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International Organization for Standardization, 2004

ISO GUM Suppl. 1 (DGUIDE 99998)

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INTERNATIONAL ORGANIZATION FOR STANDARDIZATION ORGANISATION INTERNATIONALE DE NORMALISATION

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

Guide pour l'expression de l'incertitude de mesure (GUM) Supplment 1: Mthodes numriques pour lapropagation de distribution

ICS 17.020


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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-

ber organizations of the JCGM and National Mea-

surement Institutes for review.

Contents Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Notation and definitions . . . . . . . . . . . . . . 4

3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Assignment of probability density func-

tions to the values of the input quantities . . 74.1 Probability density function assignment

for some common circumstances . . . . . . . 74.2 Probability distributions from previous

uncertainty calculations . . . . . . . . . . . . . 8

5 The propagation of distributions . . . . . . . 8

6 Calculation using Monte Carlo simulation 96.1 Rationale and overview . . . . . . . . . . . . . . 96.2 The number of Monte Carlo trials . . . . . . 126.3 Sampling from probability distributions . . 126.4 Evaluation of the model . . . . . . . . . . . . . 126.5 Distribution function for the output

quantity value . . . . . . . . . . . . . . . . . . . . 126.6 The estimate of the output quantity value

and the associated standard uncertainty . 136.7 Coverage interval for the output quantity

value . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.8 Reporting the results . . . . . . . . . . . . . . . 146.9 Computation time . . . . . . . . . . . . . . . . . 156.10 Adaptive Monte Carlo procedure . . . . . . . 15

7 Validation of the law of propagation of un-

certainty using Monte Carlo simulation . . . . 16

8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.1 Simple additive model . . . . . . . . . . . . . . 17

8.1.1 Normally distributed input quan-tities . . . . . . . . . . . . . . . . . . . . . . 17

8.1.2 Rectangularly distributed inputquantities with the same width . . 18

8.1.3 Rectangularly distributed inputquantities with different widths . . 198.2 Mass calibration . . . . . . . . . . . . . . . . . . . 20

8.2.1 Formulation . . . . . . . . . . . . . . . . . 208.2.2 Calculation . . . . . . . . . . . . . . . . . 20

8.3 Comparison loss in microwave power me-ter calibration . . . . . . . . . . . . . . . . . . . . 228.3.1 Formulation . . . . . . . . . . . . . . . . . 228.3.2 Calculation: uncorrelated input

quantities . . . . . . . . . . . . . . . . . . 228.3.3 Calculation: correlated input

quantities . . . . . . . . . . . . . . . . . . 25

Annexes

A Historical perspective . . . . . . . . . . . . . . . . 29

B Sensitivity coefficients . . . . . . . . . . . . . . . . 30

C Sampling from probability distributions . 31C.1 General distributions . . . . . . . . . . . . . . . 31C.2 Rectangular distribution . . . . . . . . . . . . . 31

C.2.1 Randomness tests . . . . . . . . . . . . 31C.2.2 A recommended rectangular ran-

dom number generator . . . . . . . . . 32

C.3 Gaussian distribution . . . . . . . . . . . . . . . 32C.4 tdistribution . . . . . . . . . . . . . . . . . . . . . 32C.5 Multivariate Gaussian distribution . . . . . 33

D The comparison loss problem . . . . . . . . . . 35D.1 The analytic solution for a zero value of

the voltage reflection coefficient . . . . . . . . 35D.2 The law of propagation of uncertainty ap-

plied to the comparison loss problem . . . . 35D.2.1 Uncorrelated input quantities . . . . 35D.2.2 Correlated input quantities . . . . . 36

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Foreword

This Supplement is concerned with the concept of thepropagation of distributions as a basis for the evaluationof uncertainty of measurement. This concept constitutesa generalization of the law of propagation of uncertaintygiven in the Guide to the Expression of Uncertainty in

Measurement (GUM) [3]. It thus facilitates the provi-sion of uncertainty evaluations that are more valid thanthose provided by the use of the law of propagation ofuncertainty in circumstances where the conditions forthe application of that law are not fulfilled. The prop-agation of distributions is consistent with the generalprinciples on which the GUM is based. An implemen-tation of the propagation of distributions is given thatuses Monte Carlo simulation.

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

created by the seven International Organizations thathad prepared the original versions of the GUM and theInternational Vocabulary of Basic and General Termsin Metrology (VIM). The Committee had the task ofthe ISO Technical Advisory Group 4 (TAG4), which haddeveloped the GUM and the VIM. The Joint Commit-tee, as was the TAG4, is formed by the BIPM withthe International Electrotechnical Commission (IEC),the International Federation of Clinical Chemistry andLaboratory Medicine (IFCC), the International Orga-nization for Standardization (ISO), the InternationalUnion of Pure and Applied Chemistry (IUPAC), theInternational Union of Pure and Applied Physics (IU-PAP) and the International Organization of LegalMetrology (OIML). A further organization joined theseseven international organizations, namely, the Interna-tional Laboratory Accreditation Cooperation (ILAC).Within JCGM two Working Groups have been estab-lished. Working Group 1, Expression of Uncertaintyin Measurement, has the task to promote the use ofthe GUM and to prepare supplements for its broad ap-plication. Working Group 2, Working Group on In-ternational Vocabulary of Basic and General Terms inMetrology (VIM), has the task to revise and promotethe use of the VIM. The present Guide has been pre-

pared by Working Group 1 of the JCGM.

Introduction

This Supplement is concerned with the concept of

the propagation of probability distributions through amodel of measurement as a basis for the evaluation ofuncertainty of measurement, and its implementation byMonte Carlo simulation. The treatment applies to amodel having any number of input quantities, and a sin-gle (scalar-valued) output quantity (sometimes knownas the measurand). A second Supplement, in prepa-ration, is concerned with arbitrary numbers of outputquantities. In particular, the provision of the probabil-ity density function for the output quantity value per-mits the determination of a coverage interval for thatvalue corresponding to a prescribed coverage probabil-ity. The Monte Carlo simulation technique in general

provides a practical solution for complicated models ormodels with input quantities having large uncertain-ties or asymmetric probability density functions. Theevaluation procedure based on probability distributionsis entirely consistent with the GUM, which states inSubclause 3.3.5 that . . . a Type A standard uncertaintyis obtained from a probability density function derivedfrom an observed frequency distribution, while a Type Bstandard uncertainty is obtained from an assumed prob-ability density function based on the degree of beliefthat an event will occur . . . . It is also consistent inthe sense that it falls in the category of the other an-

alytical or numerical methods [GUM Subclause G.1.5]permitted by the GUM. Indeed, the law of propagationof uncertainty can be derived from the propagation ofdistributions. Thus, the propagation of distributions isa generalization of the approach predominantly advo-cated in the GUM, in that it works with richer infor-mation than that conveyed by best estimates and theassociated standard uncertainties alone.

This supplement also provides a procedure for the vali-dation, in any particular case, of the use of the law ofpropagation of uncertainty.

This document is a supplement to the use of the GUMand is to be used in conjunction with it.

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1 Scope

This Supplementprovides guidance on the evaluation ofmeasurement uncertainty in situations where the con-

ditions for the applicability of the law of propagationof uncertainty and related concepts are not fulfilled orit is unclear whether they are fulfilled. It can also beused in circumstances where there are difficulties in ap-plying the law of propagation of uncertainty, because ofthe complexity of the model, for example. This guidanceincludes a general alternative procedure, consistent withthe GUM, for the numerical evaluation of measurementuncertainty, suitable for implementation by computer.

In particular, this Supplement provides a procedure fordetermining a coverage interval for an output quantityvalue corresponding to a specified coverage probability.

The intent is to determine this coverage interval to aprescribed degree of approximation. This degree of ap-proximation is relative to the realism of the model andthe quality of the information on which the probabil-ity density functions for the model input quantities arebased.

It is usually sufficient to report the measurement un-certainty to one or perhaps two significant decimal dig-its. Further digits would normally be spurious, becausethe information provided for the uncertainty evaluationis typically inaccurate, involving estimates and judge-

ments. The calculation should be carried out in a wayto give a reasonable assurance that in terms of this in-formationthese digits are correct. Guidance is given onthis aspect.

NOTE This attitude compares with that in mathematicalphysics where a model (e.g., a partial differential equation) isconstructed and then solved numerically. The constructioninvolves idealizations and inexactly known values for geo-metric quantities and material constants, for instance. Thesolution process should involve the application of suitablenumerical methods in order to make supported statementsabout the quality of the solution obtained to the posed prob-lem.

