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CUSUM Scheme for Autocorrelated Observations 391
25.4 CUSUM SCHEME FOR AUTOCORRELATED OBSERVATIONS
The issues associated with monitoring autocorrelated processes have been presented
in Chapters 22 and 23, and some of them which have been addressed further in this and
the previous chapter. Here, using the characteristics of BCUSUM, a CUSUM scheme
for detecting a step change in process mean that directly utilizes the autocorrelated
process observations is presented.

25.4.1 Formulation
Assuming that the process is stable, {y t } can be described by a stationary ARMA
process as given in equation (25.1). In SPC, our objective is to detect changes in mean
as early as possible. Recall that the sum or average of all the measurements since the
change occurred is the best indicator of process change, if we know or can guess the
time of occurrence. This intuitive claim can be supported by the hypothesis testing
problem shown in (25.2) and (25.3).
In the presence of autocorrelation, under H 0 in (25.2), we have

|h|

¯ y n− j+1 n − j + 1 ∼ N 0, 1 − γ (h) , (25.16)
n − j + 1
|h|<n− j+1
where

n
i= j y i
¯ y n− j+1 =
n − j + 1
and γ (·) is the autocovariance function of {y t , t =1, 2, . . . }. 26 The variance of
√ 2
¯ y n− j+1 n − j + 1 approaches υ = ∞ γ (h)as n becomes large. Using (25.16),
h=−∞
we can therefore test (25.2) against (25.3) using the test statistic
√
¯ y n− j+1 n − j + 1
. (25.17)
z =
(1 − |h|/n − j + 1) γ (h)

|h|<n− j+1
With the inclusion of the autocorrelation coefficient, the development is similar to
that of the i.i.d. case. We reject the null hypothesis (25.2) in favor of (25.3) when the
z-statistic in (25.17) exceeds a certain critical value z α , that is,
√
¯ y n − j + 1
> z α . (25.18)

(1 − |h|/n − j + 1) γ (h)

|h|<n− j+1
As the exact time of occurrence of change is not known a priori, to implement the
scheme on an on-line basis, we need to calculate n z-statistics,
√
¯ y n− j+1 n − j + 1
, j = 1, 2,..., n. (25.19)
z j =
(1 − |h|/n − j + 1) γ (h)

|h|<n− j+1
Similarly, when a z j exceeds a pre-specified control limit, say z * (to be obtained from
simulation with a desired ARL value) we conclude that there has been a change in