# Forecasting Process Details

### Predictions and Prediction Errors

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.