Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) — Supplement 1: Propagation of distributions using a Monte Carlo method

ISO/IEC Guide 98-3/Suppl.1:2008 provides a general numerical approach, consistent with the broad principles of the Guide to the expression of uncertainty in measurement (GUM), for carrying out the calculations required as part of an evaluation of measurement uncertainty. The approach applies to arbitrary models having a single output quantity where the input quantities are characterized by any specified probability density functions (PDFs). ISO/IEC Guide 98-3/Suppl.1:2008 is primarily concerned with the expression of uncertainty in the measurement of a well-defined physical quantity—the measurand—that can be characterized by an essentially unique value. It provides guidance in situations where the conditions for the GUM uncertainty frameworkare not fulfilled, or it is unclear whether they are fulfilled. It can be used when it is difficult to apply the GUM uncertainty framework, because of the complexity of the model, for example. Guidance is given in a form suitable for computer implementation. ISO/IEC Guide 98-3/Suppl.1:2008 can be used to provide (a representation of) the PDF for the output quantity from which (a) an estimate of the output quantity, (b) the standard uncertainty associated with this estimate, and (c) a coverage interval for that quantity, corresponding to a specified coverage probability, can be obtained. For a prescribed coverage probability, it can be used to provide any required coverage interval, including the probabilistically symmetric coverage interval and the shortest coverage interval. ISO/IEC Guide 98-3/Suppl.1:2008 applies to input quantities that are independent, where each such quantity is assigned an appropriate PDF, or not independent, i.e. when some or all of these quantities are assigned a joint PDF. Detailed examples illustrate the guidance provided.

Incertitude de mesure — Partie 3: Guide pour l'expression de l'incertitude de mesure (GUM:1995) — Supplément 1: Propagation de distributions par une méthode de Monte Carlo

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Publication Date
19-Nov-2008
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6060 - International Standard published
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10-Sep-2010
Completion Date
20-Nov-2008
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GUIDE 98-3/Suppl.1

Uncertainty of measurement
Part 3:
Guide to the expression of
uncertainty in measurement
(GUM:1995)
Supplement 1:
Propagation of distributions
using a Monte Carlo method



First edition 2008
©
 ISO/IEC 2008

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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
Contents Page
Foreword .v
Introduction.vi
1 Scope.1
2 Normative references.2
3 Terms and definitions .2
4 Conventions and notation .6
5 Basic principles .8
5.1 Main stages of uncertainty evaluation .8
5.2 Propagation of distributions .9
5.3 Obtaining summary information.9
5.4 Implementations of the propagation of distributions.10
5.5 Reporting the results .11
5.6 GUM uncertainty framework .12
5.7 Conditions for valid application of the GUM uncertainty framework for linear models .13
5.8 Conditions for valid application of the GUM uncertainty framework for non-linear models .14
5.9 Monte Carlo approach to the propagation and summarizing stages .15
5.10 Conditions for the valid application of the described Monte Carlo method .16
5.11 Comparison of the GUM uncertainty framework and the described Monte Carlo method .17
6 Probability density functions for the input quantities.18
6.1 General .18
6.2 Bayes’ theorem.19
6.3 Principle of maximum entropy.19
6.4 Probability density function assignment for some common circumstances .20
6.4.1 General .20
6.4.2 Rectangular distributions.20
6.4.3 Rectangular distributions with inexactly prescribed limits .20
6.4.4 Trapezoidal distributions.22
6.4.5 Triangular distributions .23
6.4.6 Arc sine (U-shaped) distributions.24
6.4.7 Gaussian distributions.25
6.4.8 Multivariate Gaussian distributions .25
6.4.9 t-distributions.26
6.4.10 Exponential distributions .28
6.4.11 Gamma distributions.28
6.5 Probability distributions from previous uncertainty calculations .29
7 Implementation of a Monte Carlo method.29
7.1 General .29
7.2 Number of Monte Carlo trials .29
7.3 Sampling from probability distributions.29
7.4 Evaluation of the model.30
7.5 Discrete representation of the distribution function for the output quantity.30
7.6 Estimate of the output quantity and the associated standard uncertainty.31
7.7 Coverage interval for the output quantity.31
7.8 Computation time.32
7.9 Adaptive Monte Carlo procedure.32
7.9.1 General .32
7.9.2 Numerical tolerance associated with a numerical value.32
7.9.3 Objective of adaptive procedure.33

