1. Lesson 4 -Part A Forecasting Quantitative Approaches to Forecasting Components of a Time Series Measures of Forecast Accuracy Smoothing Methods Trend Projection Trend and Seasonal Components Regression Analysis

2. Forecasting Methods ForecastingMethodsForecastingMethods QuantitativeQuantitativeQualitativeQualitative CausalCausal Time Series SmoothingSmoothingTrendProjectionTrendProjection Trend Projection Adjusted for Seasonal Influence Trend Projection Adjusted for Seasonal Influence

3. Quantitative Approaches to Forecasting Quantitative methods are based on an analysis of historical data concerning one or more time series. A time series is a set of observations measured at successive points in time or over successive periods of time. If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method. If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.

4. Time Series Methods Three time series methods are: –smoothing –trend projection –trend projection adjusted for seasonal influence

5. Components of a Time Series The pattern or behavior of the data in a time series has several components. The four components we will study are: TrendTrendCyclicalCyclicalSeasonalSeasonal IrregularIrregular

6. Components of a Time Series Trend Component –The trend component accounts for the gradual shifting of the time series to relatively higher or lower values over a long period of time. - Trend is usually the result of long-term factors such as changes in the population, demographics, technology, or consumer preferences.

7. Components of a Time Series –Any regular pattern of sequences of values above and below the trend line lasting more than one year can be attributed to the cyclical component. Cyclical Component –Usually, this component is due to multiyear cyclical movements in the economy.

8. Components of a Time Series The seasonal component accounts for regular patterns of variability within certain time periods, such as a year. The seasonal component accounts for regular patterns of variability within certain time periods, such as a year. Seasonal Component The variability does not always correspond with the seasons of the year (i.e. winter, spring, summer, fall). The variability does not always correspond with the seasons of the year (i.e. winter, spring, summer, fall). There can be, for example, within-week or within- day “seasonal” behavior. There can be, for example, within-week or within- day “seasonal” behavior.

9. Components of a Time Series The irregular component is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. The irregular component is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. Irregular Component It is unpredictable. It is unpredictable. This component is the residual, or “catch-all,” factor that accounts for unexpected data values. This component is the residual, or “catch-all,” factor that accounts for unexpected data values.

10. Measures of Forecast Accuracy The average of the squared forecast errors for the historical data is calculated. The forecasting method or parameter(s) that minimize this mean squared error is then selected. The mean of the absolute values of all forecast errors is calculated, and the forecasting method or parameter(s) that minimize this measure is selected. (The mean absolute deviation measure is less sensitive to large forecast errors than the mean squared error measure.) Mean Squared Error Mean Absolute Deviation

11. Smoothing Methods Three common smoothing methods are: In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular component of the time series. Exponential Smoothing Weighted Moving Averages Moving Averages

12. Smoothing Methods The moving averages method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period. Moving Averages

13. Sales of Comfort brand headache medicine for the past ten weeks at Rosco Drugs are shown on the next slide. If Rosco Drugs uses a 3-period moving average to forecast sales, what is the forecast for Week 11? Example: Rosco Drugs Smoothing Methods: Moving Averages

14. Example: Rosco Drugs Smoothing Methods: Moving Averages 12345678910110115125120125120130115110130 WeekWeekSalesSales

15. Time Series Plot using MINITAB

16. Time series appears to be stable over time, though random variability is present. Thus, smoothing methods are applicable.

17. Smoothing Methods: Moving Averages1 2 3 4 5 6 7 8 9 10 11 110 115 125 120 125 120 130 115 110 130 116.7 120.0 123.3 121.7 125.0 121.7 118.3 118.3 123.3 121.7 125.0 121.7 118.3 118.3 116.7 120.0 WeekSales3MAForecast (110 + 115 + 125)/3

18. Moving Averages using MINITAB

19. You can specify this i.e. 4, 6, 12, … & its depend on the problem # of future forecasts needed # of data points

21. Moving Average for Sales Moving Average: Length 3 Time Sales MA Predict Error 1 110 * * * 2 115 * * * 3 125 116.667 * * 4 120 120.000 116.667 3.3333 5 125 123.333 120.000 5.0000 6 120 121.667 123.333 -3.3333 7 130 125.000 121.667 8.3333 8 115 121.667 125.000 -10.0000 9 110 118.333 121.667 -11.6667 10 130 118.333 118.333 11.6667 Forecasts Period Forecast 95% CI 11 118.333 101.954 134.713

23. MSE # of values to be included in the moving average is chosen such that it minimizes the MSE

24. Smoothing Methods Weighted Moving Averages – The more recent observations are typically given more weight than older observations. – For convenience, the weights usually sum to 1. –To use this method we must first select the number of data values to be included in the average. –Next, we must choose the weight for each of the data values.

25. Smoothing Methods Weighted Moving Averages –An example of a 3-period weighted moving average (3WMA) is: 3WMA =.2(110) +.3(115) +.5(125) = 119 Most recent of the three observations Weights (.2,.3, and.5) sum to 1

26. Smoothing Methods Exponential Smoothing –This method is a special case of a weighted moving averages method; we select only the weight for the most recent observation. –The weights for the other data values are computed automatically and become smaller as the observations grow older. –The exponential smoothing forecast is a weighted average of all the observations in the time series.

