DSCI 5340: Predictive Modeling and Business Forecasting Spring 2013 – Dr. Nick Evangelopoulos

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

2. Review of Homework in Textbook Page 341 Ex 7.1

3. Data for Exercise 7.1 page 341, 342

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

5. Ex 7.1 part c. • Yhat17 = S1 * Trend(17) = 1.191 *(220.53893 + 19.949897*17) = 666.5874 • Yhat18 = S2 * Trend(18) = 1.521*(220.53893 + 19.949897*18) =88 • Yhat19 = S3 * Trend(19) = .804*(220.53893 + 19.949897*19) =482.0679 • Yhat20 = 300 • Note Forecasts are in Column B on spreadsheet

6. 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.

7. 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. 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. Exponential Smoothing

10. Exponential Smoothing

11. 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. Exponential Smoothing

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

14. Prediction Intervals for Exp Smoothing

15. Note Weights Decrease Exponentially

16. 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. 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. Holt’s Exponential Smoothing Suppose that the time series y1,y2, …,yn 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. Prediction Intervals for Holt’s Expo

20. Holt’s Trend Corrected Expo

21. Holt’s Error Correction Form Standard Form

22. Additive Holt Winters’ Method

23. Holt Winters Error Correction Form

24. Holt Winters

25. State Equations

26. Look for Minimum RMSE 1.45 1.4 1.35 RMSE 1.3 1.25 1.2 1.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Alpha

27. Might look good, but is it?

30. Series and Forecast using Alpha=0.9 2 1.5 1 Forecast 0.5 0 -0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Period

31. A Possibility: Forecast RMSE vs Alpha - large alpha 0.67 0.66 0.65 0.64 0.63 Forecast RMSE 0.62 Series1 0.61 0.6 0.59 0.58 0.57 0 0.2 0.4 0.6 0.8 1 Alpha

32. 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. Homework in Textbook Page 398 Ex 8.1, Ex 8.2 parts a and b Ex 8.6 parts a and b Ex 8.11