control charts to forecast errors arising from various autocorrelated processes. ... In this paper we propose the reverse moviig average control chart as a new ...
The Reverse Moving Average Control Chart for Monitoring Autocorrelated Processes JOHN N. DYER Georgia Southern University, Statesboro, GA 30460
BENJAMIN M. ADAMS and MICHAEL D. CONERLY University of Alabama, Tuscaloosa, AL 35487 Forecast-based monitoring schemes have been researched extensively in regards to applying traditional control charts to forecast errors arising from various autocorrelated processes. The dynamic response and behavior of forecast errors after experiencing a shift in the process mean make it difFicult to choose a suitable control chart. In this paper we propose the reverse moviig average control chart as a new forecast-based monitoring scheme, compare the new control chart to traditional methods applied to various ARMA(1,1), AR(1), and MA(1) processes, and make recommendations concerning the most appropriate control chart to use in a variety of situations when charting autocorrelated processes.
M\MrH
Introduction OST
traditional control charts have been de-
signed to monitor output fromi independent and identically distributed (iid) processes. When output data are autocorrelated, the traditional charts applied to the data have been shown to be unreliable (Maragalh aind Woodall (1992) and Harris and Ross (1991)). Recent advances have provided forecastbased monitoring schemes to address this problem. A forecast-based monitoring scheme involves identifying the proper time-series model which characterizes a process, obtaining the appropriate BoxJeilkins one-step-ahead (OSA) forecast of process observations, and thenl applying traditional control clharts to forecast errors (Alwan and Roberts (1988), WVardell, Moskowitz, and Plante (1994), Lin and Adams (1996), Lu and Reynolds (1999a, 1999b, 2001)). If the assumed time-series model is correct, the forecast errors are iid normal random variables. Hence, ini-control performance of the traditioilal conlDr. Dyer is an Assistant Professor of Decision Sciences. He is a Member of ASQ. His email address is jdyer©gasou.edu. Dr. Adams is an Associate Professor of Statistics. He is a Member of ASQ. His email address is badamsC"cba.ua.edu. Dr. Conerly is a Professor of Statistics. His emnail address is mconerlyLQcba.ua.edu.
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trol charting techniques is predictable, enabling monitoring for detection of shifts in the process mean level. Two types of shifts in the process can be considered. A step (level) shift is said to occur if the process mean suddenly changes to a new level. A trend occurs when there is a gradual shift to a new process level. Step shifts alone are considered in this paper. The miagnitude of the shift in the process can be measured either in terms of the white noise variance, 72, or in terms of the process variance, u% Process shifts are reported here in termis of the white noise variance since it remains constant, whereas the process variaince varies depending upon the assumed model. For a given stationary model and white noise variance, the shift size in terms of the process variance can be determined if one so chooses.
Performance Criteria When control chart performance has been evaluated, the average run lenigth (ARL) has typically been used to quantify performance of the chart. The ARL is defined as the expected niumber of time per:.ods until the control chart signals. An alternative performance criterion is the cumulative distribution finctioin (CDF) of the run lengths. The CDF is tLe cumulative proportion or percent of siginals given by the ith period following the shift. It should be
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JOHN N. DYER, BENJAMIN M. ADAMS, AND MICHAEL D. CONERLY
noted that the CDF completely characterizes the run length distribution, while the ARL does not. Additionally, the median run length (MRL) can be used in conjunction with the ARL and CDF, since it is a better measure of central tendency for skewed distributions such as the run length distribution.
Forecast Recovery One problematic characteristic of forecast-based monitoring schemes is the phenomenon of forecast recovery; that is, the process forecasts quickly adapt to a change in the process mean level. Hence, the expected forecast errors decrease in magnitude over tirne following a shift, and, depending on the model, converge to a new mean level. The impact of forecast error recovery on ARLs has been discussed (Tseng and Adams(1996), Hu and Roan(1996), Adams, Woodall, and Superville (1994), and Superville and Adamis (1994)), and the CDF technique has been recomnrended as a meaningful criterion for evaluating the performance of charts based on forecast errors. Lin and Adams (1996) found that when applied to AR(1) process forecast errors, the Individuals chart provides relatively high ARLs and CDF values, the exponentially weighted moving average (EWMA) chart provides low ARLs and CDF values, and the combined EWMA-Shewhart (CES) chart has the best properties from botlh charts (low ARLs and hiigh CDF values). High (low) CDF values are defined as those exhibiting a high (low) probability of initial shift detection relative to competing control charts. Initial shift detection refers to detection of a shift in the first few periods following the shift. The Individuals chart is a Shewhart chart based on forecast errors from individual observations. The EWMA chart is also based on forecast errors from individual observations. The design of the EWMA chart and discussion of the smoothing parameter A are addressed by Lin and Adains (1996) and Crowder (1989).
Paper Focus While other studies have focused on such techniques as removing autocorrelation from process data or autocorrelation-adjusted control limits for traditional charts, in this paper we focus on the performance of various control charts applied to forecast errors generated from a number of different timeseries models. The paper is in part an extension of recent research (Dyer (1997), Lin and Adams (1996), and Wardell, Moskowitz, and Plante (1994)) investigating forecast recovery and using traditional control charting schemes, as well as an evaluation of
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our proposed reverse moving average (RMA) control chart. Performance criteria are evaluated based on ARL, MRL, and CDF measures for charts designed to detect a specified shift in a process mean level. The standard deviation of the run length distribution (SRL) is also provided. The following two major issues will be addressed: * The relationship between thie expected forecast errors and the control clhart performance will be examined for traditional control charts and the RMA control chart. * Suggestions will be made concerning the most appropriate control chart for a variety of situations involving autocorrelated processes. In our paper, a description of the ARMA model with an introduction to the notation associated with various tirme-series mnodels is given. Three particular ARMA models and the concept of monitoring forecast errors are discussed. An introduction and description of traditional control charts and a proposed RMA control chart is provided. Also, an evaluation and discussioni of the ARL, MRL, and CDF performrance of traditional sclhemnes and the RMA control chart applied to forecast errors arising from the ARMA(1,1), AR(1), and MA(1) models is provided.
