1. 1 1 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES BY SLIDES BY........................ John Loucks St. Edward’s Univ.

2. 2 2 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 6 Time Series Analysis and Forecasting n Quantitative Approaches to Forecasting n Time Series Patterns n Forecast Accuracy n Moving Averages and Exponential Smoothing

3. 3 3 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Methods n Forecasting methods can be classified as qualitative or quantitative. n Such methods are appropriate when historical data on the variable being forecast are either not applicable or unavailable. n Qualitative methods generally involve the use of expert judgment to develop forecasts. n We will focus exclusively on quantitative forecasting methods in this chapter.

4. 4 4 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Methods n Quantitative forecasting methods can be used when:  past information about the variable being forecast is available,  the information can be quantified, and  it is reasonable to assume that the pattern of the past will continue into the future.

5. 5 5 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Quantitative Forecasting Methods n Quantitative methods are based on an analysis of historical data concerning one or more time series. n A time series is a set of observations measured at successive points in time or over successive periods of time. n 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. n 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.

6. 6 6 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Methods ForecastingMethodsForecastingMethods QuantitativeQuantitativeQualitativeQualitative CausalCausal Time Series Focus of this chapter

7. 7 7 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns n A time series is a sequence of measurements taken every hour, day, week, month, quarter, year, or at any other regular time interval. n The pattern of the data is an important factor in understanding how the time series has behaved in the past. n If such behavior can be expected to continue in the future, we can use it to guide us in selecting an appropriate forecasting method.

8. 8 8 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Plot n A time series plot is a graphical presentation of the relationship between time and the time series variable. n Time is on the horizontal axis, and the time series values are shown on the vertical axis. n A useful first step in identifying the underlying pattern in the data is to construct a time series plot.

9. 9 9 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example Time Series Plot

10. 10 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns n The common types of data patterns that can be identified when examining a time series plot include: HorizontalHorizontal Trend Trend SeasonalSeasonal CyclicalCyclical Trend & Seasonal

11. 11 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Horizontal Pattern A horizontal pattern exists when the data fluctuate around a constant mean. A horizontal pattern exists when the data fluctuate around a constant mean. Changes in business conditions can often result in a time series that has a horizontal pattern shifting to a new level. Changes in business conditions can often result in a time series that has a horizontal pattern shifting to a new level. A change in the level of the time series makes it more difficult to choose an appropriate forecasting method. A change in the level of the time series makes it more difficult to choose an appropriate forecasting method. Time Series Patterns

12. 12 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns n Trend Pattern A time series may show gradual shifts or movements to relatively higher or lower values over a longer period of time. A time series may show gradual shifts or movements to relatively higher or lower values over a longer period of time. Trend is usually the result of long-term factors such as changes in the population, demographics, technology, or consumer preferences. Trend is usually the result of long-term factors such as changes in the population, demographics, technology, or consumer preferences. A systematic increase or decrease might be linear or nonlinear. A systematic increase or decrease might be linear or nonlinear. A trend pattern can be identified by analyzing multiyear movements in historical data. A trend pattern can be identified by analyzing multiyear movements in historical data.

13. 13 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns Seasonal patterns are recognized by seeing the same repeating pattern of highs and lows over successive periods of time within a year. Seasonal patterns are recognized by seeing the same repeating pattern of highs and lows over successive periods of time within a year. n Seasonal Pattern A seasonal pattern might occur within a day, week, month, quarter, year, or some other interval no greater than a year. [ eg. sale during the weekend; Ice-cream sale during summer] A seasonal pattern might occur within a day, week, month, quarter, year, or some other interval no greater than a year. [ eg. sale during the weekend; Ice-cream sale during summer] A seasonal pattern does not necessarily refer to the four seasons of the year (spring, summer, fall, and winter). [ eg. sale in December ] A seasonal pattern does not necessarily refer to the four seasons of the year (spring, summer, fall, and winter). [ eg. sale in December ]

14. 14 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns Some time series include a combination of a trend and seasonal pattern. Some time series include a combination of a trend and seasonal pattern. n Trend and Seasonal Pattern In such cases we need to use a forecasting method that has the capability to deal with both trend and seasonality. In such cases we need to use a forecasting method that has the capability to deal with both trend and seasonality. Time series decomposition can be used to separate or decompose a time series into trend and seasonal components. Time series decomposition can be used to separate or decompose a time series into trend and seasonal components.

