DETERMINATION OF UNCERTAINTY ASSOCIATED WITH QUANTIZATION ERRORS USING THE BAYESIAN APPROACH

Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC XVII IMEKO World Congress Metrology n the 3rd Mllennum June 7, 3, Dubrovn, Croata DETERMINATION OF UNCERTAINTY ASSOCIATED WITH QUANTIZATION ERRORS USING THE BAYESIAN APPROACH Hrosh Sato and Kense Ehara Natonal Metrology Insttute of Japan (NMIJ / AIST, Tsuuba, Japan Abstract In practce, quantzaton of a measured quantty often sgnfcantly nfluences observaton values. A typcal example s found n measurements usng dgtal nstruments. In some cases, due to the quantzaton, no dsperson s observed among repeated measurements. The type A evaluaton then gves zero standard uncertanty. In such a case, the most common practce s to assume, as an a pror dstrbuton n type B evaluaton, a unform dstrbuton, the wdth of whch s gven by the quantzaton nterval, and tae the wdth dvded by square root of as the standard uncertanty. Ths practce, however, s ustfed only when the populaton standard devaton s exactly zero. But generally ths condton does not hold true even f the sample standard devaton appears to be zero. In the present study, we use the Bayesan approach to evaluate the uncertanty of a measurement based on quantzed data wth due consderaton to the dfference between the standard devaton of the apparent sample and the populaton standard devaton. We assume that the quantty before quantzaton obeys a normal dstrbuton havng the average and standard devaton. A measurement data corresponds to a value of the quantty after quantzaton. Based on a specfc combnaton of n repeated measurements, we can construct the probablty densty p(, usng the Bayesan method. The standard devaton of the functon, p% ( p(, d, n terms of gves the uncertanty of the measurement result. We have shown that when all of the measurement data tae the same value, the conventonal type B evaluaton descrbed the above results n an underestmate of the uncertanty, f the number of data s less than fve. Analyss s also conducted n cases n whch not all of the data tae the same value. Keywords: GUM, quantzaton uncertanty, Bayesan approach. INTRODUCTION Snce ssued, the Gude to the Expresson of Uncertanty n Measurement (GUM has been a useful common scale used n a number of felds to ndcate the relablty of measurement []. However, several common problems stll occur n these felds. One of these problems s the determnaton of uncertanty assocated wth quantzaton. Concernng the uncertanty assocated wth quantzaton, the GUM descrbes an evaluaton based on the resoluton of the ndcatng devce (scale nterval (GUM F... Accordng to ths descrpton, the type B evaluaton s often conducted usng as uncertanty for the scale nterval. In most cases, ths evaluaton s formulary. The uncertanty assocated wth quantzaton nfluences observaton values when the data s unform rather than when the data s dspersve. If all of the measurement data s dentcal, for example, f, due to the quantzaton, no dsperson s observed among repeated measurements. The type A evaluaton then gves zero standard uncertanty. In such a case, the most common practce s to conduct the type B evaluaton. Ths evaluaton, however, may not always gve approprate uncertanty because the populaton standard devaton may not be zero even f the sample standard devaton appears to be zero. Under these crcumstances, we need to revew the selecton of ndcatng devce. If the necessary accuracy s well satsfed or an actual beneft cannot be expected from a new ndcatng devce, the best estmaton of uncertanty under the current condtons s preferable. In the present study, we tae the Bayesan approach to evaluate the uncertanty of measurement data assocated wth quantzaton. By usng the Bayesan theorem, we can estmate the useful uncertanty from emprcal nformaton []. In the populaton of measurement data before quantzaton, the probablty dstrbuton havng a parameter (, s assumed as a probablty varable. By usng the Bayesan approach wth the parameter (, as the varable {, }, the average dsperson s estmated from the characterstcs of pror probablty on the varable plane and posteror probablty estmated by n - tmes repettve measurement.. QUANTIZATION.. Resoluton of ndcaton devce Measurement data can be obtaned from the ndcaton devce of some nstruments. Most nstruments

Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC almost have a dgtal ndcaton devce. Generally, the quantzaton error s not a sgnfcant problem. Ths means that the nfluence due to the resoluton of the ndcaton s neglgble compared to the combned standard uncertanty. In contrast, when the resoluton s not suffcent, errors due to quantzaton occur. Consder s(q, the expermental standard devaton of repeated observatons q of a normally dstrbuted random varable havng parameter (,. When the data s quantzed. u ( δ s ( q ( u( δ s the uncertanty of quantzaton error sq ( s the expermental standard devaton If we obtan enough data for the estmaton of the parameter, the mean ncludng error s as shown n Fgure, and the standard uncertanty assocated wth quantzaton error was estmated le as Fgure... Problems These fgures show results for deal occasons. In fact, due to lmted samplng, the estmaton value of uncertanty can be larger at the small observaton number or small expermental standard devaton. The data were sampled from a dstrbuton havng parameter (, that we cannot actually now. Especally, n the case of the expermental standard devaton under, some sgnfcant nfluence on the.3.. -. -. -.3 the resoluton of a dgtal ndcaton (quantzaton parameter (, X X X + DATA Indcatng value obtan the observatons Dn ( [ x ] x x3 x n n Fg.3 Quantzaton of observatons L : number of observatons estmaton occurs snce that the data does not dsperson due to quantzaton. 3. THE BAYESIAN APPROACH 3.. Assumptons Measurement data can be consdered as a sample from a dstrbuton havng parameter (, that we cannot now. An acqured quantty value s handled as measurement data after quantzaton. Snce the sample data s from an unnown probablty scatter, the estmaton s smplfed on the bass of the followng assumptons: Assumptons [ A ] observatons are samples from a normal dstrbuton [ A ] Ths event has ndcatng values for not more than three ranged values [ A3 ] pror probabltes of varable; p{, } are dentcal X X + X X +.6.5.4.3.. -. 5 uncertanty -.4 5 4 3 4 3 -.5 -.5 - - -.5 -.5 Fg. Error of best estmaton of.5 Fg. Error of best estmaton of.5.5.5 3.. Event The mean value from observed values s taen as a measured value. In the present study, from the number of observatons and the number of ndcated values, we obtan a mean value and ts uncertanty usng the Bayesan Approach (referred to hereafter as B.A.. From Assumpton [A], an event can be descrbed as follows: E [ a b c] ( a s the number of the ndcatng X observatons b s the number of the ndcatng X observatons c s the number of the ndcatng X observatons + Observed values obtaned thus far are dgtzed output (Fg. 3. We pursue the probablty of event E f we assume data to be acqured from a normal dstrbuton. Assumng a probablty densty functon of normal dstrbuton N (, shows the probablty mass of the quantzed value that can be obtaned as an observed value s pursued as follows:

Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC ( {, } P E δ. δ. Fg.5 Probablty mass of the axs ( { } X + /, (, X / P X N dx ( x X + / e dx X / π An event E ndcates that the types of data value are three contnuous values. And a, b, and c are the number of the contnuous values. The probablty of appearng s gven as the next equaton by the number of each ndcated value. ( { } X - + / P E, N(, dx X - / X + / X / N(, dx X + + / N(, dx X + / Now, we consder that the parameter {, } s a random varable n a set-space. The space settng affects accuracy of the fnal estmatons and should be meshed to calculate the posteror probablty. These meshed ntervals also affect the accuracy of the fnal estmatons. Fgure 4 shows an example of calculaton of the event E [ 5 ]. In the space, the posteror probabltes for each argument n the settng and meshed space