A Bayesian Approach to Information Fusion for Evaluating the Measurement Uncertainty

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1 006 IEEE Iteratoal Coferece o Mltsesor Fso ad Iterato for Itellet Sstems September 3-6, 006, Hedelber, Germa WeB0. A Baesa Approach to Iformato Fso for Evalat the Measremet Ucertat Klas-Deter Sommer, Olaf Keh, Ferado Pete Leò, Berd R. L. Sebert Abstract - The Baesa approach to certat evalato s a classcal example for formato fso. It s based o both, the owlede abot the measr process ad the pt attes. Approprate probablt dest fctos for the pt attes ma be obtaed b tlz the prcple of maxmm formato etrop ad the Baes theorem. The owlede abot the measremet process s represeted b the so-called model eato whch forms the bass for the fso of all volved pt attes. Compared to the ISO-GUM procedre, the Baesa approach to certat evalato does ot have a restrcto related to oleart ad determato of cofdece tervals. T I. ITRODUCTIO he cocept whch, accordace wth the Gde to the Expresso of Ucertat Measremet [] (here deoted as ISO-GUM), derles the moder evalato of measremet certat s based o both, the avalable formato abot the measr process ad the (pt) attes ad parameters that have flece o the vale of the measremet reslt. The owlede abot the measr process s to be codesed to the so-called model eato (fso model). It mathematcall represets the terrelato betwee the measrad ad the volved pt attes,..., ad ther vales respectvel: f,..., M, (a) f,..., M. (b) Mascrpt receved Aprl 5, 006. K.-D. Sommer s wth the Thra State Brea for Metrolo ad Verfcato (LMET), D Ilmea, Germa (correspod athor to provde phoe: +49(3677)850-0, fax: +49(3677) ; e-mal: lasdeter.sommer@lmet.de). O. Keh s wth the Thra State Brea for Metrolo ad Verfcato (LMET), D Ilmea, Germa (e-mal: olaf.eh@lmet.de). F. Pete Leò s wth the Techsche Uverstaet Meche (Mch Techcal Uverst, TUM), Dep. of Electrcal Eeer ad Iformato Techolo, D Mch, Germa (e-mal: f.pete@tm.de) B. R. L. Sebert s wth the Phsalsch-Techsche Bdesastalt (PTB), D-386 Braschwe, Germa (e-mal: berd.seber@ptb.de). Therefore, the model eato mht be derstood as fso model of the relevat pt attes. Moreover, becase t s presmed to represet the relatoshp betwee the pt attes ad the measrad e ad completel, t addtoall covers the scal of the pt attes. A formato abot these pt attes s to be wehed as more or less relevat ad relable b ass approprate probablt dest fctos (pdfs) to them, are the possble vales of. I accordace wth the Baesa cocept [-6], a pdf represets the state of owlede abot the dvdal pt att whece ever ths owlede s desceded from. The expectato vale of a pdf, x E, s tae for the best estmate of the att ad the stadard devato of the pdf s tae as the stadard certat assocated wth the above expectato, Var[ ]. II. x BAESIA DESCRIPTIO OF KOWLEDGE ABOUT IPUT UATITIES The state-of-owlede pdf for a pt att ma be obtaed b tlz the prcple of maxmm formato etrop (pme) [-3] that, for example, elds a rectalar pdf f oe ows that the vales of the att are cotaed a terval (practcal examples: ve toleraces or error lmts, dtal resolto), a Gassa (ormal) pdf f oe ows the best estmate x j = E[ j ] ad the assocated stadard certat xj of the att j (practcal examples: statemet of a calbrato reslt, reslt of a statstcal aalss expressed b a mea ad a stadard devato). If ew or addtoal formato I, partclarl terms of measred data of the measrad, s avalable, the chae of a (possbl) ve pror pdf s descrbed b the Baes theorem [-6]: The posteror pdf I, I ta accot of ew data I reslts from the pror formato I as /06/$ IEEE. 507

