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.

Incertitude de mesure — Partie 6: Élaboration et utilisation de modèles de mesure

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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 GUIDE 98-6:2021(E)
ISO/IEC 2021
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ISO/IEC GUIDE 98-6:2021(E)
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ISO/IEC GUIDE 98-6:2021(E)
ISO/IEC Foreword

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committees.
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ISO/IEC GUIDE 98-6:2021(E)
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
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ISO/IEC GUIDE 98-6:2021(E)
ii JCGM GUM-6:2020
© JCGM 2020

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ISO/IEC GUIDE 98-6:2021(E)
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 GUIDE 98-6:2021(E)
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 GUIDE 98-6:2021(E)
JCGM GUM-6:2020 v
Foreword

In 1997 a Joint Committee for Guides in Metrology (JCGM), chaired by the Director of

the Bureau International des Poids et Mesures (BIPM), was created by the seven interna-

tional organizations that had originally in 1993 prepared the ‘Guide to the expression of

uncertainty in measurement’ and the ‘International vocabulary of basic and general terms

in metrology’. The JCGM assumed responsibility for these two documents from the ISO

Technical Advisory Group 4 (TAG4).

The Joint Committee is formed by the BIPM with the International Electrotechnical Com-

mission (IEC), the International Federation of Clinical Chemistry and Laboratory Medicine

(IFCC), the International Laboratory Accreditation Cooperation (ILAC), the International

Organization for Standardization (ISO), the International Union of Pure and Applied

Chemistry (IUPAC), the International Union of Pure and Applied Physics (IUPAP), and

the International Organization of Legal Metrology (OIML).
JCGM has two Working Groups. Working Group 1, ‘Expression of uncertainty in mea-

surement’, has the task to promote the use of the ‘Guide to the expression of uncertainty

in measurement’ and to prepare documents for its broad application. Working Group 2,

‘Working Group on International vocabulary of basic and general terms in metrology’, has

the task to revise and promote the use of the ‘International vocabulary of basic and general

terms in metrology’ (the ‘VIM’).

In 2008 the JCGM made available a slightly revised version (mainly correcting minor er-

rors) of the ‘Guide to the expression of uncertainty in measurement’, labelling the docu-

ment ‘JCGM 100:2008’. In 2017 the JCGM rebranded the documents in its portfolio that

have been produced by Working Group 1 or are to be developed by that Group: the whole

suite of documents became known as the ‘Guide to the expression of uncertainty in mea-

surement’ or ‘GUM’. This document, previously known as JCGM 103, Supplement 3 to the

GUM, is the first to be published as a part of that portfolio, and is entitled and numbered

accordingly.

The present guide is concerned with the development and use of measurement models, and

supports the documents in the entire suite of JCGM documents concerned with uncertainty

in measurement. The guide has been prepared by Working Group 1 of the JCGM, and has

benefited from detailed reviews undertaken by member organizations of the JCGM and

National Metrology Institutes.
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ISO/IEC GUIDE 98-6:2021(E)
vi JCGM GUM-6:2020
Introduction

A measurement model constitutes a relationship between the output quantities or mea-

surands (the quantities intended to be measured) and the input quantities known to be

involved in the measurement. There are several reasons for modelling a measurement.

Models assist in developing a quantitative understanding of the measurement and in im-

proving the measurement. A model enables values of the output quantities to be obtained

given the values of the input quantities. Additionally, a model not only allows propaga-

tion of uncertainty from the input quantities to the output quantities; it also provides an

understanding of the principal contributions to uncertainty. This document is accordingly

concerned with the development of a measurement model and the practical use of the

model.

One of the purposes of measurement is to assist in making decisions. The reliability of these

decisions and the related risks depend on the values obtained for the output quantities and

the associated uncertainties. In turn, these decisions depend on a suitable measurement

model and the quality of information about the input quantities.

Although the development of a measurement model crucially depends on the nature of the

measurement, some generic guidance on aspects of modelling is possible. A measurement

model might be a straightforward mathematical relationship, such as the ideal gas law, or,

at the other extreme, involve a sophisticated numerical algorithm for its evaluation, such

as the detection of peaks in a signal and the determination of peak parameters.

A measurement model may take various forms: theoretical, empirical or hybrid (part-

theoretical, part-empirical). It might have a single output quantity or more than one out-

put quantity. The output quantity may or may not be expressed directly in terms of the

input quantities. The quantities in the measurement model may be real-valued or complex-

valued. Measurement models may be nested or multi-stage, in the sense that input quan-

tities in one stage are output quantities from a previous stage, as occurs, for instance,

in the dissemination of measurement standards or in calibration. Measurement models

might describe time series of observations, including drift, and dynamic measurement. A

measurement model may also take the form of a statistical model. In this document the

concept ‘measurement model’ is intended in this broader meaning.

In developing or using a measurement model there are important choices to be made.

The selection of a model that is adequate or fit for purpose is a key issue. Particularly for

empirical models, there is choice of representation (or parametrization) of the families of

functions concerned (polynomials, polynomial splines or rational functions, etc.). Certain

choices can be far superior to others in their numerical behaviour when the model is im-

plemented on a computer. The uncertainty arising from the choice of model is a necessary

consideration.

In many disciplines, a basic measurement model requires extension to incorporate effects

such as temperature corrections arising from the measurement to enable values for output

quantities and the associated uncertainties to be obtained reliably.