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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
7.9.4 Adaptive procedure .33
8 Validation of results .35
8.1 Validation of the GUM uncertainty framework using a Monte Carlo method.35
8.2 Obtaining results from a Monte Carlo method for validation purposes .35
9 Examples .36
9.1 Illustrations of aspects of this Supplement .36
9.2 Additive model .37
9.2.1 Formulation .37
9.2.2 Normally distributed input quantities.37
9.2.3 Rectangularly distributed input quantities with the same width.39
9.2.4 Rectangularly distributed input quantities with different widths .41
9.3 Mass calibration.42
9.3.1 Formulation .42
9.3.2 Propagation and summarizing .43
9.4 Comparison loss in microwave power meter calibration.45
9.4.1 Formulation .45
9.4.2 Propagation and summarizing: zero covariance.46
9.4.3 Propagation and summarizing: non-zero covariance.51
9.5 Gauge block calibration .53
9.5.1 Formulation: model .53
9.5.2 Formulation: assignment of PDFs .55
9.5.3 Propagation and summarizing .58
9.5.4 Results .59
Annex A Historical perspective.61
Annex B Sensitivity coefficients and uncertainty budgets .62
Annex C Sampling from probability distributions.63
C.1 General.63
C.2 General distributions.63
C.3 Rectangular distribution .64
C.4 Gaussian distribution.65
C.5 Multivariate Gaussian distribution.66
C.6 t-distribution.67
Annex D Continuous approximation to the distribution function for the output quantity.69
Annex E Coverage interval for the four-fold convolution of a rectangular distribution .72
Annex F Comparison loss problem .74
F.1 Expectation and standard deviation obtained analytically .74
F.2 Analytic solution for zero estimate of the voltage reflection coefficient having associated
zero covariance.75
F.3 GUM uncertainty framework applied to the comparison loss problem .76
Annex G Glossary of principal symbols.78
Bibliography .83
Alphabetical index .86



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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
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.
This first edition of Supplement 1 to ISO/IEC Guide 98-3 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. For further information, see the Introduction (0.2).
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)
The following parts are planned:
⎯ Part 2: Concepts and basic principles
⎯ Part 4: Role of measurement uncertainty in conformity assessment
⎯ Part 5: Applications of the least-squares method
ISO/IEC Guide 98-3 has one supplement.
⎯ Supplement 1: Propagation of distributions using a Monte Carlo method
The following supplements to ISO/IEC Guide 98-3 are planned:
⎯ Supplement 2: Models with any number of output quantities
⎯ Supplement 3: Modelling
Note that in this document, GUM is used to refer to the industry-recognized publication, adopted as
ISO/IEC Guide 98-3:2008. When a specific clause or subclause number is cited, the reference is to
ISO/IEC Guide 98-3:2008.


