There are other ways of forecasting future values. One commonly used forecast method is called
exponential smoothing. In Example
7.9, we showed that the approximate best linear predictor for an ARIMA(
0, 1, 1) process is given by an exponentially weighted average of past observations. The exponential smoothing method of forecasting is given by the same formula, that is, the forecast is the weighted average of past observations with more weight given to the more recent past values and exponentially decreasing weights for earlier values. If we define the
smoothing constant a=1+1 , (
7.11) becomes
This is the forecast formula using exponential smoothing. The function
gives the
for
i=1, ... , n using (
7.12) with smoothing constant
a and starting value
x1, the first entry in
data. We see that the approximate best linear predictor for ARIMA(
0, 1, 1) model corresponds to exponential smoothing with a smoothing constant
a=1+1 and
x0=0.