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Applications of the EWMA

Slow (A. = .25)

FigurÄ™ 1. Plot of Slow, Moderate, and Fast Drifts vs. Time.

previous forecast and the current data value. With successive substitutions for the previous forecast the EWMA is shown to be a weighted average of all previous data with the weights decreasing exponentially for data further back in time:

Z_t = XD_t + Ml- X)D_t__l + Ml- X)²D_t_₂ + Ml- ₃ +. ..

As shown by FigurÄ™ 2, these weights are relatively heavy on recent data for fast drifts but spread further back in time for slow drifts.

This EWMA forecast is useful from a process control standpoint because, as shown by Box and Jenkins (1976) it is the minimum mean squared error forecast of D_t+ j, the process deviation from target for the next sample if no adjustment is madÄ™. Box and Jenkins show that Z_t is the conditional expectation of D_t+ j given current and previous values of D, and that Var(D,₊ j - Z_t)= a\, which is the smallest possible forecast error variance assuming the

IMA model is correct. Therefore, if g and X are properly estimated and the IMA model fits the data, an adjustment of the form, x₍ = X,-X_M=-(Z,-Target)/*, will center the process on target at time / + 1 as follows: