Time Series Analysis Lecture 5 PDF

EIE3002/EII3002 Time Series Analysis Lecture 5 Exponential smoothing method 1  Smoothing Methods Averaging Methods – The Mean. – Moving Averages. Exponential Smoothing Methods – Single (Simple) Exponential Smoothing. – Holts Linear Method. – Holt-Winters’ Method....

EIE3002/EII3002 Time Series Analysis Lecture 5 Exponential smoothing method 1  Smoothing Methods Averaging Methods – The Mean. – Moving Averages. Exponential Smoothing Methods – Single (Simple) Exponential Smoothing. – Holts Linear Method. – Holt-Winters’ Method. 2  Averaging Methods: The Mean The method of simple averages is to take the average of all observed data as the forecast 1 t Ft 1   Yi t i 1 When a new observation Yt+1 becomes available, the forecast for time t+2 is the new mean including the previous observed data plus this new observation 1 t 1 Ft 2   t  1 i 1 Yi 3  Averaging Methods: The Mean Appropriate only if the process underlying the observed Y values is stationary, i.e. it has – no noticeable trend – no noticeable seasonality Data storage – For the first forecast, all the observed values are required – For all succeeding forecasts, only two items need to be stored as time moves on. 4  Averaging Methods: The Mean Since 1 t 1 Ft 2   t  1 i 1 Yi can be written as t Y i  Yt 1 Ft 2  i 1 t 1 t Ft 1  Yt 1 Ft  2  t 1 only the most recent forecast and the most recent observation need to be stored. # Refer additional notes for example. 5  Averaging Methods: Moving Averages One way to modify the influence of past data on the mean as a forecast is to specify how many past observations will be included in the mean. The term “moving average” is used to describe this procedure because as each new observation becomes available, a new average can be computed by dropping the oldest observation and including the newest one. This moving average will then be the forecast for the next period 6  Averaging Methods: Moving Averages A moving average of order or span k is given by 1 t Ft 1   Yi k i t  k 1 Note that this moving average is different from that used in smoothing a time series to estimate the trend-cycle, i.e. by taking an average of nearby points – Here we are forecasting the next observation by taking an average of the most recent observations 7  Averaging Methods: Moving Averages Advantages – Compared to the simple means which deals with all the past data, a moving average of order k deals only with the latest k periods of known data – The number of data points in each average does not change as time goes – Very simple to implement Disadvantages – A moving average of order k requires more storage because all of the k latest observations must be stored, not just the average – It cannot handle trend and seasonality very well although it can do better than the simple mean 8  Averaging Methods: Moving Averages The role of the order k is very important. If it is large - say 12 months - then many observations go into each average, and extreme values have relatively little effect on the forecasts. – The resulting series forecasts will be much smoother than the original series. For this reason, the moving average method is called a smoothing method. If the order is small - say 3 months, then extreme observations have a larger effect on the forecasts – the forecast series will be much less smooth 9  Averaging Methods: Moving Averages At one extreme, if the order is k = 1, there is no smoothing