ISO/DGuide 99998

Guide to the expression of uncertainty in measurement (GUM) — Supplement 1: Numerical methods for the propagation of distributions

Guide pour l'expression de l'incertitude de mesure (GUM) — Supplément 1 : Méthodes numériques pour la propagation de distribution

General Information

• Status: Withdrawn
• 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: 4098 - Project deleted
• Start Date: 19-Oct-2006
• Completion Date: 14-Feb-2026

Standards Content (Sample)

ISO GUM Suppl. 1 (DGUIDE 99998)
TMB
Voting begins on Voting terminates on

2004-05-14 2004-09-14
INTERNATIONAL ORGANIZATION FOR STANDARDIZATION • МЕЖДУНАРОДНАЯ ОРГАНИЗАЦИЯ ПО СТАНДАРТИЗАЦИИ • ORGANISATION INTERNATIONALE DE NORMALISATION

Guide to the expression of uncertainty in measurement
(GUM) — Supplement 1: Numerical methods for the propagation
of distributions
Guide pour l'expression de l'incertitude de mesure (GUM) — Supplément 1: Méthodes numériques pour la
propagation de distribution
ICS 17.020
Please see the administrative notes on page iii

In accordance with the ISO/IEC Directives, Part 1, 2001, Clause A.5, this draft is submitted to the ISO national
bodies for approval.
WARNING — THIS DOCUMENT IS NOT AN ISO GUIDE. IT IS DISTRIBUTED FOR REVIEW AND COMMENT. IT IS SUBJECT TO CHANGE WITHOUT
NOTICE AND MAY NOT BE REFERRED TO AS A GUIDE.
RECIPIENTS OF THIS DRAFT ARE INVITED TO SUBMIT, WITH THEIR COMMENTS, NOTIFICATION OF ANY RELEVANT PATENT RIGHTS OF WHICH
THEY ARE AWARE AND TO PROVIDE SUPPORTING DOCUMENTATION.
IN ADDITION TO THEIR EVALUATION AS BEING ACCEPTABLE FOR INDUSTRIAL, TECHNOLOGICAL, COMMERCIAL AND USER PURPOSES,
DRAFT GUIDES MAY ON OCCASION HAVE TO BE CONSIDERED IN THE LIGHT OF THEIR POTENTIAL TO BECOME DOCUMENTS TO WHICH
REFERENCE MAY BE MADE IN NATIONAL REGULATIONS.
©
International Organization for Standardization, 2004

ISO GUM Suppl. 1 (IDGUIDE 99998)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.

Copyright notice
This ISO document is a Draft Guide and is copyright-protected by ISO. Except as permitted under the
applicable laws of the user's country, neither this ISO draft nor any extract from it may be reproduced,
stored in a retrieval system or transmitted in any form or by any means, electronic, photocopying, recording
or otherwise, without prior written permission being secured.
Requests for permission to reproduce should be addressed to either ISO at the address below or ISO's
member body in the country of the requester.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail [e-mail]
Web www.iso.org
Reproduction may be subject to royalty payments or a licensing agreement.
Violators may be prosecuted.
ii © ISO 2004 – All rights reserved

ISO GUM Suppl. 1 (DGUIDE 99998)
For convenience in the management of the enquiry ballot, this draft is submitted under the reference
ISO DGUIDE 99998. On publication the reference ISO DGUIDE 99998 will be discarded.

In accordance with the provisions of Council Resolution 15/1993 this document is circulated in the English
language only.
Conformément aux dispositions de la Résolution du Conseil 15/1993, ce document est distribué en version
anglaise seulement.
To expedite distribution, this document is circulated as received from the committee secretariat. ISO Central
Secretariat work of editing and text composition will be undertaken at publication stage.
Pour accélérer la distribution, le présent document est distribué tel qu'il est parvenu du secrétariat du comité.
Le travail de rédaction et de composition de texte sera effectué au Secrétariat central de l'ISO au stade de
publication.
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
Guide to the Expression of Uncertainty in Measurement
Supplement 1
Numerical Methods for the Propagation of Distributions
This version is intended for circulation to the mem- 7 Validation of the law of propagation of un-
ber organizations of the JCGM and National Mea- certainty using Monte Carlo simulation . 16
surement Institutes for review.
