# 11.1 Control Charts

Recall that the data follow the change point model

 (11.1)

The observations are independent and the time point is unknown. The control chart (the SPC scheme) corresponds to a stopping time . Here we consider three different schemes - the Shewhart chart, EWMA and CUSUM schemes. There are one- and two-sided versions. The related stopping times in the one-sided upper versions are:

1. The Shewhart chart introduced by Shewhart (1931)

 (11.2)

with the design parameter called critical value.

2. The EWMA scheme (exponentially weighted moving average) initially presented by Roberts (1959)

 (11.3) (11.4)

with the smoothing value and the critical value . The smaller the faster EWMA detects small .

3. The CUSUM scheme (cumulative sum) introduced by Page (1954)

 (11.5) (11.6)

with the reference value and the critical value (known as decision interval). For fastest detection of CUSUM has to be set up with .

The above notation uses normalized data. Thus, it is not important whether is a single observation or a sample statistic as the empirical mean.

Remark, that for using one-sided lower schemes one has to apply the upper schemes to the data multiplied with -1. A slight modification of one-sided Shewhart and EWMA charts leads to their two-sided versions. One has to replace in the comparison of chart statistic and threshold the original statistic and by their absolute value. The two-sided versions of these schemes are more popular than the one-sided ones. For two-sided CUSUM schemes we consider a combination of two one-sided schemes, Lucas (1976) or Lucas and Crosier (1982), and a scheme based on Crosier (1986). Note, that in some recent papers the same concept of combination of two one-sided schemes is used for EWMA charts.

Recall, that Shewhart charts are a special case of EWMA schemes (). Therefore, we distinguish 5 SPC schemes - ewma1, ewma2, cusum1, cusum2 (two one-sided schemes), and cusumC (Crosier's scheme). For the two-sided EWMA charts the following quantlets are provided in the XploRe quantlib spc .

By replacing ewma2 by one of the remaining four scheme titles the related characteristics can be computed.

The quantlets spcewma1 ,..., spccusumC generate the chart figure. Here, we apply the 5 charts to artificial data. 100 pseudo random values from a normal distribution are generated. The first 80 values have expectation 0, the next 20 values have expectation 1, i.e. model (11.1) with , , and .

We start with the two-sided EWMA scheme and set , i.e. the chart is very sensitive to small changes. The critical value (see (11.3)) is computed to provide an in-control ARL of 300 (see Section 11.2). Thus, the scheme leads in average after 300 observations to a false alarm.

In Figure 11.1 the graph of is plotted against time . Further, the design parameter , the in-control ARL, and the time of alarm (if there is one) are printed. One can see, that the above EWMA scheme detects the change point at time point 94, i.e. the delay is equal to 14. The related average values, i.e. ARL and Average Delay (AD), for are 9.33 and 9.13, respectively. Thus, the scheme needs here about 5 observations more than average.

In the same way the remaining four SPC schemes can be plotted. Remark, that in case of ewma1 one further parameter has to be set. In order to obtain a suitable figure and an appropriate scheme the EWMA statistic (see (11.4)) is reflected at a pre-specified border zreflect , i.e.

for an upper EWMA scheme. Otherwise, the statistic is unbounded, which leads to schemes with poor worst case performance.

Further, the methods used in Section 11.2 for computing the chart characteristics use bounded continuation regions of the chart. If zreflect is small enough, then the ARL and the AD (which are not worst case criterions) of the reflected scheme are the same as of the unbounded scheme. Applying the quantlet XFGewma1fig.xpl with zreflect leads to Figure 11.2. Thereby, zreflect has the same normalization factor like the critical value (see 2.). The corresponding normalized border is printed as dotted line (see Figure 11.2). The chart signals one observation earlier than the two-sided version in Figure 11.1. The related ARL and AD values for are now 7.88 and 7.87, respectively.

In Figure 11.3 the three different CUSUM charts with are presented. They signal at the time points 87, 88, and 88 for cusum1, cusum2, and cusumC, respectively.

For the considered dataset the CUSUM charts are faster because of their better worst case performance. The observations right before the change point at are smaller than average. Therefore, the EWMA charts need more time to react to the increased average. The related average values of the run length, i.e. ARL and AD, are 8.17 and 7.52, 9.52 and 8.82, 9.03 and 8.79 for cusum1, cusum2, and cusumC, respectively.