ISO/IEC Guide 98-3:2008/Suppl 2:2011

Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) — Supplement 2: Extension to any number of output quantities

ISO/IEC Guide 98-3:2008/Suppl.2:2011 is concerned with measurement models having any number of input quantities and any number of output quantities. The quantities involved might be real or complex. Two approaches are considered for treating such models. The first approach is a generalization of the GUM uncertainty framework. The second is a Monte Carlo method as an implementation of the propagation of distributions. Appropriate use of the Monte Carol method would be expected to provide valid results when the applicability of the GUM uncertainty framework is questionable. For a prescribed coverage probability, ISO/IEC Guide 98-3:2008/Suppl.2:2011 can be used to provide a coverage region for the output quantities of a multivariate model, the counterpart of a coverage interval for a single scalar output quantiy. The provision of coverage regions includes those taking the form of a hyper-ellipsoid or a hyper-rectangle. These coverage regions are produced from the results of the two approaches described here. A procedure for providing an approximation to the smallest coverage region, obtained from results provided by the Monte Carol method, is also given. Detailed examples to illustrate the guidance are provided.

Incertitude de mesure — Partie 3: Guide pour l'expression de l'incertitude de mesure (GUM:1995) — Supplément 2: Extension à un nombre quelconque de grandeurs de sortie

General Information

• Status: Published
• Publication Date: 08-Nov-2011
• ICS: 17.020 - Metrology and measurement in general
• Technical Committee: ISO/TMBG - Technical Management Board - groups
• Drafting Committee: ISO/TMBG - Technical Management Board - groups
• Current Stage: 6060 - International Standard published
• Completion Date: 09-Nov-2011

Relations

• Child: ISO/IEC Guide 98-3:2008 - Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)
• Effective Date: 15-Apr-2008
• Amends: ISO/IEC Guide 98-3:2008 - Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)
• Effective Date: 06-Jun-2022

Standards Content (Sample)

GUIDE 98-3/Suppl.2
Uncertainty of measurement —
Part 3:
Guide to the expression of
uncertainty in measurement
(GUM:1995)
Supplement 2:
Extension to any number of
output quantities




First edition 2011
©
 ISO/IEC 2011

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ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
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ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
ISO/IEC Foreword
ISO (the International Organization for Standardization) and IEC (the International Electrotechnical
Commission) form the specialized system for worldwide standardization. National bodies that are members of
ISO or IEC participate in the development of International Standards through technical committees
established by the respective organization to deal with particular fields of technical activity. ISO and IEC
technical committees collaborate in fields of mutual interest. Other international organizations, governmental
and non-governmental, in liaison with ISO and IEC, also take part in the work.
Draft Guides adopted by the responsible Committee or Group are circulated to the member bodies for voting.
Publication as a Guide requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/IEC Guide 98-3/Suppl.2 was prepared by Working Group 1 of the Joint Committee for Guides in
Metrology (as JCGM 102:2011), and was adopted by the national bodies of ISO and IEC.
ISO/IEC Guide 98 consists of the following parts, under the general title Uncertainty of measurement:
 Part 1: Introduction to the expression of uncertainty in measurement
 Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)
 Part 4: Role of measurement uncertainty in conformity assessment
The following parts are planned:
 Part 2: Concepts and basic principles
 Part 5: Applications of the least-squares method
ISO/IEC Guide 98-3 has three supplements:
 Supplement 1: Propagation of distributions using a Monte Carlo method
 Supplement 2: Models with any number of output quantities
 Supplement 3: Modelling
Given that ISO/IEC Guide 98-3:2008/Suppl.2:2011 is identical in content to JCGM 102:2011, the decimal
symbol is a point on the line in the English version.
Annex ZZ has been appended to provide a list of corresponding ISO/IEC Guides and JCGM guidance
documents for which equivalents are not given in the text.