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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
Introduction
0.1 General
This Supplement to the Guide to the expression of uncertainty in measurement (GUM) is concerned with the
propagation of probability distributions through a mathematical model of measurement
[ISO/IEC Guide 98-3:2008, 3.1.6] as a basis for the evaluation of uncertainty of measurement, and its
implementation by a Monte Carlo method. The treatment applies to a model having any number of input
quantities, and a single output quantity.
The described Monte Carlo method is a practical alternative to the GUM uncertainty framework
[ISO/IEC Guide 98-3:2008, 3.4.8]. It has value when
a) linearization of the model provides an inadequate representation or
b) the probability density function (PDF) for the output quantity departs appreciably from a Gaussian
distribution or a scaled and shifted t-distribution, e.g. due to marked asymmetry.
In case a), the estimate of the output quantity and the associated standard uncertainty provided by the GUM
uncertainty framework might be unreliable. In case b), unrealistic coverage intervals (a generalization of
“expanded uncertainty” in the GUM uncertainty framework) might be the outcome.
The GUM [ISO/IEC Guide 98-3:2008, 3.4.8] “…provides a framework for assessing uncertainty …”, based on
the law of propagation of uncertainty [ISO/IEC Guide 98-3:2008, Clause 5] and the characterization of the
output quantity by a Gaussian distribution or a scaled and shifted t-distribution
[ISO/IEC Guide 98-3:2008, G.6.2, G.6.4]. Within that framework, the law of propagation of uncertainty
provides a means for propagating uncertainties through the model. Specifically, it evaluates the standard
uncertainty associated with an estimate of the output quantity, given
1) best estimates of the input quantities,
2) the standard uncertainties associated with these estimates, and, where appropriate,
3) degrees of freedom associated with these standard uncertainties, and
4) any non-zero covariances associated with pairs of these estimates.
Also within the framework, the PDF taken to characterize the output quantity is used to provide a coverage
interval, for a stipulated coverage probability, for that quantity.
The best estimates, standard uncertainties, covariances and degrees of freedom summarize the information
available concerning the input quantities. With the approach considered here, the available information is
encoded in terms of PDFs for the input quantities. The approach operates with these PDFs in order to
determine the PDF for the output quantity.
Whereas there are some limitations to the GUM uncertainty framework, the propagation of distributions will
always provide a PDF for the output quantity that is consistent with the PDFs for the input quantities. This PDF
for the output quantity describes the knowledge of that quantity, based on the knowledge of the input
quantities, as described by the PDFs assigned to them. Once the PDF for the output quantity is available, that
quantity can be summarized by its expectation, taken as an estimate of the quantity, and its standard
deviation, taken as the standard uncertainty associated with the estimate. Further, the PDF can be used to
obtain a coverage interval, corresponding to a stipulated coverage probability, for the output quantity.

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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
The use of PDFs as described in this Supplement is generally consistent with the concepts underlying the
GUM. The PDF for a quantity expresses the state of knowledge about the quantity, i.e. it quantifies the degree
of belief about the values that can be assigned to the quantity based on the available information. The
information usually consists of raw statistical data, results of measurement, or other relevant scientific
statements, as well as professional judgement.
In order to construct a PDF for a quantity, on the basis of a series of indications, Bayes’ theorem can be
applied [27, 33]. When appropriate information is available concerning systematic effects, the principle of
maximum entropy can be used to assign a suitable PDF [51, 56].
The propagation of distributions has wider application than the GUM uncertainty framework. It works with
richer information than that conveyed by best estimates and the associated standard uncertainties (and
degrees of freedom and covariances when appropriate).
Decimal sign: The decimal sign in the English text is the point on the line, and the comma on the line is the
decimal sign in the French text. (See 4.12)
An historical perspective is given in Annex A.
NOTE 1 The GUM provides an approach when linearization is inadequate [ISO/IEC Guide 98-3:2008, 5.1.2 Note]. The
approach has limitations: only the leading non-linear terms in the Taylor series expansion of the model are used, and the
PDFs for the input quantities are regarded as Gaussian.
NOTE 2 Strictly, the GUM characterizes the variable (Y − y)/u(y) by a t-distribution, where Y is the output quantity, y an
estimate of Y, and u(y) the standard uncertainty associated with y [ISO/IEC Guide 98-3:2008, G.3.1]. This characterization
is also used in this Supplement. [The GUM in fact refers to the variable (y − Y)/u(y).]
NOTE 3 A PDF for a quantity is not to be understood as a frequency density.
NOTE 4 “The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed
knowledge of the nature of the measurand and of the measurement method and procedure used. The quality and utility of
the uncertainty quoted for the result of a measurement therefore ultimately depends on the understanding, critical
analysis, and integrity of those who contribute to the assignment of its value.” [17].
0.2 JCGM background information
In 1997, the 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 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). A
further organization joined these seven international organizations, namely, the International Laboratory
Accreditation Cooperation (ILAC).
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. For further information on the activity of the JCGM, see
www.bipm.org.
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.


