线性回归拟合的测量不确定度评定

夏玉国; 彭友志; 刘正华; 宋潇; 曾卓

doi:10.3969/j.issn.1000-1158.2023.04.26

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计量学报 ›› 2023, Vol. 44 ›› Issue (4) : 664-670. DOI: 10.3969/j.issn.1000-1158.2023.04.26

计量学总论

线性回归拟合的测量不确定度评定

夏玉国¹,彭友志^1,2,3,刘正华^1,2,3,宋潇¹,曾卓¹

作者信息 +

The Measurement Uncertainty Evaluation of Linear Regression Fitting

XIA Yu-guo¹,PENG You-zhi^1,2,3 ,LIU Zheng-hua^1,2,3,SONG Xiao¹,ZENG Zhuo¹

Author information +

文章历史 +

摘要

利用GUM对最小二乘线性回归拟合结果进行测量不确定度评定,提出了同时顾及因变量以及系数矩阵中自变量误差影响的标准不确定度通用式。首先,阐述了线性回归拟合模型以及最小二乘估计结果;其次,考虑系数矩阵影响,推导了拟合参数与拟合值关于自变量、因变量的灵敏系数以及标准不确定度通用公式,同时考虑相关性,给出了对变量相关进行特殊处理后的标准不确定度通用公式;最后,以线位移传感器灵敏度(不需考虑相关性)和光电测距仪加乘常数(考虑相关性)为例,与蒙特卡洛方法(MCM)进行比较分析。结果表明,不需考虑相关性以及考虑相关性下,基于GUM评定方法的标准不确定度通用公式计算结果与MCM评定方法结果基本一致。

Abstract

The measurement uncertainty of the least squares linear regression fitting results was evaluated by GUM method, and a general formula of standard uncertainty was proposed to estimate the influence of error of independent variable in coefficient matrix and dependent variable simultaneously. Firstly, the linear regression fitting model and least squares estimation results were described.Secondly, considering the influence of coefficient matrix, the general formulas for fitting parameters, fitting values, sensitivity coefficients on independent and dependent variables, and standard uncertainty were derived. At the same time, considering the correlation, a general formula for standard uncertainty after special treatment of variable correlation is given. Finally, the sensitivity of linear displacement sensor (without considering the correlation) and the additive multiplying constant of geodimeter (considering the correlation) were taken as examples to compare with the Monte Carlo method (MCM).The research results showed that the calculation results of the GUM method were basically consistent with the results of the MCM method.

导出引用

夏玉国,彭友志,刘正华,宋潇,曾卓. 线性回归拟合的测量不确定度评定[J]. 计量学报. 2023, 44(4): 664-670 https://doi.org/10.3969/j.issn.1000-1158.2023.04.26

XIA Yu-guo,PENG You-zhi,LIU Zheng-hua,SONG Xiao,ZENG Zhuo. The Measurement Uncertainty Evaluation of Linear Regression Fitting[J]. Acta Metrologica Sinica. 2023, 44(4): 664-670 https://doi.org/10.3969/j.issn.1000-1158.2023.04.26

中图分类号： TB9

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