多元总体最小二乘问题的牛顿解法

doi:10.11947/j.AGCS.2016.20150246

测绘学报 ›› 2016, Vol. 45 ›› Issue (4): 411-417.doi: 10.11947/j.AGCS.2016.20150246

多元总体最小二乘问题的牛顿解法

王乐洋^1,2,3, 赵英文¹, 陈晓勇^1,2,3, 臧德彦^1,2,3

1. 东华理工大学测绘工程学院, 江西南昌 330013;
2. 流域生态与地理环境监测国家测绘地理信息局重点实验室, 江西南昌 330013;
3. 江西省数字国土重点实验室, 江西南昌 330013

收稿日期:2015-05-11 修回日期:2015-10-12 出版日期:2016-04-20 发布日期:2016-04-28
作者简介:王乐洋(1983-),男,博士,副教授,研究方向为大地测量反演及大地测量数据处理。
基金资助:
国家自然科学基金(41204003;41161069;41304020;41464001);测绘地理信息公益性行业科研专项(201512026);江西省自然科学基金(20151BAB203042);江西省教育厅科技项目(GJJ150595;KJLD12077;KJLD14049);流域生态与地理环境监测国家测绘地理信息局重点实验室开放基金(WE2015005);东华理工大学博士科研启动金(DHBK201113);江西省研究生创新专项资金(YC2015-S266;YC2015-S267);东华理工大学研究生创新专项资金(DHYC-2015005);对地观测技术国家测绘地理信息局重点实验室开放基金(K201502)

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)

摘要/Abstract

摘要： 为提高多元总体最小二乘问题参数估值的解算效率,推导了基于牛顿法的多元加权总体最小二乘算法;分析比较了基于牛顿法的多元加权总体最小二乘解和基于拉格朗日乘数法多元加权总体最小二乘解之间的关系,根据协因数传播律给出了多元总体最小二乘平差的16种协因数阵的近似计算公式。新算法能够解决观测矩阵和系数矩阵元素具有相关性的问题,并且可以把观测矩阵和系数矩阵的随机元素和常数元素纳入到一个协因数阵中进行处理。算例结果表明,本文提出的多元总体最小二乘问题的牛顿解法可行且收敛速度更快。

关键词: 多元总体最小二乘, 牛顿法, 协因数传播律, 仿射变换

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

中图分类号:

P207

王乐洋, 赵英文, 陈晓勇, 臧德彦. 多元总体最小二乘问题的牛顿解法[J]. 测绘学报, 2016, 45(4): 411-417.

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.

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多元总体最小二乘问题的牛顿解法

A Newton Algorithm for Multivariate Total Least Squares Problems

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