Stein EWMA Control Charts for Count Processes

Stein EWMA Con trol Charts for Coun t Pro cesses Christian H. W eiß ∗ † Jan uary 23, 2024 Abstract The monitoring of serially indep enden t or auto correlated coun t pro- cesses is considered, ha ving a Poisson or (negative) binomial marginal distribution under in-con trol conditions. Utilizing the corresp onding Stein iden tities, exponentially w eighted mo ving-av erage (EWMA) con- trol c harts are constructed, whic h can b e flexibly adapted to uncov er zero inflation, ov er- or underdisp ersion. The prop osed Stein EWMA c harts’ p erformance is inv estigated by sim ulations, and their usefulness is demonstrated by a real-w orld data example from health surveillance. Key words: attributes data; av erage run lengths; count time series; EWMA control charts; Stein iden tit y 1 In tro duction The sequen tial monitoring of coun t pro cesses ( X t ) t ∈ N = { 1 , 2 ,... } (i. e., where the X t ha ve a quantitativ e range contained in N 0 = { 0 , 1 , . . . } ) is of utmost imp ortance in many application areas, suc h as the quality con trol of manu- factured items (coun ts of defects or non-conformities) or health surv eillance (coun ts of infections or hospital admissions); see Montgomery (2009); W eiß (2015, 2018) for examples and references. The primary tool for managing sta- tistical pro cess control (SPC) for ( X t ) is the use of attributes control charts. These c harts inv olve plotting sp ecific statistics sequentially until signals in- dicate the necessity for corrective action. These statistics are computed in an online manner from the sequence of incoming coun ts ( X t ). The control ∗ Helm ut Sc hmidt Universit y , Departmen t of Mathematics and Statistics, Hamburg, German y . † Corresp onding author. E-Mail: weissc@hsu- hh.de . ORCID: 0000-0001-8739-6631 1 c hart triggers a signal (“alarm”) if the plotted statistic violates the speci- fied control limits (CLs). The CLs are derived from the so-called in-control mo del of ( X t ), i. e., a sto chastic mo del that assumes ( X t ) to op erate under stable conditions. Hence, an alarm (a v iolation of the CLs) is in terpreted as an indication of a p ossible pro cess deterioration (out-of-control situation). Therefore, if the alarm o ccurs when the pro cess ( X t ) is gen uinely out of con trol, it is considered a true alarm. On the other hand, an alarm under in- con trol conditions is regarded as a false alarm. It stands to reason that CLs should be c hosen to a void false alarms for as long as p ossible, and to receive a true alarm as so on as p ossible. The default metric for ev aluating these dura- tions until alarm is the a verage run length (ARL), i. e., the exp ected num b er of plotted statistics un til the first alarm. F or a more detailed description of the aforemen tioned terms and concepts as w ell as for further references, the reader ma y consult SPC textb o oks such as Montgomery (2009); Qiu (2014). V arious con trol charts for div erse count pro cesses hav e been dev elop ed during the last decades, cov ering serially indep enden t or auto correlated coun ts, and coun ts ha ving full N 0 as their range (unbounded coun ts) or just the finite subset { 0 , . . . , n } with sp ecified n ∈ N (b ounded counts). The most simple t yp e of control c hart is obtained b y plotting the coun ts X t themselv es against appropriately chosen CLs, whic h is called a c-chart if monitoring un b ounded coun ts and an np-chart for b ounded coun ts. Such types of Shewhart con trol c hart can b e classified as b eing memory-less as the t th plotted statistic do es not comprise information ab out earlier counts (at least not b ey ond the mere effect of auto correlation). Consequen tly , while Shewhart charts migh t b e quic k in detecting a sudden strong pro cess shift, they are slow in detecting small pro cess c hanges. Therefore, also sev eral memory-t yp e control c harts for coun ts hav e b een prop osed, where in the present researc h, our fo cus is on exp onen tially weigh ted mo ving-av erage (EWMA) c harts. Some recent references on EWMA-type control c harts for coun ts are Gan (1990); Borror et al. (1998); W eiß (2011); Rakitzis et al. (2015); Morais et al. (2018); Morais & Knoth (2020); Anastasop oulou & Rakitzis (2022a,b). F urther references (also on other t yp es of coun t con trol c harts) can b e found in W eiß (2015, 2018); Alevizakos & Koukouvinos (2020). The default version of the EWMA c hart plots the statistics Z 0 = µ 0 , Z t = λ · X t + (1 − λ ) · Z t − 1 for t = 1 , 2 , . . . (1) against a sp ecified low er and upp er CL (LCL and UCL, resp ectiv ely), where µ 0 > 0 denotes the in-control mean of ( X t ). The smo othing parameter λ ∈ (0 , 1] in (1) con trols the strength of the memory (the smaller λ , the stronger the memory). If λ = 1, then (1) reduces to the c- or np-chart, resp ectiv ely . 