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Vander Wiel

the probabilistic characteristics of run lengths where a run length is the number of periods between the shift and the first signal of the control chart. In particular, the average run length (ARL) has been emphasized. The ARL uses all signaling probabilities. It is equally important to detect step shifts in processes that wander as in processed that are nominally iid. If a sudden shift can be detected, it might be possible to remove the cause, eliminate a source of variability and improve the process. If enough sources of variation can be eliminated, the process might begin to wander less and look morę like iid observations than a random walk.

It is important to notę that a step change is morę difficult to see when buried in an IMA than when buried in iid noise. The middle column of plots in Figurę 3 shows the same IMAs as in the left column but with a step shift of 5cr beginning in period 150. The shift stands out in the iid (A. = 0) sequence but is less obvious as A. increases. For example, the random walk (X = 1) drops abruptly around period 45 by an amount similar to the sharp increase at period 150. The first change, however is due to several consecutive negative a's while the second is a real shift in the Ievel (or equivalently one giant a).

One way to understand why shifts are easier to see for smaller X is to consider how evidence of a shift builds as data accumulates. At period 151, each plot steps upward by about 5ct signifying a probable shift. In the iid plot all doubt about whether a shift really occurred is gone by, say, period 160, because each of the last 10 observations is about 5ct higher than the first 150. In the random walk plot, however, all the information about a shift comes in period 150 and new data contributes no new information because the process simply takes iid steps from its previbus position. For X = 0.5 some evidence of a shift accumulates in the periods just after 150, but by, say, period 200 the process has wandered enough that new observations are not relevant to what happened at period 150. For middling values of A. it is tricky to mentally judge how much of what we see is due to the 5ct shift and how much is due to autocorrelation. Looking at the affect of a shift on the residuals a, helps remove the ambiguity.

Suppose a process Z, behaves like an IMA but experiences a step shift of size p in its Ievel at some period k. That is,

Z,=N_t,    t =

= W,+H.

t = k,k +1,...