Following the introduction in 1993 of the Guide to the expression of uncertainty in mea-

surement, or GUM (also known as JCGM 100:2008), the practice of uncertainty evaluation

has broadened to use a wider variety of models and methods. To reflect this, this Guide

includes an introduction to statistical models for measurement modelling (clause 11) and

additional guidance on modelling random variation in Annex C.
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ISO/IEC GUIDE 98-6:2021(E)
JCGM GUM-6:2020 1
Guide to the expression of uncertainty in
measurement — Part 6: Developing and using
measurement models
1 Scope
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. Measure-

ment 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 in-

volved. These quantities relate i) to the phenomenon or phenomena on which the mea-

surement 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 measure-

ment.

The guidance provided is organised in accordance with a work flow that could be con-

templated 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 cal-

culations. 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.
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ISO/IEC GUIDE 98-6:2021(E)
2 JCGM GUM-6:2020
2 Normative references

The following documents are referred to in the text in such a way that some or all of

their content constitutes requirements of this document. For dated references, only the

edition cited applies. For undated references, the latest edition of the referenced document

(including any amendments) applies.

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of measurement data —

Guide to the expression of uncertainty in measurement. Joint Committee for Guides in

Metrology, JCGM 100:2008.

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of measurement data —

Supplement 1 to the ‘Guide to the expression of uncertainty in measurement’ — Propaga-

tion of distributions using a Monte Carlo method. Joint Committee for Guides in Metrology,

JCGM 101:2008.

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of measurement data —

Supplement 2 to the ‘Guide to the expression of uncertainty in measurement’ — Exten-

sion to any number of output quantities. Joint Committee for Guides in Metrology, JCGM

102:2011.

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. International vocabulary of metrol-

ogy — Basic and general concepts and associated terms. Joint Committee for Guides in

Metrology, JCGM 200:2012.
3 Terms and definitions
The terms and definitions of JCGM 100:2008, JCGM 101:2008, JCGM 102:2011 and JCGM
200:2012 apply.

ISO, IEC and IUPAC maintain terminological databases for use in standardization at the

following addresses:
— IEC Electropedia: available athttp://www.electropedia.org
— ISO Online Browsing Platform: available athttp://www.iso.org/obp
— IUPAC Gold Book: available athttp://www.goldbook.iupac.org
4 Conventions and notation
4.1 The conventions and notation in JCGM 100:2008, JCGM 101:2008 and

JCGM 102:2011 are adopted. Principal symbols used throughout the document are ex-

plained in annex A. Other symbols and those appearing in examples are explained at first

occurrence.

4.2 Most examples in this document contain numerical values rounded to a number of

decimal digits appropriate to the application. Because of rounding there are often numer-

ical inconsistencies among the values presented. An instance is the correlation coefficient

of0.817 in the example in 8.1.6. It is obtained from the computer-held values of two

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standard uncertainties and a covariance. If it were computed from the three presented

values (three significant decimal digits), its value correct to three decimal digits would be

0.822.
4.3 Links to numbered subclauses are indicated by underlining.
5 Basic principles

5.1 A measurand (see JCGM 200:2012, 2.3) is in many cases not measured directly, but

is indirectly determined from other quantities (see JCGM 200:2012, 1.1) to which it is

related by a measurement model (see JCGM 200:2012, 2.48) such as formula (1) in 5.2. The

measurement model is a mathematical expression or a set of such expressions (see JCGM

100:2008, 4.1.2), comprising all the quantities known to be involved in a measurement.

It enables a value (see JCGM 200:2012, 1.19) of the measurand to be provided and an

associated standard uncertainty to be evaluated. The measurement model may be specified

wholly or partly in the form of an algorithm. The quantities to which the measurand is

related constitute the input quantities (see JCGM 200:2012, 2.50) in the measurement

model. The measurand constitutes the output quantity (see JCGM 200:2012, 2.51).

5.2 Many measurements are modelled by a real functional relationship f between N real-

valued input quantities X ,. .. , X and a single real-valued output quantity (or measurand)

1 N
Y in the form
Y = f(X ,. .. , X ). (1)
1 N

This simple form is called a real explicit univariate measurement model; real since all

quantities involved take real (rather than complex) values, explicit because a value for Y

can be computed directly given values of X , ... , X , and univariate since Y is a single,

1 N

scalar quantity. However, it does not apply for all measurements. A measurement model

can be complex, involving complex-valued quantities (see JCGM 102:2011, 3.2). It can

be implicit where a value for Y cannot be determined directly given values of X ,. .. , X

1 N

(see 13.5). The measurement model can be multivariate where there is more than one

measurand, denoted by Y , .. ., Y ; for further information, see 13.4 and JCGM 102:2011.

1 m
EXAMPLE Volume of a cylinder
The volume of a cylinder is given by the measurement model
V = Ld

in which cylinder length L and diameter d are the N = 2 input quantities, corresponding to X and

X , and an output quantity V corresponding to Y .

5.3 The process of building a measurement model can be subdivided into the following

steps, each step being described in the indicated clause:
a) Select and specify the measurand (see clause 6).
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4 JCGM GUM-6:2020

b) Model the measurement principle, thus providing a basic model for this purpose (see

clause 7), choosing an appropriate mathematical form (see clauses 8 and 11).
c) Identify effects involved in the measurement (see clause 9).

d) Extend the basic model as necessary to include terms accounting for these effects

(see clauses 10 and 11).
e) Assess the resulting measurement model for adequacy (see clause 12).
In any one instance, a number of passes through the proce
...

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