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1. 武汉大学测绘学院，湖北 武汉 430079；
2. 中国矿业大学环境与测绘学院，江苏 徐州 221116；
3. 安徽理工大学测绘学院，安徽 淮南 232001

Newton-Gauss Algorithm of Robust Weighted Total Least Squares Model
WANG Bin1,LI Jiancheng1,GAO Jingxiang2,LIU Chao3
1. School of Geodesy and Geomatics,Wuhan University,Wuhan 430079,China;
2. School of Environment Science and Spatial Informatics,China University of Mining and Technology,Xuzhou 221116,China;
3. School of Geodesy and Geomatics,Anhui University of Science and Technology,Huainan 232001,China
First author: WANG Bin (1988-),male,PhD candi- date,majorsinsurveying data processing and GPS coordinatetimeseriesanalysis．E-mail: rainkingwang881107@163.com
Abstract: Based on the Newton-Gauss iterative algorithm of weighted total least squares (WTLS),a robust WTLS (RWTLS) model is presented. The model utilizes the standardized residuals to construct the weight factor function and the square root of the variance component estimator with robustness is obtained by introducing the median method. Therefore,the robustness in both the observation and structure spaces can be simultaneously achieved. To obtain standardized residuals,the linearly approximate cofactor propagation law is employed to derive the expression of the cofactor matrix of WTLS residuals. The iterative calculation steps for RWTLS are also described. The experiment indicates that the model proposed in this paper exhibits satisfactory robustness for gross errors handling problem of WTLS,the obtained parameters have no significant difference with the results of WTLS without gross errors. Therefore,it is superior to the robust weighted total least squares model directly constructed with residuals.
Key words: total least squares     robust estimation     Newton-Gauss method     standardized residuals     median

1 引 言

2 基于Newton-Gauss法的WTLS迭代算法

EIV模型的函数模型为

3 基于Newton-Gauss法的RWTLS的基本原理和算法 3.1 RWTLS的基本原理

EIV模型的参数估计准则写成式(7)的形式

3.2 WTLS残差协因数阵的推导

3.3 RWTLS的迭代计算步骤

(1) 将第k次迭代中参数和系数阵残差的初值(第k-1次迭代计算的结果)以及先验协因数阵QLQA代入式(25)和式(19)以更新相应的WTLS残差协因数阵。

(2) 将残差向量和步骤(1)获得的WTLS残差协因数代入式(16)和式(15)获得第k次迭代的标准化残差。

(3) 利用式(13)和式(14)更新QLQA，进而获得第k+1次迭代的参数估值和残差向量。

p> 反复进行步骤(1)—步骤(3)，直到满足||||＜ε0(ε0是给定的小正数)时停止迭代计算，此时便能获得参数的RWTLS解。

4 算例分析

 粗差个数 计算方案 RMSE(a) a最大偏差 RMSE(b) b最大偏差 1 ① 0.238 7 0.690 5 0.006 5 0.019 8 ② 0.752 2 2.974 2 0.019 5 0.088 3 ③ 0.325 7 1.761 1 0.011 5 0.088 4 ④ 0.253 0 0.669 7 0.007 2 0.032 7 2 ① 0.238 6 0.854 2 0.006 4 0.023 6 ② 0.995 6 3.787 9 0.026 7 0.103 6 ③ 0.404 0 1.993 2 0.016 2 0.095 7 ④ 0.268 8 0.979 6 0.007 8 0.037 6 3 ① 0.243 4 0.787 9 0.006 7 0.020 6 ② 1.145 1 3.819 7 0.031 4 0.124 6 ③ 0.496 3 2.046 8 0.020 4 0.097 5 ④ 0.275 8 1.022 4 0.008 4 0.036 3

 图 1 粗差个数为1时方案③和方案④的对比 Fig. 1 Comparison between schemes③ and ④ when there exists one gross error

 图 2 粗差个数为2时方案③和方案④的对比 Fig. 2 Comparison between schemes ③ and ④ when there exist two gross errors

 图 3 粗差个数为3时方案③和方案④的对比文标题 Fig. 3 Comparison between scheme ③ and ④ when there exist three gross errors

(1) 无粗差时WTLS方法(方案①)的估计结果在4种计算方案中是最优的。然而当观测数据受到粗差污染时，WTLS方法(方案②)的参数估值受到了破坏性的影响，与真值产生了很大的偏差。

(2) 方案③和方案④的估计结果较方案②在整体上均有较大的改善，但方案④较方案③明显更优。当粗差个数分别为1、2和3个时，方案④所得参数a的均方根误差分别为方案③的77.6%、66.5%和55.6%，参数b的均方根误差分别为方案③的62.6%、48.1%和41.2%。不难发现，随着粗差个数的增多，方案③获得的各参数均方根误差显著增大，方案④较方案③的优势也更为明显。在500次的模拟过程中，方案③的参数估值多次与真值产生了较大的偏差，而方案④所获得的参数最大偏差值仍然较小。

(3) 方案④所得参数估值的准确度虽不及方案①，但总的来说差异并不显著，参数解仍然准确可靠。

 粗差个数 计算方案 正确识别次数 正确率/(%) 1 ③ 114 22.8 ④ 362 72.4 2 ③ 95 19.0 ④ 315 63.0 3 ③ 73 14.6 ④ 260 52.0

5 结 语

(1) 由标准化残差构造的权因子函数同时实现了观测空间和结构空间抗差，利用中位数法获得的单位权中误差估值在迭代过程中具有更好的稳健性，较已有的稳健WTLS方法在理论上更加合理。

(2) 试验结果表明：当粗差个数分别为1、2和3个时，本文提出的RWTLS算法的粗差点识别正确率分别为文献[21]中RWTLS-IGG方法的3.2倍、3.3倍和3.6倍；所得参数a的均方根误差分别为该方法的77.6%、66.5%和55.6%，b的均方根误差分别为该方法的62.6%、48.1%和41.2%。粗差识别能力和参数估值的准确度均有显著改善。

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http://dx.doi.org/10.11947/j.AGCS.2015.20130704

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#### 文章信息

WANG Bin,LI Jiancheng,GAO Jingxiang,LIU Chao

Newton-Gauss Algorithm of Robust Weighted Total Least Squares Model

Acta Geodaeticaet Cartographica Sinica,2015,44(6): 602-608.
http://dx.doi.org/10.11947/j.AGCS.2015.20130704