CHAPTER 4 MOVING AVERAGES AND SMOOTHING METHODS (Page 107)

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: (4.1)

Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Naïve Models ST , T , S S TS 1 Simple Averages ST 30 Moving Averages 4-20 Double Moving Averages 2 Linear (Double) exponential smoothing (Holt’s) T 3 Quadratic exponential smoothing 4 Seasonal exponential smoothing (Winter’s) 2 x s Adaptive filtering 5 x s Simple regression I C 10 Multiple regression C , S 10 x V Classical decomposition Exponential trend models I , L S-curve fitting Gompertz models Growth curves Census X-12 6 x s ARIMA (Box-Jenkins) ST , T , C , S 24 3 x s Lading indicators Econometric models Time series multiple regression T , S   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.

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.

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

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

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

Forecasting Methods Based on Averaging (Page 111)

Minimal Data Requirements Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Naïve Models ST , T , S S TS 1 Simple Averages ST 30 Moving Averages 4-20 Double Moving Averages 2 Linear (Double) exponential smoothing (Holt’s) T 3 Quadratic exponential smoothing 4 Seasonal exponential smoothing (Winter’s) 2 x s Adaptive filtering 5 x s Simple regression I C 10 Multiple regression C , S 10 x V Classical decomposition Exponential trend models I , L S-curve fitting Gompertz models Growth curves Census X-12 6 x s ARIMA (Box-Jenkins) ST , T , C , S 24 3 x s Lading indicators Econometric models Time series multiple regression T , S   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.

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

Minimal Data Requirements Is used for stabilized series, and the environment is generally unchanging. Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Simple averages ST S TS 30 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.

Example 4.2 Table 4-2 Gasoline Purchases for the Spokane Transit Authority for Example 4.2 Week t Purchases 1 275 11 302 21 310 2 291 12 287 22 299 3 307 13 290 23 285 4 281 14 311 24 250 5 295 15 277 25 260 6 268 16 245 26 7 252 17 282 27 271 8 279 18 28 9 264 19 298 29 10 288 20 303 30

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

Solution

Minimal Data Requirements Moving Averages A moving average of order k is the mean value of the k most recent observations. (4.8) 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. Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Moving averages ST S TS 4-20 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.

Example 4.3 (last data) Week t Purchases 1 275 11 302 21 310 2 291 12 287 22 299 3 307 13 290 23 285 4 281 14 311 24 250 5 295 15 277 25 260 6 268 16 245 26 7 252 17 282 27 271 8 279 18 28 9 264 19 298 29 10 288 20 303 30

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

Results Week Purchases AVER1 RESI1 1 275 * * 2 291 * * * 3 307 * * 289.8 288.4 280.6 275.0 271.6 270.2 277.0 284.0 286.2 295.6 293.4 282.0 281.0 278.4 275.8 294.0 297.4 299.0 289.4 280.8 267.8 262.2 261.6 272.0 Week Purchases AVER1 RESI1 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

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

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

* * 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

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

Exponential Smoothing Methods

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)] = smoothing constant (0< <1) Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Single Exponential smoothing ST S TS 2

Comparison of Smoothing Constants Period = 0.1 =0.6 Calculations Weight t 0.1 0.6 t-1 0.1 x 0.9 0.09 0.6 x 0.4 0.24 t-2 0.1 x 0.9 x 0.9 0.081 0.6 x 0.4 x 0.4 0.096 t-3 0.1 x 0.9 x 0.9 x 0.9 0.073 0.038 t-4 0.1 x 0.9 x 0.9 x 0.9 x 0.9 0.066 0.6 x 0.4 x 0.4 x 0.4 0.015 All others 0.59 0.011 Totals 1.0

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.

Example 4.5 Year Quarters 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.

Results Year Quarters (α=0.1) 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 1) Initial value for the smoothed series = first observation = 500 2) 250-485 =-235 3) 4) =0.1(250)+0.9(485)=461.5

Results 5) The calculation for the first quarter of 2007 Year Quarters (α=0.1) (α=0.6) 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 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) 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

Single Exponential Smoothing (α=0.1)

Single Exponential Smoothing (α=0.6)

α = 0.266

Comparison Optimization α = 0.1 MAPE 38.9 MAD 127.0 MSD 24261.7 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 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

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.

Minimal Data Requirements 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. Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Linear (Double) exponential smoothing (Holt’s) T S TS 3

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

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.

Example 4.9 (Last data) Year Quarters 2000 1 500 * 2 350 * 3 250 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 Year Quarters Example 4.9 (Last data)

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

LBQ (36.33) shows that the series is nonrandom

Minimal Data Requirements Exponential Smoothing Adjusted for Trend and Seasonal Variations: Winters’ Method Method Pattern of Data Time Horizon Type of Model Minimal Data Requirements Nonseasonal Seasonal Seasonal exponential smoothing (Winter’s) S TS 2 x s

The equations used : 1. The exponential smoothed series: 2. The trend estimate: 3. The seasonality estimate: 4. Forecast p periods into the future: 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

Choosing the Weights , and can be selected subjectively or by minimizing an error such as MSE. A common approach: a nonlinear optimization algorithm to find optimal constants.

Starting the Procedure One approach is to set the first estimate equal to the first observation, the trend is then estimated to equal zero, and the seasonal indices are set to 1. A second approach is to use the average of the first season or s observations , the trend is the slope of a line fit to these observations, and the seasonal indices are:

Minitab develops a regression equation, and uses constants from the equation as initial estimates for the level and the trend. The seasonal components are obtained from a dummy variable regression using detrended data.

Example 4.10 (Last data) Year Quarters 2000 1 500 * 2 350 * 3 250 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 26 600 27 450 28 700 Year Quarters Example 4.10 (Last data)

Minitab Instructions: STAT > TIME SERIES > WINTERS’ METHOD. Better than the other 2 models in terms of minimizing MSE.

Autocorrelation Functions for the Residuals None of the coefficients appear to be significantly larger than zero, and the small value of LBQ (5.01) shows that the series is random.

THE END