2 All the aforemen tioned coun t EWMA charts are mainly designed to detect shifts in the process mean, although they may sometimes (“acciden tally”) also react to increases in v ariance or changes in the autocorrelation struc- ture. In the present research, ho w ever, our fo cus is on “more sophisticated” out-of-con trol scenarios, namely where the mean c hanges together with fur- ther distributional prop erties (i. e., where the pro cess c hange cannot b e traced bac k to a sole change in the mean parameter), or where “purely distributional c hanges” (not affecting the mean) happ en. Suc h “distributional changes” migh t b e increases or decreases in dispersion compared to the in-control mo del (o v erdisp ersion or underdisp ersion, resp ectively), or an excessiv e n um- b er of zero counts (zero inflation), to men tion those being most relev ant in practice. The basic idea of our approach is as follows. Man y common count distributions can b e characterized by a type of momen t iden tity , referred to as Stein identity , which has to hold for a large class of functions (e. g., all b ounded functions on N 0 ). The idea to dev elop suc h identities dates back to Stein (1972, 1986), and further con tributions and references can b e found in Sudheesh (2009); Sudheesh & Tibiletti (2012); Landsman & V aldez (2016). Recen tly , suc h Stein identities w ere successfully used to develop p ow erful go o dness-of-fit (GoF) tests for coun ts, see Betsch et al. (2022); W eiß et al. (2023) for Stein-t yp e GoF-tests for indep enden t and identically distributed (i. i. d.) coun ts, and Aleksandro v et al. (2022a,b) for tests for count time se- ries. Thus, it suggests itself to utilize these Stein iden tities also for developing sequen tial test pro cedures, namely count con trol c harts for relev ant t yp es of in-con trol mo del. In a first researc h (see W eiß, 2023), this idea w as tried for the sp ecial case of i. i. d. Poisson counts, and the achiev ed ARL p erformance w as quite app ealing. This motiv ates to dev elop and in v estigate Stein-based con trol charts for count data on a m uc h broader scale, namely for v arious dif- feren t count distributions and not only for i. i. d. but also for time series data. More precisely , w e fo cus on the three most common count distributions in practice, namely Poisson (Poi), negative binomial (NB), and binomial (Bin), and w e include first-order autoregressiv e (AR(1)) mo dels ha ving either P oi-, NB-, or Bin-distributed marginal distributions in our research. The outline of this c hapter is as follows. In Section 2, we briefly presen t the count mo dels used for this research, and we pro vide the Stein iden tities for the P oi-, NB-, and Bin-distribution. These are used in Section 3 to con- struct no v el EWMA-t yp e con trol c harts for coun ts, where the c hart design with resp ect to diverse out-of-control scenarios is discussed in detail. In Sec- tion 4, results from a simulation study are presented, which allow to analyze the Stein EWMA c harts’ ARL performance. Section 5 then in vestigates a real-w orld data example on registrations in the emergency departmen t of a 3 c hildren’s hospital, which illustrates the application and in terpretation of the no vel Stein EWMA charts in practice. Finally , Section 6 concludes the article and discusses p ossible directions for future research. 2 Coun t Mo dels and Stein Iden tities The tw o most common distributions for unbounded counts are the Poi- and NB-distribution, while the Bin-distribution is the default c hoice for b ounded counts; see Johnson et al. (2005) for details and prop erties. The P oi( µ )-distribution with mean µ > 0 is kno wn to be equidispersed, i. e., its v ariance σ 2 satisfies σ 2 = µ . By contrast, NB( ν , ν ν + µ ) with ν , µ > 0 ex- hibits ov erdisp ersion relative to the P oi-distribution, b ecause its v ariance σ 2 = (1 + µ ν ) µ alwa ys exceeds the mean. The Bin( n, µ/n )-distribution with n ∈ N and µ ∈ (0 , n ), in turn, refe rs to coun ts having the b ounded range { 0 , . . . , n } . There is a h uge v ariet y of time series models related to the Poi-, NB-, or Bin-distribution, see W eiß (2018) for a comprehensiv e ov erview. Here, many mo dels use a c onditional P oi-, NB-, or Bin-distribution (regression mo dels) while the marginal distribution do es not b elong to an y parametric model family . How ever, to b e able to apply a Stein identit y to giv en data, w e need to sp ecify the corresp onding mar ginal distribution. Regarding the count time series mo dels prop osed so far W eiß (2018), only a considerably smaller n umber of mo dels has a P oi-, NB-, or Bin- mar ginal distribution. F or the sak e of studying p ossible effects of serial dependence on the charts’ p erfor- mance, w e shall consider three of the AR(1)-type pro cesses surv ey ed by W eiß (2008), whose mo del definitions use so-called thinning op erators as integer substitutes of the m ultiplication: • the Poi-INAR(1) pro cess (integer AR) defined by X t = ρ ◦ X t − 1 + ϵ t with i. i. d. ϵ t ∼ P oi  µ (1 − ρ )  , (2) • the NB-I INAR(1) pro cess (iterated-thinning INAR) defined b y X t = ρ ⊛ π X t − 1 + ϵ t with i. i. d. ϵ t ∼ NB( ν, π ) and π = ν µ (1 − ρ )+ ν , (3) • the BinAR(1) pro cess defined by X t = α ◦ X t − 1 + β ◦ ( n − X t − 1 ) with β = (1 − ρ ) µ n , α = β + ρ. (4) 4 Definitions (2) and (4) use the binomial thinning op erator, which is defined b y requiring a conditional Bin-distribution, namely θ ◦ X | X ∼ Bin( X , θ ) for θ ∈ (0 , 1). Definition (3), in turn, uses iterated thinning defined as ρ ⊛ π X = P ( π ρ ) ◦ X i =1 Y i , where the coun ting series ( Y i ) is i. i. d. according to Y i ∼ 1 + NB(1 , π ). The crucial p oin t for the subsequent research is the follo wing: Mo dels (2)–(4) lead to stationary Marko v chains with AR(1)-lik e auto corre- lation function (A CF) ρ ( h ) = ρ h for time lags h ∈ N , where ρ ∈ (0 , 1) for all three mo dels. The stationary marginal distributions are P oi( µ ), NB( ν, ν ν + µ ), and Bin( n, µ/n ), resp ectively , see W eiß (2008). The parameter ρ allo ws to con trol the exten t of serial dep endence, where the boundary case ρ → 0 leads to i. i. d. counts. Indep enden t of the v alue of ρ , we are alwa ys concerned with a Poi-, NB-, or Bin-marginal distribution. Any of these three distributions is uniquely c haracterized by a corresp onding