Controlling the EWMA S^2 control chart false alarm behavior when the in-control variance level must be estimated
CC ONTROLLING THE
EWMA S CONTROL CHART FALSE ALARMBEHAVIOR WHEN THE IN - CONTROL VARIANCE LEVEL MUST BEESTIMATED
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Sven Knoth
Department of Mathematics and StatisticsHelmut Schmidt UniversityPO Box 70082222008 Hamburg, Germany [email protected]
January 10, 2021 A BSTRACT
Investigating the problem of setting control limits in the case of parameter uncertainty is moreaccessible when monitoring the variance because only one parameter has to be estimated. Simplyignoring the induced uncertainty frequently leads to control charts with poor false alarm performances.Adjusting the unconditional in-control (IC) average run length (ARL) makes the situation even worse.Guaranteeing a minimum conditional IC ARL with some given probability is another very popularapproach to solving these difficulties. However, it is very conservative as well as more complexand more difficult to communicate. We utilize the probability of a false alarm within the plannednumber of points to be plotted on the control chart. It turns out that adjusting this probability producesnotably different limit adjustments compared to controlling the unconditional IC ARL. We thendevelop numerical algorithms to determine the respective modifications of the upper and two-sidedexponentially weighted moving average (EWMA) charts based on the sample variance for normallydistributed data. These algorithms are made available within an R package. Finally, the impacts of theEWMA smoothing constant and the size of the preliminary sample on the control chart design and itsperformance are studied. Keywords
Control charting; S EWMA; phase I/II; false alarm probability
Applying a surveillance scheme to monitor the stability of dispersion (homogeneity, scale or other related notions) isa common task used in industry to maintain, for example, the repeatability level of gauges, the uniformity of certainentities over time or space, the risk level of some financial asset, the stability of the variance underlying the controllimits of a mean control chart and so forth. To provide an explicit example, we look at a scanning electron microscope(SEM) at a semiconductor company, where a battery of daily measurements is executed for the sake of repeatabilitymonitoring. Typically, well-defined features (lines, spaces and so on) on a wafer are measured n = 5 times, and theresulting sample standard deviation is recorded on a Shewhart S chart — see Figure 1. Hence, it is not surprising thatShewhart, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) variance control charts arepresented in popular textbooks, including Montgomery [2009], pages 259, 414 and 426, respectively, and Qiu [2013],pages 74, 146 and 198, respectively. Here, we wish to investigate methods for calibrating EWMA schemes based on thesample variance S when the in-control (IC) level of the variance must be estimated based on a preliminary sample(phase I) of the IC data. The EWMA control chart was introduced by Roberts [1959] and gained much attention withand after Lucas and Saccucci [1990]. The initial works on using EWMA charts for dispersion monitoring include a r X i v : . [ s t a t . M E ] J a n WMA S for unknown σ A P
REPRINT l l lllllll l l l l ll l l ll ll ll l l l l lll l ll l l l l l l l l l l l l l l l l l l l l l l l l lll l l l l l l . . . . . Metrology/MET121/SPC_HT
WinSPC timestamp S pa c e_3 s _ Y _ S ho r t T e r m (0 points in WinSPC excluded)creation time: 03/18/09 16:42 time span: 21−Jan−09 −− 18−Mar−09 Figure 1: Shewhart S chart for monitoring the short-term repeatability of a scanning electron microscope (SEM);sample size n = 5 ; ordinate scale nm ; EWMA ( λ = 0 . , blue) added.Wortham and Ringer [1971], Sweet [1986], Domangue and Patch [1991], Crowder and Hamilton [1992], MacGregorand Harris [1993] and Chang and Gan [1994] — see Knoth [2005, 2010] for more details.With regard to EWMA charts, Jones et al. [2001] started the analysis of using estimated parameters instead of merelyassuming known ones. It became, for control charts in general, an important topic in the statistical process control(SPC) literature over the course of the last 20 years, and Jensen et al. [2006] and Psarakis et al. [2014] have provideddetailed surveys on this subject. Typically, it is assumed that the parameters are estimated through m phase I samples,each of size n , which means that nearly all the performance measures used for control charts become uncertain. Forexample, the well-known average run length (ARL; expected number of samples until the chart signals) becomes arandom variable. Two frameworks are commonly used to deal with this additional uncertainty. In the first framework,an unconditional form is calculated by applying a total probability mechanism. We will refer to this form as theunconditional ARL — others use notions such as the marginal or mean ARL. Below, we will illustrate that controllingthe unconditional IC ARL induces some puzzling side effects. The second framework started with Albers and Kallenberg[2001, 2004a,b], who considered probabilistic bounds for performance measures, such as the conditional ARL. Morerecent contributions from, for instance, Capizzi and Masarotto [2010], Jones and Steiner [2011] and Gandy and Kvaløy[2013], have stimulated a series of additional publications employing this approach. A popular motivation for thisguarantee-a-minimum IC ARL is that it incorporates appropriately the so-called practitioner-to-practitioner variability.This framework has been discussed extensively for Shewhart charts monitoring the normal variance in Epprecht et al.