2 a) Measrad possble vales? 0mm vae pror owlede Pror pdf: I Measremet process Assemet: Lelhood: l, I observatos,..., Measremet reslt Calcls, Test Report Posteror pdf:, I b) Lelhood Posteror pdf Pror pdf,, l I h I (freec dstrbto for [5]) F.. Illstrato of Baesa ferece measremet: a) Measremet process.b) Probablstc descrpto of the state of owlede. Smbols: I pror pdf represet vae pror owlede l, I lelhood represet the measr process wth the observed att ;, I owlede ferred from the lelhood ad the pror pdf; h, I I abot the measrad, e.. the omal vale ad ve error lmts for the measrad; freec dstrbto for the att posteror pdf represet the avalable prodct of a ormatve costat C, the Lelhood l I, I ad the pror pdf: I, I d C l I, I I d. () For a repeatedl observed att whch reflects the measrad, the Baesa approach s eerall llstrated F.. It shold be oted that ths wa to certat evalato s ot (et) part of the ISO-GUM procedre [] bt the Jot Commttee for Gdes Metrolo of the Brea Iteratoal de Pods et Mesres (BIPM) s prepar frther docmets that are based cosstetl o the Baesa probablt theor [7]. III. REPEATED OBSERVATIOS Toda, practce, the pror owlede abot the measrad tself s sall elected. I case of repeated observatos,..., of the att, a Gassa probablt model for a ve datm elds [3-4, 8]: (, ) exp, (3) represets the possble vales of, ad are the possble vales of the stadard devato assocated wth. The above pdf ma be assmed to be eal to the freec dstrbto for the observed data ad s sall terpreted as be proportoal to the Lelhood fcto l that s, (,, ) l(, ) exp s, s., ad B mltpl eato (4) wth the o-formatve Jeffre s pror, the jot posteror pdf s obtaed [3; 8]:,, ( ), exp Iterato to d leads to the formato perta to the expectato for : (4) (5) 508

3 , / s /, (6) / s s. Sce the rht-had sde of ths eato correspods to a Stdet-t dstrbto, the ew varable T s s trodced [8]. Oe obtas t t / t are the possble Vales of T. Therefore, the best estmate for,.e. for the measrad, becomes E. (7) De to VarT 3,, the Baesa certat cotrbto assocated wth the expectato of the repeatedl observed att becomes [3, 8] 3 s. (8) It shold be metoed that, for small mbers of observatos ( 0), ths certat cotrbto sfcatl exceeds the so-called tpe-a certat calclated accordace wth the ISO-GUM []. The GUM-tpe-A certat, therefore, ma be derstood as approxmato for a sffcetl lare mber of observatos [0]. IV. FUSIO OF IFORMATIO GIVE FOR THE IPUT UATITIES It s the trsc prpose of the Baesa approach to certat aalss to develop the jot posteror pdf for the otpt att (measrad) whch s compatble wth the ve formato abot the (vales of the) pt attes ad the measr process.,, I l I I, (9) represets the state of owlede abot the vales of the pt attes,...,. Becase of the terrelato of the pt attes (ve b the eatos (a) ad (b)), a posteror the pt attes caot be acowleded as be depedet. Ths fact ca be tae to cosderato b wrt the jot pror pdf as I [4, 8], (0) M s the pror of the pt attes ad M the so-called model pror [9]. The model eato tself ma be obtaed b sstematcall aalz the case-ad-effect cha of the measremet cld all relevat fleces ad dstrbaces [-]. For models of the form M f, the M 0 above model pror s eal to Drac s delta fcto [8]: M M, () that taes care that ol meafl combatos of the possble vales of the pt attes are tae to cosderato ( flter fcto ). Therefore the jot posteror pdf for the otpt att becomes, I...,...,,...,. () fm,..., d,..., d Eato () s ow as Marov formla. Sce t ca be aaltcall compted farl smple cases ol, moder certat evalato tlzes Mote-Carlo teches as terato teches for pdf propaato [3-5]. F. llstrates the fll Baesa cocept for evalat the measremet reslt ad ts assocated certat. V. EPECTATIO,UCERTAIT AD EPADED UCERTAIT FOR THE OUTPUT UATIT 509

4 Measrad 0mm vae (or o) owlede I Pror pdf I possble vales Measremet process, Assmet Observatos,, fleces, parameters Addtoal owlede I e.. - temperatre - maetc feld - dtal resolto (sstematc effects) Itermedate reslt, I ( ) Addtoal pror pdfs: Fso calcls (e.. Baes),..., Fal reslt: Jot posteror pdf reres to ow the fso model: f,..., M E,I Var ths path s ot (et) part of the ISO-GUM F.. Geeralzed measremet process as Baesa ferece b meas of model-based fso of the state-of-owlede pdfs for the pt attes. Smbols: see text (depcted accord to Beerer [9]) From the pdf for the otpt att ( I, ), the expectato vale of the measrad E[ ] ad ts assocated certat ca be derved: ( ) d, ad (3) ( )( ) d. (4) Sce the Baesa approach does provde the pdf for the otpt att, the expaded certat,.e. a d of cofdece terval for ths pt att, ca easl be derved as the mmm terval U ; U P P that meets the follow coverae probablt codto: U U P P d d P. (5) Usall, ths coverae probablt P s set p to 0.95 at mmm []. VI. LIEAR FUSIO MODELS (LIEAR MODEL EUATIOS) I practce, sers of a measremet reslt wll ofte ot be terested the pdf for the otpt att bt rather ts expectato vale ad the assocated measremet certat (see eatos (4) ad (5)). I case of lear sstems or sstems that ca be learzed, e.. b frst-order Talor seres expaso, these parameters ma also be compted accordace wth the ISO-GUM method [] whch s based o Gassa certat propaato: (see eatos (3) ad (4)). f x x M (,..., ), (6) f f f M M M xxj j x (7) x j xj = E[]; r( ; ) s the estmated xxj x xj j covarace of the attes ad, ad r( ; ) s j j the respectve correlato coeffcet. It s a commo experece that, the majort of practcal certat evalatos, the ISO-GUM procedre wll provde satsf reslts. Bt besdes oleart, the calclato of the expaded measremet certat s a real wea pot of the stadard cocept. The problem s cased b the fact that the stadard procedre does ot provde the pdf for the otpt att ad, therefore, the coverae factor eeded to calclate the expaded certat s to be 50

