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CUMULATIVE SUM (CUSUM) CONTROL SCHEMES James M. Lucas E. I. du Pont de Nemours and Co., Inc. Wilmington, DE the production of nonconforming product. Because CUSUM procedures give an early ABSTRACT Cumulative Sum (CUSUM) quality indication of process changes, control sistent schemes are becoming widely used in industry because they are powerful, versatile, and easy to use. They cumul ate recent process data to quickly They detect also tool. out-of-control serve as a 19898 There are now more than they are con- philosophy encourages doing it right the first time. use of CUSUM procedures situations. powerful a management with ;s that The also consistent with a management by exception philosophy as the CUSUM will point out areas needing diagnostic attention. 10,000 CUSUM control schemes in use daily ;n Du Pont. This talk describes design and implementation procedures for CUSUM control schemes with emphasis on properties that are recorded as counts. The talk will describe recent developments which make CUSUM procedures more useful and more powerful. Cumulative sum control schemes are current ly used more for the control of vari ab 1es than for the control of counts (attributes). To help remedy this, we will give counts more emphasis here. We will show that the design and implementation procedure for Counted Data CUSUMs (Lucas 1985) is very similar to the The recent developments described are: procedure for CUSUMs for variables (Lucas 1976). We will use a Poisson distribution in o Fast Initial Response (FIR) CUSUM which our examples. gives extra sensitivity to out-ofcontrol situations at start up or after a (possibly ineffective) control action. is administratively convenient to record the number of counts in a given sampling interval. When the number of counts per interval follows a Poisson distribution, the time between counts follows an exponential distribution. A time-between-events CUSUM which we discuss elsewhere (Lucas 1985) should be used if it ;s convenient to update the CUSUM with each new count, and it is possible to record the time since the last count. o A Combined Shewhart-CUSUM which com- bines the key features of CUSUM schemes and Shewhart Schemes by adding Shewhart Control Limits to a CUSUM scheme. o ROBUST CUSUM schemes whi ch are not unduly influenced by a few outliers or A Poisson CUSUM is used when it fl iers occurring in the stream of data. We discuss the use of CUSUMs for the detection of increases and/or decreases in count rate; we discuss both one and two-sided The philosophy of continual improvement of a process is very compatible with CUSUM procedures. As CUSUM procedures gi ve much more responsive control, a CUSUM signal does not mean that the process is producing bad product. Rather it means that action should be taken so that the process does not produce bad product. CUSUMs. I NTRODUCTI ON Cumulative sum (CUSUM) quality While the most widely used applica- tion of counted data CUSUMs will probably be the control of nonconformances, they can also be used to monitor changes in count level in other situations such as accident rate or the rate of occurrence of congenital malformations. We use the terminology counts, rather than nonconformances, as counts has a broader applicability. control schemes for variables are widely used in industry. CUSUM procedures will usually give tighter process control than classical quality control schemes, such as Shewhart schemes. With the tighter control that is available with CUSUM schemes, there will be more emphasis placed on keeping the process on-aim rather than allowing it to drift in limits. Because of the tighter control, an out-ofcontrol (Signal will seldom indicate that the process is. produci ng nonconforming product; rather an out-of-control Signal will indicate that control action should be taken to prevent The recognition that a CUSUM control scheme is a sequence of Wald Sequential Probability Ratio Tests (SPRT) allows the optimal ity properties of CUSUM procedures to be developed. Johnson and Leone (1962) 9ave an early discussion of CUSUM procedures using the relationship between SPRT1s and CUSUM. Lorden (1971) showed the asymptotic optimality of CUSUM procedures for detect i ng a change in distribution. Kenett and Pollak (1983) showed the superiority of a CUSUM scheme for detecting a rare event over a non-CUSUM scheme proposed by Chen (1978). 