As was previously observed, Single Smoothing does not excel in following the data when there is a trend. This situation can be improved by the introduction of a second equation with a second constant, γ, which must be chosen in conjunction with α.
Here are the two equations associated with Double Exponential Smoothing.
Note that the current value of the series is used to calculate its smoothed value replacement in double exponential smoothing.
As in the case for single smoothing, there are a variety of schemes to set initial values for St and bt in double smoothing.
S1 is in general set to y1. Here are three suggestions for b1.
The first smoothing equation adjusts St directly for the trend of the previous period, bt−1, by adding it to the last smoothed value, St−1. This helps to eliminate the lag and brings St to the appropriate base of the current value.
The second smoothing equation then updates the trend, which is expressed as the difference between the last two values. The equation is similar to the basic form of single smoothing, but here applied to the updating of the trend.
The values for α and γ can be obtained via non-linear optimization techniques, such as the Marquardt Algorithm.