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CHAPTER 4

CHAPTER 4. MOVING AVERAGES AND SMOOTHING METHODS. (Page 107). (4.1). NAÏVE MODELS. They are based solely on the most recent information available. Is sometimes called the “no change” forecast. Suitable for very small data sets. The simplest model is:.

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CHAPTER 4

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  1. CHAPTER 4 MOVING AVERAGES AND SMOOTHING METHODS (Page 107)

  2. (4.1) NAÏVE MODELS • They are based solely on the most recent information available. • Is sometimes called the “no change” forecast. • Suitable for very small data sets. • The simplest model is:

  3. Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

  4. Example 4.1 (Page 108) • Table 4-1 Sales of Saws for Acme Tool Company, 2000 – 2006. • Initialization (Fitting) Part: 2000 – 2005. • Test Part: 2006.

  5. (4.2) • The technique can be adjusted to take trend into consideration: • The rate of change might be more appropriate than the absolute amount of change:

  6. (4.4) (4.5) • An appropriate forecast equation for quarterly data: • The analyst can combine seasonal and trend estimates using: • For monthly data:

  7. Example 4.1 (cont.) • Using Equations (4.3, 4.4, and 4.5) (Pages 110 – 111)

  8. Forecasting Methods Based on Averaging (Page 111)

  9. Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

  10. Simple Averages Uses the mean of all the relevant historical observations as the forecast of the next period. (4.6) (4.7) • New observation is added: (Page 111)

  11. Is used for stabilized series, and the environment is generally unchanging. Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

  12. Example 4.2 Table 4-2 Gasoline Purchases for the Spokane Transit Authority for Example 4.2

  13. Time Series Graph Figure 4-3 Time Series Plot of Weekly Gasoline Purchases for the Spokane Transit Authority The data seems stationary.

  14. Solution

  15. (4.8) Moving Averages • A moving average of order k is the mean value of the k most recent observations. K = number of terms in the moving average. The method does not handle trend or seasonality very well, although it does better than the simple average method. Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical. Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.

  16. Example 4.3 (last data)

  17. Calculations Using a five-week moving average (Page 114, 115) Minitab can be used (See the Minitab Applications section for instructions, Pages: 159-161)

  18. * * * * * 289.8 288.4 280.6 275.0 271.6 270.2 277.0 284.0 286.2 295.6 293.4 282.0 281.0 278.4 275.8 281.0 294.0 297.4 299.0 289.4 280.8 267.8 262.2 261.6 272.0 Results WeekPurchasesAVER1RESI1 1 275 * * 2 291 * * 3 307 * * 4 281 * * 5 295 289.8 * 6 268 288.4 -21.8 7 252 280.6 -36.4 8 279 275.0 -1.6 9 264 271.6 -11.0 10 288 270.2 16.4 11 302 277.0 31.8 12 287 284.0 10.0 13 290 286.2 6.0 14 311 295.6 24.8 15 277 293.4 -18.6 16 245 282.0 -48.4 17 282 281.0 0.0 18 277 278.4 -4.0 19 298 275.8 19.6 20 303 281.0 27.2 21 310 294.0 29.0 22 299 297.4 5.0 23 285 299.0 -12.4 24 250 289.4 -49.0 25 260 280.8 -29.4 26 245 267.8 -35.8 27 271 262.2 3.2 28 282 261.6 19.8 29 302 272.0 40.4 30 285 277.0 13.0

  19. Minitab Results Minitab Instructions Stat > time Series > Moving averages Note: (MSE is called MSD on Minitab output) (Page 115) FIGURE 4 - 4

  20. The series is nonrandom Task: Try a nine-week moving average, it would be better, because the large-order moving average pays very little attention to the large fluctuations in the data series

  21. * * * * * * * * * * 289.8 -21.8 288.4 -36.4 280.6 -1.6 275.0 -11.0 271.6 16.4 270.2 31.8 277.0 10.0 284.0 6.0 286.2 24.8 295.6 -18.6 293.4 -48.4 282.0 0.0 281.0 -4.0 278.4 19.6 275.8 27.2 281.0 29.0 294.0 5.0 297.4 -12.4 299.0 -49.0 289.4 -29.4 280.8 -35.8 267.8 3.2 262.2 19.8 261.6 40.4 272.0 13.0

  22. Double Moving Average • Equations (4.9) – (4.12) (Pages 116-117) • Example 4-4 • (Pages 118-119)

  23. Exponential Smoothing Methods

  24. = smoothing constant (0< <1) 2.1 Simple (Single) Exponential Smoothing Based on averaging (smoothing) past values of a series in a decreasing exponential manner, with more weight being given to the more recent observations. New forecast = [α x (new observation)] + [(1- α) x (old forecast)]

  25. Comparison of Smoothing Constants

  26. Starting the algorithm • An initial value for the old smoothed series must be set: • To set the first estimate = the first observation. • Another method: To use the average of the first 5 or 6 observations.

  27. Example 4.5 Quarters Year 2000 1 500 * 2 350 * 3 250 * 4 400 2001 5 450 * 6 350 * 7 200 * 8 300 2002 9 350 * 10 200 * 11 150 * 12 400 2003 13 550 * 14 350 * 15 250 * 16 550 2004 17 550 * 18 400 * 19 350 * 20 600 2005 21 750 * 22 500 * 23 400 * 24 650 2006 25 850 • The actual sales for a Company for the years 2000 to 2006 are demonstrated in the Table. • The data for the first quarter of 2006 will be used as the test part to help determine the best value of α among the two considered.

