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. With time series regression models, forecasting (prediction) is a natural step and forecasting limits (intervals) can be constructed With Classical decomposition, forecasting may be done, but estimation of accuracy lacks and no forecasting limits are produced Classical decomposition is usually combined with Exponential smoothing methods

3. 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

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

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

6. 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 ?

7. 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

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

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. Thumb rule: 0.05 < α < 0.3 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 the updates of estimates of β 0 are done on all historical data i.e. for T=1,…, n and l 0 =

15. Analysis of example data with MINITAB 

17. MTB > Name c3 "FORE1" c4 "UPPE1" c5 "LOWE1" MTB > SES 'Sales values'; SUBC> Weight 0.1; SUBC> Initial 8; SUBC> Forecasts 3; SUBC> Fstore 'FORE1'; SUBC> Upper 'UPPE1'; SUBC> Lower 'LOWE1'; SUBC> Title "SES alpha=0.1". Single Exponential Smoothing for Sales values Data Sales values Length 16 Smoothing Constant Alpha 0.1

18. Accuracy Measures MAPE 2.2378 MAD 3.2447 MSD 14.4781 Forecasts Period Forecast Lower Upper 17 146.043 138.094 153.992 18 146.043 138.094 153.992 19 146.043 138.094 153.992

19. MINITAB uses smoothing from 1st value!

20. 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 (A bit exaggerated)

21. Single Exponential Smoothing for Sales values Data Sales values Length 16 Smoothing Constant Alpha 0.5 Accuracy Measures MAPE 1.9924 MAD 2.8992 MSD 13.0928 Forecasts Period Forecast Lower Upper 17 147.873 140.770 154.976 18 147.873 140.770 154.976 19 147.873 140.770 154.976

22. Slightly wider prediction intervals

23. 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.

25. Single Exponential Smoothing for Sales values --- Smoothing Constant Alpha 0.567101 Accuracy Measures MAPE 1.7914 MAD 2.5940 MSD 12.1632 Forecasts Period Forecast Lower Upper 17 148.013 141.658 154.369 18 148.013 141.658 154.369 19 148.013 141.658 154.369 Yet, wider prediction intervals

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

27. Example: Real Estate Price Index for Weekend Cottages in Sweden YearREPI_C 1993226 1994241 1995239 1996240 1997268 1998303 1999336 2000414 2001472 2002496 2003505 2004546 2005591 Trend but no seasonal variation

28. Applying Holt’s method with MINITAB (denoted Double exponential smoothing in Minitab)

29. 2 smoothing parameters, one for level and one for trend. Option to let Minitab calculate optimal parameters. Smoothing parameters should still be kept low (0.05,0.3)

30. Double Exponential Smoothing for REPI_C Data REPI_C Length 13 Smoothing Constants Alpha (level) 0.2 Gamma (trend) 0.2 Accuracy Measures MAPE 9.78 MAD 30.15 MSD 1160.79 Forecasts Period Forecast Lower Upper 14 611.411 537.537 685.286 15 646.167 570.753 721.581

31. 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

32. Applying Winter’s multiplicative method with MINITAB

33. 3 smoothing parameters, one for level, one for trend an one for seasonal variation. No option to calculate optimal parameters. Choices have do be based on visual inspection of the times series

35. Winters' Method for sales Multiplicative Method Data sales Length 20 Smoothing Constants Alpha (level) 0.2 Gamma (trend) 0.2 Delta (seasonal) 0.2 Accuracy Measures MAPE 2.6446 MAD 3.8808 MSD 23.7076 Forecasts Period Forecast Lower Upper Q3-2013 135.625 126.117 145.133 Q4-2013 174.430 164.773 184.087 Q1-2014 194.667 184.844 204.490 Q2-2014 136.933 126.928 146.939