Uncertainty Analysis for Uncorrelated Input Quantities and a Generalization

Transcription

1 WHITE PAPER Ucertity Alysis for Ucorrelted Iput Qutities d Geerliztio Welch-Stterthwite Formul Abstrct The Guide to the Expressio of Ucertity i Mesuremet (GUM) hs bee widely dopted i the differet fields of the idustry d sciece. This guide estblished geerl rules for evlutig d expressig ucertity i the mesuremets. I this pper we will give overview o how to use it for ucorrelted iput qutities. We will lso itroduce correlted mgitudes d correltio types due to the importt issue i the evlutio of mesuremet ucertity s cosequece of the correltio betwee qutities. We will idetify situtios ot icluded ito the GUM, whe the mesurd c be expressed s fuctio of qutities with commo sources. So the issue ppers whe we use the typicl Welch-Stterthwite formul used to clculte the effective umber of degrees of freedom whe the mesuremet errors re ot with fiite degrees of freedom d ucorrelted. We will itroduce geerliztio of the Welch-Stterthwite formul for correlted compoets with fiite degrees of freedom. This pper will lso iclude other methods for computig cofidece limits d expded ucertities such s usig covolutio bsed o mthemticl methods or evlutig the mesuremet ucertity bsed o the propgtio of distributios usig Mote Crlo simultio. Pge

2 Itroductio I geerl mesuremet is ot mesured directly, but is determied from other qutities through fuctiol reltioship. Y = f X, X, X3 X I cses where the iput qutities re idepedet, the combied stdrd ucertity is the positive squre root of the combied vrice which is give by: f C = i= xi u y u xi Mutul depedeces i the kowledge bout the iput qutities c be expressed s covrice or correltio coefficiet d c be used durig the propgtio. u y f f u x x f u x f f u x x = (, ) = + (, ) C i j i i j i= j= xi xj i= xi i= j= i+ xi xj The degree of correltio betwee correltio coefficiet. ( i, j) r x x = u x u( xi, xj) ( i) * u( xj) x i d = + (, ) u y c u x ccu x u x r x x C i i i j i j i j i= i= j= i+ x j is chrcterized by the estimted The expded ucertity of mesuremet is obtied by multiplyig the stdrd ucertity of the output estimte by coverge fctor k which is chose o the bsis of the desired level of cofidece to be ssocited with the iterl defied by U = k u y : * c Whe orml distributio c be ttributed to the mesurd, d the stdrd ucertity ssocited with the output estimte hs sufficiet relibility, the stdrd coverge fctor k = shll be used. Pge

3 The ssumptio is tht the combied error follows orml (ifiite degrees of freedom) or t -Studet distributio (fiite degrees of freedom) results from the Cetrl Limit Theorem. This theorem demostrtes tht the combied error distributio coverges towrd the orml distributio s the umber of costituet errors icreses, regrdless of their uderlyig distributios (Figure ). σ ε σ ε ε Y = f X, X, X3 X f C i= xi = ( i) U y u x f ( ε T) σε T ε T ε Figure. Combied error distributio A first pproch to determie the expded ucertity for cofidece level is to use coverge fctor of orml distributio, k : A B A B U = U + U = k u + u If the umber of rdom redigs is smll, so the vlue of the u A c be ot correct, d the distributio of the rdom compoet is better to represet it by t -Studet distributio, but ow we could overvlue the ucertity, especilly if the umber of mesuremets is smll d the ua d ub vlues re similr i size U = t * u + k * u A B So the best wy to solve this problem is usig the pproch of the Welch Stterwite formul. Pge 3

4 If Norml distributio c be ssumed, but the stdrd ucertity ssocited with the output estimte is with isufficiet relibility d it is ot possible to icrese the umber of repeted mesuremets, we will use the Welch Stterthwite formul. I such cse, the relibility of the stdrd ucertity ssiged to the output estimte is determied by its effective degrees of freedom. Cosiderig direct mesuremet of idepedet mesuremet errors, the distributio my be pproximted by t -distributio with effective degrees of freedom ν eff obtied from the Welch Stterthwite formul: ν eff = u 4 C i= ( y) 4 ( y) u i ν i The coverge k( p ) fctor will be obtied from the t -Studet distributio evluted for coverge probbility of %. Degrees of freedom Frctio i percet ~ t -Studet distributio tble Pge 4