15. 15 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Patterns A cyclical pattern exists if the time series plot shows an alternating sequence of points below and above the trend line lasting more than one year. A cyclical pattern exists if the time series plot shows an alternating sequence of points below and above the trend line lasting more than one year. n Cyclical Pattern Often, the cyclical component of a time series is due to multiyear business cycles. Often, the cyclical component of a time series is due to multiyear business cycles. Business cycles are extremely difficult, if not impossible, to forecast. Business cycles are extremely difficult, if not impossible, to forecast. In this chapter we do not deal with cyclical effects that may be present in the time series. In this chapter we do not deal with cyclical effects that may be present in the time series.

16. 16 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Selecting a Forecasting Method n The underlying pattern in the time series is an important factor in selecting a forecasting method. n Thus, a time series plot should be one of the first things developed when trying to determine what forecasting method to use. n If we see a horizontal pattern, then we need to select a method appropriate for this type of pattern. n If we observe a trend in the data, then we need to use a method that has the capability to handle trend effectively.

17. 17 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy n Measures of forecast accuracy are used to determine how well a particular forecasting method is able to reproduce the time series data that are already available. n By selecting the method that has the best accuracy for the data already known, we hope to increase the likelihood that we will obtain better forecasts for future time periods. n Measures of forecast accuracy are important factors in comparing different forecasting methods.

18. 18 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy n The key concept associated with measuring forecast accuracy is forecast error. n A positive forecast error indicates the forecasting method underestimated the actual value. Forecast Error = Actual Value  Forecast n A negative forecast error indicates the forecasting method overestimated the actual value.

19. 19 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy A simple measure of forecast accuracy is the mean A simple measure of forecast accuracy is the mean or average of the forecast errors. Because positive and or average of the forecast errors. Because positive and negative forecast errors tend to offset one another, the negative forecast errors tend to offset one another, the mean error is likely to be small. Thus, the mean error mean error is likely to be small. Thus, the mean error is not a very useful measure. is not a very useful measure. This measure avoids the problem of positive and negative errors offsetting one another. It is the mean of the absolute values of the forecast errors. This measure avoids the problem of positive and negative errors offsetting one another. It is the mean of the absolute values of the forecast errors. n Mean Error n Mean Absolute Error (MAE)

20. 20 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy This is another measure that avoids the problem This is another measure that avoids the problem of positive and negative errors offsetting one another. It is the average of the squared forecast errors. errors. The size of MAE and MSE depend upon the scale of the data, so it is difficult to make comparisons for different time intervals. To make such comparisons we need to work with relative or percentage error measures. The MAPE is the average of the absolute percentage errors of the forecasts. The size of MAE and MSE depend upon the scale of the data, so it is difficult to make comparisons for different time intervals. To make such comparisons we need to work with relative or percentage error measures. The MAPE is the average of the absolute percentage errors of the forecasts. n Mean Squared Error (MSE) n Mean Absolute Percentage Error (MAPE)

21. 21 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Forecast Accuracy n To demonstrate the computation of these measures of forecast accuracy we will introduce the simplest of forecasting methods. n The naïve forecasting method uses the most recent observation in the time series as the forecast for the next time period. F t+1 = Actual Value in Period t

22. 22 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sales of Comfort brand headache medicine for the Sales of Comfort brand headache medicine for the past 10 weeks at Rosco Drugs are shown below. past 10 weeks at Rosco Drugs are shown below. n Example: Rosco Drugs 12345678910110115125120125120130115110130 WeekWeekSalesSales Forecast Accuracy If Rosco uses the naïve forecast method to forecast method to forecast sales for weeks forecast sales for weeks 2 – 10, what are the 2 – 10, what are the resulting MAE, MSE, resulting MAE, MSE, and MAPE values? and MAPE values?

