8.2.1 Shewhart Charts for i.i.d. Counts

If monitoring a count process with a Shewhart chart, the counts are commonly directly plotted on the chart as they arrive in time; that is, the plotted statistics are . If the range of the counts is unlimited, such a chart is referred to as a chart ( for “count”). For the case of the finite range , a chart for is said to be an chart, while a chart plots the relative quantities . This terminology is obviously motivated by the binomial distribution (Example A.2.1) and by the idea of a sample-based monitoring, with or expressing the absolute number or relative proportion, respectively, of “successes” in the sample being collected at time . For simplicity, we shall always consider the case of an unlimited range (and hence charts) in this section, but the presented concepts apply to charts and charts as well; see Section 9.1.1. A truly two-sided chart has control limits with . One-sided charts are obtained by either setting (upper-sided chart) or (lower-sided chart).

For the rest of this section, let us assume that is serially independent. If the process is in-control, it is i.i.d. and has the in-control marginal distribution . As an out-of-control scenario, we restrict to the case of a sudden shift; that is, at a certain time (called change point), the marginal distribution becomes . This leads to the following (unconditional) change point model (Knoth, 2006):

For , the process is in control,

while it is out of control for if .

For a change point , the process is out of control right from the beginning. If the control chart triggers an alarm at time (rule (8.2)), we stop monitoring and conclude that the process might have run out of control. If indeed and , the alarm was correct; otherwise, it was a false alarm. In the first case, the difference expresses the delay in detecting the change point. Here, the “” is used since even in the case of immediate detection (), we have one out-of-control observation (say, one defective item in a production process).

At this point, it is important to study the run length of the control chart – see (8.2) – in more detail. If the process is in control, we wish the run length to be large (a robust chart), since then the run length expresses the time until the first false alarm. In contrast, it should be small for an out-of-control process, since the run length then goes along with the delay in detecting the process change. As for a significance test, the approach to designing the chart is to choose the control limits in such a way that a certain degree of robustness against false alarms is guaranteed. For this purpose, one looks at properties of the in-control run length distribution. These could be quantiles, such as the median, but the main approach (although one that is sometimes criticized; see Kenett & Pollak (2012) as an example) is to consider the mean of the run length ; that is, the average run length (ARL).¹ If there are several candidate designs leading to (roughly) the same in-control ARL, abbreviated as , then one compares the out-of-control ARL performances of these charts to select the final chart design.

So the question is how to compute the ARL given a specific chart design . If the process is in-control (that is, i.i.d. with marginal distribution ), then a signal is triggered at time with probability

Because of the independence of the plotted statistics, the distribution of is a shifted geometric distribution (Example A.1.5), so it follows immediately that

8.3

Note that this formula also includes the one-sided cases by setting and .

If the distribution becomes out of control at time , then the delay in detecting this change is (see above). Again, because of the independence of the plotted statistics (the non-aging property), this delay can still be described by a shifted geometric distribution, but using instead of . Therefore, the out-of-control ARL for the considered i.i.d.-scenario is defined by setting (since the true position of the change point does not affect the delay anyway); that is,

8.4

Note that ARLs should be interpreted with caution in practice, since the shifted geometric distribution is strongly skewed and has a large dispersion (the standard deviation nearly equals the mean; see Example A.1.5). This is illustrated by Figure 8.2, which shows the run length distribution corresponding to . Although an alarm is triggered in the mean after 250 plotted statistics, the median, for instance, equals only 173; that is, in 50% of all cases, the actual run length is not larger than 173. The quartiles range from 72 to 346, so again in 50% of all cases, the actual run length is outside even this region. For a further critical discussion, see Kenett & Pollak (2012).

A possible way to achieve an ARL-unbiased chart design that is close to any prespecified -level was proposed by Paulino et al. (2016), and relies on a randomization of the emission of an alarm.

Usually, a control chart is designed as if the true in-control model is known precisely. In reality, however, the in-control model has to be estimated from given data (believed to stem from the presumed in-control model). Due to the uncertainty of parameter estimation, the true performance of the chart will usually deviate from the “believed” one, and this difference might be rather large. In view of (8.3) and the typically large values for (as in Example 8.2.1.1), the control limits correspond to rather extreme quantiles of the (estimated) in-control distribution. So already moderate misspecifications of the model parameters may lead to strong effects on the control limits and ARLs. Hence, for the data examples below, we should be aware that we always consider some kind of conditional ARL performance, conditioned on the fitted model.

A comprehensive literature review of the effect of estimated parameters on control chart performance is provided by Jensen et al. (2006). Probably the first such work in the attributes case is by Braun (1999), who considers the and the charts, while Testik (2007) investigates the effect of estimation on the CUSUM chart for i.i.d. Poisson counts; this chart is discussed in Section 8.3.1.

