A MULTIVARIATE MAGNITUDE ROBUST CONTROL CHART FOR MEAN SHIFT DETECTION AND CHANGE POINT ESTIMATION THESIS. Ryan M. Harrell, Captain, USAF

Transcription

1 A MULTIVARIATE MAGNITUDE ROBUST CONTROL CHART FOR MEAN SHIFT DETECTION AND CHANGE POINT ESTIMATION THESIS Ryn M. Hrrell, Cptin, USAF AFIT/GOR/ENS/07-09 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Ptterson Air Force Bse, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

2 The views expressed in this thesis re those of the uthor nd do not reflect the officil policy or position of the United Sttes Air Force, Deprtment of Defense, or the United Sttes Government.

3 AFIT/GOR/ENS/07-09 A MULTIVARIATE MAGNITUDE ROBUST CONTROL CHART FOR MEAN SHIFT DETECTION AND CHANGE POINT ESTIMATION THESIS Presented to the Fculty Deprtment of Opertionl Sciences Grdute School of Engineering nd Mngement Air Force Institute of Technology Air University Air Eduction nd Trining Commnd In Prtil Fulfillment of the Requirements for the Degree of Mster of Science in Opertions Reserch Ryn M. Hrrell, BS Cptin, USAF Mrch 007 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

4 A MULTIVARIATE MAGNITUDE ROBUST CONTROL CHART FOR MEAN SHIFT DETECTION AND CHANGE POINT ESTIMATION Ryn M. Hrrell, BS Cptin, USAF Approved: Dr. ~hf;f ~elouk (Member) /9 4 0'7 dte

5 AFIT/GOR/ENS/07-09 Abstrct Sttisticl control chrts re often used to detect chnge in n otherwise stble process. This process my contin severl vribles ffecting process stbility. The gol of ny control chrt is to detect n out-of-control stte quickly nd provide insight on when the process ctully chnged. This reduces the off-line time the qulity engineer spends ssigning cuslity. In this reserch, multivrite mgnitude robust chrt (MMRC) ws developed using chnge point model nd likelihood-rtio pproch. Here the process is considered in-control until one or more normlly distributed process vribles permnently nd suddenly shifts to out-of-control, stble vlue. Using verge run length (ARL) performnce nd the reltive men index (RMI), the MMRC is compred to the multivrite cumultive sum (MC) nd the multivrite exponentilly weighted moving verge (MEWMA). These results show the MMRC performs fvorbly to the MC nd MEWMA when the process is initilly in-control before shifting out-of-control. Additionlly, the MMRC provides n estimte for the chnge point nd out-of-control men vector. This chnge point estimtor is shown effective for medium to lrge sudden men shifts. iv

6 AFIT/GOR/ENS/07-09 Dediction To my wonderfully ptient wife v

7 Acknowledgments Professionlly, there re severl people I would like to thnk for their support nd guidnce. First off, I d like to thnk my dvisor, Dr. Mrcus Perry. His cndor, ptience, guidnce nd instruction mde this reserch effort possible. Furthermore, I would like to thnk Dr. Shrif Melouk for providing insightful feedbck. Finlly, I would like to thnk Drs. Jmes Simpson nd Joseph Pigntiello whose reserch this thesis is bsed upon. Personlly, I wnt to thnk Brin Crwford, Jon Hudson, Kristen Cvllro nd the rest of the GOR-07M clss for the numerous times they distrcted me just enough to keep the four foot by six foot window of grdution in front of me. Lst but certinly most, I hve unending grtitude to my wife, best friend nd soulmte. I love you, Chicky. Ryn M. Hrrell vi

8 Tble of Contents Pge Abstrct... iv List of Figures... ix List of Tbles...x List of Acronyms nd Abbrevitions... xii I. Introduction.... Wht is Qulity?.... Phse I Phse II Shewhrt X Control Chrt Averge Run Length Multivrite vs. Multiple Univrite Chrts Shift Detection Differences Vrible Correltion Problem Definition Reserch Objectives nd Assumptions Thesis Orgniztion... 4 II. Literture Review...6. Introduction Univrite Control Chrts Multivrite Control Chrts....4 Overview Evlutions Conclusion... 7 III. Methodology Introduction The Univrite Mgnitude Robust Chrt The Multivrite Mgnitude Robust Chrt MMRC Exmple Finding MMRC Control Limits Flse Alrms MC nd MEWMA Chrting Sttistics MC MEWMA Reltive Men Index Conclusion IV. Results...47 vii

9 4. Introduction ARL Performnce Simultion Implementtion Simultion Inputs Min Simultion Loop Single Run Simultion ARL Performnce (τ = 0) RMI Comprison Tble Comprison ARL Performnce (τ = 50) RMI Summry Tble Comprison MMRC Chnge Point Performnce MMRC CL (B) Regression Anlysis MEWMA Wrm-Up Effect Conclusion V. Recommendtions nd Future Reserch Introduction Anlysis Recommendtion Future Reserch Recommendtions Appendix A... 8 Appendix B Bibliogrphy...99 viii

10 List of Figures Pge Figure.: Fctors Clssifiction... 3 Figure.: X Control Chrt Exmple... 7 Figure.3: Eucliden Distnce... Figure.4: Mhlnobis Distnce... Figure 3.: Single run MMRC chrt with τ = 50, p = 0 nd men shift δ = [.474,.474,...,.474] Figure 3.: Find B Heuristic Psuedocode Figure 3.3: Find B Heuristic Execution for Trget ARL = Figure 3.4: Actul vs. Stedy Stte Covrince Mtrix Weights Figure 4.: Simultion Inputs Figure 4.: Min Simultion Loop Figure 4.3: Single Run Simultion... 5 Figure 4.4: Simulted B Vlues Figure 4.5: Regressed ˆB Vlues... 7 Figure 4.6: Norml Probbility Plot... 7 Figure 4.7: Regression vs. Residuls... 7 Figure 4.8: ARL 0 with r =.05 nd p = Figure 4.9: Wrm-Up Effect of τ on h 4 for ARL 0 = ix

11 List of Tbles Tble.: ARL Comprison Exmple... 9 Tble 3.: ARL vlues from Pigntiello nd Simpson [], Tble Tble 4.: RMI Summry for τ = 0 from Tbles B. through B Tble 4.: τ = 0, p =, μ = (δ,0) Tble 4.3: τ = 0, p = 3, μ = (δ,0,0) Tble 4.4: τ = 0, p = 0, μ = (δ,0,0,,0) Tble 4.5: τ = 0, p =, μ = (δ,δ) Tble 4.6: τ = 0, p = 3, μ = (δ,δ,δ) Tble 4.7: τ = 0, p = 0, μ = (δ,δ,δ,,δ) Tble 4.8: RMI Summry for τ = 50 from Tbles B.7 through B Tble 4.9: τ = 50, p =, μ = (δ,0)... 6 Tble 4.0: τ = 50, p = 3, μ = (δ,0,0) Tble 4.: τ = 50, p = 0, μ = (δ,0,0,,0) Tble 4.: τ = 50, p =, μ = (δ,δ) Tble 4.3: τ = 50, p = 3, μ = (δ,δ,δ) Tble 4.4: τ = 50, p = 0, μ = (δ,δ,δ,,δ) Tble 4.5: Chnge Point Estimtor Performnce when τ = Tble 4.6: Chnge Point Estimtor Performnce when τ = Tble 4.7: Absolute Difference Between B nd ˆB... 7 Tble 4.8: EWMA Inerti Comprison for ARL 0 = 00 with 0,000 Monte Crlo runs Tble B.: τ = 0, p =, μ = (δ,0) Tble B.: τ = 0, p = 3, μ = (δ,0,0) Tble B.3: τ = 0, p = 0, μ = (δ,0,0,,0) x

12 Tble B.4: τ = 0, p =, μ = (δ,δ) Tble B.5: τ = 0, p = 3, μ = (δ,δ,δ)... 9 Tble B.6: τ = 0, p = 0, μ = (δ,δ,δ,,δ)... 9 Tble B.7: τ = 50, p =, μ = (δ,0) Tble B.8: τ = 50, p = 3, μ = (δ,0,0) Tble B.9: τ = 50, p =, μ = (δ,0,0,,0) Tble B.0: τ = 0, p =, μ = (δ,δ) Tble B.: τ = 0, p = 3, μ = (δ,δ,δ) Tble B.: τ = 0, p = 0, μ = (δ,δ,δ,,δ) xi

13 List of Acronyms nd Abbrevitions ARL...Averge Run Length ARL 0...In-control Averge Run Length ASQ... Americn Society for Qulity ASQC...Americn Society for Qulity Control B...UCL for the MMRC CL... Control Limit COT...CUSUM of T CUSUM...Cumultive Sum EMWA... Exponentilly Weighted Moving Averge h...cl for the MC h 4...CL for the MEWMA LCL...Lower Control Limit MC...Multivrite CUSUM MEWMA... Multivrite EWMA MLE...Mximum Likelihood Estimtion MMRC... Multivrite Mgnitude Robust Chrt QE... Qulity Engineer RMI...Reltive Men Index SPC... Sttisticl Process Control UCL... Upper Control Limit UMRC... Univrite Mgnitude Robust Chrt xii

14 A MULTIVARIATE MAGNITUDE ROBUST CONTROL CHART FOR MEAN SHIFT DETECTION AND CHANGE POINT ESTIMATION I. Introduction. Wht is Qulity? Everybody hs personl understnding of wht qulity is, but very few hve concrete definition of it. The cliché, I'll know qulity when I see it, sums this up nicely. For exmple, let s tke look t the utomotive industry. I could rgue Mercedes-Benz is higher qulity vehicle becuse of its bigger engine, lether interior nd higher price, nd Toyot is lower qulity vehicle for its smller engine, cloth interior, nd lower price. If this is true, why do both J.D. Power nd Assocites nd Consumer Reports conclude Mercedes-Benz produces poor qulity crs? They simply surveyed recent owners bout defects nd problems nd the Toyot owners found fewer problems with their crs. This is the difference between the qulittive nd quntittive spects of qulity. While one cn like the qulittive feel of lether nd surge of power driving high performnce cr, one lso understnd better qulity mterils do not trnslte into superior engineering nd mnufcturing. In 95, Wlter Shewhrt of Bell Lbortories (Bell Lbs) ws grppling with the sme problems. Bell Lbs ws joint venture between AT&T nd Western Electric to conduct reserch for both compnies. At the time, the phone compnies mnufctured the phones nd lesed them to customers. When phone mlfunctioned the phone

15 compny hd to fix it t no chrge to the consumer. The fewer mlfunctioned phones ment the more money the phone compny mde. Qulity control then consisted of testing phone fter mnufcture, nd fixing the phones with defects. The ide of qulity monitoring throughout the mnufcturing process did not exist. Shewhrt [6] took his trining in sttistics nd outlined his mnifesto on wht defines qulity in mnufcturing setting. He believed the highest gol in mnufcturing ws to crete vrition-free products identicl to the engineering specifictions. In ddition, he broke down the quntittive sources of vribility even further into controllble nd uncontrollble vribles (see Figure.). Most relevnt to qulity engineer (QE) re the controllble vribles, such s stir rte, pressure nd feed rte. By properly setting these vribles, QE cn mintin the product or process in sttisticl control. Unfortuntely, known, uncontrollble vribles such wether nd different btches of rw mteril cn directly ffect the vribility of product. Since these vribles re known, they re mesurble nd often QE cn compenste by djusting the controllble vribles. There re fctors ffecting our process we cnnot mesure becuse we re not wre of them, nd therefore these fctors re unknown. If the correct known vribles re incorported into the process, these unknown vribles re ssumed to be white noise distributed ccording to some sttisticl distribution (i.e. Norml). Although these ides re reltively stndrd tody, they were revolutionry for chnging the view of qulity in mnufcturing.

