VERIFICATION OF THE CONVENTIONAL MEASURING UNCERTAINTY EVALUATION MODEL WITH MONTE CARLO SIMULATION

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1 ISSN Int. j. smul. model. 4 (2005) 2, Orgnal scentfc paper VERIFICATION OF THE CONVENTIONAL MEASURING UNCERTAINTY EVALUATION MODEL WITH MONTE CARLO SIMULATION Acko, B. & Godna, A. Unversty of Marbor, Faculty of Mechancal Engneerng, Smetanova 17, SI-2000 Marbor, Slovena E-mal: bojan.acko@un-mb.s Abstract Calbraton of gauge blocks by mechancal comparson s wdely used for assurng traceablty of ndustral measurements for the physcal quantty length. In order to dmnsh uncertanty of calbraton, we shall control all nfluence parameters very precsely. We must be able to predct ntervals of changes of these parameters n short and long term ntervals and nclude those changes n the uncertanty budget. Mathematcal model of measurement s used for calculatng standard uncertanty of the output value (calbraton result) from the uncertantes of the nput (nfluence) quanttes. However, ths calculaton s relable only f the uncertantes of the nfluence quanttes are evaluated accurately. Therefore, we decded to verfy the uncertanty calculatons from the past by Monte Carlo smulaton, whch are also proposed by the new draft of the Gude to the Expresson Uncertanty n Measurement (GUM). The approach and the results of ths verfcaton are presented n the artcle. Key Words: Measurng Uncertanty Model, Calbraton, Gauge Blocks, Monte Carlo Smulaton 1. INTRODUCTION The natonal standard for length n Slovena s materalzed wth gauge blocks. These gauge blocks are traceable to the prmary standard by calbraton n dfferent European natonal laboratores. The standards from ndustry and calbraton laboratores are calbrated by mechancal comparson wth the natonal standard. The best measurement capablty s not comparable wth the capabltes of other European natonal laboratores havng nterferometrc calbraton systems but s coverng the ndustral needs very well. However, our am s to reduce uncertanty of calbratons wthout nvolvng nterferometrc system, whch would be too expensve for the ndustry. Research work n ths feld s focused n crtcal uncertanty contrbutons lke calbraton of the equpment, as well as envronmental and human nfluences. As a result of ths research the conventonal uncertanty evaluaton procedure has been supplemented by applyng Monte Carlo smulaton for evaluatng uncertantes of crtcal nfluence quanttes. Smulaton parameters have been chosen accordng to our extensve experence and knowledge n the feld gauge block calbraton [1]. Varaton ntervals for the nfluence parameters and statstcal dstrbutons were defned by studyng measurement results of the past ten years. 2. CONVENTIONAL PROCEDURE FOR UNCERTAINTY EVALUATION Conventonal procedure for evaluatng uncertanty n measurement s descrbed n [2, 3, 4]. Ths procedure s used worldwde n calbraton laboratores, research and natonal metrology nsttutes as well as n sophstcated ndustral measurements [5, 6, 7, 8]. The am of ths DOI: /IJSIMM04(2)

