气体测量中的线性拟合

王德发; 李琪; 叶菁; 周枫然; 曾武

doi:10.12338/j.issn.2096-9015.2022.0100

气体测量中的线性拟合

doi: 10.12338/j.issn.2096-9015.2022.0100

王德发^{1, 2,},
李琪,
叶菁^{1, 2},
周枫然¹,
曾武³

1.
中国计量科学研究院，北京 100029
2.
国家气体产品质量监督检验中心，北京 100029
3.
中计华量环境科技河北有限公司，石家庄 050000

基金项目: 中国计量科学研究院基本科研业务费重点领域项目（AKYZD2008）

详细信息

作者简介:
王德发（1977-），中国计量科学研究院副研究员，研究方向：气体计量，邮箱：wangdf@nim.ac.cn

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出版历程
- 收稿日期: 2022-05-02
- 录用日期: 2022-08-15
- 网络出版日期: 2022-09-19
- 刊出日期: 2022-10-01

Linear Fitting in Gas Measurement

WANG Defa^{1, 2
,},
LI Qi,
YE Jing^{1, 2},
ZHOU Fengran¹,
ZENG Wu³

1.
National Institute of Metrology, Beijing 100029, China
2.
National Gas Product Quality Supervision and Inspection Center, Beijing 100029, China
3.
Zhongjihualiang Environmental Technology Hebei Co., Ltd., Shijiazhuang 050000, China

摘要

摘要: 气体成分测量中可使用单点、双点、线性等多种校准方法，线性校准可分为普通线性最小二乘法、加权双变量最小二乘法线性拟合等多种细分方法。对不同文献介绍的上述两种方法进行了比较分析，并对加权双变量最小二乘法线性拟合中应注意的事项进行了介绍。当测量信号的不确定度已经与气体标准物质的不确定度水平接近，甚至低于标准物质的不确定度时，采用加权双变量最小二乘法线性拟合的方法处理测量结果更加合理。使用加权双变量最小二乘法线性拟合时，权重为标准不确定度平方的倒数，可采用二步迭代法计算截矩和斜率的数值；采用数值微分法计算截矩和斜率的标准不确定度。分析函数和校准函数都可以用于加权双变量最小二乘法线性拟合，对于被测样品浓度及其不确定度的计算，采用这两种函数所获得的结果是相同的，但是在计算不确定度时，采用分析函数更为简便。
- 气体 /
- 测量 /
- 线性拟合 /
- 最小二乘法 /
- 标准物质
Abstract: Various calibration methods such as single-point, double-point, and linear can be used in gas composition measurements, and linear calibration can be divided into various subdivision methods such as ordinary linear least squares and weighted bivariate least squares linear fitting. In this paper, the above two methods introduced in different literatures are compared and analyzed, and the matters that should be noted in the weighted bivariate least squares linear fitting are introduced. When the uncertainty of the measured signal is already close to or even lower than the uncertainty level of the gas standard, it is more reasonable to use the weighted bivariate least squares linear fitting method to process the measurement results. When using the weighted bivariate least squares linear fit, the weights are the reciprocal of the squared standard uncertainty, and the values of the intercept and slope can be calculated using the two-step iterative method; the standard uncertainty of the intercept and slope is calculated using the numerical differentiation method. Both analytical and calibration functions can be used for weighted bivariate least squares linear fitting. For the calculation of the measured sample concentration and its uncertainty, the results obtained by using these two functions are the same, but it is easier to use the analytical function when calculating the uncertainty.
- gas /
- measurement /
- linear fitting /
- least square method /
- reference materials

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图 1 加权双变量最小二乘法线性拟合软件界面

Figure 1. Software interface of weighted bivariate least squares linear fitting

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表 1 ISO 6143:2001中举例的数据

Table 1. Data exemplified in ISO 6143:2001

参考标准	$ {x}_{i} $	$ u\left({x}_{i}\right) $	$ {y}_{i} $	$ u\left({y}_{i}\right) $
1	4.5	0.045	0.1969	0.003938
2	18.75	0.1875	0.7874	0.015748
3	50	0.5	2.0228	0.040456

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表 2 两种方法拟合结果

Table 2. Fitting results by two methods

方法	截矩$ {b}_{0} $	斜率$ {b}_{1} $	${x}_{1}'$	${x}_{2}'$	${x}_{3}'$
A	0.02492	0.04003	4.30	19.05	49.91
B	0.01452	0.04063	4.49	19.02	49.43
C	0.02453	0.04005	4.30	19.05	49.89
D	0.01640	0.04053	4.45	19.02	49.50

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表 3 截矩和斜率的标准不确定度

Table 3. Standard uncertainty of intercept and slope

	方法1	方法2	方法3
$ u\left({b}_{0}\right) $	0.157157874	0.049594768	0.129015968
$ u\left({b}_{1}\right) $	0.480487597	0.480268458	0.394377572
注：方法1为ISO 6143:2001的方法；方法2为文献[19]的方法；方法3为文献[16]的方法。

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表 4 ISO 6143:2001附录中被测样品的测量数据

Table 4. Measurement data of the tested sample in the appendix of ISO 6143:2001

样品序号	$ {y}_{i} $	$ u\left({y}_{i}\right) $
1	2.5800×10⁻¹	5.1600×10⁻³
2	6.0000×10⁻¹	1.2000×10⁻²
3	1.8000	3.6000×10⁻²

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表 5 分析函数和校准函数线性拟合的比较

Table 5. Comparison of linear fitting between analytical function and calibration function

	拟合参数	分析函数	校准函数
线性拟合	$ {b}_{0} $	−0.357467596	0.014524401
	$ {b}_{1} $	24.611520886	0.040631378
	$ u\left({b}_{0}\right) $	0.157157874	0.006174543
	$ u\left({b}_{1}\right) $	0.480487597	0.000793225
	$ u\left({{b}_{0},b}_{1}\right) $	−0.056922921	−3.59326×10⁻⁶
被测样品的计算结果	$ {x}_{1} $	5.99230	5.99230
	$ u\left({x}_{1}\right) $	0.16377	0.16377
	$ {x}_{2} $	14.40944	14.40944
	$ u\left({x}_{2}\right) $	0.35599	0.35599
	$ {x}_{3} $	43.94327	43.94327
	$ u\left({x}_{3}\right) $	1.16310	1.16309

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参考文献(19)

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