相关观测PEIV模型的最小二乘方差分量估计
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摘要
针对相关观测的部分变量误差(partial errors-invariables,PEIV)模型并考虑平差时随机模型的不准确,将函数模型作为非线性最小二乘并进行泰勒展开迭代求解,结合最小二乘方差分量估计方法,推导了相关观测PEIV模型的最小二乘方差分量估计公式,并给出了相应的迭代算法,通过公式推导得到相关观测PEIV模型的最小二乘方差分量估计与已有方差分量估计方法等价。实验结果表明,相关观测下对随机模型进行修正的方差分量估计方法可以得到更加合理的参数估值,该方法更具一般性。
Considering the partial errors-in-variables(PEIV)model with correlated observations and the inaccuracy of the stochastic model,the method of Taylor expansion is used when the PEIV model was regarded as a non-linear least squares.The formulas of least square variance components estimation of PEIV model with correlated observations are derived and the iterative algorithm is presented in this paper.The equivalence between the least square variance components estimation of PEIV model with correlated observations and the existed variance components estimation method has been proved.Experiments show that the variance components estimation method can obtain more reasonable parameter estimation through correcting the stochastic model under the correlated observations,the method of this paper is more general than the exist methods.
Considering the partial errors-in-variables(PEIV)model with correlated observations and the inaccuracy of the stochastic model,the method of Taylor expansion is used when the PEIV model was regarded as a non-linear least squares.The formulas of least square variance components estimation of PEIV model with correlated observations are derived and the iterative algorithm is presented in this paper.The equivalence between the least square variance components estimation of PEIV model with correlated observations and the existed variance components estimation method has been proved.Experiments show that the variance components estimation method can obtain more reasonable parameter estimation through correcting the stochastic model under the correlated observations,the method of this paper is more general than the exist methods.
引文
[1]Xu Peiliang,Liu Jingnan,Shi Chuang.Total Least Squares Adjustment in Partial Errors-in-Variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675
[2]Golub G H,van Loan C F.An Analysis of the Total Least Squares Problem[J].SIAM Journal on Numerical Analysis,1980,17(6):883-893
[3]Schaffrin B,Wieser A.On Weighted Total Least Squares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421
[4]王乐洋.基于总体最小二乘的大地测量反演理论及应用研究[J].测绘学报,2012,41(4):629
[5]王乐洋,许才军.总体最小二乘研究进展[J].武汉大学学报·信息科学版,2013,38(7):850-856
[6]Fang Xing.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[7]Fang Xing.Weighted Total Least Squares Solutions for Applications in Geodesy[D].Germany:Leibniz Universitt Hannover,2011
[8]Wang Leyang,Zhao Yingwen.Unscented Transformation with Scaled Symmetric Sampling Strategy for Precision Estimation of Total Least Squares[J].Studia Geophysica et Geodaetica,2017,61(3):385-411