2 procedure s to make uncertanty evaluaton nternatonally comparable. As a consequence, partes n tradng relatons are able to evaluate qualty of measurng protocols among each other [9]. Fgure 1 comprses only the most mportant steps to be performed. Mathematcal model of measurement: y = f(x 1, x 2,..., x,..., x N ) Evaluaton of standard uncertantes of nput values: Type A (statstcal evaluaton of experment results): s(x u(x ) = s(x ) = n Tvpe B (non-statstcal estmaton - prevous measurement data, experences, general knowledge, manufacturer's specfcatons, calbraton and other certfcates, data from handbooks),k ) Calculaton of senstvty coeffcents: y c(x ) = x Calculaton of combned standard uncertanty u 2 N 2 f 2 = c (y) u (x ) = 1 x Calculaton of expanded uncertanty U = k u c (y) Fgure 1: Procedure for evaluatng uncertanty of measurement accordng to [2]. 3. MONTE CARLO SIMULATION IN UNCERTAINTY CALCULATION 3.1 Backgrounds and the procedure Monte Carlo smulaton [10, 11] provdes a general approach to numercally approxmatng the dstrbuton functon G(η) for the value of the output quantty Y = f(x). An estmate of the measurand Y, denoted by y, s obtaned from mathematcal model of measurement usng nput estmates x 1, x 2,..., x N for the values of the N nput quanttes X 1, X 2,..., X N. Thus the output estmate y, whch s the result of the measurement, s gven by: y = f(x 1, x 2,..., x N ) (1) That s, y s taken as the arthmetc mean or average of n ndependent determnatons Y k of Y, each determnaton havng the same uncertanty and each beng based on a complete set of observed values of the N nput quanttes X obtaned at the same tme. Ths way of averagng may be preferable when f s a nonlnear functon of the nput quanttes X 1, X 2,..., X N, but the two approaches are dentcal f f s a lnear functon of the X [2]. 77

3 Although the formula (1) need not provde the most meanngful estmate of the output quantty value, t plays a relevant role wthn Monte Carlo smulaton as an mplementaton of the propagaton of dstrbutons. Monte Carlo smulaton for uncertanty calculatons [12] s based on the premse that any value drawn at random from the dstrbuton of possble values of an nput quantty s as legtmate as any other such value. Thus, by drawng for each nput quantty a value accordng to ts assgned probablty densty functon, the resultng set of values s a legtmate set of values of these quanttes. The value of the model correspondng to ths set of values consttutes a possble value of the output quantty Y. Fgure 2 llustrates the propagaton of dstrbutons. Fgure 2: Illustraton of the propagaton of dstrbutons. The model nput quanttes are X = (X 1,X 2,X 3 ) T. The probablty densty functons g (ξ ), for X, = 1, 2, 3, are Gaussan, trangular and Gaussan, respectvely. The probablty densty functon g(η) for the value of the output quantty Y s ndcated as beng asymmetrc, as can arse for nonlnear models. Consequently, a large set of model values so obtaned can be used to provde an approxmaton to the dstrbuton of possble values for the output quantty. Monte Carlo smulaton can be regarded as a generalzaton of [2] to obtan the dstrbuton for Y, rather than the expectaton of Y. Monte Carlo smulaton operates as follows: - A sample of sze N s generated by ndependently samplng at random from the probablty densty functon for each X, = 1,..., N. Ths procedure s repeated a large number of tmes (M) to yeld M ndependent samples of sze N of the set of nput quanttes. For each such ndependent sample of sze N the resultng model value of Y s calculated; - These M values of Y are used to provde Ĝ(η), an approxmaton to the dstrbuton functon G(η) for the value of Y; - Produce any requred statstcal quantty from Ĝ(η). Partcularly relevant quanttes are (a) the expectaton of Ĝ(η) as the estmate y of the output quantty value, (b) the standard devaton of Ĝ(η) as the assocated standard uncertanty u(y), and (c) two quantles of Ĝ(η) as the endponts of a coverage nterval I p (Y ) for a stpulated coverage probablty p. The effectveness of Monte Carlo smulaton to determne a coverage nterval for the output quantty value depends on the use of an adequately large value of M. Gudance on obtanng such a value and generally on mplementng Monte Carlo smulaton s avalable [13]. 78