effect at all – this method simply forecasts next month’s value to be the same as the current month’s value. i.e Ft+1 = Yt – this is called the naïve forecasting model – e.g. the forecast of tomorrow’s closing price of IBM stock is today’s closing stock. At the other extreme, if the order in k = n, the number of observations in the time series, the mean of all the observations is used as a forecast – this is equivalent to mean forecast method discussed earlier. Example (refer the additional notes): – Forecasting of the series for electric can openers 10  Averaging Methods: Moving Averages A moving average forecasting system of order k will require k data points to be stored in memory at any one time – If k is small, say at 4 then the storage requirements are not severe. – If k is large, then storage may become a problem. In practice, the technique of moving averages as a forecasting procedure is not used very often because the methods of exponential smoothing are generally superior. 11  Exponential Smoothing Methods There are two criticisms of the moving average method – it puts equal weight on each value in a typical moving average when making a forecast and it requires a lot of data storage Exponential smoothing is a method that addresses both of these criticisms – it bases it forecasts on a weighted average of past observations, with more weight put on the more recent observations and it require very little storage 12  Exponential Smoothing Methods Different versions of exponential smoothing – Single (Simple) exponential smoothing - is relevant when there is no pronounced trend or seasonality in the series – Holt’s linear method - is relevant when trend but not seasonality is present – Holt-Winter’s method - is relevant when both trend and seasonality are present 13  Single Exponential Smoothing Terminology – Every exponential smoothing method has at least one smoothing constant, which is usually between 0 and 1. – Single exponential smoothing has a single smoothing constant denoted by α. – Lt is the level of the series at time t. This value is not observable but can only be estimated. 14  Single Exponential Smoothing Single exponential smoothing is defined by the following equations Lt  Yt  1    Lt  1 (1) Ft 1  Lt (2) Ft+1 is the one-step-ahead forecast Note that these two equations are equivalent to Ft 1  Yt  1    Ft By replacing Ft with its components Ft 1 Yt  1   Yt  1  1   Ft  1  Yt   1   Yt  1  1    Ft  1 2 15  Single Exponential Smoothing If this substitution process is repeated by replacing Ft-1 by its components, Ft-2 by its components, and so on, the results is Ft 1 Yt   1   Yt  1   1    Yt  2   1    Yt  3 2 3   1   4Yt  4     1    Y1  1    F1 t 1 t So, Ft+1 represents a weighted moving average of all past observations. The weights for all past data sum approximately to one. Greater value of α gives more weight to recent observations and less for the past observations. 