相关观测的L_1范数最小化方法的比较分析
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摘要
在抵御粗差影响方面,L_1范数最小化方法比最小二乘更具可靠性。求解L_1范数最小化问题,主要有选权迭代法和线性规划法两种方法。针对相关观测,通常采用权阵的对角线元素来构造L_1范数最小化问题的目标函数,这种处理方法容易忽略观测值之间的相关性。如果采用Cholesky分解消去观测值之间的相关性,则容易造成粗差的转移,进而影响抗差功效。本文对上述两种方法进行了比较分析,数值实验结果表明将相关观测转换为独立等权观测,有利于增强线性规划的稳健性,而在探测粗差方面则具有等价性。由于基于选权迭代的方法收敛性较差,故不适合求解L_1范数最小化问题。
The L_1 norm minimization method is more reliable for resisting outliers compared with the least square method. There are two methods for obtaining the L_1 norm solution based on iteratively reweight and linear programming respectively. For correlated observations, the diagonal elements are used to construct the objective function of L_1 norm minimization problem, but the dependence of observations is ignored. By adopting the Cholesky decomposing, it can be disposed with the dependent observation with equal weight. The robustness will greatly affected due to the transformations of outliers. The paper compares the two methods for L_1 norm minimization. The results show that the way to transform the correlated observations to dependent observations is more robust than the original method for linear programming, but identical in terms of detecting outliers. However, the iteratively reweight method shows the bad convergence, so it is improper for solving the L_1 norm minimization problem.
The L_1 norm minimization method is more reliable for resisting outliers compared with the least square method. There are two methods for obtaining the L_1 norm solution based on iteratively reweight and linear programming respectively. For correlated observations, the diagonal elements are used to construct the objective function of L_1 norm minimization problem, but the dependence of observations is ignored. By adopting the Cholesky decomposing, it can be disposed with the dependent observation with equal weight. The robustness will greatly affected due to the transformations of outliers. The paper compares the two methods for L_1 norm minimization. The results show that the way to transform the correlated observations to dependent observations is more robust than the original method for linear programming, but identical in terms of detecting outliers. However, the iteratively reweight method shows the bad convergence, so it is improper for solving the L_1 norm minimization problem.
引文
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[11] Xu P L.On Robust Estimation with Correlated Observations[J].Bull Geod,1989,63(3):237-252
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[13] 郭建锋.模型误差理论若干问题研究及其在GPS数据处理中的应用[D].武汉:中国科学院测量与地球物理研究所,2007
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[16] Yang Y,Song L,Xu T.Robust Estimator for Correlated Observations based on Bifactor Equivalent Weightes[J].Journal of Geodesy,2002,76(6):353-358
[17] Guo J F,Ou J K,Wang H T.Robust Estimation for Correlated Observations:Two Local Sensitivity-based Downweighting Strategies[J],Journal of Geodesy,2010,84(4):243-250
[18] Ghilani C.Adjustment Computations Spatial Data Analysis[M].5th ed.New Jersey:John Wiley & Sons,2010
[2] Hekimlglu S,Erdogan B,Soycan M,et al.A Univariate Approach for Detecting Outliers in Geodetic Networks[J].Journal of Surveying Engineering,2014,140(2):1-8
[3] Baselga S.Noneistence of Rigorous Tests for Multiple Outlier Detection in Least-Squares Adjustment[J].Journal of Surveying Engineering,2011,137(3):109-112
[4] Keein I,Matsuoka M T,Guaztto M P,et al.On Evaluation of Different Methods for Quality Control of Correlated Observations[J].Survey Review,2015,47(340):28-35
[5] 黄维彬.近代平差理论及其应用[M].北京:解放军出版社,1992
[6] Koch K R.Parameter Estimation and Hypothesis Testing in Linear Models[M].2nd ed.Berlin:Springer Verlag,1999
[7] Marshall J,Bethel J.Basic Concepts of L1 Norm Minimization for Surveying Applications[J].Journal of Surveying Engineering,1996,122(4):168-179
[8] Amirir-Simkooei A.Formulation of L1 Norm Minimization in Gauss-Markov Models[J].Journal of Surveying Engineering,2003,129(1):37-43
[9] Yetkin M,Inal C.L1 Norm Minimization in GPS Networks[J].Survey Review,2011,43(323):523-532
[10] Khodabadeh A,Amiri-Simkooei A R.Recursive Algorithm for L1 Norm Estimation in Linear Models[J].Journal of Surveying Engineering,2011,137(1):1-8
[11] Xu P L.On Robust Estimation with Correlated Observations[J].Bull Geod,1989,63(3):237-252
[12] Xu P.L.Sign-constrained Robust Least Squares,Subjective Break-down Point and the Effect of Weights of Observations On Robustness[J].Journal of Geodesy,2005,79(1):146-159
[13] 郭建锋.模型误差理论若干问题研究及其在GPS数据处理中的应用[D].武汉:中国科学院测量与地球物理研究所,2007
[14] Farebrother R W.L1-Norm and L-Norm Estimation:An Introduction to the Least Absolute Residuals,the Minimax Absolute Residual and Related Fitting Procedures[M].New York:Springer,2013
[15] Snow K.Applications of Parameter Estimation and Hypothesis Testing to GPS Network Adjustments[D].Columbur:Ohio State University,2002
[16] Yang Y,Song L,Xu T.Robust Estimator for Correlated Observations based on Bifactor Equivalent Weightes[J].Journal of Geodesy,2002,76(6):353-358
[17] Guo J F,Ou J K,Wang H T.Robust Estimation for Correlated Observations:Two Local Sensitivity-based Downweighting Strategies[J],Journal of Geodesy,2010,84(4):243-250
[18] Ghilani C.Adjustment Computations Spatial Data Analysis[M].5th ed.New Jersey:John Wiley & Sons,2010