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Applications of the EWMA

The Control Algorithm In this algorithm, Y_t is the value of the quality measurement at time t, Z_t is the EWMA forecast of the ąuality measurement for time t + 1, anc\A_t = a₍ + the observed forecast error at time t. Controller parameters to be estimated from prior data include ct_z and cs_a, while

L defines the EWMA limits in multiples of a_z , and K defines the Shewhart limits on A_t in multiples of ct₀ .

Step 1 - Calculate the forecast error:

A_t = (T_rTarget) - Z_M Step 2 - Update the EWMA forecast:

lf\A_t\ źK <ją , then Z_t = Z_t_ j Elsę Z_t= X (Y_{ - Target) + (1 - X )Z^.j

Explanation: Y₍ - Target is substituted for the D_t in Step 2 because it is the observed control error. Justification for this will be given following Step 4. If the forecast error is out of limits, interpret this as a signal of a special cause (S_f * 0) and not a regular process disturbance (D_t = D_t_j). In this case do not update the EWMA forecast. This prevents large adjustments from being madę which could move the process to the other side of target if S_t+ j = 0. Instead, emphasis is placed on finding and fixing the special cause.

Step 3 - Calculate the process adjustment:

If \Z_t\ > L <7% , then x₍ = -Z/g Elsę x_t = 0 Step 4 - Reset the EWMA forecast to zero if an adjustment is madę:

If x_t* 0 , then Z_t = 0

Explanation: This re-initialization of the EWMA forecast is based on Eąuations 2 and 3. The next adjustment after time t will be madę to compensate for the change in Z over a series of sample intervals where there is no change in X. If S = 0 during this time period, Eąuation 1 shows that the change in Z is unaffected by using ł^-Target instead of D₍, because Y_t - D_t is a constant.

Tuning Considerations: Because most of the parameters are obtained from an analysis of prior data as described above, L and K are the only tuning