Perfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Key Wrds: Autregressive, Mving Average, Runs Tests, Shewhart Cntrl Chart

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1 Perfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Sandy D. Balkin Dennis K. J. Lin y Pennsylvania State University, University Park, PA Sandy Balkin is a graduate student in the Department f Management Science and Infrmatin Systems. y Dr. Lin is an Assciate Prfessr in the Department f Management Science and Infrmatin Systems and the Department f Statistics. He is a Senir Member f the ASQ. 1

2 Perfrmance f Sensitizing Rules n Shewhart Cntrl Charts with Autcrrelated Data Key Wrds: Autregressive, Mving Average, Runs Tests, Shewhart Cntrl Charts, Statistical Prcess Cntrl, Time Series. Abstract Sensitizing Rules are cmmnly applied t Shewhart Charts t increase their eectiveness in detecting shifts in the mean that may therwise g unnticed by the usual \ut-f-cntrl" signals. Since mst cntrl chart data are cllected as time series, it is f interest t examine the perfrmance f Shewhart's x Chart using autcrrelated data. In this paper, measurements arising frm autregressive (AR), mving average (MA) and autregressive mving average (ARMA) prcesses are examined using Shewhart Cntrl charts in cnjunctin with several sensitizing rules (Western Electric Cmpany, 1956). The results indicate that the rules wrk well when there are strng autcrrelative relatinships, but are nt as eective in recgnizing small t mderate levels f crrelatin. A questin then arises whether the results are due t the fact that lnger series are mre prne t false signal ccurrences. We investigate this by examining in-cntrl series f varius length and applying the sensitizing rules. The results suggest that the prbability f false psitives increases with series length resulting in a high number f false psitive ccurrences. We cnclude with the recmmendatin t practitiners that they use a mre denitive measure f autcrrelatin such as the Sample Autcrrelatin Functin crrelgram t detect dependency. 2

3 Intrductin The primary gal f this study is t assess the ability f cntrl chart sensitizing rules t detect dependency, bth individually and in cmbinatins, by simulating autcrrelated data and applying the tests. A secnd part f this study invlves simulating varius lengths f in-cntrl r \nrmal" data and testing whether the length f the series has an eect n the false signal rate f the sensitizing rules. The standard analysis and interpretatin f a Shewhart x chart assumes that the data are nrmally and independently distributed (N ID) with mean and standard deviatin which remain cnstant ver time. It is cmmn t apply runs tests in the analysis t increase the chart's eectiveness in detecting small changes in the prcess mean. Such tests are referred t as sensitizing rules. Sme f these tests are fund in Table 1 (see als, Mntgmery (1996) r Western Electric Cmpany (1956)). ** Insert Table 1 abut here ** The sensitizing rules make use f exclusive and exhaustive znes which divide the area between the upper and lwer cntrl limits int three regins. The znes refer t the regin between the center line and the 1 sigma limits as zne C; between the 1 sigma limits and 2 sigma limits as zne B; and between the 2 and 3 sigma limits as zne A. Figure 1 displays the znes graphically. ** Insert Figure 1 abut here ** The drawback t using these rules is that while they increase the chance f detecting changes in the prcess mean, they lead t a greater Type I errr rate. Wrk n Type I errr rates when applied t NID data can be fund in Champ and Wdall (1987), Davis and Wdall (1988), Wheeler (1983), and Chang, Tia, and Chen (1988). Type I errrs rates fr the sensitizing rules applied 3