8 Examples. 17
8.1 Simple additive model . 17
8.1.1 Normally distributed input quan-
tities . 17
8.1.2 Rectangularly distributed input
quantities with the same width . . 18
8.1.3 Rectangularly distributed input
quantities with different widths . . 19
8.2 Mass calibration . 20
8.2.1 Formulation. 20
8.2.2 Calculation . 20
8.3 Comparison loss in microwave power me-
Contents Page
ter calibration . 22
8.3.1 Formulation. 22
Foreword. 2
8.3.2 Calculation: uncorrelated input
quantities . 22
Introduction. 2
8.3.3 Calculation: correlated input
quantities . 25
1 Scope . 3
Annexes
2 Notation and definitions . 4
A Historical perspective . 29
3 Concepts . 5
B Sensitivity coefficients . 30
4 Assignment of probability density func-
tions to the values of the input quantities . 7
C Sampling from probability distributions . 31
4.1 Probability density function assignment
C.1 General distributions . 31
for some common circumstances . 7
C.2 Rectangular distribution . 31
4.2 Probability distributions from previous
C.2.1 Randomness tests . 31
uncertainty calculations . 8
C.2.2 A recommended rectangular ran-
dom number generator . . 32
5 The propagation of distributions . 8
C.3 Gaussian distribution . 32
C.4 t–distribution . 32
6 Calculation using Monte Carlo simulation 9
C.5 Multivariate Gaussian distribution . 33
6.1 Rationale and overview. 9
6.2 The number of Monte Carlo trials. 12
D The comparison loss problem . 35
6.3 Sampling from probability distributions . . 12
D.1 The analytic solution for a zero value of
6.4 Evaluation of the model . 12
the voltage reflection coefficient. 35
6.5 Distribution function for the output
D.2 The law of propagation of uncertainty ap-
quantity value . 12
plied to the comparison loss problem . . . . 35
6.6 The estimate of the output quantity value
D.2.1 Uncorrelated input quantities. 35
and the associated standard uncertainty . 13
D.2.2 Correlated input quantities . 36
6.7 Coverage interval for the output quantity
value . 14
6.8 Reporting the results . 14
6.9 Computation time . 15
6.10Adaptive Monte Carlo procedure. 15
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
Foreword
This Supplement is concerned with the concept of the
propagation of distributions as a basis for the evaluation
Introduction
of uncertainty of measurement. This concept constitutes
a generalization of the law of propagation of uncertainty
This Supplement is concerned with the concept of
given in the Guide to the Expression of Uncertainty in
the propagation of probability distributions through a
Measurement (GUM) [3]. It thus facilitates the provi-
model of measurement as a basis for the evaluation of
sion of uncertainty evaluations that are more valid than
uncertainty of measurement, and its implementation by
those provided by the use of the law of propagation of
Monte Carlo simulation. The treatment applies to a
uncertainty in circumstances where the conditions for
model having any number of input quantities, and a sin-
the application of that law are not fulfilled. The prop-
gle (scalar-valued) output quantity (sometimes known
agation of distributions is consistent with the general
as the measurand). A second Supplement, in prepa-
principles on which the GUM is based. An implemen-
ration, is concerned with arbitrary numbers of output
tation of the propagation of distributions is given that
quantities. In particular, the provision of the probabil-
uses Monte Carlo simulation.
ity density function for the output quantity value per-
mits the determination of a coverage interval for that
In 1997 a Joint Committee for Guides in Metrology
value corresponding to a prescribed coverage probabil-
(JCGM), chaired by the Director of the BIPM, was
ity. The Monte Carlo simulation technique in general
created by the seven International Organizations that
provides a practical solution for complicated models or
had prepared the original versions of the GUM and the
models with input quantities having “large” uncertain-
International Vocabulary of Basic and General Terms
ties or asymmetric probability density functions. The
in Metrology (VIM). The Committee had the task of
evaluation procedure based on probability distributions
the ISO Technical Advisory Group 4 (TAG4), which had
is entirely consistent with the GUM, which states in
developed the GUM and the VIM. The Joint Commit-
Subclause 3.3.5 that “. .a Type A standard uncertainty
tee, as was the TAG4, is formed by the BIPM with
is obtained from a probability density function derived
the International Electrotechnical Commission (IEC),
from an observed frequency distribution, while a Type B
the International Federation of Clinical Chemistry and
standard uncertainty is obtained from an assumed prob-
Laboratory Medicine (IFCC), the International Orga-
ability density function based on the degree of belief
nization for Standardization (ISO), the International
that an event will occur . .”. It is also consistent in
Union of Pure and Applied Chemistry (IUPAC), the
the sense that it falls in the category of the “other an-
International Union of Pure and Applied Physics (IU-
alytical or numerical methods” [GUM Subclause G.1.5]
PAP) and the International Organization of Legal
permitted by the GUM. Indeed, the law of propagation
Metrology (OIML). A further organization joined these
of uncertainty can be derived from the propagation of
seven international organizations, namely, the Interna-
distributions. Thus, the propagation of distributions is
tional Laboratory Accreditation Cooperation (ILAC).
a generalization of the approach predominantly advo-
Within JCGM two Working Groups have been estab-
cated in the GUM, in that it works with richer infor-
lished. Working Group 1, “Expression of Uncertainty
mation than that conveyed by best estimates and the
in Measurement”, has the task to promote the use of
associated standard uncertainties alone.
the GUM and to prepare supplements for its broad ap-
plication. Working Group 2, “Working Group on In-
This supplement also provides a procedure for the vali-
ternational Vocabulary of Basic and General Terms in
dation, in any particular case, of the use of the law of
Metrology (VIM)”, has the task to revise and promote
propagation of uncertainty.
the use of the VIM. The present Guide has been pre-
pared by Working Group 1 of the JCGM.
