Statistics for Business and Economics chapter 18 Time Series Analysis and Forecasting

Chapter 18 Time Series Analysis and Forecasting Learning Objectives Be able to construct a time series plot and identify the underlying pattern in the data Understand how to measure forecast accuracy Be able to use smoothing techniques such as moving averages and exponential smoothing to forecast a time series with a horizontal pattern Know how simple linear regression and Holt’s linear exponential smoothing can be used to forecast a time series with a linear trend Be able to develop a quadratic trend equation and an exponential trend equation to forecast a time series with a curvilinear or nonlinear trend Know how to develop forecasts for a time series that has a seasonal pattern Know how time series decomposition can be used to separate or decompose a time series into season, trend, and irregular components Be able to deseasonalize a time series Know the definition of the following terms: time series time series plot horizontal pattern stationary time series trend pattern seasonal pattern cyclical pattern mean absolute error mean squared error mean absolute percentage error moving average weighted moving average smoothing constant time series decomposition additive model multiplicative model 18 - © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 Solutions: The following table shows the calculations for parts (a), (b), and (c) Week a MAE = 22/5 = 4.4 b MSE = 104/5 = 20.8 c MAPE = 159.38/5 = 31.88 d Forecast for week is 14 Squared Forecast Error 5 22 Percentage Error 25 25 36 104 -38.46 18.75 -45.45 35.29 -21.43 -51.30 Absolute Value of Percentage Error 38.46 18.75 45.45 35.29 21.43 159.38 The following table shows the calculations for parts (a), (b), and (c) Week Time Series Forecast Value Forecast Error 18 13 18 -5 16 13 11 16 -5 17 11 14 17 -3 Totals Absolute Value of Forecast Error Time Series Forecast Value Forecast Error 18 13 18.00 -5.00 16 15.50 0.50 11 15.67 -4.67 17 14.50 2.50 14 15.00 -1.00 Totals Absolute Value of Forecast Error Squared Forecast Error 5.00 0.50 4.67 2.50 1.00 13.67 Percentage Error 25.00 0.25 21.81 6.25 1.00 54.31 a MAE = 13.67/5 = 2.73 b MSE = 54.31/5 = 10.86 c MAPE = 105.89/5 = 21.18 d Forecast for week is (18 + 13 + 16 + 11 + 17 + 14) / = 14.83 -38.46 3.13 -42.45 14.71 -7.14 -70.21 The following table shows the measures of forecast error for both methods MAE MSE MAP E Exercise 4.40 20.80 31.88 18 ­ 2 Exercise 2.73 10.86 21.18 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Absolute Value of Percentage Error 38.46 3.13 42.45 14.71 7.14 105.86 Time Series Analysis and Forecasting For each measure of forecast accuracy the average of all the historical data provided more accurate forecasts than simply using the most recent value a Month Squared Time Series Forecast Forecast Value Forecast Error Error 24 13 24 -11 121 20 13 49 12 20 -8 64 19 12 49 23 19 16 15 23 -8 64 Total 363 MSE = 363/6 = 60.5 Forecast for month = 15 b Week Squared Time Series Forecast Forecast Value Forecast Error Error 24 13 24.00 -11.00 121.00 20 18.50 1.50 2.25 12 19.00 -7.00 49.00 19 17.25 1.75 3.06 23 17.60 5.40 29.16 15 18.50 -3.50 12.25 Total 216.72 MSE = 216.72/6 = 36.12 Forecast for month = (24 + 