Double exponential smoothing

(Holt-Winters exponential smoothing forecasting model)

Many business forecasting procedures are based on extensions of simple exponential smoothing. Double exponential smoothing (also called the Holt-Winters exponential smoothing procedure) allows for trend (seasonality also) in time series. In double exponential smoothing, a second smoothing constant, , is included to account for the trend. We will consider a nonseasonal time series.

We denote the observed value as and as the estimate of the level. The trend estimate is represented as . In Holt-Winters exponential smoothing forecasting model these two variables will be estimated as

;

;

where and are smoothing constants whose values lie between 0 and 1.

To apply double exponential smoothing, we begin the computations by setting

and

Then the above equations are applied for . Standing at time

We obtain forecasts of future values, , of the series by

;

Example:

The sales manager needs to determine a monthly forecast for the number of men’s sweaters that will be sold so he can order an appropriate amount of packing boxes. Data for the past 8 months are given below:

Month	Sales

Develop Holt-Winters double exponential smoothing model using and as smoothing constants to forecast sales for the next three months.

Solution:

The initial estimates of level and trend in month 2, are

and

This smoothing application will use and and the equations

;

Then for :

and in addition

Then for :

and in addition

For :

For :

For :

For :

In general for h periods forecasting is

The most recent level and trend estimates are

;

Then the forecasts for the next three months are

The results of these calculations are shown below:

Month	Sales		MINITAB solution
		-- 157.48 163.23 157.7 171.64 192.2	146.150 161.457 174.503 156.590 163.157 157.823 171.735 192.106

The last column of the table above contains MINITAB solution.

According to MINITAB solution the predictions are

; ;

The values calculated by the MINITAB program differ slightly from those in the third column of the table above. The MINITAB procedures will generally provide slightly better forecasts compared to the more simplified procedure we have shown. The observed time series and forecasts are shown in Figure 7.3.

Figure 7.3

Remark:

To use MINITAB menu follow the following instructions:

1. Select Stat>Time series>Double exponential smoothing

2. Enter time series variable (for example, C1)

3. Select

4. Enter -for level

5. Enter - for trend

6. Select “generate forecasts”

7. Enter number of forecasting

8. Enter number of starting point for forecasting

9. Select Options

10. Select graphics, outputs

11. Click OK.

Дата добавления: 2015-08-05