部分变量误差模型的整体抗差最小二乘估计

摘要

The weighted total least-squares (WTLS)estimate for the partial errors-in-variables (EIV) model is very susceptible to outliers.Because the observations and coefficient matrix in the partial EIV model may be contaminated with outliers simultaneously,a totalrobustified least squares (TRLS) estimation for the partial EIV model is proposed by combining a two-step iterated algorithm of the WTLS estimate with the equivalent weight method of robust M-estimation.And the uniformly most powerful test statistics are constructed to determine the down-weighting factors.For the characteristics of the two-step iterated method,two different down-weighting schemes are presented.In the first scheme down-weighting is only implemented for the coefficient matrix and not for the observations when some elements of the coefficient matrix are estimated,and the second scheme is contrary.A simulated two-dimensional affine transformation and a linear fitting with real data are analyzed.The results show that the TRLS with the first scheme is superior to one with the second scheme,and it outperforms the existing robust methods with residual and posterior estimate of variance of unit weight and existing robust methods for the general EIV model.%部分变量误差模型(partial EIV model)的加权整体最小二乘(weighted total least-squares,WTLS)估计不具备抵御粗差的能力.鉴于粗差可能同时出现在观测值和系数矩阵中,本文在提出部分变量误差模型WT L S估计的两步迭代解法的基础上,运用抗差M估计的等价权方法,发展了一种整体抗差最小二乘(TRLS)估计方法,并采用一致最大功效统计量确定降权因子.针对WTLS估计两步迭代解法的特点,设计了两个不同的降权方案:第1个方案是在估计系数矩阵元素时,不对观测值降权,仅对系数矩阵降权;第2个方案是在估计系数矩阵元素时,既对系数矩阵降权,同时也对观测值降权.通过对模拟2D仿射变换和线性拟合实例进行计算和分析,结果表明第1方案优于第2方案,并且优于基于残差和验后单位权方差的抗差估计和现有的变量误差模型抗差估计.