23. 23 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 12345678910110115125120125120130115110130 125120130115110 125 120 WeekSales NaïveForecast -5 -5 10 10-15 -5 -5 20 20 -5 -5 5 Forecast Error ErrorAbsolute Squared 5 10 10 15 15 5 20 20 80 80 5 5 25 25100125 400850 25 25 Abs.%Error 4.17 4.17 7.69 7.69 13.04 13.04 4.55 4.55 15.38 15.38 65.35 65.35 4.17 4.00 Forecast Accuracy 110 115 5 10 5 10 100 254.35 8.00 Total

24. 24 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Naive Forecast Accuracy Forecast Accuracy

25. 25 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moving Averages and Exponential Smoothing n Now we discuss three forecasting methods that are appropriate for a time series with a horizontal pattern: Exponential Smoothing Weighted Moving Averages Moving Averages n They are called smoothing methods because their objective is to smooth out the random fluctuations in the time series. n They are most appropriate for short-range forecasts.

26. 26 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moving Averages The moving averages method uses the average of the most recent k data values in the time series. As the forecast for the next period. where: F t +1 = forecast of the time series for period t + 1 F t +1 = forecast of the time series for period t + 1 Each observation in the moving average calculation receives the same weight.

27. 27 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moving Averages n The term moving is used because every time a new observation becomes available for the time series, it replaces the oldest observation in the equation. n As a result, the average will change, or move, as new observations become available.

28. 28 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moving Averages n If more past observations are considered relevant, then a larger value of k is better. n A smaller value of k will track shifts in a time series more quickly than a larger value of k. n To use moving averages to forecast, we must first select the order k, or number of time series values, to be included in the moving average.

29. 29 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. If Rosco Drugs uses a 3-period moving average to If Rosco Drugs uses a 3-period moving average to forecast sales, what are the forecasts for weeks 4-11? forecast sales, what are the forecasts for weeks 4-11? n Example: Rosco Drugs Example: Moving Average 1234512345678910678910110115125120125110115125120125120130115110130120130115110130 WeekWeekWeekWeekSalesSalesSalesSales

30. 30 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 12345678910111234567891011110115125120125120130115110130110115125120125120130115110130 WeekWeekSalesSales Example: Moving Average 123.3121.7125.0121.7118.3118.3 116.7 120.0 3MA Forecast (110 + 115 + 125)/3

31. 31 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Moving Average 1234567891012345678910110115125120125120130115110130110115125120125120130115110130 123.3121.7125.0121.7118.3123.3121.7125.0121.7118.3 116.7116.7 120.0120.0 WeekWeekSalesSales 3MA Forecast Forecast3MA -3.3 -3.3 8.3 8.3-10.0-11.7 11.7 11.7 6.6 6.6 3.3 5.0 Forecast Error ErrorAbsolute Squared 3.3 3.3 8.3 8.3 10.0 10.0 11.7 11.7 53.3 53.3 3.3 5.0 10.89 10.89 68.89 68.89100.00136.89136.89489.45 10.89 25.00 Abs.%Error 2.75 2.75 6.38 6.38 8.70 8.7010.64 9.00 9.0044.22 2.75 4.00 Total

32. 32 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Moving Average n 3-MA Forecast Accuracy The 3-week moving average approach provided The 3-week moving average approach provided more accurate forecasts than the naïve approach. more accurate forecasts than the naïve approach. The 3-week moving average approach provided The 3-week moving average approach provided more accurate forecasts than the naïve approach. more accurate forecasts than the naïve approach.