While (appropriately chosen) Shewhart charts are generally quite sensitive to very large shifts in the process (and they are also generally recommended for application in Phase I (Montgomery, 2009)), Example 8.2.1.1 has already demonstrated that these charts are not particularly well-suited to detecting small-to-moderate shifts. For this reason, in Section 8.3 we shall consider advanced control schemes that are more sensitive to small shifts, because these charts are designed to have an inherent memory.

8.2.2 Shewhart Charts for Markov-Dependent Counts

In this section, we skip the i.i.d.-assumption and allow the count process to be a Markov chain, say, an INAR(1) process as in Section 2.1, a binomial AR(1) process as in Section 3.3, or an INARCH(1) process as in Example 4.1.6. But still, our aim is to plot the observed counts directly on the chart with limits ; that is, we choose again .

Because of the serial dependence of the plotted statistics, now the run length (8.2) no longer follows a simple geometric distribution. In addition, if we now look at the detection delay , the corresponding distribution generally depends on the position of the change point . Therefore, more refined ARL concepts have to be considered; a detailed survey of different ARL concepts is provided by Knoth (2006). In this book, the following ARL concepts are used:

• the zero-state ARL
  8.5
• the conditional expected delay
  8.6
• the (conditional) steady-state ARL
  8.7

where denotes the expectation related to the change point .

As before, we refer to the computed ARL value as the in-control ARL (out-of-control ARL) if (), and the in-control ARL is signified by adding the index “0”.

Obviously, the zero-state ARL is nothing other than . For the case of serial independence, as considered in Section 8.2.1, we have , but otherwise these ARLs may differ. The essential questions are:

• How can we compute these ARLs?
• Which one should be used in a specific application?

Let us start with the second question. When designing a chart, one first looks at the in-control behavior. In this context, it is reasonable to use the in-control zero-state ARL, , as a measure of robustness against false alarms. Then, in a second step, one analyzes the out-of-control behavior. If there are reasons to expect, say, that the change will probably happen quite early, then it would be reasonable to evaluate the out-of-control performance of an with sufficiently small . In many applications, however, one will not have such information. However, we shall see below that often converges rather quickly to . This implies that the steady-state ARL, , might serve as a reasonable approximation for the true mean delay of detection after the unknown change point. Therefore, in this book, we shall evaluate the out-of-control performance in terms of .

The first question is how to compute the different types of ARL. Certainly, it is always possible to approximate the ARLs through simulations; to simulate , one will simulate with a large as a substitute. But if considering a chart applied to an underlying Markov chain, a numerically exact solution is also possible by using the Markov chain (MC) approach proposed by Brook & Evans (1972). Since this approach can also be used for the advanced control charts to be introduced below, we provide a rather general description in the sequel following Weiß (2011b).

In view of decision rule (8.1), we can assume a slightly simplified range for the plotted statistics : their range is partitioned into the set of “no-alarm states” (because no alarm is triggered by the chart if takes a value in ) and the set consisting of a single “alarm state” ‘’. This is justified since any kind of violation of the control limits will lead to the same action: stop the process and search for an assignable cause. Therefore, ‘’ is an absorbing state; that is, it is no longer possible to leave this state. The set is equal to for the case of a two-sided chart.

The MC approach now assumes a conditional change point model (Weiß, 2011b), as given in Definition 8.2.2.1; see also the survey about Markov chains in Appendix B.2.

If for all , then the whole process is stationary according to the in-control model. Furthermore, since ‘’ is an absorbing state by definition, we have for all , where denotes the Kronecker delta. The requirement that consists of inessential states guarantees, among other things, that the probability of reaching ‘’ in finite time equals 1.

Let us now describe the procedures for computing the different types of ARL; derivations and more details can be found in Brook & Evans (1972) and Weiß (2011b). For this purpose, define to be the transpose of the transition matrix for the states in ; that is, . Analogously, we set . The requirement that consist of inessential states guarantees that the fundamental matrices and exist, where denotes the identity matrix. Since ‘’ is an absorbing state, the transition matrices of before and after the change point, respectively, are given by

8.8

To compute the out-of-control zero-state ARL (for the in-control ARL, we just have to replace by ), we first compute the unique solution of the equation

8.9

Here, the entries express the mean time to reach ‘’ if . After having specified the initial probabilities for (see Remark 8.2.2.2), we collect these probabilities in the vector (note that the change already happened at time 1). Then

8.10

If , then there exist in-control observations. So the entries of the solution to (8.9) express the mean delay to reach ‘’ if . If the vector consists of the probabilities and if refers to the (Markov property, in control), then the conditional expected delay equals

8.11

Finally, to compute the steady-state ARL, we need to take the limit according to (8.7). To be able to apply the Perron–Frobenius theorem – see Remark B.2.2.1 in Appendix B.2 for a summary – we have to assume that the non-negative matrix is primitive. For the corresponding Perron–Frobenius eigenvalue , there exists a strictly positive right eigenvector ; is the normed version of . Then

8.12

where the rate of convergence of for is determined by the second largest eigenvalue, which satisfies .

