COMPUTER AIDED EVALUATION OF MEASUREMENT UNCERTAINTY BY CONVOLUTION OF PROBABILITY DISTRIBUTIONS

Transcription

1 Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata XVII IMEKO World Congress Metrology n the 3rd Mllennum June 22 27, 23, Dubrovnk, Croata COMPUTER AIDED EVALUATION OF MEASUREMENT UNCERTAINTY BY CONVOLUTION OF DISTRIBUTIONS Govann B. Ross, Francesco Crenna, Mchele Codda Unversty of Genova, DIMEC, Dept. of Mechancs and Machne Desgn, Genova, Italy Abstract The evaluaton of measurement uncertanty, from the fnal user s standpont, nvolves several ssues whch are only partally addressed by the GUM []. An evaluaton procedure asssted by a computer program can be of great support and provde the user wth detaled results useful for further analyss. In ths paper a code s presented for the asssted evaluaton of measurement uncertanty, based upon the convoluton of probablty dstrbutons. So the method allows expresson of the fnal result of measurement as a probablty dstrbuton, on whch t s possble to evaluate useful parameters such as expanded uncertanty, at whatever confdence level. It also permts to evaluate, n probablstc terms, the rsk assocated to matchng tolerances. Moreover t s possble to manage the uncertanty on dsperson parameters (sometmes called uncertanty of uncertanty ) consderng a modfed probablty dstrbuton. In the paper the code s presented together wth some case studes showng the support avalable to the operator, the GUM compatblty and the applcaton mportance of the fnal probablty dstrbuton. Keywords: Measurement uncertanty, Probablty dstrbutons, Computer-asssted uncertanty evaluaton. INTRODUCTION Research studes on computer aded evaluaton of measurement uncertanty and related software codes have been and are currently carred out. Man topcs regard the mplementaton of the GUM procedure, smulaton for general purpose or specfc complex measurng problems [2-7]. The am of the software presented n ths paper s twofold: to support the user creatng a smple model for the measurement procedure; to evaluate the measurement uncertanty on the same hypothess of the GUM but gvng as a result a probablty dstrbuton together wth values from a strct GUM procedure. The former provde the user wth an n depth nformaton regardng the behavour of hs measurement procedure: t s possble to use the fnal probablty dstrbuton to evaluate an expanded uncertanty at whatever coverage level s requested, eventually comparng the results wth the classcal GUM expanded values. Moreover gven an expanded uncertanty at certan coverage level t s possble to use the probablty dstrbuton to evaluate the acceptance rsk when comparng the measurng result wth a tolerance level [8-9]. 2. THEORETICAL BACKGROUND The expresson of the result of a measurement by a probablty dstrbuton over the set of the possble values of the measurand may be founded over the followng consderatons. Let us consder the measurement task consstng n the measurement of a quantty x, wth values n X, under the assumpton t has a constant value for an nterval of tme T of nterest. The measurement process wll nclude an observaton sub-process, gvng rse to a vector of y = y,...y, produced by a measurng observatons [ ] N system, under a selected measurement strategy. The probablstc relaton holdng between such observatons and the unknown (unobservable) value of the measurand, x, may be expressed, n the most general way, as [9]: ( y ) ( y ) pθ x p x; θ () where θ Θ s a vector of parameters representng nfluence quanttes. Then measurement resttuton may be performed accordng to [9]: ( ) ( ) ( ) Θ p x y pθ y x p θ d θ ; x X (2) Formula (2) provdes the requred foundaton for expressng the measurement result as a probablty dstrbuton over the set X of the possble values of the measurand. If we now defne: ˆx E ( x ) x y (3) t s possble to obtan from formula (3) the followng condtonal dstrbuton: ( ˆ) p xx ; x,xˆ X (4)