16  Single Exponential Smoothing Equation (1) shows how to update the estimate of the level – it is the weighted average of the current observation Yt and the previous level Lt-1 with respective weights α and (1-α) Equation (2) shows how forecast one period ahead is obtained – the one-period-ahead forecast Ft+1 is equal to the most recently estimated level Lt 17  Single Exponential Smoothing The initial value of the level L1 is often taken as the first value of the time series Y1 Since the first forecast F1 is unknown, it is taken as F1 = Y1. – Hence the second forecast is F2 = Y1 Another possibility for an initial value of level L1 and hence F1 is to take an average of the first four or five values of the data set. 18  Single Exponential Smoothing What value of α should we use? – No universally accepted answer – Some practitioner recommend a value between 0.1 and 0.2. – Others recommend experimenting with different values of α until a measure such as MSE is minimised That is optimise MSE – When a small value of α is chosen, the initial forecast plays a more prominent role than when a larger α is used. Examples : Forecasting electric can opener shipments (refer the additional notes). 19  Single Exponential Smoothing Notice that the forecast function is flat for long range forecasts – When the forecast horizon is more than one period ahead Ft+m = Ft+1 m = 2, 3,... A flat forecast function is used because single exponential smoothing works best for time series that have no trend, no seasonality, or other underlying pattern. 20  Example: Number of Dwelling Units We examine the time series: Number of private dwelling units approvals in Victoria Sept 1983 – Oct 2001 (Source: DX database) The time series plot shows that there is no apparent trend but some cyclical behaviour and much random variation Does a simple exponential smoothing model track these data well? How do the forecasts depend on the smoothing parameter? Num ber of Dw elling Units Approved: Vic 4000 3500 3000 2500 2000 1500 1000 500 Sep-83 May-85 Sep-88 May-90 Sep-93 May-95 Sep-98 May-00 Nov-87 Nov-92 Nov-97 Mar-86 Mar-91 Mar-96 Mar-01 Jan-87 Jan-92 Jan-97 Jul-84 Jul-89 Jul-94 Jul-99 0 Month 21 Applying Exponential to the time series from Jan 1990 to Oct 2001 A B C D E F G H I 1 Single Exponential Smoothing Sum of square errors 2 a1 SSE Target cell for Solver MSE 167010 3 0.1 23548414.58 MAPE 15.91 4 Smoothing parameter between 0 and 1 5 6 7 8 Month Unit (Yt) Level (Lt) Forecast(Ft) Error (Et) |Et/Yt x 100| 9 10 Jan-90 1800 1800 11 Feb-90 1988 1819 1800 188 9.46 12 Mar-90 2096 1847 1819 277 13.23 13 Apr-90 1756 1837 1847 -91 5.15 14 May-90 2350 1889 1837 513 21.81 15 Jun-90 1871 1887 1889 -18 0.95 16 Jul-90 1843 1883 1887 -44 2.38 17 Aug-90 1930 1887 1883 47 2.46 18 Sep-90 1633 1862 1887 -254 15.57 19 Oct-90 1913 1867 1862 51 2.67 20 Nov-90 2018 1882 1867 151 7.48 21 Dec-90 1343 1828 1882 -539 40.14 A B C D E F G H I 149 Aug-01 3632 2479 2351 1281 35.28 150 Sep-01 2961 2527 2479 482 16.28 151 Oct-01 3309 2605 2527 782 23.63 152 Nov-01 2605 153 Dec-01 2605 22  Using Solver to Optimize the Smoothing Constant Excel:Tools/Solver A B C D E F G H I 1 Single Exponential Smoothing 2 a1 SSE Sum of square errors MSE 97250.91 3 0.560914062 13712377.79 Target cell for Solver MAPE 11.41 4 Optimized smoothing 5 parameter between 0 and 1 6 7 8 Month Unit (Yt) Level (Lt) Forecast(Ft) Error (Et) |Et/Yt x 100| 9 10 Jan-90 1800 1800 11 Feb-90 1988 1905 1800 188 9.46 12 Mar-90 2096 2012 1905 191 9.09 13 Apr-90 1756 1869 2012 -256 14.60 14 May-90 2350 2139 1869 481 20.49 15 Jun-90 1871 1989 2139 -268 14.30 16 Jul-90 1843 1907 1989 -146 7.89 17 Aug-90 1930 1920 1907 23 1.20 18 Sep-90 1633 1759 1920 -287 17.57 19 Oct-90 1913 1845 1759 154 8.05 20 Nov-90 2018 1942 1845 173 8.56 21 Dec-90 1343 1606 1942 -599 44.62 A B C D E F G H I 149 Aug-01 3632 3212 2677 955 26.31 150 Sep-01 2961 3071 3212 -251 8.49 151 Oct-01 3309 3205 3071 238 7.18 152 Nov-01 3205 153 Dec-01 