then, the EIV model is reformulated in the form of the Gauss-Markov model. The solution to EIV model and its approximate dispersion matrix are derived using the standard least squares theory, which is equivalent to the existing results. Finally, the statistical properties of the estimation of observations and residuals are derived and the system of parameter estimation and accuracy assessment of EIV model are established.
引文
[1] 武汉大学测绘学院测量平差学科组.误差理论与测量平差基础[M].武汉:武汉大学出版社,2014(Surveying Adjustment Group in School of Geodesy and Geomatics of Wuhan University.Error Theory and Foundation of Surveying Adjustment[M].Wuhan:Wuhan University Press,2014)
[2] 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
[3] Shen Y Z,Li B F,Chen Y.An Iterative Solution of Weighted Total-Least Squares Adjustment[J].Journal of Geodesy,2011,85(4):229-238
[4] Xu C J,Wang L Y,Wen Y M,et al.Strain Rates in the Sichuan-Yunnan Region Based upon the Total Least Squares Heterogeneous Strain Model from GPS Data[J].Terrestrial Atmospheric and Oceanic Sciences,2011,22(2):133-147
[5] Huffel S V,Lemmerling P.Total Least Squares and Errors-in-Variables Modeling,Analysis,Algorithms and Applications[M].Alphen:Kluwer Academic Publishers,2001
[6] Golub G H,Loan C F V.An Analysis of the Total Least Squares Problem[J].SIAM,1980,17(6):883-893
[7] Mahboub V.On Weighted Total Least Squares for Geodetic Transformations[J].Journal of Geodesy,2012,86(5):359-367
[8] Amiri-Simkooei A,Jazaeri S.Weighted Total Least Squares Formulated by Standard Least Squares[J].Journal of Geodetic Sciences,2012,2(2):113-124
[9] Fang X.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[2] 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
[3] Shen Y Z,Li B F,Chen Y.An Iterative Solution of Weighted Total-Least Squares Adjustment[J].Journal of Geodesy,2011,85(4):229-238
[4] Xu C J,Wang L Y,Wen Y M,et al.Strain Rates in the Sichuan-Yunnan Region Based upon the Total Least Squares Heterogeneous Strain Model from GPS Data[J].Terrestrial Atmospheric and Oceanic Sciences,2011,22(2):133-147
[5] Huffel S V,Lemmerling P.Total Least Squares and Errors-in-Variables Modeling,Analysis,Algorithms and Applications[M].Alphen:Kluwer Academic Publishers,2001
[6] Golub G H,Loan C F V.An Analysis of the Total Least Squares Problem[J].SIAM,1980,17(6):883-893
[7] Mahboub V.On Weighted Total Least Squares for Geodetic Transformations[J].Journal of Geodesy,2012,86(5):359-367
[8] Amiri-Simkooei A,Jazaeri S.Weighted Total Least Squares Formulated by Standard Least Squares[J].Journal of Geodetic Sciences,2012,2(2):113-124
[9] Fang X.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749