2 Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Incdentally, from a theoretcal standpont, t should be noted that dstrbuton (4), we wll be usng n the followng, actually logcally derves from formula (2), whch n turn ncludes the model of the observaton process expressed by (). Ths metrologcal background, n our opnon, should not be forgotten, n developng good evaluaton procedures. Let us now consder the followng addtonal hypotheses, conformal the GUM (estmaton) model:. t possble to express the effect of all nfluence quanttes on the fnal measurement result as a lnear functon of the varatons n the quanttes: δ x = cδv, wth known coeffcents c ; 2. for each of the nput quanttes, the followng nformaton s avalable: a) ether an estmated standard devaton σ ˆ, wth ν degrees of freedom, whch may be fnte or nfnte; b) or an estmate of the range of varablty, say v v,v + v, wth lmt ether certan or subject to a [ ] relatve uncertanty, say α = v v ; 3. t s possble to dentfy a subset of mutually uncorrelated nput quanttes, say, wthout loss of generalty, v ' = v, v,..., v, m n, for whch the correspondng [ ] 2 m correlaton matrx s known R = r j. Under the assumed hypotheses, we have: where p ( x) ( ˆ) = ( ˆ) n = p xx p v x x (5) v may be expressed as the composton, through a convoluton rule, of the dstrbuton accountng for the global nfluence of the subset of uncorrelated nput p x, wth the dstrbuton quanttes (f not null), ( ) v, uncorr accountng for the effect of complementary subset of p x mutually correlated nput quanttes (f not null), ( ) v, corr. So we have: p v( x) = p v, uncorr ( x) p v, corr ( x) = p v, uncorr ( x ξ) p v, corr ( ξ) dξ (6) The numercal calculaton of such dstrbuton wll be detaled n the next paragraphs. 3. THE ASSISTED EVALUATION PROCEDURE The user who needs to evaluate a measurement uncertanty has to face some dffcultes form the clear dentfcaton of the devces used n the measurng chan to the collecton of data and fnally computatonal problems to evaluate the fnal uncertanty. The proposed code s based on a schematsaton of the measurng chan as a sequence of standard blocks each representng a component of the measurng chan. Each block s characterzed by a set of propertes that descrbes the overall behavour of the component as an element of the measurng chan and moreover as a contrbutor to the measurng uncertanty. In ths phase of development the block represents a lnear transfer functon wth uncertanty on the senstvty and on the offset: a proper combnaton of blocks can provde a model even for a complex measurng procedure. Each nfluence quantty contrbutng to the overall uncertanty s characterzed by ts proper dsperson parameter, by the knd of underlyng probablty dstrbuton, by the number of degrees of freedom or other quantfer of the uncertanty on dsperson parameters. For mutually correlated quantse, the correlaton coeffcent must also be specfed. Such nformaton are also conformal to the requrements of the GUM. All the nformaton referrng to each block n the measurng chan s treated as defnng an object, n the object orented envronment n whch the code s developed. The evaluaton of the standard and expanded uncertanty proceeds accordng to two ndependent methods: the GUM recommended evaluaton prncples, and the convoluton of the probablty denstes of each contrbutor. 3.. Avalable optons The Asssted procedure can manage several dfferent cases, schematcally t s possble to dstngush: - type A or type B uncertanty values; - correlated and uncorrelated quanttes and mxed combnatons of them; - presence of uncertanty on the dsperson parameters of the quantty. - Type A evaluatons are treated wth the proper degrees of freedom usng t-student dstrbutons; type B evaluatons are treated wth probablty denstes such as unform, trangular and other, wth a proper dsperson parameter. Correlated quanttes are frst of all combned as specfed by the GUM and then they are assocated to a normal dstrbuton that s consdered as another uncorrelated quantty. If they present an uncertanty on the dsperson parameter, ther probablty denstes are prevously computed accordng to (), then ther new dsperson fgures are computed and combned as here descrbed. The uncertanty on the dsperson parameter s managed by assocatng to the quantty a proper probablty dstrbuton. The new probablty dstrbuton depends on the orgnal shape: n case of normal PD a t Student s used. PDF Method Data nput GUM Method Data nput Convoluton of PDF Combnaton of correlated quanttes Standard uncertanty Dscretsaton Measurement result PDF New uncorrelated quantty (normal) Equvalent DoF (Welch- Satterwhte) Compute PDFs (also wth lmted DoF) Compute u and U Expanded uncertanty Compute σs for correlated quanttes Combnaton of σ s New uncorrelated quantty (normal) Standard and Expanded uncertanty Fg.. Schematc flowchart for the codes mplementng the GUM and the PD methods