16 Quntittive Uncontrollble Controllble Known Unknown Figure.: Fctors Clssifiction. Phse I Phse I in sttisticl process control is n experimentl study on the nture of the process you re trying to monitor. A process is defined s something with desired nd mesurble trget vlue evolving over time. Process exmples rnge from count dt like scrtches on desk to the width of lumber t mill nd intngible dt like hert rte nd microprocessor switching frequency. These trget vlues re generlly either trget men or trget vrince, lthough one cn use trget medin or liner/nonliner model. As result, this thesis uses the generic term process to include ll possibilities. Phse I generlly occurs prior to process is coming online. In this phse the QE s job is to determine the nture of the process he/she wnts to monitor. This is usully ccomplished through experimenttion nd/or the use of historicl dt. The preferred method involves series of designed experiments to understnd wht fctors ffect the process, nd their sttisticl distribution. Once the fctors re discovered, response surfce study is conducted to optimize the controllble fctors to the desired outcome. Since the QE is involved in every step of dt collection, he/she receives exctly the dt required, nd possesses in-depth knowledge of the dt 3

17 collection techniques nd understnds the overll dt qulity. Unfortuntely, this process generlly involves non-trivil expense of time nd money. If the QE hs neither the time or money resources vilble, he/she cn nlyze historicl dt in lieu of experimenttion. While this dt is usully free becuse it does not cost ny dditionl time or money to collect, the QE often hs little to no knowledge bout the process used to collect this dt. As result of this limited knowledge, the dt qulity is utomticlly in question. The idel sitution is when the QE hs both historicl dt nd the budget to experiment. If the historicl dt is verified nd vlidted by experimenttion, then the QE hs lrger dt set to work with thn just experimenttion lone. Clerly, hving lrge mount of relible dt gives the QE considerble insight into the studied process. Regrdless of the dt collection method, prticulr control chrt is selected or developed to best mintin the trget vlue. Then, once Phse I is complete, Phse II monitoring cn begin..3 Phse II In Phse II, the process hs completed enough tests nd experiments to begin monitoring using control chrt. Here the process is well understood nd with ech new set of observtions, we re trying to nswer three questions. The first question is: Did the process chnge? This question then begets two other questions: If the process did ctully chnge, when did it chnge nd wht ws the cuse? Bsiclly, just becuse the chrt gives process out-of-control signl does not men the QE knows when or why. Unless the chrt used hs chnge point estimtor developed for it, he/she is often left 4

18 looking to the chrt nd mking n educted guess bout the chnge point. In fct, even with n ccurte chnge point estimtor the QE still hs to determine cuslity. This involves investigting the sequence of events, such s chnge in rw mteril supplier or tool wer, cusing the chrt to signl..3. Shewhrt X Control Chrt The X control chrt developed by Shewhrt [7] monitors whether the process men is in or out of sttisticl control. Sttisticlly speking, this trnsltes into the following hypotheses H : μ = μ 0 0 H : μ μ 0 where μ 0 is the desired in-control men nd μ is the true men. The null hypothesis, denoted by denoted by H 0 H, sttes the process is currently in-control nd the lterntive hypothesis,, sttes the process is currently out-of-control. The sttistic used to test H0 versus H is the smple subgroup men t n n i = X = x t. i Here X t is the verge of n mesurements t time or observtion point t rnging from one to the most recent observtion. Additionlly, the X control chrt ssumes ll x it vlues re normlly distributed with men, μ, nd vrince, ( ) x N μσ. σ, it ~ (, ) Since the observtions re normlly distributed, the QE cn test whether within plus or minus L stndrd devitions, σ, of μ ( X t Lσ ) ±. If ny X t X t t ech t is > Lσ or 5

19 X t < Lσ, then H is rejected in fvor H nd the process is potentilly out-of-control. 0 Essentilly, the X control chrt is series of sequentil hypothesis tests. ( it ~ ( 0,) How does the QE set the vlue for L? If the dt is distributed stndrd norml, x N ), L corresponds to the norml distribution inverse of one-hlf of the probbility of declring process out-of-control when it is ctully in-control. For exmple, probbility of.007 corresponds to the stndrd norml inverse of.0035 resulting in L = 3. This equtes to rte of /.007 or n verge of one flse lrm for every 370 observtions. These stndrd norml inverse vlues re esily obtinble from ny stndrd sttistics text or softwre pckge. For exmple, Figure. shows single run of the X control chrt. The green dots re from n in-control distribution nd the red dots re from n out-of-control process shifted by one positive stndrd devition. The brown line is the ctul chnge point, denoted s τ. The dshed line fter t = 0 indictes the one stndrd devition sudden men shift, or out-of-control vlue of the men. The two prllel blue lines re the upper nd lower control limits (UCL/LCL) corresponding to ± Lσ. If the chrting sttistic exceeds the UCL/LCL, then the chrt hs signled indicting potentil out-ofcontrol process. Ech dot represents the discrete clcultion of the chrting sttistic. 6

20 4 3 Chrting Sttistic, xx τ τ -3-4 Time, t Figure.: X Control Chrt Exmple For this X chrt, the chrting sttistic is the stndrdized smple men, X t ˆ μ, where ˆ σ ˆμ nd ˆ σ re the estimtes obtined from Phse I nd the subgroup size n =. Thus the men should equl zero nd the stndrd devition/vrince should equl one. As one cn see, the green line vries rndomly bout zero wheres the red line brely goes below zero just once. Finlly, the 6 th observtion exceeds the UCL nd the chrt signls. The QE then hs to nswer the questions of when nd why. After cuslity is discovered nd the problem is rectified, the chrt is restrted nd monitoring continues until the next signl..3. Averge Run Length The verge run length (ARL) is the expected number of observtions required until the chrt signls. The two types of ARLs re the in-control ARL (ARL 0 ) nd the out-of-control ARL. ARL 0 is the verge time to flse lrm when the process remins in-control given specified UCL/LCL. The out-of-control ARL, on the other hnd, is the 7

21 time it tkes for the chrt to signl fter chnge hs occurred. In n idel sitution, ARL 0 is infinity while the out-of-control ARL is the first observtion fter the chnge occurred. In relity, flse lrms occur in every process, nd the in-control nd out-ofcontrol ARLs re mnipulted on the bsis of the UCL/LCL. In order to compre control chrts, ARLs re used. By clibrting ll the chrts to the sme ARL 0, the different chrts re compred side by side to determine which signls quicker under different chnge mgnitudes. Thus, competing chrts re comprble using n pples to pples pproch. Tble. shows n exmple ARL comprison between the mgnitude robust nd the cumultive sum (CUSUM) chrts. The qulity chrcteristic in both chrts is distributed stndrd norml nd the results re for sudden men shift or out-of-control vlue of the men. The other detils of these chrts re unimportnt for this exmple nd re discussed in lter chpters. The top row of Tble. gives the men shift on the left nd the ARL performnce for ech chrt on the right corresponding to the prticulr men shift. Note the men shift equling 0.00 is the in-control ARL, ARL 0. The bottom row of the tble contins the control limit (CL) for ech chrt. Unlike the X chrt, these two chrts only hve one CL. Notice the CUSUM requires fewer observtions to detect for the.00 nd.50 men shift, but for ll other shifts, the mgnitude robust tkes fewer observtions to detect. Unless the bility to detect men shift of.00 or.50 is prticulrly import to the QE, the mgnitude robust chrt is the superior chrt in terms of ARL performnce. This type of evlution is used to compre mny types of chrts. 8

22 Tble.: ARL Comprison Exmple men shift mgnitude robust CUSUM (k=.5) *from: Pigntiello nd Simpson [].4 Multivrite vs. Multiple Univrite Chrts CL = 4.87 CL = 4.00 As systems become more complex, there is need to monitor more nd more vribles within system. The simplest nd most strightforwrd wy to ccomplish this is to use multiple univrite chrts nd stop when one of the chrts signls. Although this does provide qulity control, it does hve two mjor drwbcks. First, the ARL 0 of group of control chrts running in prllel is lower thn single chrt becuse when one chrt in the group signls, the whole process signls. This leds to the QE chsing down flse lrms more frequently. Consequently, either you hve to live with higher flse lrm rte or increse the UCL/LCL width nd scrifice quick detection. Both of these re highly undesirble. Secondly, cross-correltion between vribles is not considered when employing severl univrite control chrts simultneously. This is highly dubious omission becuse vribles of process re often correlted. Both problems re eliminted with multivrite control chrts..4. Shift Detection Differences 9

23 Previous studies show the generl superiority of multivrite chrts to multiple univrite chrts in terms of ARL performnce. Tke the multivrite CUSUM chrt by Pigntiello nd Runger [] versus the multiple univrite chrt by Woodll nd Ncube [8]. Pigntiello nd Runger show their chrt is more efficient thn the Woodll nd Ncube s even though the vribles re simulted s independent nd the covrince mtrix is the identity mtrix (see discussion below). As result, QE should not equte the performnce of multiple univrite chrt to its multivrite extension..4. Vrible Correltion To begin, suppose there re two vribles to keep in sttisticl control nd the incontrol men is zero, μ0 = [ 0,0 ]. If we ssume no correltion then the distnce from the center is: D e ( ) x x i (.) i = where the e stnds for the Eucliden distnce from the center, ech x i is the observtion of the i th vrible in the process nd x is the vector contining the x i vlues. Thus, the x vectors [ 0, ], [, 0 ],, nd, re ll equidistnt from μ 0 t distnce equl to one. These distnces re depicted s contour plot in Figure.3. 0

24 x x Figure.3: Eucliden Distnce. If x nd x re correlted by the covrince mtrix Mhlnobis distnce:.5 Σ=.5, then one must use the D m = x-μ Σ x-μ (.) ( ) ( ) 0 0 Note D equls D when e m Σ is the identity mtrix. In this two-vrible illustrtion, D m simplifies to x + x + x x. When grphed, we obtin the series of ellipsoids in Figure.4.

25 x x Figure.4: Mhlnobis Distnce Since the vlues re correlted, chnges with like direction hve greter distnce, nd chnges in opposite directions hve shorter distnce. Clerly, if you ssume Figure.3 is true in your chrting sttistic when Figure.4 is relity, then your ARL 0 will vry depending on the men shift direction. This is known s directionlly vrint chrt. However if the covrince is included, then the correct distnce is clculted regrdless of the men shift direction. This mens the chrt hs the desirble property of directionl invrince with stble, constnt ARL 0. The min disdvntge of using multivrite chrt is the difficulty in finding cuslity when the chrt signls. Nmely, the difficulty is in pinpointing the exct out-ofcontrol vrible(s) even with n ccurte chnge point estimtor. To get round this problem, most QE s run the univrite chrts in prllel. Then when the multivrite chrt signls, he/she cn look t the univrite chrts to find the vrible(s) cusing the signl.