Models for Autocorrelated Data Two ARMA(p, q) models have been found to have wide application in statistical process control. The first model of interest is the ARMA(1,1) model. Wardell, Moskowitz, and Plante (1992) addressed the ARMA(1,1) model, as it is a reasonable fit to data for some manufacturing processes. The second model of interest is the ARMA(1,0), also known as the AR(1). Montgomnery and Mastrangelo (1991) and Alwan and Roberts (1988) addressed the importance of the AR(1) model in manufacturing processes. Atienza, Tang, and Ang (1998) discussed a time series approach to detecting level shifts in AR(1) processes. Lastly, the ARMA(O, 1), also known as the MA(1), is considered for the sake of completion of all possible first order ARMA(p, q) models. In the next section, we briefly discuss process shifts associated with the various time-series models before description of the models.
The ARMA(1,1), AR(1), and MA(1) Models In building an empirical model of an actual timeseries process, the inclusion of both autoregressive
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THE REVERSE MOVING AVERAGE CONTROL CHART FOR MONITORING AUTOCORRELATED PROCESSES
and mnoving average terms sometimes leads to a more parsinmonious model than could be achieved with either the pure autoregressive or pure moving average alone. This results in the mixed autoregressivemoving average model.!When both ternis are mixed in first order, the resulting model is the ARMA(1, 1) model. The models for in-control ARMA(1, 1), AR(1), an(l MA(1) processes are given by Equations (1), (2), and (3), respectively: Yt =
+ 0Yt_j + Et - OEt__I!
(1)
Yt = ( + oyt_i + Et,
(2)
Yt =
(3)
(-
Ot-l + Et,
where ( is a constant and the sequence of Et (t = 1, 2, ...) values are independent N(O, ,2 ) randomii variables. The ARMA(1, 1) process is stationary for Wl < 1 and 101< 1, the AR(1) process is stationary for 101 < 1, and the MA(1) process is stationary for all values of 0. Now, suppose a step shift of size c occurs in any of the ARMA(1,1), AR(1), or MA(1) processes between time periods r - 1 and r; that is, for all observations after the step shift, we observe Yt + c instead of Yt. The Box-Jenkins one-step-ahead (OSA) forecasts are defined by Yt = +^Y_ 1 -Ocut-, for the ARMA(1,1) process, Y =t + oYt-1 for the AR(1) process, and Yt= ± -Oct-, for the MA(1) process. The OSA forecast errors are calculated as et = Yt - Yt for all processes. The expected OSA forecast errors for an ARMA(1,1) process can be described mathematically as 0 c
t =1,2, . ... r-1 t=r
E(et) =
(4) 11
1-0
I c
t =r~ + kc, Ic = 1,2,...
Similar results for the AR(1) and MA(1) processes can be obtained by setting 0 = 0 or 56 = 0, respectively, in Equation (4). This general representation is consistent with the special cases of the ARMA(1,1) model presented in Atienza, Tang, and Ang (1998), Lin and Adams (1996), and Wardell, Moskowitz, and Plante (1994). When an ARMA(p,q) process undergoes a step shift in the mean, the expected value of the forecast of the process varies for a time and then converges to a new equilibrium level (Wardell et al. (1994)), referred to in this paper as the sustained level of the shift. This sustained level of the shift, denoted
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by E(e*), may be obtained as the limiting value (as t c* no) of E(et) in Equation (4). The response of the forecasts also causes the forecast errors to respond dynanmically, hence making the choice of contlrol chart selection difficult. An extensive discussion of the behavior of forecast errors arising frorm various ARMA(1,1), AR(1), and MA(1) processes is given by Dyer (1997). Only ARMA(1,1), AR(1), and l\IA(1) processes exhibiting forecast recovery are considered in this paper. The models to be further considered are those that were previously investigated by Wardell et al. (1994). The perfornmance of several traditional control charts applied to the forecast errors are examined in this paper as well as a new control chart, the RMA control chart. The traditional control charts and the RMA are discussed in the next two sections.
Traditional Control Charts Three traditional control charts have been recommended for monitoring the forecast errors arising from various autocorrelated processes. They include the Individuals (Ind) control chart, the EWMA control clhart, and the CES control chart. It is assumed that the underlying forecast errors, et, are iid normal random variables with mean po and variance a? Stepshifts iIn the process mean are to be detected. For an extensive discussion of these control charts applied to forecast errors, see Lin and Adams (1996) and Lu and Reynolds (1999a, 1999b). A brief description of the EWMA and CES control charts follows. The Exponentially Weighted Moving Average (Control Chart The EWMA control chart was first introduced as the geometric moving average chart (Roberts (1959)). The EWMA statistic is dlefined as Zt =Axt +(I- A)Zt-1
t =1,2,3,....
vwhere xt is the individual observed value at time t, 0 < A < 1 is a smoothing constant, and the starting value at t = 1 is Zo = i, or the mean of the underlying process if known. The EWMA comitrol chart signals if the EWMA statistic falls beyond the asymptotic control limits given by /lo mLo
~
-A 2-A'
where the value of L is a function of A, the desired inecntrol ARL, and the anticipated shift size deemed important. Selected values of L and A have been tabled for independent processes when the desired
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142
ARL and anticipated mean shift are stated (Lucas and Saccucci (1990) and Crowder (1989)). In most cases, the paranieters pqo and ao are unknown and must be estimated. The Combined EWMA-Shewhart Control Chart The combined EWMA-Shewhart (CES) control chart (Lucas and Saccucci (1990)) is designed by adding Shewhart control limits to an EWMA control chart. The chart signals if either the EWMA statistic falls beyond its control limits or if the most recent observation falls beyond the Shewhart control limits. The Shewhart limits are recommended to be wider than those of a Shewhart chart alone to avoid a reduction in the desired in-control ARL. Simulation can provide the optimal L, A, aiid Shewhart limnits for a specific desired in-control ARL for a specified shift in the mean. The advantage of the Shewhart control chart alone is its ability to detect large shifts quickly, hence providing low out-of-control ARLs. Small shifts are detected more rapidly with the EWMA control chart. Since the size of the shift is unknown beforehand, the CES control chart provides for relatively rapid detection of both large and small shifts in the process mean.
The Reverse Moving Average Control Chart The reverse moving average (RMA) control chart is based on the idea that if one can guess the time period r when forecast accuracy changed due to a step shift in the process mean level, then the average of all the forecast errors since the change occurred provides a good indicator of the presence of a process disturbance. This idea originated with Harrison and Davies (1964) and was further investigated by Atienza, Tang, and Ang (2002) for forecast monitoring utilizing a backward CUSUM. Unfortunately, in practical applications r is unknown. One possible way to overcome this problem is to calculate a series of moving averages at each time t, consisting of partial averages of the last n forecast errors. If we consider n forecast errors, we will calculate n moving averages. Consider the case where we arbitrarily choose n = 3. The series of three reverse moving averages (RMAs) at time t are defined as follows: RMAt,j = ct,
RMAt 2 =
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2t
and RMA,
3
=
3t+Ct-1+et-2
In general, i E et_j+1
RMAt,i = i
i
for i = 1.2,3,...n.
