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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

- represents the time-varying mean term
- represents the time-varying slope
*s*_{p}(*t*)- represents the time-varying seasonal contribution
for one of the
*p*seasons - are disturbances

- represents the time-varying mean term

For smoothing models without trend terms, ;and for smoothing models without seasonal terms,

At each time

*L*_{t}is a smoothed level that estimates .*T*_{t}is a smoothed trend that estimates .*S*_{t-j},*j*= 0, ... ,*p*-1, are seasonal factors that estimate*s*_{p}(*t*).

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,
*Y*_{n}.

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
*S*_{t-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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