部分变量误差模型(partial EIV model)的加权整体最小二乘(weighted total least-squares, WTLS)估计不具备抵御粗差的能力。鉴于粗差可能同时出现在观测值和系数矩阵中,本文在提出部分变量误差模型WTLS估计的两步迭代解法的基础上,运用抗差M估计的等价权方法,发展了一种整体抗差最小二乘(TRLS)估计方法,并采用一致最大功效统计量确定降权因子。针对WTLS估计两步迭代解法的特点,设计了两个不同的降权方案:第1个方案是在估计系数矩阵元素时,不对观测值降权,仅对系数矩阵降权;第2个方案是在估计系数矩阵元素时,既对系数矩阵降权,同时也对观测值降权。通过对模拟2D仿射变换和线性拟合实例进行计算和分析,结果表明第1方案优于第2方案,并且优于基于残差和验后单位权方差的抗差估计和现有的变量误差模型抗差估计。
The weighted total least-squares (WTLS) estimate for the partial errors-in-variables (EIV) model is very susceptible to outliers. Because the observations and coefficient matrix in the partial EIV model may be contaminated with outliers simultaneously, a total robustified least squares (TRLS) estimation for the partial EIV model is proposed by combining a two-step iterated algorithm of the WTLS estimate with the equivalent weight method of robust M-estimation. And the uniformly most powerful test statistics are constructed to determine the down-weighting factors. For the characteristics of the two-step iterated method, two different down-weighting schemes are presented. In the first scheme down-weighting is only implemented for the coefficient matrix and not for the observations when some elements of the coefficient matrix are estimated, and the second scheme is contrary. A simulated two-dimensional affine transformation and a linear fitting with real data are analyzed. The results show that the TRLS with the first scheme is superior to one with the second scheme, and it outperforms the existing robust methods with residual and posterior estimate of variance of unit weight and existing robust methods for the general EIV model.
[1] FANG X. Weighted Total Least Squares Solutions for Application in Geodesy[D]. Hanover: Leibniz University Hanover, 2011.
[2] SNOW K B. Topics in Total Least-squares Adjustment within the Errors-in-variables Model: Singular Cofactor Matrices and Prior Information[D]. Ohio: Ohio State University, 2012.
[3] SCHAFFRIN B, WIESER A. On Weighted Total Least-squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7): 415-421.
[4] SHEN Yunzhong, LI Bofeng, CHEN Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238.
[5] MAHBOUB V. On Weighted Total Least-squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86(5): 359-367.
[6] AMIRI-SIMKOOEI A R, JAZAERI S. Weighted Total Least Squares Formulated by Standard Least Squares Theory[J]. Journal of Geodetic Science, 2012, 2(2): 113-124.
[7] FANG Xing. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749.
[8] JAZAERI S, AMIRI-SIMKOOEI A R, SHARIFI M A. Iterative Algorithm for Weighted Total Least Squares Adjustment[J]. Survey Review, 2014, 46(334): 19-27.
[9] XU Peiliang,LIU Jingnan,SHI Chuang.Total Least Squares Adjustment in Partial Errors-in-variables Models: Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8): 661-675.
[10] SHI Yun, XU Peiliang, LIU Jingnan, et al. Alternative Formulae for Parameter Estimation in Partial Errors-in-variables Models[J]. Journal of Geodesy, 2015, 89(1): 13-16.
[11] ZENG Wenxian, LIU Jingnan, YAO Yibin. On Partial Errors-in-variables Models with Inequality Constraints of Parameters and Variables[J]. Journal of Geodesy, 2015, 89(2): 111-119.
[12] SCHAFFRIN B, UZUN S. Errors-in-variables for Mobile Mapping Algorithms in the Presence of Outliers[J]. Archives of Photogrammetry, Cartography and Remote Sensing, 2011, 22: 377-387.
[13] SCHAFFRIN B, UZUN S. On the Reliability of Errors-in-variables Models[J]. Acta et Commentationes Universitatis Tartuensis De Mathematica, 2012, 16(1): 69-81.
[14] 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. CHENG Yi, LU Jue. Performing 3D Similarity Transformation by Robust Total Least Squares[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(5): 715-722.
[15] MAHBOUB V, AMIRI-SIMKOOEI A R, SHARIFI M A. Iteratively Reweighted Total Least Squares: A Robust Estimation in Errors-in-variables Models[J]. Survey Review, 2013, 45(329): 92-99.
[16] LU J, CHEN Y, LI B F, FANG X. Robust Total Least Squares with Reweighting Iteration for Three-dimensional Similarity Transformation[J]. Survey Review, 2014, 46(334): 28-36.
[17] TAO Y Q, GAO J X, YAO Y F. TLS Algorithm for GPS Height Fitting Based on Robust Estimation[J]. Survey Review, 2014, 46(336): 184-188.
[18] ROUSSEEUW P J, LEROY A M. Robust Regression and Outlier Detection[M]. New York: John Wiley & Sons, 1987.
[19] KOCH K R. Parameter Estimation and Hypothesis Testing in Linear Models[M]. 2nd ed. New York: Springer, 1999.
[20] HUBER P J. Robust Statistics[M]. New York: John Wiley & Sons, 1981.
[21] 周江文. 经典误差理论与抗差估计[J]. 测绘学报, 1989, 18(2): 115-120. ZHOU Jiangwen. Classical Theory of Errors and Robust Estimation[J]. Acta Geodaetica et Cartographica Sinica, 1989, 18(2): 115-120.
[22] YANG Y, SONG L, XU T. Robust Estimator for Correlated Observations Based on Bifactor Equivalent Weights[J]. Journal of Geodesy, 2002, 76(6-7): 353-358.
[23] YANG Y. Robust Estimation for Dependent Observations[J]. Manuscripta Geodaetica, 1994, 19(1): 10-17.
[24] YANG Y. Robust Estimation of Geodetic Datum Transformation[J]. Journal of Geodesy, 1999, 73(5): 268-274.
[25] XU Peiliang. 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.
[26] GUO Jianfeng, OU Jikun, WANG Haitao. Robust Estimation for Correlated Observations: Two Local Sensitivity-Based Downweighting Strategies[J]. Journal of Geodesy, 2010, 84(4): 243-250.
[27] 黄维彬. 近代平差理论及其应用[M]. 北京: 解放军出版社, 1992. HUANG Weibin. Theory and Applications of Modern Adjustment[M]. Beijing: PLA Publishing House, 1992.
[28] 郭建锋, 赵俊. 粗差探测与识别统计检验量的比较分析[J]. 测绘学报, 2012, 41(1): 14-18. GUO Jianfeng, ZHAO Jun. Comparative Analysis of Statistical Tests Used for Detection and Identification of Outliers[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(1): 14-18.
[29] KELLY G. The Influence Function in the Errors in Variables Problem[J]. The Annals of Statistics, 1984, 12(1): 87-100.
