The Use of Principal Components Analysis in the Assessment of Process Capability Indices

Transcription

1 Jont Statstca Meetngs - Secton on Physca & Engneerng Scences (SPES) The Use of Prnca omonents Anayss n the Assessment of Process aabty Indces Evdoka Xekaak Mchae Peraks Deartment of Statstcs Athens Unversty of Economcs Busness 76 Patson st Athens GREEE e-ma: exek@aueb.gr Keywords Phrases: rocess caabty ndces; mutvarate rocesses; rnca comonent anayss.. Introducton Process caabty ndces are measures of the caabty of a rocess to roduce accordng to some assgned secfcatons reated to some characterstcs of the tems roduced. In most of the cases t s assumed that the studed rocess s descrbed through ony one characterstc. However sometmes the need arses for measurng the caabty of rocesses descrbed through two or more characterstcs. For such mutvarate rocesses some ndces have been ntroduced n the terature based on the mutvarate jont dstrbuton of the characterstcs of the rocess. Among the suggested aroaches for measurng the caabty of mutvarate rocesses one may note those by han et a. (99) Pearn et a. (99) Kocherakota Kocherakota (99) Taam et a. (993) hen (994) Shahrary et a. (995) Bernardo Irony (996) Wang hen ( ) Nverth Dey (000). Extensve revews of the above mentoned some other roosas for caabty ndces for mutvarate rocesses are rovded by Kotz Johnson (993) Kotz Loveace (998) Kotz Johnson (00). Ths aer consders the use of rnca comonents anayss n measurng the caabty of a mutvarate rocess. The use of ths mutvarate technque n connecton wth rocess caabty assessment robems s an ssue ntay examned by Wang hen ( ). The beneft of usng ths method s that n some cases t may smfy substantay the anayss of a very comex mutvarate data set by reducng ts dmensonaty. The dea s that f some of the frst rnca comonents exan a arge ercentage of the tota varabty the assessment of the caabty of the rocess can be based on them as one does not oose much nformaton by omttng the remanng rnca comonents. In the seque some aternatve ways of mementng ths technque are roosed through the defnton of certan new ndces for mutvarate rocesses n terms of ndces reated to the rnca comonents to be used. In artcuar n secton a descrton of the nta dea of Wang hen ( ) s rovded. In secton 3 extendng Wang hen s aroach some ndces are ntroduced for mutvarate rocesses wth secfcaton mts that are unatera of the same tye. Fnay n secton 4 some new ndces are roosed for mutvarate rocesses wth unatera or batera secfcatons. These take account of the ossby unequa contrbutons of the ndvdua rnca comonents to the tota varabty.. The Aroach of Wang hen ( ) Wang hen ( ) consdered basng the caabty assessment on a set of uncorreated rnce comonents extracted from the r ntay seected varabes ( r) usng one of the ndces / M = M= = = M m = = / m = = / /. Here m denote the vaues of the unvarate ndces m resectvey for the -th rnca comonent. In the comutaton of the ndex vaues for the -th rnca comonent the secfcatons (L U T) nvoved are near combnatons of the nta secfcatons wth the eements of the egenvector corresondng to the -th rnca comonent as coeffcents. The number of rnca comonents used for the assessment of the ndces s determned through Anderson s (963) test. By ther defnton these ndces are geometrc means of the vaues of the basc PIs for batera toerances for each of the r rnca comonents. Wang hen rovde three exames a rea data acaton that carfy the assessment of the ndces M M M m M. As the authors ont out the obtaned vaues of the ndces n these 389