This document is a supplement to the use of the GUM
and is to be used in conjunction with it.
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
1 Scope description of the approach as the propagation of distri-
butions. Unlike the GUM, it does not make use of the
law of propagation of uncertainty. That approach oper-
This Supplement provides guidance on the evaluation of
ates with the best estimates (expectations) of the val-
measurement uncertainty in situations where the con-
ues of the input quantities and the associated standard
ditions for the applicability of the law of propagation
uncertainties (and where appropriate the corresponding
of uncertainty and related concepts are not fulfilled or
degrees of freedom) in order to determine an estimate of
it is unclear whether they are fulfilled. It can also be
the output quantity value, the associated standard un-
used in circumstances where there are difficulties in ap-
certainty, and a coverage interval for the output quantity
plying the law of propagation of uncertainty, because of
value. Whereas there are some limitations to that ap-
the complexity of the model, for example. This guidance
proach, (a sound implementation of) the propagation of
includes a general alternative procedure, consistent with
distributions will always provide a probability density
the GUM, for the numerical evaluation of measurement
function for the output quantity value that is consistent
uncertainty, suitable for implementation by computer.
with the probability density functions for the values of
the input quantities. Once the probability density func-
In particular, this Supplement provides a procedure for
tion for the output quantity value is available, its ex-
determining a coverage interval for an output quantity
pectation is taken as an estimate of the output quantity
value corresponding to a specified coverage probability.
value, its standard deviation is used as the associated
The intent is to determine this coverage interval to a
standard uncertainty, and a 95 % coverage interval for
prescribed degree of approximation. This degree of ap-
the output quantity value is obtained from it.
proximation is relative to the realism of the model and
the quality of the information on which the probabil-
NOTE — Some distributions, such as the Cauchy distribu-
ity density functions for the model input quantities are
tion, which arise exceptionally, have no expectation or stan-
dard deviation. A coverage interval can always be obtained,
based.
however.
It is usually sufficient to report the measurement un-
The approach here obviates the need for “effective de-
certainty to one or perhaps two significant decimal dig-
grees of freedom” [GUM G.6.4] in the determination of
its. Further digits would normally be spurious, because
the expanded uncertainty, so avoiding the use of the
the information provided for the uncertainty evaluation
Welch-Satterthwaite formula [GUM G.4.2] and hence
is typically inaccurate, involving estimates and judge-
the approximation inherent in it.
ments. The calculation should be carried out in a way
to give a reasonable assurance that in terms of this in-
The probability density function for the output quantity
formation these digits are correct. Guidance is given on
value is not in general symmetric. Consequently, a cov-
this aspect.
erage interval for the output quantity value is not nec-
essarily centred on the estimate of the output quantity
NOTE — This attitude compares with that in mathematical
physics where a model (e.g., a partial differential equation) is
value. There are many coverage intervals corresponding
constructed and then solved numerically. The construction
to a specified coverage probability. This Supplement can
involves idealizations and inexactly known values for geo-
be used to provide the shortest coverage interval.
metric quantities and material constants, for instance. The
solution process should involve the application of suitable
NOTE — Sensitivity coefficients [GUM 5.1.3] are not an
numerical methods in order to make supported statements
inherent part of the approach and hence the calculation or
about the quality of the solution obtained to the posed prob-
approximation of the partial derivatives of the model with
lem.
respect to the input quantities is not required. Values akin
to sensitivity coefficients can, however, be provided using a
This Supplement provides a general numerical pro-
variant of the approach (Appendix B).
cedure, consistent with the broad principles of
the GUM [GUM G.1.5], for carrying out the calcula-
Typical of the uncertainty evaluation problems to which
tions required as part of an evaluation of measurement
this Supplement can be applied include
uncertainty. The procedure applies to arbitrary mod-
els having a single output quantity where the values of
— those where the contributory uncertainties may
the input quantities are assigned any specified probabil-
be arbitrarily large, even comparable to the uncer-
ity density functions, including asymmetric probability
tainty associated with the estimate of the output
density functions [GUM G.5.3].
quantity value;
The approach operates with the probability density
functions for the values of the input quantities in or- — those where the contributions to the uncertainty
der to determine the probability density function for associated with the estimate of the output quan-
the output quantity value. This is the reason for the tity value are not necessarily comparable in magni-
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
tude [GUM G.2.2]; General Terms in Metrology (VIM) [4] and ISO 3534,
Part 1 [20] apply.