33. 33 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Weighted Moving Averages n Weighted Moving Averages The more recent observations are typically The more recent observations are typically given more weight than older observations. given more weight than older observations. For convenience, the weights should sum to 1. For convenience, the weights should sum to 1. To use this method we must first select the number of data values to be included in the average. 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. Next, we must choose the weight for each of the data values.

34. 34 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Weighted Moving Averages An example of a 3-period weighted moving average (3WMA) is: An example of a 3-period weighted moving average (3WMA) is: 3WMA =.2(110) +.3(115) +.5(125) = 119 125 is the most recent observation Weights (.2,.3, and.5) sum to 1 Weights (.2,.3, and.5) sum to 1 Weighted Moving Averages

35. 35 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Weighted Moving Average 1234567891012345678910110115125120125120130115110130110115125120125120130115110130 123.5121.5126.0120.5115.5123.5121.5126.0120.5115.5 119.0119.0 120.5120.5 WeekWeekSalesSales 3WMA Forecast Forecast3WMA -3.5 -3.5 8.5 8.5-11.0-10.5 14.5 14.5 3.5 3.5 1.0 4.5 Forecast Error ErrorAbsolute Squared 3.5 3.5 8.5 8.5 11.0 11.0 10.5 10.5 14.5 14.5 53.5 53.5 1.0 4.5 12.25 12.25 72.25 72.25121.00110.25210.25547.25 1.00 1.00 20.25 Abs.%Error 2.75 2.75 6.38 6.38 8.70 8.7010.64 9.00 9.0044.15 2.75 4.00 Total

36. 36 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Weighted Moving Average n 3-WMA Forecast Accuracy The 3-WMA approach (with weights of.2,.3, and.5) The 3-WMA approach (with weights of.2,.3, and.5) provided accuracy very close to that of the 3-MA provided accuracy very close to that of the 3-MA approach for this data. approach for this data. The 3-WMA approach (with weights of.2,.3, and.5) The 3-WMA approach (with weights of.2,.3, and.5) provided accuracy very close to that of the 3-MA provided accuracy very close to that of the 3-MA approach for this data. approach for this data.

37. 37 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Smoothing This method is a special case of a weighted moving averages method; we select only the weight for the most recent observation. 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 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. The exponential smoothing forecast is a weighted average of all the observations in the time series. The term exponential smoothing comes from the exponential nature of the weighting scheme for the historical values. The term exponential smoothing comes from the exponential nature of the weighting scheme for the historical values.

38. 38 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Smoothing n Exponential Smoothing Forecast 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) and let: F 2 = Y 1 (to initiate the computations)

39. 39 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. With some algebraic manipulation, we can rewrite F t +1 =  Y t + (1 –  ) F t as: With some algebraic manipulation, we can rewrite F t +1 =  Y t + (1 –  ) F t as: Exponential Smoothing n Exponential Smoothing Forecast F t +1 = F t +  Y t – F t ) 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. 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.

40. 40 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Smoothing Desirable Value for the Smoothing Constant . Desirable Value for the Smoothing Constant . n If the time series contains substantial random variability, a smaller value (nearer to zero) is preferred. If there is little random variability present, forecast errors are more likely to represent a change in the level of the time series …. and a larger value for  is preferred. If there is little random variability present, forecast errors are more likely to represent a change in the level of the time series …. and a larger value for  is preferred.

41. 41 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. If Rosco Drugs uses exponential smoothing to forecast sales, which value for the smoothing constant ,.1 or.8, gives better forecasts? to forecast sales, which value for the smoothing constant ,.1 or.8, gives better forecasts? n Example: Rosco Drugs Example: Exponential Smoothing 1234512345678910678910110115125120125110115125120125120130115110130120130115110130 WeekWeekWeekWeekSalesSalesSalesSales

