ISO/IEC GUIDE 98-6:2021

Uncertainty of measurement - Part 6: Developing and using measurement models

This document provides guidance on developing and using a measurement model and also covers the assessment of the adequacy of a measurement model. The document is of particular interest to developers of measurement procedures, working instructions and documentary standards. The model describes the relationship between the output quantity (the measurand) and the input quantities known to be involved in the measurement. The model is used to obtain a value for the measurand and an associated uncertainty. Measurement models are also used in, for example, design studies, simulation of processes, and in
engineering, research and development.
This document explains how to accommodate in a measurement model the quantities involved. These quantities relate i) to the phenomenon or phenomena on which the measurement is based, that is, the measurement principle, ii) to effects arising in the specific measurement, and iii) to the interaction with the artefact or sample subject to measurement.
The guidance provided is organised in accordance with a work flow that could be contemplated when developing a measurement model from the beginning. This work flow starts with the specification of the measurand (clause 6). Then the measurement principle is modelled (clause 7) and an appropriate form of the model is chosen (clause 8). The basic model thus obtained is extended by identifying (clause 9) and adding (clause 10) effects arising from the measurement and the artefact or sample subject to measurement. Guidance on assessing the adequacy of the resulting measurement model is given in clause 12. The distinction between the basic model and the (complete) measurement model in the work flow should be helpful to those readers who already have a substantial part of the measurement model in place, but would like to verify that it contains all effects arising from the measurement so that it is fit for purpose.
Guidance on the assignment of probability distributions to the quantities appearing in the measurement model is given in JCGM 100:2008 and JCGM 101:2008. In clause 11, this guidance is supplemented by describing how statistical models can be developed and used for this purpose.
When using a measurement model, numerical problems can arise including computational effects such as rounding and numerical overflow. It is demonstrated how such problems can often be alleviated by expressing a model differently so that it performs well in calculations. It is also shown how a reformulation of the model can sometimes be used to eliminate some correlation effects among the input quantities when such dependencies exist.
Examples from a number of metrology disciplines illustrate the guidance provided in this document.

General Information

• Status: Published
• Publication Date: 08-Feb-2021
• ICS: 17.020 - Metrology and measurement in general
• Technical Committee: IEC/TC - IEC/TC
• Current Stage: PPUB - Publication issued
• Start Date: 13-Jun-2022
• Completion Date: 09-Feb-2021

Standards Content (Sample)

GUIDE 98-6
First edition
2021-02
Uncertainty of measurement —
Part 6:
Developing and using measurement
models
Incertitude de mesure —
Partie 6: Élaboration et utilisation de modèles de mesure
Reference number
©
ISO/IEC 2021
© ISO/IEC 2021
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© ISO/IEC 2021 – All rights reserved iii

Joint Committee for Guides in Metrology
Guide to the expression of uncertainty in measurement
— Part 6: Developing and using measurement models
Guide pour l’expression de l’incertitude de mesure — Partie 6:
Élaboration et utilisation des modèles de mesure
JCGM GUM-6:2020
© JCGM 2020 – All rights reserved
© ISO/IEC 2021 – All rights reserved

ii JCGM GUM-6:2020
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JCGM GUM-6:2020 iii
Contents
Page
Foreword v
Introduction vi
1 Scope 1
2 Normative references 2
3 Terms and definitions 2
4 Conventions and notation 2
5 Basic principles 3
6 Specifying the measurand 5
7 Modelling the measurement principle 9
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.2 Theoretical, empirical and hybrid measurement models . . . . . . . . . . . . . . . . . . 9
7.3 Differential equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
8 Choosing the form of the measurement model 13
8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.2 Fitness for purpose and approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3 Representation and transformation of models . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.2 Re-parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3.3 Use in regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3.4 Simple transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.3.5 Non-linear relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.3.6 Impact on uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.3.7 Explicit and implicit forms of measurement model . . . . . . . . . . . . . . . . 22
8.4 Multi-stage measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.5 Uncertainty associated with choice of model . . . . . . . . . . . . . . . . . . . . . . . . . 24
8.6 Loss of numerical accuracy and its compensation . . . . . . . . . . . . . . . . . . . . . . 24
9 Identifying effects arising from the measurement 28
10 Extending the basic model 29
10.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.2 Adding effects to the basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
10.3 Modelling well-understood effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
10.4 Modelling poorly understood effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10.5 Shared effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10.6 Drift and other time-dependent effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
11 Statistical models used in metrology 39
11.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
11.2 Observation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
11.3 Specification of statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
11.4 Models for calibration and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.5 Models for homogeneity studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
11.6 Models for the adjustment of observations . . . . . . . . . . . . . . . . . . . . . . . . . . 45
11.7 Models for time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
11.8 Bayesian statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
11.9 Estimation and uncertainty evaluation for statistical models . . . . . . . . . . . . . . . 50
11.10 Model selection and model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
12 Assessing the adequacy of the measurement model 57
13 Using the measurement model 59
13.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
13.2 Use of a model beyond the range for which it has been validated . . . . . . . . . . . . 61
13.3 Explicit univariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
13.4 Explicit multivariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13.5 Implicit univariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13.6 Implicit multivariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
13.7 Measurement models involving complex-valued quantities . . . . . . . . . . . . . . . . 64
A Glossary of principal symbols 66
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© ISO/IEC 2021 – All rights reserved

iv JCGM GUM-6:2020
B Modelling of dynamic measurements by linear time-invariant systems 67
B.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B.2 Continuous-time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B.3 Discrete-time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
C Modelling random variation 71
C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C.1.1 Random variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C.1.2 Considerations in modelling random variation . . . . . . . . . . . . . . . . . . . 71
C.2 Including random variation in a measurement model . . . . . . . . . . . . . . . . . . . 71
C.2.1 Options for including random variation . . . . . . . . . . . . . . . . . . . . . . . 71
C.2.2 Random variation associated with an existing input quantity . . . . . . . . . . 72
C.2.3 Random variation as an effect associated with the measurand . . . . . . . . . 73
C.3 Multiple sources of random variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
C.4 Asymmetrically distributed effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C.5 Use of reproducibility studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D Representing polynomials 78
E Cause-and-effect analysis 80
E.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
E.2 5M method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
E.3 Measurement System Analysis (MSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
F Linearizing a measurement model and checking its adequacy 84
Bibliography 86
Alphabetical index 94
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© ISO/IEC 2021 – All rights reserved

JCGM GUM-6:2020 v
Foreword
In 1997 a Joint Committee for Guides in Metrology (JCGM), chaired by the Director of
t
...