抗差加权整体最小二乘模型的牛顿-高斯算法

doi:10.11947/j.AGCS.2015.20130704

测绘学报 ›› 2015, Vol. 44 ›› Issue (6): 602-608.doi: 10.11947/j.AGCS.2015.20130704

抗差加权整体最小二乘模型的牛顿-高斯算法

王彬¹, 李建成¹, 高井祥², 刘超³

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

收稿日期:2013-12-12 修回日期:2014-11-19 出版日期:2015-06-20 发布日期:2015-07-28
作者简介:王彬(1988—),男,博士生,研究方向为测量数据处理与GPS坐标时间序列分析。E-mail: rainkingwang881107@163.com
基金资助:
国家973计划(2013CB733300);国家自然科学基金青年基金(41404004);中国博士后科学基金(2014M551790)

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

Received:2013-12-12 Revised:2014-11-19 Online:2015-06-20 Published:2015-07-28
Supported by:
The National Basic Research Program of China(973 Program)(No. 2013CB733300);The National Natural Science Foundation of China(No. 41404004);The China Postdoctoral Science Foundation(No. 2014M551790)

摘要/Abstract

摘要： 基于加权整体最小二乘的牛顿-高斯迭代算法,提出了一种抗差加权整体最小二乘模型。利用标准化残差构造权因子函数,并采用中位数法获得具有抗差性的单位权中误差估值,能同时实现观测空间和结构空间抗差。为获得标准化残差,利用线性近似的协因数传播律推导了加权整体最小二乘残差协因数阵的表达式,并给出模型的迭代计算方法。试验结果表明:对于加权整体最小二乘的粗差处理问题,本文提出的方法具有良好的抗差性能,参数估值与不含粗差时加权整体最小二乘的结果没有显著的差异,性能优于直接由残差构造的稳健加权整体最小二乘模型。

关键词: 整体最小二乘, 抗差估计, 牛顿-高斯法, 标准化残差, 中位数

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

中图分类号:

P207

王彬, 李建成, 高井祥, 刘超. 抗差加权整体最小二乘模型的牛顿-高斯算法[J]. 测绘学报, 2015, 44(6): 602-608.

WANG Bin, LI Jiancheng, GAO Jingxiang, LIU Chao. Newton-Gauss Algorithm of Robust Weighted Total Least Squares Model[J]. Acta Geodaetica et Cartographica Sinica, 2015, 44(6): 602-608.

参考文献

[1] FELUS Y A. Application of Total Least Squares for Spatial Point Process Analysis[J]. Journal of Surveying Engineering, 2004, 130(3): 126-133.
[2] SCHAFFRIN B, LEE I, FELUS Y, et al. Total Least-squares (TLS) for Geodetic Straight-line and Plane Adjustment[J]. Bollettino di Geodesia e Scienze Affini, 2006, 65(3): 141-168.
[3] MARKOVSKY I, VAN HUFFEL S. Overview of Total Least Squares Methods[J]. Signal Processing, 2007, 87(10): 2283-2302.
[4] SCHAFFRIN B, FELUS Y A. On the Multivariate Total Least-squares Approach to Empirical Coordinate Transformations: Three Algorithms[J]. Journal of Geodesy, 2008, 82(6): 373-383.
[5] CHEN Yi, LU Jue, ZHENG Bo. Application of Total Least Squares to Space Resection[J]. Geomatics and Information Science of Wuhan University, 2008, 33(12): 1271-1274. (陈义, 陆珏, 郑波. 总体最小二乘方法在空间后方交会中的应用[J]. 武汉大学学报: 信息科学版, 2008, 33(12): 1271-1274.)
[6] SCHAFFRIN B, WIESER A. On Weighted Total Least-squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7): 415-421.
[7] NEITZEL F. Generalization of Total Least-squares on Example of Unweighted and Weighted 2D Similarity Transformation[J]. Journal of Geodesy, 2010, 84(12): 751-762.
[8] SHEN Y Z, LI B F, CHEN Y. An Iterative Solution of Weighted Total Least Squares Adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238.
[9] XU P L, LIU J N, SHI C. Total Least Squares Adjustment in Partial Errors-in-variables Models: Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8): 661-675.
[10] BAARDA W. A Testing Procedure for Use in Geodetic Networks[J]. Netherlands Geodetic Commission Publication on Geodesy: New Series, 1968, 2(5): 45-53.
[11] OU Jikun. A New Method to Detect Gross Errors—Quasi-accurate Detection Method[J]. Chinese Science Bulletin, 1999, 44(16): 1777-1781. (欧吉坤. 一种检测粗差的新方法——拟准检定法[J]. 科学通报, 1999, 44(16): 1777-1781.)
[12] ROUSSEEUW P J, LEROY A M. Robust Regression and Outlier Detection[M]. New York: John Wiley and Sons, 1987.
[13] ZHOU Jiangwen, HUANG Youcai, YANG Yuanxi, et al. Robust Least Squares Method[M]. Wuhan: Huazhong University of Science and Technology Press, 1997. (周江文, 黄幼才, 杨元喜, 等. 抗差最小二乘法[M]. 武汉: 华中理工大学出版社, 1997.)
[14] YANG Y. Robust Estimation of Geodetic Datum Transformation[J]. Journal of Geodesy, 1999, 73(5): 268-274.
[15] YANG Yuanxi, SONG Lijie, XU Tianhe. Robust Parameter Estimation for Geodetic Correlated Observations[J]. Acta Geodaetica et Cartographica Sinica, 2002, 31(2): 95-99. (杨元喜, 宋力杰, 徐天河. 大地测量相关观测抗差估计理论[J]. 测绘学报, 2002, 31(2): 95-99.)
[16] YANG Y X, SONG L J, XU T H. Robust Estimator for Correlated Observations Based on Bifactor Equivalent Weights[J]. Journal of Geodesy, 2002, 76(6-7): 353-358.
[17] XU P L. Sign-constrained Robust Least Squares, Subjective Breakdown Point and the Effect of Weights of Observations on Robustness[J]. Journal of Geodesy, 2005, 79(1-3): 146-159.
[18] CHEN Yi, LU Jue. Performing 3D Similarity Transformation by Robust Total Least Squares[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(5): 715-722. (陈义, 陆珏. 以三维坐标转换为例解算稳健总体最小二乘方法[J]. 测绘学报, 2012, 41(5): 715-722.)
[19] LU J, CHEN Y, LI B F, et al. Robust Total Least Squares with Reweighting Iteration for Three-dimensional Similarity Transformation[J]. Survey Review, 46(334): 28-36.
[20] GONG Xunqiang, LI Zhilin. A Robust Mixed LS-TLS Based on IGGII Scheme[J]. Geomatics and Information Science of Wuhan University, 2014, 39(4): 462-466. (龚循强, 李志林. 一种利用IGG Ⅱ方案的稳健混合总体最小二乘方法[J]. 武汉大学学报: 信息科学版, 2014, 39(4): 462-466.)
[21] GONG Xunqiang, LI Zhilin. A Robust Weighted Total Least Squares Method[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(9): 888-894. (龚循强, 李志林. 稳健加权总体最小二乘法[J]. 测绘学报, 2014, 43(9): 888-894.)