13 + 20 + 12 + 19 + 23 + 15) / = 18 c The average of all the previous values is better because MSE is smaller a 18 ­ 3 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 The data appear to follow a horizontal pattern b Three-week moving average Week Squared Time Series Forecast Forecast Value Forecast Error Error 18 13 16 11 15.67 -4.67 21.78 17 13.33 3.67 13.44 14 14.67 -0.67 0.44 Total 35.67 MSE = 35.67/3 = 11.89 The forecast for week = (11 + 17 + 14) / = 14 c Smoothing constant = Week Squared Time Series Forecast Forecast Value Forecast Error Error 18 13 18.00 -5.00 25.00 16 17.00 -1.00 1.00 11 16.80 -5.80 33.64 17 15.64 1.36 1.85 14 15.91 -1.91 3.66 Total 65.15 MSE = 65.15/5 = 13.03 The forecast for week is 2(14) + (1 - 2)15.91 = 15.53 d e The three-week moving average provides a better forecast since it has a smaller MSE Smoothing constant = 18 ­ 4 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting Week Squared Time Series Forecast Forecast Value Forecast Error Error 18 13 18.00 -5.00 25.00 16 16.00 0.00 0.00 11 16.00 -5.00 25.00 17 14.00 3.00 9.00 14 15.20 -1.20 1.44 Total 60.44 MSE = 60.44/5 = 12.09 The exponential smoothing forecast using α = provides a better forecast than the exponential smoothing forecast using α = since it has a smaller MSE a The data appear to follow a horizontal pattern Three-week moving average Week Squared Time Series Forecast Forecast Value Forecast Error Error 24 13 20 12 19.00 -7.00 49.00 19 15.00 4.00 16.00 23 17.00 6.00 36.00 15 18.00 -3.00 9.00 Total 110.00 MSE = 110/4 = 27.5 The forecast for week = (19 + 23 + 15) / = 19 18 ­ 5 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 b Smoothing constant = Week Squared Time Series Forecast Forecast Value Forecast Error Error 24 13 24.00 -11.00 121.00 20 21.80 -1.80 3.24 12 21.44 -9.44 89.11 19 19.55 -0.55 0.30 23 19.44 3.56 12.66 15 20.15 -5.15 26.56 Total 252.87 MSE = 252.87/6 = 42.15 The forecast for week is 2(15) + (1 - 2)20.15 = 19.12 c The three-week moving average provides a better forecast since it has a smaller MSE d Smoothing constant = Time Series Forecast Value Forecast Error 24 13 24.00 -11.00 20 19.60 0.40 12 19.76 -7.76 19 16.66 2.34 23 17.59 5.41 15 19.76 -4.76 Total MSE = 238.72/6 = 39.79 Week 7 a 121.00 0.16 60.22 5.49 29.23 22.62 238.72 The exponential smoothing forecast using α = provides a better forecast than the exponential smoothing forecast using α = since it has a smaller MSE Week 10 11 12 b Squared Value of Forecast Error Time-Series Value 17 21 19 23 18 16 20 18 22 20 15 22 4-Week Moving Average Forecast (Error)2 20.00 20.25 19.00 19.25 18.00 19.00 20.00 18.75 Totals 4.00 18.06 1.00 1.56 16.00 1.00 25.00 10.56 77.18 5-Week Moving Average Forecast (Error)2 19.60 19.40 19.20 19.00 18.80 19.20 19.00 MSE(4-Week) = 77.18 / = 9.65 18 ­ 6 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 12.96 0.36 1.44 9.00 1.44 17.64 9.00 51.84 Time Series Analysis and Forecasting MSE(5-Week) = 51.84 / = 7.41 c For the limited data provided, the 5-week moving average provides the smallest MSE a Week 10 11 12 b Time-Series Value 17 21 19 23 18 16 20 18 22 20 15 22 Weighted Moving Average Forecast 19.33 21.33 19.83 17.83 18.33 18.33 20.33 20.33 17.83 Forecast Error (Error)2 3.67 -3.33 -3.83 2.17 -0.33 3.67 -0.33 -5.33 4.17 Total 13.47 11.09 14.67 4.71 0.11 13.47 0.11 28.41 17.39 103.43 MSE = 103.43 / = 11.49 Prefer the unweighted moving average here; it has a smaller MSE c You could always find a weighted moving average at least as good as the unweighted one Actually the unweighted moving average is a special case of the weighted ones where the weights are equal The following tables show the calculations for = .1  Week 10 11 12 Time Series Value Forecast 17 21 17.00 19 17.40 23 17.56 18 18.10 16 18.09 20 17.88 18 18.10 22 18.09 20 18.48 15 18.63 22 18.27 Absolute Value of Forecast Error Forecast Error 4.00 1.60 5.44 -0.10 -2.09 2.12 -0.10 3.91 1.52 -3.63 3.73 Totals 4.00 1.60 5.44 0.10 2.09 2.12 0.10 3.91 1.52 3.63 3.73 28.24 Squared Forecast Error 16.00 2.56 29.59 0.01 4.37 4.49 0.01 15.29 2.31 13.18 13.91 101.72 Absolute Value of Percentage Error Percentage Error 19.05 8.42 23.65 -0.56 -13.06 10.60 -0.56 17.77 7.60 -24.20 16.95 65.67 19.05 8.42 23.65 0.56 13.06 10.60 0.56 17.77 7.60 24.20 16.95 142.42 The following tables show the calculations for = .2 Week Time Series Forecast Value Forecast Error Absolute Value of Forecast 18 ­ 7 Squared Forecast Error Percentage Error © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Absolute Value of Percentage Error Chapter 18 Error 10 11 12 17 21 19 23 18 16 20 18 22 20 15 22 17.00 17.80 18.04 19.03 18.83 18.26 18.61 18.49 19.19 19.35 18.48 4.00 1.20 4.96 -1.03 -2.83 1.74 -0.61 3.51 0.81 -4.35 3.52 Totals 4.00 1.20 4.96 1.03 2.83 1.74 0.61 3.51 0.81 4.35 3.52 28.56 16.00 1.44 24.60 1.06 8.01 3.03 0.37 12.32 0.66 18.92 12.39 98.80 19.05 6.32 21.57 -5.72 -17.69 8.70 -3.39 15.95 4.05 -29.00 16.00 35.84 19.05 6.32 21.57 5.72 17.69 8.70 3.39 15.95 4.05 29.00 16.00 147.44 a MSE for = .1 = 101.72/11 = 9.25 MSE for = .2 = 98.80/11 = 8.98 = .2 provides more accurate forecasts based upon MSE b.  MAE for = .1 = 28.24/11 = 2.57 MAE for = .2 = 28.56/11 = 2.60 = .1 provides more accurate forecasts based upon MAE; but, they are very close c MAPE for = .1 = 142.42/11 = 12.95% MAPE for = .2 = 147.44/11 = 13.40% = .1 provides more accurate forecasts based upon MAPE 10 a F13 = 2Y12 + 16Y11 + 64(.2Y10 + 8F10) = 2Y12 + 16Y11 + 128Y10 + 512F10 F13 = 2Y12 + 16Y11 + 128Y10 + 512(.2Y9 + 8F9) = 2Y12 + 16Y11 + 128Y10 + 1024Y9 + 4096F9 F13 = 2Y12 + 16Y11 + 128Y10 + 1024Y9 + 4096(.2Y8 + 8F8) = 2Y12 + 16Y11 + 128Y10 + 1024Y9 + 08192Y8 + 32768F8 b The more recent data receives the greater weight or importance in determining the forecast The moving averages method weights the last n data values equally in determining the forecast 11 a 18 ­ 8 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting The first two time series values may be an indication that the time series has shifted to a new higher level, as shown by the remainig 10 values But, overall, the time series plot exhibits a horizontal pattern b Month 10 11 12 Yt 80 82 84 83 83 84 85 84 82 83 84 83 3-Month Moving Averages Forecast 82.00 83.00 83.33 83.33 84.00 84.33 83.67 83.00 83.00 Totals (Error)2 1.00 0.00 0.45 2.79 0.00 5.43 0.45 1.00 0.00 11.12 α=2 Forecast 80.00 80.40 81.12 81.50 81.80 82.24 82.79 83.03 82.83 82.86 83.09 MSE(3-Month) = 11.12 / = 1.24 MSE(α = 2) = 39.06 / 11 = 3.55 A 3-month moving average