26 .5 Problem Definition Currently, the two multivrite chrts with the best ARL performnce re the multivrite CUSUM (MC) developed by Pigntiello nd Runger [] nd the multivrite exponentilly weighted moving verge (MEWMA) chrt developed by Lowry et l. [8] (described in Sections 3.7. nd 3.7.). Both chrts re true multivrite chrts becuse they consider cross-correltion by incorporting the covrince mtrix. However, the min limittion of these chrts is they require tuning to specific men shift with the MC or nrrow rnge of men shifts with the MEWMA. Furthermore, neither pper gives n estimte for the chnge point. The problem becomes to develop true multivrite chrt robust to wide rnge of men shifts, nd once the chrt signls, provide n estimte for the unknown chnge point..6 Reserch Objectives nd Assumptions The objectives of this reserch re s follows:. Derive the multivrite mgnitude robust chrt (MMRC) by extending the univrite mgnitude robust chrt (UMRC) developed by Pigntiello nd Simpson [] into the multivrite relm.. Using the method of mximum likelihood estimtion (MLE), derive chnge point estimtors for the true chnge point nd out-of-control men vector in the MMRC. 3. Develop heuristic progrm nd regression eqution to provide estimtes for MMRC control limits. 3

27 4. Use Monte Crlo simultion nd the Reltive Men Index (RMI) to compre the MMRC to the MC developed by Pigntiello nd Runger [] nd MEWMA developed by Lowry et l. [8] in terms of ARL performnce. 5. Use Monte Crlo simultion to present nd evlute the performnce of the derived MMRC chnge point estimtor. This reserch effort will investigte the ARL performnce of the MMRC, MC nd MEWMA under the following ssumptions:. All simulted observtions re ssumed to be tken from the multivrite norml distribution nd hve known or properly estimted in-control men vector nd covrince mtrix.. The process is ssumed to hve sudden shift of the in-control men vector, lso clled step chnge. In other words, the process hs stedy in-control men vector from zero up to some point in time, τ, where the process men suddenly shifts to stedy out-of-control men vector nd remins this wy until corrected by the QE..7 Thesis Orgniztion This thesis is divided in to five chpters. Chpter I presented the history nd bckground for SPC with specil emphsis on control chrts. Next, the dded complexities of multivrite versus multiple univrite chrts were discussed nd the current problem for this thesis effort ws introduced. Chpter II will review relevnt literture in univrite control chrts, multivrite control chrts nd comprisons mong multiple chrts. Chpter III gives the mthemticl foundtion for the UMRC nd 4

28 MMRC, nd it discusses the MC developed by Pigntiello nd Runger [] nd the MEWMA developed by Lowry et l. [8]. Chpter IV presents the simultion model nd the results bsed on the mthemticl foundtion derived in Chpter III. Chpter V gives conclusions nd recommendtions for the QE nd presents some res for future reserch. 5

29 II. Literture Review. Introduction Most of the reserch flls into one of three ctegories. The first ctegory is univrite SPC where process is ssumed to only hve one vrible. These re the ppers forming the bsis of ll SPC nd s result, the univrite cse is well studied in mny top journls (Technometrics, Journl of Qulity Technology, etc.). The next ctegory is the extension of univrite SPC into the multivrite relm. Often there re multiple pproches for single extension. For exmple, both Crosier [] nd Pigntiello nd Runger [] developed multivrite extension to the CUSUM control chrt. Finlly there re published rticles summrizing nd compring current multivrite pproches to process monitoring. In these ppers, uthors pick prticulr SPC topic such s ARL performnce nd compre newer SPC reserch with older more estblished SPC reserch. This chpter reviews ll three ctegories of reserch in chronologicl order within ech ctegory.. Univrite Control Chrts Shewhrt [6] published three pge pper on the desire of mnufcturer to crete uniform items free of vrition. Even if the mnufcturer cn crete uniform product initilly, chnges in wether, people, rw mteril, etc. cn vry the uniformity of the product. To rectify this problem, he suggested setting tolernce specifictions within L stndrd devitions round specified trget men. Additionlly, he stted without proof, one could clculte the probbility of producing unit within these tolernce specifictions bout the men. 6

30 After this probbility is discussed, he stted, A constnt system of cuses lwys exists fter cuses which re function of time hve been eliminted. In other words, he suggested once ll of the non-rndom fctors re ccounted for, the process follows known probbility distribution. Once this distribution is found, goodness of fit test could be used t ech smple observtion to test whether this observtion follows the known distribution. This sttisticl test tests two hypotheses. Under the null hypothesis the observtions follow the known probbility distribution indicting n in-control process, nd under the lterntive hypothesis the observtions do not follow the known probbility distribution indicting n out-of-control process. If the test fils the null hypothesis, then the process is considered out-of-control nd the QE must find the cuslity. Ultimtely, this pper ws the first to suggest pplying sttisticl techniques to qulity control. After six yers, Shewhrt [7] completed book on the why nd how of sttisticl process control. Hlf of the book is his motivtion for qulity control. The rest of the book is dedicted to his control chrt, now known s the Shewhrt X control chrt. Here Shewhrt hd the chrt signl when the most recent observtion or men of simultneous observtions went beyond specified number of stndrd devition units from specified smple men. This simplifiction llowed non-sttisticin to dminister nd interpret the control chrt nd report when the chrt signls. In 954, Pge [9] sked nd nswered the question: Cn the previous observtions help us crete better chrt? In his pper, Pge estblished new univrite control crt clled the cumultive sum (CUSUM), nd then lter refined it (Pge [0]). 7

31 Insted of relying solely on the most recent observtion, Pge s chrt used the following two-sided cumultive sum scheme t + mx ( S + t,0) St = xi k where ( ) to detect positive out-of-control shifts of the men nd ( t ) min,0 i= S where S = ( x + k) to detect negtive out-of-control shifts of the men. Here S t nd S + t t t i= i re the CUSUM sttistics t time t, k is tuning prmeter tuned to the men shift the QE wnts to detect, nd x i is the i th observtion of the qulity chrcteristic with. If either S t < h or S + t > h occurs, the chrt signls indicting possible out-of-control process. Once the chrt signls, the chnge point estimtor is the lst time the chrting sttistic ws zero. Tuned properly, the stndrd norml CUSUM (i.e. k = 0.50, h = 4) chrt detects smller men shifts more quickly thn the Shewhrt X control chrt, but is slower to detect lrge men shifts of two stndrd devitions or greter. Unfortuntely, most of the time the QE will hve no knowledge bout this men shift. As result, if one incorrectly clibrtes the CUSUM, then the smll shift detection cpbilities cn become degrded. Even with these shortcomings, the CUSUM is widely regrded s significnt dvncement in the field of control chrts. Roberts [3] lter cme up with different pproch to using previous dt, the geometric moving verge chrt. This pproch is so clled becuse he used geometric moving verge to develop the chrting sttistic: ( ) ( ) ( ) Z r = rx + r Z r (.) i i i 8

32 where r is the mount between zero nd one nd the i th (i = {0,, n}) subgroup men, X, is weighted ginst the previous clcultion,. Furthermore, i Z i Z i 0 Z is the incontrol men, n is the most recent observtion nd the chrt signls when the first exceeds the CL. Note the Shewhrt X chrt is specil cse of the EWMA chrt when r =. To illustrte the chrt s effectiveness with regrds to chnge detection, Roberts studied ARL performnce for different vlues of r. While completely different pproch thn the CUSUM, this chrt is well known to provide strikingly similr performnce. Over time, the chrt s nme chnged to the exponentilly weighted moving verge (EWMA) chrt, nd the chrt is commonly used tody. The fourth mjor chrt is the UMRC by Pigntiello nd Simpson []. Unlike the CUSUM nd EWMA, the UMRC does not use weighted or cumultive sum in its clcultion. Insted, it is bsed on the fmily of generlized likelihood-rtio (GLR) tests. Z n The GLR uses the rtio of the likelihood under H over the likelihood under H 0 where H sttes the process is in-control nd H 0 sttes the process is out of control. If this GLR test exceeds some defined threshold, H 0 is rejected. The term generlized in the GLR mens some of the input vribles re unknown nd estimted using MLE. For SPC control chrts, these unknown input vribles re usully the in- nd out-of-control vrince nd men nd the chnge point. The pper by Hwkins, Qiu nd Kng [] (reviewed lter in this chpter) ssumes the in- nd out-of-control vrince nd men nd the chnge point re unknown. However, the UMRC is less generl becuse Pigntiello nd Simpson ssumed only the out-of-control men nd the chnge re unknown. Additionlly, they included chnge point model into their chrt. The chnge point model ssumes the 9

33 process men is in-control up to point, denoted by τ, when the process men shifts some time between τ nd τ + with τ + s the first observtion of the out-of-control process. In their pper, Pigntiello nd Simpson use Monte Crlo simultion to compre the ARL performnce of the UMRC to the CUSUM. They show the ARL performnce of the UMRC is superior ny one CUSUM chrt over ll tested men shift mgnitudes even though n ccurtely tuned CUSUM chrt is superior to the UMRC. Additionlly, Pigntiello nd Simpson s chrt lso provides the MLE for the time of the step chnge. This is in contrst to the X where the qulity control engineer hs to look t the chrt nd tke n educted guess on when the chnge occurred. Hving n estimte for the process chnge point sves expensive down time s result of serching for nd correcting the source of the process chnge. The min disdvntge to the UMRC is the incresed level of computtion becuse the log-likelihood-rtio sttistic is evluted for ll possible chnge points. However, simple computer progrm on modest computer cn esily perform the required clcultions. The UMRC is discussed in Section 3.. Using generlized likelihood-rtios, Hwkins, Qiu nd Kng [] proposed univrite unknown prmeter chnge point model. Like Pigntiello nd Simpson [], they ssumed chnge point model. However, the difference is the process in-control nd out-of-control mens nd the process vrince re ssumed unknown by Hwkins, Qiu nd Kng. As result, the trditionl sequence of plugging Phse I vlues into Phse II process monitoring control chrt is not needed. In fct, they stted monitoring cn begin on the third observtion. However, they pointed out three observtions will contin lot of vrince nd is not enough dt to vlidte the normlity ssumptions the 0

34 chrt is bsed on. They recommended running the process until the vrince stbilizes nd the normlity ssumption is verifible. Thus, they presented their control chrt s semless trnsition from Phse I to Phse II. To test the null hypothesis the process is in-control versus the lterntive hypothesis the process is out-of-control, Hwkins, Qiu, nd Kng used two-smple t- test sttistic, denoted s T j, n, where j is the cndidte chnge point nd n is the most recent observtion. Then T j, n is mximized over ll possible j from one to n - to obtin the mximum seprtion between the in-control nd out-of-control mens, nd this mximum is denoted s. The chrt signls when exceeds the pproprite CL, T mx, n T mx, n denoted by hn, where the vlue of h n is dependent on the vlue of n. Since there ws no closed-form solution for h n, the pper provided both closed-form pproximtion nd tbles obtined through simultion for h n. Once the chrt signls, the j mximizing the MLE for the true chnge point. T j, n is To counter the need for tuning in the EWMA chrt developed by Roberts [3], Hn nd Tsung [3] proposed the generlized EWMA control chrt (GEWMA). This ws ccomplished by modifying the EWMA to: TGE ( c) = inf n : mx Wn c k n k Here TGE ( c) represents the time the chrt signls for the infinium of the n th observtion where the bsolute vlue of chrting sttistic, Wn, mximized over ll vlues of k (k k = {,,,n}) exceeds the CL c. Wn k is defined s

35 ( / k ) Wn = W / ( / ) n, k n k k k Wn = Xn + Wn k k k k where ( / k ) ( k) / k / n is the stndrd devition of Wn k. This Wn k is the sme eqution s Robert s EMWA in (.) except Hn nd Tsung only considered individul observtions, not subgroup mens. In their pper, Hn nd Tsung used Monte Crlo simultion to show the ARL performnce of the GEWMA is superior to the optiml EWMA developed by Wu [9] for ll men shifts nd the CUSUM developed by Pge [9] for ny men shift not between the intervl (0.784δ,.3798δ ) where δ is the predicted men shift. Essentilly, the uthors took the EWMA nd mde it robust to wider rry of potentil men shifts..3 Multivrite Control Chrts Hotelling [5] developed the first multivrite chrt in 947 to improve the testing of bombsights. Using Shewhrt s X control chrt s bsis, Hotelling modified the chrt to llow for vector of observtions. Additionlly, he recognized the possibility of correlted qulity mesures, nd therefore included covrince mtrix into his T sttistic. The use of T in the T sttistic is significnt becuse it the squre of the studentt sttistic. Like the X control chrt, the T chrt signled when the most recent T clcultion went beyond some user specified upper or lower control limit. The drwbcks to the T control chrt were twofold. First off, it suffered the sme poor ARL

