Double Exponential Smoothing

A Double Exponential Smoothing is a Trend Estimation that ...

• AKA: Exponential Smoothing.
• See: Trend Estimation, SAP AG.

---

---

References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Exponential_smoothing#Double_exponential_smoothing Retrieved:2017-8-23.
  • Simple exponential smoothing does not do well when there is a trend in the data, which is inconvenient. In such situations, several methods were devised under the name "double exponential smoothing" or "second-order exponential smoothing.", which is the recursive application of an exponential filter twice, thus being termed "double exponential smoothing". This nomenclature is similar to quadruple exponential smoothing, which also references its recursion depth. The basic idea behind double exponential smoothing is to introduce a term to take into account the possibility of a series exhibiting some form of trend. This slope component is itself updated via 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 {x_t}, beginning at time t = 0. We use {s_t} to represent the smoothed value for time t, and {b_t} is our best estimate of the trend at time t. The output of the algorithm is now written as F_t+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 : [math]\displaystyle{ \begin{align} s_1& = x_1\\ b_1& = x_1 - x_0\\ \end{align} }[/math] And for t > 2 by : [math]\displaystyle{ \begin{align} s_{t}& = \alpha x_{t} + (1-\alpha)(s_{t-1} + b_{t-1})\\ b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\ \end{align} }[/math] 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 : [math]\displaystyle{ \begin{align} F_{t+m}& = s_t + mb_t \end{align} }[/math] 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.  : [math]\displaystyle{ \begin{align} s'_0& = x_0\\ s''_0& = x_0\\ s'_{t}& = \alpha x_{t} + (1-\alpha)s'_{t-1}\\ s''_{t}& = \alpha s'_{t} + (1-\alpha)s''_{t-1}\\ F_{t+m}& = a_t + mb_t, \end{align} }[/math] where a_t, the estimated level at time t and b_t, the estimated trend at time t are: : [math]\displaystyle{ \begin{align} a_t& = 2s'_t - s''_t\\ b_t& = \frac \alpha {1-\alpha} (s'_t - s''_t). \end{align} }[/math]

---