Stein identit y , see Sudheesh & Tibiletti (2012): • X ∼ P oi( µ ) if and only if E  X f ( X )  = µ E  f ( X + 1)  (5) holds for all b ounded functions f : N 0 → R ; • X ∼ NB( ν, ν ν + µ ) if and only if ( ν + µ ) E  X f ( X )  = µ E  ( ν + X ) f ( X + 1)  (6) holds for all b ounded functions f : N 0 → R ; • X ∼ Bin( n, µ/n ) if and only if ( n − µ ) E  X f ( X )  = µ E  ( n − X ) f ( X + 1)  (7) holds for all b ounded functions f : N 0 → R . Iden tity (5) is commonly referred to as the Stein–Chen identit y (see Chen, 1975). Note that the iden tities (5)–(7) constitute non-trivial statements only if f is not constan t on N 0 , and if f is not identical to zero on N . In the subsequen t Section 3, we shall deriv e con trol charts from the iden tities (5)–(7). As these identities ha ve to hold for all bounded functions f under in-con trol conditions, w e can select an y c hoice of f for defining the con trol c harts. This degree of freedom can b e used to achiev e particular sensitivity regarding a sp ecified t yp e of out-of-control scenario. 5 3 Stein EWMA Charts for Coun ts The three identities (5)–(7) alw a ys dep end on three types of momen t: the mean µ , the momen t E  X f ( X )  , and a momen t in v olving f ( X + 1). The idea of the GoF-tests in Aleksandrov et al. (2022a,b); W eiß et al. (2023) as w ell as of the P oisson EWMA charts in W eiß (2023) w as to deriv e a statistic b y solving the identities in a certain wa y , and to substitute the inv olv ed p op- ulation moments by appropriate types of sample moments. In W eiß (2023), t wo types of constructing an EWMA con trol c hart w ere analyzed, and it turned out that one of these types is clearly sup erior, namely the one called “ABC-EWMA chart”. Building on this exp erience, w e prop ose the following Stein EWMA charts for sequen tially monitoring Poi-, NB-, or Bin-coun ts, resp ectiv ely , where E 0 [ · ] expresses that the exp ectation is computed with re- sp ect to the in-con trol mo del. F or all three Stein EWMA charts, w e compute A 0 = E 0  X f ( X )  , A t = λ · X t f ( X t ) + (1 − λ ) · A t − 1 , C 0 = µ 0 , C t = λ · X t + (1 − λ ) · C t − 1 , for t = 1 , 2 , . . . (8) F urthermore, we compute for the • P oi( µ 0 ) in-control mo del: B 0 = E 0  f ( X + 1)  , B t = λ · f ( X t + 1) + (1 − λ ) · B t − 1 , Z S 0 = 1 , Z S t = A t B t C t , for t = 1 , 2 , . . . ; (9) • NB( ν, ν ν + µ 0 ) in-control mo del: B 0 = E 0  ( ν + X ) f ( X + 1)  , B t = λ · ( ν + X t ) f ( X t + 1) + (1 − λ ) · B t − 1 , Z S 0 = 1 , Z S t = ( ν + C t ) A t B t C t , for t = 1 , 2 , . . . ; (10) • Bin( n, µ 0 /n ) in-control mo del: B 0 = E 0  ( n − X ) f ( X + 1)  , B t = λ · ( n − X t ) f ( X t + 1) + (1 − λ ) · B t − 1 , Z S 0 = 1 , Z S t = ( n − C t ) A t B t C t , for t = 1 , 2 , . . . (11) 6 W e define the resp ective Stein EWMA chart by plotting the statistics Z S t against appropriately c hosen LCL < 1 < UCL, where we exp ect Z S t to v ary closely around 1 under in-control assumptions. The actual c hart design now comprises t wo steps. First, w e hav e to select the function f inv olv ed in (8)–(11). Here, one could generally choose an y b ounded function o n N 0 , recall (5)–(7), except trivial c hoices lik e f b eing constan t on N 0 , or f ≡ 0 on N . But not any c hoice of f will lead to an ap- p ealing c hart p erformance. Instead, f has to b e chosen with resp ect to the an ticipated out-of-con trol scenario, in an analogous w a y as prop osed b y Alek- sandro v et al. (2022a,b); W eiß et al. (2023) and W eiß (2023). The idea is to in terpret f as a weigh t function within the “A- and B-moments”, which puts unequal w eight on the in tegers in N 0 . F or a giv en anticipated out-of-control scenario, one should put most w eight on those regions of N 0 where one gets the strongest departures from the in-con trol mo del. If w e wan t to uncov er zero inflation, for example, w e should c ho ose an f with relativ ely large w eight close to zero counts, whereas for “general” o verdispersion (i. e., if the proba- bilit y mass function (PMF) is flattened compared to the in-control model), it is adv an tageous to put more w eigh t on large counts. So in accordance to the recommendations in W eiß (2023), the follo wing c hoices of f are considered in these tw o cases: • for uncov ering o v erdisp ersion relativ e to the in-con trol mo del, we use the linear weigh ts f ( x ) = | x − 1 | ; • for uncov ering zero inflation relativ e to the in-con trol model, we use the ro ot weigh ts f ( x ) = | x − 1 | 1 / 4 . The case of underdisp ersion has not been in v estigated by W eiß (2023). Th us, as a starting p oint, we follow the findings of W eiß et al. (2023) on GoF-tests and • for uncov ering underdisp ersion relative to the in-con trol mo del, we use the inv erse w eights f ( x ) = 1 / ( x + 1). The describ ed approach for choosing the weigh t function is later illustrated in some more detail when discussing Figure 1. The ARL p erformance of the aforemen tioned choices of f is analyzed in Section 4 b elo w. As we shall see, the underdisp ersion scenario is muc h more demanding than ov erdisp ersion or zero inflation. Therefore, further choices for f shall b e considered later in Section 4.2. After having sp ecified f , the second step of chart design is the choice of the triple ( λ, LCL , UCL). Assume for the moment that the smo othing param- eter λ has already b een fixed. Then, LCL and UCL are determined based 7 on ARL considerations. While differen t ARL concepts exist in the litera- ture (Knoth, 2006), it is common to use the zero-state ARL for ev aluating the in-con trol p erformance. Th