This Supplement provides a general numerical pro-cedure, consistent with the broad principles ofthe GUM [GUM G.1.5], for carrying out the calcula-tions required as part of an evaluation of measurementuncertainty. The procedure applies to arbitrary mod-els having a single output quantity where the values ofthe input quantities are assigned any specified probabil-ity density functions, including asymmetric probabilitydensity functions [GUM G.5.3].

The approach operates with the probability densityfunctions for the values of the input quantities in or-der to determine the probability density function forthe output quantity value. This is the reason for the

description of the approach as the propagation of distri-butions. Unlike the GUM, it does not make use of thelaw of propagation of uncertainty. That approach oper-ates with the best estimates (expectations) of the val-ues of the input quantities and the associated standard

uncertainties (and where appropriate the correspondingdegrees of freedom) in order to determine an estimate ofthe output quantity value, the associated standard un-certainty, and a coverage interval for the output quantityvalue. Whereas there are some limitations to that ap-proach, (a sound implementation of) the propagation ofdistributions will always provide a probability densityfunction for the output quantity value that is consistentwith the probability density functions for the values ofthe input quantities. Once the probability density func-tion for the output quantity value is available, its ex-pectation is taken as an estimate of the output quantity

value, its standard deviation is used as the associatedstandard uncertainty, and a 95 % coverage interval forthe output quantity value is obtained from it.

NOTE Some distributions, such as the Cauchy distribu-tion, which arise exceptionally, have no expectation or stan-dard deviation. A coverage interval can always be obtained,however.

The approach here obviates the need for effective de-grees of freedom [GUM G.6.4] in the determination ofthe expanded uncertainty, so avoiding the use of theWelch-Satterthwaite formula [GUM G.4.2] and hencethe approximation inherent in it.

The probability density function for the output quantityvalue is not in general symmetric. Consequently, a cov-erage interval for the output quantity value is not nec-essarily centred on the estimate of the output quantityvalue. There are many coverage intervals correspondingto a specified coverage probability. This Supplementcanbe used to provide the shortest coverage interval.

NOTE Sensitivity coefficients [GUM 5.1.3] are not aninherent part of the approach and hence the calculation orapproximation of the partial derivatives of the model withrespect to the input quantities is not required. Values akinto sensitivity coefficients can, however, be provided using avariant of the approach (Appendix B).

Typical of the uncertainty evaluation problems to whichthis Supplement can be applied include

those where the contributory uncertainties maybe arbitrarily large, even comparable to the uncer-tainty associated with the estimate of the outputquantity value;

those where the contributions to the uncertaintyassociated with the estimate of the output quan-tity value are not necessarily comparable in magni-

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tude [GUM G.2.2];

those where the probability distribution for theoutput quantity value is not Gaussian, since re-liance is not placed on the Central Limit Theo-

rem [GUM G.2.1];

those where the estimate of the output quan-tity value and the associated standard uncertainty arecomparable in magnitude, as for measurements at ornear the limit of detection;

those in which the models have arbitrary degreesof non-linearity or complexity, since the determinationof the terms in a Taylor series approximation is notrequired [GUM 5.1.2];

those in which asymmetric distributions for thevalues of the input quantities arise, e.g., when dealingwith the magnitudes of complex variables in acousti-cal, electrical and optical metrology;

those in which it is difficult or inconvenient toprovide the partial derivatives of the model (or ap-proximations to these partial derivatives), as neededby the law of propagation of uncertainty (possiblywith higher-order terms) [GUM 8].

This Supplementcan be used in cases of doubt to checkwhether the law of propagation of uncertainty is appli-

cable. A validation procedure is provided for this pur-pose. Thus, the considerable investment in this use ofthe GUM is respected: the law of propagation of un-certainty procedure remains the main approach to thecalculation phase of uncertainty evaluation, certainly incircumstances where it is demonstrably applicable.

Guidance is given on the manner in which the propaga-tion of distributions can be carried out, without makingunquantified approximations.

This Supplement applies to mutually independent in-put quantities, where the value of each such quantityis assigned an appropriate probability density function,or mutually dependent input quantities, the values ofwhich have been assigned a joint probability densityfunction.

Models with more than one output quantity are thesubject of a further Supplement to the GUM that isin preparation.

2 Notation and definitions

For the purposes of this Supplement the definitions ofthe GUM [3], the International Vocabulary of Basic and

General Terms in Metrology (VIM) [4] and ISO 3534,Part 1 [20] apply.

JCGM-WG1 has decided that the subscript c[GUM 2.3.4, 5.1.1] for the combined standard uncer-

tainty is redundant. The standard uncertainty associ-ated with an estimate y of an output quantity value Ycan therefore be written simply as u(y), but the useof uc(y) remains acceptable if it is helpful to empha-size the fact that it represents a combined standard un-certainty. Moreover, the qualifier combined in com-bined standard uncertainty is also regarded as super-fluous and may be omitted. One reason for the deci-sion is that the argument (here y) already indicates theestimate of the output quantity value with which thestandard uncertainty is associated. Another reason isthat frequently the results of one or more uncertainty

evaluations become the inputs to a subsequent uncer-tainty evaluation. The use of the subscript c and thequalifier combined are inappropriate in this regard.

This Supplement departs from the symbols often usedfor probability density function and distribution func-tion. The GUM uses the generic symbol f to refer to amodel and a probability density function. Little confu-sion arises in the GUM as a consequence of this usage.The situation in this Supplement is different. The con-cepts of model, probability density function and distri-bution function are central to following and implement-ing the procedure provided. Therefore, in place of the

symbols f and F to denote a probability density func-tion and a distribution function, the symbols g and G,respectively, are used. The symbol f is reserved for themodel.

Citations of the form [GUM 4.1.4] are to the indicated(sub)clauses of the GUM.

The decimal point is used as the symbol to separate theinteger part of a decimal number from its fractional part.A decimal comma is used for this purpose in continentalEurope.

In this Supplement the term law of propagation of un-certainty applies to the use of a first-order Taylor seriesapproximation to the model. The term is qualified ac-cordingly when a higher-order approximation is used.Sometimes the term is extended to apply also to the as-sumption of the applicability of the Central Limit The-orem as a basis for providing coverage intervals. Thecontext makes clear the usage in any particular case.

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3 Concepts

A model of measurement having any number of inputquantities and a single (scalar-valued) output quantityis considered. For this case, the main stages in the de-

termination of an estimate of the output quantity value,the associated standard uncertainty, and a coverage in-terval for the output quantity value are as follows.

a) Define the output quantity, the quantity requiredto be measured.

b) Decide the input quantities upon which the out-put quantity depends.

c) Develop a model relating the output quantity tothese input quantities.

d) On the basis of available knowledge assign proba-bility density functions [GUM C.2.5] Gaussian (nor-mal), rectangular (uniform), etc.to the values of theinput quantities.

NOTES

1 Assign instead a joint probability density function tothe values of those input quantities that are mutually de-pendent.

2 A probability density function for the values of morethan one input quantity is commonly called joint even

if the probability density functions for the values of all theinput quantities are mutually independent.

e) Propagate the probability density functions forthe values of the input quantities through the modelto obtain the probability density function for the out-put quantity value.

f) Obtain from the probability density function forthe output quantity value

1) its expectation, taken as the estimate of theoutput quantity value;

NOTE The expectation may not be appropriate forall applications (Clause 6.1, [GUM 4.1.4]).

2) its standard deviation, taken as the standarduncertainty associated with the estimate of the out-put quantity value [GUM E.3.2];

3) an interval (the coverage interval) containingthe unknown output quantity value with a specifiedprobability (the coverage probability).

Stages a)d) are regarded in this Supplement as for-mulation, and Stages e) and f) as calculation. Theformulation stages are carried out by the metrologist,

perhaps with expert support. (Advice on formulationstages a)c) will be provided in a further Supplementto the GUM on modelling that is under development.)Guidance on the assignment of probability density func-tions (Stage d) above) is given in this Supplement for

some common cases. The calculation stages, e) and f ),for which detailed guidance is provided here, require nofurther metrological information, and in principle canbe carried out to any required degree of approximation,relative to how well the formulation stages have beenundertaken.

A measurement model [GUM 4.1] is expressed by a func-tional relationship f:

Y = f(X), (1)

where Y is a single (scalar) output quantity (the out-

put quantity) and X represents the N input quanti-ties (X1, . . . , X N)

T.

NOTES

1 It is not necessary that Y is given explicitly in termsof X, i.e., f constitutes a formula. It is only neces-sary that a prescription is available for determining Ygiven X [GUM 4.1.2].

2 In this Supplement, T in the superscript position denotestranspose, and thus X represents X1, . . . , XN arranged asa column (vector) of values.

The GUM provides general guidance on many aspects ofthe above stages. It also contains a specific procedure,the law of propagation of uncertainty [GUM 5.1, 5.2],for the calculation phase of uncertainty evaluation.

