Process Monitoring of Exponentially Distributed Characteristics. Through an Optimal Normalizing Transformation

Transcription

1 Process Montorng of Exponentally Dstrbuted Characterstcs Through an Optmal Normalzng Transformaton Zhenln Yang Department of Statstcs and Appled Probablty Natonal Unversty of Sngapore, Kent Rdge, Sngapore 96 Mn Xe Department of Industral and Systems Engneerng Natonal Unversty of Sngapore, Kent Rdge, Sngapore 96 Abstract: Many process characterstcs follow exponental dstrbuton and control charts based on such a dstrbuton have attracted a lot of attenton. Tradtonal control lmts may be not approprate because of the lack of symmetry. In ths paper, process montorng through a normalzng power transformaton s studed. The tradtonal ndvdual measurements control charts can be used based on the transformed data. The propertes of ths control chart are nvestgated. Comparson wth the chart usng probablty lmts s also carred out for the cases of known and estmated parameter. Wthout losng much accuracy even compared wth the exact probablty lmts, the power transformaton approach can easly be used to produce charts that can be nterpreted when the normalty assumpton s vald.

2 . Introducton A basc assumpton n usng the tradtonal Shewhart-type control charts s that the qualty characterstc nvolved follows a normal dstrbuton. The Shewhart charts may stll be applcable to non-normal stuatons when the sample sze s large as guaranteed by the Central Lmt Theorem. However, n modern manufacturng envronment, tems are often checked one by one and control charts for ndvdual measurements are desrable, see e.g., Shel (995), Gong et al. (997) and Xe et al. (998). The normalty assumpton for ndvdual value s usually questonable and the normalty should be tested before the mplementaton of tradtonal technque. In case the normalty s rejected, t can be convenent to frst transform a non-normal varable to near normal and then apply the tradtonal control charts on the transformed values. In ths paper, process montorng for exponentally dstrbuted qualty characterstc s dscussed and propertes for such a chart based on transformaton are studed. One usefulness of a control chart for the exponental measurement s that t can be used as a control chart for parts-per-mllon nonconformng tems (Nelson, 994 and Radaell, 998). The dea s that f the number of nonconformng tems s assumed to be a Posson random varable then the 'tme' untl a defectve tem has an exponental dstrbuton. Usually ths 'tme' can be real tme, length or volume, etc. If the producton process has a constant speed, then obvously the count and the tme untl a defectve tem are equvalent quanttes (Nelson, 994) that can both be modeled by an exponental dstrbuton. Nelson (994) proposed a normalzng transformaton whch may be not an optmal () one and the propertes of the chart based on ths transformaton have not been nvestgated. A control chart based on an optmal normalzng transformaton s proposed () It should be noted that the optmalty depends on the adopted optmalty crteron.

3 3 n ths paper. The property of ths chart s nvestgated. At the same tme t s compared wth the chart for exponental dstrbuton wth probablty lmts. Both the case of known parameter and estmated one are dscussed. The nvestgaton or comparson for parameter unknown case are done va an extensve Monte Carlo smulaton. Ths paper s organzed as follows. Secton dscusses the best transformaton wthn the power famly. Secton 3 nvestgates the propertes of the control charts when the model parameter s known, whereas Secton 4 consders the case of unknown parameter where extensve Monte Carlo smulaton results are presented. Two llustratve examples are also presented n Secton 5.. The Best Normalzng Power Transformaton.. Background In studyng the large sample behavor of transformatons to normalty, Hernandez and Johnson (98) consdered an nformaton number approach to transform a known dstrbuton to near normalty, to serve as benchmarks for the maxmum amount of mprovement achevable through Box-Cox transformaton (Box and Cox, 964). The Box-Cox transformaton technque ams to transform the nonnegatve data to near normal through a smple power transformaton so that the normal theory procedures can be appled to the transformed data. It s seen to have many applcatons n relablty, qualty control and lfetme data analyss. See for example, Hnkle and Emptage (99), Fearn and Nebenzahl (995) and Yang (999). Let the probablty densty functon (pdf) of a contnuous nonnegatve random varable X be g ( x; ) where s the mean. Denote by Y = X the power transformed observaton and f ( y;, ) ts pdf. Let ( y ;, ) be a normal pdf wth mean and standard devaton (SD). The basc dea of fndng the `best` normalzng transformaton

