基于中位参数法相关观测的抗差加权整体最小二乘算法
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摘要
在抗差加权整体最小二乘算法中,抗差模型的抗差性与初值的好坏关系极大,若以最小二乘或整体最小二乘估值作为初值,必定会受到粗差污染而影响其抗差性。考虑到观测向量和系数矩阵存在相关性,首先推导了部分变量误差(partial errors-in-variables, Partial EIV)模型的加权整体最小二乘算法,在此基础上提出了一种利用中位参数法求解抗差迭代初值的相关观测抗差加权整体最小二乘算法。然后采用中位参数法确定抗差初值,考虑到可能出现的粗差对观测空间与结构空间的综合影响,基于标准化残差构造权因子函数,实现其抗差解法。仿真实验结果表明,此算法具有良好的抗差性能,其参数估计结果比传统算法精度更高,且随着粗差个数的增加,其抗差稳定性较好。
In the robust weighted total least squares(RWTLS) algorithm, its robustness of the robust model is highly related to the initial values. If the least squares or total least squares estimates is used as the initial value, it will be affected by gross error, and certainly impacted the robust characteristics of RWTLS estimates. Considering the correlation between the observed vector and the coefficient matrix,we first deduce the weighted least-squares solution of Partial-EIV model, and a new RWTLS algorithm of correlated observation is proposed to solve the initial values of robust iterations by using the median parameter method. Then the median parameter method is used to determine the initial va-lue, and on this basis we propose a new robust estimated method, which is based on the standardized residual error and considered the influence of gross error both on observation and structure spaces. The experiment results show that the proposed estimated method has a good performance to resist gross error, and the presented solution is more accurate than the traditional method for line fitting, and with the increase of the number of gross errors, the stability of the algorithm is superior to the traditional method.
In the robust weighted total least squares(RWTLS) algorithm, its robustness of the robust model is highly related to the initial values. If the least squares or total least squares estimates is used as the initial value, it will be affected by gross error, and certainly impacted the robust characteristics of RWTLS estimates. Considering the correlation between the observed vector and the coefficient matrix,we first deduce the weighted least-squares solution of Partial-EIV model, and a new RWTLS algorithm of correlated observation is proposed to solve the initial values of robust iterations by using the median parameter method. Then the median parameter method is used to determine the initial va-lue, and on this basis we propose a new robust estimated method, which is based on the standardized residual error and considered the influence of gross error both on observation and structure spaces. The experiment results show that the proposed estimated method has a good performance to resist gross error, and the presented solution is more accurate than the traditional method for line fitting, and with the increase of the number of gross errors, the stability of the algorithm is superior to the traditional method.
引文
[1] Adcock R J. Note on the Method of Least Squares[J]. The Analyst, 1877, 4(6): 183-184
[2] Golub H G, Vanloan F C. An Analysis of the Total Least Squares Problem[J]. SIAM Journal on Numerical Analysis, 1980, 17(6): 883-893
[3] Schaffrin B, Felus Y A. On Total Least-Squares Adjustment with Constraints[J]. IAG-Symp, 2005, 128: 417-421
[4] Schaffrin B, Felus Y A. On the Multivariate Total Least-Squares Approach to Empirical Coordinate Transformations[J]. Journal of Geodesy, 2008, 82(6): 373-383
[5] 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
[6] 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
[7] Fang X. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Germany: Leibniz University Hannover, 2011
[8] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749
[9] Deng Caihua, Zhou Yongjun, Zhu Jianjun, et al. General Weighted Total Least Squares Method for a Type of New Functional Model[J]. Journal of China University of Mining and Technology, 2015, 44(5): 952-958 ( 邓才华, 周拥军, 朱建军, 等. 一类新函数模型及通用加权总体最小二乘平差方法[J]. 中国矿业大学学报, 2015, 44(5): 952-958)
[10] 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 Geo-desy, 2012, 86(8): 661-675
[11] Ou Jikun. Quasi-Accurate Detection of Gross Errors (QUAD)[J]. Acta Geodaetica et Cartographica Sinica, 1999, 28(1): 15-20 ( 欧吉坤. 粗差的拟准检定法(QUAD法)[J]. 测绘学报, 1999, 28(1): 15-20)