[2015], Guo and Wang [2017], Goedhart et al. [2017], Faraz et al. [2015], Faraz et al. [2018], Aparisi et al. [2018], andJardim et al. [2019]. Most of the latter works derived numerical procedures for calculating the limit adjustments, whichis much more difficult for CUSUM and EWMA control charts, where Monte Carlo simulations (bootstrapping for thephase I dataset) are typically used. Thus, it is not surprising that nothing has been published on monitoring the normalvariance for CUSUM and EWMA charts as of yet. Moreover, there are further problems with this framework. First, it istruly difficult to communicate the probabilistic bound for the random (conditional) IC ARL to a practitioner. Second,the modified limits are commonly quite wide, resulting in a prolongation of the detection delays. Therefore, Capizziand Masarotto [2020] proposed to re-estimate the modification regularly during phase II to tighten the limits. Third,the calculations of the actual modifications for CUSUM and EWMA charts are involved and time consuming. Whilethe last problem will be probably be solved soon, the other problems persist. Hence, we propose a different approach.We widen the limits of an EWMA S chart by assuring a certain unconditional IC run length (RL) quantile. Later,we will see that aiming at an unconditional IC RL quantile leads to a widening of the limits, whereas deploying theunconditional IC ARL can tighten them.Contrary to the case of the Shewhart variance chart, there are only a few contributions dealing with EWMA variancecharts under parameter uncertainty, namely, Maravelakis and Castagliola [2009], and more recently, Zwetsloot et al.[2015] and Zwetsloot and Ajadi [2019]. All together control the unconditional IC ARL. Maravelakis and Castagliola[2009] investigated EWMA charts utilizing ln S . From their unconditional out-of-control (OOC) ARL results wepick a few, in order to discuss what we call the unconditional ARL puzzle. In their Table 2, unconditional OOC ARLnumbers for an upper EWMA ln S with an unconditional IC ARL of 370.4 were given. We provide these numbersin Table 1, for λ = 0 . , n = 4 and m ∈ { , , , } as well as m = ∞ (known parameter case). These resultsinclude a non-remediable issue, namely, the favorable ARL values for small m suggest that small phase I samplesshould be utilized. In other words, the more reliable the estimate of the unknown σ (including in the known parameter2WMA S for unknown σ A P
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10 20 40 80 ∞ ARL 13.1 16.3 18.9 20.5 22.6Table 1: Side effects of using the unconditional ARL as a calibration target (IC 370.4): Unconditional OOC ARLvalues (standard deviation increased by 20%) for several phase I sizes, m ; sample size n = 5 ; EWMA ln S chart with λ = 0 . .case), the longer one has to wait to detect this specific increase, that is, controlling the unconditional IC ARL tightensthe limits substantially, resulting in this uncommon improvement in the detection behavior. However, this tighteninggreatly increases the probability of early false alarms. The heavy tail of the unconditional IC RL distribution enlargesthe corresponding mean (the unconditional IC ARL), while the probability of low RL values becomes larger at the sametime. Later on, we will discuss this issue in more detail. Zwetsloot et al. [2015] utilized the unconditional ARL toadjust their limits and obtained similar OOC ARL anomalies. Neither paper discussed these patterns. Yet, Chakraborti[2007] indicated that focusing on the unconditional ARL is dangerous. In sum, controlling the unconditional IC ARL isnot the way to go.It should be added that the ARL paradigm has been criticized apart from studying the estimation uncertainty influenceon control charts limits. For example, Yashchin [1985] wrote: “Though ARL is probably meaningful in the off-targetsituation, it can be highly misleading when the on-target case is under study (primarily because the set of possibleCUSUM paths includes ‘too many’ extremely ‘short’ members)” . For a more recent critique, see Mei [2008] or Kuhnet al. [2019] and the references therein. Yashchin [1985] also reported (for the competing CUSUM control chart): “In general, the user of a CUSUM scheme probably feels uneasy about specifying a particular ARL for the on-targetsituation; what he typically wants is that the scheme will not generate a false alarm within a certain period of time (say,a shift) with probability of at least, say, 0.99.” . This last passage refers to our design principle.In sum, we study and propose two key features: (i) A novel control chart design rule for incorporating estimationuncertainty that uses neither the misleading unconditional ARL nor the too conservative guarantee-a-minimum condi-tional ARL. We control the unconditional false alarm probability via an unconditional IC RL quantile. (ii)We utilize anumerical procedure that is more accurate than the Markov chain approximation [Maravelakis and Castagliola, 2009]and much quicker than Monte Carlo-based procedures [Zwetsloot et al., 2015].The paper proceeds as follows: In Section 2, we introduce the EWMA S chart in detail and illustrate the peculiaritiesthat emerge when some estimate of the IC level is simply plugged in. In addition, we elucidate the deceptive conceptof adjusting the unconditional IC ARL. Afterwards, in Section 3, we describe our novel approach and the numericalalgorithm used to obtain the unconditional RL quantiles. Eventually, in Section 4, we use this machinery to study theimpact of the actual EWMA design (smoothing constant) and of the phase I size m on both the resulting control limitmodification and the detection performance. In the last section, we present our concluding discussion. S under in-control level uncertainty EWMA schemes utilizing the sample variance S are one type of EWMA chart monitoring dispersion. Competitorsinclude