Evaluation of Measurement Uncertainty Using Bayesian Inference

Transcription

1 3 th IMEKO TC4 Symposium on Mesurement or Reserch nd Industry Appliction (Athens-Greece- Septemer 29 th -Octoer st 24 Evlution o Mesurement Uncertinty Using Byesin Inerence Getno Iuculno, Griell Pellegrini 2, Andre Znoini Deprtment o Electronics nd Telecommuniction, University o Florence, Vi di S. Mrt, 3 Florence I 539 phone: , , e-mil: iuculno@ingi.ing.unii.it, willis@ingi.ing.unii.it 2 Deprtment o Applied Mthemtics, Engineering Fculty, University o Florence, Vi di S. Mrt, 3 Florence I 539 phone: , : , e-mil: gultieri@dm.unii.it Astrct-The Byesin inerence provide nturl nd consistent wy to mke est use o ll relevnt historicl inormtion which chrcterizes the mesurnd in mesurement process. In this pper the use o Byesin method is descried to include prior inormtion in the evlution o the mesurement uncertinty. The proposed pproch gives n implementtion on the evlution o the conidence region with respect the usul techniques descried in the ISO Guide to the Epression o Uncertinty in Mesurement (GUM []. Eperimentl model is studied nd numericl results re reported to ssess the vlidity o the proposed ormule. I. Introduction In the orthodo Neymnn-Person pproch we re given set o oserved dt rom which we re to decide whether some hypothesis out the rel world (in prticulr out prmeters o primry interest is true (or, put more cutiously, whether to ct s i it were true. The irst thing orthodo sttistics does is to imed the oserved dt set in "smple spce" which is n imginry collection contining other dt sets tht one thinks might hve een oserved ut were not. Then one introduces the "smpling distriution" deined y the proility p( D H i tht the generic dt set D i would e oserved i the hypothesis H were true; this proility is interpreted s the theoric strction o the requency with which the dt set D i would e oserved in the long run i the mesurement were mde repetedly with H constntly true. When one sserts the long run results o n ritrrily long sequence o mesurements tht hve not een perormed it would pper tht he is on rther lrge hidden und o prior knowledge out the nlysed phenomenon. I we re not told wht tht knowledge is nd how it ws otined, we might e ecused or douting its eistence. Further, in the long run, how oten would it led us to correct conclusion, or how lrge would the verge error o estimtion e? Finlly when mking inerences out set o prmeters o interest we must lso tke into ccount o nuisnce or incidentl prmeters. Ecept when suitle suicient sttistics eist or ll the prmeters diiculties rise in deling with nuisnce prmeters y orthodo pproch. Suicient sttistics ply vitl role in smpling theory. For, i inerences out ied prmeters re to e mde using the distriutionl properties o sttistics which re unctions o the dt, then, to void ineiciency due to the lekge o inormtion, it is essentil tht smll minimlly suicient set o sttistics e ville contining ll the inormtion out the prmeters. By hppy mthemticl ccident such site o suicient sttistics do eist or numer o importnt distriutions nd, in prticulr, or the Norml distriution. However, serious diiculties cn ccompny the eplortion o less restricted models which my e motivted y scientiic interest, ut or which no convenient set o suicient sttistics hppens to e ville. Becuse Byesin nlysis is concerned with the distriution o prmeters, given known (ied dt, it does not suer rom this rtiicil constrint. It does not mtter whether or not the distriution o interest hppens to hve the specil orm which yields suicient sttistics. Furthermore, even when suicient sttistics re ville, emples cn occur in smpling theory where there is diiculty in eliminting nuisnce prmeters. Using smpling theory it is diicult to tke ccount o constrints which occur in the speciiction o the inerence-prmeter spce.

