不等式约束加权整体最小二乘的凝聚函数法
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摘要
误差向量的方差-协方差阵是一般对称正定矩阵下的附不等式约束加权整体最小二乘平差模型,研究了其参数估计和精度评定问题。首先,将残差平方和极小化函数在整体最小二乘准则下转化为只包含模型参数的目标函数,同时将所有的不等式约束表示成一个等价的凝聚约束函数,并运用乘子罚函数策略将不等式约束加权整体最小二乘平差问题转化为相应的无约束最优化问题,并用BFGS方法求解。然后,将误差方程和约束函数线性展开,推导了最优解和观测量间的近似线性函数关系,运用方差-协方差传播律得到了最优解的近似方差。最后,用数值实例验证了方法的有效性和可行性。
When the variance-covariance matrix of the error is fairly symmetric and positively definite,parameter estimation and accuracy assessment of weighted total least squares adjustment with inequality constraints(ICWTLS)are investigated in this paper.First,the problem of minimizing the residual sum of squares are converted to an optimization problem with only the model parameters under the total least squares,and all the inequality constraints are transformed into an equivalent aggregate constraint.Accordingly,the ICWTLS problem is converted to unconstrained optimization problem by a penalty function approach and is then solved by BFGS optimization method.Then,the observation equation and constraints function are expanded to the first order Taylor series and the linear approximation between the solution and observations is derived as well as the approximate dispersion matrix of the solution under the variance propagation law.Finally,a numerical simulation is given to indicate the validity and feasibility of this method.
When the variance-covariance matrix of the error is fairly symmetric and positively definite,parameter estimation and accuracy assessment of weighted total least squares adjustment with inequality constraints(ICWTLS)are investigated in this paper.First,the problem of minimizing the residual sum of squares are converted to an optimization problem with only the model parameters under the total least squares,and all the inequality constraints are transformed into an equivalent aggregate constraint.Accordingly,the ICWTLS problem is converted to unconstrained optimization problem by a penalty function approach and is then solved by BFGS optimization method.Then,the observation equation and constraints function are expanded to the first order Taylor series and the linear approximation between the solution and observations is derived as well as the approximate dispersion matrix of the solution under the variance propagation law.Finally,a numerical simulation is given to indicate the validity and feasibility of this method.
引文
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[7]Liew C K.Inequality Constrained Least-Squares Estimation[J].Journal of the American Statistical Association,1976,71(355):746-751
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[9]Xu P L,Liu J N,Shi C.Total Least Squares in Partial Errors-in-Variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675
[10]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
[11]Schaffrin B,Felus Y A.An Algorithmic Approach to the Total Least-Squares Problem with Linear and Quadratic Constraints[J].Studia Geophysica Geodaetica,2009,53(1):1-16
[12]Fang X.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[13]de Moor B.Total Linear Least Squares with Inequality Constraints[R].ESAT-SISTA Report1990-02,Department of Electrical Engineering,Katholieke Universiteit Leuven,Belgium,1990
[14]Zhang S L,Tong X H,Zhang K.A Solution to EIV Model with Inequality Constraints and Its Geodetic Applications[J].Journal of Geodesy,2013,87(1):23-28
[15]Fang X.On Non-combinatorial Weighted Total Least Squares with Inequality Constraints[J].Journal of Geodesy,2014,88(8):805-816
[16]Zeng Wenxian,Fang Xing,Liu Jingnan,et al.Weighted Total Least Squares Algorithm with Inequality Constraints[J].Acta Geodaetica et Cartographica Sinica,2014,43(10):1 013-1 018(曾文宪,方兴,刘经南,等.附有不等式约束的加权整体最小二乘算法[J].测绘学报,2014,43(10):1 013-1 018)
[17]Zeng W X,Liu J N,Yao Y B.On Partial Errors-inVariables Models with Inequality Constraints of Parameters and Variables[J].Journal of Geodesy,2015,89(2):111-119
[18]Wang Leyang,Xu Caijun,Lu Tieding.Ridge Estimation Method in Ill-posed Weighted Total Least Squares Adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1 346-1 350(王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报·信息科学版,2010,35(11):1 346-1 350)
[2]Mahboub V,Sharifi M A.On Weighted Total Least-Squares with Linear and Quadratic Constraints[J].Journal of Geodesy,2013,87:279-286
[3]Zhu J J,Santerre R,Chang X W.A Bayesian Method for Linear Inequality Constrained Adjustment and Its Application to GPS Positioning[J].Journal of Geodesy,2005,78(9):528-534
[4]Dai Wujiao,Zhu Jianjun.Single Epoch Ambiguity Resolution in Structure Monitoring Using GPS[J].Geomatics and Information Science of Wuhan University,2007,32(3):234-237(戴吾蛟,朱建军.GPS建筑物震动变形监测中的单历元算法[J].武汉大学学报·信息科学版,2007,32(3):234-237)
[5]Rao C R,Toutenburg H.Linear Model:Least Squares and Alternatives[M].Heidelberg:Springer,2005
[6]Lu G,Krakiwsky E J,Lachapelle G.Application of Inequality Constraint Least Squares to GPS Navigation Under Selective Availability[J].Manuscripta Geodaetica,1993,18:124-130
[7]Liew C K.Inequality Constrained Least-Squares Estimation[J].Journal of the American Statistical Association,1976,71(355):746-751
[8]Roese-Koerner L,Devaraju B,Sneeuw N,et al.A Stochastic Framework for Inequality Constrained Estimation[J].Journal of Geodesy,2012,86(11):1 005-1 018
[9]Xu P L,Liu J N,Shi C.Total Least Squares in Partial Errors-in-Variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675
[10]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
[11]Schaffrin B,Felus Y A.An Algorithmic Approach to the Total Least-Squares Problem with Linear and Quadratic Constraints[J].Studia Geophysica Geodaetica,2009,53(1):1-16
[12]Fang X.Weighted Total Least Squares:Necessary and Sufficient Conditions,Fixed and Random Parameters[J].Journal of Geodesy,2013,87(8):733-749
[13]de Moor B.Total Linear Least Squares with Inequality Constraints[R].ESAT-SISTA Report1990-02,Department of Electrical Engineering,Katholieke Universiteit Leuven,Belgium,1990
[14]Zhang S L,Tong X H,Zhang K.A Solution to EIV Model with Inequality Constraints and Its Geodetic Applications[J].Journal of Geodesy,2013,87(1):23-28
[15]Fang X.On Non-combinatorial Weighted Total Least Squares with Inequality Constraints[J].Journal of Geodesy,2014,88(8):805-816
[16]Zeng Wenxian,Fang Xing,Liu Jingnan,et al.Weighted Total Least Squares Algorithm with Inequality Constraints[J].Acta Geodaetica et Cartographica Sinica,2014,43(10):1 013-1 018(曾文宪,方兴,刘经南,等.附有不等式约束的加权整体最小二乘算法[J].测绘学报,2014,43(10):1 013-1 018)
[17]Zeng W X,Liu J N,Yao Y B.On Partial Errors-inVariables Models with Inequality Constraints of Parameters and Variables[J].Journal of Geodesy,2015,89(2):111-119
[18]Wang Leyang,Xu Caijun,Lu Tieding.Ridge Estimation Method in Ill-posed Weighted Total Least Squares Adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1 346-1 350(王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报·信息科学版,2010,35(11):1 346-1 350)
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