National Physical Laboratory

Guide to the Expression of Uncertainty in Measurement (GUM) and its supplemental guides

Maurice CoxNational Physical Laboratory, UKmaurice.cox@npl.co.uk

http://www.npl.co.uk/ssfm/index.html

IAJapan, Tokyo, March 2003

OutlineOutline

JCGMJCGM

Current GUMCurrent GUM

GUM revisionGUM revision

Two phases of uncertainty evaluationTwo phases of uncertainty evaluation

Supplemental guidesSupplemental guides

Joint Committee for Guides in Metrology (JCGM)Joint Committee for Guides in Metrology (JCGM)

JCGM comprises JCGM comprises Bureau International des Bureau International des PoisPois et et Mesures Mesures (BIPM)(BIPM)InternationalInternational ElectrotechnicalElectrotechnical Commission Commission (IEC)(IEC)International Federation of Clinical Chemistry International Federation of Clinical Chemistry (IFCC)(IFCC)International Organisation for Standardisation International Organisation for Standardisation (ISO)(ISO)International Union of Pure and Applied Chemistry International Union of Pure and Applied Chemistry (IUPAC)(IUPAC)International Union of Pure and Applied Physics International Union of Pure and Applied Physics (IUPAP)(IUPAP)International Organisation of Legal Metrology International Organisation of Legal Metrology (OIML)(OIML)International Laboratory Accreditation Cooperation International Laboratory Accreditation Cooperation (ILAC)(ILAC)

Body responsible for GUM revisionBody responsible for GUM revisionJCGM/WG1, Measurement UncertaintyJCGM/WG1, Measurement Uncertainty

Guide to the Expression of Uncertainty in Guide to the Expression of Uncertainty in MeasurementMeasurement

GUM published by ISO in midGUM published by ISO in mid--1990s1990s

Widely usedWidely used and and respectedrespected

Recognised by many as the Recognised by many as the master documentmaster document on on uncertaintyuncertainty

Embraces Embraces many aspectsmany aspects of uncertainty evaluationof uncertainty evaluation

But But …… somesome deficiencies and limitations identifieddeficiencies and limitations identified

Process of Process of GUM revisionGUM revision under wayunder way

Why revise the GUM?Why revise the GUM?

Improve ease of useImprove ease of use

Give added value Give added value

Address limitationsAddress limitations

PoliticsPolitics

Continue to promote current GUMContinue to promote current GUM

Enormous investment by organisationsEnormous investment by organisations

Make Make no changeno change to GUM to GUM per seper se

Added value through Added value through supplemental guidessupplemental guides

The two phases of uncertainty evaluationThe two phases of uncertainty evaluation

1.1. FormulationFormulationDevelop (mathematical or other) modelDevelop (mathematical or other) modelAssign input probability distributionsAssign input probability distributions

2.2. CalculationCalculationDerive output probability distributionDerive output probability distributionEstimate measurand and standard uncertaintyEstimate measurand and standard uncertaintyProvide coverage interval for measurandProvide coverage interval for measurand

E.g., at 95% level of confidenceE.g., at 95% level of confidence

Current GUMCurrent GUM

Commonest implementation of GUM: LPU + CLTCommonest implementation of GUM: LPU + CLT

[GUM Clause 8][GUM Clause 8]

Best estimate Best estimate yy of measurand of measurand YY

Model evaluated at best estimates Model evaluated at best estimates xxii of input quantities of input quantities XXii

LPU: Law of Propagation of UncertaintyLPU: Law of Propagation of Uncertainty

Combination of standard uncertainties Combination of standard uncertainties uu((xx) (and covariances) ) (and covariances) of input quantities of input quantities àà standard uncertainty standard uncertainty uu((yy) of ) of yy

CLT: Central Limit TheoremCLT: Central Limit TheoremAssign a Gaussian (or Assign a Gaussian (or ““tt””) distribution to the measurand) distribution to the measurand

Current GUMCurrent GUM

ImplementationsImplementations

HigherHigher--order LPU possibleorder LPU possible

Difficult to implementDifficult to implement

GUM provides formula for mutually independent GUM provides formula for mutually independent casecase

Assumes GaussianAssumes Gaussian

No GUM formula for mutually dependent caseNo GUM formula for mutually dependent case

Software packages often Software packages often apply LPU + CLTapply LPU + CLT

Checks for applicability?Checks for applicability?

