## Exponentially Weighted Moving Average

Another statistical quality control technique that is similar to CUSUM is the exponentially weighted moving average (EWMA) algorithm (Roberts, 1959). As its name suggests, the EWMA algorithm is a variation of the moving average algorithm in which we assign the current and prior observations weights such that observations that are further in the past receive an exponentially decreasing amount of weight. The parameter A, which is in the range 0<A^ 1, controls the weight we assign to the observations. EWMA is better at detecting small shifts in a process than a control chart, with its performance being similar to that of the CUSUM algorithm (Montgomery, 2001).

Let Xi be a measurement at time i. The EWMA statistic Zi is defined as:

Assuming that the Xis are independent random variables with variance a2, the variance of Zi can be computed as:

The starting value Z0 is set to the desired process mean |0 or to the average of some initial data ie. Z0 = X .If we unravel the recursion in Eq. 7 as shown below, we can see that observations further in the past are weighted more lightly.

Z. = AX. + (l -A)[AXM + (l - X)Zi2 ] = XXi + (l - A)AX;-l + (l - A)2Z;-2