摘要
线性回归分析中,一般最小二乘回归的目标函数只考虑一个方向的扰动,采用基于几何距离的正交回归能克服固定单方向最优带来的拟合稳定性差的弊端。本文分析和比较了正交回归和一般最小二乘回归的误差,并定量地给出了两者的几何误差与原始数据的方差、相关系数之间的关系,指出正交回归的几何误差小于一般最小二乘回归,并且正交回归具有旋转不变性。最后,以平面直线拟合为例验证了这个结论。
In linear regression analysis, the objective function of the ordinary least squares regression only takes the uncertainty of single variable into account, while the objective function in orthogonal regression or the regression based on geometric distance can overcome the ill stability incurred in the ordinary regression caused by single variable optimization. In this paper, the errors are analyzed and compared between orthogonal regression and ordinary least squares regression, which are expressed as the functions of raw data＇s covariances and correlation coefficients. It indicates that the error in the orthogonal regression is less than that in ordinary least squares regression. Furthermore, it is rotationally invariant. The results are proved by the 2D line fitting examples.
出处
《数理统计与管理》
CSSCI
北大核心
2010年第2期248-253,共6页
Journal of Applied Statistics and Management
关键词
正交回归
最小二乘法
几何误差
直线拟合
几何距离
orthogonal regression, least squares algorithm, geometric error, line fitting, geometric distance