2 3 th IMEKO TC4 Symposium on Mesurement or Reserch nd Industry Appliction (Athens-Greece- Septemer 29 th -Octoer st 24 By contrst, in Byesin nlysis, inerences re sed on proilities ssocited with dierent vlues o prmeters which could hve given rise to the ied set o dt which hs ctully occurred. In clculting such proilities we must mke ssumptions out prior distriutions, ut we re not dependent upon the eistence o suicient sttistics, nd no diiculty occurs in tking ccount o prmeters constrints. In other terms the Byesin pproch sets us ree rom the joke o suiciency. In the pure minimum cross-entropy (noiseless Byes methods, our resoning ormt is lmost the opposite o smpling theory. Insted o considering the clss o ll dt sets { D, L,D N } consistent with hypothesis H, we consider the clss o ll hypotheses { H, L,H N } consistent with the one dt set tht ws ctully oserved. In ddition we use prior inormtion I tht represents our knowledge D os (rom physicl lw o the possile wys in which Nture could hve generted the vrious H i. Out o the clss C o hypotheses consistent with our dt, we pick the one voured y prior inormtion, which mens, usully, hving the minimum cross-entropy. Ech successive piece o dt tht one otins is new constrint tht restricts the possiilities permitted y our previous inormtion. At ny stge n honest description o wht we know must tke into ccount (tht is, ssign non zero proility to every possiility tht is not ruled out y our prior inormtion nd dt. It is not possile to etrct ll tht detil rom the dt lone. The minimum cross-entropy gives us more inormtion only ecuse we hve put more inormtion into it. Oten prior inormtion is ville out nuisnce prmeters, ut orthodo ideology does not recognize it ecuse it does not consist o requencies in ny rndom eperiment. In the prolems where pure minimum cross-entropy is pproprite we re concerned, not with requencies in ny rndom eperiment, ut with rtionl thinking in sitution where our inormtion is incomplete. In orthodo thinking requency is considered "ojective" nd thereore respectle, while mere stte o knowledge is sujective nd unscientiic. But in the rel-world prolems o mesurement ced y every eperimenter it is evidently his stte o knowledge tht determines the qulity o the decisions he is le to mke in situtions tht will never e repeted; nd it is the requencies tht re pigments o the imgintions. Interestingly, the smpling distriution tht orthodo theory does llow us to use is nothing more thn wy o descriing our prior knowledge out the "noise". Thus, orthodo thinking is in the curious position o holding it decent to use prior inormtion out noise, ut indecent to use prior inormtion out the signl o interest. I we hve oth noise nd prior inormtion neither o smpling theory nd MINCENT principle is dequte. But oth re only limiting cses o more generl method tht pplies in ll cses. Adding prior inormtion cpilities to orthodo methods, or noise cpilities to MINCENT, we rrive in either cse t the Byes method, which is ctully simpler conceptully nd older historiclly thn either o these specil cses. The proposed evlution o mesurement uncertinty is sed completely on Byesin nlysis nd on the principle o minimum cross-entropy. The theory is universlly pplicle to most mesurement tsks including comple non liner djustment nd, in prticulr, in cse where the well-estlished lest-squres or mimum likelihood techniques il. II. Generl remrks Assume tht in mesurement process ll the sources o uncertinty re chrcterized y proility distriution unctions, the orm o which is ssumed to either e known rom eperience or unknown nd so conjectured. In Byesin pproch prior to otining the mesurement results the eperimenter considers his degrees o elie nd individul eperience or the possile models nd represents them in the orm o initil proilities (clled "prior". Once the results re otined Byes' theorem enles the eperimenter to clculte new set o proilities which represents revised degrees o elie in the possile models, tking into ccount the new inormtion provided y the mesurements results. The results re the linkge etween "posterior" nd "prior". In rie, results do not crete elies, rther they modiy eisting ones. The revised proilities re reerred s "posterior". For the Byesin eperimenter it is nturl to regrd sujective elies s primry nd "ojective consensus" s specil cse rising when the mount o the dt ville nd the eplicit sttement o the dt structure is dominnt enough to overwhelm every one's prior elies orcing them into the sme posterior shpe nd loction. The Byesin inerence is ppeling or its sequentil nture lterntion etween conjecture nd eperiment, since it llows us to continully updte inormtion

