A Newton Algorithm for Multivariate Total Least Squares Problems

doi:10.11947/j.AGCS.2016.20150246

Acta Geodaetica et Cartographica Sinica ›› 2016, Vol. 45 ›› Issue (4): 411-417.doi: 10.11947/j.AGCS.2016.20150246

Previous Articles Next Articles

A Newton Algorithm for Multivariate Total Least Squares Problems

WANG Leyang^1,2,3, ZHAO Yingwen¹, CHEN Xiaoyong^1,2,3, ZANG Deyan^1,2,3

1. Faculty of Geomatics, East China University of Technology, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, NASG, Nanchang 330013, China;
3. Jiangxi Province Key Lab for Digital Land, Nanchang 330013, ChinaAbstract

Received:2015-05-11 Revised:2015-10-12 Online:2016-04-20 Published:2016-04-28
Supported by:
The National Natural Science Foundation of China(Nos.41204003;41161069;41304020;41464001);The National Department Public Benefit Research Foundation(Surveying,Mapping and Geoinformation)(No.201512026);The Natural Science Foundation of Jiangxi Province(No.20151BAB203042);The Science and Technology Project of the Education Department of Jiangxi Province(Nos.GJJ150595;KJLD12077;KJLD14049);The Found of Key Laboratory of Watershed Ecology and Geographical Environment Monitoring(No.WE2015005);The Scientific Research Foundation of ECIT(No.DHBK201113);Innovation Fund Designated for Graduate Students of Jiangxi Province(Nos.YC2015-S266;YC2015-S267);Innovation Fund Designated for Graduate Students of ECIT(No.DHYC-2015005);The Found of the Key Laboratory of Mapping from Space, NASG(No.K201502)

Abstract

Abstract: In order to improve calculation efficiency of parameter estimation, an algorithm for multivariate weighted total least squares adjustment based on Newton method is derived. The relationship between the solution of this algorithm and that of multivariate weighted total least squares adjustment based on Lagrange multipliers method is analyzed. According to propagation of cofactor, 16 computational formulae of cofactor matrices of multivariate total least squares adjustment are also listed. The new algorithm could solve adjustment problems containing correlation between observation matrix and coefficient matrix. And it can also deal with their stochastic elements and deterministic elements with only one cofactor matrix. The results illustrate that the Newton algorithm for multivariate total least squares problems could be practiced and have higher convergence rate.

Key words: multivariate total least squares, Newton method, propagation of cofactor, affine transformation

CLC Number:

P207

WANG Leyang, ZHAO Yingwen, CHEN Xiaoyong, ZANG Deyan. A Newton Algorithm for Multivariate Total Least Squares Problems[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(4): 411-417.