ln S , which is used in Crowder and Hamilton [1992]; S , as in, for example, Mittag et al. [1998]; the samplerange R , which is found in Ng and Case [1989]; and a + b ln( S + c ) , as in Castagliola [2005]. Note that all thesepapers, including ours, consider normally distributed data. There are several reasons to prefer S . First, it is an unbiasedestimator of the variance. Second, EWMA S frequently exhibits the best detection performance – refer to Knoth [2005,2010]. Third, the calculation is more feasible if all the estimation and monitoring is done with S .Now, let { X ij } be a sequence of subgroups of independent and normally distributed data. Each subgroup i consists of n > observations X i , . . . , X in . As usual, we assume that the phase I data come from a stable process and that thevariance change occurs at the beginning of the monitoring period or never. Calculating the running sample variance S i , i = 1 , , . . . , S i = 1 n − n (cid:88) j =1 (cid:0) X ij − ¯ X i (cid:1) , ¯ X i = 1 n n (cid:88) j =1 X ij , we feed the EWMA iteration sequence in the usual way: Z i = (1 − λ ) Z i − + λS i , Z = z = σ . The EWMA smoothing constant, λ , is in the interval (0 , and controls the detection sensitivity. The EWMA sequence { Z i } is initialized with the IC variance level, σ . Here, we must estimate σ anyway.3WMA S for unknown σ A P
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We want to detect increases or two-sided changes in the variance level. Hence, the following (alarm) stopping times areutilized: L upper = min { i ≥ Z i > c u } ,L two = min { i ≥ Z i > c u or Z i < c l } . Note that introducing a lower reflection barrier to the upper scheme would diminish the inertia effects — see Woodalland Mahmoud [2005] for more details. It would, however, also increase the complexity and dismantle the rollingestimate feature of the plain EWMA sequence Z i . Moreover, the inertia effect is less pronounced for a S -based chartwith the intrinsic lower limit 0 compared to a mean chart, which would be unbounded from below. Therefore, we preferthe simpler design without a lower barrier.Typically, the control limits are chosen to provide a pre-defined IC ARL, for example, by aiming for E ( L ) = 500 . Inthe case of a known IC variance σ , there is a rich body of literature on calculating the ARL and solving the inversetask of determining control limits for a given IC ARL value. In this paper, we use algorithms from Knoth [2005, 2007]to compute the ARL, RL quantiles, and RL distribution for an EWMA S control chart. The related R package spc offers functions that make this calculation easy.We will start with a typical situation: Sample size n = 5 , EWMA constant λ = 0 . and target IC ARL . Thissetup calls for thresholds c l = 0 . and c u = 1 . for the two-sided EWMA alarm design, and the threshold c u = 1 . for the upper EWMA alarm design. For the former, we decided to use an ARL-unbiased design. Thisnotion was introduced by Pignatiello et al. [1995] and Acosta-Mejía and Pignatiello [2000], but the phenomenonwas discussed earlier in Uhlmann [1982] and Krumbholz [1992] (both in German), in Champ and Lowry [1994] and,presumably, in further publications. For more details, refer to the more recent Knoth [2010] and Knoth and Morais[2015]. In a nutshell, ARL-unbiased designs render the ARL maximum at the IC level; in this case, σ . For theunknown parameter case, Guo and Wang [2017] provided results recently for ARL-unbiased Shewhart S charts whileguaranteeing a minimum conditional IC ARL.In this paper, we use a phase I reference dataset consisting of m samples of size n and build the estimate, that is, thepooled sample variance ˆ σ = 1 m m (cid:88) i =1 s i , (1)where s i denotes the sample variance of the pre-run sample i = 1 , , . . . , m . For a discussion on appropriate estimatorsof the unknown σ , we refer to Mahmoud et al. [2010], Zwetsloot et al. [2015] and Saleh et al. [2015]. Here, we focuson the above “natural” estimator because it is unbiased (no further corrections are needed) and its distribution is readilyavailable, that is, a χ distribution with m × ( n − degrees of freedom. Zwetsloot et al. [2015] mentioned that “underin-control data the EWMA control charts show similar performance across all estimators,” where “in-control” refers toan uncontaminated phase I. Replacing this “natural” estimator with a more robust [Zwetsloot et al., 2015] or otherwisemore suitable estimator does not change the framework described below, but doing so makes the calculations morecomplicated. We want to emphasize, however, that all our theoretical and numerical results make use of the pooledvariance estimator (1). To use the aforementioned estimate means that the observed X ij are standardized, resultingin ˜ X ij = X ij / ˆ σ . In consequence, we run an EWMA chart design for σ = 1 = z with the limits mentioned above.Now, we wish to study the impact on the unconditional cumulative distribution function (CDF) P ( L ≤ l ) dependingon the phase I sample size m . Note that P ( L ≤ l ) covers two sources of uncertainty: (i) phase I estimation, and (ii)phase II monitoring. Applying the numerical algorithms described in the next section, in Figure 2, we illustrate thisCDF for several phase I sizes assuming a known σ during setup. The graphs in the two-sided case feature simplepatterns, namely, the smaller the phase I sample size m , the higher the probability that a false alarm is flagged by l for any l ∈ { , , . . . } . In the case of the upper chart, this relation remains valid for early values of l ≤ (roughlythe original RL median) only. For large values, it is reversed. For small phase I sample sizes, such as m ≤ , theunconditional IC RL distribution has heavy tails. For instance, the unconditional probability P ( L > ) is roughly0.1 and 0.02 for m = 10 and m = 30 , respectively. Given that, for example, Zwetsloot et al. [2015] truncate theirMonte Carlo simulations at l = 30 000 , some potential problems with the unconditional IC ARL and even some ICRL quantiles might be hidden. Nonetheless, using the unconditional IC ARL to adjust the limits is dangerous becausethe particular tails distort the expectation and explains the peculiar numbers in Table 1 taken from Maravelakis andCastagliola [2009]. Hence, calibrating upper EWMA variance charts by aiming for a certain unconditional IC ARL ismisleading. 4WMA S for unknown σ A P