The law of propagation of uncertainty has been adoptedby many organizations, is widely used and has been im-plemented in standards and guides on measurement un-certainty and also in computer packages. In order toapply this law, the values of the model input quantitiesare summarized by the expectations and standard devi-ations of the probability density functions for these val-

ues. This information is propagated through a first-order Taylor series approximation to the model to pro-vide an estimate of the output quantity value and theassociated standard uncertainty. That estimate of theoutput quantity value is given by evaluating the modelat the best estimates of the values of the input quanti-ties. A coverage interval for the output quantity value isprovided based on taking the probability density func-tion for the output quantity value as Gaussian.

The intent of the GUM is to derive the expectation andstandard deviation of the probability density functionfor the output quantity value, having first determinedthe expectations and standard deviations of the proba-bility density functions for the values of the input quan-

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tities.

NOTES

1 The best estimates of the values of the input quantities

are taken as the expectations of the corresponding probabil-ity density functions [GUM 4.1.6].

2 The summaries of values of the input quantities also in-clude, where appropriate, the degrees of freedom of the stan-dard uncertainties associated with the estimates of the valuesof the input quantities [GUM 4.2.6].

3 The summaries of the values of the input quantities alsoinclude, where appropriate, covariances associated with theestimates of the values of input quantities [GUM 5.2.5].

4 The GUM [Note to GUM Subclause 5.1.2] states thatif the non-linearity of the model is significant, higher-order

terms in the Taylor series expansion must be included in theexpressions for the standard uncertainty associated with theestimate of the output quantity value.

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-tomatic differentiation can be used. Alternatively, thesederivatives can be calculated numerically using finite differ-ences [GUM 5.1.3]. Care should be taken, however, becauseof the effects of subtractive cancellation when forming dif-ferences in values of the model for close values of the inputquantities.

6 The most important terms of next highest order to be

added to those of the formula in GUM Subclause 5.1.2 for thestandard uncertainty are given in the Note to this subclause.Although not stated in the GUM, this formula applies whenthe values ofXi are Gaussian. In general, it would not applyfor 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. Noguidance is given in the GUM if they are mutually depen-dent, but it is taken that the same statement would apply.

8 A probability density function related to a tdistributionis used instead of a Gaussian probability density function if

the effective degrees of freedom associated with the estimateof the standard deviation of the probability density functionfor the output quantity value is finite [GUM G].

The calculation stages (Stages e) and f) above) ofthe GUM that use the law of propagation of uncertaintyand the abovementioned concepts can be summarized asthe following computational steps. Also see Figure 1.

a) Obtain from the probability density functions forthe values of the input quantities X1, . . . , X N, respec-tively, the expectation x = (x1, . . . , xN)

T and thestandard deviations (standard uncertainties) u(x) =

(u(x1), . . . , u(xN))T

. Use the joint probability den-sity function for the value ofX instead if the Xi aremutually dependent.

NOTE The Xi are regarded as random variables withpossible values i and expectations xi.

b) Take the covariances (mutual uncertain-ties) [GUM C] u(xi, xj) as Cov(Xi, Xj), the

covariances of mutually dependent pairs (Xi, Xj) ofinput quantities.

c) Form the partial derivatives of first order of fwith respect to the input quantities.

d) Calculate the estimate y of the output quantityvalue by evaluating the model at x.

e) Calculate the model sensitivity coefficients[GUM 5.1] as the above partial derivatives evaluatedat x.

f) Determine the standard uncertainty u(y) by com-bining u(x), the u(xi, xj) and the model sensitivitycoefficients [GUM Formulae (10) and (13)].

g) Calculate , the effective degrees of freedom ofy,using the Welch-Satterthwaite formula [GUM For-mula (G.2b)].

h) Compute the expanded uncertainty Up, andhence a coverage interval for the output quantityvalue (having a stipulated coverage probability p),by forming the appropriate multiple of u(y) throughtaking the probability distribution of (y Y)/u(y)as a standard Gaussian distribution ( = ) or tdistribution ( < ).

x1, u(x1) -

x2, u(x2) -

x3, u(x3) -

Y = f(X) - y, u(y)

Figure 1 Illustration of the law of propagation of

uncertainty. The model has mutually independent

input quantities X = (X1, X2, X3)T, whose values are

estimated by xi with associated standard uncertain-

ties u(xi), for i = 1, 2, 3. The value of the output

quantity Y is estimated by y, with associated stan-

dard uncertainty u(y).

The computational steps above require the followingconditions to hold:

a) the non-linearity off to be insignificant [Note toGUM 5.1.2];

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b) the Central Limit Theorem [GUM G.2.1, G.6.6]to apply, implying the representativeness of the prob-ability density function for the output quantity valueby a Gaussian distribution or in terms of a tdistribution;

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.

When these three conditions hold, the results from thesound application of the law of propagation of uncer-tainty are valid. These conditions apply in many cir-cumstances. The approach is not always applicable,however. This Supplement provides a more general ap-

proach that does not require these conditions to hold.

4 Assignment of probability density

functions to the values of the input

quantities

In the first phaseformulationof uncertainty eval-uation, the probability density functions for the val-ues of the input quantities of the model are as-signed [GUM 2.3.2, 3.3.5] based on an analysis of series

of observations or based on scientific judgement [GUM2.3.3, 3.3.5] using all the relevant information [38], suchas historical data, calibrations and expert judgement.

The probability density function for the possible val-ues i of the ith input quantity Xi is denoted by gi(i)and that for the possible values of the output quantityvalue Y by g(). The distribution function for Xi isdenoted by Gi(i) and that for Y by G(). The prob-ability density functions and the distribution functionsare related by gi(i) = G

i(i) and g() = G().

When the input quantities are mutually dependent,

in place of the N individual probability density func-tions gi(i), i = 1, . . . , N , there is a joint probabilitydensity function g(). See Notes 1 and 2 at the end ofSection 5. Intermediate to these extremes, groups of theinput quantities may have values with joint probabilitydensity functions.

Clauses 4.2 and 4.3 of the GUM contain much relevantinformation on the assignment of probability densityfunctions.

NOTES

1 The Principle of Maximum Information Entropy can beapplied to assist in the assignment [7, 39, 40].

2 It may be possible to remove some mutual dependenciesby re-expressing some or all of the input quantities in termsof more fundamental mutually independent input quantitieson which the original input quantities depend [GUM F1.2.4,GUM H.1]. Such changes can simplify both the applicationof the law of propagation of uncertainty and the propagation

of distributions. Details and examples are available [13].

4.1 Probability density function assign-

ment for some common circumstances

Assignments of probability density functions to the in-put quantities are given in Table 1 for some commoncircumstances.

All available informationconcerning quantity X

Probability density function(PDF) assigned to the valueof X

The estimate x and theassociated standard uncer-tainty u(x)

The Gaussian PDFN(x, u2(x))

The estimate x (> 0), andX is known to be nonneg-ative

The exponential PDFwith expectation x, viz.,exp(/x)/x, for 0, andzero otherwise.

Independent observations

of a quantity value takento follow a normal lawwith unknown expectationequal to the value of X.From a sample of size n,an arithmetic mean x anda standard deviation s havebeen calculated

Product of

n/s and the

tdistribution with argument( x)/(s/n) and n 1 de-grees of freedom and where xand s are known constants

The estimate x of the valueof a multivariate quan-tity X and the correspond-ing uncertainty matrix (co-variance matrix) V

The multivariate GaussianPDF N(x,V) is assigned tothe value of X (Section 5,Note 2)

The endpoints a and a+of an interval containingthe value of X

The rectangular PDF withendpoints a and a+

The lower and upper lim-its a and a+ of an inter-val within which the valueof X is known to cycle si-nusoidally

The scaled and shifted arcsinePDF with endpoints a anda+, viz., (2/)/{(a+ a)2 (2 a+ a)2}1/2, for a < < a+, and zero otherwise[14], [17, Section 3.5]

Table 1 The assignment of a probability density

function to the value of an input quantity X based

on available information for some common circum-

stances.

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4.2 Probability distributions from previous

uncertainty calculations

A previous uncertainty calculation may have provided aprobability distribution for the value of an output quan-

tity that is to become an input quantity for a further un-certainty calculation. This probability distribution maybe available analytically in a recognized form, e.g., asa Gaussian probability density function, with values forits expectation and standard deviation. It may be avail-able as an approximation to the distribution function fora quantity value obtained from a previous application ofMonte Carlo simulation, for example. A means for de-scribing such a distribution function for a quantity valueis given in Clause 6.5.

