Diane K. Michelson, SAS Institute Inc, Cary, NC Annie Dudley Zangi, SAS Institute Inc, Cary, NC

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1 ABSTRACT Paper DK-02 SPC Daa Visualizaion of Seasonal and Financial Daa Using JMP Diane K. Michelson, SAS Insiue Inc, Cary, NC Annie Dudley Zangi, SAS Insiue Inc, Cary, NC JMP Sofware offers many ypes of Saisical Process Conrol (SPC) chars, including Shewhar, Cusum, and Moving Average chars. SPC char feaures in JMP can be accessed hrough he menus and dialogs, scriping, and now in version 10.0, hrough a drag and drop inerface. The periodic naure of some financial daa makes i unsuiable in is original form for deecing anomalies using a SPC char. One viable mehod for presening his daa on a SPC char is o apply ime series echniques firs, hen char he oupu. This paper invesigaes wo case sudies applying hese echniques. SPC chars work very well under he ideal condiions of daa independence and normaliy. SPC has radiionally been used in manufacuring, where hese condiions are ofen saisfied. However, SPC is beginning o be used more ouside of manufacuring, in areas like insurance claims processing, banking, healh care, and survey research. In many of hese environmens, he desired mean may be shifing up or down, or he responses may be cyclic in naure. In his paper, we examine some of he problems wih ploing ime series daa on conrol chars and sugges remedies. BACKGROUND Saisical Process Conrol (SPC) is a mehodology for monioring a ime series process o deec shifs in eiher he locaion or scale of he process. SPC has been used for almos 100 years in manufacuring. In he recen pas, i has become more prevalen in oher indusries, including finance, healh care, insurance, and survey research. (see Robers, 2006). Daa for conrol chars is colleced in subgroups. When he subgroup size is one, an Individual-Moving Range (X- MR) char is used o conrol he process. An X-MR conrol scheme consiss of wo relaed run chars. An Individual (or X) char plos he daa in ime order, along wih conrol limis. Changes in he mean are deeced when any poin is ouside of he conrol limis. The absolue value of successive differences he moving ranges are ploed on an MR char, along wih conrol limis o deec changes in he process spread. Figure 1 is an example of a ypical X-MR char commonly seen in manufacuring. When he subgroup size is greaer han one, Xbar-S conrol schemes are used. The Xbar char plos he mean of he subgroup; he S char plos he sandard deviaion of he subgroup. The upper conrol limi and lower conrol limi are drawn a a disance of hree sandard deviaions ( 3 ) of he esimae of he parameer from he cenerline. Tha is, we assume he process has been sable for long enough o esimae wihou error, and draw conrol limis a 3. Conrol limis are calculaed based on hisoric subgroup mean and sandard deviaion values. (see Wheeler (2010)) 1

2 Figure 1. Typical X-MR char. One of he assumpions in using conrol chars is ha he daa come from an independen process. Tha is, knowing he value of he ime series a any poin gives no informaion abou fuure rends of he process, excep for he locaion and scale. In manufacuring, his assumpion can ofen be me by appropriae choice of subgrouping. For example, if daa is highly auocorrelaed, sampling can be done less frequenly. Financial or oher business daa is ofen colleced daily, weekly or monhly and he sampling plan canno be changed. Using radiional conrol chars on posiively auocorrelaed daa causes false signals. (see Michelson, 1994) One remedy is o model he auocorrelaion using auoregressive-moving average (ARMA) ime series models. The residuals from he model should be uncorrelaed and hus are suiable for displaying wih a conrol char. If a shif occurs in he process mean a ime 0, he same shif will occur in he firs residual value afer 0. The residuals will no see he full shif afer ime 0, bu hey will see a dampened shif, leading o longer imes unil signal for posiively correlaed daa. In his paper, we give an inroducion o ARMA models hrough wo examples. In he firs, an equipmen company who had successfully used SPC chars in heir manufacuring deparmen in he pas was ineresed in monioring monhly revenue. Ofen revenue figures are cyclic in naure, dropping off a he beginning of each year and peaking in December. This company, however, produced elescopes and he cycles corresponded direcly o asronomy evens like eclipses, meeor showers and plane sighings. While hey could visually see he rend, weeks wih unusual paerns were masked by all he weeks riggering alarms, as radiional conrol charing mehods failed o work. In anoher company, conrol chars were being used on waer qualiy daa in a waer purificaion scheme. Ciy waer was pumped hrough a series of filers. A he end of he filering process, he qualiy of he waer was measured wice per second. The frequency of measuremen was imporan o caching diry waer before i arrived a processing equipmen, bu he resuling posiive serial correlaion led o an increase in he false alarm rae of he conrol char. EXAMPLE 1 Background 2