36 performnce for detecting smll shifts s the univrite X control chrt. Secondly, the T sttistic ws more computtionlly intensive with mtrix multipliction, mtrix inversion, nd the required development of the covrince mtrix. Although now trivil issue with computers, these computtionl issues were big hindrnce to dpttion in 947. In 985, Woodll nd Ncube [8] creted the first multivrite CUSUM control chrt entitled the MCUSUM chrt. Although clled MCUSUM, the chrt ctully consists of multiple univrite CUSUM chrts run simultneously. Ech of these CUSUM chrts independently monitors ech vrible within process. As the process runs, the first of these independent CUSUM chrts to signl cuses the MCUSUM to signl. Using Mrkov chin pproch, they showed the MCUSUM is often superior to Hotelling s T chrt when the monitored qulity chrcteristic is bivrite norml rndom vrible. The obvious disdvntge of Woodll nd Ncube s chrt is the more univrite CUSUMs implemented, the worse the ARL performnce becomes. Additionlly, the MCUSUM chrt suffers the sme tuning issues s the univrite CUSUM, nd cnnot incorporte correltion between process vribles. Most importntly, Woodll nd Ncube s rticle strted renewed interest in multivrite chrts. Inspired by Woodll nd Ncube, Crosier [] creted two true multivrite CUSUM chrts. The first chrt tkes the squre root of Hotelling s T sttistic nd used this to generte univrite CUSUM chrt. He entitles this the COT or CUSUM of T. The second sttistic clcultes directly from the vector of observtions. Unfortuntely, Crosier uses limited number of runs ( 400) in his Monte Crlo studies to come up with 3

37 his results. Despite this, Crosier showed the direct multivrite CUSUM chrt yields superior ARL performnce reltive to both the COT nd MCUSUM. Around the sme time, Pigntiello nd Runger [] creted their own pir of multivrite CUSUM chrts, entitled the MC nd MC. Although they took different pproch thn Crosier [], Pigntiello nd Runger discovered the MC produces similr results to Crosier s direct multivrite CUSUM. Unlike Crosier, Pigntiello nd Runger decided to employ run size of 6,000 in their Monte Crlo experiments to generte more ccurte results. The MC chrt, while better thn Hotelling s T chrt, produces inferior ARL performnce to Woodll nd Ncube [8]. Additionlly, the pper contins lengthy discussion on the subject of directionl invrince, detiling how directionl invrince mintins consistent ARLs, but hinders dignosis efforts when the chrt signls. The MC chrting sttistic is discussed lter in Section After the multivrite CUSUMs were developed by Woodll nd Ncube [8], Crosier [] nd Pigntiello nd Runger [], Lowry et l. [8] extended the EWMA into the multivrite EWMA, denoted by them s the MEWMA. Unlike the multivrite CUSUM schemes, the MEWMA is direct extension from the univrite cse. By incorporting the covrince mtrix into its chrting sttistic clcultion, the MEWMA llowed for correltion mong vribles in process. Lowry et l. ctully creted two chrts bsed on the derivtion of the covrince mtrix. One pproch clculted the exct covrince, while the other pproch used the stedy-stte covrince. Of the two vrince clcultions, Lowry et l. showed in their results the superiority of using the exct covrince mtrix with regrds to ARL performnce. Like Pigntiello nd Runger, Lowry et l. used Monte Crlo simultion with the sme run size of 6,000 to evlute 4

38 the ARL performnce. They concluded their chrt is superior, in terms of ARL performnce, to Hotelling s T nd the MCUSUM nd comprble to Crosier [] nd the MC. The MEWMA is discussed lter in Section An untested prmeter in n otherwise comprehensive pper ws their choice of the simulted chnge point. They stted the process is likely to be in-control for while nd then go out-of-control, yet they strted ll of their simultions s out-of-control from time zero. As this thesis shows in Section 4.7, the choice of control limit nd the resulting in-control ARL is dependent on when the chnge point ctully occurs. Runger nd Prbhu [4] (99) used Mrkov chin pproch to evlute the ARL of the MEWMA control chrt. Using symmetry nd orthogonl invrince, they generte results within 4% of Lowry et l. Since their pproch ws nlyticl, none of the error ssocited with the Monte Crlo simultions is present. Within the lst yer, Zmb nd Hwkins [0] developed multivrite extension to Hwkins, Qiu nd Kng []. Like Hwkins, Qiu nd Kng, they mde no ssumptions bout in- nd out-of-control men vlues nd vrince. The method proposed by these uthors essentilly eliminted the Phse I study required by trditionl pproches. In order to ccomplish this, Zmb nd Hwkins split the n observtions into two subgroups. Then using mximum likelihood estimtion to find the mximum men distnce between ll possible pired subgroups, they clculted T sttistic. Let n equl the number of observtions, p equl the number of vribles nd α equl the specified flse lrm rte. If the T sttistic is beyond specified hnp,, α, then the chrt signls. Differing from other chrts, hnp,, α chnges over time s function of n. Since there is no 5

39 closed-form solution for the control limits hnp,, α, for n =,,, Zmb nd Hwkins obtined these vlues using simultion. While this chrt could redily replce other chrts, the uthors pointed out the existence of situtions where complete Phse I study is necessry nd the use of their chrt s supplement. In ddition to men shift detection, their model lso gve n estimte of the chnge point τ. Overll, this pper ws welcome ddition to multivrite sttisticl process control, which this thesis is specil cse of Zmb nd Hwkins pper..4 Overview Evlutions For univrite chrts, Li [6] provided theoreticl overview of chnge point estimtion in-control chrts developed up to 995. The lck of built-in chnge point estimtor in the X nd other chrts presented mjor chllenge. Li surveyed ll ttempts up to publiction nd combined them into unified theory. While Li s purpose ws to present nd prove his theory, he presented n extensive bibliogrphy nd discussion of existing chrts. In 995, Lowry nd Montgomery [7] presented review of the stte of the rt in multivrite control chrting. Their pper includes thorough overview of the forementioned multivrite ppers with the exception of Zmb nd Hwkins [0], since their pper ws not yet published. Some discussion is provided on ech control chrt from n implementtion point of view. In conducting their review, Lowry nd Montgomery pointed out couple of res for future reserch. The first is the difficulty in interpreting out-of-control signls, nd the second is the lck of multivrite chrts for 6

40 monitoring uto correlted dt. Like Lowry nd Montgomery, this thesis will compre the ARL performnce of the multivrite mgnitude robust chrt to the MC nd MEWMA..5 Conclusion This chpter presented n overview of some relevnt pst nd present ppers in SPC control chrts. For univrite chrts, both the cnonicl chrts like Pge s CUSUM nd Robert s EWMA nd more recent dvnces like the GEWMA developed by Hn nd Tsung were reviewed. Moreover, the review showed the reserch evolution from considering only the most recent observtion in Shewhrt s X control chrt to using likelihood-rtio test in Pigntiello nd Simpson s mgnitude robust control chrt. Then these univrite chrts were extended into multivrite spce with the exception of Pigntiello nd Simpson s control chrt. Zmb nd Hwkins proposed generlized pproch to the mgnitude robust in the multivrite relm, but they do not consider the cse where the in-control men nd covrince mtrix is known. As result, this thesis extends the univrite mgnitude robust control chrt of Pigntiello nd Simpson, which is specil cse of Zmb nd Hwkin s method. Severl of these control chrts re interrelted. First off, Zmb nd Hwkin s pper is generliztion of the pper by Hwkins, Qiu, nd Kng. Pigntiello nd Simpson s pper is specil cse of the method described in Hwkins, Qiu, nd Kng s pper. Agin, this thesis is specil cse of the method developed by Zmb nd Hwkins. Additionlly, Pigntiello nd Simpson s pper is specil cse of this thesis. 7

41 III. Methodology 3. Introduction This thesis will fill the gp between the Pigntiello nd Simpson [] UMRC nd the Zmb nd Hwkins [0] unknown prmeter chnge point chrt. In this chpter, the UMRC is directly extended into the multivrite relm. Unlike Zmb nd Hwkins, this chrt will ssume known in-control men nd covrince mtrix, which requires Phse I study. In other words, the MMRC is specil cse of the Zmb nd Hwkins method. To develop the MMRC, the UMRC is derived first nd then followed by the derivtion of the MMRC. In the interest of brevity, these derivtions will leve out some intermedite steps. For those interested in the lgebric detils of the derivtion, refer to Appendix A. These derivtions re highly similr becuse they both involve likelihood-rtio test nd MLE. After deriving both chrts, single run of the MMRC is presented nd explined. To wrp up the MMRC, sections 3.5 nd 3.6 will del with the issue of specifying the vlue for the control limit (CL) nd the hndling of flse lrms. Since the ARL performnce of the MMRC is compred to the ARL performnce of the MC nd MEWMA in Chpter IV, both the MC nd MEWMA re discussed in this chpter. Finlly, the RMI by Hn nd Tsung [4] is introduced. RMI ids the QE in determining the chrt possessing superior ARL performnce over rnge of men shift mgnitudes. 8

42 3. The Univrite Mgnitude Robust Chrt The UMRC is bsed on chnge point model nd the likelihood-rtio test nd ws developed by Pigntiello nd Simpson []. This mens the process is ssumed incontrol up until the chnge point τ where the process hs sudden shift in the men between τ nd τ +. As result, the first observtion smpled from the chnged process is obtined t τ +. Throughout this derivtion nd the derivtion in Section 3.3, the index vrible denoted by t lwys refers to discrete point in time rnging from one to the most recent observtion T. Moreover, the index vrible c is cndidte chnge point rnging from zero to the observtion T -. The likelihood-rtio is used to test the hypotheses H μ = μ for t T 0 : t 0 H : μt = μ0 for t T nd μt = μ for τ + t T (3.) where μ 0 is the in-control men nd μ is the out-of-control men. Both μ nd τ re ssumed unknown. Additionlly, the UMRC ssumes the observtions re independent nd normlly distributed. To begin, ssume the rndom vrible x represents observtions from the process. These observtions re normlly distributed with the density given by f x πσ σ x ( ; μσ, ) = exp ( μ) (3.) where μ nd σ re the prmeter men nd stndrd devition respectively. Next, the likelihood function is formed under the distribution specified by H 0 : T L0 x t μ0 t= πσ σ x x ( x) = exp ( ) 9

43 As you cn see x nd σ from the Eqution (3.) were replced with prmeter estimtes x t nd σ x. The br bove the x represents the possibility of using subgroup verges t ech t point in time. In other words, over time you should hve vector of mens [ x x x ] x =,,..., T up to the most recent observtion T. The likelihood function under the distribution specified by H is given by L τ T x t 0 t= πσ σx t x = τ + πσ σ t x x ( τ ) = exp ( x μ ) exp ( x μ ) Tking the rtio of to L : L 0 L L ( τ x) ( x) 0 T exp t= τ + σ = T exp t= τ + σ x x ( x μ ) t ( x μ ) t 0 (3.3) From this point one could theoreticlly use the rtio to L s defined. However, nturl log trnsformtion simplifies the mth: R ( τ x) ( τ x) ( x) L 0 L T T = loge = ( xt μ0 ) ( ) L0 σ xt μ (3.4) x t= τ+ t= τ+ Clerly, R is greter when the subgroup x t is closer to the lterntive men T μ due to n increse in ( xt μ0 ). Thus, greter R indictes we re more likely to reject H 0. t= τ + Even with likelihood-rtio test in plce, the problem of finding estimtes for nd τ still exist. Although the true vlues for μ nd τ re ssumed unknown, they re estimble. The MLE for μ given τ is μ 30

44 ˆ μ T = x τ = x t T. (3.5) ( τ) T, τ t = τ + Essentilly ˆ μ ( τ ) is the overll men for the T τ most recent subgroups. Substituting xt, τ into (3.4) nd simplifying: This derivtion is the MLE for μ ( ) T ( ) ( ) T ( ) R τ x = x t μ0 xt, τ σ x T x t= τ+ t= τ+ T τ = ( x ) T, τ μ0 (3.6) σ ˆ x τ nd its use further simplifies R. The tsk now is to mximize Eqution (3.6) by evluting it over ll possible vlues of c: Here ˆ T c x. (3.7) ( τ ) ( μ ) RT = R = mx x T, c 0 0 c< T σ x τ is the vlue of τ mximizing ( ) R τ x in Eqution (3.6). Eqution (3.7) is equivlent to the originl function in (3.3) with log trnsformtion nd n estimte for μ mximized over the entire rnge of τ. When you implement R T s control chrt, the process is ssumed in-control until R T exceeds some CL B ( R T > B). At this point, H 0 is rejected in fvor of H from the hypotheses in (3.). Assuming the chrt hs signled, n estimte for τ is needed. This estimte is the sme ˆ τ from (3.7). When computed seprtely, Next we substitute ˆ τ for τ into (3.5): ˆ rg mx T c τ = ( x ), μ 0. 0 c< T σ Tc (3.8) x 3