5 determed o the bass of ol vae formato abot ths pdf: P U. (8) VII. COCLUSIO It becomes clear that, depedet o the calcls sed (Gassa or Baesa), for practtoers the e steps of moder certat evalato are the complato ad descrpto of the owlede abot the measremet, the modell of the measremet ad the assato of a approprate pdf to each of the volved pt attes. It ca be coclded that the Baesa approach allows for stretl evalat the measremet certat. There are o restrctos related to oleart ad determato of the expaded certat. O the other had, wth the exceptos of evalat the expaded certat ad calclat the stadard certat from ol a few repeated observatos, the ISO-GUM procedre s (for learzable sstems) cosstet wth the Baesa cocept (see also [0]). [4] Sebert, B.R.L.; Carl, P.; Sbold, D.: Mote Carlo Std o Local ad Statstcal Correlato Advaced Mathematcal & Comptatoal Tools Metrolo V, edted b Carl, P.; Flpe, E.; Forbes, AB.; Pavese, F.; Perrchet, C. ad Sebert, B.R.L., Seres o Advacesrd Mathematcs for appled Sceces, Vol. 57, World Scetfc, ew Jerse, 006, S [5] Sebert B.R.L.; Sommer, K.-D.: Weteretwcl des GUM d Mote-Carlo-Teche / ew Developemets of the GUM ad Mote Carlo Teches. tm Techsches Messe 7 (004), (Febrar 004), (ISS 078-3) REFERECES [] Gde to the Expresso of Ucertat Measremet (GUM), Frst edto, 993, corrected ad reprted 995, Iteratoal Orazato for Stadardzato (ISO), Geeva, 993 ad 995. [] Wese, K.; Woeer, W.: A Baesa theor of measremet certat. Meas. Sc. Techol. 3(99), - [3] Wese, K.; Woeer, W.: Messscherhet d Messdateaswert der Metroloe. WILE-VCH, Wehem, 999 [4] Lra, I.; Woeer, W.: Baesa evalato of the stadard certat ad coverae probablt a smple measremet model. Meas. Sc. Techol. (00), 7-79 [5] Sva, D.S.: Data Aalss A Baesa Ttoral. Claredo Press, Oxford, 996 [6] Estler, T.W.: Measremet as Iferece: Fdametal Ideas. Aals of the CIRP 48 (999), 6 63 [7] Bch, W.; Cox, M.G.; Harrs, P.M.: Evalato of the Gde to the Expresso of Ucertat Measremet. Metroloa 43 (006) 4 [8] Lra, I.: Evalat the Measremet Ucertat. Isttte of Phscs Pblsh Brstol ad Phladelpha Ltd, 00 [9] Beerer, J.: The vale of addtoal owlede measremet a Baesa approach, Measremet 5 (999) -7 [0] Kacer, R.; Joes, A.: O Use of Baesa Statstcs to Mae the Gde to the Expresso of Ucertat Measremet Cosstet. Metroloa 40 (003), [] Sommer, K.-D.; Sebert, B.R.L.; Kochse, M.; Wecema, A.: A sstematc approach to the modell of measremets for certat evalato. Joral of Phscs: Coferece Seres 3 (005), 7 th Iteratoal Smposm o Measremet Techolo ad Itellet Istrmets ISMTII 005, Isttte of Phscs Pblsh, 4-7 [] Sommer, K.-D.; Sebert, B.R.L.: Sstematc Approach the Modell of Measremets for Ucertat Evalato. Metroloa 43 (006) 4 [3] Cox, M.G.; Sebert, B.R.L.: The Use of Mote-Carlo Method for Evalat Ucertat ad Expaded Ucertat. Metroloa 43 (006) 4 5