2 Jont Statstca Meetngs - Secton on Physca & Engneerng Scences (SPES) exames are n accordance to the ndces by han et a. (99) Taam et a. (993) Shahrar et a. (995) hen (994). An addtona aeang feature of ths technque s ts sme mementaton n comarson to the other aroaches suggested for measurng the caabty of mutvarate rocesses. 3. Some New Indces for Mutvarate Processes wth Unatera Secfcatons The above defned ndces are acabe ony to stuatons where the secfcatons of a the examned characterstcs are batera. However one may be faced wth a rocess whose characterstcs have ony a ower or an uer secfcaton mt snce ther sma or arge vaues may be desrabe. In such cases Wang hen s ( ) ndces can no onger be used. Moreover the assessment of the most wdey used rocess caabty ndces such as those suggested by han et a. (99) Taam et a. (993) Shahrar et a. (995) hen (994) becomes troubesome for rocesses wth unatera secfcatons. To overcome ths mtaton we roose foowng Wang hen's ratonae usng the ndces MPL= = MPU= = PL / / PU where PL PU denote the vaues of the ndces PL PU for unvarate rocesses for the -th rnca comonent. As n the unvarate case the ndex MPL s acabe n cases where the examned characterstcs of the rocess have ony ower secfcaton mts whe the ndex MPU s acabe n stuatons where the examned characterstcs of the rocess have ony uer secfcaton mts. The reason why absoute vaues of the unvarate ndces are consdered s connected to the fact that sometmes due to the rotaton of the axes that takes ace n rnca comonent anayss the resutng near combnaton of the nta ower (uer) secfcaton mts for some rnca comonents may take the ace of an uer (ower) secfcaton mt thus eadng to negatve ndex vaues. Ths s carfed n the exames gven beow. Note that the same hods for the ndces M M M m M athough Wang hen ( ) do not use the absoute vaues of the unvarate ndces n ther defnton. The ntroducton of the ndces MPL MPU s meanngfu ony f the r varabes have unatera secfcatons of the same tye -ether ower or uer. If some of the varabes have ony a ower secfcaton mt the remanng varabes have ony an uer secfcaton mt the assessment of the secfcatons of the rnca comonents consequenty the measurement of the caabty of the whoe rocess usng rnca comonent anayss becomes troubesome. The estmaton of the ndces MPL MPU s ustrated n the seque usng the exames gven by Wang hen ( ) by further assumng that the secfcatons of the examned rocesses are unatera. Wang hen ( ) used han et a.'s (99) data that refer to a bvarate rocess reresent measurements on the brne hardness (X ) tense strength (X ). The secfcatons are gven by =(.7 3.7) u=( ) t=(77 53) the same sze s 5. The same data are gven beow: x x From these data one may easy obtan estmates of the mean vector the varance covarance matrx. In artcuar ( ) x = s = resectvey. Moreover the estmates of the egenvaues of S that foow are ˆ = ˆ =.333 those of the corresondng egenvectors are u =( ) u ˆ =( ). Wang hen ( ) argue that accordng to Anderson's (963) test ony the frst rnca comonent shoud be used for ths reason they assess the vaues of ther ndces wthout takng nto account the second rnca comonent. Let us now assume that the two varabes of the exames have ony ower secfcaton mts. In ths case the assessment of the vaues of the ndces by Wang hen ( ) or of the other ndces that have been suggested n the ˆ 380

3 Jont Statstca Meetngs - Secton on Physca & Engneerng Scences (SPES) terature s not ossbe. However we may estmate the vaue of the ndex MPL defned above. Actuay usng agan the frst rnca comonent ony an estmate of the vaue of MPL can be obtaned through µ PL MPL = = σ L µ L 3σ wth denotng the mean the stard devaton the ower secfcaton mt for the frst rnca comonent estmated by ther same counterarts f unknown. In ths case the estmated vaues of µ σ L are x s = ˆ = 7.3 =8.977 resectvey. Thus the vaue of the ndex MPL s estmated to be MPL =.8. Let us now consder the second exame gven by Wang hen ( ) n whch the same data are used but the secfcatons have become =( ) u=( ). Assumng agan that ony ower secfcaton mts have been assgned a hgher ndex vaue s exected snce the secfcaton mts have become ess strct the rocess data reman unchangeabe. Actuay n ths case the estmated vaues of µ are σ L resectvey. Hence the vaue of the ndex MPL s estmated to be equa to MPL =.669. Fnay the assessment of the ndex MPL s ustrated usng a rea data set consdered by Wang hen ( ). The data have been coected from a astcs manufacturer n Tawan. The varabes consdered are the deth (X ) the ength (X ) the wdth (X 3 ) of a astc roduct. The vectors of the secfcaton mts are gven by =( ) u=( ) resectvey. On the bass of a rom same of 50 eces of the roduct the same mean vector the varance covarance matrx are estmated to be x = ( ) s = resectvey. Wang hen have memented Anderson's test concuded that ony the frst two rnca comonents shoud be used. The estmated egenvaues the corresondng egenvectors of the frst two rnca comonents are ˆ = ˆ =0.005 u ˆ =( ) û =( ). Assumng as n the case of the revous exames that ony the ower secfcaton mts for the three varabes have been assgned we obtan x = = s = ˆ x = ˆ = s = Therefore MPL = PL =.43. = 4. Some New Indces for Mutvarate Processes wth Unatera or Batera Secfcatons Based on Unequay Weghtng the ontrbuton of the Prnca omonents Used In the revous secton some ndces suggested by Wang hen ( ) were resented. These ndces are acabe to mutvarate rocesses are defned as the geometrc means of some unvarate ndces for every rnca comonent. As ustrated n ther exames Wang hen suggest the use of ony the rnca comonents that are found to be sgnfcant accordng to Anderson's (963) test n the assessment of the ndces. However the ndces defned by Wang hen assgn the same mortance to a the rnca comonents that are found to be sgnfcant thus are nvoved n ther assessment. In order to ustrate ths statement et us reconsder Wang hen s rea-data acaton used n the revous secton n connecton wth the ndex MPL. In ths exame the data have been coected from a astcs manufacturer n Tawan the consdered varabes are the deth (X ) the ength (X ) the wdth (X 3 ) of a astc roduct. The egenvaues the corresondng egenvectors of the three rnca comonents are estmated as ˆ = ˆ =0.005 ˆ 3 = =( ) =( û ) û 3 =( ) resectvey. Snce % of the tota varabty was exaned by the frst two rnca comonents Wang hen based the assessment of ther ndces on them. From the assessment of the ndces M M M m M as gven by Wang hen ( ) t s obvous that û 38

