6.4.3.3. Double Exponential Smoothing

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

6.4.3.3. Double 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:

S(t) = alpha*y(t) + (1-alpha)*(S(t-1)+b(t-1) 0 < alpha < 1;
b(t) = gamma*(S(t) - S(t-1)) + (1-gamma)*b(t-1) 0 < gamma < 1

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.

S₁ is in general set to y₁. Here are three suggestions for b₁:

₁

₂

₁

b₁ = [(y₂ - y₁) + (y₃ - y₂) + (y₄ - y₃)]/3

b₁ = (y_n - y₁)/(n - 1)

Comments

Meaning of the smoothing equations

The first smoothing equation adjusts S_t directly for the trend of the previous period, b_t-1, by adding it to the last smoothed value, S_t-1. 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.

Non-linear optimization techniques can be used

The values for alpha

and

can be obtained via non-linear optimization techniques, such as the Marquardt Algorithm.