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JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
Joint Committee for Guides in Metrology JCGM

102
2011
Evaluation of measurement data | Supplement 2
to the \Guide to the expression of uncertainty in
measurement" | Extension to any number of
output quantities

Evaluation des donnees de mesure | Supplement 2 du \Guide pour l'expression
de l'incertitude de mesure" | Extension a un nombre quelconque de grandeurs
de sortie

c JCGM 2011| All rights reserved

© ISO/IEC – JCGM 2011 – All rights reserved                                            i

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ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
c

JCGM 2011

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JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
Contents Page
Foreword::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: v
Introduction:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: vi
1 Scope ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1
2 Normative references:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 2
3 Terms and denitions ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 2
4 Conventions and notation :::::::::::::::::::::::::::::::::::::::::::::::::::::: 8
5 Basic principles::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 10
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Main stages of uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Probability density functions for the input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3.2 Multivariate t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3.3 Construction of multivariate probability density functions . . . . . . . . . . . . . . . . . . . . . . . . 12
5.4 Propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.5 Obtaining summary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.6 Implementations of the propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 GUM uncertainty framework ::::::::::::::::::::::::::::::::::::::::::::::::::: 14
6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2 Propagation of uncertainty for explicit multivariate measurement models . . . . . . . . . . . . . . . . . . 15
6.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3 Propagation of uncertainty for implicit multivariate measurement models . . . . . . . . . . . . . . . . . . 17
6.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 Propagation of uncertainty for models involving complex quantities . . . . . . . . . . . . . . . . . . . . . . 19
6.5 Coverage region for a vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.5.2 Bivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.5.3 Multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.5.4 Coverage region for the expectation of a multivariate Gaussian distribution . . . . . . . . . . . 22
7 Monte Carlo method :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 23
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.2 Number of Monte Carlo trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.3 Making draws from probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.4 Evaluation of the vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.5 Discrete representation of the distribution function for the output quantity . . . . . . . . . . . . . . . . 27
7.6 Estimate of the output quantity and the associated covariance matrix . . . . . . . . . . . . . . . . . . . . 27
7.7 Coverage region for a vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.7.2 Hyper-ellipsoidal coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.7.3 Hyper-rectangular coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.7.4 Smallest coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.8 Adaptive Monte Carlo procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.8.2 Numerical tolerance associated with a numerical value . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.8.3 Adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Validation of the GUM uncertainty framework using a Monte Carlo method::::::::::: 34
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ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
9 Examples:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 35

9.1 Illustrations of aspects of this Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9.2 Additive measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9.2.2 Propagation and summarizing: case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9.2.3 Propagation and summarizing: case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9.2.4 Propagation and summarizing: case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3 Co-ordinate system transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3.2 Propagation and summarizing: zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.3.3 Propagation and summarizing: non-zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.4 Simultaneous measurement of resistance and reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.4.2 Propagation and summarizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.5 Measurement of Celsius temperature using a resistance thermometer . . . . . . . . . . . . . . . . . . . . . 55
9.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5.2 Measurement of a single Celsius temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5.3 Measurement of several Celsius temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Annexes
A (informative) Derivatives of complex multivariate measurement functions ::::::::::::: 59
B (informative) Evaluation of sensitivity coecients and covariance matrix for multivariate
measurement models::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 61
C (informative) Co-ordinate system transformation :::::::::::::::::::::::::::::::::: 62
C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
C.2 Analytical solution for a special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
C.3 Application of the GUM uncertainty framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
D (informative) Glossary of principal symbols ::::::::::::::::::::::::::::::::::::::: 65
Bibliography:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 69
Alphabetical index :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 71


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JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
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 international organizations that had originally in 1993
prepared the \Guide to the expression of uncertainty in measurement" (GUM) and the \International vocabulary
of basic and general terms in metrology" (VIM). 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 Commission (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 measurement", has the task
to promote the use of the GUM and to prepare Supplements and other documents for its broad application.
Working Group 2, \Working Group on International vocabulary of basic and general terms in metrology (VIM)",
has the task to revise and promote the use of the VIM.
Supplements such as this one are intended to give added value to the GUM by providing guidance on aspects of
uncertainty evaluation that are not explicitly treated in the GUM. The guidance will, however, be as consistent
as possible with the general probabilistic basis of the GUM.
The present Supplement 2 to the GUM has been prepared by Working Group 1 of the JCGM, and has beneted
from detailed reviews undertaken by member organizations of the JCGM and National Metrology Institutes.

