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

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
Given are historical values y1,y2,…yT
Assume data contains no trend

4. Algorithm for forecasting:
where
is a smoothing parameter with value between 0 and 1
The forecast procedure is a recursion formula
How shall we choose α?
Where should we start, i.e. which is the initial value ?

5. For long length time series:
Use a part (usually first half) of the historical data
and calculate their average:
Set
Update with the rest of the historical data
using the recursion formula

6. Example: Sales of everyday commodities
Year Sales values
Note! This time series is short but we use it for illustration purposes!

7. Calculate the average of the first 8 observations of the series:
Set
Assume first that the sales are very stable, i.e. during the period the background mean value 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 using the next 8 values of the historical data

9. Forecasts:

10. For short length time series:
Calculate the average of all historical data i.e.
Update from the beginning of the time series:
There are a lot of alternatives:
Average of all data, update from the middle of the series
Average of the first half, update from beginning
etc.

12. Analysis of example data with MINITAB

14. 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

15. Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper

16. MINITAB uses smoothing from 1st value!

17. Assume now that the sales are less stable, i. e
Assume now that the sales are less stable, i.e. during the period the background mean value is possibly changing.
(Note that a change means an occasional “level shift” , not a systematic trend)
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)

18. Single Exponential Smoothing for Sales values
Data Sales values
Length 16
Smoothing Constant
Alpha 0.5
Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper

19. Slightly narrower prediction intervals

20. 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.

22. Yet, narrower prediction intervals
Single Exponential Smoothing for Sales values
---
Smoothing Constant
Alpha
Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper
Yet, narrower prediction intervals

23. 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)

24. Example: Real Estate Price Index for Weekend Cottages in Sweden
Year REPI_C
Trend but no seasonal variation

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

26. 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)

27. Double Exponential Smoothing for REPI_C
Data REPI_C
Length 13
Smoothing Constants
Alpha (level) 0.2
Gamma (trend) 0.2
Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper

28. Example: Quarterly sales data
year quarter sales

29. Applying Winter’s multiplicative method with MINITAB

30. 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

32. Winters' Method for sales
Multiplicative Method
Data sales
Length 20
Smoothing Constants
Alpha (level)
Gamma (trend)
Delta (seasonal) 0.2
Accuracy Measures
MAPE
MAD
MSD
Forecasts
Period Forecast Lower Upper Q Q Q Q