Forecasting Process Details

Predictions and Prediction Errors

Predictions are made based on the last known smoothing state. Predictions made at time ${t}$ for ${k}$ steps ahead are denoted ${\hat{Y}_{t}(k)}$ and the associated prediction errors are denoted ${e_{t}(k) = Y_{t+k} - \hat{Y}_{t}(k)}$. The prediction equation for each smoothing model is listed in the following sections.

The one-step-ahead predictions refer to predictions made at time ${t-1}$ for one time unit into the future—that is, ${\hat{Y}_{t-1}(1)}$. The one-step-ahead prediction errors are more simply denoted ${e_{t} = e_{t-1}(1) = Y_{t} - \hat{Y}_{t-1}(1)}$. 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: ${{var}({\epsilon }_{t})}$ is estimated by the mean square of the one-step-ahead prediction errors.