Approximated Uncertainty Propagation of Correlated Independent Variables Using the Ordinary Least Squares Estimator
For chemical measurements, calibration is typically conducted by regression analysis. In many cases, generalized approaches are required to account for a complex-structured variance–covariance matrix of (in)dependent variables. However, in the particular case of highly correlated independent variabl...
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| Vydané v: | Molecules (Basel, Switzerland) Ročník 29; číslo 6; s. 1248 |
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| Abstract | For chemical measurements, calibration is typically conducted by regression analysis. In many cases, generalized approaches are required to account for a complex-structured variance–covariance matrix of (in)dependent variables. However, in the particular case of highly correlated independent variables, the ordinary least squares (OLS) method can play a rational role with an approximated propagation of uncertainties of the correlated independent variables into that of a calibrated value for a particular case in which standard deviation of fit residuals are close to the uncertainties along the ordinate of calibration data. This proposed method aids in bypassing an iterative solver for the minimization of the implicit form of the squared residuals. This further allows us to derive the explicit expression of budgeted uncertainties corresponding to a regression uncertainty, the measurement uncertainty of the calibration target, and correlated independent variables. Explicit analytical expressions for the calibrated value and associated uncertainties are given for straight-line and second-order polynomial fit models for the highly correlated independent variables. |
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| AbstractList | For chemical measurements, calibration is typically conducted by regression analysis. In many cases, generalized approaches are required to account for a complex-structured variance-covariance matrix of (in)dependent variables. However, in the particular case of highly correlated independent variables, the ordinary least squares (OLS) method can play a rational role with an approximated propagation of uncertainties of the correlated independent variables into that of a calibrated value for a particular case in which standard deviation of fit residuals are close to the uncertainties along the ordinate of calibration data. This proposed method aids in bypassing an iterative solver for the minimization of the implicit form of the squared residuals. This further allows us to derive the explicit expression of budgeted uncertainties corresponding to a regression uncertainty, the measurement uncertainty of the calibration target, and correlated independent variables. Explicit analytical expressions for the calibrated value and associated uncertainties are given for straight-line and second-order polynomial fit models for the highly correlated independent variables. For chemical measurements, calibration is typically conducted by regression analysis. In many cases, generalized approaches are required to account for a complex-structured variance-covariance matrix of (in)dependent variables. However, in the particular case of highly correlated independent variables, the ordinary least squares (OLS) method can play a rational role with an approximated propagation of uncertainties of the correlated independent variables into that of a calibrated value for a particular case in which standard deviation of fit residuals are close to the uncertainties along the ordinate of calibration data. This proposed method aids in bypassing an iterative solver for the minimization of the implicit form of the squared residuals. This further allows us to derive the explicit expression of budgeted uncertainties corresponding to a regression uncertainty, the measurement uncertainty of the calibration target, and correlated independent variables. Explicit analytical expressions for the calibrated value and associated uncertainties are given for straight-line and second-order polynomial fit models for the highly correlated independent variables.For chemical measurements, calibration is typically conducted by regression analysis. In many cases, generalized approaches are required to account for a complex-structured variance-covariance matrix of (in)dependent variables. However, in the particular case of highly correlated independent variables, the ordinary least squares (OLS) method can play a rational role with an approximated propagation of uncertainties of the correlated independent variables into that of a calibrated value for a particular case in which standard deviation of fit residuals are close to the uncertainties along the ordinate of calibration data. This proposed method aids in bypassing an iterative solver for the minimization of the implicit form of the squared residuals. This further allows us to derive the explicit expression of budgeted uncertainties corresponding to a regression uncertainty, the measurement uncertainty of the calibration target, and correlated independent variables. Explicit analytical expressions for the calibrated value and associated uncertainties are given for straight-line and second-order polynomial fit models for the highly correlated independent variables. |
| Audience | Academic |
| Author | Lim, Jeong Sik Woo, Jin-Chun Kim, Yong Doo |
| AuthorAffiliation | 3 Division of Chemical and Material Metrology, Korea Research Institute of Standard and Science (KRISS), Gajeong-ro 267, Yuseong-gu, Daejeon 34113, Republic of Korea jcwooo52@gmail.com (J.-C.W.) 1 Strategic Technology Research Institute, Korea Research Institute of Standards and Science (KRISS), Gajeong-ro 267, Yuseong-gu, Daejeon 34113, Republic of Korea 2 Science of Measurement, University of Science and Technology (UST), Gajeong-ro 217, Yuseong-gu, Daejeon 34113, Republic of Korea |
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| Cites_doi | 10.1088/0026-1394/50/6/654 10.1039/b002171g 10.1007/BF02108667 10.1021/acs.analchem.9b00119 10.1016/j.chemolab.2010.07.003 10.1016/S0021-9673(96)00976-4 10.1088/0957-0233/18/11/025 10.1016/j.compgeo.2020.103868 10.1021/acs.analchem.3c00128 10.1088/0957-0233/22/3/035101 10.1039/b411054d 10.1016/j.measurement.2021.110340 10.1021/acs.analchem.0c02178 10.1039/c0an00192a 10.1007/s13369-018-03693-7 10.1016/j.enggeo.2022.106899 10.1016/j.chemolab.2004.05.005 10.1088/0026-1394/43/4/S17 10.1021/ac951217s 10.1093/imanum/1.1.3 10.1039/b701696d 10.1061/(ASCE)SU.1943-5428.0000153 10.1021/acs.analchem.2c02904 10.1088/0026-1394/44/6/009 10.1016/j.sigpro.2007.04.004 10.1016/S0003-2670(97)00205-5 |
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| Keywords | uncertainty evaluation ordinary least squares calibration Monte Carlo simulation correlated independent variables |
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| Title | Approximated Uncertainty Propagation of Correlated Independent Variables Using the Ordinary Least Squares Estimator |
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