Page 399 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

OTE/SPH
OTE/SPH
3:9
Char Count= 0
JWBK119-25
August 31, 2006
384 CUSUM and Backward CUSUM for Autocorrelated Observations
purposes is that contained in the last row. The BCUSUMs in rows 1--5 are of no use
at all unless we are trying to detect change not on an on-line or real-time basis. For
example, it is easy to see that when there is a BCUSUM value in row 4 that exceeds the
specified control limits, one does not need the ensuing observations (i.e., observations
5 and 6) to conclude that there has been a change in mean of the forecast errors. At
period 4, the forecaster already knows that there has been a change or intervention
affecting the system. Thus, if one needs to implement a BCUSUM considering only
the last few forecast errors, one need only calculate the following BCUSUM:
i

S i = e n− j+1 , i = 1, 2,..., m,
j=1
where m represents the number of latest forecast errors being monitored.
One drawback in basing a monitoring scheme only on the latest few m observations
is that small process shifts will be difficult to detect. In detecting a change in the mean
of a normal random variable, for example, the ARL of a two-sided CUSUM chart
(h = 4.8, k = 0.5, ARL 0 = 390)fora0.25σ meanshiftisapproximately120.Monitoring
only the latest 6--10 observations will not provide enough sensitivity to detect such a
change. This ARL figure may not have any practical importance in forecasting since
forecasters are usually dealing with few samples. However, as pointed out by Atienza
et al., 20 forecast monitoring techniques are also useful in process monitoring. Thus, in
this chapter, we are also concerned with the ability of BCUSUM to detect small shifts
in process mean.

25.2.1 A likelihood ratio scheme
Gardner 19 noted that although the BCUSUM is considered to be the most thorough
forecast monitoring procedure available, it has not been widely used in practice be-
cause of the lack of published control limits and studies analyzing its performance
in comparison with the other established techniques. In this section, we derive the
BCUSUM scheme using the likelihood ratio principle. Using this approach, we can
easily determine the appropriate form of the control limits.
We assume that errors arising from a given forecasting model are independent and
2
normally distributed with mean μ ε = 0 and constant variance σ . Note that with this
ε
assumption, we can easily view the forecast errors as the data analogous to the pro-
cess measurements {y t } used in SPC which are often assumed i.i.d. normal when
no external sources of variation are affecting the process. In what follows, we dis-
cuss the BCUSUM scheme in the context of SPC measurements {y t }. Without loss of
generality, we assume that when the process is in control, {y t } has mean μ = 0 and
2
constant variance σ . Thus, in monitoring {y t }, our interest is in sequentially testing
the hypothesis

H 0 : μ = 0, for y 1 , y 2 ,..., y n , (25.2)
versus the alternative hypothesis

μ = 0, for y 1 , y 2 ,..., y j−1 ,
H j : (25.3)
μ = μ 1 (μ 1 > 0), for y j , y j+1 ,..., y n .