REPRINT upper two-sided l P ( L ≤ l ) m = ∞ MRL m = ∞ l P ( L ≤ l ) m = ∞ MRL m = ∞ Figure 2: Unconditional IC RL CDF for the EWMA ( λ = 0 . ) S ( n = 5 ), phase I size m , unadjusted case. We want to adjust the EWMA control limits c u and c l (just the two-sided case) in order to achieve P ( L ≤ ¯ l ) = α , (2)where P () is the unconditional IC CDF of the RL L . In other words, we alter the limits so that ¯ l becomes theunconditional IC RL α quantile. Recall that there is a direct link between the ARL and an RL quantile of order α forShewhart charts with known IC parameters: RL α = (cid:6) ln(1 − α ) / ln (cid:0) − / ARL (cid:1)(cid:7) . This simple formula remainsapproximately valid for EWMA S charts in the IC case if σ is known: The median RL (MRL) is equal to and ≈
347 = (cid:100) ln(0 . / ln (cid:0) − / (cid:1) (cid:101) with an ARL = 500 for the upper and two-sided EWMA S charts,respectively, in Figure 2. However, this simple relationship is lost, if we deal with the unconditional IC CDF. In thebeginning of Section 4, we provide some illustrations of this phenomenon. This behavior is not surprising becausethe unconditional IC CDF is very different from the simple geometric distribution we exploited for the Shewhart chartRL statement. Using our rule (2), we tackle the problems observed in Figure 2 directly. Appropriate choices of α arethose that are smaller than 0.5 (we will use α = 0 . ), while ¯ l could be either derived from the ARL and RL quantilerelationship for known σ or set manually, as in ¯ l = 1 000 , which is used for a typical control chart in practice. Next, wedevelop an algorithm to calculate the unconditional CDF and use the result to solve the implicit function (2) numericallywith the secant rule.For calculating the unconditional CDF, we adhere to Waldmann [1986], who proposed the idea for EWMA controlcharts being used to monitor a normal mean with known IC parameters. We start with the upper chart, which onlyrequires an adjustment to its upper limit c u . Let p l ( z ) = P ( L > l | Z = z ) denote the survival function (SF) of theRL for known σ and starting values Z = z . Adding further arguments, such as the actual variance σ and controllimit c u , we produce its unconditional version p l, unc. ( z ; σ , c u ) = ∞ (cid:90) f ˆ σ ( s ) p l ( z ; σ /s , c u ) ds , l = 1 , , . . . (3)The formula (3) is related to (16) in Jones et al. [2001]. Note that only one integral is needed and that we consider theSF instead of the probability mass function of the RL L . To increase the computational speed for (3), the geometrictail behaviors [Waldmann, 1986] of p l ( z ; σ /s i , c u ) at each quadrature node s i are (for large l ) exploited individually.Unfortunately, it is lacking for p l, unc. ( z ; σ , c u ) , as has been mentioned previously in, for example, Psarakis et al. [2014].The density f ˆ σ () is roughly the probability density function (PDF) of a chi-square distribution (multiply the degrees offreedom). The numerical implementation for p l () is taken from Knoth [2007]. Because its presentation is not easilyaccessible, we provide some necessary details here. We begin with the transition (from z to z ) density of the EWMA S sequence as follows: δ ( z , z ) = 1 λ f χ ; n − (cid:18) n − σ (cid:20) z − (1 − λ ) z λ (cid:21)(cid:19) n − σ , S for unknown σ A P
REPRINT where f χ ; n − () and F χ ; n − () denote the PDF and CDF, respectively, of a chi-square distribution with n − degreesof freedom. Then we obtain the following recursions for the SF p l ( z ; . . . ) := p l ( z ; σ /s , c u ) : p ( z ; . . . ) = c u (cid:90) (1 − λ ) z δ ( z , z ) dz = F χ ; n − (cid:18) n − σ (cid:20) c u − (1 − λ ) z λ (cid:21)(cid:19) , (4) p l ( z ; . . . ) = c u (cid:90) (1 − λ ) z p l − ( z ; . . . ) δ ( z , z ) dz , l = 2 , , . . . . (5)A common approach to approximating the integral recursions is to replace the integrals by quadratures [Jones et al.,2001]. Fixed quadrature grids, however, have lower integral limit issues since this limit, (1 − λ ) z , depends on theargument p l ( z ; . . . ) , see Knoth [2005] for a more thorough analysis. Another idea is to apply a collocation type ofprocedure, as in Knoth [2007], Shu et al. [2013] and Huang et al. [2013], who transferred the collocation principle fromthe ARL integral equation in Knoth [2005] to integral recursions. Essentially, for every l = 2 , , . . . , we approximate p l ( z ; . . . ) ≈ N (cid:88) s =1 g ls T ∗ s ( z ) , with N suitably shifted Chebyshev polynomials T ∗ s ( z ) , s = 1 , , . . . , N (the T s () are the unit versions), T ∗ s ( z ) = T s − (cid:0) (2 z − c u ) /c u (cid:1) , z ∈ [0 , c u ] ,T s ( z ) = cos (cid:0) s arccos( z ) (cid:1) , z ∈ [ − , . Then we pick N nodes z r defined as (roots of T N ( z ) shifted to the interval [0 , c u ] ) z r = c u (cid:20) (cid:18) (2 i − π N (cid:19)(cid:21) , r = 1 , , . . . , N , and consider the following recursion on the grid { z r } , l = 2 , , . . . , N (cid:88) s =1 g ls T s ( z r ) = N (cid:88) s =1 g l − ,s c u (cid:90) (1 − λ ) z r T s ( z ) δ ( z r , z ) dz . These definite integrals must be determined numerically. Because they do not depend on l (only on s and r ), we calculatethem once and store