摘要
利用截断奇异值解法处理了病态加权总体最小二乘模型,详细推导了参数的截断奇异值解及其偏差、方差以及均方误差公式,该算法无需迭代求解,易于实现。将截断奇异值解的均方误差与最小二乘解的方差进行比较,发现当奇异值由大到小依次变化时,截掉奇异值所造成的解的均方误差下降量的符号由负逐渐变正,由此导出了确定截断参数的公式。数值算例和病态测边网算例分析结果表明,受模型病态性的影响,最小二乘解和总体最小二乘解的精度较差;截断奇异值解能够有效地削弱模型的病态性同时又顾及了系数阵的误差,其解的精度最高。
The truncated singular value decomposition method is used to solve the ill-posed weighted total least squares model.The truncated singular value solution and its bias,variance and mean square error are derived in detail and the new algorithm is easy to implement without computing iteratively.Comparing the mean square error of the truncated singular value solution with the variance of the least squares solution,it can be found that the sign of the decrease of the mean square error of the solution caused by truncating the singular value changes from negative to positive gradually when the singular value changes in descending,from which the formula for determining the truncated parameter is derived.The numerical example and ill-posed trilateration network example are used to verify the correctness and validity of the new algorithm,the results show that the solutions of least squares and total least squares differ greatly from the true value due to the ill-posedness of the model,while the solution of truncated singular value decomposition is more accurate than the other two solutions for that the new algorithm can effectively weaken the ill-posedness of the model and take into account the error of the coefficient matrix.
作者
刘云彤
陈琳
何宽
秦奋
LIU Yuntong;CHEN Lin;HE Kuan;QIN Fen(College of Environment and Planning, Henan University, Kaifeng 475004, China;Yellow River Water Conservancy Technical Institute, Kaifeng 475004, China)
出处
《测绘科学技术学报》
北大核心
2020年第2期111-116,共6页
Journal of Geomatics Science and Technology
关键词
病态模型
总体最小二乘
测边网
正则化
截断奇异值
ill-posed model
total least squares
trilateration network
regularization
truncated singular value decomposition
