PEIV模型WTLS估计的Fisher-Score算法
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摘要
考虑系数矩阵含非随机元素和不同位置含相同随机元素的结构化特征,PEIV (partial errors-in-variables)模型较一般的EIV模型更为严格。现有PEIV模型加权整体最小二乘(weighted total least squares,WTLS)估计算法需多次迭代,影响计算效率。通过利用观测值误差和系数矩阵误差的统计性质构造非线性目标函数,并以此推导了新的PEIV模型WTLS估计的计算公式,同时设计了相应的Fisher-Score算法。算例分析结果表明,相比较而言,Fisher-Score算法迭代次数较少,计算效率得到大大提升。
Considering the non-random elements and the same random elements in different locations of coefficient matrix, the partial errors-in-variables(PEIV) model is stricter than general EIV model. However, the existing weighted total least squares(WTLS) algorithm for PEIV model requires more iteration so that the computation efficiency is reduced. Making using of statistical properties of observations errors and coefficient matrix error, this paper deduces a new computational formula of WTLS estimation for PEIV model based on Fisher-Score algorithm. The numerical results show that the Fisher-Score algorithm takes less iterations and the computation efficiency is enormously promoted.
Considering the non-random elements and the same random elements in different locations of coefficient matrix, the partial errors-in-variables(PEIV) model is stricter than general EIV model. However, the existing weighted total least squares(WTLS) algorithm for PEIV model requires more iteration so that the computation efficiency is reduced. Making using of statistical properties of observations errors and coefficient matrix error, this paper deduces a new computational formula of WTLS estimation for PEIV model based on Fisher-Score algorithm. The numerical results show that the Fisher-Score algorithm takes less iterations and the computation efficiency is enormously promoted.
引文
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[3] Shen Y Z, Li B F, Chen Y. An Iterative Solution of Weighted Total Least-Squares Adjustment[J]. Journal of Geodesy, 2011, 85:229-238
[4] Mahboub V. On Weighted Total Least-Squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86:359-367
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[7] 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, 86:661-675
[8] Zeng Wenxian. Effect of the Random Design Matrix on Adjustment of an EIV Model and Its Reliability Theory[D]. Wuhan:Wuhan University, 2013(曾文宪. 系数矩阵误差对EIV模型平差结果的影响分析研究[D]. 武汉:武汉大学,2013)
[9] Zeng Wenxian, Fang Xing, Liu Jingnan, et al. Weighted Total Least Squares Algorithm with Inequality Constraints[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(10):1 013-1 018(曾文宪,方兴,刘经南,等. 附有不等式约束的加权整体最小二乘算法[J]. 测绘学报,2014,43/(10):1 013-1 018)
[10] Zeng W X, Liu J N, Yao Y B. On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J]. Journal of Geodesy, 2015, 89:111-119
[11] Xu P L, Liu J N. Variance Components in Errors-in-Variables Models: Estimability, Stability and Bias Analysis[J]. Journal of Geodesy, 2014, 88:719-734
[12] Xu P L, Liu J N, Zeng W X, et al. Effects of Errors-in-Variables on Weighted Least Squares Estimation[J]. Journal of Geodesy, 2014, 88:705-716
[13] Wang L Y, Xu G X. Variance Component Estimation for Partial Errors-in-Variables Models[J]. Stuid Geophysica et Geodaetica, 2016, 60(1):35-55
[14] Zhao J,Gui Q M. Outlier Detection in Partial Errors-in-Variables Model[J]. Boletim de Ciencias Geodesicas, 2017, 23(1):1-20
[15] Zhao J, Gui Q M, Guo F X. L1 Norm Minimization in Partial Errors-in-Variables Model[J]. Acta Geodaetica et Geophysica, 2017,52(3):389-406
[16] Wang Songgui, Shi Jianhong, Ying Suju, et al. An Introduction of Linear Model[M]. Beijing: Science Press, 2004(王松桂, 史建红, 尹素菊, 等. 线性模型引论[M]. 北京:科学出版社,2004)
[17] Magnus J R, Neudecker H. Matrix Differential Calculus with Applications[M].3rd ed. England:John Wiley & Sons, 2007
[18] Yao Yibin, Huang Shuhua, Zhang Liang,et al. A New Method of TLS for Solving the Parameters of Three-dimensional Coordinate Transformation[J]. Geomatics and Information Science of Wuhan University, 2015, 40(7): 853-857(姚宜斌, 黄书华, 张良,等. 求解三维坐标转换参数的整体最小二乘新方法[J]. 武汉大学学报·信息科学版, 2015, 40(7): 853-857)
[2] Neitzel F. Generalization of Total Least-Squares on Example of Unweighted and Weighted 2D Similarity Transfromation[J]. Journal of Geodesy, 2010, 84:751-762
[3] Shen Y Z, Li B F, Chen Y. An Iterative Solution of Weighted Total Least-Squares Adjustment[J]. Journal of Geodesy, 2011, 85:229-238
[4] Mahboub V. On Weighted Total Least-Squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86:359-367
[5] 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
[6] Fang X. Weighted Total Least-Squares:Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87:733-749
[7] 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, 86:661-675
[8] Zeng Wenxian. Effect of the Random Design Matrix on Adjustment of an EIV Model and Its Reliability Theory[D]. Wuhan:Wuhan University, 2013(曾文宪. 系数矩阵误差对EIV模型平差结果的影响分析研究[D]. 武汉:武汉大学,2013)
[9] Zeng Wenxian, Fang Xing, Liu Jingnan, et al. Weighted Total Least Squares Algorithm with Inequality Constraints[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(10):1 013-1 018(曾文宪,方兴,刘经南,等. 附有不等式约束的加权整体最小二乘算法[J]. 测绘学报,2014,43/(10):1 013-1 018)
[10] Zeng W X, Liu J N, Yao Y B. On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J]. Journal of Geodesy, 2015, 89:111-119
[11] Xu P L, Liu J N. Variance Components in Errors-in-Variables Models: Estimability, Stability and Bias Analysis[J]. Journal of Geodesy, 2014, 88:719-734
[12] Xu P L, Liu J N, Zeng W X, et al. Effects of Errors-in-Variables on Weighted Least Squares Estimation[J]. Journal of Geodesy, 2014, 88:705-716
[13] Wang L Y, Xu G X. Variance Component Estimation for Partial Errors-in-Variables Models[J]. Stuid Geophysica et Geodaetica, 2016, 60(1):35-55
[14] Zhao J,Gui Q M. Outlier Detection in Partial Errors-in-Variables Model[J]. Boletim de Ciencias Geodesicas, 2017, 23(1):1-20
[15] Zhao J, Gui Q M, Guo F X. L1 Norm Minimization in Partial Errors-in-Variables Model[J]. Acta Geodaetica et Geophysica, 2017,52(3):389-406
[16] Wang Songgui, Shi Jianhong, Ying Suju, et al. An Introduction of Linear Model[M]. Beijing: Science Press, 2004(王松桂, 史建红, 尹素菊, 等. 线性模型引论[M]. 北京:科学出版社,2004)
[17] Magnus J R, Neudecker H. Matrix Differential Calculus with Applications[M].3rd ed. England:John Wiley & Sons, 2007
[18] Yao Yibin, Huang Shuhua, Zhang Liang,et al. A New Method of TLS for Solving the Parameters of Three-dimensional Coordinate Transformation[J]. Geomatics and Information Science of Wuhan University, 2015, 40(7): 853-857(姚宜斌, 黄书华, 张良,等. 求解三维坐标转换参数的整体最小二乘新方法[J]. 武汉大学学报·信息科学版, 2015, 40(7): 853-857)