4 The nputs to the Monte Carlo smulaton procedure for the calculaton stage are: - The model Y = f(x) (provded at the formulaton stage of uncertanty evaluaton). - The jont probablty densty functon for X (provded at the formulaton stage; also see Secton 4). NOTE Ths jont probablty densty functon reduces to the collecton of the ndvdual probablty densty functons for X 1,...,X N n the case of N mutually ndependent nput quanttes. - The requred coverage probablty p (e.g., 0.95, or 95 %). - The number M of Monte Carlo trals. The prmary output from the Monte Carlo smulaton procedure s a numercal approxmaton Ĝ(η) to the dstrbuton functon G(η) for the output quantty value, from whch the requred quanttes can be determned. Monte Carlo smulaton as an mplementaton of the propagaton of dstrbutons s shown dagrammatcally n Fg. 3 and can convenently be stated as a step-by-step procedure: MCS INPUTS Model Y = f(x) Probablty densty functons g(ξ) Number M of Monte Carlo trals Coverage probablty p PRIMARY MCS OUTPUT: DISTRIBUTION FUNCTION FOR THE OUTPUT QUANTITY VALUE M samples x 1,..x M of X from g(ξ) M model values y = (y 1,..y M ) = (f(x 1 ) f(x M )) Approxmate dstrbuton functon?(η) MCS RESULTS Estmate y of the output quantty value and assocated standard uncertanty u(y) Coverage nterval [y low, y hgh ] for the output quantty value Fgure 3: The calculaton phase of uncertanty evaluaton usng Monte Carlo smulaton (MCS) to mplement the propagaton of dstrbutons. a) The number M of Monte Carlo trals to be made shall be selected; b) M samples of the (set of N) nput quanttes shall be generated; c) A model gvng the correspondng output quantty value shall be evaluated for each sample; d) The values of the output quantty shall be sorted nto non-decreasng order, usng the sorted values to approxmate the dstrbuton functon for the output quantty value; e) The estmate of the output quantty value and the assocated standard uncertanty from the dstrbuton functon shall be formed; 79

5 f) The shortest 95 % coverage nterval for the output quantty value from ths dstrbuton functon shall be formed. 3.2 The number of Monte Carlo trals A value of M, the number of Monte Carlo trals to be made, needs to be selected. It can be chosen a pror, n whch case there wll be no drect control over the degree of approxmaton delvered by the Monte Carlo procedure. The reason s that the number needed to provde a prescrbed degree of approxmaton wll depend on the shape of the probablty densty functon for the output quantty value and on the coverage probablty requred. Also, the calculatons are stochastc n nature, beng based on random samplng. However, a value of M = 10 6 can often be expected to delver a 95 % coverage nterval, havng a length wth a degree of approxmaton of one or two sgnfcant decmal dgts, for the output quantty value. Because there s no guarantee that ths or any specfc number wll suffce, t s recommended to use a process that selects M adaptvely,.e., as the trals progress. Some gudance n ths regard s avalable [13]. A property of such a process s that t takes a number of trals that s economcally consstent wth the requrement to acheve the requred degree of approxmaton. 3.3 Evaluaton of the model The model s evaluated for each of the M samples from each of the probablty densty functons for the values of the N nput quanttes. Specfcally, denote the M samples by x 1,, x M, where the r th sample x r contans values x 1,r,, x N,r, wth x,r a draw from the probablty densty functon for X. Then, the model values are: 3.4 Dstrbuton functon for the output quantty value y r = f(x r ); r = 1,..,M (2) An approxmaton Ĝ(η) to the dstrbuton functon G(η) for the output quantty value Y can be obtaned by sortng the values y r, r = 1,, M, of the output quantty provded by Monte Carlo smulaton nto non-decreasng order and denotng the sorted values by y (r), r = 1,..., M. Unformly spaced cumulatve probabltes pr = (r 1/2)/M, r = 1,..., M, to the ordered values shall be assgned [12]. Ĝ(η) s the pecewse-lnear functon jonng the M ponts (y (r), p r ), r = 1,, M: r 1/ 2 η y(r) Ĝ( η) = + M M(y y y (r) η y (r+ 1), (r+ 1) (r) r = 1,...,M 1 ) (3) 3.5 The estmate of the output quantty value and the assocated standard uncertanty The expectaton by of the functon Ĝ(η) (3) approxmates the expectaton of the probablty densty functon g(η) for the value of Y and s taken as the estmate y of the output quantty value. The standard devaton u(ŷ) of Ĝ(η) approxmates the standard devaton of g(η) and s taken as the standard uncertanty u(y) assocated wth y. ŷ can be taken as the arthmetc mean 80

