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OTE/SPH
OTE/SPH
August 31, 2006
3:8
JWBK119-24
Char Count= 0
374 Cumulative Sum Charts with Fast Initial Response
for each unit of the distance along the x-axis between consecutive points plotted in
the V-mask scheme (see Figure 24.2).
24.2 FAST INITIAL RESPONSE

−
During the start-up stage, CUSUM and CUSUM are given a value of zero. However,
+
0 0
Lucas and Crosier 11 showed that the CUSUM may not reset to zero during start-up
or after an out-of-control situation is detected. This non-zero start-up for CUSUM is
called the fast initial response (FIR).
24.2.1 FIR for tabular CUSUM

The FIR (tabular) CUSUM for upward/downward shift is defined by

+
CUSUM = max 0, CUSUM + + y i − k , CUSUM = H ,
+
+
i i−1 0

−
−
CUSUM = max 0, CUSUM − − y i − k , CUSUM = H , (24.4)
−
i i−1 0
and signals if CUSUM > h or CUSUM > h.
+
−
i i
11
The quantities H and H are the FIR. Lucas and Crosier recommended an FIR
+
−
value of h/2. The motivation for the FIR CUSUM is that, if the process starts in an out-
of-control or off-target state, then starting the CUSUM part way toward the threshold
value h will hasten the out-of-control signal. If, however, the process is not out of
control, then it is likely that the CUSUM will soon drop back to zero, after which the
FIR CUSUM behaves like a conventional zero-start CUSUM. Thus, it has little effect
when the process is in control or on target.
24.2.2 FIR for V-mask CUSUM
While the head-start feature is well known for tabular CUSUM, its counterpart for
the V-mask CUSUM was not available until 1999 the work of Atienza et al. 12 As a
result, standard references 5,8 do not cover the FIR feature for the V-mask CUSUM.
Nevertheless, practitioners seem to prefer the ease of interpretation and the visual
effect of V-mask CUSUM. It is thus instructive to give a detailed presentation of the
FIR feature for the V-mask CUSUM.
In order to implement the FIR feature in the V-mask scheme, we need to understand
the importance of the point at the origin (0,0) which represents a CUSUM value of
when i = 0. In a tabular CUSUM without an FIR, the first observation y 1 signals an
out-of-control situation when its value exceeds h + k. Under the V-mask scheme, this
situation is represented by the point at the origin (0,0) that is outside the arms of
the mask. The origin (0,0) is thus the key consideration for head start feature during
the start-up stage. In using the tabular CUSUM with FIR, the first observation y 1
signals an out-of-control situation when its value exceeds h + k− FIR. In the V-mask
scheme, this is precisely indicated by (0,0) outside the arms of the mask with the first
+
point plotted being CUSUM = y 1 + FIR. In implementing the FIR feature under the
1