4 t varius cntrl charts by simulating NID data can be fund in Walker, Philpt, and Clement (1991). Since the data fr Shewhart's x chart are cllected as a time series, we test the ability f the sensitizing rules t identify a vilatin f the independency assumptin using linearly autcrrelated data generated frm cnventinal time series mdels. This paper will describe the autcrrelatin structures which are used in the simulatin testing f the sensitizing rules, prvide an interpretatin f the results f the simulatin fllwed by a study f the impact f series length n the prbability f false psitives. We cnclude with a discussin f the utcmes and recmmendatins practitiners. Autcrrelated Data The standard assumptins assciated with the use f cntrl charts include the data being generated by a N ID(; ) prcess with the parameters xed but unknwn (Mntgmery, 1996). This assumptin is ften invalid as time series data is frequently crrelated. When a series drifts ver time, it is said t be autcrrelated. autcrrelatin functin: and estimated using: r k = The level f autcrrelatin is measured using the k = Cv(x t; x t?k ) ; k = 0; 1; : : : V ar(x t ) P N?k t=1 (x t? x)(x t?k? x) P N t=1(x t? x) 2 ; k = 0; 1; : : : ; K where N is the length f the time series. As a general rule, the rst K N=4 sample are cmputed (Mntgmery, Jhnsn, & Gardiner, 1990). In this study, autcrrelated data are simulated using Linear Gaussian Mdels as the generating prcess. Linear Gaussian mdels are frequently used in time series analysis t explain the mvement f a series as a functin f its past perfrmance plus randm shcks. We will use the Linear Gaussian mdels described belw t induce crrelatin in the data. 4

5 The rst type f linear mdel studied will be the autregressive prcess f rder p (AR(p)) and is characterized by Y t = c + 1Y t?1 + 2Y t?2 + + p Y t?p + t The AR(p) which is a weighted average f the past perfrmance, with weights i 's and a nrmal errr term t N(0; 2 ). Such a mdel is used when the change in the series at any pint in time is linearly crrelated with previus changes. A secnd type f linear mdel will be used in the analysis is the mving average prcess f rder q (MA(q)) and is characterized by Y t = + t? 1 t?1? 2 t?2?? q t?q : The MA(q) is a weighted average (with weights i ) f randm shcks (i.e., i ) spanning q perids. Each f the i 's is assumed t fllw a nrmal distributin with mean 0 and standard deviatin 1. A mving average mdel is used when there is a linear dependence n past perfrmance. It is interesting t nte that the system has a q?perid memry meaning that a randm shck persists fr exactly q perids. Cmbining the tw mdels abve results in the mixed autregressive-mving average (ARM A(p; q)) prcess characterized by Y t = c + 1Y t?1 + + p Y t?p + t? 1 t?1?? q t?q : This type f scheme is used when bth mving average and autregressive tendencies are present. Simulatin Prcedure Our gal is t evaluate the ability f the sensitizing rules t detect dependency in a series f bservatins, nt t decide n an ptimal batch size. Thus, we will nly lk at series f individual bservatins (batch size f 1). Fr each mdel specied in Table 2, a series f 100 data pints was generated with Nrmal(0, 1) errr terms. The NID case ccurs when all parameter values f 5

6 the AR, MA r ARMA mdel are set t zer and will serve as a \benchmark" fr cmparisn. Shewhart Cntrl limits are then determined using the mean f the series as the center line and the mving range f successive bservatins t determine the cntrl limits. The mving range is dened in Mntgmery (1996) as MR i = jx i? x i?1j. The mean f the mving range is used t estimate the prcess variability. The interpretatin f the chart is then similar t that f the rdinary Shewhart-x cntrl chart. ** Insert Table 2 abut here ** All eight tests were then perfrmed n the cntrl chart nting when each rule was vilated. Ten thusand (10,000) sets f 100 data pints were generated via this prcess fr the dierent linear mdels. The values reprted are the fractin f generated series fund in vilatin f each rule and the percentage f series which vilated at least ne f the rules. The series were generated and tested using the sftware package S-plus versin 3.3. Results and Discussin Tables 3 thrugh 6 shw the results frm the simulatins. In the fllwing sectin we study the results f each mdel simulatin, examining each rule and its perfrmance under the varius mdels. ** Insert Tables 3, 4, 5, 6 Here ** Rule 1 : A pint falls utside the 3 sigma limit Rule 1 crrespnds t having an bservatin fall relatively far frm the prcess mean. Vilatin f this rule can indicate an ut f cntrl pint r dependency f the prcess. This rule is typically vilated when the generating prcess has a large autregressive cecient in abslute value r 6