— those where the probability distribution for the
output quantity value is not Gaussian, since re-
JCGM-WG1 has decided that the subscript “c”
liance is not placed on the Central Limit Theo-
[GUM 2.3.4, 5.1.1] for the combined standard uncer-
rem [GUM G.2.1];
tainty is redundant. The standard uncertainty associ-
ated with an estimate y of an output quantity value Y
— those where the estimate of the output quan-
cantherefore be writtensimplyas u(y), but the use
tity value and the associated standard uncertainty are
of u (y) remains acceptable if it is helpful to empha-
c
comparable in magnitude, as for measurements at or
size the fact that it represents a combined standard un-
near the limit of detection;
certainty. Moreover, the qualifier “combined” in “com-
bined standard uncertainty” is also regarded as super-
— those in which the models have arbitrary degrees
fluous and may be omitted. One reason for the deci-
of non-linearity or complexity, since the determination
sion is that the argument (here y) already indicates the
of the terms in a Taylor series approximation is not
estimate of the output quantity value with which the
required [GUM 5.1.2];
standard uncertainty is associated. Another reason is
that frequently the results of one or more uncertainty
— those in which asymmetric distributions for the
evaluations become the inputs to a subsequent uncer-
values of the input quantities arise, e.g., when dealing
tainty evaluation. The use of the subscript “c” and the
with the magnitudes of complex variables in acousti-
qualifier “combined” are inappropriate in this regard.
cal, electrical and optical metrology;
This Supplement departs from the symbols often used
— those in which it is difficult or inconvenient to
for probability density function and distribution func-
provide the partial derivatives of the model (or ap-
tion. The GUM uses the generic symbol f to refer to a
proximations to these partial derivatives), as needed
model and a probability density function. Little confu-
by the law of propagation of uncertainty (possibly
sion arises in the GUM as a consequence of this usage.
with higher-order terms) [GUM 8].
The situation in this Supplement is different. The con-
cepts of model, probability density function and distri-
This Supplementcanbeusedincasesofdoubttocheck
bution function are central to following and implement-
whether the law of propagation of uncertainty is appli-
ing the procedure provided. Therefore, in place of the
cable. A validation procedure is provided for this pur-
symbols f and F to denote a probability density func-
pose. Thus, the considerable investment in this use of
tion and a distribution function, the symbols g and G,
the GUM is respected: the law of propagation of un-
respectively, are used. The symbol f is reserved for the
certainty procedure remains the main approach to the
model.
calculation phase of uncertainty evaluation, certainly in
circumstances where it is demonstrably applicable.
Citations of the form [GUM 4.1.4] are to the indicated
(sub)clauses of the GUM.
Guidance is given on the manner in which the propaga-
tion of distributions can be carried out, without making
The decimal point is used as the symbol to separate the
unquantified approximations.
integer part of a decimal number from its fractional part.
A decimal comma is used for this purpose in continental
This Supplement applies to mutually independent in-
Europe.
put quantities, where the value of each such quantity
is assigned an appropriate probability density function,
In this Supplement the term law of propagation of un-
or mutually dependent input quantities, the values of
certainty applies to the use of a first-order Taylor series
which have been assigned a joint probability density
approximation to the model. The term is qualified ac-
function.
cordingly when a higher-order approximation is used.
Sometimes the term is extended to apply also to the as-
Models with more than one output quantity are the
sumption of the applicability of the Central Limit The-
subject of a further Supplement to the GUM that is
orem as a basis for providing coverage intervals. The
in preparation.
context makes clear the usage in any particular case.
2 Notation and definitions
For the purposes of this Supplement the definitions of
the GUM [3], the International Vocabulary of Basic and
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
3 Concepts perhaps with expert support. (Advice on formulation
stages a)–c) will be provided in a further Supplement
A model of measurement having any number of input to the GUM on modelling that is under development.)
quantities and a single (scalar-valued) output quantity Guidance on the assignment of probability density func-
is considered. For this case, the main stages in the de- tions (Stage d) above) is given in this Supplement for
termination of an estimate of the output quantity value, some common cases. The calculation stages, e) and f),
the associated standard uncertainty, and a coverage in- for which detailed guidance is provided here, require no
terval for the output quantity value are as follows. further metrological information, and in principle can
be carried out to any required degree of approximation,
a) Define the output quantity, the quantity required relative to how well the formulation stages have been
to be measured. undertaken.
b) Decide the input quantities upon which the out- A measurement model [GUM 4.1] is expressed by a func-
put quantity depends. tional relationship f:
Y = f(X), (1)
c) Develop a model relating the output quantity to
these input quantities.
where Y is a single (scalar) output quantity (the out-
put quantity) and X represents the N input quanti-
d) On the basis of available knowledge assign proba-
T
ties (X ,.,X ) .
1 N
bility density functions [GUM C.2.5] —Gaussian (nor-
mal), rectangular (uniform), etc.—to the values of the
NOTES
input quantities.
