Accepted Solutions
The initialization and the smoothing process are calculated by PROC ESM (Exponential Smoothing Model). According to the procedure document, here are answers to your questions. You may find more details through links below.
- Initialization (http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_tffordet_sect0...)
“An appropriate choice for the initial smoothing state is made by backcasting from time 
to 
to obtain a prediction at 
. 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, 
.”
- Smoothing (http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_tffordet_sect0...)
Multiplicative Seasonal Smoothing
In order to use the multiplicative version of seasonal smoothing, the time series and all predictions must be strictly positive.
The model equation for the multiplicative version of seasonal smoothing is
 |
The smoothing equations are
 | |||
 |
The error-correction form of the smoothing equations is
 | |||
 |
(Note: For missing values, 
.)
The k-step prediction equation is
 |
The multiplicative version of seasonal smoothing does not have an ARIMA equivalent; however, when the seasonal variation is small, the ARIMA additive-invertible region of the additive version of seasonal described in the preceding section can approximate the stability region of the multiplicative version.
The variance of the prediction errors is estimated as
 |
where 
are as described for the additive version of seasonal method, and 
for 
.
The initialization and the smoothing process are calculated by PROC ESM (Exponential Smoothing Model). According to the procedure document, here are answers to your questions. You may find more details through links below.
- Initialization (http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_tffordet_sect0...)
“An appropriate choice for the initial smoothing state is made by backcasting from time to to obtain a prediction at . 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, .”
- Smoothing (http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_tffordet_sect0...)
Multiplicative Seasonal Smoothing
In order to use the multiplicative version of seasonal smoothing, the time series and all predictions must be strictly positive.
The model equation for the multiplicative version of seasonal smoothing is
|
The smoothing equations are
| |||
|
The error-correction form of the smoothing equations is
| |||
|
(Note: For missing values, .)
The k-step prediction equation is
|
The multiplicative version of seasonal smoothing does not have an ARIMA equivalent; however, when the seasonal variation is small, the ARIMA additive-invertible region of the additive version of seasonal described in the preceding section can approximate the stability region of the multiplicative version.
The variance of the prediction errors is estimated as
|
where are as described for the additive version of seasonal method, and for .
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Use this tutorial as a handy guide to weigh the pros and cons of these commonly used machine learning algorithms.
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