Double Exponential Smoothing
Simple exponential smoothing does not do well when there is a trend in the data. In such situations, several methods were devised under the name "double exponential smoothing".
One method, sometimes referred to as "Holt-Winters double exponential smoothing" works as follows:
Again, the raw data sequence of observations is represented by {xt}, beginning at time t = 0. We use {st} to represent the smoothed value for time t, and {bt} is our best estimate of the trend at time t. The output of the algorithm is now written as Ft+m, an estimate of the value of x at time t+m, m>0 based on the raw data up to time t. Double exponential smoothing is given by the formulas
And for t > 1 by
where α is the data smoothing factor, 0 < α < 1, and β is the trend smoothing factor, 0 < β < 1.
To forecast beyond x_t
Setting the initial value b0 is a matter of preference. An option other than the one listed above is (xn - x0)/n for some n > 1.
Note that F0 is undefined (there is no estimation for time 0), and according to the definition F1=s0+b0, which is well defined, thus further values can be evaluated.
A second method, referred to as either Brown's linear exponential smoothing (LES) or Brown's double exponential smoothing works as follows.
where at, the estimated level at time t and bt, the estimated trend at time t are:
Read more about this topic: Exponential Smoothing
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