45 ˆ μ T T ˆ τ = x = x t. (3.9) ( ˆ τ) ˆ T, τ σ x t = ˆ τ + Both point estimtes, ˆ τ nd ˆ μ, re mximum likelihood estimtors for τ nd μ. Currently, no closed-form solution for finding B exists. Pigntiello nd Simpson use Monte Crlo simultion to obtin in-control ARL vlues for specific levels of B in Tble 3.. B ARL Tble 3.: ARL vlues from Pigntiello nd Simpson [], Tble 4 Applying log trnsformtion nd using ordinry lest squres on the dt shown in Tble 3. genertes regression estimte for B bsed on desired ARL 0 : Bˆ log ARL e 0 = (3.0) By inputting n ARL 0 rnging from to , UMRC implementer cn obtin n pproprite CL estimte, ˆB. Implementing the UMRC is firly strightforwrd. First, the QE needs to select n pproprite B vlue from Eqution (3.0). Then s observtions come in, compute R T using Eqution (3.7). Once the chrt signls the vlue of τ mximizing R T becomes ˆ τ nd is equivlent to Eqution (3.8). As result, there is no need to explicitly clculte 3

46 (3.8). Now input ˆ τ into Eqution (3.9) to obtin n MLE for ˆ μ given ˆ τ. Repet this process t every chrt signl nd the chrt is fully implemented. 3.3 The Multivrite Mgnitude Robust Chrt Hving derived the UMRC, this chrt is now extended to the multivrite cse using similr process with p vribles where τ, c, t nd T re s previously defined. Agin, multivrite norml distribution is ssumed for rndom vrible vector x with the following probbility density function f ' x; μ, S = exp p ( ) ( ) x μ S x μ S ( ) ( π ) (3.) where x = x, x,..., x p is by p vector of observtions, μ = μ, μ,..., μ p is by p vector of mens nd S is p by p covrince mtrix. In the literture, S is generlly symbolized by Σ, but to reduce confusion with the summtion symbol, S is used insted. The MMRC hs the sme hypothesis test s (3.); however, prmeter vectors re considered insted of sclrs: H : 0 μt = μ 0 for t T H : μ = μ 0 for t τ nd μ t = μ for τ + t T (3.) t Like the UMRC, the MMRC employs chnge point model nd likelihood-rtio test to derive the test sttistic. Using (3.), the likelihood function under H 0 is L S T 0 ( X) = exp p xt μ0 S xt μ0 t= ( π ) ( ) ( ) where X is the mtrix contining the x vectors. Under H, the likelihood function is t 33

47 L τ = p t 0 t 0 t= ( τ ) Now tke the rtio of to L : ( π ) ( ) ( ), μ X exp x μ S x μ S T exp ( ) p ( xt μ S xt μ) t= τ + ( π ) S L 0 L ( τ, μ X ) L ( X ) 0 = T t= τ + T t= τ + ( π ) ( π ) exp t t S p p ( x μ ) S ( x μ ) exp t t S ( x μ ) S ( x μ ) 0 0 (3.3) Eqution (3.3) is the multivrite equivlent of (3.3). Likewise, this function is simplified using log trnsformtion: R L X x 0 T e t t t t L t= τ + ( τ ) = log = ( μ ) S ( x μ ) ( x μ ) S ( x μ ) ( ) T T = T τ t t ( T τ) μ S μ μ S x + μ S x μ S μ t= τ+ t= τ+ Agin, n estimte for for μ is needed. In order to obtin mximum likelihood estimtor μ, given ny τ, we tke the prtil derivtive of R with respect to Setting this equl to zero yields: T t ( T τ ) t= τ + μ : R = S x S μ. (3.4) μ T t ( T τ ) t= τ + S x S μ = 0 τ t = τ + T μˆ ( τ ) = xt = x T, τ. (3.5) T Thus x T,τ is the MLE for μ given τ. Substituting x T,τ in for μ in R nd simplifying: 34

48 T τ ( τ ) = ( 0 T, ) ( 0 T, ) X μ x S μ x. (3.6) R τ τ As with Eqution (3.7), the MMRC chrting sttistic is found by mximizing R over ll possible vlues of c: where ˆ R = R τ X = μ x S μ x (3.7) R τ X in Eqution (3.6). When implemented, T c ( ) mx ( ) 0 Tc ( 0 Tc) mx,, 0 c< T τ is the vlue of τ mximizing ( ) the process is ssumed in-control until R mx exceeds specified CL B (R mx > B). Section 4.6 gives regression bsed pproch for estimting B for specific ARL 0 vlues. Once the chrt signls, H 0 is rejected in fvor of H from the hypotheses in (3.). After the chrt signls, n estimte for τ is computed using: ( 0, ) ( 0, ) T c ˆ τ = rg mx Tc Tc 0 c< T μ x S μ x (3.8) Moreover, this ˆ τ is the sme ˆ τ from Eqution (3.7). Substituting ˆ we obtin the MLE for μ ˆ given ˆ τ : τ t = ˆ τ + τ for τ in ˆ ( τ ) μ, T μˆ ( ˆ τ ) = xt = x T, ˆ τ (3.9) T ˆ where both point estimtes, ˆ τ nd ˆ μ, re mximum likelihood estimtors for τ nd The MMRC implementtion is bsiclly identicl to the UMRC. Both chrts clculte the mximum R vlue over ll possible τ chnge points. After signling, the τ mximizing R mx becomes ˆ τ nd is n input to μ ˆ to generte the MLE for μ ˆ given ˆ τ. This process is repeted fter every chrt signl. The min difference between the μ. 35

49 MMRC nd the UMRC is the dded complexity in clculting S - nd the subsequent mtrix multipliction in Eqution (3.7). 3.4 MMRC Exmple Figure 3. shows grphicl exmple of ten vrible MMRC chrt with τ = 50 nd n observtion mtrix X pulled from stndrd multivrite norml distribution. The men shift δ = [.474,.474,...,.474] corresponds with vector norm distnce of.5. The top chrt shows R plotted over t tking ten observtions to signl fter τ = 50. The mx blue line corresponds to the CL, B = 4.75, clibrted to ARL 0 = 00. For the bottom chrt, insted of chrting ten individul μ ˆ vectors, the men of ech μ ˆ vector is chrted over time. This works becuse the verge of μ 0 nd μ is known nd equl to 0 nd.474 respectfully. As you cn see, μ ˆ hovers round δ fter the second out-ofcontrol observtion until the chrt signls. However, this chrt is for demonstrtion purposes only becuse in prctice μ 0 nd μ re unknown. Lstly, using the MLE for ˆ τ in Function (3.8), we find the estimte is coincidentlly equl to the ctul known vlue for τ. 36

50 0 5 R mx Time, t ˆ τ = 50 Men of μ Vector Time, t τ =, Figure 3.: Single run MMRC chrt with 50 p = 0 nd men shift δ = [.474,.474,...,.474] 3.5 Finding MMRC Control Limits While the MMRC elimintes the need to tune the control chrt for specific step chnge mgnitude, one still needs wy to find CLs, B. To develop heuristic for estimting B, the fct higher B vlue results in higher ARL 0 ws utilized. The pseudocode in Figure 3. shows how the process works. The code strts with trget ARL 0 vlue with n error of % on ech side. In the cse of Figure 3. this equtes to desired error rge for ARL 0 of 303 to 97. Essentilly, the heuristic uses loop to generte cndidte ARL 0 vlues from the MMRC simultion. A rnge is generted using ech cndidte ARL 0 vlue plus/minus the stndrd error, nd then this rnge mtched up to the specified error rnge. 37

51 To mtch the desired nd MMRC simultion error rnges, the heuristic uses four conditions lbeled (), (), (3) nd (4) in Figure 3.. The progrm strts with run size, denoted by N, equl to ten, the CL B, denoted by B, equl to one nd B increment, denoted s B_shift, equl to one. Condition (3) keeps incrementing B by B_shift until the simultion returns cndidte ARL 0 within the desired upper nd lower trget rnge (Condition ()) or the cndidte ARL 0 is within one stndrd error of the trget (Condition ()). Typiclly Condition () is met first, nd B_shift is then divided by three nd N is multiplied by ten. For resons of computtionl speed, Condition () trunctes the heuristic if the size of N is greter thn 0,000. If B is incremented greter thn one stndrd error from the trget, then Condition (4) decrements B by B_shift/3. This synchronous incrementing nd decrementing of B gurntees the sme B vlue is not used twice within ech run of the heuristic. The heuristic ends when either the desired rnge in (Condition ()) is met or Condition () trunctes becuse B is greter thn 0,

52 trget = 300; %trget vlue upper = trget + trget *.0; %upper error limit lower = trget + trget *.0; %lower error limit B=; %control limit B_shift = ; %mount to chnge B by N = 0; %number of simultion runs while done == flse get ARL, se {stndrd error} from control chrt; () if ARL plus/minus se within upper nd lower error bounds then %h found, so stop progrm done = true; () else if rnge of ARL + se AND ARL - se contin trget %nrrow focus B_shift = B_shift/3; %increse number of simultion runs N = N * 0; if (N > 0000) then %0000 runs yields smll se, so stop for %computtionl nd time resons done = true; end if (3) else if ARL + se <= trget then %increse h B = B + B_shift; (4) else if ARL - se > trget then %decrese h by smller mount B = B - B_shift/4; end if end while Figure 3.: Find B Heuristic Psuedocode In prctice, depending on the trget ARL 0 nd p, the heuristic tkes two to eight hours to run on.8 GHz Pentium 4 processor running Mtlb 7. As one might expect, the higher ARL 0 nd p input into the heuristic, the longer it tkes to converge. Figure 3.3 shows single execution of the find B heuristic with p = 0 nd trget ARL 0 =

53 5.47 h B 0 5 ARL Number of Runs, N Number of Runs, N ARL + se ARL ARL - se Figure 3.3: Find B Heuristic Execution for Trget ARL = 300 The top grph shows the chnge in B nd the bottom grph shows the chnge in ARL nd stndrd error from the MMRC simultion. Red lines indicte the finl vlues for B nd ARL 0. While B does not seem to vry much once it reches round 5, the ARL is highly sensitive (in terms of vribility) to the vlue of B nd the number of runs. You cn see on the bottom grph the effect of over-shooting ARL 0 nd then bcktrcking to the trget vlue. This prticulr exmple truncted with rnge of versus the desired rnge. Even though this is not the exct rnge desired, this heuristic produces highly ccurte estimtes nd is more thn dequte for the purpose of this thesis. 40

54 3.6 Flse Alrms For resons of consistency nd fir comprison, this thesis will follow the conventions of Pigntiello nd Simpson [] for the simultion modeling of flse lrms. A flse lrm occurs when τ > 0 nd the chrt signls t ny time t, where t τ. If flse lrm occurs, then the control chrt is zeroed out nd the process is restrted t time t +. However, the step chnge will still occur t time τ with τ + s the first out-ofcontrol vlue. This models the rel world where n investigtive study finds true flse lrm. In the rel world, finding flse lrm indictes the process is truly in-control, the process is then restrted t time t + without chnging the process inputs. Furthermore, this rel world flse lrm would not impct the ctul but unknown chnge point. For exmple, let τ = 50 nd the chrt signls t t = 30. This signl is considered flse lrm becuse t τ. The control chrt is then zeroed out nd restrted s if the next observtion ws the first. However insted of 50 in-control observtions, the control chrt hs only 0 observtions until the step chnge occurs t τ = 50. Thus on the st observtion, step chnge is first recorded on the restrted control chrt. 3.7 MC nd MEWMA Chrting Sttistics Since Chpter 4 will compre the MMRC with the MC nd MEWMA, it is necessry to show nd discuss these control chrting strtegies prior to discussing results MC The MC ws developed by Pigntiello nd Runger []. It is bsed on the cumultive sum 4