8.3.1 CUSUM Charts for i.i.d. Counts

The traditional cumulative sum (CUSUM) control chart, being applied directly to the observations of the process, is perhaps the most straightforward advanced candidate for monitoring processes of counts, because it preserves the discrete nature of the process by only using addition (but no multiplications). Initialized by a starting value , the upper-sided CUSUM is defined by

8.13

that is, by accumulating the deviations from the reference value . Because of this accumulation, the plotted statistic at time is not solely based on but also incorporates the process in the past: If the CUSUM statistic becomes negative, the construction resets the CUSUM to zero.

The starting value is commonly chosen as ; a value is referred to as a fast initial response (FIR) feature, and it may help to detect an initial out-of-control state more quickly; see also the discussion below formulae (8.5)–(8.7). If and are taken as integer values, then also is integer-valued. As another example, if then so is , but in any case, we have a discrete range. In the sequel, we shall concentrate on integer-valued . An alarm is triggered if violates the upper control limit (typically, ).

While the upper-sided CUSUM is designed to detect increases in the process mean, the lower-sided CUSUM, defined by

8.14

aims at uncovering decreases in the mean. If are monitored simultaneously, then this chart combination is referred to as a two-sided CUSUM chart. A book with a lot of background information about CUSUM charts is the one by Hawkins & Olwell (1998).

In this section, we assume the monitored count process to be i.i.d. in its in-control state, a situation that was also considered in the article by Brook & Evans (1972). Because of the accumulation according to (8.13), however, the statistics are no longer i.i.d., but constitute a Markov chain (analogous arguments apply to the lower-sided CUSUM (8.14)) with transition probabilities

and the initial statistic satisfies . Therefore, the MC approach as described in Section 8.2.2 is applicable, with . In fact, Brook & Evans (1972) introduced their MC approach for exactly this type of control chart and considered the application to i.i.d. Poisson counts.

We conclude this section by pointing out the relationship between the CUSUM scheme (8.13) and the sequential probability ratio test (SPRT); see Sections 6.1, 6.2 in Hawkins & Olwell (1998) for more details. The likelihood function (see Remark B.2.1.2) for i.i.d. counts is given by

so we obtain the likelihood ratio (LR) as

The SPRT now monitors the logarithmic likelihood ratio (log-LR)

8.15

for increasing . This procedure can be rewritten recursively by accumulating the contributions to the log-LR at times , thus leading to a type of one-sided CUSUM scheme:

Note the relation to the random walk in Example B.1.6. Comparing this type of CUSUM recursion with the one given in (8.13), we see that the construction is missing, so this CUSUM is not reset to zero if the CUSUM statistic becomes negative. As pointed out by Lorden (1971), the CUSUM (8.13) is equivalent to monitoring a slight modification of (8.15):

8.16

If this statistic was not positive at time but , then the statistic at time just equals , which corresponds to the above resetting feature.

8.3.2 CUSUM Charts for Markov-dependent Counts

Now, let us turn back to the case of a Markov-dependent count process , as in Section 8.2.2. If we apply the upper-sided CUSUM scheme (8.13) to such a process, then the statistics no longer constitute a Markov chain, so the MC approach of Brook & Evans (1972) is not directly applicable. But, as shown in Weiß & Testik (2009) and Weiß (2011b), ARL computations are possible by considering the bivariate process , which is a bivariate Markov chain with transition probabilities

8.17

In view of the CUSUM decision rule, it is clear that the set of “no-alarm states” is contained in . However, since values of larger than will always push beyond , the set is indeed finite. Excluding impossible transitions (say, from an alarm state back to a no-alarm state), Weiß & Testik (2009) showed that

8.18

which is of size . So the matrices required for the MC approach (8.8) are of dimension , which will often be a rather large number. It should be noted, however, that many entries of will be equal to 0 according to (8.17); that is, are sparse matrices. Therefore, the MC approach for ARL computation can be implemented efficiently using sparse matrix techniques; see Section 3 in Weiß (2011b) for possible software solutions.

The idea of applying the MC approach to the bivariate process of observed counts and CUSUM statistics also essentially applies to the lower-sided CUSUM scheme (8.14), but the set then becomes infinite; that is, ARLs can only be computed approximately (see Yontay et al. (2013) for details). If using a two-sided scheme, then the MC approach has to be applied to the trivariate Markov chain . Although here, the set is finite again, computations become very slow because of the immense matrix dimensions; more details and feasible approximations are presented by Yontay et al. (2013).

Although we exemplified the log-LR approach for Markov count processes here, it can also be used for completely different types of count process. As an example, Höhle & Paul (2008) derived such a log-LR CUSUM chart for counts stemming from the seasonal log-linear model (5.6), which proved to be useful for the surveillance of epidemic counts. A related study is the one by Sparks et al. (2010).