N(p = 0, cr) for the in-control Recall that et case. Since the phenomenon of forecast recovery is quickly manifested througlh the forecast errors, we are only interested in the first few values of RMA (n averages involving the last n forecast errors). Hence, the choice of n need not be too large and can be determined through simulation and knowledge of the degree of forecast recovery. For the purpose of process monitoring, each RMA statistic at time t can be compared against its own control limits, ±Li. As long as each RMAt,i statistic is within ±Li, the process is deemed in-control. Since each Li is a function of the forecast error variance., as the number of errors averaged increases, we can expect L1 > L2 > ... > L,,. The general form of
the control limnits can be calculated as Li = kord/i, where k is chosen to provide the desired in-control ARL. The choice of k can be obtained through simulation for a given value of n. It should be noted that each RMA statistic can be standardized using RMAt,i/(>/v7/) and more easily compared against the constant control limits ±k. The examples provided for constructing an RMA control chart are based on the assumption that constant control limits will be used with standardized RMA statistics. The notation RMAn implies the use of an RMA control chart of size n. Consider an example for the case where n = 6. Ta-
ble 1 column (a) contains 10 consecutive data points assumed to have occurred at time periods well after process startup. Points 1-6 are simulated N(0,1) random variables and points 7-10 have a mean shifted to 3. For time periods t = 1 to t = 10, columns (b) through (g) contain the n = 6 calculated RMA
values, while the last row of Table 1 contains the control limits for eaclh RMA component. At each time period, one may construct a new control chart containing each of the six RMAt,i values relative to the control limits. An example of control charts constructed at time t = 6, 7, 8 is provided in Figure 1. Note that each chart corresponds to a plot of the RMA values in a single row of Table 1. At time period 6 all plotted RMA values lie within the control limits, whereas the shift is indicated at time periods Vol. 35, No. 2, April 2003
THE REVERSE MOVING AVERAGE CONTROL CHART FOR MONITORING AUTOCORRELATED PROCESSES
143
TABLE 1. Data and RMA Statistics for the RMA Control Charts.
Time t
(a) Observed et
RMAt,1
RMA,, 2
RMAt, 3
RMA,, 4
RMAt.5
RMA,, 6
0.686 -1.362 0.040 -0.602 -1.159 -1.129 3.083 3.900 2.100 2.892
0.686 --1.362 0.040 -0.602 -1.159 --1.129 3.083 3.900 2.100 2.892
-0.338 -0.661 -0.281 -0.881 -1.144 0.977 3.492 3.000 2.496
-0.212 -0.641 -0.574 -0.964 0.265 1.951 3.028 2.964
-0.310 -0.771 -0.713 0.048 1.174 1.989 2.994
-0.480 -0.843 0.047 0.819 1.359 2.169
-0.588 -0.188 0.689 1.032 1.615
L5 1.34
L6 1.22
1 2 3 4 5 6 7 8 9 10 Control Limits
L1 to L6
(b)
(c)
(d)
(e)
LI
L2
L3
L4
3.00
2.12
1.73
1.50
7-8. In the next section we relate the performance of the traditional control charts and the RMA control chart applied in the case of the pure ARMA(1,1), AR(1), and MA(1) inodels.
(f)
(g)
in Table 2. The simulation program code is available on request from the authors. A description of the simulations follows.
Simulation Description Table 2 provides simulated ARL, MRL, and CDF performance of RMA control charts applied in normal processes using n = 5, 10, 15, 20, 25, and 30. If only one value of n is to be considered as having greater overall performance, it appears to be n = 15. Hence, we recommend n = 15 due to the RMA's performance across shifts of size c = 1lu, 2u6 , and 3o(J. The RMA chart (n = 15) is also competitive. in regards to performance measures, with the CES control chart designed to detect a shift of c = la, (A = 0.15). The RMA chart is less likely to signal at lag 1 in most cases, but far surpasses the CES chart CDF values thereafter, hence providing ARLs, MRLs, and CDF values which are generally equal to or better than the CES control charts. In general, when a shift has occurred, lower values of n provide for higher initial detection probabilities, while larger values of n provide for lower ARLs. An RMA control chart with 5 < n < 10 is recommended in cases where large shifts are to be detected and the sustained level of forecast recovery is not significant, while an RMA control chart with n = 15 is recommended in cases where small shifts are to be detected or extreme levels of forecast recovery are expected. RMA control charts with n = 15 are used throughout the remainder of this paper. RMA control limits based on standardized RMA statistics are provided
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The simulation programs were designed, compiled, and run in Microsoft FORTRAN PowerStarion for Windows, Version 4.0, utilizing FORTRAN 90. The program for finding ARLs was also used to estimate the appropriate control limits through trial and error. The simulations were conducted as folows:
1. A series of 4,100 ARMA(1,1) variates were generated by FORTRAN MSIMSL subroutine RNARM. These variates were the simulated observations, Yi's, for each of the models investigated. 2. The first 100 observations were used to allow the time series to stabilize. 3. A step shift was induced in the simulated observations at the 10 1st observation. The magnitudes of shift range from 0 to 3a, in increments of 1f.4. The appropriate Box-Jenkins OSA forecast and forecast errors were calculated. 5. The forecast errors were monitored by the control chart. The run lengths for the specified shift size were recorded. 6. Steps (1) through (5) were repeated 10,000
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144
When simulating to obtain control limits for the incontrol ARL, MRL, and CDF performance, the programs were run from zero state. When simulating the ARL, MRL, and CDF performance for a process that has experienced a shift, the programs were run from steady state. For all simulationis we set ( = 0 and o> = 1, without loss of generality. The results of these simulatioins are shown in Table 3 for twelve different choices of the parameters 0 and q. For ease of discussion, each different model was assigned a number that is listed in the first column of the table. The first 8 rnodels showxn in Table 3 are AR.MA(1,1) models, Models 9-10 are AR(1) models, and the last two models are MA(1) processes. The models chosen for inclusion in Table 3 are illustrative examples of the behavior of these models for other choices of the parameters.