55 where t i= t nt + ( ) Ct = Xi μ 0 (3.0) X i is the observtion vector t time i, μ 0 is the in-control men vector, n t n + if MC > 0 if otherwise t t (3.) = nd the chrting sttistic is - { kn } MC = mx C S C t,0. (3.) t t t Here S represents the covrince mtrix nd k is the tuning prmeter used to dmpen the Mhlnobis distnce - Ct S C t t time t. If the MC exceeds the CL h, then the chrt signls. Furthermore, k corresponds to one-hlf of the Eucliden distnce the QE wishes to detect. For exmple, tke the two vrible cse with = {, } X X X. Suppose the.5.5 / expected shift in X is {.5,.5}. The resulting vlue of k is ( ) + ( ) = Like the univrite CUSUM, the MC renews itself (i.e. zeros out) periodiclly when the process is in-control. This is controlled in the MC by the counter vrible n t in (3.), nd n t = 0 upon strtup. For more informtion on the MC see Pigntiello nd Runger [] MEWMA The MEWMA developed by Lowry et l. [8] is nturl extension of the EMWA. The MEWMA chrting sttistic is 4

56 T Z S Z t (3.3) - t = t Zt where T t is the MEWMA sttistic t the t th observtion nd - S Z t is the weighted covrince mtrix defined by either (3.6) or (3.7). When T t exceeds the control limit h4, the chrt signls. Note the subscript of 4 in h 4 hs no other significnce other thn to differentite h 4 from other CLs using h, such s the MC. Eqution (3.3) is vrition of Hotelling s T sttistic with the replcement of Z for X. Z t is clculted s Z = RX + ( I R) Z. (3.4) t t t Here R is digonl mtrix of weights greter thn zero nd less thn one used to clibrte the chrt nd I is the identity mtrix. The vlue of Z t t strtup is zero. If r = r =... = rp = r in R, then (3.4) is Zt = rxt + ( r) Z t. (3.5) The lst clcultion required is the weighted covrince mtrix - S Z t of the stndrd covrince mtrix S. Assuming the weights re equl, this mtrix is clculted one of two wys. Lowry et l. [8] found n eqution to find the exct or ctul covrince mtrix bsed on the t th observtion: ( r) t r S Z = S. (3.6) t r The other method is to ssume the chrt is fully wrmed up nd in stedy-stte. This stedy stte covrince mtrix is r SZ = S. (3.7) t r 43

57 Note, in either ssumption when r =, we re left with Hotelling s T chrt, which is specil cse of the MEWMA. Figure 3.4 illustrtes the convergence of the ctul weight to the stedy stte weight. As the grph shows, the two converge on nd observtion weight i Actul Covrince Mtrix Weight Stedy Stte Covrince Mtrix Weight Figure 3.4: Actul vs. Stedy Stte Covrince Mtrix Weights For more informtion on the MEWMA, consult the pper by Lowry et l. [8]. 3.8 Reltive Men Index When conducting n ARL comprison of control chrts, some chrts hve different detection cpbilities t different men shift mgnitudes, denoted by δ. For exmple, consider two control chrts, sy, Chrt nd Chrt. Chrt shows superior performnce when δ, but poor performnce when δ >. Chrt performs in n opposite fshion with poor performnce when δ, but superior performnce when δ >. The question becomes: Which chrt is better? To nswer this question, the RMI developed by Hn nd Tsung [4] is used. The RMI is simple weighted verge of the compred chrts for desired rnge of δ. The clcultion of the RMI score is: ( ) RMI c * ( ) ARL n ARL c δ δ i i =, δ 0 * i > (3.8) n i= ARLδ i 44

58 where c is the desired chrt, δ i is the i th men shift mgnitude, ARL( c) δ is the ARL t i chrt c nd δ i nd * ARL δ is the smllest ARL over ll chrts t i i δ. This gives single vlue of the chrt s overll detection performnce, nd the lower the RMI score, the better the performnce. An RMI of zero indictes the chrt hs the quickest out-ofcontrol detection of the entire rnge of tested men shifts. Note, if n inpproprite rnge of δ were chosen, sy [ 0.,0.,0.3,0.4,0.5,.0,.0 ], then this rnge of δ would skew the RMI towrd chrts with superior detection performnce for smll δ. Overll, when relistic δ rnge is chosen, the RMI is n excellent tool when conducting n ARL performnce comprison of control chrts. 3.9 Conclusion The MMRC is much needed ddition to the field of SPC. By deriving nd then extending the UMRC into the multivrite relm, mgnitude robust control chrt for multivrite men shift detection ws developed. Mgnitude robust mens the QE no longer hs to clibrte the chrt to specific men shift mgnitude. This ws ccomplished using log-likelihood-rtio nd MLE to test for n out-of-control process. Another significnt dvntge of this method is the MMRC outputs n MLE for the ctul chnge point τ, denoted by ˆ τ. Use of ˆ τ cn significntly reduce QE s serch for cuslity when the chrt signls. This results in less down time nd llows for greter productivity for the process. Furthermore, the MMRC is identicl to the UMRC when p =, nd therefore cn supplnt the UMRC. 45

59 In ddition, simultion-bsed serch heuristic ws presented to find the control limit B for the MMRC. This heuristic llows QE to quickly (within few hours) obtin n estimte for B. Next, the hndling of flse lrms within simultion ws discussed. Essentilly, once flse lrm is detected the control chrt resets itself, but the chnge point is not ltered. This models the rel world where flse lrm is independent of the ctul chnge point. Finlly, this chpter presented the competing chrts (MC nd MEWMA) nd the RMI, method to compre ARL performnce of these chrts with the MMRC. The MC nd MEWMA represent the current stte of the rt in multivrite control chrts. As result, successful ARL performnce comprison of the MMRC with the MC nd MEWMA is criticl to its cceptnce in the rel world. For these comprisons, the RMI ws used to mesure control chrt performnce. 46

60 IV. Results 4. Introduction In Chpter III, control chrt using chnge point model nd likelihood-rtios ws derived. In this chpter, the results of this chrt, the MMRC, re compred to the MC nd MEWMA. Note Hotelling s T is not directly considered becuse it is specil cse of the MEWMA. This comprison ws ccomplished through ARL evlutions over severl tuning prmeters nd two chnge points. To evlute the ARL performnce of the MMRC, MC nd MEWMA, Monte Crlo simultion is used. This simultion will llow the QE to specify the ctul chnge point, control limit, in-control men vector, out-of-control men vector, covrince mtrix, tuning prmeter (MC nd MEWMA) nd the run size or number of simulted runs to collect. All of these inputs give the QE gret flexibility to evlute the MMRC, MC, nd MEWMA. For outputs, the simultion gives the overll verge ARL nd stndrd error, nd for the MMRC, it gives the verge estimte for the chnge point. Although there is n infinite combintion of input prmeters for the simultion, this reserch will focus only on select few to nswer three questions. These three questions ll refer to the comprison of the ARL performnce mong the MMRC, MC, nd MEWMA. First off, Wht effect does the chnge point position hve? To nswer this question, the results will be split between those where the process is simulted s outof-control from the beginning nd simulted s in-control for 50 observtions nd then suddenly shifting out-of-control. Within these results the second nd third questions re sked: Wht effect does the number of out-of-control vribles in the out-of-control 47

61 vector hve? nd Wht effect does the number of vribles in the process hve? With regrd to the number of out-of-control vribles question, two situtions will be considered: one where the one vrible in the out-of-control men vector shifts nd nother where ll vribles in the out-of-control men shift. Finlly, three different vrible sizes will nswer the lst question. In ddition to the ARL performnce comprison, regression will be used to estimte vlues of B in the MMRC. These vlues will llow for n ARL 0 rnging from 50 to 300 with up to 0 vribles. Next, the robustness of the MMRC chnge point estimtor is evluted. These estimtors will be pulled from ech of the ARL performnce simultions nd compred to the known simulted chnge point. Finlly, preliminry results from this reserch effort hve shown the MEWMA hs different in-control ARL vlues when the process is out-of-control from the beginning versus when the process is initilly in-control nd then suddenly shifts out-ofcontrol some time lter. Since this phenomenon could dversely effect the ARL performnce evlution, this chpter will reserch it in depth. 4. ARL Performnce Simultion Implementtion Although control chrts re progrmmble on mny different lnguges nd sttisticl pckges, Mtlb 7 ws selected for its rpid coding cpbilities nd wide vriety of built-in sttisticl functions. However, in this section, pseudocode is used insted of Mtlb code in order to ese understnding. The gret dvntge to progrmming the simultion model used for evluting ARL performnce of control 48

62 chrts is the ctul code is very similr from chrt to chrt. The only rel difference is clculting the chrting sttistic. As result, the sme code is slightly modified to clculte ARL performnce results for the MMRC, MC nd MEWMA. Moreover, since the simultion code is quite long, we will brek it up into three prts: simultion inputs, min simultion loop, nd the single run simultion of the chrt itself. 4.. Simultion Inputs Figure 4. shows the stndrd simultion inputs for multivrite control chrt. The min difference between the chrts is Hotelling s T nd MMRC do not need the tuning prmeter k. Vribles tu nd delt re the induced nd therefore they re the known chnge point nd step chnge from mu0. If you re using stndrdized dt, mu0 is column vector of zeros from where the number of vribles, p, is equl to mu0 s row dimension. % B = upper control limit % tu = chnge point % N = number of times to run the simultion % mu0 = in-control verge vector % delt = step chnge vector % sigm = covrince mtrix % k = tuning prmeter (for non-mmrc chrts) Figure 4.: Simultion Inputs 4.. Min Simultion Loop The loop in Figure 4. is responsible for running the simultion N times where N is the run size of the simultion. As one cn see, there re four internl vribles here nd three output vribles. At the strt of ech run the chrt hs not signled, thus chrt_signls is flse before going into the single run simultion. The other two input 49

63 vribles re used to hndle flse lrms. The vrible t_const dvnces monitors the time when the ctul shift occurs while t is reset to zero when flse lrms occur. All three of these re fed into the single run simultion, which outputs the sum of ARL, the sum of ARL, nd the estimte for the chnge point, tu_ht. These vlues re used to clculte the verge ARL, stndrd error, nd tu_ht. Note tu_ht is only used with the MMRC nd not the MC or MEWMA. for i = to N %outer simultion loop chrt_signls = flse; %boolen test for loop termintion t = 0; %counter reset fter flse lrms t_const = 0; %counter not reset fter flse lrms end get sum of ARL, squred ARL nd tu_ht from the single run _ simultion; clculte nd output verge ARL clculte nd output stndrd error clculte nd output verge tu_ht Figure 4.: Min Simultion Loop 4..3 Single Run Simultion The third component of the simultion is the single run simultion. This section in Figure 4.3 uses while loop to run until non-flse lrm is detected. To strt, both t nd t_const re incremented to the first observtions. The if then structure determines whether the p by t mtrix X receives n in or out-of-control rndom vector of observtions. This set of observtions is sent to control chrt function, such s R mx in (3.7), which outputs the chrting sttistic, chrt_stt, nd for the MMRC, the potentil chnge point estimte, cp. The next couple of if sttements determine the flse lrm sttus when the chrt signls. If flse lrm occurs, then X nd t re reset to zero nd the entire chrt is restrted. Otherwise the chrt truly signls, chrt_signls is set to 50