6 ŷ = 1 M M y r r = 1 formed from the M values y 1,..., y M, and the standard devaton u(ŷ) determned from: (4) 2 1 u (ŷ) = M 1 M 2 (yr ŷ) r = 1 (5) 4. CONVENTIONAL MEASURING UNCERTAINTY EVALUATION IN MECHANICAL GAUGE BLOCK CALIBRATION 4.1 Mathematcal model of the measurement The length L of the gauge block beng calbrated s gven by the expresson [2]: L = Le + ΔL- Le (δα θ + αe δθ) (6) where: L - length of the gauge block B (gauge beng calbrated) at 20 C L e - length of the standard gauge A, gven n the calbraton certfcate at 20 C ΔL - measured dfference between the gauge blocks A and B δα - dfference of the thermal expanson coeffcents of gauges A and B θ - devaton of the temperature of the gauge block B from 20 C αe - thermal expanson coeffcent of the gauge block A δθ - dfference of temperatures of gauge blocks A and B Measured dfference between the gauge blocks A and B s calculated as by the expresson: ΔL = ΔL am + k 1 + k 2 (7) where: ΔL am - arthmetc mean of the measured dfference (expermental) k 1 - comparator's random error - comparator's systematc error k 2 Devaton of the temperature of the gauge block B from 20 C conssts of the readng from the temperature measurement system, of the dfference between the table temperature and the mean temperature of the standard and of temperature varaton: where: θ 1 Δθ θ 2 θ = θ 1 + Δθ + θ 2 (8) - readng from the temperature measurement system - dfference between the table temperature and the mean temperature of the standard - temperature varaton 81

7 4.2 Calculaton of combned standard uncertanty All evaluated standard uncertantes of the nput value estmates for 100 mm GB are shown n Table I [14]. Table I: Standard uncertantes of the nput value estmates for the 100 mm GB. Quantty Estmated value Standard X uncertanty Dstrbuton Senstvty coeffcent Uncertanty contrbuton Le 100 mm 13.3 nm normal nm ΔL am < 1 nm 1.88 nm normal nm k 1 < 1 nm 15 nm normal 1 15 nm k 2 < 1 nm 5 nm normal 1 5 nm δα 0 C -1 0, C -1 rectangular -L e θ 5.8 nm θ 1 0 C 2, C normal -L e δα 0.25 nm Δθ 0 C C normal -L e δα 3.3 nm θ 2 0 C C rectangular -L e δα 3.5 nm αe C C -1 rectangular - L e δθ 2.9 nm δθ 0 C C rectangular -L e α e nm Total: nm In our case, combned standard uncertanty (after roundng t up) for the 100 mm GB under the best possble condtons s u = 26 nm. 4.3 Expanded uncertanty When usng factor k = 2, absolute expanded uncertanty of calbraton for the 100 mm GB s: U = 52 nm. 5. VERIFICATION OF THE CONVENTIONAL UNCERTAINTY EVALUATION BY USING MONTE CARLO SIMULATION For the Monte Carlo smulaton of the uncertanty evaluaton n mechancal gauge block calbraton the same mathematcal model of the measurement (6) as n conventonal procedure of uncertanty evaluaton, together wth assgned probablty densty functons (Table II), was used. The number of Monte Carlo trals was set to M = 100,000 [15]. Fgure 4 shows the Monte Carlo smulaton's resultng hstogram wth correspondng normal ft and the shortest 95 % coverage nterval. 82

