Recently, some papers investigated the steady-state performance of the control charts. They claimed that steady-state performance should be preferred over zero-state performance. In steady-state, some in-control samples (ICS) are generated before a shift has occurred in the process and conditional expected delay is calculated.
Questions
1. Is conditional expected delay=ICS+ARL1? where ARL1 is the average run length of zero state. If ICS=0,conditional expected delay=ARL1???
2. We can say that zero-state is a special case of steady-state??
3. Do Zero state control charts have no benefits?
Your feedback will be highly appreciated
Muhammad Aslam
1- The conditional expected delay (CED) is a metric that is used to evaluate the performance of a control chart in steady-state. The CED is defined as the expected number of additional in-control samples that must be generated after the first out-of-control sample is detected before a signal is declared. The formula for the CED depends on the specific control chart and the assumptions made about the process distribution. In general, the CED is not equal to the sum of the in-control samples (ICS) and the ARL1. However, if ICS is 0, the CED will be equal to the ARL1.
Muhammad Aslam
2- Yes, zero-state can be considered a special case of steady-state. In zero-state, the process is assumed to be in control at the start of the monitoring period, and the control chart is used to detect the first out-of-control signal. In steady-state, the process is assumed to be in a stable state, and the control chart is used to detect any changes in the process mean or variance. In both cases, the performance of the control chart is evaluated using metrics such as the ARL and CED.
Muhammad Aslam
3- Zero-state control charts have some benefits, particularly in cases where the process is assumed to be in control at the start of the monitoring period. Zero-state control charts can be used to quickly detect the first out-of-control signal, which can be useful in preventing the production of defective products. However, zero-state control charts may not be as effective in detecting smaller shifts or changes in the process mean or variance, which is where steady-state control charts may be more useful. Therefore, the choice of control chart depends on the specific application and the goals of the monitoring process.
Let ARL1 =20.
1. Suppose that ICS=20 and samples are generated from the shifted process is 25. In this case, the process will be out of control at 40th sample. In this case CED>ICS+ARL1
2. Suppose that ICS=20 and 20 samples are generated from the shifted process. The process will be out of the control at 35th sample. In this case, CED=ICS+ARL1
3. Suppose that ICS=0 and process is shifted at 20th sample, then CED=ARL1
Please give feedback
The conditional expected delay, CED(t), is the expected number of samples to obtain an out-of-control signal given that the process shift occurs at time t and there were no previous false alarms.
The zero state ARL is CED(1). The steady state ARL is the limit of CED(t) as t increases.
The zero state ARL can be very misleading. If the process shift is assumed to occur immediately as monitoring begins then all data values could be weighted equally. This leads to essentially the progressive mean chart.
For delayed shifts, data should not be weighted equally. Recent data should be given more weight as with the EWMA statistic,
Because the zero state ARL performance can be very misleading, the steady-state performance is preferred. It is unrealistic to assume any shift must occur immediately.
I try a short answer. If one uses the zero-state ARL to quantify the detection speed, then he/she is implicitly assuming to know the actual value of the control chart statistic right before the change happens. Mostly, we do not know when the change happens (otherwise we would not apply a control chart) and in consequence we do not know the position of the chart statistic. One popular way of dealing with this uncertainty is to consider the (conditional) steady-state distribution of this statistic. Others use the worst case (Lorden 1971; Moustakides in 1986 proved that CUSUM is optimal). Looking at the CED(t) profile tells us whether the steady-state ARL does its job.
Most of the confusion stems from numerical surprises. For two-sided EWMA charts with constant limits, the zero-state and steady-state ARL values nearly coincide. For the usual CUSUM chart (one-sided) w/o head-start, the zero-state ARL exactly coincides with the worst case measure due to Lorden. For other control charts (in particular some of the recently proposed ones), this is not valid anymore so that judging the full CED(t) profile is "mandatory".