64 true nd cp becomes the ctul chnge point estimte, tu_ht. See Section 3.6 for more informtion on flse lrms. At the bottom, we tlly the ARL nd ARL used in Figure 4.. This informtion is then sent bck to the min simultion loop in Section 4... while chrt_signls == flse t = t + ; t_const = t_const + ; end end if t_const <= tu then %if process in-control X(row of obs.,col t) = multivrite norml with mu0; else %if process out-of-control X(row of obs.,col t) = multivrite norml with mu0 _ + delt; end get chrt_stt, cp from the pproprite function; if (chrt_stt > B) nd (t_const <= tu) %flse lrm occured zero out X; t = 0; end if (t_stt > B) nd (t_const > tu) %chrt signls chrt_signls = true; tu_ht = cp; end sum the ARL; sum the ARL^; Figure 4.3: Single Run Simultion 4.3 ARL Performnce (τ = 0) In their ppers, both Pigntiello nd Runger [] nd Lowry nd Montgomery [7] provide CL vlues for the MC nd MEWMA respectively. These vlues re recomputed using the method in section 3.5 s both n error check nd to compenste for different pseudorndom number genertors. The proceeding ARL performnce evlution will use these recomputed CL vlues. 5

65 Assuming the process ws out-of-control from the beginning, n evlution of the ARL performnce ws conducted using Monte Crlo simultion to compre the ARL performnces of the MMRC with those of the MC nd MEWMA. The run size for ech ARL simultion ws 0,000. Additionlly, the covrince mtrix ws ssumed equivlent to the identity mtrix (i.e. process vribles ssumed to be uncorrelted). This study will strt with n RMI summry cross ll tuning prmeter settings contined in Tbles 4. through 4.7. After this RMI comprison, the simulted ARL vlues re presented with their corresponding stndrd errors. The ARL performnce is evluted when p = {,3,0}. Furthermore, two different types of men shifts re considered: one where single vrible in the process suddenly shifts ( μ = [ δ,0,0,...,0]) nd nother where ll vribles in the process suddenly shift simultneously with identicl mgnitude ( = [ δ, δδ,,..., δ] ) μ RMI Comprison Tble 4. gives summry of the RMI scores contined in Tbles B. through B.6 from Appendix B. The RMI compres the ARL performnce of the MMRC, MC, nd MEWMA (both using the ctul covrince mtrix nd the stedy stte covrince mtrix) cross the rnge of tested chnge mgnitudes. These chnge mgnitudes re the sme s those in the D e columns from TblesB. through B.6 (see Section 4.3. for n explntion). The top row of Tble 4. gives the chrt type, the tuning prmeter (if needed), the RMI vlues when p = {,3,0} for = [ ] when p = {,3,0} for = [,,,..., ] μ δ δδ δ. μ δ,0,0,...,0 nd the RMI vlues 5

66 Here lower RMI vlues equte to better ARL performnce nd higher RMI vlues equte to lesser ARL performnce over the rnge of tested chnge mgnitudes. An RMI of zero indictes this chrt nd tuning prmeter ws superior over ll chnge points for given number of vribles p nd the prticulr μ shift. Tble 4.: RMI Summry for τ = 0 from Tbles B. through B.6 μ = (δ,0,0,,0) μ = (δ,δ,δ,,δ) Type Tuning Prmeter p = p = 3 p = 0 p = p = 3 p = 0 MMRC MC MEWMA Actul Covrince Mtrix MEWMA Stedy- Stte Covrince Mtrix k = k = k = r = r = r = r = r = r = r = r = r = r = Clerly, the MEWMA using the ctul covrince mtrix nd tuning prmeter of r = 0.05 is superior to ll other chrts due to its ner zero RMI score. Moreover, ll RMI scores under the MEWMA stedy-stte covrince mtrix clcultion re inferior to the MEWMA ctul covrince mtrix, MC, nd MMRC RMI scores. Likewise, the vlues of r t 0.5 nd 0.8 lso hve reltively poor performnce to the MC nd MMRC. As result, Section 4.3. will eliminte these dt points in the interest of brevity nd defer the complete tbles to Appendix B. However, the RMI vlues re not recomputed for these bbrevited tbles to mintin consistency with the Appendix B tbles Tble Comprison In this section, Tbles 4. through 4.4 ssume = [ ] through 4.7 ssume μ = [,,,..., ] μ δ,0,0,...,0, nd Tbles 4.5 δ δδ δ. Although there exists n infinite number of wys 53

67 to define μ, using extremes mkes interpoltion esier nd gives good look t ech chrt s performnce. The tbles re rrnged with the estimted ARL on top nd the corresponding stndrd error in prentheses long the bottom. The heder row is orgnized left to right with D e s the Eucliden distnce from the centroid (Eqution(.)), δ s the individul men shift contined within μ, the MMRC column, the MC CUSUM columns with tuning prmeter settings k = { 0.5,0.50,.00} nd the MEWMA using the ctul covrince mtrix (Eqution (3.6)) with tuning prmeter settings r = { 0.05,0.0,0.5}. The second to the bottom row contins the CL used, nd the bottom row presents the RMI vlues. 54

68 Tble 4.: τ = 0, p =, μ = (δ,0) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.88) (.05) (.0) (.) (.08) (.96) (0.69) (0.54) (0.87) (.8) (0.56) (0.7) (0.8) (0.) (0.5) (0.6) (0.55) (0.7) (0.0) (0.5) (0.) (0.07) (0.0) (0.3) (0.08) (0.09) (0.) (0.06) (0.04) (0.05) (0.) (0.05) (0.05) (0.06) (0.04) (0.03) (0.03) (0.06) (0.03) (0.03) (0.04) (0.03) (0.0) (0.0) (0.03) (0.0) (0.0) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 6.66 h = 7.5 h = 4.78 h =.69 h4 = 7.7 h4 = 8.79 h4 = 9.36 RMI: In Tble 4., the MEWMA with r =.05 is superior to both the MMRC nd MC for every mgnitude of chnge tested. The distinction between the MMRC nd MC is less cler. As typicl with most CUSUM bsed chrts, the MC hs difficulty detecting lrge shifts unless it is tuned with lrge k vlue, which, in turn, increse the number of observtions to detect smll shifts. As expected, n exctly tuned MC t k = { 0.5,0.50,.00} corresponding to { 0.50,.00,.00} δ = redily outperforms the MMRC. However, considering the entire rnge of possible shifts, the MMRC is superior to the MC over ll three tuning prmeters with n RMI =

69 Tble 4.3: τ = 0, p = 3, μ = (δ,0,0) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.96) (.89) (.99) (.98) (.7) (.04) (.00) (0.74) (0.59) (0.96) (.37) (0.63) (0.8) (0.9) (0.4) (0.7) (0.9) (0.6) (0.9) (0.4) (0.9) (0.) (0.07) (0.) (0.6) (0.09) (0.0) (0.) (0.07) (0.04) (0.05) (0.) (0.05) (0.06) (0.06) (0.04) (0.03) (0.03) (0.06) (0.03) (0.04) (0.04) (0.03) (0.0) (0.0) (0.04) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 7.94 h = 8.79 h = 5.55 h = 3.5 h4 = 9.8 h4 = 0.99 h4 =.57 RMI: As p increses, the number of smples required to detect one vrible shift increses, especilly under smller mgnitude shift. Otherwise, the sme conclusion for Tble 4. holds for Tble

70 Tble 4.4: τ = 0, p = 0, μ = (δ,0,0,,0) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.06) (.05) (.94) (.6) (.08) (.0) (0.94) (0.78) (.9) (.6) (0.9) (.8) (.8) (0.33) (0.7) (0.4) (0.89) (0.8) (0.39) (0.50) (0.5) (0.08) (0.3) (0.39) (0.3) (0.6) (0.) (0.09) (0.05) (0.06) (0.7) (0.07) (0.08) (0.0) (0.06) (0.03) (0.03) (0.08) (0.05) (0.05) (0.06) (0.04) (0.0) (0.0) (0.04) (0.03) (0.04) (0.04) (0.03) (0.0) (0.0) (0.0) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 4.75 h = 5.33 h = 9.58 h = 5.53 h4 =.4 h4 =.98 h4 = 3.7 RMI: Agin, s p increses to ten, the number of smples required to detect one vrible shift increses. However, this smple increse ppers nonliner nd decreses s p increses for ll chrts nd tuning prmeters. For exmple the MMRC t δ =.5 goes from 87. (p = ) to (p = 3) nd then to 8.79 (p = 0). This is positive result if one hs mny vribles within their process. 57

71 Tble 4.5: τ = 0, p =, μ = (δ,δ) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.88) (.05) (.0) (.7) (.0) (.0) (0.68) (0.55) (0.87) (.8) (0.56) (0.70) (0.83) (0.) (0.5) (0.5) (0.55) (0.6) (0.) (0.5) (0.) (0.07) (0.0) (0.3) (0.08) (0.09) (0.0) (0.06) (0.04) (0.05) (0.) (0.05) (0.05) (0.06) (0.04) (0.03) (0.03) (0.06) (0.03) (0.03) (0.04) (0.03) (0.0) (0.0) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 6.66 h = 7.5 h = 4.78 h =.69 h4 = 7.7 h4 = 8.79 h4 = 9.36 RMI: Here in Tble 4.5, ll process vrible mens hve shifted insted of simply one. In order to mintin the sme Eucliden distnce from the centroid, ech δ is clculted s follows: De = pδ δ = D e p where δ is the men shift contined in μ. For exmple, with p = t distnce D e = results in totl shift of δ = =.35. Interestingly, this does not seem to hve n impct on the ARL detection cpbilities when compred to the single shift μ becuse 58

72 the Eucliden distnces re the sme. In fct, this tble is lmost indistinguishble from Tble 4.. Tble 4.6: τ = 0, p = 3, μ = (δ,δ,δ) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.96) (.89) (.99) (.98) (.8) (.09) (.0) (0.75) (0.6) (0.95) (.4) (0.6) (0.78) (0.97) (0.5) (0.7) (0.9) (0.6) (0.9) (0.4) (0.30) (0.) (0.07) (0.) (0.6) (0.09) (0.0) (0.) (0.07) (0.04) (0.05) (0.) (0.05) (0.06) (0.06) (0.04) (0.03) (0.03) (0.06) (0.03) (0.04) (0.04) (0.03) (0.0) (0.0) (0.04) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 7.94 h = 8.79 h = 5.55 h = 3.5 h4 = 9.8 h4 = 0.99 h4 =.57 RMI: Likewise, Tble 4.6 nd Tble 4.7 require more smples to detect when dditionl vribles re dded, nd hve similr performnce to Tble 4.3 nd Tble

73 Tble 4.7: τ = 0, p = 0, μ = (δ,δ,δ,,δ) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.06) (.05) (.94) (.9) (.07) (.03) (0.94) (0.8) (.0) (.6) (0.9) (.7) (.9) (0.33) (0.7) (0.4) (0.89) (0.8) (0.38) (0.50) (0.6) (0.08) (0.3) (0.4) (0.3) (0.6) (0.) (0.09) (0.05) (0.06) (0.8) (0.07) (0.08) (0.0) (0.06) (0.03) (0.03) (0.08) (0.05) (0.05) (0.06) (0.04) (0.0) (0.0) (0.04) (0.03) (0.04) (0.04) (0.03) (0.0) (0.0) (0.03) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 4.75 h = 5.33 h = 9.58 h = 5.53 h4 =.4 h4 =.98 h4 = 3.7 RMI: ARL Performnce (τ = 50) At τ = 50, the MEWMA is no longer the superior chrt nd is outperformed by the MMRC. Furthermore, the sme underlying ssumptions from Section 4.3 hold in this section with the exception of τ = 50 insted of τ = RMI Summry Tble 4.8 is nother RMI summry similr to Tble 4. from Section 4.3. except τ = 50 nd the scores re from Tbles B.7 through B.. Moreover, the sme ssumptions from Section 4.3. pply here s well. 60