Period #6 4-
32-
1~~ -, -2 -3-
1
---
0_
3
4
2
-4
Period #7 4
0~~~~~~~~~
-3-
Control Chart Recommendations _
_
____
_
-2
Period #8 4
7 -21 >
1
2
-4 FIGURE 1. RMA Control Charts for Time Periods t = 6, 7, and 8.
timnes for each model and process shift. The run length for the control chart was recorded for each simulation repetition and the ARL was obtained based on 10,000 repetitions. For the CDF programs, the percentages of runs producing a signal within each of the first 300 observations following the shift were obtained. One issue concerning the simulation should be addressed. Each program can be run to simulate a process in zero state or steady state. Zero state provides for simulating a process from start-up, while steady state provides for simulating a process that has been running in an in-control state for some time.
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The degree and rate of forecast recovery, as well as the time until the sustained level of forecast recovery occurs, provide a source of conflict in choosing among control charts for monitoring forecast errors (Dyer (1997)). Althouglh the behavior of the recovering forecast errors has a tremendous effect on control chart performance, the sustained level of the average forecast recovery error is equally important. As shown by Dyer (1997), the sustained level of forecast recovery and the first order autocorrelation structuire inherenlt in ARMA(1,1), AR(1), and MA(1) models are highly related. In general, if the first order autocorrelation is known, the sustained level of forecast recovery is also known, and vice versa. Extreme forecast recovery results in poor performance of all control charts relative to the monitoring of independent processes. Extreme forecast recovery is considered to be a sustained level of recovery of approximately E(ef) < 0.40c in this paper. This corresponds to ARMA(1,1), AR(1), and MA(1) processes with first order autocorrelation of approximately Pi > .60. The inodels in Table 3 that are considered to exhibit extreme forecast recovery are Models 2, 3, 4, 6, 7, and 9. Moderate recovery is considered to be a sustained level of recovery in the range 0.40c < E(ef) < 0.60c, which includes Models 1, 10, and 12 in Table 3. Low recovery is considered a sustained level of recovery of approximately E(ef) > 0.60c, and includes Models 5, 8, and 11 in Table 3. In the case of models exhibiting extreme levels of sustained forecast recovery, the CES control chart is
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THE REVERSE MOVING AVERAGE CONTROL CHART FOR MONITORING AUTOCORRELATED PROCESSES
145
TABLE 2. ARLs, MRLs, and CDF Values for Various RMA Control Charts for an Independent Process with Step Shifts.
n
Control Limits ±k
5 10 15 20 25 30
3.215 3.268 3.295 3.298 3.304 3.308
Shift c/Ua
0
1
2
3
Cumuilative Percentage of Signals Following Shift 2 nd
3 rd
4 th
5 th
6 th
7ti
ARL
MRL
SRL
1 st
300 300 300 300 300 300
213 212 211 211 212 212
296 299 297 296 295 293
0.10 0.09 0.09 0.09 0.09 0.09
0.29 0.25 0.24 0.23 0.22 0.22
0.55 0.46 0.44 0.42 0.41 0.41
0.87 0.73 0.69 0.66 0.65 0.65
1.16 0.97 0.93 0.90 0.88 0.88
1.50 1.29 1.23 1.18 1.16 1.16
1.83 1.53 1.46 1.41 1.38 1.37
5 10 15 20 25 30
13.47 10.55 9.78 9.54 9.47 9.43
10 8 8 8 8 8
11.77 7.83 6.47 6.01 5.78 5.71
2.19 2.17 2.23 2.30 2.34 2.43
6.38 6.20 6.18 6.27 6.31 6.42
12.59 12.22 12.16 12.22 12.23 12.32
19.93 19.92 19.76 19.82 19.82 19.89
27.39 27.75 27.66 27.71 27.74 27.80
33.35 35.74 35.71 35.72 35.75 35.81
38.80 43.78 43.84 43.75 43.72 43.76
5 10 15 20 25 30
3.11 3.12 3.14 3.14 3.15 3.15
3 3 3 3 3 3
1.69 1.54 1.54 1.54 1.54 1.54
13.41 12.68 12.46 12.35 12.32 12.37
41.17 39.72 39.35 39.08 38.94 38.89
67.48 66.05 65.68 65.47 65.26 65.23
84.59 83.76 83.48 83.29 83.17 83.16
92.89 92.84 92.72 92.62 92.54 92.52
96.05 97.04 97.00 96.96 96.92 96.90
97.74 98.80 98.80 98.79 98.78 98.78
5 10 15 20 25 30
1.70 1.74 1.75 1.76 1.76 1.76
2 2 2 2 2 2
0.72 0.73 0.74 0.74 0.74 0.74
43.69 41.95 41.51 41.35 41.24 41.17
87.39 86.35 86.07 85.93 85.76 85.72
98.29 98.05 98.03 97.97 97.93 97.93
99.86 99.85 99.84 99.85 99.83 99.83
99.99 99.99 100.00 100.00 99.99 99.99
99.99 100.00 100.00 100.00 100.00 100.00
100.00 100.00 100.00 100.00 100.00 100.00
recommended because of its overall ARL, MRL, and CDF performance across all shift levels. For small shifts, thie CES chart maintains ARLs similar to, but larger than, those of the EWMA control chart. For most larger shift sizes, the CES chart maintains ARLs that are nearly equal to or better thian those of the EWMA chart. The CES chart always provides ARLs lower than those of the Individuals chart. The CES chart consistently maintains good CDF and MRL performance relative to both the EWMA and Individuals charts. It should be noted if one is primarily interested in detecting small shifts, that the EWMA chart does perform better for ARMA(1,1) processes exhibiting extreme forecast recovery characteristics.
c = l,,, however, the EWMA chart consistently provides ARL, MRL, and CDF performance better than that of the CES cllart. and better ARL andl MRL performance than the RMA chart. The RMA chart mainltains the better initial CDF values in this case. The CES chiart in this case is no longer a good compromise between the Individuals and EWMA charts, since the EWMA chart perforins sliglhtly better in all aspects. For small shifts irl these moderately recovering models, the RMA control chart seems to I)ehave much like tlle CES chart behaves wlhen app)lied to an independent process, providing relatively low ARLs and high CDF values. When the CES and JIMA charts are compared thouglh, the CES chart still provides slightly lower ARLs and MRLs.