42. 42 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Exponential Smoothing Using Smoothing Constant Value  =.1 Using Smoothing Constant Value  =.1 F 2 = Y 1 = 110 F 3 =.1 Y 2 +.9 F 2 =.1(115) +.9(110) = 110.50 F 4 =.1 Y 3 +.9 F 3 =.1(125) +.9(110.5) = 111.95 F 5 =.1 Y 4 +.9 F 4 =.1(120) +.9(111.95) = 112.76 F 6 =.1 Y 5 +.9 F 5 =.1(125) +.9(112.76) = 113.98 F 7 =.1 Y 6 +.9 F 6 =.1(120) +.9(113.98) = 114.58 F 8 =.1 Y 7 +.9 F 7 =.1(130) +.9(114.58) = 116.12 F 9 =.1 Y 8 +.9 F 8 =.1(115) +.9(116.12) = 116.01 F 10 =.1 Y 9 +.9 F 9 =.1(110) +.9(116.01) = 115.41

43. 43 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Exponential Smoothing Using Smoothing Constant Value  =.8 Using Smoothing Constant Value  =.8 F 2 = = 110 F 2 = = 110 F 3 =.8(115) +.2(110) = 114.00 F 3 =.8(115) +.2(110) = 114.00 F 4 =.8(125) +.2(114) = 122.80 F 4 =.8(125) +.2(114) = 122.80 F 5 =.8(120) +.2(122.80) = 120.56 F 5 =.8(120) +.2(122.80) = 120.56 F 6 =.8(125) +.2(120.56) = 124.11 F 6 =.8(125) +.2(120.56) = 124.11 F 7 =.8(120) +.2(124.11) = 120.82 F 7 =.8(120) +.2(124.11) = 120.82 F 8 =.8(130) +.2(120.82) = 128.16 F 8 =.8(130) +.2(120.82) = 128.16 F 9 =.8(115) +.2(128.16) = 117.63 F 9 =.8(115) +.2(128.16) = 117.63 F 10 =.8(110) +.2(117.63) = 111.53 F 10 =.8(110) +.2(117.63) = 111.53

44. 44 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 1234567891012345678910110115125120125120130115110130110115125120125120130115110130 WeekWeekSalesSales 113.98114.58116.12116.01115.41113.98114.58116.12116.01115.41 111.95111.95 112.76112.76  =.1 Forecast Forecast 110.00110.00 110.50110.50 Example: Exponential Smoothing (  =.1) 6.02 6.02 15.42 15.42 -1.12 -1.12 -6.01 -6.01 14.59 14.59 8.05 8.05 12.24 12.24 Forecast Error ErrorAbsolute Squared 6.02 6.02 15.42 15.42 1.12 1.12 6.01 6.01 14.59 14.59 82.95 82.95 8.05 12.24 36.25 36.25237.73 1.26 1.26 36.12 36.12212.87974.22 64.80 149.94 Abs.%Error 5.02 5.02 11.86 11.86 0.97 0.97 5.46 5.46 11.22 11.22 66.98 66.98 6.71 9.79 5.00 5.00 14.50 5.00 14.50 210.25 25.004.35 11.60 Total

45. 45 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Exponential Smoothing (  =.1) n Forecast Accuracy Exponential smoothing (with  =.1) provided Exponential smoothing (with  =.1) provided less accurate forecasts than the 3-MA approach. less accurate forecasts than the 3-MA approach. Exponential smoothing (with  =.1) provided Exponential smoothing (with  =.1) provided less accurate forecasts than the 3-MA approach. less accurate forecasts than the 3-MA approach.

46. 46 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 1234567891012345678910110115125120125120130115110130110115125120125120130115110130 124.11120.82128.16117.63111.53124.11120.82128.16117.63111.53 122.80122.80 120.56120.56 WeekWeekSalesSales  =.8 Forecast Forecast 110.00110.00 114.00114.00 Example: Exponential Smoothing (  =.8) -4.11 -4.11 9.18 9.18-13.16 -7.63 -7.63 18.47 18.47 -2.20 -2.20 4.44 4.44 Forecast Error ErrorAbsolute Squared 4.11 4.11 9.18 9.18 13.16 13.16 7.63 7.63 18.47 18.47 75.19 75.19 2.20 4.44 16.91 16.91 84.23 84.23173.30 58.26 58.26341.27847.52 7.84 7.84 19.71 Abs.%Error 3.43 3.43 7.06 7.06 11.44 11.44 6.94 6.94 14.21 14.21 61.61 61.61 1.83 3.55 5.00 5.00 11.00 5.00 11.00 121.00 25.004.35 8.80 Total

