6.
Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis 6.4.3. What is Exponential Smoothing?


Forecasting formula 
The oneperiodahead forecast is given by:
$$ F_{t+1} = S_t + b_t \, . $$
The \(m\)periodsahead forecast is given by: $$ F_{t+m} = S_t + m b_t \, . $$ 

Example  
Example 
Consider once more the data set:
For comparison's sake we also fit a single smoothing model with \(\alpha = 0.977\) (this results in the lowest MSE for single exponential smoothing).
The MSE for double smoothing is 3.7024.


Forecasting results for the example 
The smoothed results for the example are:


Comparison of Forecasts  
Table showing single and double exponential smoothing forecasts 
To see how each method predicts the future, we computed the first five
forecasts from the last observation as follows:


Plot comparing single and double exponential smoothing forecasts 
A plot of these results (using the forecasted double smoothing
values) is very enlightening.
This graph indicates that double smoothing follows the data much closer than single smoothing. Furthermore, for forecasting single smoothing cannot do better than projecting a straight horizontal line, which is not very likely to occur in reality. So in this case double smoothing is preferred. 

Plot comparing double exponential smoothing and regression forecasts 
Finally, let us compare double smoothing with linear regression:
This is an interesting picture. Both techniques follow the data in similar fashion, but the regression line is more conservative. That is, there is a slower increase with the regression line than with double smoothing. 

Selection of technique depends on the forecaster  The selection of the technique depends on the forecaster. If it is desired to portray the growth process in a more aggressive manner, then one selects double smoothing. Otherwise, regression may be preferable. It should be noted that in linear regression "time" functions as the independent variable. Chapter 4 discusses the basics of linear regression, and the details of regression estimation. 