1 It is not necessary that Y is given explicitly in terms
NOTES
of X,i.e., f constitutes a formula. It is only neces-
sary that a prescription is available for determining Y
1 Assign instead a joint probability density function to given X [GUM 4.1.2].
the values of those input quantities that are mutually de-
pendent.
2 In this Supplement, T in the superscript position denotes
“transpose”, and thus X represents X ,.,X arranged as
1 N
2 A probability density function for the values of more a column (vector) of values.
than one input quantity is commonly called “joint” even
if the probability density functions for the values of all the
The GUM provides general guidance on many aspects of
input quantities are mutually independent.
the above stages. It also contains a specific procedure,
the law of propagation of uncertainty [GUM 5.1, 5.2],
e) Propagate the probability density functions for
for the calculation phase of uncertainty evaluation.
the values of the input quantities through the model
to obtain the probability density function for the out-
The law of propagation of uncertainty has been adopted
put quantity value.
by many organizations, is widely used and has been im-
plemented in standards and guides on measurement un-
f) Obtain from the probability density function for
certainty and also in computer packages. In order to
the output quantity value
apply this law, the values of the model input quantities
are summarized by the expectations and standard devi-
1) its expectation, taken as the estimate of the
ations of the probability density functions for these val-
output quantity value;
ues. This information is “propagated” through a first-
order Taylor series approximation to the model to pro-
NOTE — The expectation may not be appropriate for
vide an estimate of the output quantity value and the
all applications (Clause 6.1, [GUM 4.1.4]).
associated standard uncertainty. That estimate of the
output quantity value is given by evaluating the model
2) its standard deviation, taken as the standard
uncertainty associated with the estimate of the out- at the best estimates of the values of the input quanti-
put quantity value [GUM E.3.2]; ties. A coverage interval for the output quantity value is
provided based on taking the probability density func-
3) an interval (the coverage interval) containing tion for the output quantity value as Gaussian.
the unknown output quantity value with a specified
probability (the coverage probability). The intent of the GUM is to derive the expectation and
standard deviation of the probability density function
Stages a)–d) are regarded in this Supplement as for- for the output quantity value, having first determined
mulation, and Stages e) and f) as calculation.The the expectations and standard deviations of the proba-
formulation stages are carried out by the metrologist, bility density functions for the values of the input quan-
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
tities. NOTE — The X are regarded as random variables with
i
possible values ξ and expectations x .
i i
NOTES
b) Take the covariances (mutual uncertain-
1 The best estimates of the values of the input quantities
ties) [GUM C] u(x ,x)asCov(X ,X ), the
i j i j
are taken as the expectations of the corresponding probabil-
covariances of mutually dependent pairs (X,X )of
i j
ity density functions [GUM 4.1.6].
input quantities.
2 The summaries of values of the input quantities also in-
c) Form the partial derivatives of first order of f
clude, where appropriate, the degrees of freedom of the stan-
dard uncertainties associated with the estimates of the values with respect to the input quantities.
of the input quantities [GUM 4.2.6].
d) Calculate the estimate y of the output quantity
3 The summaries of the values of the input quantities also
value by evaluating the model at x.
include, where appropriate, covariances associated with the
estimates of the values of input quantities [GUM 5.2.5].
e) Calculate the model sensitivity coefficients
[GUM 5.1] as the above partial derivatives evaluated
4 The GUM [Note to GUM Subclause 5.1.2] states that
if the non-linearity of the model is significant, higher-order at x.
terms in the Taylor series expansion must be included in the
expressions for the standard uncertainty associated with the
f) Determine the standard uncertainty u(y) by com-
estimate of the output quantity value.
bining u(x), the u(x ,x ) and the model sensitivity
i j
coefficients [GUM Formulae (10) and (13)].
5 If the analytic determination of the higher derivatives,
required when the non-linearity of the model is significant,
is difficult or error-prone, suitable software systems for au- g) Calculate ν, the effective degrees of freedom of y,
tomatic differentiation can be used. Alternatively, these
using the Welch-Satterthwaite formula [GUM For-
derivatives can be calculated numerically using finite differ-
mula (G.2b)].
ences [GUM 5.1.3]. Care should be taken, however, because
of the effects of subtractive cancellation when forming dif-
h) Compute the expanded uncertainty U,and
ferences in values of the model for close values of the input p
quantities. hence a coverage interval for the output quantity
value (having a stipulated coverage probability p),
6 The most important terms of next highest order to be
by forming the appropriate multiple of u(y) through
added to those of the formula in GUM Subclause 5.1.2 for the
taking the probability distribution of (y − Y )/u(y)
standard uncertainty are given in the Note to this subclause.
as a standard Gaussian distribution (ν = ∞)or t–
Although not stated in the GUM, this formula applies when
distribution (ν<∞).
the values of X are Gaussian. In general, it would not apply
i
for other probability density functions.