3 A elescope company who had successfully used SPC chars in heir manufacuring deparmen in he pas was ineresed in monioring revenue. Their revenue figures were cyclic in naure, peaking several weeks before asronomical aciviies (oal eclipses, meeor showers and planeary sighings), hen dropping during periods of lower aciviy. The cyclical naure mean ha no only did he daa have large swings raher han spikes, bu he swings gradually ascended and descended. Firs Aemp a Conrol Chars To graph he daa in JMP, Selec Analyze Qualiy and Process Conrol Char Builder Drag Revenue o he Y drop zone Drag Dae o he X region Figure 2. Individual and Moving Range char of weekly revenues Figure 2 shows en years of weekly oal revenue for he company. The swings were so large ha nearly 50% of he daa on he Individual char fell ouside he limis. Turning on he ess for poins ou of conrol would be meaningless for his char as so many of he poins would signal. The limis, on he oher hand, were narrow because he relaive change from week o week was ypically small, so he moving range char didn widen or deec many odd spikes. While hey could visually see he rend, weeks wih unusual paerns were masked by all he weeks riggering alarms as radiional conrol charing mehods failed o work. Look for Auocorrelaion using Lag and Bivariae To evaluae he possibiliy of auocorrelaion, look a he relaionship beween each consecuive poin. To do his in JMP: 3

4 Creae a new column wih he formula, Lag(:Revenue, 1). Selec Analyze Fi Y by X, using Revenue as he Y and Lag(Revenue) as he X. In he Bivariae repor, selec Densiy Ellipse 0.95 Table 1. Fi Y by X correlaion oupu In his oupu, we see ha he correlaion is high a 0.97 and significanly differen from zero, wih a p-value < From his es, we can rejec he hypohesis ha he lag of Revenue is uncorrelaed wih Revenue and conclude here is an auocorrelaion effec beween consecuive weeks. Applying a Time Series Model JMP offers many opions for modeling auocorrelaion hrough he Time Series plaform. To ry ou some modeling opions: Selec Analyze Modeling Time Series Selec Revenue as he Y, Time Series and Dae as he X, Time ID Click OK In he oupu window, selec ARIMA model group wih he inpus shown in Dialog 1: Dialog 1. Model Group dialog for specifying he ARIMA Model The resuls show ha here are a few models ha fi very well. The simples model is he AR(1) model, which has he lowes value of Schwarz s Bayesian Crierion (SBC), as well as a low value of Akaike s Informaion Crierion (AIC). Nex, selec oupu from he AR(1) model and selec Save Columns o save he residuals. Oupu 1. Time Series Model comparison oupu Second aemp a Conrol Chars Because he significan auocorrelaion has been removed from he residuals of he Time Series model, we can use hese residuals o idenify unusual daes on he Conrol Char. To run a Conrol Char using he residuals saved in a new daa able from he AR(1) model: Selec Analyze Qualiy and Process Conrol Char Builder Drag Residual Revenue o he Y-axis drop region 4

