病态加权总体最小二乘模型的正则化抗差解法  

Regularized Robust Solution for Ill-Posed Weighted Total Least Squares Model

作　　者：邹时林[1,2] 吴星 王奉伟 ZOU Shilin;WU Xing;WANG Fengwei(Faculty of Geomatics,East China University of Technology,Nanchang 330013,China;Institute of Survey and Design,East China University of Technology,Fuzhou 344000,China;College of Surveying and Geo-Informatics,Tongji University,Shanghai 200092,China)

机构地区：[1]东华理工大学测绘工程学院,南昌市330013 [2]东华理工大学勘察设计研究院,江西省344000 [3]同济大学测绘与地理信息学院,上海市200092

出　　处：《大地测量与地球动力学》2021年第11期1106-1110,1126,共6页Journal of Geodesy and Geodynamics

摘　　要：针对EIV模型系数阵病态且系数阵和观测值精度不同的情形,基于拉格朗日乘数法导出病态加权总体最小二乘模型的正则化解法,并证明已有的等权病态总体最小二乘模型的正则化解法是其特例。在此基础上,进一步提出基于中位数法的病态加权总体最小二乘模型的正则化抗差解法,并用第一类Fredholm积分方程和病态测边网两个算例验证算法的有效性。结果表明,受系数阵病态性以及粗差的影响,最小二乘解和总体最小二乘解精度较差,严重偏离真值;正则化解法在顾及系数阵和观测值误差的同时可有效削弱模型的病态性,其精度较最小二乘解和总体最小二乘解有所提升;而正则化抗差解法在正则化解的基础上,利用等价权函数重构权阵,能有效抵御粗差的影响,其精度最高。The coefficient matrix of the error-in-variables(EIV) model is ill-posed, and the precision of the coefficient matrix and the observation are not equal, so we derive the regularized solution of the ill-posed weighted total least squares model using the Lagrangian multiplier method. We prove the existing regularized solution of the ill-posed total least squares model to be a special case of the new model. On this basis, we propose the regularized robust solution based on the median method for ill-posed weighted total least squares model, and the effectiveness of the new algorithm is verified by two examples of the first kind Fredholm integral equation and ill-posed trilateration network. The results show that the least squares solution and the total least squares solution have poor accuracies and seriously deviate from the true value due to the influence of ill-posedness and the outliers of the model, while the accuracy of the regularized solution has been improved for weakening the ill-posedness of the model taking into account the errors of coefficient matrix and the observations. On the basis of regularized solution, the regularized robust solution reconstructs the weight matrix using the equivalent weight function, which can effectively resist the influence of gross error and has the highest accuracy.

关 键 词：病态模型 总体最小二乘法 正则化法 稳健估计 

分 类 号：P207[天文地球—测绘科学与技术]