guide to the expression of uncertainty in … to the expression of uncertainty in measurement gum...
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National Physical Laboratory
Guide to the Expression of Uncertainty in Measurement (GUM) and its supplemental guides
Maurice CoxNational Physical Laboratory, [email protected]
http://www.npl.co.uk/ssfm/index.html
IAJapan, Tokyo, March 2003
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OutlineOutline
JCGMJCGM
Current GUMCurrent GUM
GUM revisionGUM revision
Two phases of uncertainty evaluationTwo phases of uncertainty evaluation
Supplemental guidesSupplemental guides
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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
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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
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Why revise the GUM?Why revise the GUM?
Improve ease of useImprove ease of use
Give added value Give added value
Address limitationsAddress limitations
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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4. Modelling4. Modelling
Multistage modelsMultistage modelsManaging multistage modelsManaging multistage modelsExample: calibration curvesExample: calibration curves
Measurementdata
Calibrationcurve
parameters
Calibrationcurvevalues
Area undercalibration
curve
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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
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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
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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
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Optional slidesOptional slides
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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
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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
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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
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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
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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
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AudienceAudience
Many bodies, includingMany bodies, including
AcademiaAcademia
IndustryIndustry
Accreditation bodiesAccreditation bodies
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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
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DiscussionDiscussion