In the Holt Winters Method (aka Triple Exponential Smoothing), we add a seasonal component to the Holt’s Linear Trend model.
Let c be the length of a seasonal cycle. Thus c = 12 for months in a year, c = 7 for days in a week and c = 4 for quarters in a year. The model takes the following form for all i > c
where 0 < α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ γ ≤ 1–α. The initial values, i.e. where 1 ≤ i ≤ c, are given by
An alternative form of these equations is given by
Note that if γ = 0, then the Holt-Winters model is equivalent to the Holt model and if β = 0 and γ = 0, then the Holt-Winters model is equivalent to the Single Exponential Smoothing model.
Example 1: Calculate the forecasted values of the time series shown in range C4:C19 of Figure 1 using the Holt-Winter method with α = .5, β = .5 and γ = .5.
The result is shown in Figure 1. First we calculate s1, s2, s3, s4, where c = 4, as shown in range F4:F7. We do this by inserting the formula =C4/AVERAGE(C$4:C$7) in cell F4, highlighting the range F4:F7 and pressing Ctrl-D.
Next, we calculate uc and vc by placing the formula =C7/F7 in cell D7 and the value 0 in cell E7.
We now insert the formula =C$22*C8/F4+(1-C$22)*(D7+E7) in cell D8, the formula =D$22*(D8-D7)+(1-D$22)*E7 in cell E8, =E$22*(C8/D8)+(1-E$22)*F4 in cell F8 and the formula =(D7+E7)*F4 in cell G8, and then highlight the range D8:F19 and press Ctrl-D.
Figure 1 – Holt-Winters Method
Observation: For any value of i, the forecast at time i+h is given by the formula
Example 2: Forecast the y values for 2014 from Example 1 (i.e. the next 4 quarters).
The result is shown in Figure 2. The values through 2013 are copied from Figure 1. The forecasted value for Q1 of 2014 is 35.36211 (cell N20), as calculated by the following formula with reference to cells in Figure 1.
The other three forecasted values are calculated by highlighting the range N20:N23 and pressing Ctrl-D.
Figure 2 – Holt-Winters’ Forecast
Observation: As we have done in Example 2 of Holt’s Linear Trend, we can use Solver to determine which values of alpha, beta and gamma yield the best Holt-Winters’ fit for the data in Example 1. In this case you need to assign the value α+γ to some cell and add a constraint to Solver making this value less than 1 (as well as constraining α and β to be less than 1). This turns out to be α = .184124, β = .439756 and γ = .417974, which yields a value for MSE of 19.9562.