Ridge estimation method for linearized general EIV adjustment model
-
摘要: 通用EIV(errors-in-variables)平差模型作为经典平差模型的一般化形式，具有同时顾及多种随机误差的优势. 在通用EIV平差模型加权总体最小二乘(WTLS)的线性化估计基础上，引入正则化准则. 正则化矩阵为单位矩阵时为岭估计，添加目标函数，通过建立拉格朗日目标函数的最小化求解，导出加权通用EIV平差模型对应的岭估计解式，给出了确定岭参数的U曲线法和L曲线法. 计算了通用EIV平差模型的线性化估计、两种岭估计及其对应的方差分量值；验证岭估计对通用EIV模型的线性化估计具有促进性，可减少迭代次数，使得参数方差分量更快趋于平稳，降低参数估计的计算量.
-
关键词:
- 通用EIV(errors-in-variables)模型 /
- 总体最小二乘(TLS) /
- 线性化估计 /
- 岭估计 /
- L曲线 /
- U曲线
Abstract: As a general form of classical adjustment model, general errors-in-variables (EIV) adjustment model has the advantage of taking into account multiple random errors. Based on the linear estimation of the weighted total least squares of the general EIV adjustment model, the regularization criterion is introduced. When the regularization matrix is the unit matrix, it is called the ridge estimation. The objective function is then added. By establishing the minimization solution of the Lagrange objective function, the ridge estimation solution corresponding to the weighted general EIV adjustment model is derived. The U curve method and L curve method for determining ridge parameters are given. The linear estimation, two ridge estimations and their corresponding variance components of the general EIV adjustment model are calculated. It is validated that ridge estimation can promote the linearization estimation of general EIV model, reduce the times of iterations, make the parameter variance component more stable and reduce the calculation of parameter estimation. -
表 1 含有随机误差的模拟数据矩阵
$ {\boldsymbol{B}}_{4\times 2} $ $ {\boldsymbol{A}}_{4\times 4} $ $ {\boldsymbol{L}}_{0} $ $ \boldsymbol{e} $ 10.409 7 17.543 9 12.466 9 11.096 3 15.871 7 11.725 4 27.543 0 −1425.323 18.033 4 15.171 3 8.882 6 10.291 8 2.928 7 3.665 7 20.726 8 −852.619 18.631 3 15.855 2 12.321 0 1.108 5 6.391 9 15.808 5 20.838 9 −1142.913 15.670 5 11.878 4 3.551 0 12.104 0 3.866 5 1.256 7 25.032 5 −658.407 
下载: 导出CSV
表 2 参数解估计值及其方差估计值
参数解/中误差 LTLS算法 LTLS岭估计（L 曲线法） LTLS岭估计（U 曲线法） $ {\hat{X}}_{1} $ 5.012 550 72 5.010 468 13 5.008 176 42 $ {\hat{X}}_{2} $ 9.994 963 66 9.995 139 46 9.996 834 92 $ {\sigma }_{{\hat{X}}_{1}}^{2} $ 4.272 016$ \times {10}^{-25} $ 4.109 261$ \times {10}^{-25} $ 4.001 347$ \times {10}^{-25} $ $ {\sigma }_{{\hat{X}}_{2}}^{2} $ 7.245 756$ \times {10}^{-25} $ 4.532 423$ \times {10}^{-25} $ 4.076 249$ \times {10}^{-25} $ $ {\sigma }_{{\hat{X}}_{1}{\hat{X}}_{2}} $ −5.044 943$ \times {10}^{-25} $ −4.141 588$ \times {10}^{-25} $ −3.738 164$ \times {10}^{-25} $ ${\rm {tr} }\left[D\left(\hat{\boldsymbol{X} }\right)\right]$ 1.151 777$ \times {10}^{-24} $ 8.641 684$ \times {10}^{-25} $ 8.077 596$ \times {10}^{-25} $ $ {\parallel \mathrm{\Delta }\boldsymbol{X}\parallel }^{2} $ 1.828 853$ \times {10}^{-4} $ 1.332 065$ \times {10}^{-4} $ 7.687 158$ \times {10}^{-5} $ $ \mathrm{迭}\mathrm{代}\mathrm{次}\mathrm{数} $ 11 8 8
下载: 导出CSV
表 3 坐标观测值及相应权值
点号 观测数据 权值 $ {y}_{2} $ $ {x}_{2} $ $ {P}_{y} $ $ {P}_{x} $ 1 5.9 0.0 1.0 1 000.0 2 5.4 0.9 1.8 1 000.0 3 4.4 1.8 4.0 500.0 4 4.6 2.6 8.0 800.0 5 3.5 3.3 20.0 200.0 6 3.7 4.4 20.0 80.0 7 2.8 5.2 70.0 60.0 8 2.8 6.1 70.0 20.0 9 2.4 6.5 100.0 1.8 10 1.5 7.4 500.0 1.0