8 Table II: Input values and probablty densty functons for the 100 mm GB. Quantty X Probablty densty functon g(x ) Le normal dstrbuton (M; 100 mm; 13.3 nm) ΔL am normal dstrbuton (M; 0 mm; 1.88 nm) k 1 normal dstrbuton (M; 0 mm; 15 nm) k 2 normal dstrbuton (M; 0 mm; 5 nm) δα rectangular dstrbuton (M; C -1 ; C -1 ) θ 1 normal dstrbuton (M; 0 C; 2, C) Δθ normal dstrbuton (M; 0 C; C) θ 2 rectangular dstrbuton (M; C; 0.03 C) αe rectangular dstrbuton (M; C -1 ; C -1 ) δθ rectangular dstrbuton (M; C; 0.02 C) Y 0025 = mm Y 0975 = mm mean(l) = mm stdev(l) = 21.0 nm Y 0025 Y Hstogram Normal ft Fgure 4: Probablty densty functon g(l) for mechancal calbraton of the 100 mm GB. 6. CONCLUSIONS Mathematcal model of measurement was used for calculatng standard uncertanty of the gauge block calbraton by mechancal comparson. By Monte Carlo smulaton we wanted to verfy the correctness and completeness of uncertantes of the nfluence quanttes, together wth ther accurate evaluaton. 83

9 Comparson of uncertanty, calculated by conventonal measurng uncertanty evaluaton, wth the result of Monte Carlo smulaton, shows small dfference of only 4,5 nm. Agreement n results n such an extent verfes our mathematcal model of mechancal gauge block calbraton, evaluaton of ts standard uncertantes and performed calculatons. Intensons for our future work are mplementaton of Monte Carlo smulaton for verfcaton of more complex mathematcal models lke models of form calbraton. REFERENCES [1] Acko, B.; Sostar, A. (2002). Modfcaton of the model for measurement uncertanty evaluaton n a gauge-block calbraton based on measurement automaton, Strojnsk vestnk, Vol. 48, No. 1, 9-16 [2] Internatonal Standardzaton Organzaton (1995). ISO Gude to the Expresson of Uncertanty n Measurement [3] Acko, B. (2003). A unversal model for evaluatng measurng uncertanty n calbraton, Internatonal Journal of Smulaton Modellng, Vol. 2, No. 4, [4] Doron, T.; Beers, J. (2005). The Gauge Block Handbook, NIST Monograph 180, accessed on [5] Thalmann, R.; Baechler, H. (2003). Issues and advantages of gauge block calbraton by mechancal comparson, Proceedngs of SPIE, Vol. 5190, [6] Taylor, B. N.; Kuyatt, C. E. (1993). Gudelnes for Evaluatng and Expressng the Uncertanty of NIST Measurement Results, NIST Techncal Note 1297 [7] Faust, B.; Stoup, J.; Stanfeld, E. (1998). Mnmzng error sources n gage block mechancal comparson measurements, Proceedngs of SPIE, Vol. 3477, [8] Kacker, R.; Jones, A. (2003). On use of Bayesan statstcs to make the Gude to the Expresson of Uncertanty n Measurement consstent, Metrologa, Vol. 40, No. 5, [9] Detrch, C. F. (1991). Uncertanty, Calbraton and Probablty, Adam Hlger, Brstol [10] Fsher, G. S. (1996). Monte Carlo. Sprnger-Verlag, New York [11] Robert, C. P., Casella, G. (1999). Monte Carlo Statstcal Methods, Sprnger-Verlag, New York [12] Gude to the Expresson of Uncertanty n Measurement, Supplement 1: Numercal Methods for the Propagaton of Dstrbutons (Draft), accessed on [13] Cox, M. et al (2001). Best Practce Gude No. 6. Uncertanty and statstcal modelng, Techncal report, Natonal Physcal Laboratory, Teddngton [14] Godna, A.; Acko, B. (2004). Influence of the Measurement Force on the Uncertanty of the Gauge Block Comparator, VDI-Berchte 1860 (Duesseldorf: VDI) [15] Druzovec, M. et al (2001). Dagnostcs and a qualtatve model, Internatonal journal of medcal nformatcs, No. 63,