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August 31, 2006
JWBK119-23
358 Statistical Process Control for Autocorrelated Processes
To detect an AO, IO or LS at unknown position, we calculate the following statistics:

λ j,max = max
λ j,t
j = AO, IO, LS. (23.8)
1≤t≤T
The null hypothesis of no AO, IO or LS is rejected when λ j,max exceeds a pre-specified
critical value. Note that it is difficult to determine the exact repeated sampling distri-
bution of equation (23.8). Chen and Liu 35 considered a broad spectrum of π weight
patterns in estimating the percentiles of (23.8) using Monte Carlo simulation. They
noted that the percentiles associated with (23.6) are distinctively lower than those
associated with equations (23.4) and (23.5). For an AR(1) process, for example, when
T = 200 the estimated 1st percentile for λ LS,max , based on Chen and Liu’s results, is
between 3.3 and 3.5, while for λ AO,max and λ IO,max it is between 4.0 and 4.2.
In SPC, we are primarily concerned with the detection of the presence of external
sources of variation manifested by changes in process level or variance. If we can
determine whether the change in process behavior is due to an AO, IO or LS, we can
narrow down its most probable cause. This will facilitate the quick discovery and
correction of problems associated with the detected signal of process change. In the
following sections, we can see that an SPC scheme based on the outliers-and-level-
shift-detection statistics can be established for process monitoring purposes. Although
we focus our discussion on LS detection for an AR(1) process, one can easily extend
the application of the proposed procedures in effectively identifying an LS, IO, or AO
under the general autogressive integrated moving average model.

23.3 BEHAVIOR OF λ LS,t
It is known that, when an outlier or level shift occurs at some known point, the statis-
tics (23.4), (23.5), and (23.6) are normally distributed with means ω AO , ω IO , and ω LS ,
2
2
2
and variances ρ 2 σ , ρ 2 σ and ρ 2 σ , respectively. In this section, we study the
AO,d IO,d LS,d
behavior of {λ LS,t :1 ≤ t ≤ T} when a level shift or outlier occurs. Doing so may lead
to ways of making the procedure described in the preceding section more sensitive to
the presence of exogenous interventions or special causes when implemented on-line.
23.3.1 Effect of a level shift

In studying the effect of outliers and level shifts on λ LS,t :1 ≤ t ≤ T , we focus on
the AR(1) model
Z t − μ = φ(Z t−1 − μ) + ε t , t = 1, 2,..., T, (23.9)
2
where −1 <φ < 1 and the random errors ε t are i.i.d N(0,σ ). For the AR(1) process
2
given by equation (23.9), π(B) = 1 − φB and η(B) = 1 − (φ − 1)B − (φ − 1)B − ... .
Now assume that at time t = d (1 ≤ t ≤ T) the mean shifts from 0 to ω LS . Using Tsay’s
33
notation , we have Y t = Z t for t < d and Y t = Z t + ω LS for t ≥ d. It can be easily shown
that
⎧
⎨ ε t , t < d,
y t = ω LS +ε t , t = d, (23.10)
(1 − φ)ω LS + ε t , t > d.
⎩