4 4 s to fnd, and such that f ( y;, ) and ( y;, ) are closest n a certan sense. Hernandez and Jonhson (98) suggested the so-called Kullback-Lebler (KL) nformaton number as a measure of closeness: f ( y;, ) I ( f, ) = f ( y;, )log dy. (.) ( y;, ) Thus, the optmal transformaton value s found by mnmzng (.). That s, mnmze I [ f ( ;, ), ( ;, )] over (,, ). (.) Ths can be done by usng the general result of Hernandez and Johnson (98). For an exponental dstrbuton, a drect algebrac manpulaton can be easer... Dervaton of the best power transformaton for exponental varables If X s an exponental random varable, then ts pdf has the form g ( x; ) = exp( x ), >. The pdf of Y = X s thus f ( y;, ) = ( ) y exp( y ) by change of varable technque. Indeed, t can be notced that f ( y;, ) s a Webull pdf of the form y exp[ ( y ) ] wth = and =. If ( y ;, ) s a normal pdf wth mean and SD, then the Kullback-Lebler nformaton between f ( y;, ) and ( y;, ) becomes I ( f, ) = y exp y log dy y exp exp( y ( y ) ) = log + dy y y log exp y exp y y ( y ) y dy

5 5 By ntroducng x y, the frst ntegral becomes y log y exp y dy = log x exp xdx = (log ), where =.5776, the Euler-Gamma constant and the second ntegral becomes y y ( y ) exp y dy exp = x x x xdx = ( ) + ( ). Fnally, we have I ( f, ) = log + ()(log) + ( ) ( ) +. Now, settng the partal dervatves of I ( f, ) wth respect to and to zero and solvng the equatons, t can be shown that I ( f, ) s partally mnmzed at () = ( ) and ( ) = [ ( ) ( )]. Substtutng () and ( ) back nto I ( f, ) gves the partally mnmzed Kullback- Lebler nformaton number I() = log( ) + () log + log[ ( ) ( ) ]. Fnally, mnmzng I() gves the best transformaton. Ths can be done by any of numercal maxmzaton procedure. The resulted optmal parameter values are: =.654, =.934 and =.675

6 6 and the mnmzed dstance s I f ( ;, ), ( ;, )] =.78. [ Notce that I() s ndependent of the parameter, whch makes the transformaton very attractve to the practtoners. It s nterestng to pont out that the orgnal exponental pdf ( = ), havng a KL number of.489, s very 'far' from a normal pdf. Comparson of the mnmzed KL nformaton numbers shows therefore a great mprovement to normalty by such a smple power transformaton. It can be further noted that usng the power transformaton proposed n Nelson (994), by matchng the frst few moments, =.777, we have a KL number of.93, that s approxmately 5% larger than that from the transformaton =.654, showng that =.777 s not optmal n terms of KL mnmzng crteron. Hence, although the two transformatons perform smlarly, the former s recommended as t has a sound theoretcal bass. Table. presents some tal probabltes of the optmally transformed Y wth the related normal varable. It has seen that the tal probabltes of Y are very close to those of the correspondng normal random varable. Table.. Tal Probabltes for the Transformed Exponental Random Varable K P( Y k ) P( Y k ) Normal Tal Prob

7 7 3. Control Chart when s Known In ths secton the operatng characterstc (OC) functon of the chart for Y s nvestgated for the case when the n-control exponental mean,, s known and compared wth the chart for X based on the exact probablty lmts. It should be ponted out that for asymmetrcally dstrbuted characterstcs the exact probablty lmts should be usually preferable to the tradtonal 3-sgma lmts. Indeed, t can be noted that for the exponental dstrbutons, 3-sgma lmts would lead to a negatve LCL. The two-sded control chart for the orgnal X, wth allowed false alarm probablty or Type I rsk s of the form: LCL = log( ), CL =.693, (3.) UCL = log( ). In accordance wth the usual approach, the two-sded control chart for Y can be defned as follows: LCL = k, CL =, (3.) UCL = + k, where =.934, =.675 and k s the number of SDs allowed n the control lmts. The ablty to detect shfts n process qualty s an mportant factor to be consdered n desgnng a control chart. Such ablty s often measured by the OC functon of the control chart, whch s defned as the probablty of not detectng a shft under an out-ofcontrol stuaton. Ths probablty represents the type II rsk. As we are nterested n

8 8 detectng a shft n exponental mean, the -rsk for an r SDs shft n the mean of X s gven (under the X chart) by X (r) = P log( ) X log( ) r (3.3) = ( ) (r ) /(r ) ( ). For the Y chart, the OC functon s: (r) = P k Y k r Y = exp r k k exp r, (3.4) where k > s so that PY k + PY k =. Notce that one SD shft n X scale causes less than one SD shft n Y scale. The X chart centers at the medan poston whle the Y chart centers at the mean poston. Ths s not unexpected snce the dstrbuton of Y s almost symmetrc, wth mean and medan almost the same, whch are.934 and (.693 ) =.973, respectvely. Fgure 3. shows the OC functon of X and Y charts, for two typcal values of k,.e., k = (equvalently =.455) and k = 3 ( =.7). b rsk, k= b rsk, k= Fgure 3.. OC Curves for X chart (sold lne) and Y chart (dashed lne).

9 9 As expected, for k = two charts perform very smlarly, whereas for k = 3 the exact probablty lmts wll perform slghtly better than the transformed Y chart. However, the chart based on the transformaton s able to provde a smlar level of senstvty, especally for the case of small to medum shft. Lastly, t can be remarked that, as s known, comparson between the Average Run Lengths (ARLs) of X and Y charts s readly dervable from the OC curves. 4. Control Chart when s Unknown When s unknown, t can be commonly estmated on earler process data. In ths case, the propertes of the two charts are not obvous. In ths secton, the two approaches are compared for the case that s unknown and supposed to be estmated from n past observatons X, X,, X n. 4.. Dervaton of the control lmts For the X chart, the sample mean X can be used to estmate, whch gves the X chart wth estmated control lmts: LCL X X log( ), CL X. 693X, (4.) UCL X X log( ). For the Y chart, one can work drectly wth the transformed data and construct a control chart n the usual way as f the transformed data are exactly normal. A typcal control chart would be the one whch uses the mean of past data as an estmator or and the mean of the movng ranges as an estmator of (Nelson, 994):

10 LCL Y Y k MR d, CL Y Y, (4.) UCL Y Y k MR d. where MR s the mean of the movng ranges of two observatons and d =.8. There are other ways to estmate the control lmts of the Y chart. However, we wll focus our study on (4.) as t s more common for ndvdual chart that s the case here. 4.. Comparson of the alarm rate Snce the control lmts are estmated n ths case, t s dffcult to evaluate the -rsk, the -rsk and the ARL. Monte Carlo smulaton s helpful and convenent for the nvestgatons. There are three ssues that have to be addressed for the Y chart wth estmated control lmts: ) the effect of nonnormalty as the transformed data are stll not exactly normal, ) the effect of estmated control lmts, and ) the relatve performance compared wth the X chart. As the comparson of the Y chart wth X chart s the am of ths paper, we wll concentrate on the last ssue. The smulaton process can be smply descrbed as follows. Let B be the event that the next th observaton falls ether below LCL or above UCL. Then P( B ) represents the probablty that the next th observaton sgnals a process out of control. When process s n control, P( B ) s the -rsk whle when the process s shfted, then P( B ) s the -rsk. Notce that n contrast to the known control lmt case, the B 's are not ndependent anymore. Hence the ARL does not equal to / or /(). We frst smulate P( B ) and then ARL for certan values of n and r.

11 To smulate P( B ) for the X chart, n values from Exp() are generated to calculate LCL X and UCL X. Then a value from Exp(+ r) s generated and s counted by one f ths value s ether below LCL X or above UCL X. Repeat ths process many tmes. The proporton of tmes that the extra value falls ether below Monte Carlo estmate of P( B ). LCL X or above UCL X gves a To smulate P( B ) for the Y chart, generate n values from Exp(), rase these values to a power of, and then calculate LCL Y and UCL Y. Generate another value from Exp(+ r), and rase t to a power of, and then check to see f ths value falls below LCL Y or above UCL Y. Repeat ths process many tmes. The proporton of tmes that the extra value falls outsde of the control lmts gves a Monte Carlo estmate of P( B ) for the Y chart. All the smulaton results n ths paper were based on, runs, that ensured a good accuracy of the smulated probabltes. For example, the estmated standard error of the estmated probablty. s only.4. Some results are presented n Table 4.. It can be seen that the alarm rate s smlar for both methods. Table 4.. Smulated probabltes of acceptance at r sgma shft Upper entry s for the X chart; lower entry s for the Y chart. r= k = =.455) n = k = 3 ( =.7) n =