For models exhibiting moderate levels of sustained forecast recovery, the CES control chart fully embodies the characteristics provided to it by the Individuals and EWMA control charts. For a shift of size
For a shift of size c = 2oam, the CES control chart again performs as a compromise between the Individuals and EWMA control charts. The CES chart provides ARLs and MRLs equal to or slightly larger
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JOHN N. DYER, BENJAMIN M. ADAMS, AND MICHAEL D. CONERLY
than those of the EWMA control chart, and high CDF values almost equal to those of the Individuals control chart. The RMA control chart provides ARLs and MRLs lower than the other control charts, and CDF values that exceed the other control charts values by the second or third period following the shift, hence resulting in lower MRLs. For a shift of size c = 3a,8 the CES and RMA control charts perform almost identically, and both outperform the Individuals and EWMA control charts in nearly all respects. Both the CES and RMA control charts are recommended for models exhibiting moderate levels of sustained forecast recovery. The CES chart is a suitable choice for models exhibiting more moderate, nearing extreme, levels of recovery, while the RMA chart is suitable for models exhibiting levels of recovery approaching wlhat is considered to be a low degree of forecast recovery. For models exhibiting low levels of sustained forecast recovery, all of the control charts provide performance results similar to the case of the monitoring of an independent process. For models exhibiting low levels of sustained forecast recovery over shifts of size c = in, and 2a6 , the CES control chart again performs as a compromise between the Individuals and EWMA control charts. For the smaller shift, the EWMA chart provides ARLs and MRLs lower than that of the CES chart, but the CES chart's values are only slightly larger. The initial CDF values of the CES chart are also lower than those of the EWMA chart. The RMA control chart also outperforms the CES chart in all respects, and performs almost identically to the EWMA chart, but with slightly larger initial CDF values. For shifts of size c = 2o> and 3(7,X however, the CES chart provides ARL and MRL performance equal to or better than the EWMA chart, and provides CDF values that are always better. The RMA chart consistently outperforms the CES chart by all performance criteria. For these models, the RMA control chart is recommended because of its overall better performance over a wide range of shifts.
Control Chart Selection In this section, Figure 2 is provided to enable the practitioner to readily identify a suitable control chart when monitoring forecast errors arising from various ARMA(l,1), AR(1), and MA(1) processes. The CES and RMA control charts are the only recommended control charts, given their performance characteristics over a broad range of models and shift sizes. Although control chart simulation
Journal of Quality Technology
performance results are provided only for the models in Table 3, Figure 2 provides a guide for control chart selection when monitoring forecast errors arising from a wide variety of ARMA(l,1), AR(1), and MA(1) processes. Only processes exhibiting forecast recovery are considered. The recommendations in Figure 2 are based on simulation results obtained over every parameter combination contained in the given margins. Knowledge of forecast recovery was used in the design of each CES control chart simulated as recommended by Lin and Adams (1996), while all RMA control charts were simulated using n = 15. All recommen(lations are also based on the ARL, MRL, and CDF performance of each chart over shifts of size c = 1a6. 2cr6, and 3cr 6 . The CES control chart is recommended for use in processes where 'C' is given in Figure 2. These processes produce forecast errors which experience extreme levels of sustained forecast recovery. The RMA control chart is recommended for use in processes where 'R' is given in Figure 2. These processes produce forecast errors which experience low levels of sustained forecast recovery. Both the CES and RMA control charts are equally recommended for use in processes where 'B' is given. These processes produce forecast errors which experience moderate levels of sustained forecast recovery. In many cases where both charts are recommended, the CES chart has better performance than the RMA chart in detecting small shifts, while the RMA chart is better for large shifts. In any event, they perform practically the same for moderate size shifts (c = 2Cr6 ).
Constructing the RMA Control Chart The degree of forecast recovery and the magnitude of the shift to be detected should be considered in the construction of the RMA control chart applied to forecast errors. We recommend using an RMA control chart with n = 15 for models exhibiting extreme forecast recovery, or when small shifts are to be detected with any model, since larger values of n provide a greater probability of eventually detecting a shift. RMA control charts with smaller values of n, usually between 5 and 10, are suitable for models that exhibit moderate to small forecast recovery and when large shifts are to be detected, since smaller values of n generally result in lower ARLs for a given shift. We also recommend using standardized RMA statistics and constant control limits. At control chart startup, for time periods t =