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ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
Introduction

The \Guide to the expression of uncertainty in measurement" (GUM) [JCGM 100:2008] is mainly concerned
with univariate measurement models, namely models having a single scalar output quantity. However, mod-
els with more than one output quantity arise across metrology. The GUM includes examples, from electrical
metrology, with three output quantities [JCGM 100:2008 H.2], and thermal metrology, with two output quan-
tities [JCGM 100:2008 H.3]. This Supplement to the GUM treats multivariate measurement models, namely
models with any number of output quantities. Such quantities are generally mutually correlated because they
depend on common input quantities. A generalization of the GUM uncertainty framework [JCGM 100:2008 5]
is used to provide estimates of the output quantities, the standard uncertainties associated with the estimates,
and covariances associated with pairs of estimates. The input or output quantities in the measurement model
may be real or complex.
Supplement 1 to the GUM [JCGM 101:2008] is concerned with the propagation of probability distributions
[JCGM 101:2008 5] through a measurement model as a basis for the evaluation of measurement uncertainty,
and its implementation by a Monte Carlo method [JCGM 101:2008 7]. Like the GUM, it is only concerned with
models having a single scalar output quantity [JCGM 101:2008 1]. This Supplement describes a generalization of
that Monte Carlo method to obtain a discrete representation of the joint probability distribution for the output
quantities of a multivariate model. The discrete representation is then used to provide estimates of the output
quantities, and standard uncertainties and covariances associated with those estimates. Appropriate use of the
Monte Carlo method would be expected to provide valid results when the applicability of the GUM uncertainty
framework is questionable, namely when (a) linearization of the model provides an inadequate representation, or
(b) the probability distribution for the output quantity (or quantities) departs appreciably from a (multivariate)
Gaussian distribution.
Guidance is also given on the determination of a coverage region for the output quantities of a multivariate
model, the counterpart of a coverage interval for a single scalar output quantity, corresponding to a stipulated
coverage probability. The guidance includes the provision of coverage regions that take the form of hyper-
ellipsoids and hyper-rectangles. A calculation procedure that uses results provided by the Monte Carlo method
is also described for obtaining an approximation to the smallest coverage region.


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JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
Evaluation of measurement data | Supplement 2
to the \Guide to the expression of uncertainty in
measurement" | Extension to any number of output
quantities
1 Scope
This Supplement to the \Guide to the expression of uncertainty in measurement" (GUM) is concerned with
measurement models having any number of input quantities (as in the GUM and GUM Supplement 1) and
any number of output quantities. The quantities involved might be real or complex. Two approaches are
considered for treating such models. The rst approach is a generalization of the GUM uncertainty framework.
The second is a Monte Carlo method as an implementation of the propagation of distributions. Appropriate
use of the Monte Carlo method would be expected to provide valid results when the applicability of the GUM
uncertainty framework is questionable.
The approach based on the GUM uncertainty framework is applicable when the input quantities are summarized
(as in the GUM) in terms of estimates (for instance, measured values) and standard uncertainties associated
with these estimates and, when appropriate, covariances associated with pairs of these estimates. Formul
and procedures are provided for obtaining estimates of the output quantities and for evaluating the associated
standard uncertainties and covariances. Variants of the formul and procedures relate to models for which the
output quantities (a) can be expressed directly in terms of the input quantities as measurement functions, and
(b) are obtained through solving a measurement model, which links implicitly the input and output quantities.
The counterparts of the formul in the GUM for the standard uncertainty associated with an estimate of
the output quantity would be algebraically cumbersome. Such formul are provided in a more compact form
in terms of matrices and vectors, the elements of which contain variances (squared standard uncertainties),
covariances and sensitivity coecients. An advantage of this form of presentation is that these formul can
readily be implemented in the many computer languages and systems that support matrix algebra.
The Monte Carlo method is based on (i) the assignment of probability distributions to the input quantities in
the measurement model [JCGM 101:2008 6], (ii) the determination of a discrete representation of the (joint)
probability distribution for the output quantities, and (iii) the determination from this discrete representation of
estimates of the output quantities and the evaluation of the associated standard uncertainties and covariances.
This approach constitutes a generalization of the Monte Carlo method in Supplement 1 to the GUM, which
applies to a single scalar output quantity.
For a prescribed coverage probability, this Supplement can be used to provide a coverage region for the output
quantities of a multivariate model, the counterpart of a coverage interval for a single scalar output quantity.
The provision of coverage regions includes those taking the form of a hyper-ellipsoid or a hyper-rectangle. These
coverage regions are produced from the results of the two approaches described here. A procedure for providing
an approximation to the smallest coverage region, obtained from results provided by the Monte Carlo method,
is also given.
This Supplement contains detailed examples to illustrate the guidance provided.
This document is a Supplement to the GUM and is to be used in conjunction with it and GUM Supplement 1.
The audience of this Supplement is that of the GUM and its Supplements. Also see JCGM 104.

