总体最小二乘若干拓展研究及其在变形数据处理中的应用

The deformation data analysis and processing is the base of physical explanation and disaster prediction for the deformable body, and the higher requirement is put forward to the theory and method of data processing because of its high precision. According to the actual situation of deformation data processing model and based upon the total least squares criterion, the model parameters estimation method is investigated when considering the error of coefficient matrix. The research emphases of the project are total least squares algorithms for the problem with generalized weight or equality and inequality constraint condition, the model construction for the ill-posed situation, the choice of the optimum ridge parameter, the determination of variable truncation threshold value and the proposition of anti-error equivalent weight function for the errors of coefficient matrix and observation data. Take into account the independent variable error of linear regression, the relationship between the solutions of total least squares estimation and other criteria is demonstrated in theory, and the differences and relations between the ordinary least squares and total least squares estimates is also demonstrated, then the suitability of the total least squares is evaluated. At last, the theories of the project will be used in the deformation data analysis and processing, and its effectiveness is to be verified. A more rigorous parameter estimation method for the deformation data processing will be given by the project research, and the theoretical basis for surveying data processing with similar problems will be laid.

变形观测数据处理是变形体物理解释和灾害预报的基础，其高精度的特点对数据处理理论和方法提出了更高的要求。本课题根据变形观测数据处理模型的实际情况，基于总体最小二乘准则，探讨适用于同时顾及系数矩阵误差的模型参数估计方法。重点研究一般加权、附有等式和不等式约束条件的总体最小二乘解算方法，病态情形下的模型构建及其最优岭参数选择和可变截断阀值的确定，并且提出构建系数矩阵误差与观测误差的抗差等价权函数；以研究线性回归中顾及自变量误差为基础，在理论上论证总体最小二乘估计与其他准则解的关系，论证和普通最小二乘估计的区别与联系，进而对总体最小二乘的适宜性进行评价。最后将理论研究成果应用于测量数据实践，验证其有效性。本课题的研究将为变形观测数据处理提供一种更严密的参数估计方法，同时为测绘数据处理类似问题奠定理论基础。

变形观测数据处理是变形体物理解释和灾害预报的基础，其高精度的特点对数据处理理论和方法提出了更高的要求。项目基于总体最小二乘准则，探讨适用于同时顾及系数矩阵误差的模型参数估计方法。重点研究了混合总体最小二乘的解算算法，推导了相关解算公式和精度评定；针对平差时 EIV 模型中所含粗差的问题，给出了 EIV 模型的粗差探测法；选择PEIV模型加权总体最小二乘，引入两个备选假设下的可靠性理论，探讨了总体最小二乘中粗差的可区性；研究了稳健总体最小二乘Helmert方差分量估计；推导了部分不确定性平差模型的直接迭代算法，并讨论了迭代不收敛时的算法；推导给出了测量中几种常用模型计算问题，诸如球面拟合的改进总体最小二乘算法，空间直线拟合的PEIV模型的解算，圆曲线拟合的PEIV模型解算，球面函数的总体最小二乘解算，F-范数的不确定性平差模型的解算等。本课题的研究将为变形观测数据处理提供一种更严密的参数估计方法，同时为测绘数据处理类似问题奠定理论基础。项目资助发表学术论文24篇，培养硕士研究生7名，其中6人已经获得学位，1人在读。