9.1.1 Sample-based Monitoring: Binary Case

In view of its practical importance, let us first focus on the special case of a binary process , with the range coded as as in Example 6.3.2. Having available the binary segment , one commonly determines either the sample sum (say, counts of non-conforming items) or the corresponding sample fraction of ‘1’s. Since the sample fraction differs from the count just by a factor , we shall always consider the resulting count process in the sequel. The original binary process is now monitored by monitoring this derived count process . At this point, the fundamental premise of this section should be remembered: although might exhibit serial dependence, due to taking sufficiently distant segments, we shall assume to be serially independent, and hence i.i.d. in its in-control state.

For monitoring (being i.i.d. in its in-control state), any of the concepts discussed in Chapter 8 can be used, it just has to be adapted to the finite range of . This difference sometimes manifests itself in the name of the resulting control charts. If the counts are plotted directly on a Shewhart-type chart, for instance, it is no longer referred to as a chart, but as an chart; see also the discussion in Section 8.2.1, as well as Montgomery (2009). If the sample fractions are plotted, it is called a chart. Despite this different terminology, the chart still has two control limits satisfying , which includes the one-sided charts as boundary cases (upper-sided if ). Also ARLs are computed as before; see (8.3) and (8.4).

Concerning the distribution of the sample counts, the serial dependence structure of the underlying binary process is of importance. If is i.i.d. with (say, the probability of a non-conforming item), then each sample sum is binomially distributed according to (Example A.2.1). So the statistics constitute themselves as an i.i.d. process of binomial counts. But if exhibits serial dependence, in contrast, the distribution of will deviate from a binomial one.

In Deligonul & Mergen (1987), Bhat & Lal (1990) and Weiß (2009f), the case of being a binary Markov chain with success probability and autocorrelation parameter was considered (Example 7.1.3); that is, with the transition matrix given by (7.6). In this case, follows the so-called Markov binomial distribution (which coincides with iff ). While the mean of is not affected by the serial dependence, the variance in particular changes (extra-binomial variation if , see the discussion in the context of Equation (2.3)):

9.1

The pmf is given by (Kedem, 1980, Corollary 1.1)

9.2

for (zero inflation if ; see the discussion in Appendix A.2). If the time distance between successive segments from is sufficiently large, the resulting process of counts can still be assumed to be approximately i.i.d. (note that the correlation between and decays exponentially with ), but with a marginal distribution different from a binomial one. This difference in the distribution of certainly has to be considered when designing a corresponding control chart (Weiß, 2009f).

In addition, advanced control schemes, such as the EWMA or CUSUM charts discussed in Section 8.3, can be used for monitoring . Assuming the counts to be binomially distributed in their in-control state (that is, is assumed to be i.i.d.), Gan (1993) applied the CUSUM scheme described in Section 8.3.1 for process monitoring, while Gan (1990b) used the modified EWMA chart from Section 8.3.3 (with rounding operation as in (8.20)) for this purpose. The application of such an EWMA chart to the case of being a binary Markov chain – that is, with following the Markov binomial distribution – was considered by Weiß (2009f). The computation of ARLs is done in the same way as described in Sections 8.3.1 and 8.3.3, by just using the pmf of the (Markov) binomial distribution. A completely different approach for a sample-based monitoring of an underlying binary Markov chain was recently proposed by Adnaik et al. (2015), who did not compute the sample sums as the charting statistics, but instead used some kind of likelihood ratio statistic for each of the successive segments. Finally, Höhle (2010) proposed a log-LR CUSUM chart for monitoring the under the assumption that these counts follow a marginal (beta-)binomial logit regression model; see Section 7.4.

9.1.2 Sample-based Monitoring: Categorical Case

Let us return to the truly categorical case; that is, where the range of consists of more than two states, with . As in Section 6.2, we denote the time-invariant marginal probabilities by . If the number of different states, , is small, it would be feasible to monitor the process with simultaneous binary charts; say, by using the -tree method described in Duran & Albin (2009). However, here we shall concentrate on charting procedures in which the information about the process is comprised in a univariate statistic: After having taken a segment from the process, we first compute the resulting frequency distribution as a summary, which then serves as the base for deriving the statistic to be plotted on the control chart. To keep it consistent with the binary case from Section 9.1.1, we concentrate on absolute frequencies: with being the absolute frequency of the state in the sample , such that . With denoting the binarization of , we may express .