5 The propagation of distributions

Several approaches can be used for the second phase calculation of uncertainty evaluation:

a) analytical methods;

b) uncertainty propagation based on replacing themodel by a first-order Taylor series approxima-tion [GUM 5.1.2] the law of propagation of un-certainty;

c) as b), except that contributions derived fromhigher-order terms in the Taylor series approximationare included [Note to GUM 5.1.2];

d) numerical methods [GUM G.1.5] that implementthe propagation of distributions, specifically MonteCarlo simulation (Section 6).

NOTE Analytical methods are ideal in that they do notintroduce any approximation. They are applicable in simplecases only, however. A treatment and examples are avail-able [7, 12]. These methods are not considered further inthis Supplement, apart from in the examples section (Sec-tion 8.1.1) for comparison purposes.

Techniques other than the law of propagation of uncer-tainty are permitted by the GUM [GUM G.1.5]. Theapproach advocated in this Supplement, based on thepropagation of distributions, is general. For linear orlinearized models and input quantities with values forwhich the probability density functions are Gaussian,the approach yields results consistent with the law ofpropagation of uncertainty. But in cases where the lawof propagation of uncertainty cannot be applied the ad-vocated approach still gives correct uncertainty state-ments.

In terms of the calculations required, there are threeclasses of uncertainty evaluation problem:

a) those for which a general approach is needed;

b) those for which uncertainty propagation based ona first-order Taylor series approximation is applicable;

c) those for which it is unclear which approachshould be followed.

For Class a), this Supplementprovides a generic, broadlyapplicable approach based on the propagation of distri-butions. With respect to Class b), this Supplementdoesnot provide new material. For Class c), this Supplementprovides a procedure for validating in any particular cir-cumstance the use of the law of propagation of uncer-tainty (possibly based on a higher-order Taylor seriesapproximation).

The propagation of the probability density func-tions gi(i), i = 1, . . . , N , for the values of the inputquantities through the model to provide the probabilitydensity function g() for the output quantity value isillustrated in Figure 2 for the case N = 3. This figureis the counterpart of Figure 1 for the law of propaga-tion of uncertainty. Like the GUM, this Supplement isconcerned with models having a single output quantity.

-

g3(3)

-

g2(2)

-

g1(1)

Y = f(X) -

g()

Figure 2 Illustration of the propagation

of distributions. The model input quantities

are X = (X1, X2, X3)T. The probability density func-

tions gi(i), for Xi, i = 1, 2, 3, are Gaussian, triangular

and Gaussian, respectively. The probability density

function g() for the value of the output quantity Y is

indicated as being asymmetric, as can arise for non-linear models. (An asymmetric g() can also arise

when the probability density functions for the val-

ues of the input quantities are asymmetric.)

NOTES

1 The only joint probability density functions consideredin this Supplement are multivariate Gaussian.

2 A multivariate Gaussian probability density functionwith expectation x = (x1, . . . , xN)

T and uncertainty ma-trix V is given by

g() =1

((2)NdetV)1/2exp

1

2( x)TV1( x)

.

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This probability density function reduces to the product of Nunivariate Gaussian probability density functions when thereare no covariance effects, for the following reason. In thatcase

V= diag(u2(x1), . . . , u2(xN)),

whence

g() =1

(2)N/2u(x1) u(xN) exp

12

Ni=1

(i xi)2u2(xi)

=

Ni=1

gi(i),

with

gi(i) =1

2u(xi)exp

(i xi)

2

2u2(xi)

.

6 Calculation using Monte Carlo simu-

lation

6.1 Rationale and overview

Monte Carlo simulation provides a general approachto numerically approximating the distribution func-tion G() for the value of the output quantity Y =f(X). The following GUM Subclause is relevant to theconcept embodied in Monte Carlo simulation:

An estimate of the measurand Y, denoted by y,is obtained from equation (1) [identical to ex-pression (1) of this Supplement] using input esti-mates x1, x2, . . . , xN for the values of the N quan-tities X1, X2, . . . , X N. Thus the output estimate y,which is the result of the measurement, is given by

y = f(x1, x2, . . . , xN) . . . (2)

NOTE In some cases the estimate y may be obtainedfrom

y = Y =1

n

nk=1

Yk =1

n

nk=1

f(X1,k, X2,k, . . . , XN,k)

That is, y is taken as the arithmetic mean or aver-age (see 4.2.1) of n independent determinations Ykof Y, each determination having the same uncertainty

and each being based on a complete set of observed

values of the N input quantities Xi obtained at the

same time. This way of averaging, rather than y =

f(X1, X2, . . . , XN), where Xi = (n

k=1Xi,k)/n is the

arithmetic mean of the individual observations Xi,k,

may be preferable when f is a nonlinear function of

the input quantities X1, X2, . . . , XN, but the two ap-

proaches are identical if f is a linear function of the Xi(see H.2 and H.4). [GUM 4.1.4]

Although the GUM formula (2) need not provide themost meaningful estimate of the output quantity value,

as stated in the note to GUM Subclause 4.1.4, repeatedabove, it plays a relevant role within Monte Carlo sim-ulation as an implementation of the propagation of dis-tributions.

Monte Carlo simulation for uncertainty calculations [7,9] is based on the premise that any value drawn at ran-dom from the distribution of possible values of an inputquantity is as legitimate as any other such value. Thus,by drawing for each input quantity a value according toits assigned probability density function, the resultingset of values is a legitimate set of values of these quanti-ties. The value of the model corresponding to this set ofvalues constitutes a possible value of the output quan-tity Y. Figure 3 is similar to Figure 2 except that itshows a value sampled from each of the three proba-bility density functions for the input quantities and the

resulting value of the output quantity.

-

?

-

?

-?

Y = f(X) -?

Figure 3 As Figure 2 except that what is shown

is a value sampled from each of the three probabilitydensity functions for the values of the input quanti-

ties, and the resulting output quantity value.

Consequently, a large set of model values so obtainedcan be used to provide an approximation to the distri-bution of possible values for the output quantity. MonteCarlo simulation can be regarded as a generalization ofGUM 4.1.4 (above) to obtain the distribution for Y,rather than the expectation of Y. In particular, theabove drawing of a value from the probability density

function for each input quantity corresponds to a com-plete set of observed values of the N input quantities Xiobtained at the same time in GUM 4.1.4.

Monte Carlo simulation operates as follows:

Generate a sample of size N by independentlysampling at random from the probability densityfunction for each Xi, i = 1, . . . , N (or in the case ofmutually dependent input quantities, from the jointprobability density function for X). Repeat this pro-cedure a large number, M, say, of times to yield M in-dependent samples of size N of the set of input quan-tities. For each such independent sample of size N,calculate the resulting model value of Y, yielding M

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values of Y in all.

NOTES

1 Appendix C provides information on sampling fromprobability distributions.

2 Clause 6.4 gives more explicit information concerningthe sample taken.

3 According to the Central Limit Theorem [30, p169],the arithmetic mean of the M values of the output quan-tity obtained in this manner converges as 1/M1/2 to theexpectation of the probability density function for thevalue of Y = f(X).

Use these M values of Y to provide G(), anapproximation to the distribution function G() forthe value of Y.

Produce any required statistical quantityfrom G(). Particularly relevant quantities are(a) the expectation of G() as the estimate y of theoutput quantity value, (b) the standard deviation

of G() as the associated standard uncertainty u(y),and (c) two quantiles of G() as the endpoints ofa coverage interval Ip(Y) for a stipulated coverageprobability p [7]. For further details see Clause 6.7.

The effectiveness of Monte Carlo simulation to deter-mine a coverage interval for the output quantity value

depends on the use of an adequately large value of M.Guidance on obtaining such a value and generally onimplementing Monte Carlo simulation is available [7, 8].Also see Clauses 6.2 and 6.10.

The inputs to the Monte Carlo simulation procedure forthe calculation stage are

the model Y = f(X) (provided at the formula-tion stage of uncertainty evaluation);

the joint probability density function for the val-ues ofX (provided at the formulation stage also see

Section 4);

NOTE This joint probability density function reducesto the collection of the individual probability density func-tions for X1, . . . , XN in the case of N mutually indepen-dent input quantities.

the required coverage probability p (e.g., 0.95,or 95 %);

the number M of Monte Carlo trials.

The primary output from the Monte Carlo simulationprocedure is a numerical approximation G() to the dis-tribution function G() for the output quantity value,from which the required quantities can be determined.

Monte Carlo simulation as an implementation of thepropagation of distributions is shown diagrammaticallyin Figure 4 and can conveniently be stated as a step-by-step procedure:

a) Select the number M of Monte Carlo trials to bemade. See Clause 6.2.

b) Generate M samples of the (set ofN) input quan-tities. See Clause 6.3.

c) For each sample, evaluate the model to give thecorresponding output quantity value. See Clause 6.4.

d) Sort these values of the output quantity into non-decreasing order, using the sorted values to approxi-mate the distribution function for the output quantity

value. See Clause 6.5.

e) Form the estimate of the output quantity valueand the associated standard uncertainty from this dis-tribution function (or directly from the set of valuesof the output quantity). See Clause 6.6.

f) Form the shortest 95 % coverage interval for theoutput quantity value from this distribution function.See Clause 6.7.