1
VoL 35, No. 2, April 2003
THE REVERSE MOVING AVERAGE CONTROL CHART FOR MONITORING AUTOCORRELATED PROCESSES
147
TABLE 3. ARLs, MRLs, and CDF values for the ARMA(1,1) Process with Step Shifs.
ARMA Model
ARMA Parameters
=
0
p
=
0=
p
=
IND EWMA CES RMA
115 15 21 23
70 12 16 12
128 12 19 30
2
IND EWMA CES RMA
16 5 5 4
5 5 5 3
27 2 3 2
3
IND EWMA CES RMA
2 3 2 2
1 3 2 2
1
IND EWMA CES RMA
279 136 184 245
IND
2.52 2.03 2.41 2.23
4.67 6.81 4.04 6.95 4.65 7.05 5.97 10.95
8.61 10.75 9.55 16.59
10.06 11.70 13.10 15.48 21.03 26.94 12.28 15.48 18.91 22.24 27.82 32.98
39.19 26.43 39.73 59.89
45.92 44.39 49.90 76.33
51.24 61.61 60.31 86.26
55.71 75.18 70.22 92.00
59.32 84.96 78.17 95.26
3 1 1 1
51.74 7.26 49.74 41.51
73.98 29.61 72.48 83.61
84.65 61.16 84.60 96.49
90.34 84.01 92.22 99.36
93.62 95.02 96.49 99.90
95.44 98.74 98.54 99.97
96.74 99.68 99.55 99.98
191 96 136 166
284 141 177 259
2.52 7.15 3.03 2.23
3.23 8.32 3.88 3.87
3.62 9.35 4.41 4.93
3.98 10.05 4.86 5.76
4.28 10.74 5.27 6.46
4.62 11.31 5.66 7.06
5.00 11.91 6.13 7.57
RMA
209 63 84 130
124 49 63 65
255 61 86 172
17.43 8.03 16.74 12.46
19.56 10.19 18.98 20.55
20.28 11.86 19.99 24.90
20.74 21.03 21.38 21.77 13.18 14.37 15.33 16.20 20.62 21.11 21.65 22.27 27.42 28.89 29.93 30.73
3
IND EWMA CES RMA
98 35 29 35
1 28 2 2
185 34 44 81
51.74 55.39 9.43 13.56 49.97 53.69 41.51 58.37
56.17 16.34 54.75 64.46
56.56 18.57 55.31 67.07
56.81 20.34 55.76 68.39
57.07 22.16 56.24 69.63
57.33 23.80 56.65 70.49
1
IND EWMA CES RMA
290 217 259 283
199 145 178 198
295 239 265 286
2.52 7.01 3.03 2.23
3.23 7.92 3.74 2.71
3.60 8.63 4.19 3.12
3.93 9.22 4.54 3.53
4.23 9.83 4.90 3.92
4.53 10.39 5.24 4.32
4.88 10.86 5.64 4.55
2
IND EWMA CES RMA
237 164 184 236
143 113 118 151
286 174 213 269
17.43 8.02 16.73 12.46
18.81 8.94 18.02 13.24
19.39 9.67 18.67 13.88
19.70 10.11 19.00 14.33
19.97 10.69 19.36 14.74
20.22 11.16 19.68 15.14
20.54 11.70 20.03 15.45
3
IND EWMA CES RMA
130 121 85 142
1 87 2 34
236 124 141 224
51.74 9.45 49.94 41.51
53.66 10.33 51.83 42.68
54.22 11.09 52.38 43.39
54.38 54.55 54.70 11.54 12.11 12.67 52.60 52.84 53.02 43.73 44.10 44.39
54.90 13.22 53.22 44.68
2
EWMA CES
0.95 0 = -0.45 p = 0.971 =
Vol. 35, No. 2, April 2003
Cumulative Percentage of Signals Following Shift St 1 2 nd 3 rd 4 th 5 th 6 th 7 th
17.43 29.98 3.74 12.02 16.25 28.83 12.46 36.94
0.95 0.45 0.824
3
MRL SRL
1 0.95 0.90 0.072
2 =
Shift Size Control c/ag Chart ARL
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JOHN N. DYER, BENJAMIN M. ADAMS, AND MICHAEL D. CONERLY
148
TABLE 3. Continued
ARMA ARMA Model Parameters 4
Shift Size Control Chart c/a6
=
8.93 10.71 9.08 6.19
9.75 11.22 9.86 6.70
177 150 165 190
293 252 276 289
2.52 7.03 3.03 2.23
4.34 8.16 4.77 3.21
2
IND EWMA CES RMA
156 189 140 197
15 126 28 97
259 206 220 268
17.43 8.03 16.74 12.46
26.50 9.59 25.05 17.53
33.68 37.66 41.13 43.11 44.92 10.48 10.95 11.67 12.07 12.67 31.61 35.24 38.45 40.23 41.97 22.00 24.23 26.26 27.50 28.85
3
IND EWMA RMA
29 148 26 68
1 102 2 2
119 158 94 178
51.74 9.44 49.94 41.51
68.79 11.33 66.69 54.72
78.14 12.51 76.01 63.89
82.37 85.53 87.09 88.43 12.91 13.80 14.10 14.79 80.21 83.44 85.05 86.43 67.71 70.77 72.38 73.68
1
IND EWMA CES RMA
42 9 12 10
29 8 11 9
42 5 7 7
2.52 1.51 2.34 2.23
4.80 4.28 4.93 6.07
7.25 8.73 8.09 11.59
9.44 15.38 12.14 18.51
11.43 23.33 16.89 25.81
13.63 32.56 22.85 33.16
15.80 41.75 29.38 40.66
2
IND EWMA
6 4 4 3
4 4 4 3
6 2 2 2
17.43 4.24 16.32 12.46
30.89 18.82 30.74 38.13
41.74 44.32 46.58 63.24
50.64 68.66 63.18 81.04
58.27 85.34 78.25 90.96
64.76 94.00 88.84 95.89
70.25 97.80 94.86 98.22
2 3 2 2
1 2 2 2
1 1 1 1
51.74 11.04 49.81 41.51
75.35 51.05 75.07 84.81
87.20 86.69 89.87 97.42
93.42 97.91 97.43 99.69
96.51 99.79 99.53 99.98
98.12 99.02 100 99.99 100 99.94 100 100
177 35 51 75
122 30 43 52
180 26 39 74
2.52 2.59 2.45 2.23
2.94 3.38 2.95 2.98
3.68 4.34 3.71 3.93
4.13 5.13 4.30 4.91
4.59 6.09 4.95 6.05
5.10 7.28 5.68 7.18
5.67 8.46 6.53 8.43
61 14 16 16
37 13 15 12
71 8 11 15
17.43 17.81 5.36 4.11 16.27 16.82 12.46 14.25
20.08 7.89 19.26 18.79
20.99 10.34 20.61 22.34
22.11 13.87 22.34 26.41
23.21 17.61 24.32 30.27
24.30 22.03 26.44 34.33
16 8 6 5
1 8 2 3
26 5 6 5
51.74 7.03 49.76 41.51
52.02 8.72 50.20 44.41
55.60 14.63 54.03 52.78
56.52 20.22 55.49 58.49
58.30 28.43 58.19 64.99
59.56 37.38 60.84 70.72
60.94 46.64 64.03 75.96
0.475 0.025
CES RMA 3
IND EWMA
CES RMA 1
6
8.10 10.16 8.32 5.63
7.14 9.48 7.39 4.91
270 227 252 278
0 = 0.45
p
5.97 8.96 6.32 4.26
IND EWMA CES RMA
CES
=
1 st
1 0.95 9 = -0.90 p = 0.975 =
5
Cumulative Percentage of Signals Following Shift 7 th 6 th 5 th 4 th 3 rd 2 nd
ARL MRL SRL
0.475 0 = -0.45 p = 0.689 =
2
IND EWMA CES RMA IND
EWMA CES RMA 3
IND EWMA
CES RMA
Journal of Quality Technology
Vol. 35, No. 2, April 2003
THE REVERSE MOVING AVERAGE CONTROL CHART FOR MONITORING AUTOCORRELATED PROCESSES
149
TABLE 3. Continued
ARMA Model
ARMA Parameters
7
Shift Size c/l 6 1
