Statistics for Environmental Engineers

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FIGURE 12.3 Cusum chart for the single observations in the top panel (also the top panel of Figure 12.2). The mean level shifts up by 0.5 units from day 50-75, it is back to normal from days 76-92, it shifts down by 0.5 units from days 93-107, and is back to normal from day 108 onward. The increase is shown by the upward trend that starts at day 50, the decrease is shown by the downward trend starting just after day 90. The periods of normal operation (days 1 -50, 76-92, and 108-150) are shown by slightly drifting horizontal pieces.


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Specialized Control Charts


KEY WORDS AR model, autocorrelation, bump disturbance, control chart, Cusum, Cuscore, cyclic variation, discrepancy vector, drift, EWMA, IMA model, linear model, moving average, process monitoring, random variation, rate of increase, serial correlation, Shewhart chart, sine wave disturbance, slope, spike, weighted average, white noise.


Charts are used often for process monitoring and sometimes for process control. The charts used for these different objectives take different forms. This chapter deals with the situation where the object is not primarily to regulate but to monitor the process. The monitoring should verify the continuous stability of the process once the process has been brought into a state of statistical control. It should detect deviations from the stable state so the operator can start a search for the problem and take corrective actions.


The classical approach to this is the Shewhart chart. A nice feature of the Shewhart chart is that it is a direct plot of the actual data. Humans are skilled at extracting information from such charts and they can sometimes discover process changes of a totally unexpected kind. However, this characteristic also means that the Shewhart chart will not be as sensitive to some specific deviation from randomness as another specially chosen chart can be. When a specific kind of deviation is feared, a chart is needed that is especially sensitive to that kind of deviation. This chart should be used in addition to the Shewhart chart. The Page-Barnard cumulative sum (Cusum) chart is an example of a specialized control chart. It is especially sensitive to small changes in the mean level of a process, as indicated by the change in slope of the Cusum plot. The Cusum is one example of a Cuscore statistic.

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