47. 47 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Exponential Smoothing (  =.8) n Forecast Accuracy Exponential smoothing (with  =.8) provided Exponential smoothing (with  =.8) provided more accurate forecasts than ES with  =.1, but more accurate forecasts than ES with  =.1, but less accurate than the 3-MA. less accurate than the 3-MA. Exponential smoothing (with  =.8) provided Exponential smoothing (with  =.8) provided more accurate forecasts than ES with  =.1, but more accurate forecasts than ES with  =.1, but less accurate than the 3-MA. less accurate than the 3-MA.

48. 48 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Optimal Smoothing Constant Value We choose the value of  that minimizes the mean squared error (MSE). We choose the value of  that minimizes the mean squared error (MSE). Determining the value of  that minimizes MSE is a nonlinear optimization problem. Determining the value of  that minimizes MSE is a nonlinear optimization problem. n These types of optimization models are often referred to as curve fitting models.

49. 49 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Time Series Analysis and Forecasting  Linear Trend Projection  Seasonality

50. 50 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend Projection  If a time series exhibits a linear trend, the method of least squares may be used to determine a trend line (projection) for future forecasts.  Least squares, also used in regression analysis, determines the unique trend line forecast which minimizes the mean square error between the trend line forecasts and the actual observed values for the time series.  The independent variable is the time period and the dependent variable is the actual observed value in the time series.

51. 51 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend Projection  Using the method of least squares, the formula for the trend projection is: where: T t = linear trend forecast in period t where: T t = linear trend forecast in period t b 0 = Y-intercept of the linear trend line b 0 = Y-intercept of the linear trend line b 1 = slope of the linear trend line b 1 = slope of the linear trend line t = the time period t = the time period T t = b 0 + b 1 t

52. 52 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend Projection  For the trend projection equation T t = b 0 + b 1 t = mean value of t = average values of the time series where: Y t = actual value of the time series in period t n = number of periods in the time series n = number of periods in the time series

53. 53 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend Projection The number of plumbing repair jobs performed by The number of plumbing repair jobs performed by Auger's Plumbing Service in the last nine months is listed on the right. listed on the right.  Example: Auger’s Plumbing Service Month Jobs March 353 May 342 April 387 July 396 June 374 August 409 September 399 October 412 November 408 Month Jobs Forecast the number of repair jobs Auger's will repair jobs Auger's will perform in December perform in December using the least squares using the least squares method. method.

54. 54 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend Projection Sum 45 60 3480 444.00 Sum 45 60 3480 444.00 (Nov.) 9 4 16 408 21.33 85.32 (Oct.) 8 3 9 412 25.33 75.99 (Sep.) 7 2 4 399 12.33 24.66 (Aug.) 6 1 1 409 22.33 22.33 (July) 5 0 0 396 9.33 0 (June) 4 -1 1 374 -12.67 12.67 (May) 3 -2 4 342 -44.67 89.34 (Apr.) 2 -3 9 387 0.33 -0.99 (Mar.) 1 -4 16 353 -33.67 134.68 (month) t Y t  Example: Auger’s Plumbing Service

55. 55 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend Projection T 10 = 351.07 + (7.12)(10) = 422.27  Example: Auger’s Plumbing Service

56. 56 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend  To the extent that seasonality exists, we need to incorporate it into our forecasting models to ensure accurate forecasts.  We will first look at the case of a seasonal time series with no trend and then discuss how to model seasonality with trend.

57. 57 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 112515310688 2118161133102 313814411380 4109137125109 513016512896  Example: Umbrella Sales  Sometimes it is difficult to identify patterns in a time series presented in a table.  Plotting the time series can be very informative.

