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382 CUSUM and Backward CUSUM for Autocorrelated Observations
even better than the V-mask but gave no analysis of the properties of such tests. More
6
recently, Wiklund described a parabolic CUSUM based on Lorden’s results. 7
On the other hand, with the widespread use of high-speed data collection schemes,
process measurements are often serially correlated even when there are no special
causes of variation affecting the system. The presence of autocorrelation violates the
independence assumption in many statistical process control (SPC) charting proce-
dures and degrades the performance of classical SPC procedures. 8,9 When the process
measurements are serially correlated, a popular monitoring approach is to model
the inherent autocorrelation using the Box--Jenkins autoregressive moving average
(ARMA) model

(25.1)
φ p (B)y t = φ 0 + θ q (B)ε t
where {y t } is a stationary time series representing the process measurements,
2
φ p (B) = 1 − φ 1 B − φ 2 B −· · ·− φ p B p is an autoregressive polynomial of order p,
2
q
θ q (B) = 1 − θ 1 B − θ 2 B −· · ·− θ q B is a moving average polynomial of order q, B
is the backshift operator, and {ε t } is a sequence of normally and independently dis-
2
tributed random errors with mean zero and constant variance σ . If we let ˆy t rep-
ε
resent the predicted value from an appropriately identified ARMA model, then the
sequence of residuals {e 1 = y 1 − ˆy 1 , e 2 = y 2 − ˆy 2 ,...} will behave like an independent
and identically distributed (i.i.d.) random variable. 10 In detecting process changes,
classical SPC charts can therefore be applied to the residuals of the appropriately cho-
sen ARMA model. However, the performance of these control charts is very different
from the i.i.d. case. 11
11
Wardell et al. showed that the Shewhart chart on residuals (SCR) has poor sensitiv-
ity when the process is highly positively autocorrelated. To address this issue, Runger
12
et al. suggested the use of CUSUM in monitoring the residuals (CUSUMR). Yaschin 13
proposed a transformation that generates an i.i.d. sequence of random variable when
the process is in control. He suggested monitoring the resulting transformed series
14
using a CUSUM chart. Wardell et al. explained that Yaschin’s approach is essentially
equivalent to monitoring the residuals using a CUSUM chart.
In this chapter, a purely parabolic mask for CUSUM, proposed by Atienza et al., 15
is presented. This mask, which is based on a uniformly most powerful (UMP) test, is
motivated by constructing the dual relationship of CUSUM and backward CUSUM
(BCUSUM). In addition, an alternative CUSUM scheme that directly utilizes the au-
tocorrelated observations in monitoring changes in process mean is presented.

25.2 BACKWARD CUSUM

16
The BCUSUM scheme was originally developed by Harrison and Davies for routine
forecast monitoring. It is based on the idea that if we can guess the time j at which the
mean of the forecast errors (e 1 , e 2 ,. . . ) shifted, then the sum or the average of {e t } since
the change occurred would be the best indicator of the presence of a special cause. In
practice, the time j when the shift in mean occurred is unknown a priori. One way to
overcome this problem is to calculate a series of partial sums of all the past errors as