Control Chart ARL IND EWMA
Cumulative Percentage of Signals Following Shift
MRL SRL
1 st
2 nd
3 rd
4 th
5 th
6 th
7 th
3.18 5.19 3.48 2.71
5.08 6.27 5.34 4.19
5.46 6.72 5.76 4.67
6.70 7.80 6.97 5.99
7.03 8.31 7.33 6.49
8.15 9.32 8.53 7.77
34.03 12.02 32.40 27.06
34.32 12.65 32.73 27.75
37.88 15.73 36.53 32.30
78.14 78.31 19.38 20.24 75.88 76.08 68.40 68.89
82.07 27.95 80.13 74.68
82.15 29.23 80.27 75.31
0 =-0.90
CES
205 48 67
p
RMA
111
138 41 57 77
IND EWMA CES RMA
70 21 20 25
25 19 17 15
103 13 18 28
17.43 5.70 16.52 12.46
18.66 28.03 28.60 6.36 8.84 9.30 17.70 26.54 27.12 13.20 20.91 21.57
IND
11 13 5 6
1 12 1 2
29 8 8 8
51.74 8.32 51.50 41.51
70.17 13.10 67.81 59.04
70.56 13.45 68.21 59.49
43 11 15 12
63 8 11 12
2.52 1.61 2.36 2.23
3.38 3.06 3.40 4.13
5.84 5.99 6.11 7.99
= -
0.475 0.737 2
3
EWMA CES RMA 8
1
IND EWMA
218 37 52 113
2.52 4.46 2.79 2.23
=-0.90
CES
p= 0.255
RMA
64 13 17 16
2
IND EWMA CES RMA
11 5 5 4
7 5 5 4
11 2 3 2
3
IND EWMA CES RMA
3 3 2 2
1 3 2 2
1
IND EWMA CES RMA
290 194 242 276
2
IND EWMA CES RMA
3
IND EWMA CES
-0.475
=b
9 0.95 0 = 0.00 p = 0.95 =
RMA
Vol. 35, No. 2, April 2003
6.85 9.03 7.77 11.59
8.78 9.92 11.85 14.30 19.20 26.22 11.08 13.76 18.36 16.54 20.92 26.41
17.43 20.51 3.87 9.35 16.29 19.93 12.46 22.54
32.86 35.77 26.65 40.40 34.42 41.17 45.96 58.59
44.62 62.21 57.00 74.93
47.53 73.96 66.07 82.55
54.28 86.39 78.47 90.28
3 1 2 1
51.74 9.18 49.76 41.51
57.20 24.42 56.08 62.42
77.32 80.52 63.99 81.61 78.76 85.39 90.16 95.83
89.16 95.87 95.47 99.27
90.80 98.72 97.83 99.75
94.58 99.82 99.57 99.96
199 131 167 192
295 209 246 281
2.52 7.05 3.03 2.23
2.93 7.95 3.51 2.95
3.24 8.72 3.93 3.44
3.56 9.26 4.30 3.86
3.84 9.89 4.66 4.28
4.14 10.44 5.03 4.64
4.51 10.97 5.44 4.96
235 125 150 212
143 90 104 134
277 129 164 248
17.43 8.06 16.72 12.46
17.81 9.03 17.22 14.08
18.10 9.89 17.61 15.04
18.39 10.49 17.99 15.73
18.64 11.16 18.31 16.33
18.92 11.78 18.68 16.86
19.27 12.34 19.09 17.34
130 81 65 115
1 62 2 18
232 80 100 186
51.74 9.42 49.92 41.51
52.00 52.17 52.35 52.53 10.69 11.68 12.52 13.36 50.33 50.63 50.85 51.08 44.08 45.07 45.77 46.30
52.72 14.12 51.31 46.65
52.96 14.84 51.63 47.04
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JOHN N. DYER, BENJAMIN M. ADAMS, AND MICHAEL D. CONERLY
150
TABLE 3. Continued
ARMA ARMA Model Parameters 10
Shift Size Control Chart c/o-,
5.60 8.10 6.32 9.78
6.41 10.16 7.59 11.98
7.21 12.67 8.98 14.26
17.43 3.89 16.25 12.46
19.80 22.27 24.47 7.41 12.14 18.39 18.99 22.34 25.67 21.13 29.95 38.90
26.67 26.51 29.68 46.85
28.87 35.53 34.70 54.80
31.00 44.88 40.26 61.91
59.79 63.25 27.64 42.76 59.26 64.43 72.90 82.99
66.35 58.44 70.37 89.49
69.36 72.07 71.68 82.29 76.88 82.73 93.67 96.26
IND EWMA CES RMA
119 21 29 36
81 19 26 25
120 14 20 34
2.52 2.24 2.44 2.23
2
IND EWMA CES
29 8 9 7
18 8 9 6
33 5 6 6
RMA
=
4.91 6.24 5.22 7.68
1 0.475 0 = 0.00 p = 0.475 =
11
ARL MRL SRL
Cumulative Percentage of Signals Following Shift 7 th 6 th 5 th 4 th 3 rd 2 nd 1 st
3.27 3.51 3.28 3.97
4.08 4.82 4.18 5.69
3
IND EWMA CES RMA
6 5 4 3
1 5 2 2
9 3 3 2
51.74 7.06 49.78 41.51
55.91 15.30 54.32 59.61
1
IND EWMA CES RMA
79 15 20 20
54 13 18 15
79 9 14 17
2.52 1.70 2.38 2.23
3.34 3.11 3.35 4.04
4.77 5.11 5.04 6.70
5.93 7.85 6.68 9.97
7.01 11.34 8.60 13.32
8.29 15.65 10.99 17.18
9.51 20.50 13.83 20.95
2
IND EWMA CES RMA
15 6 6 5
10 6 6 4
16 3 4 3
17.43 20.16 8.76 3.84 16.26 19.48 12.46 21.89
26.32 19.70 27.30 37.24
30.21 33.24 34.64 50.99
34.50 48.75 44.15 63.53
38.40 62.96 54.76 73.89
42.34 74.97 64.65 81.56
3
IND EWMA CES RMA
4 4 3 2
1 3 2 2
4 2 2 1
51.74 8.63 49.76 41.51
56.53 67.23 22.05 50.15 55.24 68.57 61.04 82.60
72.83 72.45 78.22 92.42
78.59 88.51 87.71 97.03
82.53 95.85 94.02 98.88
85.99 98.76 97.48 99.65
1
IND EWMA
78 19 26 25
120 14 20 33
2.52 2.24 2.44 2.23
2.93 3.05 2.91 2.99
5.13 4.63 4.99 5.37
5.49 5.43 5.45 6.53
7.04 7.64 7.32 9.32
7.45 9.02 8.01 10.85
8.92 11.96 10.07 13.76
0.00
0 =-0.45 p = 0.374
0 =-0.90
CES
0.497
RMA
117 22 29 36
2
IND EWMA CES RMA
23 9 9 8
11 9 8 6
30 5 6 6
17.43 3.90 16.26 12.46
17.84 5.18 16.82 14.44
28.82 11.68 27.93 29.40
29.27 14.53 28.87 33.11
36.50 25.32 37.38 45.80
36.98 30.32 39.11 50.16
42.35 43.22 47.21 60.44
3
IND EWMA CES RMA
4 5 3 3
1 5 2 3
6 3 3 2
51.74 7.07 49.78 41.51
52.04 9.08 50.27 44.78
71.81 26.80 70.32 72.42
72.11 32.78 71.08 76.08
81.75 56.74 81.89 89.11
82.08 63.85 83.26 91.24
86.98 80.86 89.98 96.09
12 p = 0.00 p
Journal of Quality Technology
Vol. 35, No. 2, April 2003
THE REVERSE MOVING AVERAGE CONTROL CHART FOR MONITORING AUTOCORRELATED PROCESSES ,\,
75
-. 95 -. dl -. -. 65 -. 55 .45 -. 35 -. 25 -. 15 -. 05 .00 .05 .15 .25 .35 .45 .55 .65 .75 .65
C C C C C C asCCCCCCC
.5 .75
C
C
C
C
C
C
C
.45
C
C
C
C
C
C
C
.45 .5 .25 .55 .05 .00
R
R
R
C C B B
BBB B C C C C C C |B B R: Rs B R R R R R R R R B R R R R R
-A
R
-. 05 .15 .'25 .35 -45 .55 .65
R R R R R R
C C CC B B B B
R R R R R R
R R R R R R
R R R R R
R R R R R
R R R R
C
R R R
C C
C C
B B
R R
R
C BC C C C B C C CBC C C C C C JB R B
B
R
R
R
B
R
R
R R e no Ch ar ReR oC R R RCR R R R R R R
B=Bt
heCSadRM
oto
Legend C = CES Control Chart Recommended R = RMA Control Chart Recommended B = Both the CES and RMA Control Charts Equally Recommended.
