靶向谱修正迭代法
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摘要
病态的存在对测量数据处理影响较大,会降低参数估计的可靠性。谱修正迭代法是修正法矩阵所有谱并迭代计算,结果是无偏估计。而一般病态问题是法矩阵的几个谱奇异,因而存在谱多余修正的问题。针对该问题,文中提出靶向谱修正迭代法,即只修正法矩阵奇异的谱。将迭代矩阵中的单位阵变成靶向矩阵,靶向矩阵由法矩阵较小特征值对应的特征向量构造的对称矩阵,并用算例验证该方法的有效性与可行性。结果表明,随谱修正参数增大,结果更优且迭代次数更少。
The presence of morbid effects on the measurement data processing will reduce the reliability of the parameter estimation.The iteration by correcting characteristic value corrects all the characteristic value of matrix and iterate.The result is unbiased estimation.The general morbid problem is a few characteristic value strange of the matrix and thus there exists a problem of characteristic value superfluous amendments.For this problem,the paper proposes an iteration method by targeted correcting characteristic value,that is,only corrects strange characteristic value of the matrix.The unit matrix in the iterative matrix is transformed into target matrices,which is the symmetric matrix constructed by the eigenvector corresponding to the smaller eigenvalues of the matrix and then used examples to verify the validity and feasibility of the method.The result shows that the results of the method will be better and the number of iterations will be fewer with the characteristic value correction parameters increase.
The presence of morbid effects on the measurement data processing will reduce the reliability of the parameter estimation.The iteration by correcting characteristic value corrects all the characteristic value of matrix and iterate.The result is unbiased estimation.The general morbid problem is a few characteristic value strange of the matrix and thus there exists a problem of characteristic value superfluous amendments.For this problem,the paper proposes an iteration method by targeted correcting characteristic value,that is,only corrects strange characteristic value of the matrix.The unit matrix in the iterative matrix is transformed into target matrices,which is the symmetric matrix constructed by the eigenvector corresponding to the smaller eigenvalues of the matrix and then used examples to verify the validity and feasibility of the method.The result shows that the results of the method will be better and the number of iterations will be fewer with the characteristic value correction parameters increase.
引文
[1]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.
[2] LI Bofeng,SHEN Yunzhong,FENG Yanming.Fast GNSS Ambiguity Resolution as an Ill-posed Problem[J].Journal of Geodesy,2010,84(11):683-698.
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[4]顾勇为,归庆明,张璇,等.大地测量与地球物理中病态性问题的正则化迭代解法[J].测绘学报,2014(43):331-336.
[5]孙文,吴晓平,王庆宾,等.航空重力数据向下延拓的波数域迭代Tikhonov正则化方法[J].测绘学报,2014,43(6):566-574.
[6]李伟,许才军.航空重力梯度数据向下延拓的正则化方法及参数选取研究[J].大地测量与地球动力学,2017,37(2):154-159.
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[14]黄德伦,史经俭,廉琦,等.基于法方程病态的谱修正迭代算法的探讨[J].测绘科学,2013,38(2):35-37.
[15]潘朝毅.谱修正迭代法的收敛分析及其改进[J].成都师范学院学报,2009,25(5):112-113.
[16]邓兴升,孙虹虹.自适应谱修正LU分解法解算高病态法方程[J].大地测量与地球动力学,2014,34(6):135-139.
[17]刘斌,龚健雅,江万寿,等.基于岭参数的谱修正迭代法及其在有理多项式参数求解中的应用[J].武汉大学学报(信息科学版),2012,37(4):399-402.
[18]于冬冬,王乐洋.病态总体最小二乘问题的谱修正迭代法[J].大地测量与地球动力学,2015,35(4):702-706.
[19]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.
[20]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.
[2] LI Bofeng,SHEN Yunzhong,FENG Yanming.Fast GNSS Ambiguity Resolution as an Ill-posed Problem[J].Journal of Geodesy,2010,84(11):683-698.
[3]王振杰,欧吉坤,柳林涛.单频GPS快速定位中病态问题的解法研究[J].测绘学报,2005,34(3):197-201.
[4]顾勇为,归庆明,张璇,等.大地测量与地球物理中病态性问题的正则化迭代解法[J].测绘学报,2014(43):331-336.
[5]孙文,吴晓平,王庆宾,等.航空重力数据向下延拓的波数域迭代Tikhonov正则化方法[J].测绘学报,2014,43(6):566-574.
[6]李伟,许才军.航空重力梯度数据向下延拓的正则化方法及参数选取研究[J].大地测量与地球动力学,2017,37(2):154-159.
[7] TIKHONOV A.Solution of Incorrectly Formulated Problems and the Regularization Method[J].Soviet Math Dokl,1963,5(4).
[8] TIKHONOV A N,ARSENIN V Y.Solutions of Illposed Problems[J].Mathematics of Computation,1977,32(144):491-491.
[9] HOERL A E,KENNARD R W.Ridge Regression:Biased Estimation for Nonorthogonal Problems[J].Technometrics,1970,12(1):55-67.
[10] HOERL A E,KENNARD R W.Ridge Regression:Applications to Nonorthogonal Problems[J].Technometrics,1970,12(1):69-82.
[11]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.
[12]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,2009.9
[13]王新洲,刘丁酉,张前勇,等.谱修正迭代法及其在测量数据处理中的应用[J].黑龙江工程学院学报,2001,15(2):3-6.
[14]黄德伦,史经俭,廉琦,等.基于法方程病态的谱修正迭代算法的探讨[J].测绘科学,2013,38(2):35-37.
[15]潘朝毅.谱修正迭代法的收敛分析及其改进[J].成都师范学院学报,2009,25(5):112-113.
[16]邓兴升,孙虹虹.自适应谱修正LU分解法解算高病态法方程[J].大地测量与地球动力学,2014,34(6):135-139.
[17]刘斌,龚健雅,江万寿,等.基于岭参数的谱修正迭代法及其在有理多项式参数求解中的应用[J].武汉大学学报(信息科学版),2012,37(4):399-402.
[18]于冬冬,王乐洋.病态总体最小二乘问题的谱修正迭代法[J].大地测量与地球动力学,2015,35(4):702-706.
[19]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.
[20]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.