are estmated based on Bayes' theorem. Expectaton of mean and dsperson of mean when event E occurred can be decded based on the characterzed probablty densty functon estmated from ths space. We assume that all the pror probablty s equal, so that the expectaton of mean s not a complcated equaton. We meshed the space by δ and δ. The probablty densty of the posteror probablty p( {, } E from whch one populaton s selected s shown as follows: p, E : ({ } P( E {, } p( {, } P( E {, } p( {, } δ δ a b c : 3 Fg.4 Probablty mass of the calculaton-space (3 (4 (5 The probablty functon ({, } p s obtaned from the pror probablty densty functon from whch the - varable {, } s selected. 3.3. Expectaton of Mean and Varance We calculated the expectaton of mean from the event of the observed data. Usng the probablty densty functon of the posteror probablty p( E ntegral for the axs, we estmate the expectaton. Fgure 5 shows the probablty mass of the axs of the example. P( E {, } p( {, } δ p( E P( E {, } p( {, } δ δ (6 [ E] p( E P( E, E δ δ (7 δ where, P( E, ( {, } ({, P E p } δ (8 f we set the constant δ, the equaton s as follows. ( [ ] P E, E E (9 Usng the mean expectaton estmated from (9, we can descrbe the varance of mean as follows: V E E E p E δ [ ] ( [ ] ( ( E [ E] P( E, δ δ ( From (, a standard uncertanty of mean expectaton E[ E] ˆ follows as u ( [ ] ( ( E E P E, ˆ 4. ALGORITHM 4.. Categorzaton of events (

Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC Add t onal Grds Add t onal Grds X- -Δ / X- -Δ / X- 3-Δ / Based Gr ds Fg.6 Calculaton-space For a space havng the varable { }, we calculate the posteror probablty densty changng contnuously n the space and estmate the uncertanty from the calculated value. Fgure 6 shows an outlne of the algorthm. δ. axs max.5 δ. axs mn. max 3 The above ntal value s expanded by n the drecton from the axs and by n the and drectons from the axs. The space s expanded untl the followng crteron of convergence s satsfed. 4.. Crteron of convergence If the number of expansons s h, the space s expanded untl the estmated mean uncertanty satsfes the followng crteron: u ˆ u ˆ 5. ( ( 5 X X++Δ / X++Δ / X+3+Δ / ( h+ h Based on the mean uncertanty estmated under the crteron, an accuracy up to the thrd decmal place can be expected. If the number of measurements n s small, however, a convergence falure s antcpated. In such a case, the maxmum range of expanson s max <. 5. ESTIMATIONS 3 5.. Categorzaton of events Based on these assumptons, the event E s classfed nto the followng three cases: [ Case ] All of the ndcated values are dentcal. [ Case ] Two types of adacent ndcated values are dentcal. Standard uncertanty usng B.A. standard uncertanty.5.45.4.35.3.5..5..5 N4 Type B 4 6 8 Number of data Fg.7 Standard uncertanty usng B.A.; Case.5.4.3.. 4 6 Expandng of Fg. 8 Standard devaton of mean usng B.A. at N4, Case [ 4 ] [ Case 3 ] Three types of contnuous ndcated values are dentcal. The uncertanty assocated wth quantzaton s dscussed for each of the above cases. For numerc operatons, we used general-purpose calculaton software, MATLAB Ver. 6. (Cybernet Systems Co., Ltd.. 5.. Case In some cases, no apparent dsperson s observed among repeated measurements on account of the quantzaton. The type A evaluaton then gves zero standard uncertanty. In such a case, the most common practce s to assume, as an a pror dstrbuton n type B evaluaton, a unform dstrbuton, the wdth of whch s gven by the quantzaton nterval, and tae the wdth dvded by square root of as the standard uncertanty n []. Therefore, we estmated the standard uncertanty of ths case usng the B.A. The results are shown n Fg.7. Dependng on measurement data number n, an uncertanty of mean shows the estmaton becomng small. When the measurement data number s less than fve accordng to the estmaton, the uncertanty estmaton may be underestmated when usng the conventonal type B estmaton. When N4, convergence crtera; equaton. (, was not attaned, and the value gven n Fg. 7 shows ust the

Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC value at the maxmum calculaton-space. Fgure 8 shows the ncrease of the estmaton value dependng on the algorthm at N4. 5.3. Case Smlar to Case, the dfference from these estmatons of mean s gradually extendng. These results are shown n Fg. 9. By ncreasng N, the dfference of mean s gettng large gradually. Fgure shows the expermental standard devaton of mean wth respect to the standard uncertanty usng B.A. for Case, from N4 to 5,, 3, 5 and. The standard uncertanty s estmated to be larger than the expermental STD of mean, as n the case for the estmaton of mean. Partcularly, at N4, the estmaton was very large. When N4, the convergence crtera was not attaned, and the value gven n Fg. shows ust the value at the maxmum calculaton-space. 5.4. Case 3 It was not able to fnd the dfference of the mean values between these estmaton methods, gven n Fg.. Fgure shows the expermental standard devaton of the mean wth respect to the standard uncertanty usng B.A. for Case 3, from N4 to 5,, 3 and 5. The standard uncertanty of N4 usng B.A. was estmated for very large uncertanty. As n the case for N4 n Case and Case, the estmaton of uncertanty was estmated to be very large. When N4, the convergence crtera was not attaned, and the value gven n Fg. shows ust the value at the maxmum calculaton-space. 5.5. Dscusson We assumed the pror probabltes of varable; p{, } are dentcal. The convergent crtera was not attaned at N4 n any cases. It was due to the calculaton algorthm dd not use the goodness of Bayesan approach postvely. We consder that the algorthm should nclude some nformaton of the pror probabltes as a varable. After ths consderaton, t s possble that the uncertanty s estmated more exactly. Though numerous problems reman unresolved at N<5, t s consdered that the approach s a new method to estmate of quantzaton uncertanty. Mean usng B.A.6.5.4.3.. N N5 N4; 5 N; 3...3.4.5.6 Arthmetc mean Fg. 9 Arthmetc mean vs. mean usng B.A.; Case Mean usng B.A.6.5.4 N;.3 N4; 5.....3.4.5.6 Arthmetc mean Fg. Arthmetc mean vs. mean usng B.A.; Case 3 standard uncertanty usng B.A.9.8.7.6.5.4.3.. N N5 N; 3 N4 N5; 5...3.4.5 Exp. STD Fg. Expermental standard devaton of mean vs. standard uncertanty usng B.A.; Case standard uncertanty usng B.A..4..8.6.4. N; N5...3.4.5.6.7 Exp. STD N6; 5 N4 Fg. Expermental standard devaton of mean vs. standard uncertanty usng B.A.; Case 3

Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC 6. CONCLUSIONS We calculated the estmaton usng the Bayesan approach from N4 to 5,, 3, 5. In all cases the estmaton of standard uncertanty as N4 was larger than the expermental standard devaton of mean or the conventonal type B evaluaton. We have shown that when the values of all of the measurement data are dentcal, the conventonal type B evaluaton descrbed above results n underestmaton of the uncertanty, f the number of data s less than fve. In estmaton of the uncertanty assocated wth quantzaton, N4 showed a long-tal posteror probablty densty functon, where the mean value s not stable. Consequently, the mean uncertanty cannot be obtaned n a space n whch a soluton of convergence s set. Therefore, we propose to obtan at least fve data sets, not only when evaluatng the uncertanty assocated wth quantzaton, but for all cases n whch expermental data showng lttle dsperson s antcpated. REFERENCES [] Gude to the Expresson of Uncertanty n Measurement nd edton; BIPM, IEC, IFCC, ISO, IUPAC, IUPAC, IUPAP, OIML, Internatonal Organzaton for Standardzaton, 995 [] Calculaton of Measurement Uncertanty Usng Pror Informaton, S. D. Phllps, Journal of Research of the NIST Vol. 3-6 65-63, 998 Authors: Dr. Hrosh Sato, Metrologcal Statstcs and Partcle Measurement Secton, NMIJ / AIST Address: ---c3, Umezono Tsuuba, Ibara, 35 8563, Japan Phone: +8 9 86 45, Fax: +8 9 86 47 e-mal: sato-hrosh@ast.go.p Dr. Kense Ehara, Metrologcal Statstcs and Partcle Measurement Secton, NMIJ / AIST Address: ---C3, Umezono Tsuuba, Ibara, 35 8563, Japan Phone: +8 9 86 45, Fax: +8 9 86 47 e-mal: ehara.ense@ast.go.p