3 Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata In case of unform PD wth uncertanty on the dstrbuton lmts, then a new PD s assgned that s the mean among all the possble probablty denstes consderng the uncertanty on the dstrbuton parameter. Uncorrelated probablty denstes are then convolved together gvng the fnal probablty dstrbuton for the consdered measurement process. It s then possble to evaluate drectly both the standard and the expanded uncertanty, U, gven the desred coverage level, p. Note that havng the probablty dstrbuton, there s no need for the applcaton of the Welch-Satterwate formula [-]. The codes provdes also standard and expanded uncertantes computed strctly accordng to the GUM. As descrbed n the next secton, at the present development stage the asssted procedure conssts of two separate software codes for the mplementaton of the GUM and the PD methods, ther flowcharts beng presented n fgure The software development Partcular care was dedcated to the software development procedure, ntegratng ths actvty n the qualty management polcy of the laboratory. The development followed two separate lnes, one dedcated to GUM methods, the other to probablty dstrbutons methods. Successve versons of the software proceeded n a parallel way ntegratng at each step a lmted set of new capabltes. In such a way at each step a complete software valdaton was possble together wth a drect comparson of the results obtaned wth the two methods at the same development stage. The sequence of development for both methods s presented n table and at the last stage covers all the avalable optons descrbed before. Stage 2 TABLE I. Development steps and ther functonaltes Functonaltes Only uncorrelated quanttes, nfnte degrees of freedom (DoF), no uncertanty on the parameter. Only uncorrelated quanttes, nfnte or lmted DoF, uncertanty on the parameter. Correlated or uncorrelated quanttes, nfnte or lmted DoF, uncertanty on the parameter also for correlated quanttes. At each step of development a detaled documentaton was produced ncludng man functonaltes, software strateges, user manual and dscusson on valdaton polces and results. The code and the correspondng documentaton are then nserted n the laboratory archve and managed accordng to the qualty procedures of the lab. The software s wrtten n the Matlab envronment accordng to an object orented approach that s partcularly suted to our model of measurement chan and to the asssted procedure [2]. The memorsaton of the nfluence quanttes n an object form proved to be really effectve n order to have all the characterstcs of a quantty, or a block n the model, avalable at the same tme. Each verson of the software conssts of a set of separate codes or functons, each one dedcated to a specfc am. In such a way t s possble to re-use functons whenever necessary n new software versons. Moreover a separate valdaton of each functon s often much more feasble and effcent than a complete valdaton of the overall software package. The asssted procedure s ntended to have an actve place n the measurement procedure, n order to gve a fgure of the relablty of the measurement result. So the asssted procedure has to be treated wth metrologcal crtera and valdaton accordng to reference cases s necessary. The valdaton polcy was selected accordng to the dfferent functonaltes. In partcular for the overall code a set of examples was selected from GUM appendx as reference cases. Moreover, n order to test completely all the avalable optons, a set of more complex and less common cases was defned. Valdaton was consdered postve when besdes the fundamental functonaltes of the code such as data nput and output, the results were consstent wth the expected results obtaned by manual computaton accordng to the GUM, to the PD method or obtaned drectly from the GUM appendx [3]. In order to have an easer valdaton the code was developed n dfferent phases accordng to ts possble functonaltes. In the frst phase, only dscrete probablty denstes have been consdered to avod the usual problems n gong from a contnuous doman to a dscrete one. Then a dscretsaton procedure was set up and valdated, extendng the software capabltes to the contnuous case. Quantsaton was deeply nvestgated, as t s a key-pont regardng the accuracy of the computaton. Ths s moreover mportant when small dfferences due for example to the effect of the uncertanty