[12] Yang Y X. Robust Estimation of Geodetic Datum Transformation[J]. Journal of Geodesy, 1999, 73(5): 268-274
[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. 抗差估计最小二乘法[M]. 武汉: 华中理工大学出版社, 1997)
[14] Gao Jingxiang, Zhang Huahai, Yu Xuexiang. Study on Robust Estimation Model for Coordinate Transformation of GPS Network in Ming Areas[J]. Journal of China University of Mining and Technology, 1999, 28(2): 99-103(高井祥, 张华海, 余学祥. 矿区GPS网坐标转换的抗差模型[J]. 中国矿业大学学报, 1999, 28(2): 99-103)
[15] Chen Yi, Lu Jue. Peforming 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)
[16] Mahboub V, Amiri-Simkooei A R, Sharifi M A. Iteratively Reweighted Total Least Squares: A Robust Estimation in Errors-in-Variables Models[J]. Surv Rev, 2013, 45(329): 92-99
[17] 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)
[18] Lu J, Chen Y, Li B, et al. Robust Total Least Squares with Reweighting Iteration for Three-Dimensional Similarity Transformation[J]. Surv Rev, 2014, 46(334): 28-36
[19] Tao Y Q, Gao J X, Yao Y F. TLS Algorithm for GPS Height Fitting Based on Robust Estimation[J]. Surv Rev, 2014, 46(336): 184-188
[20] Wang B, Li J C, Liu C. A Robust Weighted Total Least Squares Algorithm and Its Geodetic Applications[J]. Studia Geophysica et Geodaetica, 2016, 60(2): 177-194
[21] Tao Yeqing, Gao Jingxiang, Yao Yifei. Solution for Robust Total Least Squares Estimation Based on Median Method[J]. Acta Geodaetica et Cartographica Sinica,2016, 45(3): 297-301 (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)
[2] Golub H G, Vanloan F C. An Analysis of the Total Least Squares Problem[J]. SIAM Journal on Numerical Analysis, 1980, 17(6): 883-893
[3] Schaffrin B, Felus Y A. On Total Least-Squares Adjustment with Constraints[J]. IAG-Symp, 2005, 128: 417-421
[4] Schaffrin B, Felus Y A. On the Multivariate Total Least-Squares Approach to Empirical Coordinate Transformations[J]. Journal of Geodesy, 2008, 82(6): 373-383
[5] 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
[6] 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
[7] Fang X. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Germany: Leibniz University Hannover, 2011
[8] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749
[9] Deng Caihua, Zhou Yongjun, Zhu Jianjun, et al. General Weighted Total Least Squares Method for a Type of New Functional Model[J]. Journal of China University of Mining and Technology, 2015, 44(5): 952-958 ( 邓才华, 周拥军, 朱建军, 等. 一类新函数模型及通用加权总体最小二乘平差方法[J]. 中国矿业大学学报, 2015, 44(5): 952-958)
[10] 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 Geo-desy, 2012, 86(8): 661-675
[11] Ou Jikun. Quasi-Accurate Detection of Gross Errors (QUAD)[J]. Acta Geodaetica et Cartographica Sinica, 1999, 28(1): 15-20 ( 欧吉坤. 粗差的拟准检定法(QUAD法)[J]. 测绘学报, 1999, 28(1): 15-20)
[12] Yang Y X. Robust Estimation of Geodetic Datum Transformation[J]. Journal of Geodesy, 1999, 73(5): 268-274
[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. 抗差估计最小二乘法[M]. 武汉: 华中理工大学出版社, 1997)
[14] Gao Jingxiang, Zhang Huahai, Yu Xuexiang. Study on Robust Estimation Model for Coordinate Transformation of GPS Network in Ming Areas[J]. Journal of China University of Mining and Technology, 1999, 28(2): 99-103(高井祥, 张华海, 余学祥. 矿区GPS网坐标转换的抗差模型[J]. 中国矿业大学学报, 1999, 28(2): 99-103)
[15] Chen Yi, Lu Jue. Peforming 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)
[16] Mahboub V, Amiri-Simkooei A R, Sharifi M A. Iteratively Reweighted Total Least Squares: A Robust Estimation in Errors-in-Variables Models[J]. Surv Rev, 2013, 45(329): 92-99
[17] 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)
[18] Lu J, Chen Y, Li B, et al. Robust Total Least Squares with Reweighting Iteration for Three-Dimensional Similarity Transformation[J]. Surv Rev, 2014, 46(334): 28-36
[19] Tao Y Q, Gao J X, Yao Y F. TLS Algorithm for GPS Height Fitting Based on Robust Estimation[J]. Surv Rev, 2014, 46(336): 184-188
[20] Wang B, Li J C, Liu C. A Robust Weighted Total Least Squares Algorithm and Its Geodetic Applications[J]. Studia Geophysica et Geodaetica, 2016, 60(2): 177-194
[21] Tao Yeqing, Gao Jingxiang, Yao Yifei. Solution for Robust Total Least Squares Estimation Based on Median Method[J]. Acta Geodaetica et Cartographica Sinica,2016, 45(3): 297-301 (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)