7 The statement in the GUM [Note to GUM Sub-
clause 5.1.2] concerning significant model non-linearity re-
lates to input quantities that are mutually independent. No �
x , u(x )
1 1
guidance is given in the GUM if they are mutually depen-
dent, but it is taken that the same statement would apply. � �
x , u(x ) y, u(y)
2 2
Y = f(X)
�
x , u(x )
8 A probability density function related to a t–distribution 3 3
is used instead of a Gaussian probability density function if
the effective degrees of freedom associated with the estimate
of the standard deviation of the probability density function
Figure 1 — Illustration of the law of propagation of
for the output quantity value is finite [GUM G].
uncertainty. The model has mutually independent
T
input quantities X =(X ,X ,X ) , whose values are
1 2 3
The calculation stages (Stages e) and f) above) of
estimated by x with associated standard uncertain-
i
the GUM that use the law of propagation of uncertainty
ties u(x ),for i =1, 2, 3. The value of the output
i
and the abovementioned concepts can be summarized as
quantity Y is estimated by y, with associated stan-
the following computational steps. Also see Figure 1.
dard uncertainty u(y).
a) Obtain from the probability density functions for
the values of the input quantities X ,.,X , respec-
1 N
The computational steps above require the following
T
tively, the expectation x =(x ,.,x ) and the
1 N
conditions to hold:
standard deviations (standard uncertainties) u(x)=
T
(u(x ),.,u(x )) . Use the joint probability den-
1 N
a) the non-linearity of f to be insignificant [Note to
sity function for the value of X instead if the X are
i
mutually dependent. GUM 5.1.2];
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
b) the Central Limit Theorem [GUM G.2.1, G.6.6] 2 It may be possible to remove some mutual dependencies
by re-expressing some or all of the input quantities in terms
to apply, implying the representativeness of the prob-
of more fundamental mutually independent input quantities
ability density function for the output quantity value
on which the original input quantities depend [GUM F1.2.4,
by a Gaussian distribution or in terms of a t–
GUM H.1]. Such changes can simplify both the application
distribution;
of the law of propagation of uncertainty and the propagation
of distributions. Details and examples are available [13].
c) the adequacy of the Welch-Satterthwaite for-
mula for calculating the effective degrees of free-
dom [GUM G.4.2].
NOTE — The last two conditions are required for com-
putational steps g) and h) above. 4.1 Probability density function assign-
ment for some common circumstances
When these three conditions hold, the results from the
sound application of the law of propagation of uncer-
Assignments of probability density functions to the in-
tainty are valid. These conditions apply in many cir-
put quantities are given in Table 1 for some common
cumstances. The approach is not always applicable,
circumstances.
however. This Supplement provides a more general ap-
proach that does not require these conditions to hold.
All available information Probability density function
concerning quantity X (PDF) assigned to the value
of X
4 Assignment of probability density
The estimate x and the The Gaussian PDF
functions to the values of the input associated standard uncer- N(x, u (x))
tainty u(x)
quantities
The estimate x (> 0), and The exponential PDF
In the first phase—formulation—of uncertainty eval- X is known to be nonneg- with expectation x,viz.,
ative exp(−ξ/x)/x, for ξ ≥ 0, and
uation, the probability density functions for the val-
zero otherwise.
ues of the input quantities of the model are as-
√
signed [GUM 2.3.2, 3.3.5] based on an analysis of series
Independent observations Product of n/s and the
of observations or based on scientific judgement [GUM
of a quantity value taken t–distribution with argument
√
2.3.3, 3.3.5] using all the relevant information [38], such to follow a normal law (ξ− x¯)/(s/ n) and n− 1 de-
with unknown expectation grees of freedom and where x¯
as historical data, calibrations and expert judgement.
equal to the value of X. and s are known constants
From a sample of size n,
The probability density function for the possible val-
an arithmetic mean x¯ and
ues ξ of the ith input quantity X is denoted by g (ξ )
i i i i
a standard deviation s have
and that for the possible values of the output quantity been calculated
value Y by g(η). The distribution function for X is
i
The estimate x of the value The multivariate Gaussian
denoted by G (ξ ) and that for Y by G(η). The prob-
i i
of a multivariate quan- PDF N(x, V ) is assigned to
ability density functions and the distribution functions
tity X and the correspond- the value of X (Section 5,
� �
are related by g (ξ )= G (ξ)and g(η)= G (η).
i i i ing uncertainty matrix (co- Note 2)
i
variance matrix) V
When the input quantities are mutually dependent,
The endpoints a and a The rectangular PDF with
− +
in place of the N individual probability density func-
of an interval containing endpoints a and a
− +
tions g (ξ ), i =1,.,N, there is a joint probability
i i
the value of X
density function g(ξ). See Notes 1 and 2 at the end of
Section 5. Intermediate to these extremes, groups of the The lower and upper lim- The scaled and shifted arcsine
its a and a of an inter- PDF with endpoints a and
− + −
input quantities may have values with joint probability
val within which the value a ,viz., (2/π)/{(a − a ) −
+ + −
density functions.