provides the most accurate forecast using MSE c 3-month moving average forecast = (83 + 84 + 83) / = 83.3 12 a 18 ­ 9 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part (Error)2 4.00 12.96 3.53 2.25 4.84 7.62 1.46 1.06 0.03 1.30 0.01 39.06 Chapter 18 The data appear to follow a horizontal pattern b Month 10 11 12 Time-Series Value 9.5 9.3 9.4 9.6 9.8 9.7 9.8 10.5 9.9 9.7 9.6 9.6 3-Month Moving Average Forecast 9.40 9.43 9.60 9.70 9.77 10.00 10.07 10.03 9.73 Totals (Error)2 0.04 0.14 0.01 0.01 0.53 0.01 0.14 0.18 0.02 1.08 4-Month Moving Average Forecast 9.45 9.53 9.63 9.73 9.95 9.98 9.97 9.92 MSE(3-Month) = 1.08 / = 12 MSE(4-Month) = 1.09 / = 14 Use 3-Month moving averages c Forecast = (9.7 + 9.6 + 9.6) / = 9.63 13 a 18 ­ 10 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part (Error)2 0.12 0.03 0.03 0.59 0.00 0.08 0.14 0.10 1.09 Chapter 18 c Tt  =  244.778 + 22.088t MSE  =  357.81 d 46 a Trend projection provides much better forecasts because it has the smallest MSE.  The  reason MSE is smaller for trend projection is that sales are increasing over time; as a result,  exponential smoothing continuously underestimates the value of sales.  If you look at the  forecast errors for exponential smoothing you will see that the forecast errors are positive  for periods 2 through 18 The following table shows the calculations using a smoothing constant of Month January February March April May June Sales ($1000s) 185.72 167.84 205.11 210.36 255.57 261.19 Forecast 185.72 178.57 189.18 197.65 220.82 The forecast for July is 4(261.19) + 6(220.82) = 236.97 Forecast for August, using forecast for July as the actual sales in July, is 236.97 Exponential smoothing provides the same forecast for every period in the future.  This is  why it is not usually recommended for long­term forecasting b Using Minitab’s regression procedure we obtained the linear trend equation  Tt  =  149.72 + 18.451t Forecast for July (t = 7) is 149.72 + 18.451(7) =  278.88 Forecast for August (t = 8) is 149.72 + 18.451(8) =   is 297.33 c The proposed settlement is not fair since it does not account for the upward trend in sales.   Based upon trend projection, the settlement should be based on forecasted lost sales of  $278,880 in July and $297,330 in August.  18 ­ 52 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting 47 a The time series plot indicates a linear trend b Holt’s linear exponential smoothing using α = and β = Month Cash Required Estimated Estimated Forecast ($1000s) Level Trend Forecast Error 205 205.00 7.00 212 212.00 7.00 212.00 0.00 218 218.40 6.76 219.00 -1.00 224 224.46 6.48 225.16 -1.16 230 230.38 6.25 230.95 -0.95 240 238.65 7.06 236.63 3.37 246 245.89 7.13 245.72 0.28 Total Squared Value of Forecast Error Forecast for month = 245.89 + 7.13(1) = 253 Forecast for month = 245.89 + 7.13(2) = 260 c A portion of the Minitab regression output follows The regression equation is Cash Required ($1000s) = 198 + 6.82 Month Predictor Constant Month Coef 197.714 6.8214 S = 1.19224 SE Coef 1.008 0.2253 R-Sq = 99.5% T 196.22 30.28 P 0.000 0.000 R-Sq(adj) = 99.3% Analysis of Variance Source Regression Residual Error Total DF SS 1302.9 7.1 1310.0 MS 1302.9 1.4 F 916.61 P 0.000 Forecast for month = 197.714 + 6.8214(8) = 