# Holt-Winters multiplicative method

Contents

## Holt-Winters multiplicative method

This is how the Holt-Winters multiplicative or triple smoothing method works.

## Simple exponential smoothing

In simple (aka simple) exponential smoothing, the predicted value at time i+1 is based on the value at time i and the predicted value at time i (and therefore indirectly on all previous time values). In particular, for some α where 0 ≤ α ≤ 1, for all i > 1, we define

Note that we do not include time i = 1 in the calculations of MAE and MSE.

In algebra simple, this iteration can also be expressed as

Let's look at the formula with Excel. The formula in cell C4 is =B4 and the formula in cell C5 is =C4+B\$21*(B4-C4).

The forecast for the next value in the time series is 74.0 (cell C19), using the formula =C18+B\$21*(B18-C18)

Excel provides the Exponential Smoothing data analysis tool to simplify the calculations described above.

To use this tool for example, select Data > Analysis|Data analysis and choose Exponential Smoothing from the menu that appears. A dialog box now appears. A Damping Factor field is used instead of the Interval field. If this field is left blank, its default value is 0.7.

The damping factor is only 1 – α. So, for example, you should use 0.6 as the damping factor.

The result is presented in columns D and E of the figure with the graph.

## Holt's linear trend

The data in the previous figure from simple exponential smoothing (as well as previous figures on this web page) show a clear upward trend. The moving average and single exponential smoothing methods do not model this adequately, but Holt's linear trend method (aka double exponential smoothing) does. This is accomplished by adding a second unique exponential smoothing model to capture the trend (upward or downward). The model takes the following form for all i > 1.

where 0 ≤ α ≤ 1 and 0 < β ≤ 1.

An alternative form of these equations is

Or

If beta=0 then we have simple exponentiation.

The result is presented in the figure. Here, cell C4 contains the formula =B4, cell D4 contains the value 0, cell C5 contains the formula =B\$21*B5+(1-B\$21)*(C4+D4), cell D5 contains the formula =C\$21* (C5-C4)+(1-C\$21)*D4 and cell E5 contains the formula =C4+D4.

For any value of i, the forecast at time i+h is given by the formula

## Multiplicative Holt-Winters 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.

The Holt-Winters additive model is identical to the multiplicative model, except that seasonality is considered additive. This means that the predicted value for each data element is the sum of the baseline, trend, and seasonal components. The sum of the seasonality components for c consecutive periods is approximately 1.

The recursive approach to the additive model is

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

The predictions for data elements yi are given by

For forecasts at future times, we use the form

where h′ =INT((h–1)/c)+1.An alternative version of the seasonality term is:

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

Forecast y values for 2014 (i.e. the next four quarters) using the Holt-Winter additive method based on data from the E4:F19 range of the figure using alpha values, beta and gamma which minimize the MSE statistic.

This is the same problem as the Holt-Winters multiplicative method example, except now we need to apply the additive method.

We can calculate the standard error and prediction intervals for Holt-Winters additive model forecasts as described in Exponential Smoothing Confidence Interval, except now the standard error of k steps ahead is

or

The standard errors and prediction intervals in the figure are calculated exactly as for the Holt linear trend (see Holt linear trend confidence interval) with the exception of cell J26 since it is a entry where c|i-1. The formula in cell J26 is

=22 K\$*SQRT((K25/K\$22)^2-1+(F\$30*(1+(ROW(J26)-ROW(J\$22))*G\$30+(1-F\$30)*H\$30) )^2+1)

The term (1-F\$30)*H\$30 in this formula is not found in other standard error formulas (although such a term is included in the 5 step lead, 9 step lead, 13 step d advance, etc. standard errors).

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