2 1/2
of X is known to cycle si-
(2ξ− a − a ) } , for a <
+ − −
nusoidally
ξ< a , and zero otherwise
+
Clauses 4.2 and 4.3 of the GUM contain much relevant
[14], [17, Section 3.5]
information on the assignment of probability density
Table 1 — The assignment of a probability density
functions.
function to the value of an input quantity X based
on available information for some common circum-
NOTES
stances.
1 The Principle of Maximum Information Entropy can be
applied to assist in the assignment [7, 39, 40].
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
4.2 Probability distributions from previous a) those for which a general approach is needed;
uncertainty calculations
b) those for which uncertainty propagation based on
a first-order Taylor series approximation is applicable;
A previous uncertainty calculation may have provided a
probability distribution for the value of an output quan-
c) those for which it is unclear which approach
tity that is to become an input quantity for a further un-
should be followed.
certainty calculation. This probability distribution may
be available analytically in a recognized form, e.g., as
For Class a), this Supplement provides a generic, broadly
a Gaussian probability density function, with values for
applicable approach based on the propagation of distri-
its expectation and standard deviation. It may be avail-
butions. With respect to Class b), this Supplement does
able as an approximation to the distribution function for
not provide new material. For Class c), this Supplement
a quantity value obtained from a previous application of
provides a procedure for validating in any particular cir-
Monte Carlo simulation, for example. A means for de-
cumstance the use of the law of propagation of uncer-
scribing such a distribution function for a quantity value
tainty (possibly based on a higher-order Taylor series
is given in Clause 6.5.
approximation).
The propagation of the probability density func-
5 The propagation of distributions
tions g (ξ ), i =1,.,N, for the values of the input
i i
quantities through the model to provide the probability
Several approaches can be used for the second phase —
density function g(η) for the output quantity value is
calculation — of uncertainty evaluation:
illustrated in Figure 2 for the case N =3. This figure
is the counterpart of Figure 1 for the law of propaga-
a) analytical methods;
tion of uncertainty. Like the GUM, this Supplement is
concerned with models having a single output quantity.
b) uncertainty propagation based on replacing the
model by a first-order Taylor series approxima-
tion [GUM 5.1.2] — the law of propagation of un-
�
certainty;
g (ξ )
1 1
c) as b), except that contributions derived from
� �
Y = f(X)
higher-order terms in the Taylor series approximation
g (ξ ) g(η)
are included [Note to GUM 5.1.2]; 2 2
�
d) numerical methods [GUM G.1.5] that implement
the propagation of distributions, specifically Monte
g (ξ )
3 3
Carlo simulation (Section 6).
Figure 2 — Illustration of the propagation
NOTE — Analytical methods are ideal in that they do not of distributions. The model input quantities
T
introduce any approximation. They are applicable in simple
are X =(X ,X ,X ) . The probability density func-
1 2 3
cases only, however. A treatment and examples are avail-
tions g (ξ ),for X , i=1, 2, 3, are Gaussian, triangular
i i i
able [7, 12]. These methods are not considered further in
and Gaussian, respectively. The probability density
this Supplement, apart from in the examples section (Sec-
function g(η) for the value of the output quantity Y is
tion 8.1.1) for comparison purposes.
indicated as being asymmetric, as can arise for non-
linear models. (An asymmetric g(η) can also arise
Techniques other than the law of propagation of uncer-
when the probability density functions for the val-
tainty are permitted by the GUM [GUM G.1.5]. The
ues of the input quantities are asymmetric.)
approach advocated in this Supplement, based on the
propagation of distributions, is general. For linear or
linearized models and input quantities with values for
NOTES
which the probability density functions are Gaussian,
the approach yields results consistent with the law of
1 The only joint probability density functions considered
propagation of uncertainty. But in cases where the law
in this Supplement are multivariate Gaussian.
of propagation of uncertainty cannot be applied the ad-
vocated approach still gives correct uncertainty state- 2 A multivariate Gaussian probability density function
T
with expectation x =(x ,.,x ) and uncertainty ma-
ments. 1 N
trix V is given by
� �
In terms of the calculations required, there are three
1 1
T −1
g(ξ)= exp − (ξ− x) V (ξ− x) .
classes of uncertainty evaluation problem: N 1/2
((2π) det V ) 2
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
This probability density function reduces to the product of N as stated in the note to GUM Subclause 4.1.4, repeated
univariate Gaussian probability density functions when there
above, it plays a relevant role within Monte Carlo sim-
are no covariance effects, for the following reason. In that
ulation as an implementation of the propagation of dis-
case
2 2 tributions.