3205 23  Accuracy Measures While there is a slight improvement in the MSE and MAPE after optimization, both MSE and MAPE are still fairly large However, a plot of the time series with the optimal forecasts shows that the single exponential smoothing does to some extent appear to track the series fairly well Number of Dwelling Units Approved: Vic 4000 3500 3000 2500 Unit (Yt) 2000 Forecast(Ft) 1500 1000 500 0 Dec-94 Dec-99 Aug-91 Feb-94 Aug-96 Feb-99 Aug-01 Oct-90 Oct-95 Oct-00 Jun-92 Jun-97 Apr-93 Apr-98 24 Month  Splitting Into Initialization and Test Sets A B C D E F G H I 1 Single Exponential Smoothing 2 a1 SSE Initialisation Set Initialisation Set: 3 0.1 19783074.17 MSE 151015.8 Jan-90 to Dec-00 4 Smoothing parameter MAPE 15.64 Test Set: 5 between 0 and 1 Sum of square errors Jan 01 to Oct 01 Target cell for Solver 6 Test Set 7 MSE 416847.1 8 Month Unit (Yt) Level (Lt) Forecast(Ft) Error (Et) |Et/Yt x 100| MAPE 20.60 9 10 Jan-90 1800 1800 11 Feb-90 1988 1819 1800 188 9.46 12 Mar-90 2096 1847 1819 277 13.23 13 Apr-90 1756 1837 1847 -91 5.15 14 May-90 2350 1889 1837 513 21.81 15 Jun-90 1871 1887 1889 -18 0.95 16 Jul-90 1843 1883 1887 -44 2.38 A B C D E F G H I 142 Jan-01 1649 2343 -694 42.08 143 Feb-01 2282 2343 -61 2.67 144 Mar-01 2185 2343 -158 7.23 145 Apr-01 1811 2343 -532 29.37 146 May-01 2658 2343 315 11.86 147 Jun-01 2615 2343 272 10.41 148 Jul-01 2817 2343 474 16.83 149 Aug-01 3632 2343 1289 35.49 150 Sep-01 2961 2343 618 20.88 151 Oct-01 3309 2343 966 29.20 Very poor MSE and MAPE for the test set. 25  Using Solver to Optimize the Smoothing Constant A B C D E F G H I 1 Single Exponential Smoothing 2 a1 SSE Initialisation Set Initialisation Set: Sum of square errors 3 0.538732079 11539416.02 MSE 88087.15 Jan-90 to Dec-00 Target cell for Solver 4 MAPE 11.15 Test Set: 5 Jan 01 to Oct 01 Smoothing parameter 6 between 0 and 1 Test Set 7 MSE 884699.6 8 Month Unit (Yt) Level (Lt) Forecast(Ft) Error (Et) |Et/Yt x 100| MAPE 27.08 9 10 Jan-90 1800 1800 11 Feb-90 1988 1901 1800 188 9.46 12 Mar-90 2096 2006 1901 195 9.29 13 Apr-90 1756 1871 2006 -250 14.25 14 May-90 2350 2129 1871 479 20.37 15 Jun-90 1871 1990 2129 -258 13.80 16 Jul-90 1843 1911 1990 -147 7.98 A B C D E F G H I 142 Jan-01 1649 1864 -215 13.04 143 Feb-01 2282 1864 418 18.32 144 Mar-01 2185 1864 321 14.69 145 Apr-01 1811 1864 -53 2.93 146 May-01 2658 1864 794 29.87 147 Jun-01 2615 1864 751 28.72 148 Jul-01 2817 1864 953 33.83 149 Aug-01 3632 1864 1768 48.68 150 Sep-01 2961 1864 1097 37.05 151 Oct-01 3309 1864 1445 43.67 Again very poor MSE and MAPE for the test set. 26  Accuracy Measures While the MSE and MAPE for the initialization set improve with optimization, they become very poor for the test set. Clearly exponential smoothing is not appropriate for forecasting too many steps ahead - also evident from the time plots Number of Dwelling Units Approved: Vic 4000 3500 3000 2500 Unit (Yt) 2000 Forecast(Ft) 1500 1000 500 0 Dec-94 Dec-99 Aug-91 Feb-94 Aug-96 Feb-99 Aug-01 Oct-90 Oct-95 Oct-00 Jun-92 Jun-97 Apr-93 Apr-98 27 Month  Holt’s Linear Method The single exponential smoothing method generally works well if there is no obvious trend in the series. But if there is a trend, then this method lags behind. – For example, if the series is constantly increasing, simple exponential smoothing forecast will be consistently low Holt’s linear model rectifies this by dealing with trend explicitly. 