58. 58 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend Time Series Plot Time Series Plot  Example: Umbrella Sales

59. 59 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend The time series plot does not indicate any long-term trend in sales. The time series plot does not indicate any long-term trend in sales. However, close inspection of the plot does reveal a seasonal pattern. However, close inspection of the plot does reveal a seasonal pattern.  The first and third quarters have moderate sales,  the second quarter the highest sales, and  the fourth quarter tends to be the lowest quarter in terms of sales.  Example: Umbrella Sales

60. 60 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend  We will treat the season as a categorical variable.  Recall that when a categorical variable has k levels, k – 1 dummy variables are required.  If there are four seasons, we need three dummy variables.  Qtr1 = 1 if Quarter 1, 0 otherwise  Qtr2 = 1 if Quarter 2, 0 otherwise  Qtr3 = 1 if Quarter 3, 0 otherwise

61. 61 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend General Form of the Equation is: General Form of the Equation is: Optimal Model is: Optimal Model is: The forecasts of quarterly sales in year 6 are: The forecasts of quarterly sales in year 6 are:  Quarter 1: Sales = 95 + 29(1) + 57(0) + 26(0) = 124  Quarter 2: Sales = 95 + 29(0) + 57(1) + 26(0) = 152  Quarter 3: Sales = 95 + 29(0) + 57(0) + 26(1) = 121  Quarter 4: Sales = 95 + 29(0) + 57(0) + 26(0) = 95  Example: Umbrella Sales

62. 62 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality without Trend Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 112515310688 2118161133102 313814411380 4109137125109 513016512896 Average12415212195  Example: Umbrella Sales  We could have obtained the quarterly forecasts for next year by simply computing the average number of umbrellas sold in each quarter.

63. 63 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality and Trend  We will now extend the curve-fitting approach to include situations where the time series contains both a seasonal effect and a linear trend.  We will introduce an additional variable to represent time.

64. 64 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Business at Terry's Tie Shop can be viewed as falling into three distinct seasons: (1) Christmas falling into three distinct seasons: (1) Christmas (November and December); (2) Father's Day (late (November and December); (2) Father's Day (late May to mid June); and (3) all other times. May to mid June); and (3) all other times.  Example: Terry’s Tie Shop Seasonality and Trend Average weekly sales ($) during each of the three seasons during the past four years are shown on the next slide. three seasons during the past four years are shown on the next slide.

65. 65 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality and Trend  Example: Terry’s Tie Shop Determine a forecast for the average weekly sales in year 5 for each of the three seasons. Year Season 1 2 3 1856 2012 985 1995 2168 1072 2241 2306 1105 2280 2408 1120 1234

66. 66 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality and Trend  There are three seasons, so we will need two dummy variables.  Seas1 t = 1 if Season 1 in time period t, 0 otherwise  Seas2 t = 1 if Season 2 in time period t, 0 otherwise  General Form of the Equation is:  Example: Terry’s Tie Shop  Optimal Model

67. 67 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonality and Trend  The forecasts of average weekly sales in the three seasons of year 5 (time periods 13, 14, and 15) are: Seas. 1: Sales 13 = 797 + 1095.43(1) + 1189.47(0) + 36.47(13) Seas. 1: Sales 13 = 797 + 1095.43(1) + 1189.47(0) + 36.47(13) = 2366.5 = 2366.5 Seas. 2: Sales 14 = 797 + 1095.43(0) + 1189.47(1) + 36.47(14) Seas. 2: Sales 14 = 797 + 1095.43(0) + 1189.47(1) + 36.47(14) = 2497.0 = 2497.0 Seas. 3: Sales 15 = 797 + 1095.43(0) + 1189.47(0) + 36.47(15) Seas. 3: Sales 15 = 797 + 1095.43(0) + 1189.47(0) + 36.47(15) = 1344.0 = 1344.0  Example: Terry’s Tie Shop

68. 68 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 6, Part B