252 18 ­ 53 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 0.00 1.00 1.35 0.89 11.34 0.08 14.66 Chapter 18 Forecast for month = 197.714 + 6.8214(9) = 259 d Holt’s linear exponential smoothing using α = and β = 4: MSE = 14.66/6 = 2.44 Linear trend regression: MSE = 1.4 from the regression output; recall, however, that this value of MSE is not the average sum of squared errors that is computed using Holt’s method 48 a The time series plot shows a linear trend n b t  �t t 1 n n  15 3 Y  (t  t )(Yt  Y )  150 �Y t 1 n t  200  40 (t  t )  10 n b1  �(t  t )(Y  Y ) t t 1 n �(t  t )  150  15 10 t 1 b0  Y  b1 t  40  (15)(3)  5 Tt  5  15t The slope of 15 indicates that the average increase in sales is 15 pianos per year c Forecast for year is T6  5  15(6)  85 Forecast for year is T6  5  15(7)  100 18 ­ 54 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting 49 a A portion of the Minitab regression output follows The regression equation is Sales = 7.15 - 5.59 Qtr1 - 10.9 Qtr2 - 11.5 Qtr3 + 0.937 t Predictor Constant Qtr1 Qtr2 Qtr3 t b Coef 7.150 -5.587 -10.925 -11.463 0.9375 SE Coef 2.283 2.253 2.232 2.219 0.1384 T 3.13 -2.48 -4.90 -5.17 6.77 P 0.007 0.025 0.000 0.000 0.000 Quarterly forecasts for next year correspond to t = 21, 22, 23, and 24 Forecast for Quarter (t = 21) = 7.150 -5.587(1) -10.925(0) -11.463(0) + 9375(21) = 21.25 Forecast for Quarter (t = 22) = 7.150 -5.587(0) -10.925(1) -11.463(0) + 9375(22) = 16.85 Forecast for Quarter (t = 23) = 7.150 -5.587(0) -10.925(0) -11.463(1) + 9375(23) = 17.25 Forecast for Quarter (t = 24) = 7.150 -5.587(0) -10.925(0) -11.463(0) + 9375(24) = 29.65 50 a t 10 11 12 13 14 15 16 17 18 19 20 Sales  4  2  1  5  6  4  4 14 10  3  5 16 12  9  7 22 18 10 13 35 Quarter Centered Moving Average Seasonal­Irregular Value 3.250 3.750 4.375 5.875 7.500 7.875 7.875 8.250 8.750 9.750 10.750 11.750 13.250 14.125 15.000 17.375 0.308 1.333 1.371 0.681 0.533 1.778 1.270 0.364 0.571 1.641 1.116 0.766 0.528 1.558 1.200 0.576 Seasonal­Irregular Values Seasonal Index 1.371, 1.270, 1.116, 1.200 0.681, 0.364, 0.776, 0.576 0.308, 0.533, 0.571, 0.528 1.333, 1.778, 1.641, 1.558 Total  1.2394 0.5965 0.4852 1.5774 3.8985 18 ­ 55 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 Quarter Adjusted Seasonal Index 1.2717 0.6120 0.4978 1.6185 Note: Adjustment for seasonal index = 4 / 3.8985 = 1.0260 b.  The largest effect is in quarter 4; this seems reasonable since retail sales are generally higher during  October, November, and December 51 a Note: the adjusted seasonal indexes were computed in Exercise 50 Year Quarter 4 4 Sales 4 14 10 16 12 22 18 10 13 35 Adjusted Seasonal Index 1.2717 0.6120 0.4978 1.6185 1.2717 0.6120 0.4978 1.6185 1.2717 0.6120 0.4978 1.6185 1.2717 0.6120 0.4978 1.6185 1.2717 0.6120 0.4978 1.6185 Deseasonalized Sales 3.1454 3.2680 2.0088 3.0893 4.7181 6.5359 8.0354 8.6500 7.8635 4.9020 10.0442 9.8857 9.4362 14.7059 14.0619 13.5928 14.1543 16.3399 26.1149 21.6250 A portion of the Minitab regression output for the deseasonalized sales values follows The regression equation is Deseasonalized Sales = - 0.36 + 0.997 t Predictor Constant t Coef -0.356 0.9966 S = 2.61907 SE Coef 1.217 0.1016 R-Sq = 84.3% T -0.29 9.81 P 0.773 0.000 R-Sq(adj) = 83.4% Analysis of Variance