5 A Norml Distributio Cot Be Justified I cses where the ssumptio of orml distributio cot be justified d it is ot possible to pply the Cetrl Limit Theorem, we my fid situtios where oe of the ucertity cotributios i the budget c be idetified s domit term or situtios whe two of the ucertity cotributios i the budget c be idetified s domit terms. Kowig the distributio desity ( y) ϕ we c determie the coverge probbility p, usig the followig itegrl reltio: y+ U ϕ p U k p = y U ( p) U = u y y dy As the coverge fctor my be expressed s: Cses where we hve rectgulr distributio s domit term / Figure. Rectgulr distributio Solvig this reltio for the expded ucertity U d isertig the result together with the expressio of the stdrd mesuremet ucertity relted to rectgulr distributio give by: y+ U y+ U y+ U y U U p ( U ) = ϕ ( y) dy = dy = = y U u( y ) = 3 y U Pge 5

6 Filly gives the reltio: ( p) U p. u( y) = k( p) = = = p. 3 3 u y 3 Ad for coverge probbility p = 95 %, the coverge fctor k is: k( p ) = = Cses where two of the ucertity cotributios i the budget c be idetified s domit terms This will ivolve evlutio of the coverge fctor of stted coverge probbility for the covolved distributios. So depedig o the distributio types which re covolved, the coverge fctor for coverge probbility of 95.45% my be obtied from the followig Tble depedig of the stted rtio: Tble. If two distributios re covolved, the coverge fctor k is obtied from the tble U(y ) N/R k 95.45% U(y ) N/U k 95.45% U(y ) R/U k 95.45% U(y ) R/R k 95.45% N: Norml, R: Rectgulr, U: U-Shped Pge 6

7 Distributio for Combied Error Usig Covolutio I cse where two or more errors re sttisticlly idepedet, the distributio for the combied errors c be obtied by covolutio. This method c be pplied for direct mesuremets where the mesuremet process errors re sttisticlly idepedet, so o error correltios. = ( * ) = ( ) ϕ y f g y f τ g y τ dτ Covolvig two rectgulr distributios If domit cotributios rise from rectgulr distributios of vlues, the distributio resultig from covolvig the gives symmetricl trpezoidl distributio (Figure 3). b y f ( y) g( y) /b / b, b y b f ( y) = b 0, otherwise, y g( y) = 0, otherwise y d ϕ ( y) d y -c c Figure 3. Symmetricl trpezoidl distributio Pge 7

8 Where the hlf widths of the bse d top respectively re: c= b+ d d = b d the edge prmeter: d b β = = c b+ Where the distributio desity my be coveietly expressed i the form: ϕ ( y) if y < β. c = * c( + β ) 0 if c < y y ϕ( y) = * * if β. c y c c( + β) β c The squre of the stdrd mesuremet ucertity deduced from the trpezoidl distributio is: c u y β 6 = ( + ) Kowig tht the coverge probbility is: y+ U ϕ p U k p = y U ( p) U = u y y dy Ad the coverge fctor So, the coverge fctor will be: { k( p)= + β p( + β ) 6 ( ) p β if β if β > Filly, the coverge fctor for coverge probbility of 95% pproprite to trpezoidl p p p p distributio with edge prmeter of β < 0.95 is clculted from the reltio: k p + β p β 6 = * ( )( ) Pge 8

9 k( p ) c chge from.645 to.93 depedig of β. (Tble d Figure 4) Tble. Edge prmeter vs. coverge fctor Edge prmeter (β ) Coverge fctor (k).9 k β Figure 4. Coverge fctor k vs. edge prmeter of trpezoidl distributio Covolvig two Gussi distributios The combied error distributio tkes o Gussi (Figure 5). f ( y) = * e σ π g y = * e σ π ( x µ ) σ ( x µ ) σ f * g y = * e π σ ( + σ ) t ( µ µ ) ( σ + σ ) Pge 9