3 3 th IMEKO TC4 Symposium on Mesurement or Reserch nd Industry Appliction (Athens-Greece- Septemer 29 th -Octoer st 24 out the sme mesurnd s more mesurements re undertken in dierent occsions. In ct tody's posterior or the irst mesurement occsion cn ply the role o tomorrow's prior or the second one. Let us to come ck to the mesurement process whose involved quntities re summrized in two principl sets represented y row vectors: the input quntities X in numer o n, nd the output quntities Y, in numer o m. Let ( y, the ctul reliztions o ( Y X, in prticulr occsion, they represent stte o the mesurement process in tht occsion. The process hs set D o possile sttes, y D X, Y. ( which identiy the joint domin o the rndom vriles ( The link etween the input nd output quntities is epressile through unctionl reltionships o type: Y ( i g i X, L, X n ; i,..., m with m n. The mutul ehviour etween the input quntities X nd Y is sttisticlly drwn y the joint proility (, y which cn e written s: ( y ( ( y where ( is the mrginl joint density o the input quntities X nd ( y density o the output quntities Y, given, ( X. In the Byesin pproch we hve nother importnt linkge etween ( where ( y prior joint density nd ( y l(, y ( y c ( ( y is the conditionl joint nd ( y,, tht is: (2 is the posterior joint density o the input quntities X, whose ( is interpreted s the, regrded s unction o or preied output vlues y, represents the well-known likelihood; c is "normlizing" constnt necessry to ensure tht the posterior joint density integrtes, with respect to, to one. In prcticl situtions the mesurement process represents controlled lerning process in which vrious spects on uncertinty nlysis re illuminted s the study proceeds in n up-to-dte lterntion etween conjecture nd eperiment crried out vi eperimentl design nd dt nlysis. A mesurement process is perormed i inormtion supplied y it is likely to e considerly more ccurte, stle nd relile thn the pool o inormtion lredy ville. The sustntil mount o inormtion, got with respect to the conditions prior to the result ter the mesurement process is perormed, cn e connected to the "Kullck's principle o minimum cross-entropy". This, s it is known, is correct method o inductive inerence when no suicient knowledge out the sttisticl distriutions o the involved rndom vriles is ville eore the mesurement process is crried out ecept or the permitted rnges, the essentil model reltionships nd some constrints, gined in pst eperience, vlule usully in terms o epecttions o given unctions or ounds on them. III The Entropy optimistion principles nd the Byesin Inerence The evlution o the mesurement uncertinty is nlysed y pplying the entropy optimistion, y epressed y (, tking into ccount the Byesin principles to the joint density unction ( inerence, given y (2, which discrimintes etween the prior densities o the input quntities X nd the conditionl densities o the output quntities given the input ones., y, sujected to given constrints, we introduce the In order to produce the est estimtion or ( joint cross entropy unctionl: S (, y ln (, y (, ddy E ln D where we hve pssed to the compct nottions: d ( X,Y ( X,Y Ld n,dy dy m (3 d Ldy nd where (, y, which priori must e known, is deined y Jynes [] n "invrint mesure" unction. According to the principle o minimum cross entropy introduced y Kulck under the nme o the principle o minimum discrimintion inormtion, we reserch the joint proility unction tht minimize the unctionl S using the Euler-Lgrnge method y the clculus o vrition, tking into ccount the given constrints., y is ssumed to e constnt, it is equivlent to pply the principle o mimum entropy to the I ( unctionl:

4 3 th IMEKO TC4 Symposium on Mesurement or Reserch nd Industry Appliction (Athens-Greece- Septemer 29 th -Octoer st 24 H D (, y ln (, y ddy A priori distriution: Eponentil model Beore to pply the Byesin pproch, we hve to ind the proility distriution unction (the prior density,, o quntity X whose previous dt re tken rom mnucturer's speciictions or ( hndooks. We suppose tht the vlue µ (the est ville estimte is given nd it is stted tht "the smllest possile vlue is µ nd the lrgest possile vlue is µ ", with, >. The known upper nd lower ounds, nd or the quntity X re not symmetric with respect to its est known estimte µ, ssumed to e the epecttion o X, tht is: E { X} ( d µ We consider the cross-entropy o the proility distriution unction ( prior one ( : ( ( S ( d nd ccording to the Kulck s cross-entropy principle, the est estimte o ( minimizing S. We suppose ( (4 with respect to given ln (5 is the distriution constnt; it is relly uniorm i the sic ville inormtion concerns only the ounds without involving the most plusile vlue mimize the well-known Jynes entropy: µ. Consequently, insted o (5, we must ( ( d suject to the constrint (4 nd to the normlizing condition method y the clculus o vrition. Ater simple mnipultion we otin: H ln ( λ( µ ( ce ( d, using Euler-Lgrnge (7 tht is the priori distriution o X is truncted eponentil with epecttion µ. The constnt c is such tht: λ c e d i.e. λ c λ λ e e The Lgrnge multiplier λ is determined through the constrints (4. In ct we hve: c λ ( µ ( e λ µ d c e d (9 For using the integrl's property nd tking into ccount (8we deduce: e e λ ( Sustituting ( into (8 we otin: The vrince o X my e evluted y: Vr λ λ λ λ e e c λ [ λ e e ] 2 λ ( µ { X} c ( µ e e e (8 ( 2 λ 2 λ 2 λ d c e d λ λ e e (2 Finlly, y ( nd (2 we otin: Vr { X} (3 λ ccording to the GUM [] note 2 in prgrph We will work through the emple in prgrph tken rom the GUM. The vlue o the