7 negatively large mving average term. Fr example, AR(1)-6, AR(2)-25, MA(2)-1 and ARMA-21 are all examples f mdels detected by this rule. Rule 2 : 8 pints in a rw in zne C r beynd n the same side f the center line Rule 2 crrespnds t a trend in the data. Vilatin f this rule is indicative f dependency in the data. This rule is typically vilated when 2 is large fr the AR schemes, when 1 and 2 are negatively large fr the MA schemes and when 1 is large and 1 is negatively large fr the ARMA scheme. Mdels AR(1)-6, AR(2)-20, MA(2)-1 and ARMA-21 are examples where this rule is eective. Rule 3 : 6 pints in a rw increasing r decreasing Rule 3 als crrespnds t a trend in the data. Vilatin f this rule is indicative f psitive autcrrelatin in the data. It is typically vilated by AR(2) schemes when bth cecients are large and psitive. Fr example, AR(2)-25 and ARMA-21 are schemes that cnsistently vilate this rule. Rule 4 : 14 pints in a rw alternating up and dwn Rule 4 crrespnds t a series that is mean reverting. This is characteristic f an AR(1) scheme with negative cecient. Thus, it is n surprise that this test is mst ften vilated by the AR(1) and ARMA(1, 1) schemes with largely negative autregressive cecients, by AR(2) schemes with largely negative 1 and psitive 2 and hardly ever by pure mving average schemes. Mdels AR(1)-1, AR(2)-5, and ARMA-3 are examples where this rule is eective. Rule 5 : 2 ut f 3 pints in a rw in zne A r beynd n the same side f the center line Rule 5 is an indicatr f pssible dependency. This rule is vilated when a cuple f pints clse tgether are very large, either psitively r negatively. It is typically vilated by AR(1) schemes when is large and in AR(2) schemes when j1j and 2 are large. Fr example, AR(1)-6, AR(2)-25, MA(1)-1, MA(2)-2 and ARMA-21 are schemes causing this rule t be vilated. 7

8 Rule 6 : 4 ut f 5 pints in a rw in zne B r beynd n the same side f the center line Rule 6 is similar t Rule 5 in that it states that several pints in a rw were large, either psitively r negatively. This als is indicative f dependency. This rule is typically vilated by AR(1) schemes with a large cecient and by AR(2) schemes when bth cecients are psitive. It is als frequently vilated by MA schemes with a largely negative 1 value as well as the cmbinatin f when 1 is large and 1 is negatively large fr the ARMA prcesses. This rule is vilated by mdels such as AR(1)-6, AR(2)-25, MA(1)-1, MA(2)-1 and ARMA-21. Rule 7 : 15 pints in a rw in zne C Rule 7 crrespnds t the bservatins falling t clse t the center line fr an extended perid f time. This can be interpreted as an indicatin f dependency. This rule is typically vilated when 1 is largely negative and infrequently when applied t series with mving average structure. Fr example, mdels AR(1)-1, AR(2)-5 and ARMA-3 cause this rule t be vilated. Rule 8 : 8 pints in a rw nt in zne C Rule 8 can als be used t detect dependency in the data. It is typically vilated when 2 is large fr the AR(2) schemes and smewhat less frequently when is negative fr the ARMA schemes. AR(1)-6, AR(2)-25 and ARMA-21 are examples f schemes that cnsistently vilate this rule. Overall, it appears that high levels f autcrrelatin are eectively detected. Strng negative cecient mving average structures als tends t vilate the rules frequently. It is apparent, hwever, that series with weak t mderate dependencies slip passed the rules such as schemes AR(1)-3, AR(2)-7, MA(1)-4, MA(2)-10 and ARMA-4. Prbability f False Psitives In practice, a single vilatin f any f the sensitizing rules wuld result in an investigatin int what was causing the prcess t g ut f cntrl. Walker, Philpht and Clement (1991) 8