916 Yi Recent enhancements to CUSUM quality control schemes have included the Fast Initial Response (FIR) feature (Lucas and Crosier 1982A). The FIR feature recognizes that when will the initial SPRT to test a different null hypothesis than the following SPRTs. In A Robust CUSUM (Lucas and Crosier, 19828) Robust CUSUM counts in the ith use a positive starting value. We for the initial SPRT. has been recommended when isolated outliers or extreme values occur for reasons other than a A of the process is off-aim at start-up. For a process running at the desired level, the head start will soon !Izera!!, so it has little effect. It is instructive to view the choice of So in terms of the II all and lIa n errors of the equivalent SPRT. The a-error is the probabi 1ity of declaring the process off-aim when it is not. The a-error is the probability of declaring the process on-aim when it is not. When the CUSUM is cons i dered to be a sequence of Wald SPRTs, one finds that the SPRTs have small a and large a errors. Using a head start value approximately equal to h/2 is equivalent to equating the a and a errors between-event CUSUMs see Lucas (1985). shift. number generally recommend a head start (SO) value approximately equal to h/2. With such a head start, the CUSUM will more quickly Signal if process control work, it is more likely that the process is not at the desired aim value at start-up than after the process has been The FIR running smoothly for some time. feature gives a simple procedure for more quickly detecting an out-of-control situation at start-up. If the process is initially in control, the Fast Initial Response feature has little effect while if the process starts out in an out-of-control condition, an out-ofcontrol signal is given much more quickly. In this paper, we demonstrate the benefits of the FIR feature for a Poisson CUSUM. For a discussion of the FIR feature for a time- process the A standard CUSUM will have starting value So = 0, while a Fast Initial Response CUSUM a CUSUM is started, it may be appropriate for true is interval. A Robust CUSUM (Lucas and Crosier, 19828) is obtained by using the Utwo-in-a-row rule". To robustify the CUSUM, an outl ier 1 imit is specified. A single observation outside this limit does not enter the CUSUM. However, two outliers in a row are an out-of-control Signal. will quickly detect true changes in level that occur in the process; yet it will be insensit i ve to the occurrence of an occas i on a 1 outlier or flyer. In this paper, we discuss a Robust Poisson CUSUM. CUSUM evaluated The properties of a combined ShewhartCUSUM control scheme are described by Lucas (I982). For variables, the combined scheme has proven of most value for measurement control. In this application a Shewhart signal can indicate a bad sample while a CUSUM Signal often indicates a calibration problem. length (ARL). bution of run lengths (Brook and Evans 1972). For a standard CUSUM, the ARL distribution is nearly geometric except that there ;s a lower probability of extremely short run lengths. When the FIR feature is used, the distribution is nearly geometric except that there is an increased probability of short run lengths due A cumulative sum control scheme cumulates the difference between an observed value Vi and a reference value k. If this cumulation equals or exceeds the decision interval value h, an out-of-control signal is given. The CUSUM statistics are: max (0, Yi - k + SH(i-l)) SLi max (0, k - Yi + SL(i-l)) The ARL is the average number of samples taken before an out-of-control signal is obtained. The ARL should be large when the process is at its aim level and short when the process shifts to an undesirable level. In-control ARLs are often well approximated by a geometric distribution; hence, the ARL also characterizes the distri- IMPLEMENTATION EXAMPLE SHi contro 1 schemes are usually by calculating their average run to the head start (Lucas and Crosier 1982A). Table 1 illustrates the implementation of a Poisson CUSUM with parameter values k=5 and h=lO. A CUSUM with and without a head start value SO=5 is shown. where max (a,b) is the maximum of a and b. The first formula is used to detect an increase in level, while the second formula is used to detect a decrease in level. A two sided CUSUM uses both formulas Simultaneously. a would work We wi11 find that such well if the acceptable count 1eve 1 was about 4 and it was des ired to quickly detect if the count level increased to 7. I n such a case, the i n-contro 1 average run 1engt would be approximately 397 or 422 and the out-of-control ARL would be 3.35 or 5.59 for the FIR CUSUM and the standard CUSUM, For the control of variables, the Vi in the above formulas is a standardized variable: Vi = CUSUM respectively. In Table 1, the first column is the observation number, i=1. •• 15. The second column gives the observed number of counts, Vi. The th i rd co 1umn records an i ntermediate calculation step. It calculates Vi-k. The fourth column calculates Si for a Xi - AIM S where Xi is the observed sample average and S is