一种构造正则化矩阵的新方法
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摘要
在系数矩阵病态时进行参数求解,合理地选择正则化参数和正则化矩阵可以提高参数估计的可靠性。针对正则化矩阵如何构造的问题,提出一种新的正则化矩阵构造方法。通过法矩阵较小奇异值对应的特征向量构造出一个对称矩阵,用该矩阵的主对角线元素构造出对角矩阵,然后与单位矩阵组合得出一种新的正则化矩阵。实验表明,当正则化参数小于1时,新算法的参数估值优于岭估计。
In parameter solving under the conditions of coefficient matrix,the rational selection of regularization parameters and regularization matrix can improve the reliability of parameter estimation.The symmetric matrix is constructed by the eigenvectors corresponding to the smaller singular values of the matrix.The diagonal matrix is constructed by the main diagonal elements of the matrix,and then a new regularization matrix is obtained by combining with the unit matrix.The experimental results show that when the regularization parameter is less than 1,the parameter estimation of this algorithm is better than the ridge estimation.
In parameter solving under the conditions of coefficient matrix,the rational selection of regularization parameters and regularization matrix can improve the reliability of parameter estimation.The symmetric matrix is constructed by the eigenvectors corresponding to the smaller singular values of the matrix.The diagonal matrix is constructed by the main diagonal elements of the matrix,and then a new regularization matrix is obtained by combining with the unit matrix.The experimental results show that when the regularization parameter is less than 1,the parameter estimation of this algorithm is better than the ridge estimation.
引文
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[2] Xu P L.Truncated SVD Methods for Discrete Linear IllPosed Problems[J].Geophysical Journal of the Royal Astronomical Society,1998,135(2):505-514
[3] Tikhonov A N.Solution of Incorrectly Formulated Problems and the Regularization Method[J].Soviet Math,1963,4:1 035-1 038
[4] Bell J B,Tikhonov A N,Arsenin V Y.Solutions of IllPosed Problems[J].Mathematics of Computation,1978,32(144):1 320
[5] Hoerl A E,Kennard R W.Ridge Regression:Biased Estimation for Nonorthogonal Problems[J].Technometrics,1970,12(1):55-67
[6] Hoerl A E,Kennard R W.Ridge Regression:Applications to Nonorthogonal Problems[J].Technometrics,1970,12(1):55-67
[7]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)
[8] Hansen P C.Analysis of Discrete Ill-posed Problems by Means of the L-Curve[J].SIAM Review,1992,34(4):561-580
[9] Hansen P C,O’Leary D P.The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J].SIAM Sci Comput,1993,14(6):1 487-1 503
[10]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 Method and Application[M].Hefei:Anhui Education Press,1987)
[11]Golub G H,Heath M,Wahba G.Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter[J].Technometrics,1979,21(2):215-223
[12]Sourbron S,Luypaert R,Schuerbeek V P,et al.Choice of the Regularization Parameter for Perfusion Quantification with MRI[J].Physics in Medicine&Biology,2004,49(14):3 307
[13]崔希璋,於宗俦,陶本藻,等.广义测量平差[M].武汉:武汉大学出版社,2009(Cui Xizhang,Yu Zongchou, Tao Benzao,et al.Generalized Surveying Adjustment[M].Wuhan:Wuhan University Press,2009)
[14]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)
[15]王振杰,欧吉坤,柳林涛.一种解算病态问题的方法——两步解法[J].武汉大学学报:信息科学版,2005,30(9):821-824(Wang Zhenjie,Ou Jikun,Liu Lintao.A Method for Resolving Ill-Conditioned Problems:Two Step Solution[J].Geomatics and Information Science of Wuhan University,2005,30(9):821-824)
[16]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)
[17]王振杰.测量中不适定问题的正则化解法[M].北京:科学出版社,2006(Wang Zhenjie.Research on the Regularization Solutions of Ill-Posed Problems in Geodesy[M].Beijing:Science Press,2006)
[18]杨文采.地球物理反演和地震层析成像[M].北京:地质出版社,1989(Yang Wencai.Geophysical Inversion and Seismic Tomography[M].Beijing:Geological Publishing House,1989)
[19]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)