them in an N × N matrix. Using this matrix and the starting vector g = ( g , g , . . . , g N ) (cid:48) derived from (4), we build a numerical approximation for (5) that provides a highly accurate numerical presentation ofthe SF p l ( z ; . . . ) used in (3) to determine the unconditional CDF of the RL L . The resulting SF p l, unc. ( z ; σ , c u ) is im-plemented in the R package spc as the function sewma.sf.prerun(l, lambda, 0, cu, sigma, n-1, m*(n-1),hs=z0, sided="upper") , see the Appendix for an example of an application. In Figure 3, we compare the approxi-mation accuracies of the collocation and the Markov chain framework. We investigate the EWMA S with λ = 0 . and sample size n = 5 and set ¯ l = 10 and α = 0 . , resulting in c u = 1 . . The integral in (3) is approximatedby the Gauß-Legendre quadrature with 60 nodes, while replacing the upper limit ∞ by 1.773 ( − − quantileof a chi-square distribution with m × ( n −
1) = 200 degrees of freedom divided by 200). This integral is deployedfor the SF p l, unc. ( z ; σ , c u ) and the unconditional ARL as well. The matrix dimension N indicates the size of thecollocation basis (see above) and the number of transient states of the Markov chain. From the two figures, we concludethat collocation with N = 50 yields much higher accuracy than the Markov chain with N = 500 . For calculating p l, unc. (1; 1 , c u ) , collocation needs about 1 second for N = 50 , while the Markov chain approximation requires 3, 6,15 and 22 seconds for N = 200 , , and , respectively. Eventually, we want to solve (2) as an implicit,continuous function of the upper limit c u numerically by executing a secant rule-type algorithm. The starting valueswill be slightly increased limits from the known parameter case.Turning to the two-sided case, we face additional problems. First, the numerical procedure (collocation proceeds“piece-wise” now) becomes more involved and therefore more time consuming. The general idea follows what hasbeen outlined above, so we will skip the details [for an elaborated description see Knoth, 2005]. The second problemis that we must now determine two limits, c l and c u , without an intrinsic symmetric limit design, unlike what weencountered for monitoring the normal mean. Hence, we must either deploy the symmetric design σ ± c by simplyignoring the asymmetric behavior of EWMA S or enforce something similar to the ARL-unbiased design used inthe known-parameter case. We try two concepts: (i) make the unconditional P σ ( L ≤ ¯ l ) minimal in σ = σ = 1 S for unknown σ A P
REPRINT unconditional IC SF, i. e. p l, unc. (1; 1 , c u ) = 1 − α = 0 . unconditional IC ARL N S F ( N ) Markov chaincollocation 0 100 200 300 400 500matrix dimension N A R L ( N ) / Figure 3: Approximation accuracies of the Markov chain and collocation for EWMA ( λ = 0 . ) S ( n = 5 ), phase Isize m = 50 , P IC ( L ≤ ) = 0 . . unconditional P σ ( L ≤ ) unconditional ARL σ P σ ( L ≤ ) unbiasedsymmetricquasi-unbiased σ u n c o n d i t i o n a l A R L · · · unbiasedsymmetricquasi-unbiased Figure 4: Judging “unbiasedness” in the two-sided case: The unconditional P σ ( L ≤ ) and ARL as functions of theactual standard deviation σ for EWMA ( λ = 0 . ) S ( n = 5 ), phase I size m = 50 , P IC ( L ≤ ) = 0 . .(with no loss of generality) — denoted henceforth as the “unbiased” version, and (ii) perform (i) for the knownparameter case (much faster) and expand the resulting limits ( c ∞ l , c ∞ u ) by incorporating the correction ξ > via c l = c ∞ l /ξ and c u = c ∞ u · ξ so that we achieve the unconditional P IC ( L ≤ ¯ l ) = α — we label this method as the“quasi-unbiased” method. All three approaches are illustrated in the following example: λ = 0 . , ¯ l = 10 , α = 0 . , n = 5 and m = 50 . In all cases, we determine the new limits numerically by essentially applying the secant rule (amore sophisticated implementation is the function uniroot() in R ). The resulting limits are (0 . , . (half width c = 0 . ), (0 . , . and (0 . , . (correction factor ξ = 1 . appliedto ( c ∞ l , c ∞ u ) = (0 . , . ) for the symmetric, unbiased and quasi-unbiased approaches, respectively. InFigure 4, we illustrate the resulting profiles for P ( L ≤ ) and the ARL as functions of the actual standard deviation σ . For both, we deployed the unconditional distribution. Note that the simple symmetric design exhibits profiles (SFand ARL) that are far from being unbiased. Moreover, the unconditional OOC ARL is very large for σ < σ = 1 (the lower limit is much smaller than those of the two competitors). Hence, from this point forward, we will drop thesymmetric limit design. The more sophisticated procedures feature rather equal profiles. In the sequel, we will applythe unbiased approach to be on the safe side. However, because it needs considerably more computing time than the7WMA S for unknown σ A P
REPRINT quasi-unbiased scheme, we recommend the latter for daily practice. We should note that the large unconditional ARLvalues are the result of the special setup utilized here. For instance, when σ is known, we observe an IC ARL of about , which is then inflated to about by two sources: the widened limits and enlarged tails of the unconditionalRL distribution. To achieve smaller values, ¯ l = 10 should be decreased or α = 0 . should be increased. It should benoted that that using σ = 1 does not violate the generality of our results. Hence, σ will refer to the standardized version σ = 1 . For example, σ = 1 . means that the OOC standard deviation is 20% larger than its (unknown) IC counterpart.Finally, we should emphasize that the unconditional α = 0 . RL quantile ¯ l = 10 differs substantially from measuressuch as “ Percentile marginal ” [Zhang et al., 2014], where the RL quantile for known σ replaces p l ( z ; . . . ) in (3). Thisweighted average over all conditional RL quantiles is much larger. For example, we obtain 244 325 for α = 0 . . Theexception is the unconditional ARL. To calculate it, we could utilize either (3) and plug in the conditional means orsum up p l, unc. ( z ; . . . ) over all l , which is just the expectation of the unconditional RL distribution. It remains somewhatunclear what exactly is being measured with Percentile marginal .After deriving these quite involved algorithms, we use them to illustrate the dependence of c u on the phase I size m . Utilizing our setup with ¯ l = 1 000 , α = 0 . and EWMA’s λ = 0 . , we start with the limits for known σ as phase I samples ( m ) m o d i fi e d c o n t r o lli m i t