  28. (α=0.1) 1) Initial value for the smoothed series = first observation = 500 250-485 =-235 3) =0.1(250)+0.9(485)=461.5 4) Results Year Quarters 2000 1 500 * 2 350 * 3 250 * 4 400 500.000 1)0.000 500.000 -150.000 485.000 2)-235.000 3) 461.500 4)-61.500 2)

  29. (α=0.6) Results Year Quarters (α=0.1) 500.000 1) 0.000 500.000 -150.000 485.000 2)-235.000 3) 461.500 4) -61.500 455.350 -5.350 454.815 -104.815 444.334 -244.334 419.900 -119.900 407.910 -57.910 402.119 -202.119 381.907 -231.907 358.716 41.284 362.845 187.155 381.560 -31.560 378.404 -128.404 365.564 184.436 384.007 165.993 400.607 -0.607 400.546 -50.546 395.491 204.509 415.942 334.058 449.348 50.652 454.413 -54.413 448.972 201.028 469.075 5) 2000 1 500 * 2 350 * 3 250 * 4 400 2001 5 450 * 6 350 * 7 200 * 8 300 2002 9 350 * 10 200 * 11 150 * 12 400 2003 13 550 * 14 350 * 15 250 * 16 550 2004 17 550 * 18 400 * 19 350 * 20 600 2005 21 750 * 22 500 * 23 400 * 24 650 2006 25 850 500.000 0.000 500.000 -150.000 410.000 -160.000 314.000 86.000 365.600 84.400 416.240 -66.240 376.496 -176.496 270.598 29.402 288.239 61.761 325.296 -125.296 250.118 -100.118 190.047 209.953 316.019 233.981 456.408 -106.408 392.563 -142.563 307.025 242.975 452.810 97.190 511.124 -111.124 444.450 -94.450 387.780 212.220 515.112 234.888 656.045 -156.045 562.418 -162.418 464.967 185.033 575.987 5) 5) The calculation for the first quarter of 2007

  30. Single Exponential Smoothing (α=0.1)

  31. Single Exponential Smoothing(α=0.6)

  32. α = 0.266

  33. Comparison Optimization MAPE 32.2 MAD 117.5 MSD 19447.0 The weight α is selected subjectively or by minimizing an error such as the MSE Initial smoothed Value = The first observation α = 0.1 MAPE 38.9 MAD 127.0 MSD 24261.7 α = 0.6 MAPE 36.5 MAD 134.5 MSD 22248.4 Initial Smoothed Value = The Average of the first six Observations α = 0.1 MAPE 32.1 MAD 115.5 MSD 21091.2 α = 0.6 MAPE 36.7 MAD 137.1 MSD 22152.8

  34. Large residual autocorrelations at lags 2 and 4: • Seasonal Variation in the data is not accounted for by simple exponential method. • The large value of LBQ (33.86): series is nonrandom.

  35. Exponential Smoothing Adjusted for Trend:(Holt’s Method)“Holt’s two-parameter method” Double Exponential Smoothing • Smoothes the level and slope (trend) using different constants.

  36. Lt = new smoothed value. α = smoothing constant for the data. β = smoothing constant for trend estimate. Yt = Actual value of series in period t. 1. The current level estimate: 2. The trend estimate: 3. Forecast p periods in the future. Tt = trend estimate. p = periods to be forecast into the future. = forecast for p periods into the future. 0 ≤ α and β ≤ 1. Equations Used:

  37. Starting the algorithm • The weights can be selected as in the single exponential smoothing method. • A grid of values could be developed, then selecting the ones producing the lowest MSE. • To begin the algorithm: One approach is to set the first estimate equal to the first observation, the trend is then estimated to equal zero. A second approach is to use the average of the first six observations , the trend is the slope of a line fit to these observations. • Minitab develops a regression equation, and uses constants from the equation as initial estimates for the level and the trend.

  38. Quarters Year 2000 1 500 * 2 350 * 3 250 * 4 400 2001 5 450 * 6 350 * 7 200 * 8 300 2002 9 350 * 10 200 * 11 150 * 12 400 2003 13 550 * 14 350 * 15 250 * 16 550 2004 17 550 * 18 400 * 19 350 * 20 600 2005 21 750 * 22 500 * 23 400 * 24 650 2006 25 850 Example 4.9(Last data)

  39. Gamma = β Comparison: Single: α = 0.266 Holts: α = 0.3, β = 0.1 MSE 19447 20515.5 MAPE 32.2 35.4

  40. LBQ (36.33) shows that the series is nonrandom

  41. Exponential Smoothing Adjusted for Trend and Seasonal Variations: Winters’ Method

  42. 1. The exponential smoothed series: 2. The trend estimate: 3. The seasonality estimate: Lt = new smoothed value. α = smoothed constant for the level. Yt = actual observation in period t β = smoothed constant for trend. Tt = trend estimate. = smoothing constant for seasonality. St = seasonal estimate. p = periods to be forecast in the future. s = length of seasonality. = forecast for p periods in the future 4. Forecast p periods into the future: The equations used :

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