12 4.3. Comparson of the ARL Unlke the case of a known, the ARL cannot be easly calculated as the events are no longer ndependent. Monte Carlo smulaton s used here. To obtan the ARL, n values wth zero shft from the target value are generated to calculate the estmated control lmts. Then a number of values are generated wth an r sgma shft from the target value, untl an out of control sgnal s gven. Ths number serves as one observaton drawn from the run length dstrbuton. Repeat ths process many tmes. The mean of all the observatons drawn from the run length dstrbuton serves as an estmate of the ARL. B 's Snce the orgnal measurements are exponental and the transformed measurements are Webull where both cases gve closed form cdfs, t s possble to gve an equvalent but smpler algorthm for smulatng ARL. Instead of generatng addtonal measurements at the shfted mean untl one out of control sgnal, we can smply generate a geometrc observaton wth the probablty of success (out of control) beng the condtonal probablty that a new observaton s out of the estmated control lmts. For the X chart, t s p = P( X LCL X) P( X UCL X) LCLX UCLX = exp exp, ( r) ( r) and for the Y chart, t s gven by X X p = P( Y LCL Y) P( Y UCL Y) ( LCLY ) ( UCLY ) = exp exp ( r) ( r) Y where X and Y denote the orgnal and transformed vectors of past observatons. Notce that to calculate p for the Y chart, t s necessary that Y LCL Y be nonnegatve. Ths s not always true for the Y chart usng MR. We thus use the Y chart wth ˆ X for

13 3 smulaton and comparson. The smulated ARLs are summarzed n Table 4.. Agan, each smulated ARL s based on, runs. In ths case, t can be found that all the estmated standard errors are around.4% of the correspondng smulated ARLs, showng that the smulated ARLs are very accurate. Table 4.. Smulated ARLs: Upper entry s for X chart; lower entry s for Y chart. r= k = ( =.455) n = k = 3 ( =.7) n = From Table 4. we see that, for k = ( =.455) and when the process s n control (r = ), the ARLs for both charts are all very close to the nomnal value / = /.455 =.98, though the values for the Y chart are slghtly larger. When there s a shft n the process mean, the ARLs for the X chart and Y chart are agan very close, showng that the two charts are essentally equvalent n detectng a process shft. Also from Table 4., for the charts wth =.7 or k = 3, the Y chart has much larger ARLs than the X chart when process s n control, whch s not bad. For relatvely small shfts the Y chart leads to a larger out-of-control ARL than the X chart (e.g., for n = and r = 3, Y-ARL was approxmately 5% larger than the X-ARL). However, when there s a large enough shft n process mean, the ARLs for the two charts are very close, meanng that the power transformaton-based chart wll be able to detect process shft at a comparable fast speed to the orgnal X chart (e.g., for n = and r = 6, Y-ARL was approxmately only 3% larger than the X-ARL).

14 4 5. Two Implementaton Examples In ths secton, two examples are used to llustrate the control charts dscussed above. The frst example uses a smulated data set for llustratng the known case and the second uses a real data set of Baker (996) for llustratng the unknown case. Example 5.. The smulated data. The frst 3 values are generated from an exponental populaton wth = and used n dervng control lmts, assumng =.7 (.e., k = 3). The next values are generated from an exponental populaton wth = 3 and the last wth = 5. The smulated data are lsted below and the control charts are gven n Fgures 5. where the observatons are plotted on the chart n the same order as they are lsted n the table. Table 5.. A set of smulated data. = = = From the orgnal X chart, we see that the plotted ponts cluster between LCL and CL whch are very close to each other. Ths makes the data vsualzaton dffcult. From the control chart for the transformed data, we see that the plotted ponts scatter evenly around the center lne. The two charts perform smlarly as far as the ablty of dentfyng a process shft s concerned, but the latter s easer to nterpret. In fact, suppose that run rules and other tests (Nelson, 984 and Wang et al., 998) are to be mplemented, t s much clearer f the transformed plot s used.

