2. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-2 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Time-Series Smoothing Methods 1.Moving Averages 2.Simple Exponential Smoothing 3.Holt’s exponential Smoothing 4.Winters’ exponential smoothing 5.Adaptive-response-rate Single exponential smoothing All of these methods use: Weighted average of past observations to smooth up-and-down movements, AND All are based on the concept that: There is some underling pattern to the data.

3. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-3 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Moving Averages A simple but popular forecasting method. Appropriate to use for forecasting stationary time series. A moving average of order N is simply the arithmetic average of the most recent N observations. Figure 3.1 shows the exchange rate between the Japanese yen and the U.S. Dollar from 1983Q1 through 1998Q4. Do we observe any data pattern in these data? No, we just observe irregular up-and-down movements.

4. Figure 3-1

6. Table 3-1 (continued)

8. Figure 3-2

9. Figure 3-3

10. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-10 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Moving Averages Selecting the interval for moving average: -- depends on the length of the underlying cycle or pattern in the original data. If a cycle occurs every four period, then we can calculate a four-period moving average in order to smooth out the short-run fluctuation. Note that the simplest naive model in Chapter 1 is a one-period moving average. The shortcomings of moving-average models are: 1.May not be successful in predicting peaks and troughs. 2.May not be successful when there is a trend in the series. 3.May exhibit a cycle, in fact, no cycle was present in the actual data.

11. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-11 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Simple Exponential Smoothing Like moving averages, this method is appropriate to use when there is no trend or seasonality present in the data In exponential smoothing, the current forecast is the weighted average of the last forecast and the actual value in that period. F t+1 = αX t + (1- α)F t α: smoothing constant 0 < α < 1

12. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-12 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Simple Exponential Smoothing Note that moving averages give equal weights to past values included in each average. Whereas exponential smoothing gives more weight to recent obsevations. The justification for this is that the most recent observations contain the most relevant information so that these observations with higher weights will have more influence on forecast value.

13. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-13 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Simple Exponential Smoothing If we rearrange the following equation: F t+1 = αX t + (1- α)F t we obtain this: F t+1 = F t + α(X t - F t )  error= X t - F t This equation says that the exponential smoothing model learns from past errors. If we forecast low in period t, then the error term will be positive and an adjustment will be made to incerese the current forecast. If we forecast high in period t, then the error term will be negative and an adjustment will be made to lower the current forecast.

14. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-14 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Simple Exponential Smoothing As mentioned earlier, the weights of past observations decline geometrically with the age of observation: F t+1 =αX t +(1- α)αX t-1 +(1- α) 2 αX t-2 + (1- α) 3 αF t-2 0 < α < 1 If α is chosen close to 1, in calculating forecast values, the recent values of time series will be weighted heavily relative to the distant past.

15. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-15 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Simple Exponential Smoothing If α is chosen close to 0, then the weights of the distant time series will be comparable to those given the recent values. (see pg. 105 and 106) Note that the sum of weights will be eventually 1. In choosing an appropriate value of smoothing constant, α, usually the RMSE is used as a criterion. In calculating forecast values, this method requires a limited quantity of data, so this can be considered as an advantage. However, this method does not have ability to adjust data when there is any trend or seasonality in data. (see example given on pg.107)

16. Table 3-2

17. Table 3-2 (continued)

18. Figure 3-4

19. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-19 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Holt’s Exponential Smoothing This method adjusts the smoothing model for any trend in the data by adding a growth factor (or trend factor) to the smoothing equation. It uses three equations to calculate forecast value: 1) F t+1 =αX t + (1- α)(F t +T t ) this equation adjusts F t+1 by adding a growth factor(T t ) to the smoothed value of earlier period, F t.

20. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-20 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Holt’s Exponential Smoothing 2)T t+1 = β(F t+1 – F t ) + (1 – β)T t this equation estimates the smoothed trend value by multiplying the most recent trend (F t+1 – F t ) by β and the last previous smoothed trend by (1 – β). 3)H t+m = F t+1 + mT t+1 this equation forecasts m periods into the future, in other words, it calculates Holt’s forecast value for period t+m. With these three equations, this method accurately accounts for any linear trend in the data. This method is also called as linear-trend smoothing. (see example given on pg. 110)

21. Table 3-3

22. Figure 3-5

23. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-23 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Winters’ Exponential Smoothing This method is used when data exhibit both trend and seasonality. To account for seasonality, this method adds a third parameter to the Holt’s model. (see equations given on pg.112) Winters’ exponential smoothing has been implemented using data for the production of light trucks in the United States by quarter. (see Fig. 3.6 and Table 3.4)

24. Figure 3-6

25. Table 3-4

26. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-26 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Adaptive-Response-Rate Single Exponential Smoothing (ADRES) This method is the same as simple exponential smoothing except that it uses a different α value (smoothing factor) for each period. The value of α is determined in such a way that it adapts to the changes in the basic pattern of the data. Other than its ability to adapt to changing circumstances, this method is the same as simple smoothing model, so it is applied to data which have little trend or seasonality.

27. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-27 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Adaptive-Response-Rate Single Exponential Smoothing (ADRES) F t+1 = α t X t + (1- α t )F t As you see in the formula, there may be a different α value for each period. 1) e t = X t – F t 2) S t = βe t + (1 – β)S t-1 (Smoothed error) 3) A t =β|e t| + (1 – β)A t- (Absolute smoothed error) 4) α t = |S t /A t | In most cases β is assigned a value of either 0.1 or 0.2. (See Table 3.5 on pg.116)

28. Table 3-5

29. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-29 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Using Single, Holt’s or ADRES Smoothing to Forecast a Seasonal Data Series When we observe seasonal pattern in the data, we can either use Winters’ smoothing model or apply the following steps: 1.Calculate seasonal indices 2.Deseasonalize the data 3.Apply a forecasting method 4.Reseasonalize the data (See Table 3.6 and Figure 3.7)

30. Table 3-6

31. Figure 3-7

32. McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. 3-32 Business Forecasting with Accompanying Excel-Based ForecastX™ Software Event Modeling This technique helps to improve forecast accuracy by taking into consideration special events such as promotions, natural disasters. The logic of event models is similar to seasonal models: just as each period (i.e., month or a quarter) is assigned its own index for seasonality, so, too, each event type is assigned its own index for a specific promotional activity. See the event values used by a condiment manufacturer on pg. 121.

33. Figure 3-8

34. Table 3-7

35. Table 3-7 (continued)

36. Figure 3-9

37. Table 3-8