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Coverage interval [ylow, yhigh]

for the output quantity valueClause 6.7

?

Estimate y of the output quantity valueand associated standard uncertainty u(y)

Clause 6.6

?

Approximate distribution function G()Clause 6.5

?

M model valuesy = (y1, . . . , yM) = (f(x1), . . . , f (xM))

Clause 6.4

?

M samples x1, . . . ,xMofX from g()

Clause 6.3

?

ModelY = f(X)

Concepts

?

Probability density

functions g()Section 4

?

Number M of

Monte Carlo trialsClause 6.2

?

Coverage

probability p

MCS

INPUTS

MCS CALCULATION:SAMPLES FROMTHE PROBABILITYDENSITY FUNCTIONSFOR THE VALUESOF THE INPUTQUANTITIESAND MODELEVALUATION FORTHESE SAMPLES

PRIMARY MCSOUTPUT:DISTRIBUTIONFUNCTION FORTHE OUTPUTQUANTITY VALUE

MCSRESULTS

Figure 4 The calculation phase of uncertainty evaluation using Monte Carlo simulation (MCS) to implement

the propagation of distributions.

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6.2 The number of Monte Carlo trials

A value of M, the number of Monte Carlo trials to bemade, needs to be selected. It can be chosen a priori,in which case there will be no direct control over the

degree of approximation delivered by the Monte Carloprocedure. The reason is that the number needed toprovide a prescribed degree of approximation will de-pend on the shape of the probability density functionfor the output quantity value and on the coverage prob-ability required. Also, the calculations are stochasticin nature, being based on random sampling. However,a value of M = 106 can often be expected to delivera 95 % coverage interval for the output quantity value,such that this length has a degree of approximation ofone or two significant decimal digits,.

Because there is no guarantee that this or any specificnumber will suffice, it is recommended to use a processthat selects M adaptively, i.e., as the trials progress.Some guidance in this regard is available [2, 7]. Alsosee Clause 6.10. A property of such a process is that ittakes a number of trials that is economically consistentwith the requirement to achieve the required degree ofapproximation [7].

NOTE If the model is complicated, e.g., involving the so-lution of a finite-element model or a non-linear least-squaresproblem, because of the consequently prohibitive computingtime, it may not be possible to use a sufficiently large valueofM to obtain adequate distributional knowledge of the out-put quantity value. In such a case one approach could be asfollows. A relatively small value of M, 10 or 20, perhaps,would be used. The arithmetic mean and standard devi-ation of the resulting M values of Y would be taken as yand u(y), respectively, and a coverage interval for the out-put quantity value obtained by regarding Y to be distributedas y + T u(y), where T denotes the tdistribution with M 1degrees of freedom.

6.3 Sampling from probability distribu-

tions

In an implementation of the Monte Carlo procedure, Msamples xi, i = 1, . . . , M (Clause 6.2), are obtained fromthe probability density functions for the values of theinput quantities X. Samples would be obtained from ajoint (multivariate) Gaussian probability density func-tion when appropriate. Recommendations [8] concern-ing the manner in which this sampling can be carriedout are given in Appendix C for the commonest distri-butions, viz., the rectangular, Gaussian, t and multi-variate Gaussian. It is possible to sample from anyother distribution [7]. Some such distributions wouldbe those based on Monte Carlo results from a previous

uncertainty calculation (Clauses 4.2 and 6.5).

NOTE For the results of Monte Carlo simulation to

be statistically valid it is necessary that the pseudorandomnumber generators used to sample from the distributions re-quired have appropriate properties. Tests of randomness ofthe numbers produced by a generator are indicated in Ap-pendix C.

6.4 Evaluation of the model

The model is evaluated for each of the M samples fromeach of the probability density functions for the valuesof the N input quantities. Specifically, denote the Msamples by x1, . . . ,xM, where the rth sample xr con-tains values x1,r, . . . , xN,r , with xi,r a draw from theprobability density function for Xi. Then, the modelvalues are

yr = f(xr), r = 1, . . . , M .

NOTE In Monte Carlo simulation the model is evalu-ated for each sample of the input quantities and hence forvalues that may be distanced by several standard devia-tions from the estimates of the values of the input quan-tities. For this reason some issues may arise regarding thenumerical procedure used to evaluate the model, e.g., ensur-ing its convergence (where iterative schemes are used) andnumerical stability. When applying the law of propagationof uncertainty, the model is evaluated only at the estimatesof the values of the input quantities or, if finite-differenceapproximations are used [GUM 5.1.3, Note 2], also at pointsperturbed by one standard deviation from the estimate ofthe value of each input quantity in turn. The user shouldensure that, where appropriate, the numerical methods used

to evaluate f are valid for a sufficiently large region centredon these estimates.

6.5 Distribution function for the output

quantity value

An approximation G() to the distribution func-tion G() for the output quantity value Y can be ob-tained as follows. Sort the values yr, r = 1, . . . , M ,of the output quantity provided by Monte Carlo simula-tion into non-decreasing order. Denote the sorted valuesby y(r), r = 1, . . . , M . Assign uniformly spaced cumu-lative probabilities pr = (r 1/2)/M, r = 1, . . . , M , tothe ordered values [7].

NOTES

1 The term non-decreasing rather than increasing isused because of possible equalities among the values of yr.

2 A sorting algorithm taking a number of operations pro-portional to M log M should be be used [36]. A naive al-gorithm would take a time proportional to M2, making thecomputation time unnecessarily long. See Clause 6.9.

3 The values pr, r = 1, . . . , M , are the midpoints of Mcontiguous probability intervals of width 1/M between zeroand one.

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Form G() as the piecewise-linear function joiningthe M points (y(r), pr), r = 1, . . . , M :

G() =

r 1/2M

+ y(r)

M(y(r+1) y(r)),

y(r) y(r+1), r = 1, . . . , M 1. (2)Figure 5 illustrates G() from a Monte Carlo simulationbased on M = 50 trials.

Figure 5 The piecewise-linear function, form-

ing an approximation G() to the distribution func-tion G(), derived from 50 sampled values from a

Gaussian probability density function g() with ex-

pectation 10 and standard deviation 1.

NOTES

1 G() is defined only for values of corresponding to val-ues of probability p in the interval 1/(2M) p 11/(2M).Indeed, it should not be used near the endpoints of this in-terval, e.g., for very large coverage probabilities in the caseof a symmetric or approximately symmetric distribution, be-cause it is less reliable there.

2 The values of y(r) (or yr), when assembled into a his-togram (with suitable cell widths) form a frequency distri-bution that, when normalized to have unit area, provides anapproximation

g() to the probability density function g()

for Y. Calculations are not generally carried out in terms of

this histogram, the resolution of which depends on the choiceof cell size, but in terms of the approximation G() to thedistribution function G(). The histogram can, however, beuseful as an aid to understanding the nature of the proba-bility density function, e.g., the extent of its asymmetry.

3 The very small value ofM used to provide G() in Fig-ure 5 is for purposes of illustration only.

6.6 The estimate of the output quantity

value and the associated standard un-

certainty

The expectation y of the function G() (Expression (2))approximates the expectation of the probability density

function g() for the value of Y and is taken as the esti-mate y of the output quantity value. The standard devi-ation u(y) of G() approximates the standard deviationof g() and is taken as the standard uncertainty u(y)associated with y. y can be taken as the arithmeticmean y = 1

M

Mr=1

yr, (3)

formed from the M values y1, . . . , yM, and the standarddeviation u(y) determined from

u2(y) = 1M 1

Mr=1

(yr y)2. (4)NOTES

1 Formulae (3) and (4) do not in general provide valuesthat are identical to the expectation and standard deviation,

respectively, of G(). The latter values are given byy = 1

M

Mr=1

y(r) (5)

and

u2(y) = 1M

Mr=1

(y(r) y)2 16

M1r=1

(y(r+1) y(r))2

,

(6)

where the double prime on the summation in Expression (5)and on the first summation in Expression (6) indicates thatthe first and the last terms are to be taken with weight onehalf. For a sufficiently large value ofM (104, say, or greater),the values obtained using Formulae (3) and (4) would gen-erally be indistinguishable for practical purposes from thosegiven by (5) and (6).

2 If the standard deviation is formed directly from the Mvalues y1, . . . , yM, it is important to use Formula (4) ratherthan the mathematically equivalent formula

u2(

y) =

M

M 1

1

M

M

r=1y2r

y2

.