of the dsperson parameter have to be evaluated. The polcy of dscretsaton consders the use of an odd number of ponts over the full nterval. The number s determned on half range and then extended to the full range addng the central pont. When changng from a contnuous to a dscrete probablty dstrbuton there may be two man effects due to the dscretsaton polcy: - the poston of the pont on the edges can produce a dscrete dstrbuton slghtly larger or smaller than the orgnal dstrbuton; - the truncaton of an nfnte probablty dstrbuton n partcular a normal, to a gven coverage level. A study was carred out to nvestgate the optmal dscretsaton for normal and unform dstrbuton n order to have an acceptable computatonal accuracy together wth an acceptable computaton tme. A further study wll regard the Student and the unform wth uncertanty on the lmts dstrbutons. Some care was dedcated to the desgn and realsaton of the user nterface, whch asssts the user n the organzaton of the nput data necessary for the computaton [4]. At the end of the data nput phase a check procedure s provded n whch data are verfed both by the user and partally by the supportng software. 4. TEST CASES Test cases are presented n order to show the possbltes offered by the consdered method and the dfferences wth the GUM procedure. The effect of the uncertanty on the dsperson parameter s shown together wth the effectve

4 Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata calculaton of expanded uncertantes. Test cases nclude a GUM example n order to have drect comparson of the results and drect valdaton of the fgures evaluated accordng to standard methods; a complex case to show all the avalable features and a dscrete case to show how t s possble to manage dscrete measurement systems. 4.. An example from the GUM Ths frst test case s the example H of the GUM and regards the measurement of standard reference blocks. Uncertanty evaluaton s carred out after lnearzaton the measurement model accordng to the GUM procedure, and table 2 presents the nput data. TABLE II. Input data for test case 4. Quantty PD Dsperson DoF/Uncertanty parameter on parameter Normal Normal Unform 2.9 % range or 5 DoF. 4 Unform 6.6 5% range or 2 DoF Fgure 2 presents the PDs of ths example. Note that: - a t-student PD s used for the frst two quanttes; - the unform dstrbutons are modfed accordng to the uncertanty on the dsperson parameter x -3 Quantty Quantty 2 Quantty 3 reduced by 2 Quantty 4 uncertanty based on the fnal PD s slghtly dfferent from the fgures based upon the GUM method, whch are based on the Welch-Satterwate approxmaton. TABLE III. Test case 3., comparson of GUM and PD results GUM PD Standard uncertanty, u [nm] Expanded uncertanty, 95%, U 95 [nm] Expanded uncertanty, 99%, U 99 [nm] Overall capabltes test case In ths test case all the avalable optons from the procedure are consdered. TABLE IV. Input data for test case 4.2 PD Dsperson DoF/Uncertanty Correlaton parameter on parameter coeffcent Unform.4 2% range or 25 DoF. -,5 2 Normal. 5 3 Unform.7 % range or 5 DoF - 4 Normal Quantty 3 Quantty 4 Correlated.2 x -3 FINAL DISTRIBUTION FINAL DISTRIBUTION nm Fg. 2. Test case : probablty dstrbutons of the nput quanttes and convolved fnal dstrbuton on a separate scale. Table 3 presents a comparson of the results, n whch t s possble to apprecate that the evaluaton of the expanded Fg. 3. Test case 2:probablty dstrbutons of the nput quanttes and convolved fnal dstrbuton on a separate scale. In ths case the normal probablty dstrbuton for the correlated quanttes has a dsperson parameter computed accordng to the GUM, but consderng the actual

5 Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata dspersons of the nput dstrbutons, modfed to take nto account the uncertanty on ther dstrbuton parameters. There are ether correlated and uncorrelated quanttes whch present lmted DOF or uncertanty on the dsperson parameter. Input data are presented n table 4. Input and fnal dstrbutons are presented n fgure 3. TABLE V. Test case 3.2, PD results (arbtrary unts) PD Standard uncertanty, u 2.2 Expanded uncertanty, 95%, U Expanded uncertanty, 99%, U Fnal fgures are presented n table 5. Snce the PD method evaluates the correlated quanttes dstrbuton not accordng to the GUM a comparson of the fnal fgures s unfeasble Dscrete test case In ths last example a dscrete case s consdered n order to