Fitness for Fitness for purposepurpose

Supplemental guidesSupplemental guides

1.1. Propagation of distributions Propagation of distributions

2.2. More than one measurandMore than one measurand

3.3. Conformity assessmentConformity assessment

4.4. ModellingModelling

5.5. Calibration curvesCalibration curves

1. Propagation of distributions1. Propagation of distributions

Required to ensure valid solution whenRequired to ensure valid solution when

Contributory uncertainties arbitrarily largeContributory uncertainties arbitrarily large

Contributory uncertainties not comparable in sizeContributory uncertainties not comparable in size

Very few contributory uncertainties Very few contributory uncertainties

Models highlyModels highly nonlinearnonlinear or complicatedor complicated

Distributions asymmetricDistributions asymmetric

1. Propagation of distributions1. Propagation of distributions

Recommended implementation procedure: Monte Carlo simulationVery simpleAvoids partial derivatives!

Y = fY = f((XX11, , XX22, , XX33))

XX11

XX22

XX33

YY

1. Very simple non1. Very simple non--linear model: linear model: YY = = XX1122

Y = XY = X1122

XX11 YY

-5 0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Estimate of output pdf's

Output value

Normal input does not imply normal output when Normal input does not imply normal output when model is nonmodel is non--linearlinear

Applicability of Central Limit Theorem!Applicability of Central Limit Theorem!

xx11 = 1, = 1, uu((xx11) = 1) = 1

1. Validation of uncertainty calculations1. Validation of uncertainty calculations

Use MCS to validate LPU + CLT Use MCS to validate LPU + CLT

Favourable comparisonFavourable comparisonLPU + CLT satisfactory for applicationLPU + CLT satisfactory for application

Unfavourable comparisonUnfavourable comparisonEitherEither detailed investigation requireddetailed investigation requiredOrOr use MCS instead for uncertainty calculationuse MCS instead for uncertainty calculation

2. More than one measurand2. More than one measurand

Y1 = f1(X1, X2, X3)

Y2 = f2(X1, X2 , X3)

X1

X2

X3

Y1

Y2

Input quantities Input quantities XX11, , XX22, , ……

Values, standard uncertainties and covariancesValues, standard uncertainties and covariances

Output quantities Output quantities YY11, , YY22, , ……

Values, standard uncertainties and covariancesValues, standard uncertainties and covariancesYY11, , YY22, , …… almost invariably correlatedalmost invariably correlated

2. More than one measurand2. More than one measurand

Generalisations of the GUM formulaeGeneralisations of the GUM formulae

MatrixMatrix--vector expressionsvector expressionsCompact formulaeCompact formulaeSuitable for software implementationSuitable for software implementation

Provision of best estimates of Provision of best estimates of YY11, , YY22, , ……, and , and standard uncertainties and covariancesstandard uncertainties and covariances

Coverage Coverage regionsregions under consideration!under consideration!

E.g., E.g., VVyy = = JJxxVVxxJJxxTT

3. Uncertainty in conformance testing3. Uncertainty in conformance testing

Industrial inspection: tolerance zonesIndustrial inspection: tolerance zones

Regulations: Limit valuesRegulations: Limit values

Process and inherent variabilityProcess and inherent variability

Decisions based on measurementDecisions based on measurement

3. Uncertainty in conformance testing3. Uncertainty in conformance testing

Accounting for measurement uncertaintiesAccounting for measurement uncertainties

Acceptance and rejection of componentsAcceptance and rejection of components

Repercussions of surpassing regulatory limitsRepercussions of surpassing regulatory limits

Decision rulesDecision rules

ConsumerConsumer’’s and producers and producer’’s riskss risks

4. Modelling4. Modelling

The formulation phase of uncertainty evaluationThe formulation phase of uncertainty evaluation

Physical, empirical and hybrid modelsPhysical, empirical and hybrid models

Functional and probabilistic modelsFunctional and probabilistic models

2 × 2 × 2 = 8 categories}

Classification of modelsExplicit or implicitReal or complex-valued1 or more than 1 measurand