e most simple solution (the one used here) are symmetric CLs of the form LCL = 1 − L and UCL = 1 + L , where L is c hosen suc h that the desired in-control ARL (ARL 0 ) is met in close approximation (the textb o ok choice is the target v alue 370). Alternativ ely , one could define asymmetric CLs b y also considering the out-of-control ARL p erformance, but this is only reasonable if a particular out-of-con trol scenario has b een fixed. F or example, if one assumes that the out-of-control mo dels differ from the in-con trol one solely in terms of the mean µ , then asymmetric CLs ma y allo w to obtain an un biased ARL performance with resp ect to µ , i. e., the ARL as a function of µ is maximal in the in-control mean µ 0 and decreases symmetrically around µ 0 , see Section 4 in Morais & Knoth (2020). But as w e shall consider a broad v ariety of out-of-con trol scenarios, we restrict to symmetric CLs here, i. e., µ 0 ∓ L for ordinary EWMA and 1 ∓ L for Stein EWMA charts. A t this p oint, let us recall that the ab ov e approach for choosing f assumes that we ha v e sp ecified a single relev an t out-of-control scenario, suc h as “zero inflation”. If the application context allows for various out-of-control sce- narios (e. g., if underdisp ersion could also b e p ossible), one can run multiple Stein EWMA c harts in parallel, each designed for a different t yp e of dete- rioration. This is later done in Section 5 when monitoring the emergency coun ts. There, one Stein EWMA c hart is designed to detect o v erdisp ersion, another one for zero inflation, and a third one for underdisp ersion. Then, the observ ed pattern of alarms enables a kind of targeted diagnosis. F or example, if the “zero inflation”-c hart is the first to trigger an alarm, we conclude that w e might b e confronted with zero inflation (rather than with underdisp ersion etc.). Certainly , if running man y charts in parallel, the risk increases that one of these charts gives a false alarm (“multiple testing”). Generally , the risk of false alarms is con trolled b y setting a target v alue ARL 0 for the in-con trol ARL. As describ ed b efore, in our simulation study , w e set ARL 0 ≈ 370 for eac h single chart for comparabilit y . But if m ultiple control c harts are applied sim ultaneously , it would also be p ossible to determine their CLs suc h that the joint in-con trol ARL meets a target v alue, i. e., the ARL w ould be defined as the exp ected time un til one of the charts triggers the first alarm. Another remark refers to the c hoice of the smo othing parameter λ . T o ensure a sufficien t memory , one often c ho oses small v alues for λ , suc h as λ ∈ { 0 . 25 , 0 . 10 , 0 . 05 } . The actual choice of λ migh t b e done based on out-of- con trol ARLs. F or example, we could first compute several candidate triples ( λ i , LCL i , UCL i ) having the same ARL 0 . Then, we fix a sp ecified out-of- 8 con trol scenario and select that triple where the corresp onding out-of-control ARL is closest to a given target v alue. In this w a y , one might end up with differen t λ for different Stein EWMA charts. But as we choose f ( x ) with re- sp ect to differen t out-of-control scenarios, comparable c hart designs are most easily obtained by a unique λ . F urthermore, all Stein EWMA charts follo w the same t yp e of recursive scheme, which makes it plausible to use a unique v alue for λ . Th us, in the sequel, w e follo w the practice in W eiß (2023) and set λ = 0 . 10 throughout our analyses. Let us conclude this section with a note on out-of-control ARLs. As men- tioned b efore, there are a couple of comp eting ARL concepts for ev aluating a con trol chart’s out-of-control p erformance, see Knoth (2006), which are t yp- ically defined with resp ect to differen t positions of the c hange p oin t. Here, the c hange point τ expresses the time where the pro cess turns out of con- trol, i. e., the pro cess is in control (out of con trol) for t < τ ( t ≥ τ ). The out-of-con trol zero-state ARL assumes that the process c hange happ ens righ t at the b eginning of pro cess monitoring, i. e., the c hange p oint equals τ = 1. The conditional exp ected delay CED( τ ), by con trast, assumes a later change p oin t τ > 1, where the limit τ → ∞ leads to the steady-state ARL. While τ = 1 w ould happ en for a missp ecified in-control mo del, late change p oints are often more realistic in practice (but one does not kno w the true v alue of τ in adv ance). F ortunately , it turned out in the initial analyses of W eiß (2023) that the exact p osition of τ do es not ha ve a substan tial effect on the computed out-of-control ARL v alues, i. e., the Stein EWMA c harts sho w ed roughly the same p erformance for early and late pro cess changes. Since the c harts of the present study are defined in complete analogy to those of W eiß (2023), w e exp ect the v alue of the zero-state ARL to b e representativ e also for τ > 1 (for “problematic” t yp es of con trol chart, see the discussion in Knoth et al. (2023)). F or th is reason, we restrict our subsequen t analyses to zero-state ARLs, and these are computed based on sim ulations. So one sim ulates the considered pro cess for R times (we use R = 10 4 ), applies the sp ecified con trol chart to it, and determines the time until the first alarm. In this wa y , one gets R run lengths l 1 , . . . , l R , and the sample mean thereof pro vides an appro ximate v alue of the actual ARL. 4 Sim ulation-based P erformance Analyses 4.1 Ov erdisp ersion or Zero Inflation Let us start our p erformance analyses with the case of ov erdisp ersion or zero inflation (compared