顾及自变量与因变量误差及相关性的线性回归

提出一种顾及自变量和因变量观测误差及误差相关性的线性回归新方法,并导出了求解线性回归系数的迭代公式.以一元线性回归为例,导出了与最小二乘回归系数表达形式类似的解析解,并揭示了新方法与最小二乘方法的本质区别.此外,对于含有多个自变量的多元线性回归,给出了相应的同时考虑自变量和因变量观测误差及误差相关性的回归系数求解方法.试验表明,当自变量是非随机变量时,新方法与最小二乘方法的回归效果相同;当因变量和自变量都是随机变量(自变量与因变量的观测误差相关或不相关)时,新方法的回归系数比最小二乘方法的回归系数更加接近实际值.

Linear Regression with Corrected Errors of Independent and Dependent Variables

Linear regression is aiming to establish a linear relationship between independent and dependent variables, and afterward to predict dependent variables or control the independent variables by the built relationship. The key of regression is to calculate the regression coefficients. Traditional least-squares linear regression method only considers errors of dependent variables. However, in real applications, both dependent and independent variables are observations and inevitably contaminated by random errors and even correlated errors. This paper presented a new linear regression method where the errors of dependent and independent variables and correlations of errors were adequately captured. The iteration formulae for calculating the regression parameters were derived at the same time. Taking univariate linear regression problem as an example, analytical formulas for linear regression parameters that similar to those from least-squares method were derived, with which the essential difference between least-squares method and new method were demonstrated. In addition, for the multiple linear regression that with multiple independent variables, the corresponding method, which considers the errors of both independent and dependent variables and the correlations of errors, for calculating the linear regression parameters were also shown. The experiment results shown that the new method and least-squares method were equivalent to each other when independent variables were non-random; whereas, the regressive parameters from new method were more closer to the true values than those from the least-squares method when both independent and dependent variables were all random (no matter their errors were correlated or not).