系数矩阵误差对最小二乘估计的影响及EIV模型的可靠性理论研究

The estimate of the EIV（errors-in-variables）model，in which the design matrix and observation vector are both random, is a nonlinear adjustment. It's accuracy of estimate,efficency of algorithm and the reliability of adjustment are the difficult problems in modern theory and method of surveying data processing.Although total least squares (TLS) is more rigorous than the least squares(LS) method to estimate the EIV model, it is computationally much more complicated than the LS method, even failed calculating for large amount of calculations. For some EIV problems, the TLS and LS methods have been shown to produce practically negligible differences in the estimated parameters. In order to understand under what conditions we can safely use the easy LS method to estimate the EIV model, the effects of random design matrix on the LS method is a foundamental theoretical problem of an EIV model.In this project, we will systematically investigate the effect of the random errors of the design matrix on the estimated quantities of geodetic interest, in particular, the model parameters, the variance-covariance matrix of the estimated parameters and the variance of unit weight. By using the derived formulae of bias, we can then attempt to remove the effect of the random matrix from the LS estimate and accordingly obtain the bias-corrected LS estimate for the EIV model. As the published theory of geodetic reliability of EIV model can't be used pratically, we will develop the theory and method of reliability of EIV model in the general weighted case, starting from a partial EIV model.Our research have important theoretical contribution to the adjustment of the data processing.

EIV（errors-in-variables）模型描述了系数矩阵和观测向量均含随机误差的平差模型，模型的非线性特性导致其估计理论一直是现代测量数据处理领域的难点问题。EIV模型的整体最小二乘估计比最小二乘估计理论上更为严密，但算法的复杂度远高于最小二乘估计，是制约其应用的主要因素，甚至因计算量过大导致无法解算。实际应用表明，某些EIV模型的最小二乘和整体最小二乘估计结果差异很小可忽略不计。因此，确定在哪些情况下可用简单的最小二乘求解是EIV模型估计理论的基本问题之一。本项目拟系统地研究系数矩阵误差引起的最小二乘参数估值和改正数的偏差及其对参数精度、单位权方差影响的估计公式，并提出EIV模型偏差改正的最小二乘估计新算法。同时，顾及现有EIV模型可靠性理论无法用于实际计算等缺陷，建立了普遍适用的EIV模型的可靠性理论体系。研究成果对测量平差数据处理有重要理论贡献和应用价值。

EIV（errors-in-variables）模型的系数矩阵和观测向量均含随机误差，模型的非线性特性导致其估计理论一直是现代测量数据处理领域的难点问题。EIV模型的整体最小二乘估计比最小二乘估计理论上更为严密，但算法的复杂度远高于最小二乘估计。实际应用表明，某些EIV模型的最小二乘和整体最小二乘估计结果差异很小可忽略不计。因此，确定在哪些情况下可用简单的最小二乘求解是EIV模型估计理论的基本问题之一。针对这一问题极为有限的研究成果，本项目开展了系数矩阵误差对EIV模型最小二乘估计的影响以及EIV模型的可靠性研究，研究内容和成果包括：（1）基于数理统计、矩阵微分和最优估计理论，全面系统地推导了系数矩阵误差引起的EIV模型最小二乘参数估计值与改正数的偏差公式，系数矩阵误差对单位权中误差和估计量的方差协方差阵的影响公式等，为EIV模型估计方法的选择提供了理论依据；（2）提出了EIV模型偏差改正的最小二乘估计方法，利用上述理论确定EIV模型的最小二乘估计能够满足精度要求的情况下，采用偏差改正的最小二乘估计方法可以减弱或消除估计量的偏差；（3）以Partial-EIV模型为基础，通过将系数矩阵的随机量提取出来作为虚拟观测值，从而将EIV模型从形式上转换为经典高斯马尔科夫模型，再根据可靠性理论推导了EIV模型的内部可靠性和外部可靠性公式；（4）将项目理论成果在大地测量、回归分析以及地价评估等领域进行了应用拓展。研究成果完善了EIV模型估计的理论体系，在测绘及相关领域具有应用前景，对EIV模型估计理论具有重要理论贡献和应用价值。受本基金项目资助，项目组共发表SCI和EI相关研究论文15篇，其中SCI论文8篇、EI论文4篇，项目申请人作为第一作者或者通讯作者发表SCI论文4篇以及EI论文3篇。