to the in-con trol mo del). Strictly sp eaking, a zero- 9 inflated distribution also exhibits increased disp ersion, but this increase is caused b y a single p oint mass in zero. With ov erdisp ersion, by con trast, w e refer to a flattened PMF compared to the in-control mo del’s PMF. F or un- b ounded counts, we use the NB-distribution for generating o v erdisp ersion, and the zero-inflated Poisson (ZIP) distribution for zero inflation. In the b ounded-coun ts case, w e use the b eta-binomial (BB) distribution for o verdis- p ersion and the zero-inflated binomial (ZIB) distribution for zero inflation, see App endix A in W eiß (2018) for details on these distributions. F or the sak e of a unique representation, we sp ecify any of the aforementioned mo dels by the mean µ and b y an appropriate dispersion index, namely by I P = σ 2 /µ for un b ounded counts and I B = nσ 2 /  µ ( n − µ )  for b ounded counts. Note that I P = 1 for the P oi-distribution while I P > 1 for NB and ZIP . Analogously , I B = 1 for the Bin-distribution and I B > 1 for BB and ZIB. The in-control means are set at either µ 0 = 2 (low counts) or µ 0 = 5 (medium coun ts). The data-generating pro cess (DGP) pro duces either i. i. d. coun ts ( ρ = 0) or AR(1)-lik e coun ts with ρ = 0 . 5, recall (2)–(4). The simulated ARLs rely on 10 4 replications, the target v alue for ARL 0 is 370. As the case of o verdispersion or zero inflation relative to an in-control P oi- distribution w as already analyzed b y W eiß (2023), we restrict our subsequent analyses to in-control NB- and Bin-mo dels, see T ables 1 – 2 for the obtained results. The Bin-results are quite close to the P oi-results an ywa y , which is not surprising as these distributions are related to eac h other by the P oisson limit theorem. Let us start our discussion with the Bin-case in T able 1. The design of this table (and of any further table) is as follo ws. It consists of t welv e 3 × 3-blo cks with a negativ e (p ositive) mean shift on the left (righ t) and no mean shift in the cen ter column. In the Bin-rows, the binomial distribution is preserved, whereas we hav e distributional changes in the ZIB- and BB-ro ws. Note that at the bottom of eac h table, there is a sc heme for in terpreting the ARLs within eac h blo c k. P art (a) of eac h table refers to i. i. d. coun ts, part (b) to AR(1)-lik e coun ts. Note that the BB- and ZIB- AR(1) mo del are constructed as in (3.27) of W eiß (2018) b y replacing the Bin-thinnings in (4) by BB- or ZIB-thinnings, resp ectively . Finally , the first column of blocks alw a ys refers to the ordinary EWMA chart (1), whereas the remaining tw o columns refer to Stein EWMA c harts, i. e., to (11) in case of T able 1. P art (a) of T able 1 just confirms the findings of W eiß (2023) for the Poi-case. If w e are concerned with the textb o ok situation, i. e., if there is a c hange solely in the mean parameter (the distribution is preserv ed otherwise), then the ordinary EWMA c hart is clearly the b est c hoice. But if there is an additional distributional change, or even only a distributional change while the mean is 10 T able 1: Sim ulated ARLs of ordinary EWMA and Stein EWMA c hart for Bin-coun ts with n = 10. BB and ZIB with I B = 5 / 3. µ 0 µ = µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 (a) i. i. d. counts EWMA 0.7805 EWMA S | x − 1 | 0.534 EWMA S | x − 1 | 1 / 4 0.4235 2 ZIB 69.2 87.9 51.6 22.7 26.1 29.2 16.6 19.1 21.9 Bin 171.5 370.2 99.3 240.2 369.5 550.6 191.5 370.6 671.0 BB 71.5 90.0 51.8 25.9 29.2 33.3 30.8 40.5 55.6 EWMA 0.974 EWMA S | x − 1 | 0.2115 EWMA S | x − 1 | 1 / 4 0.0511 5 ZIB 69.2 87.9 51.6 18.9 19.9 20.6 12.9 14.0 15.5 Bin 162.5 369.5 164.1 307.2 370.1 443.4 311.5 369.5 417.2 BB 66.7 88.2 66.3 25.1 26.9 29.0 27.4 28.9 30.3 (b) AR(1) counts with ρ = 0 . 5 EWMA 1.191 EWMA S | x − 1 | 0.639 EWMA S | x − 1 | 1 / 4 0.568 2 ZIB 77.9 73.8 55.8 22.9 24.3 25.3 23.8 26.9 30.0 Bin 384.8 370.1 158.0 247.1 369.7 554.4 211.9 371.2 634.3 BB 105.0 105.7 77.2 33.0 39.5 47.0 44.7 58.9 77.1 EWMA 1.493 EWMA S | x − 1 | 0.225 EWMA S | x − 1 | 1 / 4 0.0528 5 ZIB 30.8 29.9 28.0 7.9 7.7 7.3 7.1 6.9 6.5 Bin 258.0 369.1 257.3 316.7 370.9 424.1 293.9 370.9 421.8 BB 86.3 96.1 85.9 35.3 37.6 39.5 37.9 39.4 40.7 Notes: In-control ARL printed in b old font, CL L shown in italic font. “EWMA” = ordinary EWMA; “EWMA S ” = Stein EWMA, where weigh t functions f ( x ) = | x − 1 | and | x − 1 | 1 / 4 . In terpre- pure zero inflation tation: sole mean change distrib. change ov erdisp ersion k ept fixed, then the Stein EWMA c harts b ecome sup erior. More precisely , if w e are concerned with ov erdisp ersion (BB-mo del), then the weigh t function f ( x ) = | x − 1 | leads to the lo west ARLs, whereas zero inflation is b est detected using f ( x ) = | x − 1 | 1 / 4 . It should be noted that w e dra w the same conclusions from part (a) of T able 2, whic h refers to the NB-case (in-control lev el of disp ersion is I P = 5 / 3) with Stein EWMA chart (10). There, o verdispersion (relativ e to the in-control mo del) was generated by an NB-distribution with higher disp ersion, namely I P = 5 / 2, whereas w e used the ZIP-distribution with I P = 5 / 3 for causing zero inflation. Thus, we generally conclude that if monitoring i. i. d. coun ts and if b eing concerned with ov erdisp ersion under out-of-con trol conditions, the Stein EWMA chart with f ( x ) = | x − 1 | is the b est c hoice, whereas f ( x ) = | x − 1 | 1 / 4 p erforms b est under zero inflation. If the t yp e of out-of-con trol situation is not clear in adv ance, it is recommended to run all three c harts of T ables 1 – 2 sim ultaneously . As these c harts show a 11 T able 2: Sim ulated ARLs of ordinary EWMA and Stein EWMA c hart for NB-coun ts with I P = 5 / 3. ZIP with I P = 5 / 3, and ov erdisp ersed NB (“oNB”) with I P = 5 / 2. µ 0 µ = µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 (a) i. i. d. counts EWMA 1.156 EWMA S | x − 1 | 0.349 EWMA S | x − 1 | 1 / 4 0.3146 2 ZIP 506.6 462.0 149.7 139.4 257.2 481.5 54.2 81.0 124.9 NB 605.8 370.7 133.1 172.1 370.9 892.2 154.0 369.9 1001.2 oNB 171.9 135.1 75.6 45.2 67.2 103.8 52.2 86.9 163.2 EWMA 1.805 EWMA S | x − 1 | 0.1554 EWMA S | x − 1 | 1 / 4 0.0883 5 ZIP 342.6 407.5 261.9 116.2 143.9 170.2 22.7 24.7 26.8 NB 444.7 370.8 205.3 267.3 369.8 522.7 260.0 370.1 537.2 oNB 130.2 124.7 94.8 42.9 51.3 59.1 55.5 68.7 87.3 (b) AR(1) counts with ρ = 0 . 