Page 401 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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August 31, 2006
JWBK119-25
386 CUSUM and Backward CUSUM for Autocorrelated Observations
In SPC applications, the time j when the change occurred is unknown a priori. Thus,
in order to detect whether there has been a change in the process average on an on-line
basis, we need calculate the following backward standardized average (BCuZ):

n
i= j y i
BCuZ j = √ , (25.7)
σ n − j + 1
for 1 ≤ j ≤ n. When a particular BCuZ exceeds a pre-specified threshold value, say
z*, we conclude that there has been a change in the process average. Note that at every
time period t, we calculate n dependent BCuZs. Thus, unlike the z α in (25.6), z*isthe
threshold for the n dependent BCuZs calculated at time t. For the BCuZ, the α-value
in (25.6) cannot therefore be interpreted as the probability of eventually obtaining a
false alarm. One chooses z* such that the control charting scheme produces a desired
in-control ARL. This can be done using simulation.
Instead of plotting the BCuZ, one may utilize the BCUSUM with the corresponding
control limits

n √
BCUSUM j = y i ≥ z * σ n − j + 1, 1 ≤ j ≤ n. (25.8)
i= j
One can see that the control limits of the above BCUSUM resemble a parabolic mask.
This BCUSUM can also be implemented using a backward moving average (BMA)
scheme. The BMA with its control limits takes the form

n
i= j y i * σ
BMA j = ≥ z √ , 1 ≤ j ≤ n.
n − j + 1 n − j + 1
In using the BCuZ, BCUSUM or BMA, one needs to plot n points at each time period
t. If one prefers a Shewhart-like plot, the following statistic may be monitored at each
time period:

n
i= j y i
max √ .
1≤ j≤n σ n − j + 1
When this statistic exceeds z * , one concludes that the process average has shifted. We
can see that the above statistic is a special case of the level shift detection procedure
described by Tsay. 22
For a two-sided alternative for hypothesis (25.2) where j is known, we use the
following statistic to determine whether there has been a change in mean:

n

y i
i= j

√
≥ z α/2 .

σ n − j + 1
For on-line detection of mean shift, we may use the following BCUSUM scheme:

n √
BCUSUM j = y i ≥ z * σ n − j + 1
i= j

n √ , 1 ≤ j ≤ n, (25.9)
BCUSUM j = y i ≤−z * σ n − j + 1
i= j
in which the right-hand side represents a parabolic mask. Alternatively, one may
monitor the following statistic:

n

y i
i= j

*
max
√
≥ z . (25.10)
1≤ j≤n
σ n − j + 1