Page 196 - Special Topic Session (STS) - Volume 3

STS540 Zhi Song et al.
Generally, the ARL is the widely used criterion for assessing the
performance of monitoring schemes. However, since the FIR feature is more
meaningful for early detection, we use two other design criteria rather than
depending solely on the ARL performance measure. One design criterion is
cumulative unconditional false alarm probability (CUFAP), which is denoted by
CUFAP()=P( ≤ |). Another is cumulative unconditional true signal
probability (CUTSP), denoted as CUTSP()=P( ≤ |). Thereafter, we
may use a simple algorithm to optimize the design of FIR-based EWMA
schemes via the following steps:

Step-1: Given , , , and a target IC ARL ( ), say , compute for a given
0
EWMA-LS scheme without the FIR feature. The abbreviation LS stands for the
Location-Scale statistic, namely, Lepage or Cucconi, with = (LS|IC).
0

Step-2: Let ∈ [0,1] is a small positive proper fraction chosen a-priori by the
practitioner, denotes the maximum allowable CUFAP till -th test sample.
Compute CUFAP() for a pre-specified ∈ N. If CUFAP() > , increase
by a margin of 0.01 and re-compute and CUFAP( ). Note that, in
0
general, the CUFAP() is a decreasing function and the is an increasing
0
function of given other charting parameters, namely, , , , remain same.

Step-3: Repeat the Step-2, unless, CUFAP() ≤ .

Step-4: Set the CUFAP() = , and use the L obtained at the end of the
Step-3, say , to determine dynamic FIR quotient. Using and determine
0
0
0
the distance, say dLS between and UCL at the first test sample. Set FIR
0
feature ℎ ∈ [0,1], where ℎ = ℎ implies that the starting value will be reset

at 0ℎ = + ℎ × .

0

Step-5: We see, in general, for fixed , if increases to ℎ > 0, ,
0
0ℎ
0
CUFAP() increases. Therefore, to maintain the specified preapproved level
of CUFAP(), we need to increase the to and consequently, the ,
0
ℎ
will also increase. Compute for each ℎ = 0.01(0.01)1.00.
ℎ
0ℎ

Step-6: Given , , , and target shift = , = compute

CUTS () for all ℎ = 0.00(0.01)1.00.
ℎ, ,

Step-7: Select dynamic FIR as ℎ = ℎ for which CUTSP ℎ, , () is maximum.
Because of the distribution-free nature of the proposed class of monitoring
schemes when the process is IC, determination of value based on a fixed
ℎ
CUFAP() will be the same for all continuous distributions. We may generate
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