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


Double exponential smoothing uses two constants and is better at handling trends 
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, \(\gamma\),
which must be chosen in conjunction with \(\alpha\).
Here are the two equations associated with Double Exponential Smoothing. $$ \begin{eqnarray} S_t & = & \alpha y_t + (1  \alpha)(S_{t1} + b_{t1}) & & 0 \le \alpha \le 1 \\ & & \\ b_t & = & \gamma(S_t  S_{t1}) + (1  \gamma) b_{t1} & & 0 \le \gamma \le 1 \end{eqnarray} $$ Note that the current value of the series is used to calculate its smoothed value replacement in double exponential smoothing. 

Initial Values  
Several methods to choose the initial values 
As in the case for single smoothing, there are a variety of schemes
to set initial values for \(S_t\) and \(b_t\)
in double smoothing.
\(S_1\) is in general set to \(y_1\). Here are three suggestions for \(b_1\). $$ \begin{eqnarray} b_1 & = & y_2  y_1 \\ & & \\ b_1 & = & \frac{1}{3} \left[ (y_2  y_1) + (y_3  y_2) + (y_4  y_3) \right] \\ & & \\ b_1 & = & \frac{y_n  y_1}{n1} \end{eqnarray} $$ 

Comments  
Meaning of the smoothing equations 
The first smoothing equation adjusts \(S_t\)
directly for the trend of the previous period, \(b_{t1}\),
by adding it to the last smoothed value, \(S_{t1}\).
This helps to eliminate the lag and brings \(S_t\)
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. 

Nonlinear optimization techniques can be used  The values for \(\alpha\) and \(\gamma\) can be obtained via nonlinear optimization techniques, such as the Marquardt Algorithm. 