Non-negative Least Squares Variance Component Estimation of Partial EIV Model

doi:10.11947/j.AGCS.2017.20160501

Abstract

Abstract: As an extended form of the errors-in-variables (EIV) model, and the weight matrix is easy to structure, the applicability is stronger when both non-random elements and random elements exist in the coefficient matrix. According to the inaccurate stochastic model in the Partial EIV model, the coefficient matrix and observation vector are used as a kind of data respectively. The least squares variance component estimation method based on Partial EIV model is used and the relevant calculated formulas and iterative algorithm are derived. Then the corresponded variance components are estimated. The non-negative least squares is used when the negative variance appears, then add constraint condition to correct the rand model, so the estimated parameters are more reasonable. The experiments show that the results obtained by the method of this paper and other methods are equivalent.

Key words: Partial EIV model, EIV model, least squares variance component estimation, non-negative least squares

CLC Number:

P228

WANG Leyang, WEN Guisen. Non-negative Least Squares Variance Component Estimation of Partial EIV Model[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(7): 857-865.

References

[1] GOLUB G H, VAN LOAN C F. An Analysis of the Total Least Squares Problem[J]. SIAM Journal on Numerical Analysis, 1980, 17(6): 883-893.
[2] 王乐洋. 基于总体最小二乘的大地测量反演理论及应用研究[J]. 测绘学报, 2012, 41(4): 629. WANG Leyang. Research on Theory and Application of Total Least Squares in Geodetic Inversion[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(4): 629.
[3] 王乐洋, 许才军. 总体最小二乘研究进展[J]. 武汉大学学报(信息科学版), 2013, 38(7): 850-856. WANG Leyang, XU Caijun. Progress in Total Least Squares[J]. Geomatics and Information Science of Wuhan University, 2013, 38(7): 850-856.
[4] 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.
[5] 王乐洋, 余航, 陈晓勇. Partial EIV模型的解法[J]. 测绘学报, 2016, 45(1): 22-29. DOI: 10.11947/j.AGCS.2016.20140560. WANG Leyang, YU Hang, CHEN Xiaoyong. An Algorithm for Partial EIV Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1): 22-29. DOI: 10.11947/j.AGCS.2016.20140560.
[6] 刘经南, 曾文宪, 徐培亮. 整体最小二乘估计的研究进展[J]. 武汉大学学报(信息科学版), 2013, 38(5): 505-512. LIU Jingnan, ZENG Wenxian, XU Peiliang. Overview of Total Least Squares Methods[J]. Geomatics and Information Science of Wuhan University, 2013, 38(5): 505-512.
[7] 王乐洋, 于冬冬. 病态总体最小二乘问题的虚拟观测解法[J]. 测绘学报, 2014, 43(6): 575-581. DOI: 10.13485/j.cnki.11-2089.2014.0091. WANG Leyang, YU Dongdong. Virtual Observation Method to Ill-posed Total Least Squares Problem[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(6): 575-581. DOI: 10.13485/j.cnki.11-2089.2014.0091.
[8] 王乐洋, 许才军, 鲁铁定. 病态加权总体最小二乘平差的岭估计解法[J]. 武汉大学学报(信息科学版), 2010, 35(11): 1346-1350. WANG Leyang, XU Caijun, LU Tieding. Ridge Estimation Method in Ill-posed Weighted Total Least Squares Adjustment[J]. Geomatics and Information Science of Wuhan University, 2010, 35(11): 1346-1350.
[9] 王乐洋, 于冬冬, 吕开云. 复数域总体最小二乘平差[J]. 测绘学报, 2015, 44(8): 866-876. DOI: 10.11947/j.AGCS.2015.20130701. WANG Leyang, YU Dongdong, LV Kaiyun. Complex Total Least Squares Adjustment[J]. Acta Geodaetica et Cartographica Sinica, 2015, 44(8): 866-876. DOI: 10.11947/j.AGCS.2015.20130701.
[10] FANG Xing. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749.