27. Smoothing Methods Exponential Smoothing Model F t +1 =  Y t + (1 –  ) F t where F t +1 = forecast of the time series for period t + 1 Y t = actual value of the time series in period t F t = forecast of the time series for period t  = smoothing constant (0 <  < 1) To start the calculations, we let F 1 = Y 1 To start the calculations, we let F 1 = Y 1

28. –With some algebraic manipulation, we can rewrite F t+1 =  Y t + (1 –  )F t as: Smoothing Methods Exponential Smoothing Model F t +1 = F t +  Y t – F t ) –We see that the new forecast F t+1 is equal to the previous forecast F t plus an adjustment, which is  times the most recent forecast error, Y t – F t.

29. Sales of Comfort brand headache medicine for the past ten weeks at Rosco Drugs are shown on the next slide. If Rosco Drugs uses exponential smoothing to forecast sales, which value for the smoothing constant ,.1 or.8, gives better forecasts? Example: Rosco Drugs Smoothing Methods: Exponential Smoothing

30. Example: Rosco Drugs 12345678910110115125120125120130115110130 WeekWeekSalesSales Smoothing Methods: Exponential Smoothing

31. Exponential Smoothing (  =.1, 1 -  =.9) F 1 = 110 F 2 =.1Y 1 +.9F 1 =.1(110) +.9(110) = 110 F 3 =.1Y 2 +.9F 2 =.1(115) +.9(110) = 110.5 F 4 =.1Y 3 +.9F 3 =.1(125) +.9(110.5) = 111.95 F 5 =.1Y 4 +.9F 4 =.1(120) +.9(111.95) = 112.76 F 6 =.1Y 5 +.9F 5 =.1(125) +.9(112.76) = 113.98 F 7 =.1Y 6 +.9F 6 =.1(120) +.9(113.98) = 114.58 F 8 =.1Y 7 +.9F 7 =.1(130) +.9(114.58) = 116.12 F 9 =.1Y 8 +.9F 8 =.1(115) +.9(116.12) = 116.01 F 10 =.1Y 9 +.9F 9 =.1(110) +.9(116.01) = 115.41

32. Smoothing Methods: Exponential Smoothing Exponential Smoothing (  =.8, 1 -  =.2) F 1 = 110 F 2 =.8(110) +.2(110) = 110 F 3 =.8(115) +.2(110) = 114 F 4 =.8(125) +.2(114) = 122.80 F 5 =.8(120) +.2(122.80) = 120.56 F 6 =.8(125) +.2(120.56) = 124.11 F 7 =.8(120) +.2(124.11) = 120.82 F 8 =.8(130) +.2(120.82) = 128.16 F 9 =.8(115) +.2(128.16) = 117.63 F 10 =.8(110) +.2(117.63) = 111.53

33. Smoothing Methods: Exponential Smoothing [( Y 2 - F 2 ) 2 + ( Y 3 - F 3 ) 2 + ( Y 4 - F 4 ) 2 +... + ( Y 10 - F 10 ) 2 ]/9 Mean Squared Error In order to determine which smoothing constant gives the better performance, we calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10.

34. Smoothing Methods: Exponential Smoothing2345678910115125120125120130115110130 Week  =.1  =.8 FtFtFtFt FtFtFtFt ( Y t - F t ) 2 110.00110.50111.95112.76113.98114.58116.12116.01115.41110.00114.00122.80120.56124.11120.82128.16117.63111.5325.00210.2564.80149.9436.25237.731.2636.12212.8725.00121.007.8419.7116.9184.23173.3058.26341.27 Sum 974.22 Sum 847.52 Sum/9 108.25 Sum/9 94.17 MSE YtYtYtYt

35. Exponential Smoothing using MINITAB

36. # of future forecasts needed # of data points value for the smoothing constant 

39. Single Exponential Smoothing for Sales Smoothing Constant Alpha 0.1 Time Sales Smooth Predict Error 1 110 110.000 110.000 0.0000 2 115 110.500 110.000 5.0000 3 125 111.950 110.500 14.5000 4 120 112.755 111.950 8.0500 5 125 113.980 112.755 12.2450 6 120 114.582 113.980 6.0205 7 130 116.123 114.582 15.4184 8 115 116.011 116.123 -1.1234 9 110 115.410 116.011 -6.0111 10 130 116.869 115.410 14.5901 Forecasts Period Forecast Lower Upper 11 116.869 96.5445 137.193

40. Single Exponential Smoothing for Sales Smoothing Constant Alpha 0.8 Time Sales Smooth Predict Error 1 110 110.000 110.000 0.0000 2 115 114.000 110.000 5.0000 3 125 122.800 114.000 11.0000 4 120 120.560 122.800 -2.8000 5 125 124.112 120.560 4.4400 6 120 120.822 124.112 -4.1120 7 130 128.164 120.822 9.1776 8 115 117.633 128.164 -13.1645 9 110 111.527 117.633 -7.6329 10 130 126.305 111.527 18.4734 Forecasts Period Forecast Lower Upper 11 126.305 107.735 144.876

42. smoothing Value of the smoothing constant  constant  is chosen such that it minimizes the MSE