1. slide 1 DSCI 5340: Predictive Modeling and Business Forecasting Spring 2013 – Dr. Nick Evangelopoulos Lecture 5: Exponential Smoothing (Ch. 8) Material based on: Bowerman-O’Connell-Koehler, Brooks/Cole

2. slide 2 DSCI 5340 FORECASTING Page 341 Ex 7.1 Review of Homework in Textbook

3. slide 3 DSCI 5340 FORECASTING Data for Exercise 7.1 page 341, 342

4. slide 4 DSCI 5340 FORECASTING Exercise 7.1 page 341, 342 Seasonal factors – 1.191, 1.521,.804,.484 (average y/CMA values) Trend – Estimated using regression

5. slide 5 DSCI 5340 FORECASTING Ex 7.1 part c. Yhat 17 = S 1 * Trend(17) = 1.191 *(220.53893 + 19.949897*17) = 666.5874 Yhat 18 = S 2 * Trend(18) = 1.521*(220.53893 + 19.949897*18) =88 Yhat 19 = S 3 * Trend(19) =.804*(220.53893 + 19.949897*19) =482.0679 Yhat 20 = 300 Note Forecasts are in Column B on spreadsheet

6. slide 6 DSCI 5340 FORECASTING Ex 7.1 Part d & e & f Point Forecast for total tractor sales for year 5. Yhat = 667 + 882 + 482 + 300 = 2331 (use forecast in part c.) Cycle appears to be well defined. Cycle length is equal to 4. Point forecasts in column B are the same as answers in part c. 0.998 0.994 1.003 1.006Peak 0.999 0.994 0.999 1.002Peak 0.996 0.995 1 1.004Peak 0.997 0.998

7. slide 7 DSCI 5340 FORECASTING Ex 7.1 part g & h Part g – multiplicative decomposition is the same as performed in lab during last class. Multiply 95% prediction intervals by seasonal indexes to get 95% PIs for forecasts. For example, for period 17, 95% PI for forecasts is 1.191*545.64 to 1.191*573.76 which is equal to 649.86 to 683.35.

8. slide 8 DSCI 5340 FORECASTING Exponential Smoothing Exponential Smoothing is a forecasting method that is most effective when the trend and seasonal components of the time series are changing over time. It is a method for weighting time series unequally, with the more recent data weighted more heavily than more remote observations

9. slide 9 DSCI 5340 FORECASTING Exponential Smoothing

10. slide 10 DSCI 5340 FORECASTING Exponential Smoothing

11. slide 11 DSCI 5340 FORECASTING Single Smoothing (one parameter) This single exponential smoothing method is appropriate for series that move randomly above and below a constant mean with no trend nor seasonal patterns. The smoothed series is computed recursively, by evaluating:

12. slide 12 DSCI 5340 FORECASTING Exponential Smoothing

13. slide 13 DSCI 5340 FORECASTING Single Smoothing (one parameter) The forecasts from single smoothing are constant for all future observations. This constant is given by:

14. slide 14 DSCI 5340 FORECASTING Prediction Intervals for Exp Smoothing

15. slide 15 DSCI 5340 FORECASTING Note Weights Decrease Exponentially

16. slide 16 DSCI 5340 FORECASTING Single Smoothing (one parameter)...where alpha is the damping (or smoothing) factor. The smaller is the alpha, the smoother is the forecasted series. By repeated substitution, we can rewrite the recursion as

17. slide 17 DSCI 5340 FORECASTING Holt’s Exponential Smoothing Holt’s trend corrected exponential smoothing is a method to forecast time series that has a linear trend locally but a growth rate (or a slope) that is changing over time.

18. slide 18 DSCI 5340 FORECASTING Holt’s Exponential Smoothing Suppose that the time series y 1,y 2, …,y n exhibits a linear trend for which the level and growth rate may be changing with no seasonal pattern. Then the estimate lT for the level of the time series and the estimate bT for the growth rate of the time series are given by the following smoothing equations.

19. slide 19 DSCI 5340 FORECASTING Prediction Intervals for Holt’s Expo

20. slide 20 DSCI 5340 FORECASTING Holt’s Trend Corrected Expo

21. slide 21 DSCI 5340 FORECASTING Holt’s Error Correction Form Standard Form

22. slide 22 DSCI 5340 FORECASTING Additive Holt Winters’ Method

23. slide 23 DSCI 5340 FORECASTING Holt Winters Error Correction Form

24. slide 24 DSCI 5340 FORECASTING Holt Winters

25. slide 25 DSCI 5340 FORECASTING State Equations

26. slide 26 DSCI 5340 FORECASTING 1.15 1.2 1.25 1.3 1.35 1.4 1.45 00.10.20.30.40.50.60.70.80.91 Alpha RMSE Look for Minimum RMSE

27. slide 27 DSCI 5340 FORECASTING Might look good, but is it?

28. slide 28 DSCI 5340 FORECASTING

29. slide 29 DSCI 5340 FORECASTING

30. slide 30 DSCI 5340 FORECASTING

31. slide 31 DSCI 5340 FORECASTING A Possibility: Forecast RMSE vs Alpha - large alpha 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 00.20.40.60.81 Alpha Forecast RMSE Series1

32. slide 32 DSCI 5340 FORECASTING Recommended Alpha Typically alpha should be in the range 0.05 to 0.3 If RMSE analysis indicates larger alpha, exponential smoothing may not be appropriate

33. slide 33 DSCI 5340 FORECASTING Page 398 Ex 8.1, Ex 8.2 parts a and b Ex 8.6 parts a and b Ex 8.11 Homework in Textbook