Source Regression Residual Error Total DF 18 19 SS 660.52 123.47 783.99 MS 660.52 6.86 18 ­ 56 F 96.29 P 0.000 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting b The quarterly trend forecasts for next year correspond to t = 21, 22, 23, and 24 Forecast for Quarter (t = 21) = -.356 + 9966(21) = 20.57 Forecast for Quarter (t = 22) = -.356 + 9966(22) = 21.57 Forecast for Quarter (t = 23) = -.356 + 9966(23) = 22.57 Forecast for Quarter (t = 24) = -.356 + 9966(24) = 23.56 c Multiplying the quarterly trend forecasts by the adjusted seasonal indexes provides the forecasts for next year Forecast for Quarter (t = 21) = 20.57(1.2717) = 26.2 Forecast for Quarter (t = 22) = 21.57(.6120) = 13 Forecast for Quarter (t = 23) = 22.57(.4978) = 11.2 Forecast for Quarter (t = 24) = 23.56(1.6185) = 38.1 52 a A linear trend pattern appears to be present in the time series plot b A portion of the Minitab regression output follows The regression equation is Number Sold = 22.9 + 15.5 Year Predictor Constant Year S = 5.32849 Coef 22.857 15.536 SE Coef 4.503 1.007 R-Sq = 97.9% T 5.08 15.43 P 0.004 0.000 R-Sq(adj) = 97.5% Analysis of Variance 18 ­ 57 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 Source Regression Residual Error Total c 53 a DF SS 6758.0 142.0 6900.0 MS 6758.0 28.4 F 238.02 P 0.000 Forecast in year = 22.857 +15.536(8) = 147.15 or approximately 147 units A portion of the Minitab regression output follows The regression equation is Sales = 0.036 + 4.91 Qtr1 + 14.5 Qtr2 + 9.11 Qtr3 + 0.971 t Predictor Constant Qtr1 Qtr2 Qtr3 t Coef 0.0357 4.9129 14.5134 9.1138 0.97098 S = 1.61791 SE Coef 0.8648 0.8724 0.8682 0.8657 0.03822 R-Sq = 97.5% T 0.04 5.63 16.72 10.53 25.41 P 0.967 0.000 0.000 0.000 0.000 R-Sq(adj) = 97.1% Analysis of Variance Source Regression Residual Error Total b DF 23 27 SS 2385.04 60.21 2445.25 MS 596.26 2.62 F 227.79 P 0.000 The quarterly forecast for next year correspond to t = 29, 30, 31, and 32 Forecast for Quarter (t = 29) = 0.0357 + 4.9129(1) + 14.5134(0) + 9.1138(0) + 0.971(29) = 33.1 Forecast for Quarter (t = 30) = 0.0357 + 4.9129(0) + 14.5134(1) + 9.1138(0) + 0.971(30) = 43.7 Forecast for Quarter (t = 31) = 0.0357 + 4.9129(0) + 14.5134(0) + 9.1138(1) + 0.971(31) = 39.3 Forecast for Quarter (t = 32) = 0.0357 + 4.9129(0) + 14.5134(0) + 9.1138(0) + 0.971(32) = 31.1 54 a Centered 18 ­ 58 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting t 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Sales 15 10 10 18 15 14 26 23 12 19 28 25 18 22 34 28 21 24 36 30 20 28 40 35 27 Moving Average 9.250 10.125 11.125 12.125 13.000 14.500 16.500 18.125 19.375 20.250 20.750 21.750 22.875 24.000 25.125 25.875 26.500 27.000 27.500 27.625 28.000 29.000 30.125 31.625 b The centered moving average values smooth out the time series by removing seasonal effects and some of the random variability The centered moving average time series shows the trend in the data c t Sales Centered Seasonal-Irregular Moving Average Value 18 ­ 59 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Quarter 15 10 10 18 15 14 26 23 12 19 28 25 18 22 34 28 21 24 36 30 20 28 40 35 27 9.250 10.125 11.125 12.125 13.000 14.500 16.500 18.125 19.375 20.250 20.750 21.750 22.875 24.000 25.125 25.875 26.500 27.000 27.500 