多元总体最小二乘问题的牛顿解法  

A Newton Algorithm for Multivariate Total Least Squares Problems

作　　者：王乐洋[1,2,3] 赵英文[1] 陈晓勇[1,2,3] 臧德彦[1,2,3] 

机构地区：[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

出　　处：《测绘学报》2016年第4期411-417,424,共8页Acta Geodaetica et Cartographica Sinica

基　　金：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);东华理工大学研究生创新专项资金(DHYC-2015005);江西省研究生创新专项资金(YC2015-S266;YC2015-S267);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)~~

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。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.

关 键 词：多元总体最小二乘 牛顿法 协因数传播律 仿射变换 

分 类 号：P207[天文地球—测绘科学与技术]