The GUM perspective on straight-line errors-in-variables regression

Klauenberg, Katy; Martens, Steffen; Bošnjaković, Alen; Cox, M G; Veen, Adriaan van der; Elster, Clemens

doi:10.1016/j.measurement.2021.110340

Cited by 18 publications

(10 citation statements)

References 29 publications

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“…As discussed in [17], the uncertainty matrix of parameter estimates derived from ISO does not strictly follow the law of propagation of uncertainty as defined in GUM and its supplements, which could lead to obvious differences especially in situations with an implicit measurement model and large combined measurement uncertainties of the variables involved. The authors argue that the uncertainties correctly derived based on the LPU are always larger than the uncertainties based on the ISO approach proposed in [5].…”

Section: Discussionmentioning

confidence: 99%

“…where X, and Y are the input quantities and a, b, and µ are the output quantities. The measurement model ( 5) is nonlinear and implicit, see [17].…”

Section: Linear Comparative Calibration Modelmentioning

confidence: 99%

“…[13], [21], [26], [23], [22], [24], and [9]. The estimated covariance matrix is usually based on the information matrices or their observed version based on the calculated Hessian matrix, although simplifications are possible, see for example [8] and [17] for a detailed discussion of the available approaches and comparisons in nonlinear measurement models.…”

Section: Linear Comparative Calibration Modelmentioning

confidence: 99%

“…Recently, the ISO working group Statistical Methods to Support the Evaluation of Measurement Uncertainty (ISO /TC 69/ SC 6/ WG 7) has officially started the development of the new standard ISO 28037 Determination and Use of Linear Calibration Functions, which will potentially address known shortcomings of ISO/TS. In particular, the problem is that the ISO method does not correctly apply the LPU for the implicit measurement model according to GUM and its Supplements, which was recently pointed out in [17]. The ISO approach is based on a generalised (weighted) least squares estimation as described in ( 8)-( 9) and on the application of the law of uncertainty propagation (LPU), albeit approximate, using a first order linearisation of the associated statistical model.…”

Section: A Calibration According To the Iso Technical Specification 2...mentioning

confidence: 99%

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ISO Linear Calibration and Measurement Uncertainty of the Result Obtained With the Calibrated Instrument

Palenčár

Chytil

et al. 2022

Measurement Science Review

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We address the problem of linear comparative calibration, a special case of linear calibration where both variables are measured with errors, and the analysis of the uncertainty of the measurement results obtained with the calibrated instrument. The concept is explained in detail using the calibration experiment of the pressure transducer and the subsequent analysis of the measurement uncertainties. In this context, the calibration and the measurements with the calibrated instrument are performed according to ISO Technical Specification 28037:2010 (here referred to as ISO linear calibration), based on the approximate linear calibration model and the application of the law of propagation of uncertainty (LPU) in this approximate model. Alternatively, estimates of the calibration line parameters, their standard uncertainties, the coverage intervals and the associated probability distributions are obtained using the Monte Carlo method (MCM) based on the law of propagation of distributions (LPD). Here we also obtain the probability distributions and the coverage interval for the quantities measured with the calibrated instrument. Furthermore, motivated by the model structure of this particular example, we conducted a simulation study that presents the empirical coverage probabilities of the ISO and MCM coverage intervals and investigates the influence of the sample size, i.e. the number of calibration points in the measurement range, and the different combinations of measurement uncertainties. The study generally confirms the good properties and validity of the ISO technical specification within the considered (limited) framework of experimental designs motivated by real-world application, with small uncertainties in relation to the measurement range. We also point out the potential weaknesses of this method that require increased user attention and emphasise the need for further research in this area.

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Section: Discussionmentioning

confidence: 99%

“…where X, and Y are the input quantities and a, b, and µ are the output quantities. The measurement model ( 5) is nonlinear and implicit, see [17].…”

Section: Linear Comparative Calibration Modelmentioning

confidence: 99%

Section: Linear Comparative Calibration Modelmentioning

confidence: 99%

Section: A Calibration According To the Iso Technical Specification 2...mentioning

confidence: 99%

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ISO Linear Calibration and Measurement Uncertainty of the Result Obtained With the Calibrated Instrument

Palenčár

Chytil

et al. 2022

Measurement Science Review

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“…Following the Guide to the Expression of Uncertainty in Measurement (GUM), a common text for the metrological determination of the measurement value [ 33 ], the uncertainty of the calibration target can be evaluated by applying the (linear) law of propagation of uncertainty (LPU) to the abscissa. Therefore, for a generalized approach with the non-diagonalized SR matrix, an analytical process to derive the variance–covariance matrix of regression coefficients has been demonstrated only for particular cases wherein the correlation between variables does not disturb the derivation of the explicit variance–covariance formulae [ 33 ]. For chemical or biological RMs prepared from the same mother material, one may reasonably assume a strong correlation between these uncertainties [ 34 ].…”

Section: Introductionmentioning

confidence: 99%

Approximated Uncertainty Propagation of Correlated Independent Variables Using the Ordinary Least Squares Estimator

Lim,

Kim,

Woo

2024

Molecules

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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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Creating Models for Predictive Maintenance of Field Equipment in the Oil Industry Using Simulation Based Uncertainty Modelling

Ramírez-Velarde

Hervert-Escobar

Hernández-Gress

2023

Lecture Notes in Computer Science

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The GUM perspective on straight-line errors-in-variables regression

Cited by 18 publications

References 29 publications

ISO Linear Calibration and Measurement Uncertainty of the Result Obtained With the Calibrated Instrument

ISO Linear Calibration and Measurement Uncertainty of the Result Obtained With the Calibrated Instrument

Approximated Uncertainty Propagation of Correlated Independent Variables Using the Ordinary Least Squares Estimator

Creating Models for Predictive Maintenance of Field Equipment in the Oil Industry Using Simulation Based Uncertainty Modelling

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