9 experimented with series f length 20 and 30. Cnsistently, as the series gets larger, s des the prbability f btaining a false psitive. The simulatin prcess explained abve was run n dierent series lengths where the series where generated accrding t a standard nrmal prcess. Table 6 gives the series length, the prprtin ut f 10,000 each individual rule was vilated, and the percentage f times any rule was vilated. ** Insert Table 6 abut here ** It is clear that large Type I errrs are an artifact f large series lengths. In particular, rules 1, 2 and 6 appear particularly prne t falsely vilating a large series. Thus, practitiners must be careful when applying all eight rules t lng series as the prbabilities f falsely rejecting increase with series length. Recmmendatins Frm the simulatins, it is evident that the sensitizing rules are nt reliable fr determining dependency, especially as series length increase. The riginal intent fr these rules was t make it pssible fr a persn n a factry r t quickly determine if a prcess was ut-f-cntrl r nt. Hwever, with the current level f cmputer pwer, there exist mre eective techniques fr ding this jb. A simple way t shw the crrelatin structure f a series is by its the Autcrrelatin Functin (see Mntgmery et al. (1990) Chapter 10.2 fr an explanatin f autcrrelatin functins). Figures 2 and 3 shw the theretical autcrrelatin functins fr the AR(1) and M A(1) mdels. Frm crrelgrams f bserved series, we can see hw strng the crrelatin is between time s as well as hw lng it lasts. Such plts are useful in determining what, if any, autcrrelatin is inherent in a realized series f bservatins. ** Insert Figure 2 abut here ** 9

10 ** Insert Figure 3 abut here ** Figure 4 shws sme autcrrelated series and their crrespnding Sample Autcrrelatin and Partial Autcrrelatin Functin plts as described in Sectin 2. ** Insert Figure 4 abut here ** The crrelgrams eectively shw when a series' bservatins are nt independent with signicantly large spikes at sme s. Mrever, Mntgmery et al. (1990) explains hw t use the plts t identify a particular ARMA mdel. Cnclusin Each f the rules applied has its place in detecting fr structure in a time series. N ne rule is adequate t use in determining if the series is randm r nt. Fr instance, Rule 1, the easiest t apply, is nly eective fr certain types f autcrrelatin. The rules that are eective simply lk fr characteristics f AR r MA schemes. Hence, hw well a rule des is dependent n hw strng the characteristic is. Fr example, the pattern searched fr by Rule 4 is fund in AR(1) mdels with a negative cecient. The larger the negativity, the greater the prprtin f vilatins fund. In cnclusin, the sensitizing rules were nt as eective in identifying mving average prcesses as they are fr autregressive series. This is nt cmpletely surprising as mving average prcesses are nly crrelated fr a nite number q s. Mst f the runs tests rely n a multi-pint pattern as a means f vilatin detectin. There is a high level f falsely classifying a series as ut f cntrl when using the sensitizing rules n lng series. A pssible alternative t the Shewhart Chart and sensitizing rules are SACF and SPACF plts which identify signicant crrelatin between ged pints f the series. These plts are easy t btain using almst any statistical package and shuld be cnsidered fr use in practice. Acknwledgement: The authrs wish t thank the Editr and tw annymus referees fr their valuable cmments which cnsiderably added t the value f this manuscript. 10

11 References Bx, G., Jenkins, G., & Reinsel, G. (1994). Time Series Analysis Frecasting and Cntrl (Third editin). Prentice Hall : Englewd Clis, New Jersey. Champ, C., & Wdall, W. (1987). Exact Results fr the Shewhart Cntrl Charts With Supplementary Runs Rules. Technmetrics, 29, 393{399. Chang, I., Tia, G., & Chen, C. (1988). Estimatin f Time Series Parameters in the Presence f Outliers. Technmetrics, 30, 193{204. Davis, R., & Wdall, W. (1988). Perfrmance f the Cntrl Chart Trend Rule Under Linear Shift. Jurnal f Quality Technlgy, 20, 260{262. Mntgmery, D. (1996). Intrductin t Statistical Quality Cntrl (Third editin). Jhn Wiley and Sns, Inc. Mntgmery, D., Jhnsn, L., & Gardiner, J. (1990). Frecasting and Time Series Analysis (Secnd editin). McGraw-Hill, Inc. Walker, E., Philpt, J., & Clement, J. (1991). False Signal Rates fr the Shewhart Cntrl Chart with Supplementary Runs Tests. Jurnal f Quality Technlgy, 23, 247{252. Western Electric Cmpany (1956). Statistical Quality Cntrl Handbk. Western Electric Cmpany, Indianaplis, IN. (available frm the ASQ). Wheeler, D. (1983). Detercting a Shift in the Prcess Average: Tables f the Pwer Functin fr x Charts. Jurnal f Quality Technlgy, 15, 155{