V = diag(u (x ),.,u (x )),
1 N
whence
Monte Carlo simulation for uncertainty calculations [7,
� �
N
� 2
9] is based on the premise that any value drawn at ran-
1 1 (ξ − x )
i i
g(ξ)= exp −
N/2 2 dom from the distribution of possible values of an input
(2π) u(x )··· u(x ) 2 u (x )
1 N i
i=1
quantity is as legitimate as any other such value. Thus,
N
�
by drawing for each input quantity a value according to
= g (ξ ),
i i
its assigned probability density function, the resulting
i=1
set of values is a legitimate set of values of these quanti-
with ties. The value of the model corresponding to this set of
� �
1 (ξ − x )
i i
values constitutes a possible value of the output quan-
g (ξ )= √ exp − .
i i
2u (x )
2πu(x ) i
i
tity Y . Figure 3 is similar to Figure 2 except that it
shows a value sampled from each of the three proba-
bility density functions for the input quantities and the
6 Calculation using Monte Carlo simu-
resulting value of the output quantity.
lation
�
6.1 Rationale and overview
�
Monte Carlo simulation provides a general approach
� �
to numerically approximating the distribution func-
Y = f(X)
� �
tion G(η) for the value of the output quantity Y =
f(X). The following GUM Subclause is relevant to the
concept embodied in Monte Carlo simulation:
�
�
An estimate of the measurand Y,denoted by y,
Figure 3 — As Figure 2 except that what is shown
is obtained from equation (1) [identical to ex-
is a value sampled from each of the three probability
pression (1) of this Supplement]using input esti-
density functions for the values of the input quanti-
mates x ,x ,.,x for the values of the N quan-
1 2 N
ties, and the resulting output quantity value.
tities X ,X ,.,X .Thusthe output estimate y,
1 2 N
whichistheresultofthemeasurement,isgivenby
Consequently, a large set of model values so obtained
y = f(x ,x ,.,x ) .(2)
1 2 N
can be used to provide an approximation to the distri-
NOTE – In some cases the estimate y may be obtained
bution of possible values for the output quantity. Monte
from
Carlo simulation can be regarded as a generalization of
n n
� � GUM 4.1.4 (above) to obtain the distribution for Y ,
1 1
¯
y = Y = Y = f(X ,X ,.,X )
k 1,k 2,k N,k
rather than the expectation of Y . In particular, the
n n
k=1 k=1
above drawing of a value from the probability density
function for each input quantity corresponds to “a com-
That is, y is taken as the arithmetic mean or aver-
plete set of observed values of the N input quantities X
age (see 4.2.1) of n independent determinations Y i
k
obtained at the same time” in GUM 4.1.4.
of Y , each determination having the same uncertainty
and each being based on a complete set of observed
Monte Carlo simulation operates as follows:
values of the N input quantities X obtained at the
i
same time. This way of averaging, rather than y =
�
n
¯ ¯ ¯ ¯
f(X , X ,., X ), where X =( X )/n is the — Generate a sample of size N by independently
1 2 N i i,k
k=1
arithmetic mean of the individual observations X , sampling at random from the probability density
i,k
may be preferable when f is a nonlinear function of function for each X , i =1,.,N (or in the case of
i
the input quantities X ,X ,.,X , but the two ap- mutually dependent input quantities, from the joint
1 2 N
proaches are identical if f is a linear function of the X probability density function for X). Repeat this pro-
i
(see H.2 and H.4). [GUM 4.1.4] cedure a large number, M,say,oftimestoyield M in-
dependent samples of size N of the set of input quan-
Although the GUM formula (2) need not provide the tities. For each such independent sample of size N,
most meaningful estimate of the output quantity value, calculate the resulting model value of Y , yielding M
JCGM-WG1-SC1-N10
Supplement 1. 2004-03-16 : 2004
values of Y in all. Monte Carlo simulation as an implementation of the
propagation of distributions is shown diagrammatically
NOTES
in Figure 4 and can conveniently be stated as a step-by-
step procedure:
1 Appendix C provides information on sampling from
probability distributions.
a) Select the number M of Monte Carlo trials to be
made. See Clause 6.2.
2 Clause 6.4 gives more explicit information concerning
the sample taken.
b) Generate M samples of the (set of N) input quan-
3 According to the Central Limit Theorem [30, p169],
tities. See Clause 6.3.
the arithmetic mean of the M values of the output quan-
1/2
tity obtained in this manner converges as 1/M to the
c) For each sample, evaluate the model to give the
expectation of the probability density function for the
corresponding output quantity value. See Clause 6.4.
value of Y = f(X).
d) Sort these values of the output quantity into non-
�
— Use these M values of Y to provide G(η), an
decreasing order, using the sorted values to approxi-
approximation to the distribution function G(η)for
mate the distribution funct
...