00196 ÷c3e9ff6e1645b5c66f96f0e3b703fa

198

Messina, Montgomery, Keats & Runger

The residual for the t th period is defined as e_t = u_t-zt

Next the predicted error EWMA is defined as

Zt = tet + ( 1 - X )z_t_!    (5)

The standard deviation of the predicted error EWMA is defined as

n

The predicted error EWMA control limits are set at

UCL = 3 o_zJ(X / (2 - X) and

LCL =    / (2 - X)

The average of the predicted error EWMA is set at 0 sińce the Efe^) =

0.

Since a forecasting approach is being employed here, a tracking signal can be applied to the residuals in order to monitor the forecast to determine if it is unbiased. Here we employ the smoothed error tracking signal as described in Montgomery, Johnson, and Gardiner (1990). Define Ä…uantities Q(t) and A(t) asfollows:

Q(t) = a _ei(T) + ( 1 - a) Q(T - 1)

A(t) = «|e,(r)| + ( 1 - a) A(T-1)

The value of a = 0.1. The smoothed error tracking signal (S2) is defined as follows