the standard deviation of X. For Counts standard CUSUM (with SO=O) , and the fifth column calculates Si for a FIR CUSUM with 917 SO=5. In Table 1, the first AVERAGE RUN LENGTHS FOR THE MOST WIDELY USED CUSUM PROCEDURES ten observa- tions are from a process at the desired mean level while the last five are from a process with a higher mean level. The process starts out i nit i ally in contro 1, so the FI R feature has little effect; the head start soon zeroes out. The benefit of the FIR feature can be Deviation From Aim (Multiple of S) seen by implementing the CUSUM using only the last five observations. This represents a process that is out of control when the CUSUM is started. In this case, the CUSUM with the FIR feature would give an out-af-control signal at the first observation (Observation 11), while the standard CUSUM would not signal until the fourth observation (Observation 14). POISSON CUSUM EXAMPLE h=lO k=5 Y -k S No FIR o 1 2 3 7 3 4 5 6 2 0 2 8 7 4 8 9 10 13 0 2 3 10 8 4 14 15 1l 1l 12 9 -2 2 -3 -5 -3 3 -1 -5 -3 -2 5 3 -1 4 6 0 0 2 0 0 0 3 2 0 0 0 FIR 5 3 5 2 0+ 0 3 2 0 0 0 5 5 8 7 1l* 17* 8 7 1l* 17* So 4 4 5 5 .5 .5 .5 .5 0.0 2.0 0.0 2.5 0 .5 168. 26.6 150 20.1 465. 38.0 432 28.7 1. 2. 4 8.4 5.3 10.4 6.4 3.3 2.0 4.0 2.4 1.7 1.0 2.0 1.2 should be selected to be close to: This reference value is the same as the reference value for an SPRT testing the null hypotheSiS that the mean is ua and alternate hypothes i s that the mean is ud. When kp 21, the kp value will usually be rounded to the nearest integer. Then, the CUSUM computations require only integer arithmetic. After k is selected, the decision interval value (h) *Out of Control Signal + k We will give a more detailed discussion of the design procedure for a Poisson CUSUM for detecting an increase in count rate. For a Poisson CUSUM, the k value will be between the accept ab 1e proces s mean (jJ a) and the mean 1eve 1 of counts ("d) whi ch the CUSUM scheme is to detect quickly. While the desired value (or goal value) for J.la is often zero, a ).Ia value of zero requires that the CUSUM be designed with h=l and k=O. Any occurrence of a count gives a Signal; when count levels are low and a decrease in count level indicates an improved system, every count must followed up. The circumstances surrounding every count must be examined to find and remove assignable causes. In practice, ).I a is often chosen near to the current mean 1eve 1; th i s represents current system performance. The reference value for the Poisson CUSUM (k p ) TABLE 1 Y h is chosen using a table look-up procedure. The value of h Should give an appropriately large ARL when the counts are at the after the FIR CUSUM zeroes, the standard acceptable level. It should also be chosen to give an appropriately small ARL value when the process is running at the count CUSUM and the FIR CUSUM are equivalent level Which should be detected quickly. The CUSUM parameter k is determined by the a Poisson CUSUM of counts is 4 7) is ("a = 4) and a count rate of 7 ("d to be detected quickly. The Poisson CUSUM is designed with a k value close to: acceptable mean level (]Ja) and by the unacceptab 1e mean ("d) 1eve 1 whi ch the CUSUM scheme is to detect quickly. For variables the k value is chosen half way between the acceptable mean level and the unacceptable mean level. when The CUSUM parameter h is the ("d-"a)/( In("d)-ln("a)) chosen by a table look up procedure to give an acceptably long in-control ARL. In practice the parameter values h = 4 or 5 and k = .5 are often used. Consider the design of the acceptable number = (7-4)/(ln 7-ln 4)= 5.36 The FIR feature with S(O) = h/2 For ease in implementation using integer arithmetic, a kp value of 5 is recommended. usually should be used. With these parameters you get the following ARL values for a two With this kp (Lucas 1985): sided CUSUM procedure (Lucas and Crosier 1982): 918 value we find for the ARL THE FAST INITIAL RESPONSE FEATURE AVERAGE RUN LENGTH FOR THE POISSON CUSUM EXAMPLE At start up, or following a possibly ineffective control action, control of the process is less certain. Extra sensitivity is often des i red at these times. The Fast Initial Response feature for CUSUM control schemes was developed to give this extra sensitivity. For a standard CUSUM scheme SO, the initial CU5UM value, is set equal to " (multiple of k p ) kp hp 7 7 So 5 5 5 5 5 5 10 10 15 15 0 4 0 5 0 8 4(.8) 7(1.4) 108 95 422 397 3740 3630 4.09 2.37 5.59 3.35 8.09 4.36 zero. The FIR CUSUM uses an initial CUSUM value, SO, that is greater than zero. Our recommended starting value is h/2. With this starting value, a FIR CUSUM detects an initially out-at-control situation about 40 percent faster than a standard CUSUM. An appropriate Poisson CUSUM could