5 EWMA 1.855 EWMA S | x − 1 | 0.45 EWMA S | x − 1 | 1 / 4 0.4415 2 ZIP 1287.7 505.7 238.0 108.3 167.0 266.3 90.6 139.7 219.4 NB 770.9 369.7 200.4 187.4 370.7 710.6 182.7 370.7 768.8 oNB 288.9 178.8 115.2 61.1 93.5 141.8 75.4 123.1 210.1 EWMA 2.78 EWMA S | x − 1 | 0.177 EWMA S | x − 1 | 1 / 4 0.1105 5 ZIP 502.7 408.0 284.2 147.8 178.4 214.4 93.6 110.2 131.1 NB 510.8 369.6 242.5 278.0 370.8 494.2 276.7 370.2 492.8 oNB 183.8 156.4 124.4 60.1 71.6 83.0 78.5 97.1 122.0 Notes: In-control ARL printed in b old font, CL L shown in italic font. “EWMA” = ordinary EWMA; “EWMA S ” = Stein EWMA, where weigh t functions f ( x ) = | x − 1 | and | x − 1 | 1 / 4 . In terpre- pure zero inflation tation: sole mean change distrib. change ov erdisp ersion rather differen t ARL p erformance, they can be used for a kind of targeted diagnosis, i. e., one can conclude from the observ ed pattern of alarms on the type of out-of-con trol situation. F or example, if the c hart with f ( x ) = | x − 1 | 1 / 4 signals first, we conclude on zero inflation. Next, let us look at parts (b) of T ables 1 – 2, i. e., on the case of positively correlated coun ts. Note that in T able 2, the o verdispersed NB-coun ts are still generated b y the NB-I INAR(1) mo del (3), whereas the ZIP-INAR(1) mo del was used for zero inflation, i. e., mo del (2) with ZIP-inno v ations ( ϵ t ). In most cases in parts (b), CLs ha ve to b e increased and the charts’ ARL p erformances get w orse. But it still holds that the Stein EWMA charts are sup erior under o verdispersion or zero inflation. As the only difference, the ARL p erformances of b oth Stein EWMA charts are now quite similar under 12 (a) 0 2 4 6 8 0.0 0.1 0.2 0.3 0.4 0.5 x PMF µ 0 =2 I P =0.5 (Good) I P =1 (P oi) x (b) 0 2 4 6 8 10 12 0.00 0.10 0.20 0.30 x PMF µ 0 =5 I P =0.5 (Good) I P =1 (P oi) x Figure 1: PMF plots for Go o d with I P = 0 . 5 vs. Poi, for (a) µ 0 = 2 and (b) µ 0 = 5. Dotted line prop ortional to 1 / ( x + 1). zero inflation in the Bin-case, but not for the NB-case. So altogether, the conclusions done for i. i. d. counts still apply , but with generally increased out-of-con trol ARLs due to the serial dep endence. 4.2 Underdisp ersion Underdisp ersion is the opposite phenomenon to ov erdisp ersion, i. e., the PMF is concentrated more closely around the mean. While there are hardly an y mo dels for underdispersion of b ounded coun ts, the underdispersion of un- b ounded coun ts received more interest in the literature. F or the subsequen t sim ulations, we use the Go o d distribution (see W eiß, 2018, App endix A) for generating underdisp ersion relative to the Poi-distribution (see T able 3), whereas for underdisp ersion relativ e to an NB-distribution, w e use either an NB-distribution with less disp ersion or the Poi-distribution (see T able 4). It quic kly gets clear from T ables 3 – 4 that underdisp ersion is quite demanding for process monitoring. If looking first at the ordinary EWMA c hart (first columns of blo c ks), we recognize severely increasing ARLs in the presence of underdisp ersion. This is reasonable as fluctuations of the DGP are reduced suc h that small mean shifts (w e ha ve ± 0 . 25) hardly lead to a violation of the CLs. The second columns in T ables 3 – 4 refer to the Stein EWMA charts (9) and (10), where w e used f ( x ) = 1 / ( x + 1) as recommended by W eiß et al. (2023) in the context of GoF-tests. While this chart indeed p erforms w ell for the low mean µ = 2, the ARL p erformance is v ery bad for µ = 5, again demonstrating that pro cess monitoring in the presence of underdisp er- sion is quite demanding. T o b etter understand these difficulties and to find a solution, let us lo ok at Figure 1, where we visualize the effect of under- disp ersion (black) compared to the in-con trol mo del (grey) b y PMF plots. It can b e seen that the strongest deviations are in a small area b elo w µ 0 . 13 T able 3: Sim ulated ARLs of ordinary EWMA and Stein EWMA c hart for P oi-counts. Goo d with I P = 3 / 4 (first row) and I P = 1 / 2 (second row). µ 0 µ = µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 (a) i. i. d. counts EWMA 0.877 EWMA S 1 / ( x + 1) 0.223 EWMA S p P ( x + 2) 0.608 2 Poi 252.6 369.1 106.1 274.6 368.9 470.8 538.9 370.3 271.7 Goo d 622.4 948.9 158.5 96.9 142.3 249.5 90.7 71.4 60.5 Goo d 3611.8 6346.9 380.6 32.3 42.2 63.0 29.1 26.2 24.3 EWMA 1.388 EWMA S 1 / ( x + 1) 0.1775 EWMA S p P ( x + 2) 0.293 5 Poi 309.9 371.4 185.1 352.9 370.5 398.1 526.1 368.7 268.9 Goo d 1006.0 1015.9 327.2 > 10 4 > 10 4 > 10 4 228.4 149.3 106.4 Goo d 8154.5 7882.4 1168.9 > 10 4 > 10 4 > 10 4 52.5 40.1 33.2 (b) AR(1) counts with ρ = 0 . 