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Proposed Monitoring Procedure 363
the AO case, when an IO occurs at T = d, E(ˆω LS,d ) = E(ˆω IO,d ) = ω IO,d . So SPC schemes
based on ˆω LS,t may also detect the presence of an IO.
23.4 PROPOSED MONITORING PROCEDURE

The results in the preceding sections suggest that we can establish control charting
procedures for detecting process level shifts by monitoring either λ LS, max or λ LS, =
T λ LS,,t /T. In assessing the performance of the proposed control charting schemes,
t=1
we focus on detecting the LS in an AR(1) process.

23.4.1 Control chart implementation
In implementing the proposed control charting scheme, one needs to start with an
initial set of m observations taken when the process is in control. During the initial
stage of implementation, the process is most likely affected by outliers and level
shifts. Their presence may result in inappropriate model identification. This may
subsequently lead to misidentification of process excursions. Thus, at this stage, we
recommend the use of procedures that can accomplish both outlier detection and
model estimation jointly. 33,35−37 .
Analysis shows that control charts based on the statistic λ LS,t are sensitive to the
choice of m. For example, simulation results show that m = 100 gives poor average
run length (ARL) performance. A satisfactory ARL performance is achieved when
m = 200. Note that although this m is large by conventional SPC standards, it may
only correspond to a few seconds or minutes in real-time process monitoring.
As soon as the initial set of m process observations becomes available, one may start
calculating λ LS,t . In using the proposed control charting scheme, one only needs to
maintain the m latest observations. That is, when a new observation becomes avail-
able, the oldest observation is discarded. Using the m latest observations, we calculate
the m λ LS,t s. To illustrate, suppose we start with an initial set of process observations
{Z 1 , Z 2 ,..., Z m }. From this initial set, we calculate the m λ LS,t s, {λ LS,1 ,λ LS,2 ,...,λ LS,m }.
When the observation Z m+1 becomes available, we use the window of observations
{Z 2 , Z 3 ,..., Z m+1 } to calculate {λ LS,2 ,λ LS,3 ,...,λ LS,m+1 }. Thus, for any time period i
after the initial set of m observations was taken, we have {Z i+1 , Z i+2 ,..., Z m+i }, a win-
dow of m latest observations, from which we calculate {λ LS,i+1 ,λ LS,i+2 ,...,λ LS,m+i }.In
mathematical form, we therefore need to track either

λ LS,max,i = max {|λ LS,t |}, i = 1, 2, 3,...,
i+1≤t≤m+i
or
m+i
t=i+1 λ LS,t
λ LS,i = , i = 1, 2, 3,....
m
The chart signals an out-of-control situation when λ LS, max or λ LS,t exceeds some pre-
specified control limits. These control limits can be based on the sampling distribution
of λ LS,max or λ LS , which is difficult to obtain as these statistics are functions of depen-
dent variables. We therefore recommend the use of simulation in determining the