27.625 28.000 29.000 30.125 31.625 1.081 0.395 0.899 1.485 1.154 0.483 0.848 1.434 1.187 0.593 0.916 1.287 1.093 0.750 0.876 1.314 1.057 0.778 0.873 1.303 1.071 0.690 0.929 1.265 Seasonal-Irregular Component Values 0.899, 0.848, 0.916, 0.876, 0.873, 0.929 1.485, 1.434, 1.287, 1.314, 1.303, 1.265 1.081, 1.154, 1.187, 1.093, 1.057, 1.071 0.395, 0.483, 0.593, 0.750, 0.778, 0.690 Total Quarter Seasonal Index 0.890 1.348 1.107 0.615 3.960 Adjusted Seasonal Index 0.899 1.362 1.118 0.621 Note: Adjustment for seasonal index = 4.00 / 3.96 = 1.0101 d Hudson Marine experiences the largest seasonal increase in quarter Since this quarter occurs prior to the peak summer boating season, this result seems reasonable But the largest seasonal effect is the seasonal decrease in quarter This is also reasonable because of decreased boating in the fall and winter 18 ­ 60 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Time Series Analysis and Forecasting 55 a Year Quarter 4 4 4 Sales 15 10 10 18 15 14 26 23 12 19 28 25 18 22 34 28 21 24 36 30 20 28 40 35 27 Adjusted Seasonal Index 0.899 1.362 1.118 0.621 0.899 1.362 1.118 0.621 0.899 1.362 1.118 0.621 0.899 1.362 1.118 0.621 0.899 1.362 1.118 0.621 0.899 1.362 1.118 0.621 0.899 1.362 1.118 0.621 Deseasonalized Sales 6.673 11.016 8.942 6.443 11.122 13.219 13.413 11.275 15.571 19.094 20.566 19.328 21.132 20.563 22.355 28.993 24.468 24.969 25.037 33.825 26.692 26.438 26.825 32.214 31.141 29.376 31.296 43.489 A portion of the Minitab regression output for the deseasonalized sales time series follows The regression equation is Deseasonalized Sales = 6.33 + 1.05 t Predictor Constant t Coef 6.332 1.05466 S = 3.07863 SE Coef 1.195 0.07203 R-Sq = 89.2% T 5.30 14.64 P 0.000 0.000 R-Sq(adj) = 88.8% Analysis of Variance Source Regression Residual Error Total b DF 26 27 SS 2032.2 246.4 2278.6 MS 2032.2 9.5 F 214.41 P 0.000 The quarterly forecast for next year correspond to t = 29, 30, 31, and 32 Forecast for Quarter (t = 29) = 6.332 + 1.055(29) = 36.93 Forecast for Quarter (t = 30) = 6.332 + 1.055(30) = 37.98 Forecast for Quarter (t = 31) = 6.332 + 1.055(31) = 39.04 Forecast for Quarter (t = 32) = 6.332 + 1.055(32) = 40.09 18 ­ 61 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 18 c Multiplying the quarterly trend forecasts by the adjusted seasonal indexes provides the forecasts for next year Forecast for Quarter (t = 29) = 36.93(.899) = 33.2 Forecast for Quarter (t = 30) = 37.98(1.362) = 51.7 Forecast for Quarter (t = 31) = 39.04(1.118) = 43.65 Forecast for Quarter (t = 32) = 40.09(.621) = 24.9 18 ­ 62 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ... calculations for = .1  Week 10 11 12 Time Series Value Forecast 17 21 17.00 19 17.40 23 17.56 18 18.10 16 18. 09 20 17.88 18 18.10 22 18. 09 20 18. 48 15 18. 63 22 18. 27 Absolute Value of Forecast Error Forecast... accessible website, in whole or in part Time? ?Series? ?Analysis? ?and? ?Forecasting Week Squared Time Series Forecast Forecast Value Forecast Error Error 18 13 18. 00 -5.00 25.00 16 16.00 0.00 0.00 11... calculations for parts (a), (b), and (c) Week Time Series Forecast Value Forecast Error 18 13 18 -5 16 13 11 16 -5 17 11 14 17 -3 Totals Absolute Value of Forecast Error Time Series Forecast Value Forecast