5 3 th IMEKO TC4 Symposium on Mesurement or Reserch nd Industry Appliction (Athens-Greece- Septemer 29 th -Octoer st 24 coeicient o liner therml epnsion o pure copper t α, s 52 C, 2 C ( CU 2 ssuming tht the error in this vlue should not eceed. 4 C, it is resonle to ssume tht the vlue o α (CU lies with equl proility in the intervl 2 C to C So tht i we ssume 4 C nd 92 C, it is interesting to study the vriility o the stndrd uncertinty nd its reltive vlue with respect ll the possile vlues o α (CU inside the given intervl [ 2 ],. Computtionl results re reported in tle : In column ( is given the vlue µ ttriuted to α ( CU 2 inside the intervl [ ],, in column (2 the evlution o the scle prmeter λ stisying eq. (, in column (3 the stndrd uncertinty { X} σ Vr given y eq. (2 nd inlly in column (4 the coeicient c σ µ representing the reltive vriility o the stndrd uncertinty with respect the µ vlue: µ λ { } σ Vr X c σ µ A posterior distriution y Byesin pproch Tle According to Byes theorem or continuous vriles, the posterior distriution unction my e written s: y ce l, y (4 where or generic ( ( y X y is the vlue o the output Y [, otined in prticulr occsion nd l(, y ( y y is the well known likelihood. By pplying the mimum entropy principle to the likelihood (, y we otin sustituting (5 into (4 we hve: (, y dy l nd yl (, y dy E{ Y } X y (, y e l ( y l sujected to the constrints: (5 c e y λ X now the constnt c my e evluted y imposing the constrint ( y dy. So tht we cn write: nd consequently K ν ( z c e y λ d ( λy 2 Y ( c 2 (7 K eing the modiied Bessel unctions. Epecttion nd vrince or the posterior distriution re evluted respectively y: E y y K { } ( 2 λy X y c e d λ λ K ( 2 λy (8

6 3 th IMEKO TC4 Symposium on Mesurement or Reserch nd Industry Appliction (Athens-Greece- Septemer 29 th -Octoer st 24 nd { } { } { } ( 2 2 y K 2 λy X y E X y E X y y λ λ K ( 2 λy y K λ K ( 2 λy ( 2 λy Vr - (9 By reerring to the intervl,, i the prior epecttion vlue is ssumed to e µ 52, we hve the prior distriution scle prmeter λ 533. µ In Tle 2 re reported the posterior epecttion vlue µ, the posterior stndrd uncertinty { X } σ Vr y nd its reltive vlue with respect µ coming rom equtions (8 nd (9 or dierent vlues o y, representing the output Y [, otined in prticulr occsion: y µ E { X y } Vr{ X } σ µ σ y Tle 2 Conclusion Rigorous ormule or the evlution o mesurement uncertinty hve een developed. The ormule were estlished using entropy optimistion principles nd Byesin inerence. Emples or eponentil model were presented. Reerences [] Guide to the Epression o Uncertinty in Mesurement, irst edition (993, corrected nd reprinted (995, Interntionl Orgniztion or Stndrdiztion (Genev, Switzerlnd. [2] Interntionl Stndrd ISO Sttistics-Voculry nd Symols-Prt I: Proility nd Generl Sttisticl Terms, irst edition (993, Interntionl Orgniztion or Stndrdiztion (Genev, Switzerlnd. [3] Kpur J.N., Mimum-Entropy Models in Science nd Engineering, John Wiley & Sons [4] Weise K., W & oger W., A Byesin theory o mesurement uncertinty, Mes. Science Tech., 993, 4, - [5] Lir I., Kyrizis G., Byesin inerence rom mesurement inormtion, Metrologi, 999, 3, 3-9.