5 EWMA 1.351 EWMA S 1 / ( x + 1) 0.2467 EWMA S p P ( x + 2) 0.7235 2 Poi 627.2 371.0 162.0 274.8 370.0 478.4 530.0 370.5 273.2 Goo d 2177.5 814.6 261.3 153.7 242.0 439.2 143.5 108.9 89.6 Goo d > 10 4 2866.0 581.6 47.6 62.4 92.3 43.7 36.7 33.2 EWMA 2.123 EWMA S 1 / ( x + 1) 0.1707 EWMA S p P ( x + 2) 0.345 5 Poi 432.2 369.8 236.3 332.3 370.1 395.3 514.5 370.1 280.7 Goo d 1369.3 909.2 446.1 2652.2 4029.5 5159.4 226.6 161.8 122.0 Goo d > 10 4 4264.3 1402.0 1856.0 6087.1 > 10 4 69.8 53.9 44.3 Notes: In-control ARL printed in bold font, CL L shown in italic font. “EWMA” = ordinary EWMA; “EWMA S ” = Stein EWMA, where weigh t functions f ( x ) = 1 / ( x + 1) and p P ( x + 2). In terpre- sole mean change pure tation: distrib. increasing change underdispersion F or µ 0 = 2 in (a), the main deviations happen for x ∈ { 0 , 1 , 2 } . In fact, f ( x ) = 1 / ( x + 1) (symbolized by the dotted graphs in Figure 1) indeed puts most w eight to this region, which explains the go o d ARL p erformance in this case. F or µ 0 = 5 in (b), ho wev er, the main deviations are for x ∈ { 2 , . . . , 5 } . As f ( x ) = 1 / ( x + 1) puts relatively low w eight in to this region, we no w get a p o or ARL p erformance. T o get a go o d performance regarding underdisp ersion, Figure 1 suggests to put most weigh t somewhat b elo w the in-con trol mean, whic h leads to the follo wing idea. The in-con trol PMF itself (plotted in grey in Figure 1), in- terpreted as a w eight function, puts most w eight around the mean of the in-con trol distribution. Thus, we ma y just shift this PMF do wn w ards to mo ve most weigh t b elow the mean. Inspired b y Figure 1 and some sim ula- tion experiments, the idea was developed to define f ( x ) b y simply shifting the in-control PMF by 2. More precisely , for the in-control Poi-model of T a- 14 T able 4: Sim ulated ARLs of ordinary EWMA and Stein EWMA c hart for NB-coun ts with I P = 5 / 3. Underdisp ersed NB (“uNB”) with I P = 4 / 3. µ 0 µ = µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 µ 0 − 0 . 25 µ 0 µ 0 + 0 . 25 (a) i. i. d. counts EWMA 1.156 EWMA S 1 / ( x + 1) 0.2215 EWMA S p N ( x + 2) 0.4163 2 NB 605.8 370.7 133.1 240.3 371.5 590.6 367.7 370.3 325.2 uNB 1531.2 799.1 202.7 280.0 380.5 568.9 315.1 213.5 152.7 Poi 5998.4 3237.7 429.8 106.1 128.8 179.9 93.7 70.8 57.3 EWMA 1.805 EWMA S 1 / ( x + 1) 0.165 EWMA S p N ( x + 2) 0.22 5 NB 444.7 370.8 205.3 301.9 369.1 452.9 399.3 369.1 328.4 uNB 1072.6 840.9 372.4 1185.9 1531.8 1921.6 455.4 313.1 226.8 Poi 4236.6 3623.7 986.3 7006.1 > 10 4 > 10 4 132.6 96.9 76.5 (b) AR(1) counts with ρ = 0 . 5 EWMA 1.855 EWMA S 1 / ( x + 1) 0.2412 EWMA S p N ( x + 2) 0.4626 2 NB 770.9 369.7 200.4 266.3 369.8 514.0 440.5 369.8 293.9 uNB 1797.1 685.4 315.4 233.3 319.1 441.7 248.6 177.5 137.2 Poi > 10 4 2329.1 713.9 103.7 136.9 190.5 87.6 68.6 58.2 EWMA 2.78 EWMA S 1 / ( x + 1) 0.1727 EWMA S p N ( x + 2) 0.247 5 NB 510.8 369.6 242.5 308.7 370.7 442.5 414.8 370.0 306.6 uNB 1148.0 700.0 420.9 787.5 973.5 1222.5 329.4 241.2 191.5 Poi 4918.5 2317.8 1082.8 1689.4 2741.1 4019.6 124.0 96.6 78.9 Notes: In-control ARL printed in bold font, CL L shown in italic font. “EWMA” = ordinary EWMA; “EWMA S ” = Stein EWMA, where weigh t functions f ( x ) = 1 / ( x + 1) and p N ( x + 2). In terpre- sole mean change pure tation: distrib. increasing change underdisp ersion ble 3, w e use p P ( x + 2) as the w eight function, and p N ( x + 2) for the NB-model of T able 4. Here, p P ( · ) and p N ( · ) abbreviate the PMFs of the in-control Poi- and NB-distribution, resp ectively . The computed c hart designs and ARLs are summarized in the resp ectiv e third column of T ables 3 – 4. W e now get notably decreasing ARLs for increasing underdisp ersion in an y scenario, b oth under i. i. d. and AR(1)-lik e counts. 5 An Illustrativ e Data Application W eiß & T estik (2015) analyzed a large set of daily count time series referring to registrations in the emergency department of a children’s hospital. More precisely , these emergency coun ts were determined p er 5-min in terv al from 08:00:00 to 23:59:59 on a da y (so length T = 192), and the full set of time series cov ers the p erio d from F ebruary 13 to August 13, 2009. W eiß & T estik 15 (2015) used the sixteen time series from F ebruary 13 to 28 to dev elop the in-con trol mo del (Phase-I analysis), namely a Poi-INAR(1) mo del with in- con trol mean µ 0 = 2 . 1 and dep endence parameter ρ 0 = 0 . 78. Con trol charts based on this model w ere then applied to prosp ective pro cess monitoring (Phase-I I application). While most Phase-I I series did not con tradict the in-con trol mo del, W eiß & T estik (2015) recognized a few unusual days, tw o of which shall no w serve as illustrative data examples. Let us start our analyses by designing the con trol c harts based on the in- con trol mo del. T o mak e the results consisten t with Section 4, the CLs of eac h chart are chosen suc h that the in-con trol ARL is close to 370. On the one hand, we consider the c-c hart and the ordinary EWMA chart (1) as comp etitors. On the other hand, the new Stein EWMA chart (9) was used together with four w eigh t function: f ( x ) = | x − 1 | and | x − 1 | 1 / 4 for indicating p ossible o verdispersion or zero inflation, and f ( x ) = 1 / ( x + 1) and p P ( x + 2) for underdisp ersion. While the ARLs of the c-chart are computed numerically exactly b y using the Marko v-c hain approac h (see W eiß, 2018, Section 8.2.2), w e again used simulations with 10 4 replications for the remaining c harts. Here, the c-chart is most difficult to design due to discreteness; the best design has LCL = 0 and UCL = 6 (so an only one-sided design, where coun ts X t > 6 cause an alarm) and ARL 0 ≈ 326 . 2. The EWMA chart designs (again with λ = 0 . 1) are truly t wo-sided, namely • EWMA (1) with L = 1 . 851 and ARL 0 ≈ 370 . 