Assignment of probability distributions to input quantities

4. Modelling4. Modelling

Multistage modelsMultistage modelsManaging multistage modelsManaging multistage modelsExample: calibration curvesExample: calibration curves

Measurementdata

Calibrationcurve

parameters

Calibrationcurvevalues

Area undercalibration

curve

5. Least squares adjustment5. Least squares adjustment

LeastLeast--squares modellingsquares modelling

Widespread in metrologyWidespread in metrology

Associated uncertainty evaluationAssociated uncertainty evaluation

Not well understoodNot well understood

Calibration curves a very important applicationCalibration curves a very important application

Response vs. stimulusResponse vs. stimulus

Generation of curve from standards (response values)Generation of curve from standards (response values)

Use of curve to obtain stimulus valuesUse of curve to obtain stimulus values

x

y

yAu(yA)

xAu(xA)

Given stimuli (standards xi & u(xi))Measure responses yi & obtain u(yi)Derive regression line (curve) y = f(x)Form x & u(x) given y & u(y)

5. Calibration 5. Calibration curvescurves

Respect stated uncertainties and

known mutual dependencies

JCGM and ISO/TC69 link

ConclusionConclusion

Respect investment in common use of GUMRespect investment in common use of GUM

Give added value to current GUMGive added value to current GUM

Two phases of uncertainty evaluationTwo phases of uncertainty evaluation

Supplemental guidesSupplemental guides1.1. Propagation of distributionsPropagation of distributions2.2. More than one measurandMore than one measurand3.3. Uncertainty in conformance testingUncertainty in conformance testing4.4. ModellingModelling5.5. LeastLeast--squares adjustmentsquares adjustment

Optional slidesOptional slides

UncertaintyUncertainty--related standardsrelated standards

Precision of test methods

Measurement unc

GUM revision

Appln of Statl Methods

Unc of measurement

BIPM Director’s Adv Group on Uncertainties

WG: Relation ISO5725 & GUM

JCGM

ISO

CEN

BSI

Unc experts’ groupEA = convenorship

MRA

Terms of ref

Other new work items in JCGMOther new work items in JCGM

Introductory document on uncertaintyIntroductory document on uncertainty

Introduce the GUM and its supplemental guidesIntroduce the GUM and its supplemental guides

““IndustrialIndustrial”” guide to uncertaintyguide to uncertainty

Introductory document on uncertaintyIntroductory document on uncertainty

Possible scopePossible scope

An introduction to measurement uncertaintyAn introduction to measurement uncertainty

An introduction to the GUMAn introduction to the GUM

An introduction to GUM supplemental guidesAn introduction to GUM supplemental guides

Introductory document on uncertaintyIntroductory document on uncertainty

Emphasise intention of the GUMEmphasise intention of the GUM

Provide an estimate Provide an estimate (the measurement result)(the measurement result) of a of a quantity of interest quantity of interest (the measurand)(the measurand)

Provide a Provide a coverage intervalcoverage interval about the estimate that about the estimate that can be expected to contain a can be expected to contain a specified large specified large fraction of the possible values of the measurandfraction of the possible values of the measurand

Means of achieving aimMeans of achieving aim

GivenGiven

The The effectseffects on which the measurand dependson which the measurand depends

Probability distributionsProbability distributions assigned to these effectsassigned to these effects

A A rule/functional relationship/modelrule/functional relationship/model for expressing for expressing the measurand in terms of these effectsthe measurand in terms of these effects

Determine Determine probability distributionprobability distribution for the measurand for the measurand àà

the the measurement resultmeasurement result

the associated the associated standard uncertaintystandard uncertainty

a a coverage intervalcoverage interval for the measurandfor the measurand

AudienceAudience

Many bodies, includingMany bodies, including

AcademiaAcademia

IndustryIndustry

Accreditation bodiesAccreditation bodies

ApproachApproach

Describe scope of GUM and supplemental guidesDescribe scope of GUM and supplemental guides

Help to the reader to navigate through themHelp to the reader to navigate through them

Three considerationsThree considerations

ConceptsConcepts

FormulationFormulation

ImplementationImplementation

Particular GUM implementationParticular GUM implementation

LPU + CLTLPU + CLT

DiscussionDiscussion