Recall that the data
follow the change point model
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:
with the design parameter called critical value.
with the smoothing value and the critical value
. The smaller
the faster EWMA detects small
.
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.