Tikhonov正则化法引入正则化参数和稳定泛函来改善矩阵的病态性。稳定泛函表示为参数的二范约束时，正则化矩阵为单位阵的正则化法即为岭估计法。通过对岭估计的方差与偏差进行分析可知，岭估计改善矩阵病态性的同时也过度地引入了偏差，降低了解的可靠性，对较大奇异值的修正不能有效地减小估计的方差，却引入了偏差，而对较小奇异值的修正可有效地减小估计的方差。因此，选择较小奇异值特征向量构造正则化矩阵，调节各奇异值的修正，可有效减小参数估计的方差，减少偏差的引入，得到更为可靠的参数估计。通过试验证明了该方法的有效性。
Tikhonov regularization introduces regularization parameter and stable functional to improve the ill-condition. When the stable functional expressed as two-norm constraint, the regularization method is the same as ridge estimation. The analysis of the variance and bias of the ridge estimation shows that ridge estimation improved the ill-condition but introduced more bias. The estimation reliability is lowered. We get that correct the larger singular values cannot decrease the variance effectively but introduced more bias, correcting the smaller singular values can decrease the variance effectively. We choose the eigenvectors of the smaller singular values to construct the regularization matrix. It can adjust the correction of the singular values, decrease the variance and biases and finally get a more reliable estimation.
[1] GUI Qingming,HAN Songhui.New Algorithm of GPS Rapid Positioning Based on Double-k-type Ridge Estimation[J].Journal of Surveying Engineering,2007,133(4):173-178.
[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] SAVE H,BETTADPUR S,TAPLEY B D.Reducing Errors in the GRACE Gravity Solutions Using Regularization[J].Journal of Geodesy,2012,86(9):695-711.
[4] 崔希璋,於倧俦,陶本藻,等.广义测量平差[M].新版.武汉:武汉测绘科技大学出版社,2001.CUI Xizhang,YU Zongchou,TAO Benzao,et al.Generalized Surveying Adjustment[M].New Ed.Wuhan:Wuhan Technical University of Surveying and Mapping Press,2001.
[5] 朱建军.岭估计的一种新的算法[J].测绘信息与工程,1997(3):22-25.ZHU Jianjun.A New Algorithm for Ridge Estimate[J].Journal of Geomatics,1997(3):22-25.
[6] HANSEN P C.The Truncated SVD as a Method for Regularization[J].BIT Numerical Mathematics,1987,27(4):534-553.
[7] TIKHONOV A N,ARSENIN V Y.Solutions of Ill-posed Problems[M].JOHN F,trans.New York:Halsted Press,1977.
[8] 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.
[9] HANSEN P C.Analysis of Discrete Ill-posed Problems by Means of the L-curve[J].SIAM Review,1992,34(4):561-580.
[10] HANSEN P C.The Use of the L-curve in the Regularization of Discrete Ill-posed Problems[J].SIAM Journal on Scientific Computing,1993,14(6):1487-1503.
[11] 戴吾蛟,冯光财,朱建军.一种基于Helmert方差分量估计的岭参数确定方法[J].大地测量与地球动力学,2006,26(4):30-33.DAI Wujiao,FENG Guangcai,ZHU Jianjun.A Method for Selecting Ridge Parameter Based on Helmert Variance Components Estimation[J].Journal of Geodesy and Geodynamics,2006,26(4):30-33.
[12] 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.
[13] 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):196-201.WANG Zhenjie,OU Jikun,LIU Lintao.Investigation on Solutions of Ill-conditioned Problems in Rapid Positioning Using Single Frequency GPS Receivers[J].Acta Geodaetica et Cartographica Sinica,2005,34(3):196-201.
[14] 徐新禹,李建成,王正涛,等.Tikhonov正则化方法在GOCE重力场求解中的模拟研究[J].测绘学报,2010,39(5):465-470.XU Xinyu,LI Jiancheng,WANG Zhengtao,et al.The Simulation Research on the Tikhonov Regularization Applied in Gravity Field Determination of GOCE Satellite Mission[J].Acta Geodaetica et Cartographica Sinica,2010,39(5):465-470.
[15] SHEN Yunzhong,XU Peiliang,LI Bofeng.Bias-corrected Regularized Solution to Inverse Ill-posed Models[J].Journal of Geodesy,2012,86(8):597-608.
[16] 朱建军,田玉淼,陶肖静.带准则参数的平差准则及其统一与解算[J].测绘学报,2012,41(1):8-13.ZHU Jianjun,TIAN Yumiao,TAO Xiaojing.United Expression and Solution of Adjustment Criteria with Parameters[J].Acta Geodaetica et Cartographica Sinica,2012,41(1):8-13.
[17] 欧吉坤.测量平差中不适定问题解的统一表达与选权拟合法[J].测绘学报,2004,33(4):283-288.OU Jikun.Uniform Expression of Solutions of Ill-posed Problems in Surveying Adjustment and the Fitting Method by Selection of the Parameter Weights[J].Acta Geodaetica et Cartographica Sinica,2004,33(4):283-288.
[18] XU Peiliang,SHEN Yunzhong,FUKUDA Y,et al.Variance Component Estimation in Linear Inverse Ill-posed Models[J].Journal of Geodesy,2006,80(2):69-81.
[19] 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.
[20] 王振杰,欧吉坤.用L-曲线法确定岭估计中的岭参数[J].武汉大学学报(信息科学版),2004,29(3):235-238.WANG Zhenjie,OU Jikun.Determining the Ridge Parameter in a Ridge Estimation Using L-curve Method[J].Geomatics and Information Science of Wuhan University,2004,29(3):235-238.
