一种加权总体最小二乘问题的解法  

A method for weighted total least squares problem

作　　者：王乐洋[1,2] 余航[1] 李毅[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省数字国土重点实验室,江西南昌330013 [3]中国矿业大学力学与建筑工程学院,江苏徐州221116

出　　处：《中国矿业大学学报》2016年第6期1263-1270,共8页Journal of China University of Mining & Technology

基　　金：国家自然科学基金项目(41664001;41204003);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项目(WE2015005);对地观测技术国家测绘地理信息局重点实验室项目(K201502);江西省教育厅科技项目(GJJ150595);测绘地理信息公益性行业科研专项项目(201512026);东华理工大学博士科研启动基金项目(DHBK201113);东华理工大学研究生创新基金项目(DHYC-2015005);江西省杰出青年人才资助计划项目;国家重点研发计划项目(2016YFB0501405)

摘　　要：加权总体最小二乘(WTLS)方法是一种可同时顾及观测向量与系数矩阵误差的方法,其理论更为严密.基于已有总体最小二乘方法,推导了一种WTLS平差方法,该方法可顾及观测向量与系数矩阵中随机元素间的相关性;给出了一种可顾及相关观测情况下WTLS平差的新的等价平差准则,该准则具有与最小二乘(LS)平差准则相似的结构,进而可按照求解自由极值的方式进行推导,并将WTLS与LS平差思路进行了统一.针对推导过程中观测向量与系数矩阵组成的增广矩阵的协因数阵存在不可逆情况,构建的等价平差准则可避免直接对其求逆.通过直线拟合与二维仿射变换算例分析,结果表明:提出的方法能取得与已有平差方法相同的结果.并验证了该方法的正确性与可行性.As the weighted total least squares（WTLS）method considers errors both in the observation vector and the coefficient matrix simultaneously,its theory is more rigorous.Based on the existing total least squares method,a WTLS method was proposed in this paper,which takes into account a general cofactor matrix including the case of cross correlation between cofactor matrices of the observation vector and the coefficient matrix.A new equivalent objective function of the WTLS problem was also derived.It has a similar structure to the objective function of the least squares（LS）case,and relevant formulae can be obtained without using the Lagrange multipliers,which unifies the thought of WTLS and LS adjustment.During derivations,the proposed method can avoid using the inverse cofactor matrix of the augmented matrix of the observation vector and the coefficient matrix that may not necessarily exist in many problems.Through straight line fit and two-dimensional affirm transformation examples analysis,the results show that the proposed method can achieve the same results as the existingmethods,and proved to be correct and feasible.

关 键 词：变量含误差模型 相关观测 加权总体最小二乘平差 最小二乘平差 

分 类 号：P207[天文地球—测绘科学与技术]