[9]Shi Yun,Xu Peiliang,Liu Jingnan,et al.Alternative Formulae for Parameter Estimation in Partial Errorsin-Variables Models[J].Journal of Geodesy,2015,89(1):13-16
[10]Zeng Wenxian,Liu Jingnan,Yao Yibin.On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J].Journal of Geodesy,2014,89(2):111-119
[11]王乐洋,余航,陈晓勇.PEIV模型的解法[J].测绘学报,2016,45(1):22-29
[12]赵俊,归庆明,郭飞宵.基于改进目标函数的PEIV模型WTLS估计的新算法[J].武汉大学学报·信息科学版,2017,42(8):1 179-1 184
[13]许光煜.PEIV模型的总体最小二乘方法及应用研究[D].南昌:东华理工大学,2016
[14]Wang Leyang,Xu Guangyu.Variance Component Estimation for Partial Errors-in-Variables Models[J].Studia Geophysica et Geodaetica,2016,60(1):35-55
[15]王乐洋,温贵森.PEIV模型的非负最小二乘方差分量估计[J].测绘学报,2017,46(7):857-865
[16]王乐洋,温贵森.一种基于PEIV模型的多项式拟合解法[J].大地测量与地球动力学,2017,37(7):737-742
[17]王乐洋,熊露云.水准测量中尺度比参数的附加系统参数的PEIV模型解法[J].大地测量与地球动力学,2017,37(8):856-859
[18]王乐洋,许光煜,陈晓勇.附有相对权比的PEIV模型总体最小二乘平差[J].武汉大学学报·信息科学版,2017,42(6):857-863
[19]王乐洋,许光煜,温贵森.一种相关观测的PEIV模型求解方法[J].测绘学报,2017,46(8):978-987
[20]姚宜斌,黄书华,陈家君.求解自回归模型参数的整体最小二乘新方法[J].测绘学报,2014,39(12):1 463-1 466
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[22]赵辉,张书毕,张秋昭.基于加权总体最小二乘法的GPS高程拟合[J].大地测量与地球动力学,2011,31(5):88-96
[23]朱建军.方差分量的Bayes估计[J].测绘学报,1991,21(1):1-6
[24]杨元喜,张晓东.基于严密Helmert方差分量估计的动态Kalman滤波[J].同济大学学报(自然科学版),2009,37(9):1 241-1 245
[25]李博峰,沈云中.基于等效残差积探测粗差的方差-协方差分量估计[J].测绘学报,2011,40(1):10-14
[26]李博峰.无缝仿射基准转换模型的方差分量估计[J].测绘学报,2016,45(1):30-35
[27]Xu Peiliang,Liu Jingnan.Variance Components in Errors-in-Variables Models:Estimability,Stability and Bias Analysis[J].Journal of Geodesy,2014,88(8):719-734
[28]於宗俦.方差-协方差分量估计的统一理论[J].测绘学报,1991,20(3):161-171
[29]刘玉梅,李博峰.GPS双差观测值的方差-协方差分量估计[J].工程勘察,2007(7):36-38
[30]於宗俦.Helmert型方差-协方差分量估计的通用公式[J].武汉测绘科技大学学报,1991,16(2):8-17
[31]赵俊,郭建锋.方差分量估计的通用公式[J].武汉大学学报·信息科学版,2013,38(5):580-583
[32]Amiri-Simkooei A R.Application of Least Squares Variance Component Estimation to Errors-in-Variables Models[J].Journal of Geodesy,2013,87(10):935-944
[33]Amiri-Simkooei A R.Least-Squares Variance Component Estimation:Theory and GPS Applications[D].Delft:Delft University of Technology,2007
[34]Teunissen P J G,Amiri-Simkooei A R.Least-Squares Variance Component Estimation[J].Journal of Geodesy,2008,82(2):65-82
[35]Amiri-Simkooei A R.Non-negative Least-Squares Variance Component Estimation with Application to GPS Time Series[J].Journal of Geodesy,2016,90(5):451-466
[36]胡川,陈义.非线性整体最小平差迭代算法[J].测绘学报,2014,43(7):668-674
[37]崔希璋,於宗俦,陶本藻,等.广义测量平差(新版)[M].武汉:武汉大学出版社,2005
[2]Golub G H,van Loan C F.An Analysis of the Total Least Squares Problem[J].SIAM Journal on Numerical Analysis,1980,17(6):883-893
[3]Schaffrin B,Wieser A.On Weighted Total Least Squares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421
[4]王乐洋.基于总体最小二乘的大地测量反演理论及应用研究[J].测绘学报,2012,41(4):629
[5]王乐洋,许才军.总体最小二乘研究进展[J].武汉大学学报·信息科学版,2013,38(7):850-856
