总体最小二乘线性回归统一模型及解算

The unified model and algorithm of total least squares linear regression

对总体最小二乘线性回归问题进行了进一步的分析探讨,针对文献中提出的同时考虑因变量和自变量误差的条件平差模型和间接平差模型,通过理论分析和推导,当把因变量与自变量看成是等精度时其平差结果并未达到预定目的,而是与只考虑因变量误差的平差结果一致。同时,文献中所述平差模型都只限于一元线性回归,而未讨论多元线性回归的总体最小二乘问题。鉴于此,本文基于间接平差方法提出了一种总体最小二乘线性回归的统一模型,并推导了其具体解算方法,该模型同时考虑了因变量和自变量的误差且同样适用于多元线性回归。通过实例分析,说明了本文所述方法的正确性和合理性。

This paper further analyzes the problem of total least squares linear regression, especially the condition adjustment model and indirect adjustment model for simultaneous consideration of dependent variable error and independent variable error. Theoretical analysis dependent variables and independent variables are of equal precision, and deduction showed, ,when the the adjustment result didn t achieve the intended purpose. However, the result is consistent with the adjustment results in the case of only considering the dependent variable error. At the same time, the adjustment models described in the literature are limited to the simple linear regression, and failed to discuss the total least squares problem in multivariate linear regression. In view of this, based on indirect adjustment method, this paper proposes a general unified least squares linear regression model and detail algorithm. The model considers the error of the dependent variables and independent variables, and meanwhile it can also be applied to the nultivariate linear regression.. The analysis of examples proved the correctness and rationality of the method lescribed in this paper.