References

[1] 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.
[2] 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.
[3] SPRENT P. Models in Regression and Related Topics[M]. London:Methuen & Co Ltd, 1969.
[4] 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, 878. WANG Leyang, XU Caijun. Progress in Total Least Squares[J]. Geomatics and Information Science of Wuhan University, 2013, 38(7):850-856, 878.
[5] VAN HUFFEL S, VANDEWALLE J. The Total Least Squares Problem:Computational Aspects and Analysis[D]. Philadelphia, Pennsylvania:SIAM, 1991.
[6] SCHAFFRIN B, FELUS Y A. Multivariate Total Least-squares Adjustment for Empirical Affine Transformations[C]//VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy:Challenge and Role of Modern Geodesy. Berlin:Springer, 2008, 132:238-242.
[7] 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.
[8] SCHAFFRIN B, WIESER A. Empirical Affine Reference Frame Transformations by Weighted Multivariate TLS Adjustment[C]//Geodetic Reference Frames. Berlin:Springer, 2009, 134:213-218.
[9] FELUS Y A, BURTCH R C. On Symmetrical Three-dimensional Datum Conversion[J]. GPS Solutions, 2009, 13(1):65-74.
[10] FANG X. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Hanover:Leibniz University of Hanover, 2011.
[11] 孙大双, 张友阳, 黄令勇, 等. 多元总体最小二乘在大旋转角三维坐标转换中的应用[J]. 测绘科学技术学报, 2014, 31(5):481-485. SUN Dashuang, ZHANG Youyang, HUANG Lingyong, et al. Application of Multivariate Total Least Squares Method in Three-dimension Coordinate Conversion with Large Rotation Angle[J]. Journal of Geomatics Science and Technology, 2014, 31(5):481-485.
[12] 黄令勇, 吕志平, 任雅奇, 等. 多元总体最小二乘在三维坐标转换中的应用[J]. 武汉大学学报(信息科学版), 2014, 39(7):793-798. HUANG Lingyong, LV Zhiping, REN Yaqi, et al. Application of Multivariate Total Least Square in Three-dimensional Coordinate Transformation[J]. Geomatics and Information Science of Wuhan University, 2014, 39(7):793-798.
[13] 钱承军, 陈义. 多变量总体最小二乘在点云拼接中的应用[J]. 测绘与空间地理信息, 2015, 38(1):67-69, 76. QIAN Chengjun, CHEN Yi. Application of Multivariable Total Least Squares in the Registration of Point Clouds[J]. Geomatics & Spatial Information Technology, 2015, 38(1):67-69, 76.
[14] 王乐洋, 许才军, 鲁铁定. 边长变化反演应变参数的总体最小二乘方法[J]. 武汉大学学报(信息科学版), 2010, 35(2):181-184. WANG Leyang, XU Caijun, LU Tieding. Inversion of Strain Parameter Using Distance Changes Based on Total Least Squares[J]. Geomatics and Information Science of Wuhan University, 2010, 35(2):181-184.
[15] XU Caijun, WANG Leyang, WEN Yangmao, et al. Strain Rates in the Sichuan-Yunnan Region Based upon the Total Least Squares Heterogeneous Strain Model from GPS Data[J]. Terrestrial Atmospheric and Oceanic Sciences, 2011, 22(2):133-147.
[16] 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, LÜ Kaiyun. Complex Total Least Squares Adjustment[J]. Acta Geodaetica et Cartographica Sinica, 2015, 44(8):866-876. DOI:10.11947/j.AGCS.2015.20130701.
[17] MAHBOUB V. On Weighted Total Least-squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86(5):359-367.
[18] 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.
[19] 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.
[20] BOYD S, VANDENBERGHE L. Convex Optimization[M]. Cambridge:Cambridge University Press, 2004.
[21] 王乐洋, 鲁铁定. 总体最小二乘平差法的误差传播定律[J]. 大地测量与地球动力学, 2014, 34(2):55-59. WANG Leyang, LU Tieding. Propagation Law of Errors in Total Least Squares Adjustment[J]. Journal of Geodesy and Geodynamics, 2014, 34(2):55-59.
[22] ZHOU Yongjun, KOU Xinjian, ZHU Jianjun, et al. A Newton Algorithm for Weighted Total Least-squares Solution to a Specific Errors-in-variables Model with Correlated Measurements[J]. Studia Geophysica et Geodaetica, 2014, 58(3):349-375.
[23] 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.

A Newton Algorithm for Multivariate Total Least Squares Problems

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 4

Recommended Articles

Metrics

Comments

[1]	ZHENG Maoteng, ZHOU Shunping, XIONG Xiaodong, ZHU Junfeng. GPU Parallel Bundle Block Adjustment [J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(9): 1193-1201.
[2]	ZHENG Maoteng, ZHANG Yongjun, ZHU Junfeng, XIONG Xiaodong, ZHOU Shunping. A Fast and Effective Block Adjustment Method with Big Data [J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(2): 188-197.
[3]	LI Bofeng. Variance Component Estimation in the Seamless Affine Transformation Model [J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1): 30-35.
[4]	Xue shuqiang Yuanxi Yang Yamin Dang. A Closed-form of Newton Iterative Formula for Nonlinear Adjustment of Distance Equations [J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(8): 771-777.