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13

Specialized Control Charts

KEY WORDS AR model, autocorrelation, bump disturbance, control chart, Cusum, Cuscore, cyclic vari-
ation, 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.

The Cuscore Statistic
Consider a statistical model in which the y t are observations, θ is some unknown parameter, and the x t ’s
are known independent variables. This can be written in the form:
y t = fx t , θ) + a t t = 1,2,…, n
(
Assume that when θ is the true value of the unknown parameter, the resulting a t ’s are a sequence of
independently, identically, normally distributed random variables with mean zero and variance σ a = σ 2 .
2
The series of a t ’s is called a white noise sequence. The model is a way of reducing data to white noise:
(
a t = y t – fx t , θ)

The Cusum chart is based on the simplest possible model, y t = η + a t . As long as the process is in control
(varying randomly about the mean), subtracting the mean reduces the series of y t to a series of white
noise. The cumulative sum of the white noise series is the Cusum statistic and this is plotted on the
Cusum chart. In a more general way, the Cusum is a Cuscore that relates how the residuals change with
respect to changes in the mean (the parameter η).
Box and Ramirez (1992) defined the Cuscore associated with the parameter value θ = θ 0 as:
Q = ∑ a t0 d t0

© 2002 By CRC Press LLC