12 Standard Deviatins frm Mean Zne A Zne B Zne C Zne C Zne B Zne A Figure 1: Shewhart Chart with znes 12

13 AR(1); phi= -0.9 AR(1); phi= -0.5 AR(1); phi= 0 rh rh rh AR(1); phi= 0.1 AR(1); phi= 0.5 AR(1); phi= 0.9 rh rh rh Figure 2: Theretical ACF fr AR(1) Mdels 13

14 MA(1); theta= -0.9 MA(1); theta= -0.5 MA(1); theta= 0 rh rh rh MA(1); theta= 0.1 MA(1); theta= 0.5 MA(1); theta= 0.9 rh rh rh Figure 3: Theretical ACF fr MA(1) Mdels 14

15 Observatin Series : AR(1); phi=-0.9 Series : AR(1); phi=-0.9 Series : AR(1); phi=-0.9 Sample Mean ACF Partial ACF Series : AR(1); phi= Series : AR(1); phi= Series : AR(1); phi=0.1 Sample Mean ACF Partial ACF Observatin Series : NID Series : NID Series : NID Sample Mean ACF Partial ACF Observatin Series : MA(1); theta= Series : MA(1); theta= Series : MA(1); theta=-0.5 Sample Mean ACF Partial ACF Observatin Series : MA(1); theta= Series : MA(1); theta= Series : MA(1); theta=0.1 Sample Mean ACF Partial ACF Observatin Series : MA(1); theta= Series : MA(1); theta= Series : MA(1); theta=0.9 Sample Mean ACF Partial ACF Observatin Figure 4: Cntrl Chart with crrespnding SACF and SPACF plts 15

16 Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7 Rule 8 A pint falls utside the 3 sigma limit 8 pints in a rw in zne C r beynd n the same side f the center line 6 pints in a rw increasing r decreasing 14 pints in a rw alternating up and dwn 2 ut f 3 pints in a rw in zne A r beynd n the same side f the center line 4 ut f 5 pints in a rw in zne B r beynd n the same side f the center line 15 pints in a rw in zne C 8 pints in a rw nt in zne C Table 1: Sme Sensitizing Rules fr Shewhart Cntrl Charts 16

17 NID values: AR(1) values: AR(2) 1 values: values: MA(1) values: MA(2) 1 values: values: ARMA(1; 1) values: values: Table 2: Parameter values used in simulatins 17

18 Case 1 2 Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7 Rule 8 % Vilated NID AR(1)? AR(1)? AR(1)? AR(1)? AR(1)? AR(1)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? AR(2)? Table 3: Results f the AR simulatins. vilated. Numbers indicate the fractin f times the rule was 18

19 Case 1 2 Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7 Rule 8 % Vilated NID MA(1)? MA(1)? MA(1)? MA(1)? MA(1)? MA(1)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? MA(2)? Table 4: Results f the M A simulatins. vilated. Numbers indicate the fractin f times the rule was 19

20 Case 1 1 Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7 Rule 8 % Vilated NID ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? ARMA(1; 1)? Table 5: Results f the ARMA(1; 1) simulatins. Numbers indicate the fractin f times the rule was vilated. 20

21 Series Length Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7 Rule 8 % Vilated Table 6: Results f Type I errr simulatins. Numbers indicate the fractin f times the rule was vilated. 21