10 20 50 100 200 500 10000.511.52 uppertwoknown
Figure 5: Modified control limits for P IC ( L ≤ ) = 0 . , upper and two-sided EWMA ( λ = 0 . ) S ( n = 5 ), m varies.a benchmark — c u = 1 . and ( c l = 0 . , c u = 1 . for the upper and two-sided cases, respectively. Forrealistic values of phase I sample sizes m between 10 and 1 000, we obtain widened limits, as can be seen in Figure 5.From the profiles, we conclude that the widening is less pronounced than might be expected. From sizes m = 50 on,the resulting limits on the control chart device in use would not really differ from the ideal case in which σ is known.Applying these new control limits changes the CDF profiles from those in Figure 2 to the ones presented in Figure 6.All profiles go through the point (¯ l, α ) by construction, of course. However, we observe that the smaller the phase size m , the more likely the very early false alarms.Widening the limits allows poor false alarm levels to be dealt with. However, the behavior in the OOC case hasdeteriorated. Using the limits from P IC ( L ≤ . , we show the unconditional CDFs for selected OOC cases( σ ∈ { . , . } ) in Figure 7. Note the poor detection behavior for smaller values of m . For m < , it is possible thatthe variance change will remain undetected over the entire planned monitoring time span ( ¯ l = 1 000 observations). It iseven worse for the two-sided case. Based on the profiles in Figure 7, we would recommend phase I sizes of at least 100.For more details, we refer the reader to the next section.In order to provide some more familiar representations and at least get an idea of the detection speed, we add someunconditional ARL values to the OOC case in Figure 8. The differences in the benchmark case are considerably largefor m < and become negligible only for m > . Hence, there is an obvious price to pay if we account for thephase I estimation uncertainty when calibrating the chart. Because the false alarm behavior is really important forpractical control charting in industry, the calibration strategy utilizing P ( L ≤ ¯ l ) = α seems to be passable despite theseside effects. Note that the even more conservative approach of guaranteeing a minimum conditional IC ARL yieldssubstantially larger unconditional OOC ARL results. 8WMA S for unknown σ A P
REPRINT upper two-sided l P ( L ≤ l ) l α m = ∞ l P ( L ≤ l ) l α m = ∞ Figure 6: Unconditional IC RL CDF for EWMA ( λ = 0 . ) S ( n = 5 ), phase I size m , P IC ( L ≤ ) = 0 . . upper two-sided l P ( L ≤ l ) m = ∞ σ = l α l P ( L ≤ l ) m = ∞ σ = σ = l α Figure 7: Unconditional OOC RL CDFs for EWMA ( λ = 0 . ) S ( n = 5 ), phase I size m , P IC ( L ≤ ) = 0 . . To reconcile the common IC ARL user to this method, we investigate the selection of the monitoring horizon ¯ l and falsealarm probability α for a given IC ARL of, for example, 500 and its impact on the actual adjustment of the controllimits accounting for the estimation uncertainty. To begin with, we set α = 0 . to ensure that the IC median run length(MRL) 348 (349 in the two-sided case) is achieved. From Figure 9(a) , we conclude that focusing on the unconditionalARL yields the smallest c U , followed by simply utilizing the c U value for known σ and, finally, the unconditionalMRL (median RL) design. Obviously, downsizing the upper limit seems to be counter-intuitive and results in morefalse alarms than intended. The slight increase of c U from the known σ case to the MRL- conserving approach offersa cautious and effective way of dealing with the estimation uncertainty. By changing α (or ¯ l ), we can see that for all α < . , the modified c U is larger than for known σ . In addition, decreasing α (and ¯ l , accordingly) increases c U further (except for very small α ). Of course, proper choices of α are 0.5 or smaller. In the two-sided case, all designssecuring some unconditional measure widen the original limits. In Figure 9(b), we plot only the upper value c U . For α < . , the unconditional RL quantiles induce wider limits than the unconditional ARL design. In summary, decidingon a reasonable combination (¯ l, α ) provides plausible and effective limit adjustments that can overcome the estimationuncertainty distortions. 9WMA S for unknown σ A P
REPRINT upper two-sided phase I samples ( m ) u n c o n d i t i o n a l OO C A R L
50 100 200 500 1000406080100 σ = m ) u n c o n d i t i o n a l OO C A R L