5 In Figure 3, we see mos of he poins are now in conrol and we can invesigae he few ha wen ou of conrol more carefully o find he special causes influencing hese daes. Figure 3. Individual and Moving Range char of Revenue Residuals by week In fac, we can now see he value on 3/20/2010 flags as an ou of conrol poin on he graph of residuals. In preparaion of he July 2010 Toal Solar Eclipse, he markeing deparmen began running adverisemens on 3/01/2010 in popular asronomy magazines. In Figure 3, he posiive change in revenue is much more clearly idenified han in he conrol char in Figure 1. The high residual on 12/10/2005 has a similar explanaion. A markeing campaign argeed oward he Toal Solar Eclipse of March, 2006 caused a spike in sales hree monhs prior. EXAMPLE 2 Background Many indusries, including semiconducor and pharmaceuical processing, as well as power generaion, rely on waer use for many of heir seps. Ciy waer is fed ino a faciliy hrough pipes. While he waer is safe for human consumpion, he pariculaes and organic compounds in he waer can cause all sors of problems in he manufacuring process. Waer is fed hrough filers and is subjeced o chemical and oher processing o reduce levels of conaminans o pars-per-billion or pars-per-rillion levels. The resuling waer is ofen called ulrapure waer (UPW). A a cerain semiconducor manufacurer, he coun of pariculaes and percen oal oxygenaed compounds (TOC) were measured before he waer was sen from he cenral uiliy building (afer filraion) o he uni process sep. An auomaed sysem was se up o signal he engineers if he couns ever wen over a specificaion limi. The process had improved o he poin ha engineers were ineresed in moving o a daa-based decision sysem, relying on conrol limis o ell hem when he process had changed, raher han using a specificaion limi for process conrol. Daa was colleced wice per second, due o he need o quickly signal when he pariculaes or TOC were above he 5

6 specificaion limi, so ha he conaminaed waer would no reach he processing equipmen. Due o he frequency of daa collecion, boh he pariculae coun daa and TOC daa ime series were highly posiively auocorrelaed. The engineers knew his posiive correlaion would cause an increase in false alarms on he Individual char. Time Series Modeling One hour s worh of daa was colleced for modeling purposes. There were 7200 observaions of TOC. Due o he proprieary naure of he daa, he TOC values in his paper were simulaed from he same model used in he applicaion. Figure 4. Individual and Moving Range char of Toal Oxygenaed Compounds (percen). Figure 4 conains an Individual and Moving Range char of he firs 100 observaions aken in 50 seconds. The posiive serial correlaion has caused wo signals on he Individual char. Conrol limis were calculaed on he firs 1000 observaions of he TOC process, using he Conrol Char Builder. The limis were copied o a column propery ha were hen used on he conrol char for all 7200 observaions. Selec Analyze Qualiy and Process Conrol Char Builder. Drag TOC o he Y zone. Drag Time o he X zone. 6

7 Figure 5. Individual and Moving Range char on 7200 daa poin from TOC process. Figure 5 shows ha he limis look oo igh, and he daa wanders around he mean value of jus over 6%. The firs 1000 daa poins were used o fi ime series models, and he bes model was chosen based on he saisics AIC, SBC, as well as model simpliciy. An AR(5) process was deermined o be he bes fi. The predicion formula from his model was saved o he full able and residuals were calculaed for each poin. These residuals were ploed on a conrol char, shown in Figure 6. 7