show that approxmatng a dscrete PD by a contnuous one can lead to msleadng results FINAL DISTRIBUTION Fg. 4. Test case 3: probablty dstrbutons of the nput quanttes and convolved fnal dstrbuton on a separate scale. Ths s the case n whch measurement results are affected for example by a large quantsaton effect, n such a way that the repeated measurement dstrbuton appears as dscrete and not contnuous. Fgure 4 presents the nput and output dscrete PDs, together wth the fnal one. Table 7 presents a comparson of the fnal results. TABLE VI. Input data for example 3.3 PD Dsperson DOF/Uncertanty parameter on parameter Unform 3-2 Dscrete Normal (dscrete t-student).5 Note that snce the fnal PD s obtaned as a result of a convoluton process between two dscrete dstrbutons, ts shape s dscrete tself. In ths case the use of a contnuous PD such as a t-student for the evaluaton of the expanded uncertanty s not a proper choce and the evaluaton based upon the fnal PD s the only possble way of proceedng. TABLE VII. Test case 3.3, comparson of contnuous approxmaton and PD results. GUM PD Standard uncertanty, u.8 2. Expanded uncertanty, 95%, U Expanded uncertanty, 99%, U In table 7 some fgures are presented to compare the contnuous approxmaton and calculaton of dscrete dstrbuton. Note that whle the expanded uncertanty evaluated by the GUM method gves fgures that do not consder the dscrete resoluton of the nstrument, the proposed method gves a result compatble wth the nstrument s resoluton, snce t evaluates the expanded uncertanty drectly on the fnal dscrete probablty dstrbuton. 5. CONCLUSIONS A software for the evaluaton of measurement uncertanty usng probablty dstrbutons was presented, together wth a set of test cases to present ts functonalty. The use of the proposed method and asssted procedure gves effectve results to the user and addtonal nformaton regardng the fnal probablty dstrbuton. It s also possble apply the standard GUM method. Computatonal effcency s ensured by a careful dscretsaton together wth nowadays powerful PC. Data nput s guded by a proper and frendly user nterface. REFERENCES [] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Gude for the Expresson of Uncertanty n Measurement (GUM), 995. ISBN [2] K. Wese, Woeger W., A bayesan theory of measurement uncertanty, Meas. Scence Technol [3] G.B. Ross, F. Crenna, Probablstc uncertanty evaluaton adng crtcal measurement-based decsons, Proc IMEKO TC7 Symposum, Cracow, June 25-27, 22 [4] M.G.Cox, P.M. Harrs Measurement of uncertanty and the propagatons of dstrbutons, NPL, 22. [5] A. Hetman, M.J. Korczynsk Calculaton of expanded uncertanty, Proc. 2 nd on lne Workshop of Tools for Educaton of Measurement June [6] Lra I., Evaluatng the measurement uncertanty. Fundamentals and practcal gudance, IoP, 22

6 Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata Proceedngs, XVII IMEKO World Congress, June 22 27, 23, Dubrovnk, Croata [7] Lra I., Kyraks G., Bayesan nference from measurement nformaton, Metrologa, 36, 63-69, 999 [8] W. Schulz, K.D. Sommer, Uncertanty of measurement and error lmts n legal metrology, OIML Bullettn, 999, XL, 4, 5-5 [9] Ross G. B., Potentals of uncertanty evaluaton based on probablty denstes, IMEKO Workshop Evaluaton and check of traceablty: basc aspects and expermental results, Torno, - September, 998 [] P. Otomansk, The range of applyng approxmate methods of expanded uncertanty estmaton n ndrect measurements, Proc IMEKO TC7 Symposum, Cracow, June 25-27, 22 [] Hall B D, 999, Uncertanty calculatons usng objectorented computer modellng, Measurement Scence and Technology,, [2] Rchter D., Software engneerng related standards and gudelnes for metrology, Adv. Math. Comp. Tools n Metrology AMCTM 999, Oxford, Aprl 2-5, 999 [3] Ross G. B., Crenna F., Pampagnn F.: Qualty assurance for unversty research laboratores orented to measurements for P. E., Proceedngs of the st Internatonal Conference of the European Socety for Precson Engneerng and Nanotechnology, Bremen, May 3-June 4, 999, 2, [4] Ross G. B., Crenna F.: Metrology software for expressng uncertanty by probablty denstes, Proceedngs of the XVI IMEKO World Congress, Venna, September 25-28, 2, 9, Authors: Govann B. Ross, Francesco Crenna, Mchele Codda, Unversty of Genova, DIMEC, Dept. of Mechancs and Machne Desgn, Va all Opera Pa 5A, I- 645, Genova, Italy, phone: , fax: , Emal: gb.ross@dmec.unge.t