10 ( f * g)( y) f ( y) g( y) σ σ * = µ µ Figure 5. If two Gussi distributios re covolved, the result is other Gussi distributio Welch-Stterthwite Formul for Correlted Compoets The implemettio of the GUM exhibits issue relted to the effective degrees of freedom whe the mesurd is expressible s fuctio of itermedite qutities tht deped o oe or more shred iputs. The ppret issue is foud i the fct tht lier correltio coefficiet of zero does ot imply sttisticl idepedece. Therefore, the vribles cot be idepedet uless they re orml, d both degrees of freedom re ifiite. It is ot specificlly ssocited with the use of the Welch-Stterthwite formul. It rises from loose d icomplete usge of sttisticl priciples. We will exted the method described i the GUM to be pplicble with correlted compoets of ucertity with fiite degrees freedom. For this kid of coditio, we c use s geerliztio of the Welch-Stterthwite formul the expressio proposed by Howrd Cstrup. So, cosiderig ow two mesuremet errors e d e with ucertities u d u respectively, d whose correltio coefficiet is ρ, the the vrice of the totl error is give by: u = u + u + ρ uu C Usig the dditio rule, we hve ( uc ) ( u ) ( u ) 4 ( uu ) cov ( u, u ) cov ( u uu ) + ρ cov ( u uu ) σ = σ + σ + ρ σ ρ, 4, Workig bit with the cross-product d covrice terms we chieve the followig expressio: + 8 uu ρ σ u σ u { } 4 4 u u σ ( uc ) = + + σ ( u ) + u σ ( u ) + u u u + ν ν Pge 0

11 Kowig tht 4 4 ui y u u i j νeff = uc ( y) / + ρij νi + ν j + + i= νi i= j> ν i ν j ν u ui j ρijuu i j + i= j> νi ν j eff 4 u c = d usig the before expressio for mesuremet σ ( uc ) errors where two or more re correlted, so the fil expressio to clculte the effective degrees of freedom for correlted compoets will be: If ll the correltio coefficiets re zero, this equtio simplifies to the Welch- Stterthwite formul. Propgtio of Distributios Usig Mote Crlo Simultio Whe the model is o-lier or whe the probbility desity fuctio (PDF) for the output qutity deprts pprecible from Gussi distributio or scled d shifted t -Studet distributio we will lso itroduce s ltertive the Mote Crlo Method (MCM). This method dels with the propgtio of probbility distributio of the iput qutities of mthemticl method of mesuremets. MCM provides method to obti pproprite umericl represettio of the output qutity Y by mes of its distributio fuctio G, give mesuremet model equtio. G is obtied by smplig of the PDF of the iput qutities x d pplyig the model of mesuremet to obti smpled vlues for the output qutity Y. Expecttio vlues, vrice d coverge itervls of Y c be extrcted from G. The ccurcy of the resultig G icreses with the umber of trils. A dpttive Mote Crlo procedure c be used isted of usig fixed umber of trils to gurtee tht the results chieve required tolerce. This procedure ivolves crryig out icresig umber of Mote Crlo trils util me, vrice d coverge itervl hve stbilized. A umericl result c be cosidered stbilized if twice the stdrd devitio ssocited with it is less th the umericl tolerce ssocited with the stdrd ucertity u( y ). i Pge

12 Lerig Objectives This pper described the method used to estimte the mesuremet ucertity i ccordce with the priciples give i the GUM for ucorrelted iput qutities, icludig cses where it is ot possible to pply the Cetrl Limit Theorem. This pper hs lso exteded the method to be pplicble with correlted compoets of ucertity with fiite degree of freedom usig geerliztio of the Welch- Stterthwite formul. Aother possibility ws filly itroduced whe ll the qutities re correlted d ormlly distributed. The Mote Crlo Method is useful tool, to clculte ucertities whe the coditios required pplyig the GUM re ot met, d to vlidte d gi cofidece with the results obtied with GUM. The lst itroduced method c hdle correltios s log s ll qutities which re correlted re distributed ormlly or re totlly correlted. I prctice this c be importt limittio i cse the distributio of the correlted qutities differs sigifictly from orml. Ofte it is ot possible to specify the joit PDF for the iput vribles or the joit PDF my ot be i form tht is esy to umericlly simulte. Refereces ISO, Guide to the expressio of ucertity i mesuremet GUM Howrd Cstrup. Welch-Stterthwite reltio for correlted errors Willik Metrologí Supplemet to the GUM. Propgtio of distributios usig Mote Crlo method EA-4/0 Expressio of the ucertity of mesuremet i clibrtio CEM, Correltio types Ler more t: For more iformtio o Keysight Techologies products, pplictios or services, plese cotct your locl Keysight office. The complete list is vilble t: This iformtio is subject to chge without otice. Keysight Techologies, 0-08, Published i USA, December 8, 08, EN Pge