If the underlying categorical process is even serially independent (so altogether i.i.d.), then the distribution of each is a multinomial one; see Example A.3.3. This case was considered by Marcucci (1985) and Mukhopadhyay (2008), among others, who proposed plotting Pearson's -statistic on a control chart,

9.3

where refers to the in-control values of the categorical probabilities. So in the in-control case, the process is i.i.d. with a marginal distribution that might be approximated by a -distribution (see Horn (1977) concerning the goodness of this approximation). As an alternative, Weiß (2012) proposed using a control statistic that measures the relative change of categorical dispersion. As the underlying categorical dispersion measure, the Gini index (6.1) might be used. If is i.i.d., following the in-control model, then

9.4

is approximately normally distributed, with a mean of and variance ; see Section 6.2. These approximate distributions for or may be used during chart design. But since the quality of these approximations is often rather bad (note that is often quite small and that the control limit is usually chosen as an extreme quantile), the final design and evaluation of the ARL performance requires simulations in practice.

A sample-based approach is also possible if is serially dependent. But then, certainly, the distributions of and hence of and will deviate from those given above for the i.i.d. case. If, for instance, is an NDARMA process (Section 7.2), then the effect on the distribution can be quantified in terms of the constant from (7.12). Considering the complete vector , the covariance matrix from Example A.3.3 is asymptotically inflated by the factor (Weiß, 2013b). For the Gini statistic , variance and mean change (approximately) according to (7.13) and (7.14), respectively, while Weiß (2013b) showed that is approximately -distributed.

Höhle (2010) proposed a log-LR CUSUM chart if the stem from a marginal multinomial logit regression model; see Section 7.4.

9.2.1 Continuous Monitoring: Binary Case

As in Section 9.1.1, let us first focus on the special case of a binary process . Perhaps the best-known approach for (quasi) continuously monitoring a binary process is by plotting run lengths on an appropriately designed chart:

9.5

As an example,

The monitoring of such runs is a reasonable approach, especially for high-quality processes where is very small. Small implies that long runs are observed, but if increases (deterioration of quality), the runs become shorter (and vice versa). So the detection of a decrease in the run lengths is often particularly relevant. Having fixed a truly two-sided design , we stop monitoring with the th run if either for the first time, or if already zeros have been observed since the last run (because then, will necessarily become larger than , but we do not need to wait until the run is finished).

Concerning performance evaluation, Remark 9.1.1.1 should be remembered. The ARL – that is, the average number of plotted runs until the first alarm – would be quite misleading, since a single run might comprise a rather large number of original observations. Therefore, the ATS is clearly preferable as a measure of chart performance.

If is i.i.d. (Bourke, 1991; Xie et al., 2000; Weiß, 2013c), then is also i.i.d. according to the shifted geometric distribution (Example A.1.5). The ATS can be computed according to (Weiß, 2013c)

9.6

The monitoring of runs is straightforwardly extended to the Markov case; see Blatterman & Champ (1992) and Lai et al. (2000). The runs from a binary Markov chain according to Example 7.1.3 are still serially independent, but their distribution is no longer shifted geometric. While has to be treated separately, for with , we obviously have

If ‘1’s are observed more frequently, the runs become quite short on average. In such a case, the CUSUM procedure proposed by Bourke (1991) to monitor the run length in is more appropriate. This geometric CUSUM control chart is essentially equivalent to the Bernoulli CUSUM control chart, which was proposed by Reynolds & Stoumbos (1999) for an i.i.d. binary process , and which was extended to the case of a binary Markov chain, as in Example 7.1.3, by Mousavi & Reynolds (2009). These charts are constructed in an analogous way to (8.15): the CUSUM chart is defined by accumulating the contributions to the log-likelihood ratio (log-LR) at times . The contribution by the th observation equals

where refers to the relevant out-of-control parameter value of , while represents the in-control value. In the Markov case (Example 7.1.3; see also Remark 8.3.2.2), is computed as before, while

An upper-sided CUSUM chart (we restrict to this case, since usually increases in are to be detected) can now be constructed analogously to (8.13) by defining

9.7

In the i.i.d. case, the CUSUM (9.7) might be rewritten in the form

9.8

Note that the plotted statistics of these CUSUM charts go along with the observations, so . To allow for an exact ARL computation with the MC approach (Section 8.2.2), can be required to take the form with an (Reynolds & Stoumbos, 1999); a similar strategy is proposed by Mousavi & Reynolds (2009) for the case of being a binary Markov chain. In this case (with being a multiple of ), the resulting transition matrices for the MC approach are sparse matrices (see Section 8.3.2), since only a few combinations are possible at all for . We have

Note that a lower-sided CUSUM chart can be constructed in an analogous way to (9.8). For a log-LR CUSUM with respect to an underlying logit regression model (Section 7.4), see Höhle (2010).

We conclude by pointing out another approach for continuously monitoring a binary process: the EWMA chart discussed in Section 8.3.3, which was applied to binary processes by Yeh et al. (2008) and Weiß & Atzmüller (2010), among others.