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Baxley

where the a, ’s are NID(0, a]) representing random shocks to the process, and

X. is a parameter taking values between 0 and 1. In this model, the D’s are non-stationary (have no fixed mean), so use of this model implies that if no control actions are taken, the process will drift away from target with no tendency to return. The above disturbance model can often be fit to sampled data from the laboratory in the process industries, because these data are comprised of real process drifts buried in measurement noise. The EWMA model can be re-cast to reflect this behavior as follows:

D_t = R_t +b(,

where b_t is white noise , NID(0, o\), representing sampling and measurement error, and R_t is a random walk process representing the combined effects of lurking variables which tend to drive the process away from target. The random walk is an integrated moving average (0,1,1) process with X = 1:

^Rt = ^Rt-l + ^ub

where the u_t ’s are NID(0, <j²u). It is shown by Box and Jenkins (1976, p. 124) that <j²u = A² a] and that a] =(l-A)<j²a. A process with fast drifts is one where R^, which represents the true process level at time /, can move rapidly over the rangÄ™ defined by the magnitude of o_b, resulting from a relatively large variance of the shocks, u,. In other words, the forces driving the process away from target are large relative to the sampling and measurement errors. The integrated moving average parameter, X , which has values from zero to one, is closer to one for such a process. In contrast, processes with slow drifts have a smali variance of «, and larger sampling and measurement errors. They

can be modelled as an integrated moving average with X closer to zero. Simulated data from slow, moderate, and fast drifting processes are shown in FigurÄ™ 1.

The minimum mean squared error one-step-ahead forecast statistic for the integrated moving average model is given by the familiar EWMA formula,

Z( — (1 - X)Zf_ j + XDf_y

where X is usually estimated from process data by iterative procedures discussed by Hunter (1986), or by non-linear least sÄ…uares methods as discussed by Brocklebank (1990). The EWMA forecast is a weighted average of the