Predictions are made based on the last known smoothing state. Predictions made at time for
steps ahead are denoted
and the associated prediction errors are denoted
. The prediction equation for each smoothing model is listed in the following sections.
The one-step-ahead predictions refer to predictions made at time for one time unit into the future—that is,
. The one-step-ahead prediction errors are more simply denoted
. The one-step-ahead prediction errors are also the model residuals, and the sum of squares of the one-step-ahead prediction
errors is the objective function used in smoothing weight optimization.
The variance of the prediction errors are used to calculate the confidence limits (Sweet, 1985; McKenzie, 1986; Yar and Chatfield, 1990; Chatfield and Yar, 1991). The equations for the variance of the prediction errors for each smoothing model are listed in the following sections.
Note: is estimated by the mean square of the one-step-ahead prediction errors.