# Double exponential smoothing

Description and example of the double exponential smoothing

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Created: 2016-02-09 19:52:00, Last updated: 2021-02-25 08:50:07

We continue our topic of exponential smoothing started in the article Exponential smoothing

Last time we reviewed simple exponential smoothing - single exponential smoothing, today we'll be looking at double exponential smoothing.

Single exponential smoothing doesn't show quite outstanding characteristics in data depicting the presence of a clear trend. That is, it's kind of "catches" it, but not very good.

Double exponential smoothing was invented to work with data, showing a clear trend.

The idea is that - an additional component is added to a formula of exponential smoothing, see Exponential smoothing, and it changes the contribution of the previous values, depending on the trend.

Generalized formulas look as follows:

$S_t = \alpha y_t + (1-\alpha)(S_{t-1}+b_{t-1})$,
where $\alpha$ takes the value from the range [0;1)

$b_t = \gamma (S_t - S_{t-1}) + (1-\gamma)b_{t-1}$,
where $\gamma$ takes the value from the range [0;1]

Note that here for the calculation of the current S, the current value of y is used.

As in the simple exponential smoothing, there are several ways of the initial parameters selection, namely, S and b, which affect the final result.

Typically, the following methods are used (I used them too):

$S_1=y_1$
$b_1=y_2-y_1$

Also, for the calculation of the initial b there are also the following options:

$b_1=\frac{ (y_2-y_1) + (y_3-y_2) + (y_4-y_3)}{3}$
$b_1=\frac{y_n-y_1}{n-1}$

And yet again, as in the case of the simple exponential smoothing, the choice of optimal parameters $\alpha$ and $\gamma$ - is not a trivial task, and one of the methods is the exhaustive method followed by a selection of the optimal value for the criterion of the minimum mean square error.

There is one subtlety, though, since there is a current value of y in the formula, the error can not be considered as the difference between the current S and the current y. Obviously, it will be equal to 0 with $\alpha$ equal 1 (this is generally a punctured case). Therefore, as an error, the difference between the forecasted value (see below) and the following actual value.

To forecast the following value, the formula below is used.

$S_{t+1}=S_t + b_t$

to forecast several values

$S_{t+n}=S_t + nb_t$

In the calculator below, you can change the values $\alpha$ and $\gamma$, and also, you can compare the result with a simple exponential smoothing.
The following forecasted value is depicted on a graph for reference, i.e., smoothed average extended for one count further the actual data.

#### Time series

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Digits after the decimal point: 2
Double smoothing mean-square error

Single smoothing mean-square error

Double exponential smoothing
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As default data the best, as I remember, are the coefficients $\alpha$=0.6 and $\gamma$=0.05. Note that the error in the double smoothing with coefficients is by default greater than that of a simple smoothing because the double smoothing conducts itself worse on the unexpected trend reversals (see the forecast column and compare it with the following actual value) - in such cases, the difference between the forecast in contrast to the actual value gives a bigger mistake.

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