For cases in which u(y) is much smaller than |y| (in whichcase the yr have a number of leading digits in common) thelatter formula suffers numerically from subtractive cancella-tion (involving a mean square less a squared mean). Thiseffect can be so severe that the resulting value may havetoo few correct significant decimal digits for the uncertaintyevaluation to be valid [6].

3 In some special circumstances, such as when one of theinput quantities has been assigned a probability density func-tion based on the tdistribution with fewer than three de-grees of freedom, and that input quantity has a dominant ef-fect, the expectation and the standard deviation of g() maynot exist. Formulae (5) and (6) (or Formulae (3) and (4))

will not then provide meaningful results. A coverage intervalfor the output quantity value (Clause 6.7) can, however, beformed. See also the second Note in Section 1.

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4 The value of y so obtained yields the smallest meansquared deviation over all possible estimates of the outputquantity value. However, the value will not in general agreewith the model evaluated at the estimates of the values ofthe input quantities [GUM 4.1.4]. When the model is linearin the input quantities, agreement (in a practical sense) will

be achieved for a large value of M. Whether this generallack of agreement is important depends on the application.The value of y, even in the limit as M , is not in gen-eral equal to the model evaluated at the expectations of theprobability density functions for the input quantities, unlessthe model is linear [GUM 4.1.4].

6.7 Coverage interval for the output quan-

tity value

Let denote any value between zero and 1p, where pis the required coverage probability. The endpoints of

a 100p % coverage interval Ip(Y) for the output quantityvalue are G1() and G1(p+) i.e., the and (p+)quantiles of G(). The quantile is the value of suchthat G() = . The choice = 0.025 gives the coverageinterval defined by the 0.025 and 0.975quantiles, pro-viding an I0.95(Y) that is probabilistically symmetric.The probability is 2.5 % that the value of Y is smallerthan the left-hand endpoint of the interval and 2.5 %that it is larger than the right-hand endpoint. If g()is symmetric about its expectation, Ip(Y) is symmet-ric about the estimate of the output quantity value y,and the left-hand and right-hand endpoints ofIp(Y) areequidistant from y.

NOTE For a symmetric probability density function forthe output quantity value Y, the procedure would, for asufficiently large value of M, give the same coverage intervalfor practical purposes as taking the product of the standarduncertainty and the coverage factor that is appropriate forthat probability density function. This probability densityfunction is generally not known analytically.

A value of different from 0.025 would generally beappropriate were the probability density function asym-metric. Usually the shortest Ip(Y) is required, becauseit corresponds to the best possible location of the value

of the output quantity Y within a specified probabil-ity. It is given by the value of satisfying g(G1()) =g(G1(p + )), if g() is single-peaked, and in generalby the value of such that G1(p + ) G1() is aminimum. If g() is symmetric, the shortest Ip(Y) isgiven by taking = (1 p)/2.

The shortest Ip(Y) can generally be obtained computa-

tionally from G() by determining such that G1(p +) G1() is a minimum.Henceforth Ip(Y) will denote the shortest 100p % cov-erage interval. The law of propagation of uncertainty,as in the GUM [3], provides the shortest such intervalfor the Gaussian (or scaled and shifted t) distribution

assigned to the output quantity value. Therefore, it isappropriate in contrasting this approach with other ap-proaches to use the shortest 100p % coverage intervalfor the distributions they provide.

Figure 6 shows a distribution function G() for an asym-metric probability density function.

Figure 6 A distribution function G() corre-

sponding to an asymmetric probability density func-

tion. Broken lines mark the endpoints of the prob-

abilistically symmetric 95 % coverage interval and

the corresponding probability points, viz., 0.025 and

0.975. Full lines mark the endpoints of the short-

est 95 % coverage interval and the corresponding

probability points, which are 0.006 and 0.956 in this

case. The lengths of the 95 % coverage intervals in

the two cases are 1.76 units and 1.69 units, respec-

tively.

6.8 Reporting the results

The estimate y of the output quantity value and the end-points ylow and yhigh of the coverage interval Ip(Y) =[ylow, yhigh] corresponding to the coverage probabil-ity p for the output quantity value should be reportedto a number of decimal digits such that the leastsignificant digit is in the same position with respectto the decimal point as that for the standard uncer-tainty u(y) [GUM 7.2.6].

NOTES

1 One significant digit or two significant digits would beadequate to represent u(y) in many cases.

2 A factor influencing the choice of one or two significantdigits is the leading significant digit ofu(y). If this digit is 1,the relative accuracy to which u(y) is reported is low if onedigit is used. If the leading significant digit is 9, the relative

accuracy is almost an order of magnitude greater.

3 If the results are to be used within further calculations,

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consideration should be given to whether additional digitsare required. There might be a repercussion for the numberof Monte Carlo trials (Clause 6.2).

As an example, suppose that it is meaningful to state

u(y) = 0.03 V.

Then, correspondingly, values of y and I0.95(Y) mightbe

y = 1.02 V, I0.95(Y) = [0.98, 1.09] V.

I0.95(Y) is an asymmetric interval in this example. If itwere meaningful to report two significant digits in u(y),the values might be

y = 1.024 V, u(y) = 0.028 V,

I0.95(Y) = [0.983, 1.088] V.

6.9 Computation time

The computation time for the Monte Carlo calculationsummarized in Clause 6.1 is dominated by that requiredfor the following three steps.

a) Take M samples from the probability densityfunction for the values of each of the input quanti-ties.

b) Make M corresponding evaluations of the model.

c) Sort the M resulting values into non-decreasingorder.

The time taken in Step a) is directly proportional to M.The same statement applies to Step b). The time takenin Step c) is proportional to Mlog M (if an efficient sortprocedure [36] is used).

If the model is very simple, the time in Step c) domi-nates, and the overall time taken is usually a few secondsfor M = 106 on a PC operating at several GHz. Oth-

erwise, let T1 be the time in seconds taken to draw onesample from the probability density functions for theinput quantities and T2 that to make one evaluation ofthe model. Then, the overall time can be taken as es-sentially M(T1+T2) s, or, if the model is complicated,as MT2 s.

The computation times taken on a personal computeroperating at several GHz are typically a few seconds forsimple models, seconds or perhaps a few minutes formore complicated models, and minutes or hours or evenlonger for very complicated models.

NOTE An indication of the computation time requiredfor a Monte Carlo calculation can be obtained as follows.

Consider an artificial problem with a model consisting of thesum of five terms:

Y = cos X1 + sin X2 + tan1 X3 + e

X4 + X1/25 . (7)

Assign a Gaussian probability density function to each of

the input quantities Xi. Make M = 106

Monte Carlo tri-als. Computation times observed for a 1 GHz Pentium 3 PCusing MATLAB [27]1) are summarized in Table 2. This infor-mation provides a simple basis for estimating the computa-tion time for other models, other values of M and other PCs.The times quoted apply to a particular PC, a particular im-plementation, and the version of MATLAB used, and aretherefore only indicative. They do not necessarily scale wellto other configurations.

Operation Time/sGenerate 5M random Gaussian numbers 1Form M model values 1Sort the M model values 3Total 5

Table 2 Computation times for the main steps in

Monte Carlo simulation applied to the model (7),

where each Xi is assigned a Gaussian probability

density function and M = 106 Monte Carlo trials are

taken.

6.10 Adaptive Monte Carlo procedure

A basic implementation of an adaptive Monte Carlo pro-cedure can be described as follows. It is based on carry-

ing out an increasing number of Monte Carlo trials untilthe various quantities of interest have stabilized in a sta-tistical sense. A quantity is deemed to have stabilizedif twice the standard deviation associated with the esti-mate of its value is less than the degree of approximationrequired in the standard uncertainty u(y).

NOTE Typically, the more sensitive quantities are theendpoints of the coverage interval for the value of the out-put quantity Y. The expectation and the standard deviationas the estimate y of the output quantity value and the as-sociated standard uncertainty u(y), respectively, generallyconverge considerably faster.

A practical approach consists of carrying out a sequenceof Monte Carlo calculations, each containing a relativelysmall number, say, M = 104 trials. For each MonteCarlo calculation in the sequence, y, u(y) and I0.95(Y)are formed from the results obtained, as in Clauses 6.6

and 6.7. Denote by y(h), u(y(h)), y(h)low and y

(h)high the

values of y, u(y) and the left- and right-hand endpointsof I0.95(Y) for the hth member of the sequence.

After the hth Monte Carlo calculation (apart fromthe first) in the sequence, the arithmetic mean y

1)

Any identification of commercial products in this Supplementdoes not imply endorsement or that such products are b est suitedfor the purpose.

