直线拟合的技术综合法及稳健性判据

TECHNICALLY HYBRID METHOD FOR LINEAR REGRESSION AND THE ASSESSMENT OF ROBUSTNESS

直线拟合常用的最小二乘法使残差平方和极小,然而稳健性(抗粗差性)较差,直线两端的数据误差对拟合斜率的影响大。最小一乘法使残差绝对值之和极小,但算法较复杂。本文提出技术综合方法,力图部分综合几种直线拟合方法的特点,不作粗差剔除,仅在求斜率时适度降低大于临界系数的个别数据的权重,以达到如下效果:稳健性显著优于最小二乘法,也优于最小一乘法;残差绝对值和、标准偏差近似与最小一乘法相同;计算过程比最小一乘法简单,适用范围广。

The least squares method is widely applied in linear regression, imposing a linear fit to a data set by minimizing the sum of the squared residuals. However, due to the squaring of the residuals, the robustness against the existence of the outliers is not very high and the data points at both ends affect greatly the slope parameter particularly when the residues of these data points are large. The least absolute residuals method, on the other hand, minimizes the sum of the absolute residuals and is more robust because it gives equal weights to all residuals. However, quite numerical complication arises in the least absolute residuals method because it can’t be solved analytically. In this paper, the technically hybrid method(THM) is proposed. Instead of rejecting the outliers, the method sets properly reduced weights to the identified outliers. The advantages of this method include: 1) A linear regression fit is realized that is more robust than the least squares method and the least absolute residual method;2) The sum of the absolute residuals and the standard deviations of the regression is close to those of the least absolute residuals method;3) The computing procedure is much simpler than the least absolute residuals method, so it has wide range of applications.