[11] FANG Xing. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Germany: Leibniz Universität Hannover, 2011.
[12] SNOW K. Topics in Total Least-squares Adjustment within the Errors-in-variables Model: Singular Cofactor Matrices and Prior Information[D]. Ohio: The Ohio State University, 2012.
[13] SCHAFFRIN B, WIESER A. On Weighted Total Least-Squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7): 415-421.
[14] WANG Leyang, XU Guangyu. Variance Component Estimation for Partial Errors-in-Variables Models[J]. Studia Geophysica et Geodaetica, 2016, 60(1): 35-55.
[15] 崔希璋, 於宗俦, 陶本藻, 等. 广义测量平差[M]. 武汉: 武汉大学出版社, 2005. CUI Xizhang, YU Zongchou, TAO Benzao, et al. Generalized Surveying Adjustment[M]. Wuhan: Wuhan University Press, 2005.
[16] XU Peiliang, LIU Jingnan. Variance Components in Errors-in-Variables Models: Estimability, Stability and Bias Analysis[J]. Journal of Geodesy, 2014, 88(8): 719-734.
[17] 刘志平, 张书毕. 方差-协方差分量估计的概括平差因子法[J]. 武汉大学学报(信息科学版), 2013, 38(8): 925-929. LIU Zhiping, ZHANG Shubi. Variance-covariance Component Estimation Method Based on Generalization Adjustment Factor[J]. Geomatics and Information Science of Wuhan University, 2013, 38(8): 925-929.
[18] 王乐洋, 许才军, 张朝玉. 一种确定联合反演中相对权比的两步法[J]. 测绘学报, 2012, 41(1): 19-24. WANG Leyang, XU Caijun, ZHANG Chaoyu. A Two-step Method to Determine Relative Weight Ratio Factors in Joint Inversion[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(1): 19-24.
[19] AMIRI-SIMKOOEI A. Least-squares Variance Component Estimation: Theory and GPS Applications[D]. Delft: Delft University of Technology, 2007.
[20] TEUNISSEN P J G, AMIRI-SIMKOOEI A R. Least-squares Variance Component Estimation[J]. Journal of Geodesy, 2008, 82(2): 65-82.
[21] AMIRI-SIMKOOEI A R. Application of Least Squares Variance Component Estimation to Errors-in-Variables Models[J]. Journal of Geodesy, 2013, 87(10-12): 935-944.
[22] AMIRI-SIMKOOEI A R. Non-negative Least-squares Variance Component Estimation with Application to GPS Time Series[J]. Journal of Geodesy, 2016, 90(5): 451-466.
[23] 赵俊, 郭建锋. 方差分量估计的通用公式[J]. 武汉大学学报(信息科学版), 2013, 38(5): 580-583, 588. ZHAO Jun, GUO Jianfeng. Auniversal Formula of Variance Component Estimation[J]. Geomatics and Information Science of Wuhan University, 2013, 38(5): 580-583, 588.
[24] VAN HUFFEL S, VANDEWALLE J. The Total Least Squares Problem: Computational Aspects and Analysis[M]. Philadelphia: Society for Industrial and Applied Mathematic, 1991.
[25] 曾文宪. 系数矩阵误差对EIV模型平差结果的影响研究[D]. 武汉: 武汉大学, 2013. ZENG Wenxian. Effect of the Random Design Matrix on Adjustment of An EIV Model and Its Reliability Theory[D]. Wuhan: Wuhan University, 2013.
[26] 许光煜. Partial EIV模型的总体最小二乘方法及应用研究[D]. 南昌: 东华理工大学, 2016. XU Guangyu. The Total Least Squares Method and Its Application of Partial Errors-in-variables Model[D]. Nanchang: East China University of Technology, 2016.
[27] LAWSON C L, HANSON R J. Solving Least Squares Problems[M]. Philadelphia: SIAM, 1995.
[28] FRANC V, HLAVÁČ V, NAVARA M. Sequential Coordinate-wise Algorithm for the Non-negative Least Squares Problem[C]//Proceedings of the 11th International Conference on Computer Analysis of images and Patterns. Berlin Heidelberg: Springer, 2005: 407-414.
[29] MAHBOUB V. On Weighted Total Least-Squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86(5): 359-367.
[30] AMIRI-SIMKOOEI A, JAZAERI S. Weighted Total Least Squares Formulated by Standard Least Squares Theory[J]. Journal of Geodetic Science, 2012, 2(2): 113-124.