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of the values y(1), . . . , y(h) and the standard devi-ation sy associated with this arithmetic mean areformed. The counterparts of these statistics for y aredetermined for u(y), ylow and yhigh. If the largestof 2sy, 2su(y), 2sylow and 2syhigh does not exceed the de-

gree of approximation required in u(y), the overall com-putation is regarded as having stabilized. The resultsfrom the total number of Monte Carlo trials taken arethen used to provide the estimate of the output quan-tity value, the associated standard uncertainty and thecoverage interval for the output quantity value.

7 Validation of the law of propagation of

uncertainty using Monte Carlo simu-

lation

The law of propagation of uncertainty can be expectedto work well in many circumstances. However, it isgenerally difficult to quantify the effects of the approx-imations involved, viz., linearization [GUM 5.1.2], theWelch-Satterthwaite formula for the effective degreesof freedom [GUM G.4.2] and the assumption that theprobability distribution for the output quantity value isGaussian (i.e., that the Central Limit Theorem is appli-cable) [GUM G.2.1, G.6.6]. Indeed, the degree of diffi-culty of doing so would typically be considerably greaterthan that required to apply Monte Carlo simulation (as-suming suitable software were available [8]). Therefore,

since these circumstances cannot readily be tested, anycases of doubt should be validated. To this end, sincethe propagation of distributions is more general, it isrecommended that both the law of propagation of un-certainty and Monte Carlo simulation be applied and theresults compared. Should the comparison be favourable,the law of propagation of uncertainty could be used onthis occasion and for sufficiently similar problems in thefuture. Otherwise, consideration could be given to usingMonte Carlo simulation instead.

Specifically, it is recommended that the two steps below

and the following comparison process be carried out.a) Apply the law of propagation of uncertainty (pos-sibly based on a higher-order Taylor series approx-imation) [GUM 5] to yield a 95 % coverage inter-val y U0.95(y) for the output quantity value.

b) Apply Monte Carlo simulation (Section 6) to pro-vide values for the standard uncertainty u(y) and theendpoints ylow and yhigh of a 95 % coverage intervalfor the output quantity value.

NOTE The process is described here in terms of a 95 %coverage interval, but other coverage intervals can be used.

A comparison procedure is based on the following objec-

tive: determine whether the coverage intervals obtainedby the law of propagation of uncertainty and MonteCarlo simulation agree to a stipulated degree of approx-imation. This degree of approximation is assessed interms of the endpoints of the coverage intervals and

corresponds to that given by expressing the standarduncertainty u(y) to what is regarded as a meaningfulnumber of significant decimal digits (cf. Clause 6.10).The procedure is as follows:

a) Let ndig denote the number of significant digitsregarded as meaningful in the numerical value ofu(y).Usually, ndig = 1 or ndig = 2 (Clause 6.8). Expressthe value of u(y) in the form a 10r, where a is anndig-digit integer and r an integer. The comparisonaccuracy is

=1

2

10r. (8)

EXAMPLES

1 The estimate of the output quantity valuefor a nominally 100 g measurement standard ofmass [GUM 7.2.2] is y = 100.021 47 g. The stan-dard uncertainty u(y) = 0.000 35 g. Thus, ndig = 2and u(y) can be expressed as 35105 g, and so a = 35and r = 5. Take = 12 105 g = 0.000 005 g.

2 As Example 1 except that only one significantdigit in u(y) is regarded as meaningful. Thus, ndig = 1and u(y) = 0.000 4 g = 4104 g, giving a = 4 and r =4. Thus, = 12 104 g = 0.000 05 g.

3 u(y) = 2 K. Then, ndig = 1 and u(y) = 2100 K,giving a = 2 K and r = 0. Thus, = 12 100 K = 0.5 K.

b) Compare the coverage intervals obtained by thelaw of propagation of uncertainty and Monte Carlosimulation to determine whether the required num-ber of correct digits in the coverage interval providedby the law of propagation of uncertainty has been ob-tained. Specifically, determine the quantities

|y U0.95(y) ylow|, |y + U0.95(y) yhigh|, (9)

viz., the absolute values of the differences of therespective endpoints of the two coverage intervals.Then, if both these quantities are no larger than ,the comparison is successful and the law of propaga-tion of uncertainty has been validated in this instance.

NOTES

1 The validation applies for the specified coverage proba-bility only.

2 The choice of 95 % coverage interval will influence thecomparison. The shortest 95 % coverage interval should gen-

erally be taken, for consistency with the shortest (probabilis-tically symmetric) coverage interval determined following theapplication of the law of propagation of uncertainty.

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8 Examples

The examples given illustrate various aspects of thisSupplement. They show the application of the law ofpropagation of uncertainty with and without contribu-

tions derived from higher-order terms in the Taylor se-ries approximation of the model function. They alsoshow the corresponding results provided by Monte Carlosimulation, and the use of the validation procedure ofSection 7. In some instances, solutions are obtained an-alytically for further comparison.

The first example (simple additive model) demonstratesthat the results from Monte Carlo simulation agree withthose from the application of the law of propagation ofuncertainty when the conditions hold for the latter toapply. The same model, but with changes to the prob-

ability density functions assigned to the values of theinput quantities, is also considered to demonstrate somedepartures when the conditions do not hold. In the sec-ond example (mass calibration) the law of propagationof uncertainty is shown to be valid only if the contribu-tion derived from higher-order terms in the Taylor se-ries approximation of the model function are included.In the third example (electrical measurement), the lawof propagation of uncertainty is shown to yield invalidresults, even if all higher-order terms are taken into ac-count. Instances where the input quantities are uncor-related and correlated are treated.

NOTE Although it is generally recommended that a fi-nal statement of the uncertainty be made to no more thanone or two significant decimal digits (Clause 6.8), more thantwo digits are reported in this section so as to highlight dif-ferences between the results obtained from the various ap-proaches.

8.1 Simple additive model

This example considers the simple additive model

Y = X1 + X2 + X3 + X4 (10)

for three different sets of probability density func-tions gi(i) assigned to the Xi. The Xi and Y areexpressed in (common) arbitrary units. For the firstset each gi(i) is a standard Gaussian probability den-sity function (expectation zero and standard deviationunity). For the second set each gi(i) is a rectangularprobability density function with zero expectation andunit standard deviation. The third set consists of threeidentical rectangular probability density functions anda fourth rectangular probability density function with astandard deviation ten times as large.

Further information concerning additive models suchas (10), where the probability density functions are

Gaussian or rectangular or a combination of both, isavailable [12].

8.1.1 Normally distributed input quantities

Assign a standard Gaussian probability density functionto the value of each Xi. The best estimates of the valuesof the input quantities are xi = 0, i = 1, 2, 3, 4, withassociated standard uncertainties u(xi) = 1.

The law of propagation of uncertainty [GUM 5] givesthe estimate y = 0.0 of the output quantity and theassociated standard uncertainty u(y) = 2.0, using a de-gree of approximation of two significant decimal digitsin u(y) (Clause 6.8). A 95 % coverage interval for thevalue of Y [GUM 6], based on a coverage factor of 1.96,

is [3.9, 3.9].The application of the Monte Carlo procedure (Sec-tion 6) with M = 105 trials gives y = 0.0, u(y) = 2.0 andthe coverage interval [3.9, 3.9]. Repetition with M =106 trials confirms these results to the number of deci-mal digits reported. The latter case was re-run (differ-ent random samplings being made from the probabilitydensity functions) to show the variability in the resultsobtained.

The results are summarized in the first four columnsof Table 3. Figure 7 shows some of the correspond-

ing approximations obtained to the probability densityfunction for the output quantity value. For this exam-ple, such agreement would be expected for a sufficientlylarge value ofM, because all the conditions hold for theapplicability of the law of propagation of uncertainty.

Method y u(y) 95 % CI dlow, dhigh VLPU 0.00 2.00 [-3.92, 3.92]

MCS1 0.01 2.00 [-3.90, 3.92] 0.02, 0.00 YesMCS2 0.00 2.00 [-3.91, 3.94] 0.01, 0.02 YesMCS2 0.00 2.00 [-3.93, 3.91] 0.01, 0.01 Yes

Table 3 The application to the model (10), with

each Xi assigned a standard Gaussian probability

density function (arbitrary units), of (a) the law ofpropagation of uncertainty (LPU) and (b) Monte

Carlo simulation (MCS). MCS1 and MCS2 de-

note MCS with 105 and 106 trials, respectively, y

denotes the estimate of the value of the output

quantity Y, u(y) the standard uncertainty associated

with y, 95 % CI the 95 % coverage interval for the

output quantity value, dlow and dhigh the magnitudes

of the endpoint differences (9), and V whether the

results were validated to two significant digits.