3; • Stein EWMA (9) with f ( x ) = | x − 1 | , L = 0 . 848, ARL 0 ≈ 370 . 5; • Stein EWMA (9) with f ( x ) = | x − 1 | 1 / 4 , L = 0 . 829, ARL 0 ≈ 370 . 5; • Stein EWMA (9) with f ( x ) = 1 / ( x + 1), L = 0 . 2994, ARL 0 ≈ 370 . 5; • Stein EWMA (9) with f ( x ) = p P ( x + 2), L = 0 . 9594, ARL 0 ≈ 370 . 2. As a first illustrative example, w e apply these con trol c harts to the emergency coun ts collected on March 28, 2009, see Figure 2. The c-chart triggers an alarm at t = 23, the ordinary EWMA only rather late at t = 171, b oth indicating that the in-control mo del is violated. The situation gets more clear if looking at the four Stein EWMA c harts in Figure 2. The c harts in (c) and (d) b oth trigger a very early alarm at t = 6, whereas those of (e) and (f ) do not trigger an alarm at all. This indicates that w e are confron ted with o verdispersion and zero inflation. In fact, comparing the sample prop erties of the emergency series from Marc h 28, 2009, to the in-control mo del, we note a sligh t increase in the mean µ (from 2.1 to ≈ 2 . 323) and a substantial 16 (a) c-c hart: (b) Ordinary EWMA chart: 0 50 100 150 0 2 4 6 8 t c−char t t 0 50 100 150 0 1 2 3 4 5 t EWMA char t t (c) Stein EWMA with f ( x ) = | x − 1 | : (d) Stein EWMA with f ( x ) = | x − 1 | 1 / 4 : 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 t Stein EWMA char t t 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 t Stein EWMA char t t (e) Stein EWMA with f ( x ) = 1 / ( x + 1): (f ) Stein EWMA with f ( x ) = p P ( x + 2): 0 50 100 150 0.6 0.8 1.0 1.2 1.4 t Stein EWMA char t t 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t Stein EWMA char t t Figure 2: Emergency counts from Marc h 28, 2009, see Section 5: c-c hart in (a), ordinary EWMA c hart in (b), and differen t Stein EWMA c harts in (c)–(f ). CLs as dashed lines, solid center line, and first alarm at dotted line. increase in disp ersion I P (from 1 to ≈ 1 . 312). F urthermore, the n umber of zeros equals 27, b eing larger than 23.5 as exp ected under the in-con trol mo del. So the Stein EWMA c harts gav e a clear diagnosis of the t yp e of out-of-con trol situation. In addition, their alarm at t = 6 was not only muc h faster than those of c-chart and ordinary EWMA c hart, but also than those of the con trol charts in W eiß & T estik (2015) (these trigger at t = 48 and t = 50, resp ectively). 17 (a) c-c hart: (b) Ordinary EWMA chart: 0 50 100 150 0 2 4 6 8 t c−char t t 0 50 100 150 0 1 2 3 4 5 t EWMA char t t (c) Stein EWMA with f ( x ) = | x − 1 | : (d) Stein EWMA with f ( x ) = | x − 1 | 1 / 4 : 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 t Stein EWMA char t t 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 t Stein EWMA char t t (e) Stein EWMA with f ( x ) = 1 / ( x + 1): (f ) Stein EWMA with f ( x ) = p P ( x + 2): 0 50 100 150 0.6 0.8 1.0 1.2 1.4 t Stein EWMA char t t 0 50 100 150 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t Stein EWMA char t t Figure 3: Emergency coun ts from July 16, 2009, see Section 5: c-chart in (a), ordinary EWMA chart in (b), and different Stein EWMA charts in (c)–(f ). CLs as dashed lines, solid center line, and first alarm at dotted line. As the second illustrativ e example, see Figure 3, we consider the emergency coun ts from July 16, 2009, where no results are rep orted by W eiß & T estik (2015). This time, neither c-chart nor ordinary EWMA chart trigger an alarm, so the user w ould conclude that the emergency series is in control. Lo oking at the Stein EWMA c harts, ho w ever, the ones in (e) and (f ) signal at times t = 35 and t = 74, resp ectiv ely , so we seem to b e confron ted with underdisp ersion. In fact, the sample v alue of I P is ≈ 0 . 710 being notably 18 smaller than 1, but also the mean has decreased from 2.1 to ≈ 1 . 911. As we kno w from Section 4.2, the ordinary EWMA chart is hardly able to detect mean shifts in the presence of underdisp ersion, so the nov el Stein EWMA c harts constitute a welcome complement for this scenario. 6 Conclusions and F uture Researc h F or the monitoring of either P oi-, NB-, or Bin-counts, corresp onding Stein EWMA charts were prop osed, which are constructed b y utilizing the resp ec- tiv e Stein identit y . Their ARL p erformance was inv estigated by simulations, b oth for i. i. d. and AR(1)-lik e counts. It turned out that o v erdisp ersion is b est detected by using the linear weigh ts f ( x ) = | x − 1 | , and zero inflation b y the ro ot weigh ts f ( x ) = | x − 1 | 1 / 4 . Count monitoring in the presence of underdisp ersion, ho w ever, turned out to b e quite demanding. While the ordinary EWMA c harts p erforms very p o orly in this case, the Stein EWMA c hart with in verse w eights f ( x ) = 1 / ( x + 1) has appealing out-of-con trol ARLs for lo w counts. Even more promising is the weigh t function obtained b y a do wnw ard shift of the in-control PMF. These findings w ere also con- firmed by the data example on emergency counts. Nevertheless, it app ears that the monitoring of underdisp ersion requires additional researc h activity . There are further directions for future researc h. As cum ulative sum (CUSUM) c harts often show a better out-of-control performance than EWMA c harts, it w ould b e relev an t to dev elop and inv estigate Stein CUSUM charts for coun t pro cesses. Here, a residuals-based approac h in analogy to W eiß & T estik (2015) would b e attractiv e, as this w ould not only b e applicable to P oi-, NB-, or Bin-marginals, but also to in-control mo dels ha ving a Poi-, NB-, or Bin-conditional distribution. 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