基于Neumann级数的有偏估计方法  

Biased Estimation Method Based on Neumann Series

作　　者：杨秋伟 白志超 李翠红 YANG Qiuwei;BAI Zhichao;LI Cuihong(Department of Civil Engineering,Shaoxing University,508 West-Huancheng Road,Shaoxing 312000,China)

机构地区：[1]绍兴文理学院土木工程学院,浙江省绍兴市312000

出　　处：《大地测量与地球动力学》2020年第5期512-516,共5页Journal of Geodesy and Geodynamics

基　　金：国家自然科学基金(11202138)。

摘　　要：针对病态最小二乘问题,提出基于Neumann级数的有偏估计方法。由该方法建立的有偏估计公式,包含现有的最小二乘估计、岭估计和广义岭估计公式,建立有偏估计和无偏估计的本质联系,并可以据此引申出一系列新的有偏估计形式。在这一系列的有偏估计中,存在着比岭估计或广义岭估计更接近于真值的解。以一个病态方程组为例对该方法进行验证表明,当观测向量中含有误差时,由最小二乘估计所得结果和真值误差较大,而各级有偏估计的结果和真值更加接近,且二级和三级有偏估计结果比相应的岭估计更接近于真值。In this paper,we propose a biased estimation method based on Neumann series to solve the ill-conditioned least squares problem.The biased estimation formulas established by the proposed method can include the existing least squares estimation,ridge estimation and generalized ridge estimation formulas.Using the proposed method,we can establish the essential relationship between biased estimation and unbiased estimation.In addition,we can derive a series of new biased estimations from the proposed method.In the series of biased estimates,there are solutions that are closer to the true value than ridge estimates or generalized ridge estimates.An ill-conditioned system of equations is taken as an example to verify the proposed method.The results show that when the observation vectors contain noise,the errors between the results obtained by least squares estimation and the true values are very large.However,the results of biased estimates at all levels are closer to the true values.Moreover,the results of biased estimates at two and three levels are closer to the true values than the corresponding ridge estimates.It is shown that the method proposed in this paper provides a new way to solve the ill conditioned least squares problem.

关 键 词：病态方程 Neumann级数 最小二乘估计 岭估计 

分 类 号：P207[天文地球—测绘科学与技术]