15 5 3 X Chart Indvdual Value UCL= CL=6.93 LCL= Observaton Number 5 4 Y Chart Indvdual Value 3 UCL=3.43 CL=.6645 LCL= Observaton Number Fgure 5.. Control charts for the smulated data Example 5.. The nterfalure tme data. The data set gven n Table 5. represents the nterfalure tmes for 9 successve falures of a photocoper system. It was orgnally gven n Baker (996) n the form of age (n days) at the falures. Table 5.. A set of nterfalure tme data

16 6 The lkelhood rato test (Lawless, 98, p44) s performed on the orgnal as well as the transformed data and the results are consstent wth the hypotheszed dstrbutonal assumptons. The sample mean of the orgnal data s X =.337. Wth =.7 and k = 3, the control charts specfed by (4.) and (4.) can be easly constructed, see Fgure 5.. It can be seen that the plotted ponts n Y chart scatter evenly on both sdes of the centre lne and agan further nterpretatons are straghtforward. 9 Indvdual Value X Chart UCL=8.586 CL=8.558 LCL= Observaton Number 4 3 Y Chart UCL=3.557 Indvdual Value CL=.6767 LCL= Observaton Number Fgure 5.. Control charts for the nterfalure tme data

17 7 6. Conclusons The Box-Cox transformaton can be used to transform a non-normal dstrbuton to normal. In ths paper, we have studed the use of ths transformaton for exponentally dstrbuted qualty characterstc by mnmzng the Kullback-Lebler nformaton number. The study ndcates that t s easy to use and possesses a number of nterestng statstcal propertes, especally t s of great advantage that the transformaton to normalty does not depend on the specfc parameter value of the exponental dstrbuton. Smlar transformatons have been proposed n Nelson (994) and Kttltz, Jr. (999), but they may not be optmal and no nvestgaton has been gven for the propertes of the resulted control charts. Compared wth the tradtonal probablty lmts, the data transformaton approach leads to a chart that s only a lttle less accurate than the exact one. The transformed data should be used when a control chart s to be nterpreted n a tradtonal sense and especally when run rules are to be used. Ths s especally the case when the model parameter s unknown. However, t should be ponted out that data transformaton approach should be avoded when plottng actual observaton for data recordng s an mportant ssue for specfc mplementaton. Acknowledgement: The authors are grateful to the edtor and to the referee for the detaled comments that lead to a sgnfcant mprovement n the presentaton of the paper. Part of ths research s funded by a research grant from the Natonal Unversty of Sngapore for the project Some practcal aspects of SPC for automated manufacturng process (RP39865).

18 8 References Baker, R. D. (996) Some new tests of the power law process. Technometrcs, 38, Box, G. E. P. and Cox, D. R. (964) An analyss of transformaton (wth dscusson). Journal of the Royal Statstcal Socety B, 6, -5. Fearn, D.H. and Nebenzahl, E. (995) Usng power-transformatons when approxmatng quantles. Communcatons n Statstcs - Theory and Methods, 4, Gong, L.G., Jwo, W.S. and Tang, K. (997) Usng on-lne sensors n statstcal process control. Management Scence, 43, 7-8. Kttltz, Jr., R.G., (999) Transformng the exponental for SPC applcatons. Journal of Qualty Technology, 3, Hernandez, F. and Johnson, R.A. (98) The large-sample behavor of transformatons to normalty. Journal of the Amercan Statstcal Assocaton, 75, Hnkle, A.J. and Emptage, M.R. (99) Analyss of fatgue lfe data usng Box-Cox transformaton. Fatgue and Fracture of Engneerng Materals and Structures, 4, Lawless, J. F. (98) Statstcal Models and Methods for Lfetme Data. John Wley & Sons. New York. Nelson, L.S. (984) The Shewhart control chart - tests for specal causes. Journal of Qualty Technology, 6, Nelson, L.S. (994) A control chart for parts-per-mllon nonconformng tems. Journal of Qualty Technology, 6, Radaell, G. (998) Plannng tme-between-events Shewhart control charts. Total Qualty Management, 9, Shel, J. (995) An economc-approach towards usng ndvdual measurements to control the mean of contnuously montored processes. Internatonal Journal of Producton Research, 33,

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