病态数据处理的岭估计迭代解法
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摘要
岭估计通常无法单次计算使得均方误差达到最小,因此提出岭估计迭代法。将岭估计参数估值代入平差模型,更新观测向量,再次用岭估计法求解参数。依此迭代,每次迭代计算方差和偏差,当均方误差达到最小或收敛时终止。模拟算例验证结果表明,该方法有效、可行。
Ridge estimation usually cannot be counted singly to bring the mean square error to a minimum.So,this paper proposes a ridge estimation iterative method.The ridge estimation parameter is brought into the adjustment model,the observation vector is updated,and the parameters are solved again by the ridge estimation method.The iteration is used to calculate the variance and deviation of every iteration and terminates when the mean square error reaches the minimum or the convergence.The results show that the method is effective and feasible;an example is given to demonstrate the iterative method of ridge estimation.
Ridge estimation usually cannot be counted singly to bring the mean square error to a minimum.So,this paper proposes a ridge estimation iterative method.The ridge estimation parameter is brought into the adjustment model,the observation vector is updated,and the parameters are solved again by the ridge estimation method.The iteration is used to calculate the variance and deviation of every iteration and terminates when the mean square error reaches the minimum or the convergence.The results show that the method is effective and feasible;an example is given to demonstrate the iterative method of ridge estimation.
引文
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[14]林东方,朱建军,宋迎春.顾及截断偏差影响的TSVD截断参数确定方法[J].测绘学报,2017,46(6):679-688(Lin Dongfang,Zhu Jianjun,Song Yingchun.Truncation Method for TSVD with Account of Truncation Bias[J].Acta Geodaetica et Cartographica Sinica,2017,46(6):679-688)
[15]王泽文,邱淑芳,阮周生.数值分析与算法[M].北京:科学出版社,2016(Wang Zewen,Qiu Shufang,Ruan Zhousheng.Numerical Analysis and Algorithms[M].Beijing:Science Press,2016)
[16]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,2003(Department of Surveying Adjustment,School of Geodesy and Geomatics,Wuhan University.Error Theory and Foundation of Surveying Adjustment[M].Wuhan:Wuhan University Press,2003)
[17]鲁铁定.总体最小二乘平差理论及其在测绘数据处理中的应用[D].武汉:武汉大学,2010(Lu Tieding.Research on the Total Least Squares and Its Applications in Surveying Data Processing[D].Wuhan:Wuhan University,2010)
[18]郭建锋.测量平差系统病态性的诊断与处理[D].郑州:信息工程大学,2002(Guo Jianfeng.The Diagnosis and Process of Ill-Conditioned Adjustment System[D].Zhengzhou:Information Engineering University,2002)
[2] Xu P L.Truncated SVD Methods for Discrete Linear IllPosed Problems[J].Geophysical Journal International,1998,135(2):505-514
[3] Tikhonov A N.Solution of Incorrectly Formulated Problems and the Regularization Method[J].Soviet Math Dokl,1963(4):1 035-1 038
[4] Tikhonov A N,Arsenin V Y.Solutions of Ill-Posed Problems[J].Mathematics of Computation,1977,32(144):491-491
[5] Hoerl A E,Kennard R W.Ridge Regression:Biased Estimation for Nonorthogonal Problems[J].Technometrics,1970,12(1):55-67
[6]林东方,朱建军,宋迎春,等.正则化的奇异值分解参数构造法[J].测绘学报,2016,45(8):883-889(Lin Dongfang,Zhu Jianjun,Song Yingchun,et al.Construction Method of Regularization by Singular Value Decomposition of Design Matrix[J].Acta Geodaetica et Cartographica Sinica,2016,45(8):883-889)
[7] Hansen P C.Analysis of Discrete Ill-Posed Problems by Means of the L-Curve[J].SIAM Review,1992,34(4):561-580
[8] Hansen P C,O’Leary D P.The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J].SIAM J Sci Comput,1993,14(6):1 487-1 503
[9]陈希孺,王松桂.近代回归分析——原理方法及应用[M].合肥:安徽教育出版社,1987(Chen Xiru,Wang Songgui.Modern Regression Analysis[M].Hefei:Anhui Education Press,1987)
[10]Golub G H,Heath M,Wahba G.Generalized Crossvalidation as a Method for Choosing a Good Ridge Parameter[J].Technometrics,1979,21(2):215-223
[11]Sourbron S,Luypaert R,Van Schuerbeek P,et al.Choice of the Regularization Parameter for Perfusion Quantification with MRI[J].Physics in Medicine&Biology,2004,49(14):3 307-3 324
[12]Shen Y Z,Xu P L,Li B F.Bias-Corrected Regularized Solution to Inverse Ill-Posed Models[J].Journal of Geodesy,2012,86(8):597-608
[13]徐天河,杨元喜.均方误差意义下正则化解优于最小二乘解的条件[J].武汉大学学报:信息科学版,2004,29(3):223-226(Xu Tianhe,Yang Yuanxi.Condition of Regularization Solution Superior to LS Solution Based on MSE Principle[J].Geomatics and Information Science of Wuhan University,2004,29(3):223-226)
[14]林东方,朱建军,宋迎春.顾及截断偏差影响的TSVD截断参数确定方法[J].测绘学报,2017,46(6):679-688(Lin Dongfang,Zhu Jianjun,Song Yingchun.Truncation Method for TSVD with Account of Truncation Bias[J].Acta Geodaetica et Cartographica Sinica,2017,46(6):679-688)
[15]王泽文,邱淑芳,阮周生.数值分析与算法[M].北京:科学出版社,2016(Wang Zewen,Qiu Shufang,Ruan Zhousheng.Numerical Analysis and Algorithms[M].Beijing:Science Press,2016)
[16]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,2003(Department of Surveying Adjustment,School of Geodesy and Geomatics,Wuhan University.Error Theory and Foundation of Surveying Adjustment[M].Wuhan:Wuhan University Press,2003)
[17]鲁铁定.总体最小二乘平差理论及其在测绘数据处理中的应用[D].武汉:武汉大学,2010(Lu Tieding.Research on the Total Least Squares and Its Applications in Surveying Data Processing[D].Wuhan:Wuhan University,2010)
[18]郭建锋.测量平差系统病态性的诊断与处理[D].郑州:信息工程大学,2002(Guo Jianfeng.The Diagnosis and Process of Ill-Conditioned Adjustment System[D].Zhengzhou:Information Engineering University,2002)
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