Holt-Winters multiplicative method

In the Holt-Winters method (aka triple exponential smoothing), we add a seasonal component to Holt's linear trend model. We explore two of these models: the multiplicative seasonality model and the additive seasonality model. We consider the first of these models on this web page. See Holt–Winters additive model for the second model.

Let c be the duration of a seasonal cycle. Thus, c = 12 for the months of a year, c = 7 for the days of a week and c = 4 for the quarters of a year. The model takes the following recursive form for all i > c

where 0 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ γ ≤ 1.

The ui values represent the baseline, the vi values represent the trend (i.e. slope), and the si values represent the seasonal component. In the multiplicative model, for any c consecutive time periods, the sum of the si values is approximately equal to c (at least for reasonable values of α, β, γ).

Predictions for data elements yi can be expressed as

For forecasts at future times, we use the form

where h′ =INT((h–1)/c)+1.

Alternative forms
We can also use the following alternative version of the term seasonality:

Based on this version of the seasonality term, we have the following alternative form of the recursive equations:

The initial values of the model, i.e. where 1 ≤ i ≤ c, are

Alternatively, we can set the initial trend value using the average slope of the first two years, namely:

Note that if γ = 0, then the Holt-Winters model is equivalent to the Holt linear trend model, and if β = 0 and γ = 0, then the Holt-Winters model is equivalent to the simple exponential smoothing model.

Calculate the predicted values of the time series shown in the range C4:C19 of the figure using the Holt-Winter method with α = 0.5, β = 0.5, and γ = 0.5.

The result is presented in the figure. First, we calculate s1, s2, s3, s4, where c = 4, as shown in the range F4:F7. We do this by inserting the formula =C4/AVERAGE(C$4:C$7) into 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, then highlight the range D8:F19 and press Ctrl- D.

We can use Solver to determine which values of alpha, beta and gamma give the best Holt-Winters fit for the example data.

The optimization approach using Excel Solver is, however, likely to find a local minimum instead of a global minimum. For this reason, the optimized values for alpha, beta, and gamma are sensitive to the initial values used. You can ask Solver to try different initial values to find the parameter values that reduce the MAE value.

To do this, select Data > Analysis | Solver and press the Options button in the Solver dialog box. The Solver dialog box automatically contains the values collected by the Basic Forecasting data analysis tool. Now choose the Multistart option from the GRG Nonlinear tab of the Options dialog box. The solver will now run multiple times using different starting values, selecting the values that produce the best result. The cost of this option is slower execution times.