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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)

Uncertainty of measurement
Part 3:
Guide to the expression of uncertainty in measurement
(GUM:1995)
Supplement 1:
Propagation of distributions using a Monte Carlo method
1 Scope
This Supplement provides a general numerical approach, consistent with the broad principles of the GUM
[ISO/IEC Guide 98-3:2008, G.1.5], for carrying out the calculations required as part of an evaluation of
measurement uncertainty. The approach applies to arbitrary models having a single output quantity where the
input quantities are characterized by any specified PDFs [ISO/IEC Guide 98-3:2008, G.1.4, G.5.3].
As in the GUM, this Supplement is primarily concerned with the expression of uncertainty in the measurement
of a well-defined physical quantity—the measurand—that can be characterized by an essentially unique value
[ISO/IEC Guide 98-3:2008, 1.2].
This Supplement also provides guidance in situations where the conditions for the GUM uncertainty
framework [ISO/IEC Guide 98-3:2008, G.6.6] are not fulfilled, or it is unclear whether they are fulfilled. It can
be used when it is difficult to apply the GUM uncertainty framework, because of the complexity of the model,
for example. Guidance is given in a form suitable for computer implementation.
This Supplement can be used to provide (a representation of) the PDF for the output quantity from which
a) an estimate of the output quantity,
b) the standard uncertainty associated with this estimate, and
c) a coverage interval for that quantity, corresponding to a specified coverage probability
can be obtained.
Given (i) the model relating the input quantities and the output quantity and (ii) the PDFs characterizing the
input quantities, there is a unique PDF for the output quantity. Generally, the latter PDF cannot be determined
analytically. Therefore, the objective of the approach described here is to determine a), b), and c) above to a
prescribed numerical tolerance, without making unquantified approximations.
For a prescribed coverage probability, this Supplement can be used to provide any required coverage interval,
including the probabilistically symmetric coverage interval and the shortest coverage interval.
This Supplement applies to input quantities that are independent, where each such quantity is assigned an
appropriate PDF, or not independent, i.e. when some or all of these quantities are assigned a joint PDF.
Typical of the uncertainty evaluation problems to which this Supplement can be applied include those in which
⎯ the contributory uncertainties are not of approximately the same magnitude [ISO/IEC Guide 98-3:2008,
G.2.2],

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ISO/IEC GUIDE 98-3/Suppl.1:2008(E)
⎯ it is difficult or inconvenient to provide the partial derivatives of the model, as needed by the law of
propagation of uncertainty [ISO/IEC Guide 98-3:2008, Clause 5],
⎯ the PDF for the output quantity is not a Gaussian distribution or a scaled and shifted t-distribution
[ISO/IEC Guide 98-3:2008, G.6.5],
⎯ an estimate of the output quantity and the associated standard uncertainty are approximately of the same
magnitude [ISO/IEC Guide 98-3:2008, G.2.1],
⎯ the models are arbitrarily complicated [ISO/IEC Guide 98-3:2008, G.1.5], and
⎯ the PDFs for the input quantities are asymmetric [ISO/IEC Guide 98-3:2008, G.5.3].
A validation procedure is provided to check whether the GUM uncertainty framework is applicable. The GUM
uncertainty framework remains the primary approach to uncertainty evaluation in circumstances where it is
demonstrably applicable.
It is usually sufficient to report measurement uncertainty to one or perhaps two significant decimal digits.
Guidance is provided on carrying out the calculation to give reasonable assurance that in terms of the
information provided the reported decimal digits are correct.
Detailed examples illustrate the guidance provided.
This document is a Supplement to the GUM and is to be used in conjunction with it. Other approaches
generally consistent with the GUM may alternatively be used. The audience of this Supplement is that of the
GUM.
NOTE 1 This Supplement does not consider models that do not define the output quantity uniquely (for example,
involving the solution of a quadratic equation, without specifying which root is to be taken).
NOTE 2 This Supplement does not consider the case where a prior PDF for the output quantity is available, but the
treatment here can be adapted to cover this case [16].
2 Normative references
The following referenced documents are indispensable for the application of this document. For dat
...

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