28  Holt’s Linear Method In addition to the level of the series Lt , Holt’s model includes a trend term bt and a corresponding smoothing constant β. The interpretation of Lt is exactly as before - level of the series and it has to be estimated. The interpretation of bt is that it represents an estimate of the change in the series from one period to the next. 29  Holt’s Linear Method for Trend The equations for Holt’s method are Lt  Yt  1   Lt  1  bt  1  (1) bt  Lt  Lt  1  1    bt  1 (2) Ft m  Lt  mbt (3) 30  Holt’s Linear Method Equation 1 says that the updated level is a weighted average of the current observation and the previous level plus the estimated change. Equation 2 says that the updated trend term is a weighted average of the difference between two consecutive levels and the previous trend term. Equation 3 says that a m-period-ahead forecast made in period t is the estimated level plus m times the estimated change per period. 31  Holt’s Linear Method The initialisation process requires two estimates the first level L1 and the trend b1. Use L1 = Y1 Use b1 = Y2 – Y1 or b1 = (Y4 - Y1)/3 Another alternative is to use the least squares regression on the first few values of the series for finding L1 and b1. 32  Holt’s Linear Method Both α and β take on values between 0 and 1 This new smoothing constant β controls how quickly the method reacts to perceived changes in the trend. – If it is small the method reacts slowly – If it is large, the method reacts more quickly What value of α and β should we use? – Some practioners suggest using a small value of α and β (0.1 to 0.2). – Others suggest using an optimization option to select the “best” smoothing constants. Example: Forecasting inventory demand (refer the additional notes). 33  Example: Purchase Value of Housing from all Lenders in Victoria We examine the time series: Purchase Value of Housing from all Lenders in Victoria Oct 1975 – Sept 2001 (Source: ABS) The time series plot shows that there is trend, some cyclical behaviour and much random variation Does a Holt’s linear track these data well? How do the forecasts depend on the smoothing constant ? Purchase Value of Housing from all Lenders in Victoria 90000 80000 70000 60000 50000 40000 30000 20000 10000 Oct-75 Oct-76 Oct-77 Oct-78 Oct-79 Oct-80 Oct-81 Oct-82 Oct-83 Oct-84 Oct-85 Oct-86 Oct-87 Oct-88 Oct-89 Oct-90 Oct-91 Oct-92 Oct-93 Oct-94 Oct-95 Oct-96 Oct-97 Oct-98 Oct-99 Oct-00 34 0 Applying Holt’s Linear method to the time series from Jan 1990 to Sept 2001 A B C D E F G H 1 Holt's Linear trend method 2 a1 b1 SSE 3 0.1 0.1 2.319E+10 Sum of square MSE 165641399.7 errors 4 MAPE 34.09 Target cell for 5 Solver 6 7 8 Smoothing parameter 9 between 0 and 1 Smoothing parameter 10 Change for Solver between 0 and 1 11 Change for Solver 12 13 14 Month Period Value (Yt) Level (Lt) Trend(bt) Forecast(Ft) Error (Et) |Et/Yt x 100| 15 Jan-90 1 18210 18210.00 5322.00 16 Feb-90 2 23532 23532.00 5322.00 23532.00 0.00 0.00 17 Mar-90 3 25725 28541.10 5290.71 28854.00 -3129.00 12.16 18 Apr-90 4 19080 32356.63 5143.19 33831.81 -14751.81 77.32 19 May-90 5 25392 36289.04 5022.11 37499.82 -12107.82 47.68 20 Jun-90 6 19604 39140.44 4805.04 41311.15 -21707.15 110.73 21 Jul-90 7 19645 41515.43 4562.04 43945.48 -24300.48 123.70 A B C D E F G H 154 Aug-01 140 84609 65364.59 234.68 63226.32 21382.68 25.27 155 Sep-01 141 73180 66357.35 310.49 65599.27 7580.73 10.36 156 Oct-01 142 66667.84 157 Nov-01 143 66978.33 158 Dec-01 144 67288.82 35  