下载: 导出CSV
表 4 参数及其方差估计值、迭代次数
参数解/
中误差LTLS算法 LTLS岭估计
（L 曲线法）LTLS岭估计
（U 曲线法）${\;\hat{\beta } }_{1}$ −0.489 907 073 −0.489 907 072 −0.489 907 072 ${\;\hat{\beta } }_{2}$ 5.527 557 906 5.527 557 907 5.527 557 907 $ {\sigma }_{{\hat{X}}_{1}}^{2} $ 0.656 376 350 0.656 376 380 0.656 376 380 $ {\sigma }_{{\hat{X}}_{2}}^{2} $ 1.821 311 510 1.821 311 500 1.821 311 500 $ \mathrm{迭}\mathrm{代}\mathrm{次}\mathrm{数} $ 18 14 13
下载: 导出CSV
-
[1] VAN HUFFEL S, VANDEWALLE J. The total least squares problem: computational aspects and analysis [M]. Philadelphia: Society for Industrial and Applied Mathematics, 1991. [2] ADCOCK R J. A note on the method of least squares[J]. The analyst, 1877, 4(6): 183-184. DOI: 10.2307/2635777 [3] GOULUB G H, LOAN V C F. An analysis of the total least squares problem[J]. SIAM journal on numerical analysis, 1980, 17(6): 883-893. DOI: 10.1137/0717073 [4] 张俏, 徐爱功, 祝会忠, 等. BDS总体最小二乘整周模糊度解算方法[J]. 导航定位学报, 2017, 5(1): 65-69,80. [5] 武曙光, 聂桂根, 彭凤友, 等. 高精度GPS超短基线场数据处理与分析[J]. 全球定位系统, 2020, 45(2): 7-12. [6] 何安良, 张品超, 程兴保. 稳健加权整体最小二乘法在GPS高程拟合中的应用[J]. 江西科学, 2019, 37(4): 549-552,574. [7] 潘雪琛, 姜挺, 余岸竹, 等. 总体最小二乘平差下的基准影像辅助国产卫星影像定位[J]. 测绘通报, 2019(3): 57-60. [8] 曾文宪, 方兴, 刘经南, 等. 通用EIV平差模型及其加权整体最小二乘估计[J]. 测绘学报, 2016, 45(8): 890-894. DOI: 10.11947/j.AGCS.2016.20150156 [9] AMIRI-SIMKOORI A R, MORTAZAVI, ASGARI J. Weighted total least squares applied to mixed observation model[J]. Survey review, 2016, 48(349): 278-286. DOI: 10.1179/1752270615Y.0000000031 [10] 曾文宪, 刘泽邦, 方兴, 等. 通用EIV平差模型的线性化估计算法[J/OL]. (2021-01-12)[2021-07-30]. 武汉大学学报(信息科学版). https://doi.org/10.13203/j.whugis20200243 [11] 吴光明, 鲁铁定. 病态数据处理的岭估计迭代解法[J]. 大地测量与地球动力学, 2019, 39(2): 178-183. [12] 王乐洋, 许才军, 鲁铁定. 病态加权总体最小二乘平差的岭估计解法[J]. 武汉大学学报(信息科学版), 2010, 35(11): 1346-1350. [13] 陈帅, 詹本勇. 基于L曲线法的抗差岭估计模型[J]. 全球定位系统, 2014, 39(5): 41-45. [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. DOI: 10.11947/j.AGCS.2016.20150134 [15] HANSEN P C. Analysis of discrete Ill-posed problems by means of the L-Curve[J]. SIAM review, 1992, 34(4): 561-580. DOI: 10.1137/1034115 [16] 鲁洋为, 王振杰. 用U曲线法确定岭估计中的岭参数[J]. 导航定位学报, 2015, 3(3): 132-134,138. [17] KRAWCZKY-STANDO D, RUDNICKI M. Regularization parameter selection in discrete Ill-posed problems — the use of the U-Curve[J]. International journal of applied mathematics and computer science, 2007, 17(2): 157-164. DOI: 10.2478/v10006-007-0014-3 [18] 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. DOI: 10.1080/00401706.1979.10489751 [19] 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, 4(4): 30-33. [20] FIERRO R D, GOLUB G H, HANSEN P C, et al. Rcgularization by truncated total least squares[J]. SIAM journal on scientific and statistical computing, 1997, 18(4): 1223-1241. DOI: 10.1137/S1064827594263837 [21] WANG L Y, XU G Y. Variance component estimation for partial errors-in-variables models[J]. Studia geophysica et geodaetica, 2016, 60(1): 35-55. DOI: 10.1007/s11200-014-0975-2 -
点击查看大图
图(2) / 表(4)
计量
- 文章访问数: 262
- HTML全文浏览量: 106
- PDF下载量: 11
- 被引次数: 0