50 100 200 500 1000406080100 σ = σ = Figure 8: Unconditional OOC ARLs for EWMA ( λ = 0 . ) S ( n = 5 ) vs. phase I size m , P IC ( L ≤ ) = 0 . . upper two-sided . . . . . ¯ l c U c U for ARL = 500, if σ known c U aiming at unconditional ARL = 500MRL for ARL = 500, if σ known c U . . . . . . α α . . . . . . . ¯ l c U c U for ARL =500, if σ known c U aiming at unconditional ARL = 500MRL for ARL = 500, if σ known c U . . . . . . α α Figure 9: Choice of (¯ l, α ) within α = P ( L ≤ ¯ l ) and E ( L ) = 500 (all for known σ ) and its impact on the c U modification to secure α = P ( L ≤ ¯ l ) in the case of unknown σ , which will be estimated with a m = 50 × n = 5 phase I sample.Next, we wish to compare the detection behaviors of various values of the smoothing constant λ ∈ { . , . , . , . } .In all cases, we calibrate the schemes to ensure that P IC ( L ≤ ) = 0 . . Again, we consider samples of size n = 5 and a phase I study of size m = 50 . In Table 2 and Table 3, we provide some (unconditional) ARL values for the upperand the two-sided designs, respectively. To judge phase I‘s influence on the uncertainty, we compare the unconditionalARL numbers with the initial ones for a known IC variance. Interestingly, the new IC ARL results are very large butdecline with increasing λ . Similar patterns can be observed in the OOC case, where for σ = 1 . , the ARL numbersare doubled. For the medium size increase, that is, σ = 1 . , the change is much smaller. Note that the order betweenthe different EWMA designs remains stable, that is, λ = 0 . is the best for σ = 1 . , while λ = 0 . works best with σ = 1 . . The Shewhart S chart ARL results are added, which are considerably larger than all EWMA ones.Turning to the two-sided designs, we observe some slight differences. Most notably, the IC ARL values do not explode.Again, the OOC ARL results are tripled ( . ) and doubled ( . ) for small changes, while the increases are quite smallfor larger variance changes ( . , . ). For the simple Shewhart chart, the unconditional ARL values nearly coincidewith their known σ counterparts (increasing σ only). For the upper and two-sided designs, the patterns in the detectionranking remain stable. For small shifts, for example, the EWMA chart with λ = 0 . exhibits the best detection10WMA S for unknown σ A P
REPRINT λ Shewhart known σ c u E ( L ) E . ( L ) E . ( L ) m = 50 c u E ( L ) > × > ×
47 128 21 477 8 091 E . ( L ) E . ( L ) λ and P IC ( L ≤ ) = 0 . , upper chart. λ Shewhart known σ c l , c u E . ( L ) E . ( L ) E ( L ) E . ( L ) E . ( L ) m = 50 c l , c u E . ( L ) E . ( L ) E ( L )
11 240 6 803 4 961 4 386 3 500 E . ( L )
140 173 251 328 1 094 E . ( L ) λ and P IC ( L ≤ ) = 0 . , two-sided chart.performance for known and unknown IC variances. It should be stated that the Shewhart ARL performance is muchworse than the EWMA ARL performance for the changes considered here.Following the focus of this paper, we now examine the CDF profiles. Beginning with the IC versions for both designsin Figure 10, we conclude that for l ≤ ¯ l = 1 000 , the profiles look very similar. The λ = 0 . line lies above all theothers for these l , which changes for l > ¯ l , where it features the lowest values. All other curves follow according their λ values, that is, the larger the λ , the lower the l ≤ ¯ l and the higher the l > ¯ l . Comparing the results for the upper andtwo-sided EWMA designs, we observe much steeper developments for the latter ones. In summary, we conclude thatfor the interesting part, namely, l ≤ ¯ l , the IC behavior of P ( L ≤ l ) for all considered λ values and for both design typesdoes not really differ.Turning to the OOC case, we start with the upper EWMA chart and two different possible new σ values, namely, thesmall change σ = 1 . and the medium one σ = 1 . . Examining Figure 11, we observe the following stylized facts.The larger change is detected by l ≤ with probability one, while for σ = 1 . , we need the whole time span, thatis, l ≤ . Recall the corresponding expected values in Table 2, which are roughly 10 for σ = 1 . for all EWMAdesigns, while for σ = 1 . , they range from 70 for λ = 0 . to 150 for the largest λ < . Moreover, the order betweenthese λ values defined by their ARL values is reflected by the P ( L ≤ l ) profiles. Similar patterns can be recognizedfor the two-sided designs in Figure 12, where we included the results for decreased variances. Not surprisingly, thedetection performance for σ ∈ { . , . } is weaker compared to the upper chart profiles in Figure 11. However,detecting decreases of the same relative order proceeds more quickly.11WMA S for unknown σ A P
REPRINT upper two-sided l P ( L ≤ l ) λ = ¯ l α l P ( L ≤ l ) λ = ¯ l α Figure 10: IC CDFs of the RL L , P IC ( L ≤ ) = 0 . , EWMA (various λ ) S ( n = 5 ), m = 50 phase I samples. σ = 1 . σ = 1 . l P ( L ≤ l ) λ = ¯ l α l P ( L ≤ l ) λ = ¯ l α Figure 11: OOC CDFs of the RL L , P IC ( L ≤ ) = 0 . , upper EWMA (various λ ) S ( n = 5 ), m = 50 phase Isamples.After some first glimpses of the impact of the phase 1 sample size, namely, m , on the magnitude of the limit modificationin Figure 5, the resulting unconditional OOC ARLs in Figure 8 and the snapshots in Tables 2 and 3, some additionaldetails will be provided here to develop recommendations regarding some lower bound for m and the choice of thesmoothing constant λ . We start with the upper design and look at our selection of EWMA smoothing constants λ ∈ { . , . , . , . } . In the following Figure 13, the c u vs. phase I size m profiles are provided in two ways. First,the raw c u limits are presented, demonstrating the typical behavior of decreasing values if λ ↓ or m ↑ . Note that thechange from λ = 0 . to the Shewhart case ( λ = 1 ) is really pronounced. However, the amount of widening in thecontrol chart’s continuation region done to cope with estimating the IC value of the variance is not large. Comparedto the known parameter case, illustrated in Figure 13(b), we must increase the original c u by 5 to 10% (along the λ range) for small m < and by less than 3% for m > . The relative amount of change decreases with increasing λ .Similar results can be observed for the two-sided case. Next, we consider just the relative changes plotted in Figure 14.Except for the lower limit in the Shewhart chart, which is driven by the small sample size n = 5 creating difficultieswhile detecting variance decreases and is typically very close to zero, the profiles do not really differ from those for theone-sided case. Not surprisingly, the adjustment needed is larger than for the one-sided design. Overall, the widening ofthe control chart limits is about 10% or smaller for m ≥ . Even some crude rule of thumb for selected values of m could be derived, such as widen the limits by 10, 8, 5, 3, 2 and 1% for m =