8 Figure 6. Individual and Moving Range char on residuals from AR(5) model on TOC process. The reducion in false alarms is seen on he residual char. In pracice, he engineers decided o widen he limis of he residual char o avoid even hese false alarms, due o he frequen sampling of he process. The final sysem consised of specificaion limis on he raw daa, and widened limis on a residuals char. I has been successfully monioring waer qualiy, signaling qualiy shifs, for over 10 years. STATISTICAL PROPERTIES OF RESIDUAL CHARTS The use of residual chars for processes wih posiive auocorrelaion has a heoreical jusificaion. Conrol chars are measured by heir run lengh, a random variable ha represens he ime unil a signal. When he process is sable, and consiss of jus one disribuion, he run lengh should be long. When he process mean has shifed, he run lengh should be shor. I is easily shown ha he average run lengh (arl) is 1/p, where p is he probabiliy of a signal on he char. We denoe he average run lengh as arl(), where represens he size of a mean shif in sandard deviaion unis. For an Individual char wih 3 limis, assuming he process follows a normal disribuion, p= and arl(0)=370. Tha is, i akes on average 370 runs before his char falsely signals a mean shif. By conras, for his char, arl(1)=43.9. Tha is, i akes on average 43.9 runs before his char signals a mean shif of 1. Suppose now ha { X } is an auoregressive process of order 1 (AR(1)) wih a mean shif of size a ime 0. Then X 1 1 0, 0 2 ( X ), where, and ~ N(0, ). The residuals from he model, 0 e X X for all. Properies of he residuals are given as follows. are { e }, where 1 8

9 0, 0 E[ e ], 0 (1 ), 0, Var e 2 2 [ ] (1 ), Cov[ e, e ] 0, for all s. s The enire shif is seen by he residuals a ime 0. For 0, only a fracion of he shif is seen, for >0. Smaller shifs are harder o deec, resuling in higher run lenghs for posiively correlaed daa. For example, arl(1)=101.9 for an Individual char on an AR(1) process wih =0.4, and arl(1)=223.3 for an Individual char on an AR(1) process wih =0.9. For exremely high correlaion he average run lengh is low, for example, for an AR(1) process wih =0.98, arl(1)=8.56. The probabiliy of a signal on he firs observaion afer he shif is higher for highly correlaed daa han i is for independen daa. Thus, even hough he average run lengh is large, he probabiliy of a signal on he firs observaion is greaer for correlaed daa han for independen daa. For example, he probabiliy of signal on he firs observaion afer a 1 mean shif on an Individuals char on independen daa is abou 2%. The same probabiliy for a residual char on an AR(1) process wih =0.9 is 24% and for =0.95 is 58%. CONCLUSION A firs glance, saisical process conrol chars appear wholly unsuiable for seasonal or financial daa. As illusraed in he wo examples given, modeling echniques applied o he daa can sufficienly smooh ou he cyclical behavior hrough he residuals. This smoohed daa allows for boh applying saisical conrol chars o he process and furher idenificaion and invesigaion of unusual ime periods. Insead of approximaing he expeced shape and reacing o changes, by applying Time Series echniques ogeher wih saisical process conrol on he model residuals, he analys can gain considerable knowledge by boh beer undersanding he financial sae iself and idenifying which ouside indicaors or evens migh affec he sae boh posiively and negaively. While his is a very effecive mehod, i is compuaionally inensive and is bes suied for a sysem ha is conrolled by compuers. I is no suiable for paper chars filled ou by human operaors. REFERENCES Robers, L. (2006), SPC for Righ-Brain Thinkers: Process Conrol for Non-Saisicians: Qualiy Press, Milwaukee. Wheeler, D. W., and Chambers, D. S. (2010), Undersanding Saisical Process Conrol:, 3 rd Ed. SPC Press, Knoxville. Michelson, Diane K. (1994) Saisical Process Conrol for Correlaed Daa: unpublished Ph.D. disseraion, Texas A&M Universiy, College Saion. CONTACT INFORMATION Your commens and quesions are valued and encouraged. Conac he auhors a: Diane K. Michelson, Educaion Division Annie Dudley Zangi, JMP Group SAS Insiue Inc. Cary, NC , Di.Michelson@sas.com, Annie.Zangi@jmp.com SAS and all oher SAS Insiue Inc. produc or service names are regisered rademarks or rademarks of SAS Insiue Inc. in he USA and oher counries. indicaes USA regisraion. Oher brand and produc names are rademarks of heir respecive companies. 9