1. Forecasting Purpose is to forecast, not to explain the historical pattern Models for forecasting may not make sense as a description for ”physical” beaviour of the time series Common sense and mathematics in a good combination produces ”optimal” forecasts

2. Exponential smoothing Use the historical data to forecast the future Let different parts of the history have different impact on the forecasts Forecast model is not developed from any statistical theory

3. Single exponential smoothing Assume historical values y 1,y 2,…y T Assume data contains no trend, i.e.

4. Forecasting scheme: whereis a smoothing parameter between 0 and 1

5. The forecast procedure is a recursion formula How shall we choose α? Where should we start, i.e. Which is the initial value l 0 ?

6. Use a part (usually half) of the historical data to estimate β 0  Set l 0 = Update the estimates of β 0 for the rest of the historical data with the recursion formula  l T which can be used to forecast y T+τ

7. Example: Sales of everyday commodities Year Sales values 1985151 1986151 1987147 1988149 1989146 1990142 1991143 1992145 1993141 1994143 1995145 1996138 1997147 1998151 1999148 2000148

8. Time series graph

9. Assume the model: Estimate β 0 by calculating the mean value of the first 8 observations of the series  Set l 8 = =146.75

10. Assume first that the sales are very stable, i.e. during the period the mean value β 0 is assumed not to change  Set α to be relatively small. This means that the latest observation plays a less role than the history in the forecasts. E.g. Set α=0.1 Update the estimates of β 0 using the next 8 values of the historical data

12. Forecasts

13. Alternative In Bowerman/O’Connell/Koehler instead the updates of estimates of β 0 are done on all historical data i.e. for T=1,…, n and l 0 =

14. Analysis of example data with MINITAB yearsales l T y T – l T forecasts 1985151146,7504,25000* 1986151147,1753,82500* 1987147147,558-0,55750* 1988149147,5021,49825* 1989146147,652-1,65158* 1990142147,486-5,48642* 1991143146,938-3,93778* 1992145146,544-1,54400* 1993141146,390-5,38960* 1994143145,851-2,85064* 1995145145,566-0,56557* 1996138145,509-7,50902* 1997147144,7582,24188* 1998151144,9826,01770* 1999148145,5842,41593* 2000148145,8262,17433* 146,043

16. Assume now that the sales are less stable, i.e. during the period the mean value β 0 is possibly changing  Set α to be relatively large. This means that the latest observation becomes more important in the forecasts. E.g. Set α=0.5

17. Analysis with MINITAB

18. We can also use some adaptive procedure to continuosly evaluate the forecast ability and maybe change the smoothing parameter over time Alt. We can run the process with different alphas and choose the one that performs best. This can be done with the MINITAB procedure.

19. Automatic selection of smoothing parameter with MINITAB

20. Exponential smoothing for times series with trend and/or seasonal variation Double exponential smoothing (one smoothing parameter) Holt-Winter’s method (two smoothing parameters) Multiplicative Winter’s method (three smoothing parameters) Additive Winter’s method (three smoothing parameters)

21. Example: Quarterly sales data yearquartersales 19911124 19912157 19913163 19914126 19921119 19922163 19923176 19924127 19931126 19932160 19933181 19934121 19941131 19942168 19943189 19944134 19951133 19952167 19953195 19954131

23. Applying Winter’s multiplicative method with MINITAB