74 Tble 4.8: RMI Summry for τ = 50 from Tbles B.7 through B. μ = (δ,0,0,,0) μ = (δ,δ,δ,,δ) Type Tuning Prmeter p = p = 3 p = 0 p = p = 3 p = 0 MMRC MC MEWMA Actul Covrinc e Mtrix MEWMA Stedy- Stte Covrinc e Mtrix k = k = k = r = r = r = r = r = r = r = r = r = r = Looking t the MEWMA ctul covrince mtrix with r = 0.05, we see this chrt nd tuning prmeter is no longer superior in terms of the RMI score. In fct, mong ll of the MEWMA nd MC tuning prmeters, the MEWMA with r = 0.5 under the ctul covrince mtrix hs lower RMI score for p = {3,0}. Only the MEWMA stedy stte covrince mtrix with r = 0.5 nd the MC with k =.00 hve lower RMI score thn the ctul covrince mtrix MEWMA with r = 0.5 when p =. Even then the RMI scores only hve mximum difference of 0.0 between them. Regrdless, the MMRC possesses the lowest RMI score with vlues less thn or equl to 0.09 for ll p = {,3,0} nd both configurtions of μ. The strk contrst of the MMRC RMI scores versus the MC nd MEWMA RMI scores is due to the nrrow rnge of potentil men shift mgnitudes where the MC nd MEWMA hve quick detection nd the fct the chrt is in-control for 50 observtions. This is shown in Tbles B.7 through B. nd discussed in Section Tble Comprison 6

75 In this section, Tbles 4.9 through 4. ssume μ = [ ] 4. through 4.4 ssume μ = [,,,..., ] δ,0,0,...,0, nd Tbles δ δδ δ. Agin, for the sme resons in Section 4.3., Tbles 4.9 through 4.4 re bbrevited from Tbles B.7 through B. in Appendix B. Tble 4.9: τ = 50, p =, μ = (δ,0) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.88) (.05) (.0) (.33) (.) (.04) (0.67) (0.57) (0.88) (.9) (0.60) (0.73) (0.83) (0.) (0.7) (0.6) (0.54) (0.8) (0.) (0.6) (0.0) (0.08) (0.0) (0.3) (0.09) (0.09) (0.0) (0.06) (0.05) (0.06) (0.0) (0.06) (0.05) (0.06) (0.04) (0.04) (0.04) (0.06) (0.04) (0.04) (0.04) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.0) (0.03) (0.0) (0.0) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 6.66 h = 7.5 h = 4.78 h =.69 h4 = 7.86 h4 = 8.86 h4 = 9.39 RMI: Tble 4.9 illustrtes when the MC nd MEWMA re ppropritely tuned they perform much better thn the MMRC t detecting smll shifts. However, when considering the entire rnge of shifts, the MMRC is superior s indicted by the RMI score. When compred with Tble 4., the performnce of both the MC nd the 6

76 MEWMA decreses s D e increses while the MMRC performnce improves cross the vlues of D e. Tble 4.0: τ = 50, p = 3, μ = (δ,0,0) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.96) (.89) (.99) (.98) (.30) (.8) (.0) (0.74) (0.6) (0.96) (.40) (0.65) (0.85) (0.97) (0.3) (0.9) (0.9) (0.63) (0.0) (0.5) (0.30) (0.) (0.0) (0.) (0.6) (0.0) (0.) (0.) (0.07) (0.06) (0.06) (0.) (0.06) (0.06) (0.06) (0.04) (0.05) (0.04) (0.06) (0.05) (0.04) (0.04) (0.03) (0.04) (0.03) (0.04) (0.04) (0.03) (0.03) (0.0) (0.03) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 7.94 h = 8.79 h = 5.55 h = 3.5 h4 = 9.97 h4 =. h4 =.6 RMI: Tbles 4.0 through 4.4 t τ = 50 re similr to tbles 4.3 through 4.7 t τ = 0. Note the difference between Tbles 4.9 through 4. nd Tbles 4. through 4.4 is negligible when the only chnge is the out-of-control men vector conclude the choice of μ = [ ] or = [,,,..., ] δ,0,0,...,0 μ. As such, once cn μ δ δδ δ hs no effect on ARL performnce. Furthermore, the increse of p shows the sme tpering decrese in ARL performnce seen in the previous section for ll chrts nd tuning prmeters. 63

77 There re two min differences between the τ = 50 nd τ = 0 cses. The first is the consistently lower RMI of the MMRC to the other two chrts. The other difference is s p increses, the properly tuned MC requires more observtions to detect thn the MMRC. For exmple, in Tble 4., the MC ARL vlues for k = { 0.5,0.50,.00} corresponding to D = { 0.50,.00,.00} re { } e 48.30,8.,6.4 versus { 48.09,5.86,5. } for the MMRC. However, Tbles 4.9 through 4.4 continue to show properly tuned MEWMA requires fewer observtions to detect thn the MMRC over nrrow rnge of D e men shift mgnitudes. Overll, when the ARL performnce is tken over the entire rnge of tested D e men shift mgnitudes nd τ = 50, the MMRC is clerly the better chrt. 64

78 Tble 4.: τ = 50, p = 0, μ = (δ,0,0,,0) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.06) (.05) (.94) (.58) (.9) (.08) (0.9) (0.9) (.) (.57) (.0) (.3) (.37) (0.3) (0.3) (0.43) (0.88) (0.3) (0.43) (0.53) (0.5) (0.6) (0.8) (0.39) (0.5) (0.7) (0.) (0.09) (0.) (0.0) (0.8) (0.09) (0.09) (0.0) (0.06) (0.08) (0.07) (0.09) (0.06) (0.06) (0.06) (0.04) (0.06) (0.05) (0.06) (0.05) (0.04) (0.04) (0.03) (0.05) (0.04) (0.04) (0.04) (0.03) (0.03) (0.0) (0.04) (0.04) (0.03) (0.03) (0.03) (0.0) (0.0) (0.04) (0.03) (0.0) (0.03) (0.0) (0.0) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.0) B = 4.75 h = 5.33 h = 9.58 h = 5.53 h4 =.97 h4 = 3.3 h4 = 3.89 RMI:

79 Tble 4.: τ = 50, p =, μ = (δ,δ) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.88) (.05) (.0) (.8) (.3) (.03) (.94) (0.55) (0.86) (.7) (0.59) (0.73) (0.83) (0.67) (0.7) (0.6) (0.54) (0.8) (0.) (0.6) (0.) (0.08) (0.0) (0.3) (0.09) (0.09) (0.) (0.0) (0.05) (0.06) (0.) (0.06) (0.05) (0.06) (0.06) (0.04) (0.04) (0.06) (0.04) (0.04) (0.04) (0.04) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) (0.0) (0.0) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 6.66 h = 7.5 h = 4.78 h =.69 h4 = 7.86 h4 = 8.86 h4 = 9.39 RMI:

80 Tble 4.3: τ = 50, p = 3, μ = (δ,δ,δ) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.96) (.89) (.99) (.98) (.4) (.7) (.04) (0.74) (0.6) (0.96) (.40) (0.67) (0.83) (0.94) (0.4) (0.9) (0.9) (0.63) (0.0) (0.4) (0.30) (0.) (0.0) (0.) (0.6) (0.0) (0.) (0.) (0.06) (0.06) (0.06) (0.) (0.06) (0.06) (0.06) (0.04) (0.05) (0.04) (0.06) (0.05) (0.04) (0.04) (0.03) (0.04) (0.03) (0.04) (0.04) (0.03) (0.03) (0.0) (0.03) (0.0) (0.03) (0.03) (0.0) (0.0) (0.0) (0.03) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) B = 7.94 h = 8.79 h = 5.55 h = 3.5 h4 = 9.97 h4 =. h4 =.6 RMI:

81 Tble 4.4: τ = 50, p = 0, μ = (δ,δ,δ,,δ) MC CUSUM MEWMA Actul Covrince Mtrix Distnce D e δ MMRC k = 0.5 k = 0.50 k =.00 r = 0.05 r = 0. r = (.9) (.06) (.05) (.94) (.47) (.30) (.3) (0.94) (0.9) (.) (.6) (.00) (.6) (.34) (0.3) (0.3) (0.43) (0.88) (0.3) (0.4) (0.5) (0.5) (0.7) (0.8) (0.4) (0.5) (0.7) (0.) (0.09) (0.) (0.0) (0.8) (0.09) (0.09) (0.0) (0.06) (0.08) (0.07) (0.09) (0.06) (0.06) (0.06) (0.04) (0.06) (0.05) (0.06) (0.05) (0.04) (0.04) (0.03) (0.05) (0.04) (0.04) (0.04) (0.03) (0.03) (0.0) (0.04) (0.04) (0.03) (0.03) (0.03) (0.0) (0.0) (0.04) (0.03) (0.03) (0.03) (0.0) (0.0) (0.0) (0.03) (0.03) (0.0) (0.03) (0.0) (0.0) B = 4.75 h = 5.33 h = 9.58 h = 5.53 h4 =.97 h4 = 3.3 h4 = 3.89 RMI: MMRC Chnge Point Performnce Here the verges vlues of the MMRC s built-in chnge point estimtor, ˆ τ, tken re displyed in Tbles 4.5 nd 4.6. These ˆ τ verges were obtined during the ARL performnce simultions in Sections 4.3 nd 4.4 using the methodology in Section 4.. The top two rows give the configurtion for the number of vribles, p, nd the-outof control men, μ, used. The left most column of numbers corresponds to rnge of men shifts D e = {0.5,0.5,.0,,5}. The lower-right 0 by 6 mtrix gives the simulted verge vlues for ˆ τ given specific p, μ nd D e. 68

82 Tble 4.5: Chnge Point Estimtor Performnce when τ = 0 D e p μ (δ,0) (δ,0,0) (δ,0,,0) (δ,δ) (δ,δ,δ) (δ,δ,,δ) In Tble 4.5, when the process is out-of-control from the beginning, the MMRC hs incresed bis when detecting smller shifts less thn one. Also, the more vribles there re the less ccurte the chnge point becomes. Essentilly, ˆ τ is less bised when p is smll nd D e is lrge. Tble 4.6: Chnge Point Estimtor Performnce when τ = 50 D e p μ (δ,0) (δ,0,0) (δ,0,,0) (δ,δ) (δ,δ,δ) (δ,δ,,δ)

83 For Tble 4.6, when τ = 50, the verge vlue of ˆ τ is exct for D e from.5 to 5.00 nd hs slight positive bis for 0.75 to.00. Although this bis is lessened when τ = 0, the chrt still hs substntil positive bis under smll men shift chnges. Agin, ˆ τ is less bised when p is smll nd D e is lrge. 4.6 MMRC CL (B) Regression Anlysis Like the UMRC, regression nlysis ws used to provide B vlues resulting in estimted ARL 0 vlues from 50 to 300 nd p from to 0. Using the heuristic in Section 3.5, Figure 4.4 shows 3D plot of the simulted B vlues. The figure shows input ARL 0 on the x-xis, input number of fctors p on the y-xis, nd simulted response B on the z- xis. Estimtes for B were simulted t ll ARL 0 = {50,00,50,00,50,300} versus p = {,,3,,0} combintions for totl of 60 B estimtes. 3D Surfce of B Vlues B Number of Fctors p ARL Figure 4.4: Simulted B Vlues Clerly, n increse in either ARL 0 or p requires lrger B vlue. In fct, the plot resembles rising ridge in RSM nlysis. As result, the following model is postulted: 70

84 ( ) Bˆ x, x = c+ x + x + x + x + x x. (4.) Here x nd x re ARL 0 nd p respectively. Using ordinry lest squres, the following function ws computed: ( ) Bˆ x, x = x +.3 x.368 E x.338 x +.48E x x (4.) 4 3 When plotted, Eqution (4.) results in Figure 4.5 nd the bsolute difference between the simultion nd regression pproximtion is documented in Tble 4.7. Figure 4.5: Regressed ˆB Vlues Tble 4.7: Absolute Difference Between B nd ˆB p ARL While this pproximtion ppers reltively ccurte, the result is meningless without checking the normlity ssumptions. The dignostic plots in Figure 4.6 nd Figure 4.7 reveled no serious violtion(s) of the normlity ssumption. 7