be the scheme with hp : 10, kp:5, SO=5. TWO-SIDED CUSUMs For the control CUSUMs The of variables, two-sided are usually used even when shifts in FIR features. only one direction indicate a deterioration in quality. Different control actions may be taken depending on which side of aim the CUSUM scheme is usually out-of-control signal indicate need the on for recommended. the low control side when the CUSUM The CUSUM scheme with the FIR Shewhart charts provide rapid detection of large deviations, while CUSUM schemes provide rapid detection of small to moderate deviations. The Shewhart-CUSUM scheme (Lucas, 1982) incorporates the good features of both. The prime motive for Shewhart-CUSUM is to obtain faster response to 1arge shifts with only small changes in performance otherwise. With the Shewhart-CUSUM scheme. Shewhart Control Limits (SCL) are added to the CUSUM scheme. An out-of-control signal is given if the most recent sample ;s outside of the SCL or if a regular CUSUM signal is given. Since the control action is modified only for the current observation, the combined scheme is easy to implement. The performance of the combi ned Shewhart-CUSUM approach is comparab le to that of more in olved modifications of however. for seen COMBINED SHEWHART-CUSUM CONTROL PROCEDURE it is usually easier to obtain the properties of a two-sided scheme by combining results for two one-sided schemes. When the parameter values are such that one side gives an out-ofonly be to For attributes, also we generally recommend two-sided control schemes where, for example, a significant decrease in count rate might trigger an investigation which could lead to performance improvements. Properties of a two-s i ded scheme may be evaluated signal can An increase the strength, while an out-af-control control CUSUM is used both a 1arger i n-contro 1 ARL and a quicker detection of an initial out-of-control situation can be achieved. would action and Cros i er. 1982A); the FIR The ARL tab les show that when the FIR feature signal on the high side would indicate a need to go into a detective mode to find why the high strength was achieved. The investigation could indicate ways to improve the product. di rect ly (Lucas of feature usually should have a larger h value. signals. For example, it may be important to contro 1 the strength of a gi yen product property. Even if there is no prob 1em when a higher strength is achieved, a two-sided control value from the ARL tables which we have given. Compare ARLls for CUSUMls with and without the the other side has zeroed, the ARL for a two-sided scheme can be obtained directly from the ARL of two one-sided schemes. (Lucas and Crosier CUSUMs such as Parabolic CUSUM (Lucas (1973)). 1982A. Lucas 1985) as: -~ Des i gn ARL (SOH. SOL) easy. a a Shewhart-CUSUM CUSUM is scheme designed the is ARL tables are examined. The Shewhart Limits are included if they make a significant improvement in the ARL curve. The combined scheme should use the same k value and the same or slightly larger h value to compensate for decreases in in-control ARL. We generally recommend SCL of 3.5 or 4.0 rather than 3.0 as the larger limits have less effect on the incontrol ARL. The following gives ARL values LH(SOH) LL(O) + LH(O) LL(SOL) - LH(O) LL(O) LH(O) + LdO) where the subscript Hand L indicate parameter values for the high and low side respectively, and L *( SO*) is the ARL for a one-s i ded CUSUM scheme with head start SO* and L or H may be substituted for of After for h = 4 or 5 k = .5 CUSUM schemes. ditional tables are given by Lucas (1982). * 919 Ad- AVERAGE RUN LENGTH FOR A COMBINED SHEWHART-CUSUM combined with a cluster distribution having probability c. If the cluster was sampled, a large number of cou.nts (sufficient to trigger a standard Deviation From Aim (Multiple of S1 CUSUM) would h k SCL 0 .5 1 2 4 4. 4.0 5.0 5.0 5.0 .5 .5 .5 .5 •5 3.0 3.5 3.0 3.5 4.0 125. 159. 223 • 391. 459 • 25. 26. 34. 37. 38. 8.0 8.3 9.8 10.2 10.4 3.0 3.2 3.5 3.8 4.0 1.2 1.3 1.2 1.3 1.6 We to be 0, .001, and .01. Continuing our design example with h=10, k=5, 50=0,5 we find: AVERAGE RUN LENGTHS FOR A ROBUST CUSUM We often select a Shewhart-CUSUM scheme to process control observed. row rule (with and without the FIR feature). We considered the probability (c) of a cluster Average Run Lengths with 50=5(50=0) contro 1 measurement processes (test 1aboratory control)~ whereas we usually employ a standard CUSUM to control production processes. In both cases we normally include the FIR feature. When the Shewhart-CUSUM is chosen for measurement be compared the performance of a standard Poisson CUSUM and a Poisson CUSUM using the two-in-a- and a mean of: Out 1 i er Standard a Shewhart Cusum signal often indicates a bad test