Predictions and Prediction Errors

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, ${\hat{Y}_{t-1}(1)}$ ,and 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 (refer to Sweet 1985, McKenzie 1986, Yar and Chatfield 1990, and 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.

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