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Forecasting Process Details |
The descriptions and properties of various smoothing methods can be found in Gardner (1985), Chatfield (1978), and Bowerman and O'Connell (1979). The following section summarizes the smoothing model computations.
Given a time series , the
underlying model assumed by the smoothing models has the
following (additive seasonal) form:
where
The smoothing process starts with an initial estimate of the smoothing state, which is subsequently updated for each observation using the smoothing equations.
The smoothing equations determine how the smoothing state changes as time progresses. Knowledge of the smoothing state at time t-1 and that of the time-series value at time t uniquely determine the smoothing state at time t. The smoothing weights determine the contribution of the previous smoothing state to the current smoothing state. The smoothing equations for each smoothing model are listed in the following sections.
An appropriate choice for the initial smoothing state is made by backcasting from time t=n to t=1 to obtain a prediction at t=0. The initialization for the backcast is obtained by regression with constant and linear terms and seasonal dummies (additive or multiplicative) as appropriate for the smoothing model. For models with linear or seasonal terms, the estimates obtained by the regression are used for initial smoothed trend and seasonal factors; however, the initial smoothed level for backcasting is always set to the last observation, Yn.
The smoothing state at time t=0 obtained from the backcast is used to initialize the smoothing process from time t=1 to t=n (refer to Chatfield and Yar 1988).
For models with seasonal terms, the smoothing state is normalized so that the seasonal factors St-j for j = 0, ... , p-1 sum to zero for models that assume additive seasonality and average to one for models (such as Winters method) that assume multiplicative seasonality.
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