20, 30, 50, 100, 200 and 400 for the12WMA S for unknown σ A P
REPRINT σ = 1 . σ = 1 . l P ( L ≤ l ) λ = ¯ l α l P ( L ≤ l ) λ = ¯ l α σ = 0 . σ = 0 . l P ( L ≤ l ) λ = ¯ l α l P ( L ≤ l ) λ = ¯ l α Figure 12: OOC CDFs of the RL L , P IC ( L ≤ ) = 0 . , two-sided EWMA (various λ ) S ( n = 5 ), m = 50 phase Isamples.phase I samples, respectively. In conclusion, utilizing the P ( L ≤ ¯ l ) = α design yields moderate changes to the controlchart limits. To identify a minimum m rule or a λ guideline, we now consider the unconditional OOC ARL for twomagnitudes of change.Starting with the upper chart, we present the unconditional OOC ARL values for σ = 1 . and σ = 1 . in Figure 15for the previously considered configurations. In Figure 15(a), we detect two segments in the ARL profiles. For smallvalues of m < , we observe huge ARL values for all considered values of λ . In addition, the smaller the λ , thesteeper the curves, which completely changes the order of the analyzed control chart designs. Given these patterns,it can be concluded that when attempting to detect small changes with an EWMA S chart, phase I samples with m ≥ are definitely needed. Namely, avoiding too many false alarms for small m < leads inevitably to thedelayed detection of small changes. Things look much better for the medium-sized change σ = 1 . in Figure 15(b),where for all λ and roughly all m , the adjustment of the upper limit c u only mildly distorts the unconditional OOCARL. Moreover, the popular choices λ = 0 . and = 0 . produce overall decent ARL levels, indicating that a reasonableapproach would be to recommend these two values, in general. Turning now to the two-sided case in Figure 16, weconfirm the judgments made for the upper schemes, where again small changes create problems for m < . The goodnews is that for the control chart user who is interested in detecting medium-sized and large changes, the proposedadjustments of the control limits do not destroy the ability of the applied EWMA charts to detect these changes. Ifflagging smaller changes is of concern, a larger phase I sample size is needed, that is, m ≥ , to obtain a detectionperformance that is comparable to that of the known parameter case. It should be noted that the ARL values for theShewhart chart are almost always too large to be displayed in Figures 15 and 16. The one and only exception in13WMA S for unknown σ A P
REPRINT (a) raw size (b) relative change to known parameter case
20 50 100 200 500 1000123456 phase I samples ( m ) c u λ =
20 50 100 200 500 10001.001.051.101.15 phase I samples ( m ) c u ( m ) / c u ( ∞ ) λ = Figure 13: Modified c u needed to achieve P IC ( L ≤ ) = 0 . , upper EWMA (various λ ) S ( n = 5 ), phase I size m = 15 , , . . . , . (a) lower limit (b) upper limit
20 50 100 200 500 10000.850.900.951.00 phase I samples ( m ) c l ( m ) / c l ( ∞ ) λ =
20 50 100 200 500 10001.001.051.101.151.20 phase I samples ( m ) c u ( m ) / c u ( ∞ ) λ = Figure 14: Ratios of the modified control limits to the original ones (known IC variance), P IC ( L ≤ ) = 0 . ,two-sided EWMA (various λ ) S ( n = 5 ), phase I size m = 15 , , . . . , .Figure 15(a) emphasizes that detecting a small variance increase in presence of an unknown IC variance level is difficultif only m ≤ observations are available to estimate the latter value. In order to control the false alarm behavior of EWMA S charts used for monitoring a normal variance, we proposedan approach that widens the limits in a balanced way. The resulting control chart design exhibits reasonable false alarmbehavior while still being able to detect medium-sized and large changes. To detect small changes, the phase I samplesize must be increased to m ≥ to achieve a performance that is comparable to the known parameter case. Moreover,we believe that the notion that we are calibrating for a certain false alarm probability α within a given number of controlchart values (the chart horizon ¯ l ) is easier to communicate to the statistical process monitoring community than declaringthat one guarantees with probability − α that the random conditional IC ARL is at least some nominal value, whichcorresponds more or less directly to ¯ l anyway. In addition, we recommend that λ = 0 . or = 0 . be used for setting upa reasonable EWMA S chart. Finally, it should be noted that we have prepared an R package (available from CRAN:14WMA S for unknown σ A P
REPRINT (a) σ = 1 . (b) σ = 1 .
20 50 100 200 500 100050100200500 phase I samples ( m ) u n c o n d i t i o n a l OO C A R L λ =
20 50 100 200 500 10006810121416 phase I samples ( m ) u n c o n d i t i o n a l OO C A R L λ = Figure 15: Unconditional OOC ARLs of the upper EWMA (various λ ) S ( n = 5 ) charts, P IC ( L ≤ ) = 0 . , phaseI size m = 15 , , . . . , . https://cran.r-project.org/ ) that contains the functions needed to calculate the unconditional RL quantiles andARL values as well as the control charts limits (including their adjustments for a given phase I size m ). Some examplesare given in the Appendix. Moreover, the shiny app https://kassandra.hsu-hh.de/apps/knoth/s2ewmaP/ provides a more convenient access. References
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A Software implementation
The functions utilized throughout the paper are implemented in the R package spc . For most of the figures and tables,the corresponding R code is provided as supplementary material to this contribution. Here, some basic functions( sewma.***.prerun() ) are described as follows. install.packages("spc") S for unknown σ A P