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Forecasting Process Details |

Predictions are made based on the last known smoothing state.
Predictions made at time *t* for *k* 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 *t*-1 for one time unit into the future,
that is, ,and 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
(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: is estimated by the mean
square of the one-step-ahead prediction errors.

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