4 Jont Statstca Meetngs - Secton on Physca & Engneerng Scences (SPES) the frst two rnca comonents are consdered to have the same mortance snce the resutng ndces are the geometrc means of the corresondng ndces for unvarate rocesses. Nevertheess the frst rnca comonent exans 63.93% of the tota varabty the second rnca comonent exans ony 5.% of the tota varabty. Furthermore f we ay Anderson's (963) test for testng the hyothess H 0 : = t foows that the vaue of the test statstc equas χ whch exceeds =5.99. Therefore even though the frst two rnca comonents have sgnfcant dfferences n ther assocated egenvaues consequenty to the ercentage of the varance exaned by each of them the use of the ndces M M M m M does not take nto account ths fact. In the seque n order to overcome ths defcency of these ndces we suggest some new ndces that aow for otenta dfferences n the ortons of the varance exaned by the rnca comonents consdered. These dfferences are taken account of by assgnng unequa weghts to the unvarate ndex vaues corresondng to the rnca comonents used n artcuar roortona to the ortons of the varance exaned by them as determned by ther resectve egenvaues. Secfcay the foowng ndces are roosed: M = V = M m = V = M = V m = M = V = when the toerances of the r nta varabes are batera MP L = PL V = MP U = PU V = f the secfcatons of the r nta varabes are unatera consstng ony of ower or uer secfcaton mts resectvey. Here agan denotes the number of the rnca comonents used as determned through Anderson's (963) test V denotes the sum of the egenvaues assocated wth the seected rnca comonents. Obvousy the ndces M M M M MP L m MPU are weghted averages of the unvarate ndces for the seected rnca comonents wth weghts based on the roorton of varabty exaned by each of them. If the varance covarance matrx Σ s unknown the vaues of the ndces can be estmated usng the egenvaues of the same varance covarance matrx S. For Wang hen s data consdered above ˆ = ˆ = Therefore Vˆ =0.005 thus the corresondng estmates of the ndces M M M M M' are M' M' m =.94 =.6 =.6 M' =.5. In the case where the r varabes that descrbe the rocess are exressed n measurement unts wth substanta dfferences one may aternatvey assess the vaues of the ndces usng the egenvaues the egenvectors of the correaton matrx. References Anderson T.W. (963). Asymtotc Theory for Prnca omonent Anayss Annas of Mathematca Statstcs Bernardo J.M. Irony T.Z. (996). A Genera Mutvarate Bayesan Process aabty Index. The Statstcan 45(3) han L.K. heng S.W. Srng F.A. (99). A Mutvarate Measure of Process aabty Journa of Modeng Smuaton () -6. hen H. (994). A Mutvarate Process aabty Index Over a Rectanguar Sod Toerance Zone Statstca Snca Kocherakota S. Kocherakota K. (99). Process aabty Index: Bvarate Norma Dstrbuton. ommuncatons n Statstcs - Theory Methods 0(8) Kotz S. Johnson N.L. (993). Process aabty Indces haman Ha. Kotz S. Johnson N.L. (00). Process aabty Indces - a Revew Journa of Quaty Technoogy 34() -9. Kotz S. Loveace.R. (998). Process aabty Indces n Theory Practce Arnod. Nverth M. Dey D.K. (000). Mutvarate Process aabty A Bayesan Persectve. ommuncatons n Statstcs - Smuaton omutaton 9() Pearn W.L. Kotz S. Johnson N.L. (99). Dstrbutona Inferenta Proertes of Process aabty m 38

5 Jont Statstca Meetngs - Secton on Physca & Engneerng Scences (SPES) Indces Journa of Quaty Technoogy 4(4) 6-3. Shahrar H. Hubee N.F. Lawrence F.P. (995). A Mutvarate Process aabty Vector. Proceedngs of the 4th Industra Engneerng Research onference Insttute of Industra Engneers Taam W. Subbaah P. Lddy J.W. (993). A note on Mutvarate aabty Indces Journa of Aed Statstcs 0(3) Wang F.K. hen J.. ( ). aabty Index Usng Prnca omonent Anayss Quaty Engneerng () -7. Wang F.K. Hubee N.F. Lawrence F.P. Mskun J.D. Shahrar H. (000). omarson of Three Mutvarate Process aabty Indces Journa of Quaty Technoogy 3(3)