sample while a CUSUM signal indicates a calibration problem or instrument offset. If a measurement process is prone to outliers, then a standard Robust Cusum CUSUM or a Robust CUSUM scheme may be better. Probabil ity 4 7 .000 .001 .010 397 (422) 281 (298) 77 (82) 3.35 (5.59) 3.34 (5.58) 3.28 (5.43) .000 .001 .010 401 (425) 401 (425) 388 (412) 3.40 (5.72) 3.41 (5.73) 3.44 (5.78) ROBUST POISSON CUSUM More extens; ve compari son of the perforA Robust CUSUM for vari ab 1es has mance of a Robust CUSUM and a standard CU5UM proven valuable when isolated outliers or flyers occur fairly often for reasons other than a (Lucas and Crosier, true process shift 1982B). A Robustification procedure for a representat; ve set of parameter values are contained in (Lucas 1985). for variables is the IItwo-in-a-row rule ll in which a single suspected outlier does not enter a CUSUM, whi le two suspected outl iers in a row are an out-of-control signal. In evaluating the properties of a Robust CUSUM for variables, we used the widely used contaminated normal distribution (Andrews, et a1 1972; Chen and Box 1979). clusters occur (c = .001 or .01). Using this distribution, we CONCLUSION We have discussed the design and use of CUSUM procedures. We have shown that they are simple to use, versatile in that they can be specifically tailored to detect shifts in level, and powerful in that they use all i nformat; on in the data to qui ck 1y detect the shift. The design and implementation procedure for counts is essentially the same as the procedure used for the control of variables. The FIR feature, a combined Shewhart-CUSUM scheme and robustification have all proved to be valuable enhancements for CUSUM control schemes. It has been our experience that occasionally a cluster of nonconformances occurs in a single sample and that it is sometimes valid to ignore the occurrence of a single sample having a cluster of nonconformances (especially if this cluster could be due to sampling or measuring instrument problems). To model the Robust CUSUM, we assumed that the underlying distribution was a mixture of a distribution (with In these cases, a Robust CUSUM gives a very large percentage increase in the in-control ARLs and on ly a Sma 11 percentage increase in the outof-control ARLs. evaluated the performance of various CUSUM schemes. We demonstrated that the two-in-arow rule was a good robustification procedure and showed that more complicated rules could give little improvement. In this section, we evaluate the properties of the two-in-a-row rule for robustifying a Poisson CUSUM for detecting an increase in count rate. Poisson The cost of the robustification procedure can be seen in the no outl ier case (c = 0). In these cases, equivalent increases in in-control ARLs can be achieved with less inflation of the out-ofcontrol ARLs by simply increasing the h value. The benefits of the robustification procedures show up in the cases where outlier probability I-c) 920 BI BLIOGRAPHY Andrews, O. F., P. J. Bickel F. R. Hampel, P. J. Huber, W. H. Rogers, and J. W. Tukey (1972). Robust Estimates of Location: Survey and Advances. Brook, D. Approach Lorden, G. (1971). "Procedures for React ing to a Change in Distribution ll , Ann. Math. Stat., 42, 1897-1908. Princeton University Press. and D. A. Evans (1972). Lucas, J. M. (1973). "A Modified "V" Mask Control Scheme," Technometrics, 15, 833-847. "An to the Probability Distribution of CUSUM Run Lengths", Biometrika, 59, 539-549. Chen, G. and G. E. P. Box (1979). Lucas, J. M. (1976). "The Design and Use of Cumulative Sum Quality Control Schemes", Journal of Quality Technology, ~, 1-12. "Implied Assumptions for Some Proposed Robust Estimatorsll. University of Wisconsin. Math. Research Center Technical Summary Report, 1979. Chen, R. Congenital (1978). Lucas, J. M. (1982). "Combined Shewhart-CUSUM Quality Control Schemes II , Journal of Quality Technology, li, 51-59. A Surveillance System for Malformations, Journal of the Lucas, J. M. (1985). "Counted Data CUSUMs", Technometrics to appear, May 1985. Ameri.can Statistical Association, 73. 323 327. Johnson, N. L. and F. C. Leone (1962). "Cumulative Sum Control Charts - Mathematical Principles Applied to their Construction and Use" (in three parts) Industrial Quality Control, June 15-21, July 29-36, AU9ust 22-28. Lucas, J. M. and R. B. Crosier (1982A). "Fast Initial Response (FIR) for Cumulative Sum Quality Control Schemes". Technometrics, 24, 199-205. - Kenett, R. and M. Pollak (1983). "On Sequential Detection of a Shift in the Probability of a Rare Event", Journal of the Lucas, J. M. and R. B. Crosier (1982B). "Robust CUSUM'I, Corrrnunications in Statistics, Theor. Meth. 11 (23), 2669 2687. American Statistical Association. 78, 389-395. 921