抗差加权整体最小二乘模型的牛顿-高斯算法

Newton-Gauss Algorithm of Robust Weighted Total Least Squares Model

PDF

可视化

被引次数

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	杨高朝, 王庆, 蔚保国, 刘鹏飞, 李爽. 基于抗差LM的视觉惯性里程计与伪卫星混合高精度室内定位[J]. 测绘学报, 2022, 51(1): 18-30.
[2]	郭迎钢, 李宗春, 何华, 王志颖. 三维坐标转换公共点最优权值的单纯形搜索算法[J]. 测绘学报, 2020, 49(8): 1004-1013.
[3]	胡川, 方兴, 赵立都. 融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法[J]. 测绘学报, 2020, 49(7): 816-823.
[4]	尹潇, 柴洪洲, 向民志, 杜祯强. 附加运动学约束的BDS抗差UKF导航算法[J]. 测绘学报, 2020, 49(11): 1399-1406.
[5]	邓兴升, 彭思淳, 游扬声. 加权整体最小二乘EIO模型与算法[J]. 测绘学报, 2019, 48(7): 926-930.
[6]	王乐洋, 余航, 邹传义, 鲁铁定. 加权整体最小二乘EIV模型与算法——与“加权整体最小二乘EIO模型与算法”一文的讨论[J]. 测绘学报, 2019, 48(7): 931-937.
[7]	焦博, 郝金明, 刘伟平, 张辉, 温旭峰, 师一帅. 单参考星下全星座单差FCB估计与应用[J]. 测绘学报, 2018, 47(10): 1326-1336.
[8]	姚宜斌, 熊朝晖, 张豹, 张良, 孔建. 顾及设计矩阵误差的AR模型新解法[J]. 测绘学报, 2017, 46(11): 1795-1801.
[9]	李林阳, 崔阳, 王宇谱, 吕志平. 窄巷FCB估计方法改进及时变特性分析[J]. 测绘学报, 2017, 46(1): 34-43.
[10]	陈康慷, 徐天河, 杨玉国, 蔡洪亮, 陈国. iGMAS GNSS钟差产品综合与评估[J]. 测绘学报, 2016, 45(S2): 46-53.
[11]	曾文宪, 方兴, 刘经南, 姚宜斌. 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.
[12]	王宇谱, 吕志平, 王宁, 李林阳, 宫晓春. 顾及卫星钟随机特性的抗差最小二乘配置钟差预报算法[J]. 测绘学报, 2016, 45(6): 646-655.
[13]	赵俊, 归庆明. 部分变量误差模型的整体抗差最小二乘估计[J]. 测绘学报, 2016, 45(5): 552-559.
[14]	方兴, 曾文宪, 刘经南, 姚宜斌, 王勇. 基于非线性高斯-赫尔默特模型的混合整体最小二乘估计[J]. 测绘学报, 2016, 45(3): 291-296.
[15]	陶叶青, 高井祥, 姚一飞. 基于中位数法的抗差总体最小二乘估计[J]. 测绘学报, 2016, 45(3): 297-301.