Table 3 also shows the results of applying the com-parison procedure of Section 7. Using the terminologyof that section, ndig = 2, since two significant digits

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Figure 7 Approximations for the model (10),

with each Xi assigned a standard Gaussian prob-

ability density function, to the probability densityfunction for the output quantity value provided

by (a) the law of propagation of uncertainty and

(b) Monte Carlo simulation. For (b) the approxima-

tion constitutes a scaled frequency distribution (his-

togram) of the M = 106 values ofY. The endpoints of

the 95 % coverage interval provided by both meth-

ods are shown as vertical lines. The approximations

to the probability density function are visually indis-

tinguishable, as are those of the coverage intervals.

in u(y) are sought. Hence, u(y) = 2.0 = 20 101

, andso a = 20 and r = 1. Thus the comparison accuracyis

=1

2 101 = 0.05.

The magnitudes of the endpoint differences (Expres-sions ( 9)), and whether the law of propagation of uncer-tainty has been validated, are shown in the right-mosttwo columns of Table 3.

Figure 8 shows the length of the 95 % coverage interval(Clause 6.7), as a function of the probability value at its

left-hand endpoint, for the approximation to the distri-bution function provided by Monte Carlo simulation. Asexpected for a symmetric probability density function,the interval takes its shortest length when symmetricallydisposed with respect to the expectation.

In contrast, Figure 9 shows the comparable figure for theasymmetric probability density function given in Fig-ure 16 (Clause 8.3), relating to a model of comparisonloss in power meter calibration. The 95 % coverage in-terval does not take its shortest length when symmet-rically disposed with respect to the expectation in thiscase. Indeed, the shortest 95 % coverage interval is asfar-removed as possible from an equiprobabilistic cov-erage interval, the left and right tail probabilities be-

Figure 8 The length of the 95 % coverage in-

terval, as a function of the probability value at its

left-hand endpoint, for the approximation to the dis-tribution function obtained by applying Monte Carlo

simulation to the model (10), where each Xi is as-

signed a standard Gaussian probability density func-

tion.

ing 5 % and 0 %, respectively, as opposed to 2.5 %and 2.5 %. The length of the equiprobabilistic cover-age interval is 50 % greater than that of the shortestcoverage interval.

Figure 9 As Figure 8 except for a model of

comparison loss in power meter calibration (Sub-

clause 8.3.3).

8.1.2 Rectangularly distributed input quanti-

ties with the same width

Assign a rectangular probability density function havingan expectation of zero and a standard deviation of unity(cf. Subclause 8.1.1) to the value of each Xi. Again,the best estimates of the values of the input quantities

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are xi = 0, i = 1, 2, 3, 4, with associated standard un-certainties u(xi) = 1.

By following the analogous steps of Subclause 8.1.1, theresults in Table 4 were obtained. In addition, the ana-

lytic solution for the endpoints of the coverage interval,viz., 23(2 (3/5)1/4) 3.88, was determined by asimple calculation.


MCS1 0.00 2.00 [-3.88, 3.85] 0.04, 0.07 NoMCS2 0.00 2.00 [-3.89, 3.87] 0.03, 0.05 YesMCS2 0.00 2.00 [-3.85, 3.90] 0.07, 0.02 No

Analytic 0.00 2.00 [-3.88, 3.88]Table 4 As Table 3, but for rectangular proba-

bility density functions having the same expectations

and standard deviations, and with the analytic solu-

tion.

Figure 10 shows the counterpart of Figure 7 in this case.By comparison with Figure 7, some modest differencesbetween the approximations to the probability densityfunctions can be seen. The use of the law of propagationof uncertainty overestimates the value of the probabilitydensity function in the neighbourhood of the expecta-tion and to a lesser extent in the tails. It underestimatesits value in the flanks. The endpoints of the coverage in-tervals provided are again almost visually indistinguish-able, but Table 4 shows there are small differences.

The coverage interval determined on the basis of the lawof propagation of uncertainty is in this case slightly moreconservative than that obtained analytically. As forthe previous example, the comparison procedure of Sec-tion 7 was applied (Columns 5 and 6 of Table 4). As be-fore, ndig = 2, u(y) = 20101, a = 20, r = 1 and =0.05. For Monte Carlo simulation with M = 105, thelaw of propagation of uncertainty is not validated. Forone of the Monte Carlo simulations with M = 106,this law is validated and for the other it is not. Theseresults emphasize the difficulty of making an absolute

decision automatically when a stochastic process is in-volved. The user should take account of such resultsto decide whether they are fit for purpose. For a de-gree of approximation of one significant digit in u(y),for which = 0.5, the validation status is positive in allthree cases.

8.1.3 Rectangularly distributed input quanti-

ties with different widths

Consider the same example as in the previous section,except that the fourth probability density function hasa standard deviation of ten rather than unity. Table 5contains the results obtained.

Figure 10 The counterpart of Figure 7 for rectan-

gular probability density functions having the same

expectations and standard deviations.


MCS1 0.0 10.2 [-17.1, 16.8] 2.8, 3.1 NoMCS2 0.0 10.1 [-17.0, 17.0] 2.9, 2.9 NoMCS2 0.0 10.2 [-17.1, 17.0] 2.8, 2.9 No

Table 5 As Table 4, except that the probability

density function for the value of the fourth input

quantity has a standard deviation of ten rather than

unity.

Figure 11 shows the approximations obtained to theprobability density function for the output quantityvalue and of the endpoints of the shortest 95 % coverageinterval for the output quantity value. The approxima-tions have very different appearance. The dominanceof the probability density function for the fourth inputquantity is evident. To a first approximation the prob-ability density function for the output quantity valueresembles that of the fourth input quantity. There is,however, an effect in the flanks resulting from the prob-ability density functions for the other input quantities.

The coverage interval determined on the basis of thelaw of propagation of uncertainty in this case is moreconservative than that obtained using Monte Carlosimulation. Again, the comparison procedure of Sec-tion 7 was applied (Columns 5 and 6 of Table 5).Now, ndig = 2, u(y) = 10 = 10 100, a = 10, r = 0and = 1/2 100 = 0.5. In all three cases the law ofpropagation of uncertainty is not validated. For a de-gree of approximation of one significant digit in u(y),for which = 5, the validation status would be posi-tive in all three cases, the 95 % coverage intervals all

being [2 101

, 2 101

].

NOTES

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Figure 11 As Figure 10, except that the probabil-

ity density function for the fourth input quantity has

a standard deviation of ten rather than unity. Theinner pair of vertical lines indicates the endpoints

of the shortest 95 % for the output quantity value

determined by Monte Carlo simulation. The outer

pair is those from the law of propagation of uncer-

tainty and the assignment of a Gaussian probability

density function, with a coverage factor of k = 2.

1 The conditions for the Central Limit Theorem to applyare not well met in this circumstance [GUM G.6.5], becauseof the dominance of the fourth probability density function.However, because these conditions are often in practice as-

sumed to hold, especially when using proprietary softwarefor uncertainty evaluation (cf. the Note in Subclause 8.3.2),the Gaussian probability density function assigned on theassumption of the applicability of this theorem is made inthis subclause for comparison purposes.

2 The coverage interval is expressed as [2 101, 2 101]because the form [20, 20] would imply that its endpointswere given to two significant decimal digits rather than one.

8.2 Mass calibration

8.2.1 Formulation

Consider the calibration of a weight W of mass den-sity W against a reference weight R of mass density Rhaving the same nominal mass using a balance operat-ing in air of mass density a [29]. Since W and R aregenerally different it is necessary to account for buoy-ancy effects. Applying Archimedes Principle, the modeltakes the form

mW

1 a

W

= (mR + mR)

1 a

R

, (11)

where mR is the mass of a small weight of density Radded to R to balance it with W.

Since weights are generally used in air at a density close

to a particular value, it is usual to work in terms ofconventional values. The conventional mass mW,cof W is the mass of a (hypothetical) weight of den-sity 0 = 8 000 kg/m

3 that balances W in air at den-sity a0 = 1.2 kg/m3. Thus,

mW

1 a0W

= mW,c

1 a0

0

.

In terms of conventional values mW,c, mR,c and mR,c,the model (11) becomes

mW,c

1 a

W

1 a0

W

1

=

(mR,c + mR,c)

1 a

R

1 a0

R

1

, (12)

from which, approximately,

mW,c = (mR,c + mR,c)

1 + (a a0) 1W

1R

.

(13)Table 6 lists the input quantities and the probabil-ity density functions assigned to their values for themodel (13).

Xi PDF Parameter valuesmR,c Gaussian 100 g, 0.050 mgmR,c Gaussian 1.234 mg, 0.020 mga Rectangular 1.20 kg/m3, 0.10 kg/m3

W Rectangular 8 000 kg/m3, 1 000 kg/m3

R Rectangular 8 000 kg/m

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