[6]Fang Xing.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[7]Fang Xing.Weighted Total Least Squares Solutions for Applications in Geodesy[D].Germany:Leibniz Universitt Hannover,2011
[8]Wang Leyang,Zhao Yingwen.Unscented Transformation with Scaled Symmetric Sampling Strategy for Precision Estimation of Total Least Squares[J].Studia Geophysica et Geodaetica,2017,61(3):385-411
[9]Shi Yun,Xu Peiliang,Liu Jingnan,et al.Alternative Formulae for Parameter Estimation in Partial Errorsin-Variables Models[J].Journal of Geodesy,2015,89(1):13-16
[10]Zeng Wenxian,Liu Jingnan,Yao Yibin.On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J].Journal of Geodesy,2014,89(2):111-119
[11]王乐洋,余航,陈晓勇.PEIV模型的解法[J].测绘学报,2016,45(1):22-29
[12]赵俊,归庆明,郭飞宵.基于改进目标函数的PEIV模型WTLS估计的新算法[J].武汉大学学报·信息科学版,2017,42(8):1 179-1 184
[13]许光煜.PEIV模型的总体最小二乘方法及应用研究[D].南昌:东华理工大学,2016
[14]Wang Leyang,Xu Guangyu.Variance Component Estimation for Partial Errors-in-Variables Models[J].Studia Geophysica et Geodaetica,2016,60(1):35-55
[15]王乐洋,温贵森.PEIV模型的非负最小二乘方差分量估计[J].测绘学报,2017,46(7):857-865
[16]王乐洋,温贵森.一种基于PEIV模型的多项式拟合解法[J].大地测量与地球动力学,2017,37(7):737-742
[17]王乐洋,熊露云.水准测量中尺度比参数的附加系统参数的PEIV模型解法[J].大地测量与地球动力学,2017,37(8):856-859
[18]王乐洋,许光煜,陈晓勇.附有相对权比的PEIV模型总体最小二乘平差[J].武汉大学学报·信息科学版,2017,42(6):857-863
[19]王乐洋,许光煜,温贵森.一种相关观测的PEIV模型求解方法[J].测绘学报,2017,46(8):978-987
[20]姚宜斌,黄书华,陈家君.求解自回归模型参数的整体最小二乘新方法[J].测绘学报,2014,39(12):1 463-1 466
[21]王新洲,陶本藻,邱卫宁,等.高等测量平差[M].北京:测绘出版社,2006
[22]赵辉,张书毕,张秋昭.基于加权总体最小二乘法的GPS高程拟合[J].大地测量与地球动力学,2011,31(5):88-96
[23]朱建军.方差分量的Bayes估计[J].测绘学报,1991,21(1):1-6
[24]杨元喜,张晓东.基于严密Helmert方差分量估计的动态Kalman滤波[J].同济大学学报(自然科学版),2009,37(9):1 241-1 245
[25]李博峰,沈云中.基于等效残差积探测粗差的方差-协方差分量估计[J].测绘学报,2011,40(1):10-14
[26]李博峰.无缝仿射基准转换模型的方差分量估计[J].测绘学报,2016,45(1):30-35
[27]Xu Peiliang,Liu Jingnan.Variance Components in Errors-in-Variables Models:Estimability,Stability and Bias Analysis[J].Journal of Geodesy,2014,88(8):719-734
[28]於宗俦.方差-协方差分量估计的统一理论[J].测绘学报,1991,20(3):161-171
[29]刘玉梅,李博峰.GPS双差观测值的方差-协方差分量估计[J].工程勘察,2007(7):36-38
[30]於宗俦.Helmert型方差-协方差分量估计的通用公式[J].武汉测绘科技大学学报,1991,16(2):8-17
[31]赵俊,郭建锋.方差分量估计的通用公式[J].武汉大学学报·信息科学版,2013,38(5):580-583
[32]Amiri-Simkooei A R.Application of Least Squares Variance Component Estimation to Errors-in-Variables Models[J].Journal of Geodesy,2013,87(10):935-944
[33]Amiri-Simkooei A R.Least-Squares Variance Component Estimation:Theory and GPS Applications[D].Delft:Delft University of Technology,2007
[34]Teunissen P J G,Amiri-Simkooei A R.Least-Squares Variance Component Estimation[J].Journal of Geodesy,2008,82(2):65-82
[35]Amiri-Simkooei A R.Non-negative Least-Squares Variance Component Estimation with Application to GPS Time Series[J].Journal of Geodesy,2016,90(5):451-466
[36]胡川,陈义.非线性整体最小平差迭代算法[J].测绘学报,2014,43(7):668-674
[37]崔希璋,於宗俦,陶本藻,等.广义测量平差(新版)[M].武汉:武汉大学出版社,2005