151
chart with n = 15. Startup control limits can be established using one of the two methods previously stated. Select the standardized RMA, control limits for a desired in-control ARL based on siniulation. 3. At each time period, calculate the standardized RMA statistics and plot them on a control chart versus the constant control limits. An alternative is to plot the most extreme RMA statistic on the control chart instead of all the RMA statistics.
Example FIGURE 2. Control Chart Selection Guide for ARMA(1, 1), AR(1), and MA(1) Processes.
to n, where n is the desired size of the RMA statistic, control limits can be established using one of two methods. Both methods have been shown, via simulation, to provide approximately equal results for ARLs, MRLs, and CDF values in the in-control case. The two methods are as follows: 1. For tinme periods t = 1 to n - 1, use the Individual control chart control limits based on the desired in-control ARL, applied to the forecast errors. For time periods t = n and beyond, use the constant RMAn control limits based on the desired in-control ARL, applied to the RMA statistics. 2. For time period t = 1, use the Individual control chart control limits based on the desired in-control ARL, applied to forecast errors. For time periods t = 2 to n - 1, use the n - 2 different constant RMA,=t control limits, each based on the desired in-control ARL, applied to RMA statistics. For time periods t = n and beyond, use the constant RMAn control limits based on the desired in-control ARL, applied to the RMA statistics. We have obtained good performance from the RMA control chart using the following construction guidelines: 1. Identify the underlying model (ARMA or AR) and its associated parameters, and determine if the degree of forecast recovery is extreme, moderate, or small using Equation (4). 2. For models exhibiting moderate or low forecast recovery when a large shift is to be detected, select an RMA control chart with 5 < n < 10. For all other models, select an RMA control
Vol. 35, No. 2, April 2003
Suppose we wish to design an RMA control chart to detect both small and large shifts in the process mrean, the desired in-control ARL is 300, and the underlying process is known to be autocorrelated. The steps are as follows: 1. Determnine the underlying ARMA(11), AR(1), or MA(1) time series model and parameters. For example, assume the model is determined to be an ARMA(l,l) with parameters 4 = -0.95 and 0 = -0.05. Based on Equation (4), the model will exhibit moderate forecast recovery, with a sustained level of forecast recovery of approximately 54% of the initial shift. 2. Since small shifts are important, select an RMA with n = 15. Startup control limits at time periods t = 1 to 14 are k = ± 2.935, and are based on the standard normal distribution (ARL = 300). The RMA constant control limits for the standardized RMA statistics and ARL = 300 are k = ±3.295. 3. For startup time periods t = 1 to 14, plot only the individual forecast errors versus the constant control limits of k = ±2.935. Since so many RMA statistics are calculated at each time period (n = 15), for time periods t = 15 and beyond plot only the most extreme standardized RMA statistic on the control chart versus the constant control limits of k = ±3.295.
Conclusion In this paper we provided an introduction to the RMA control chart, and a performance evaluation of various control charts applied to the forecast errors arising from simulated ARMA(l,1), AR(1), and MA(1) processes. Performance criteria were based oii the ARLs, MRLs, and CDF values of each con-
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152
JOHN N. DYER, BENJAMIN M. ADAMS, AND MICHAEL D. CONERLY
trol chart for step shifts in the process mean of size C = lUg,, 2 ue, and 3u6. Tables showed the performance results for a variety of the above processes. Additionally, it was found that the sustained level of forecast recovery following a shift had a substantial effect on the performance of each control chart examinied. The rate of recovery anld the absolute magnitude and signi of forecast errors prior to attaining the sustainied level of recovery were also founid to affect the performance of the control charts. It was shown that for a givenl shift arild sustained level of recovery, the control charts generally performii worse when applied to tlle forecast errors arising front AR(1) processes. This is due to the suddeni forecast recovery characteristics inlherent in these processes. Finally, Figure 2 was provided to enable the practitionier to readily identify a suitable control chart for application to forecast errors arising from various ARMA(1,1), AR(1), and MA(1) processes. The CES arid RMA control clharts were the only recommiriended conitrol charts because of their superior performance characteristics over a wide range of models anid shift sizes.
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Journal of Quality Technology
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Key Words: Autocorrelation, Autoregressive Moving
Average, Exponentially Weighted Moving Average, Forecasting Techniques, Statistical Process Control.
Vol. 35, No. 2, April 2003
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TITLE: The Reverse Moving Average Control Chart for Monitoring Autocorrelated Processes SOURCE: J Qual Technol 35 no2 Ap 2003 WN: 0309102184002 The magazine publisher is the copyright holder of this article and it is reproduced with permission. Further reproduction of this article in violation of the copyright is prohibited.
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