© ISO/IEC – JCGM 2011 – All rights reserved 1

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ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
2 Normative references

The following referenced documents are indispensable for the application 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.
JCGM 100:2008. Guide to the expression of uncertainty in measurement (GUM).
JCGM 101:2008. Evaluation of measurement data | Supplement 1 to the \Guide to the expression of uncer-
tainty in measurement" | Propagation of distributions using a Monte Carlo method.
JCGM 104:2009. Evaluation of measurement data | An introduction to the \Guide to the expression of
uncertainty in measurement" and related documents.
JCGM 200:2008. International Vocabulary of Metrology|Basic and General Concepts and Associated Terms
(VIM).
3 Terms and denitions
For the purposes of this Supplement, the denitions of the GUM and the VIM apply unless otherwise indicated.
Some of the most relevant denitions, adapted or generalized where necessary from these documents, are
given below. Further denitions are given, including denitions taken or adapted from other sources, that are
especially important for this Supplement.
A glossary of principal symbols used is given in annex D.
3.1
real quantity
quantity whose numerical value is a real number
3.2
complex quantity
quantity whose numerical value is a complex number
NOTE A complex quantityZ can be represented by two real quantities in Cartesian form
>
Z (Z ;Z ) =Z + iZ ;
R I R I
2
where> denotes \transpose", i =1 and Z and Z are, respectively, the real and imaginary parts ofZ, or in polar
R I
form
> iZ

Z (Z ;Z ) =Z (cosZ + i sinZ ) =Z e ;
r  r   r
where Z and Z are, respectively, the magnitude (amplitude) and phase ofZ.
r

3.3
vector quantity
set of quantities arranged as a matrix having a single column
3.4
real vector quantity
vector quantity with real components
EXAMPLE A real vector quantityX containingN real quantitiesX ;:::;X expressed as a matrix of dimension N 1:
1 N
2 3
X
1
6 7
. >
X = . = (X ;:::;X ) :
4 5 1 N
.
X
N

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JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
3.5

complex vector quantity
vector quantity with complex components

EXAMPLE A complex vector quantity Z containing N complex quantities Z ;:::;Z expressed as a matrix of
1 N
dimension N 1:
2 3
Z
1
6 7
. >
Z = . = (Z ;:::;Z ) :
4 5 1 N
.
Z
N
3.6
vector measurand
vector quantity intended to be measured
NOTE Generalized from JCGM 200:2008 denition 2.3.
3.7
measurement model
model of measurement
model
mathematical relation among all quantities known to be involved in a measurement
NOTE 1 Adapted from JCGM 200:2008 denition 2.48.
NOTE 2 A general form of a measurement model is the equation h(Y;X ;:::;X ) = 0, whereY , the output quantity
1 N
in the measurement model, is the measurand, the quantity value of which is to be inferred from information about input
quantities X ;:::;X in the measurement model.
1 N
NOTE 3 In cases where there are two or more output quantities in a measurement model, the measurement model
consists of more than one equation.
3.8
multivariate measurement model
multivariate model
measurement model in which there i
...