In Weighted Moving Average, you can give more weight to recent events, but you are limited to the last m observations. Exponential Smoothing improves on Weighted Moving Average by taking all previous observations into account, while still favoring the most recent observations.
In Simple (aka Single) Exponential Smoothing, the forecasted value at time i+1 is based on the value at time i, and the forecasted value at time i (and so indirectly on all the previous time values). In particular, for some α where 0 ≤ α ≤ 1, for all i > 1, we define
Note that we don’t include time i = 1 in the calculations of MAE and MSE.
By simple algebra, this iteration can also be expressed as
For example
and similarly for all values of ŷi.
Example 1: Redo Example 1 of Simple Moving Average using exponential smoothing with α = .4.
The result is shown in Figure 1. The formula in cell I4 is =H4 and the formula in cell I5 is =I4+H$21*(H4-I4).
Figure 1 – Single Exponential Smoothing
The forecast for the next value in the time series is , by using the formula
=I18+H$21*(H18-I18)
Excel Data Analysis Tool: Excel provides the Exponential Smoothing data analysis tool to simplify the calculations described above.
To use this tool for Example 1, select Data > Analysis|Data Analysis and choose Exponential Smoothing from the menu that appears. A dialog box now appears which is similar to that shown in Figure 2 of Simple Moving Average, except that a Damping Factor field is used in place of the Interval field. If this field is left blank it defaults to .7.
The damping factor is just 1 – α. Thus for Example 1, you need to use .6 as the damping factor.
The output is shown in columns D and E of Figure 2 along with the chart.
Figure 2 – Excel’s Exponential Smoothing data analysis tool
The output takes a form similar to that shown in Figure 3 of Simple Moving Average. The standard error is calculated just as for moving averages with an interval of 3.
Example 2: Find the best fit exponential smoothing approximation to the data Example 1, using the MAE measure of accuracy.
Once again we use Solver. This time we minimize the value of MAE (cell J21 in Figure 3) by changing the value in cell H21 subject to the constraint that H21 <= 1.0.
The result shown in Figure 3 is that α = .656205 with MAE = 10.08.
Figure 3 – Optimizing Exponential Smoothing