Using Solver to Optimize the Smoothing Constant Excel:Tools/Solver A B C D E F G H 1 Holt's Linear trend method Sum of square 2 a1 b1 SSE errors 3 0.478363 0.142975 7.103E+09 Target cell for MSE 50735311.62 4 Solver MAPE 14.34 5 6 7 Optimized Smoothing Optimized smoothing 8 parameter between 0 parameter between 0 and 1 9 and 1 10 11 12 13 14 Month Period Value (Yt) Level (Lt) Trend(bt) Forecast(Ft) Error (Et) |Et/Yt x 100| 15 Jan-90 1 18210 18210.00 5322.00 16 Feb-90 2 23532 23532.00 5322.00 23532.00 0.00 0.00 17 Mar-90 3 25725 27357.20 5108.00 28854.00 -3129.00 12.16 18 Apr-90 4 19080 26062.22 4192.53 32465.20 -13385.20 70.15 19 May-90 5 25392 27928.59 3859.94 30254.74 -4862.74 19.15 20 Jun-90 6 19604 25959.90 3026.60 31788.53 -12184.53 62.15 21 Jul-90 7 19645 24517.87 2387.69 28986.50 -9341.50 47.55 A B C D E F G H 154 Aug-01 140 84609 81908.35 2555.69 79431.75 5177.25 6.12 155 Sep-01 141 73180 79066.17 1783.93 84464.04 -11284.04 15.42 156 Oct-01 142 80850.10 157 Nov-01 143 82634.02 158 Dec-01 144 84417.95 36  Accuracy Measures After optimization there is an improvement in the MSE, and MAPE but the values are still quite large. A plot of the time series with the optimal forecasts shows that the Holts Linear does appear to track the series fairly well Purchase Value of Housing from all Lenders in 120000 Victoria 100000 80000 Value (Yt) 60000 Forecast(Ft) 40000 20000 0 Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jul-90 Jul-91 Jul-92 Jul-93 Jul-94 Jul-95 Jul-96 Jul-97 Jul-98 Jul-99 Jul-00 Jul-01 Jul-02 37  Splitting Into Initialization and Test Sets A B C D E F G H I 1 Holt's Linear trend method 2 a1 b1 SSE Sum of square Initialization Set 3 0.1 0.1 1.846E+10 errors MSE 194319245 4 Target cell for MAPE 43.84 Smoothing parameter 5 Initialization Set: between 0 and 1 Solver 6 Jan90 to Dec97 Change for Solver 7 Test Set: Smoothing parameter Test Set between 0 and 1 8 Jan98 to Sept01 MSE 1048472351 Change for Solver 9 MAPE 45.17 10 11 12 13 14 Month Period Value (Yt) Level (Lt) Trend(bt) Forecast(Ft) Error (Et) |Et/Yt x 100| 15 Jan-90 1 18210 18210.00 5322.00 16 Feb-90 2 23532 23532.00 5322.00 23532.00 0.00 0.00 17 Mar-90 3 25725 28541.10 5290.71 28854.00 -3129.00 12.16 18 Apr-90 4 19080 32356.63 5143.19 33831.81 -14751.81 77.32 19 May-90 5 25392 36289.04 5022.11 37499.82 -12107.82 47.68 20 Jun-90 6 19604 39140.44 4805.04 41311.15 -21707.15 110.73 21 Jul-90 7 19645 41515.43 4562.04 43945.48 -24300.48 123.70 A B C D E F G H I 108 Oct-97 94 67866 56353.97 1267.58 55074.85 12791.15 18.85 109 Nov-97 95 60359 57895.29 1294.95 57621.54 2737.46 4.54 110 Dec-97 96 64486 59719.82 1347.91 59190.24 5295.76 8.21 111 Jan-98 97 48230 61067.73 -12837.73 26.62 112 Feb-98 98 56090 62415.64 -6325.64 11.28 113 Mar-98 99 63467 63763.55 -296.55 0.47 114 Apr-98 100 51085 65111.46 -14026.46 27.46 Very poor MSE and MAPE for the test set 38  Using Solver to Optimize the Smoothing Constant A B C D E F G H I 1 Holt's Linear trend method 2 a1 b1 SSE Initialization Set Sum of square 3 0.685597 0.130018 2835394262 errors MSE 29846255.4 4 Target cell for MAPE 14.67 Optimized smoothing 5 parameter between 0 Solver 6 and 1 7 Test Set 8 Initialization Set: Optimized smoothing MSE 998852235 parameter between 0 and 1 9 Jan90 to Dec97 MAPE 45.78 10 Test Set: 11 Jan98 to Sept01 12 13 14 Month Period Value (Yt) Level (Lt) Trend(bt) Forecast(Ft) Error (Et) |Et/Yt x 100| 15 Jan-90 1 18210 18210.00 5322.00 16 Feb-90 2 23532 23532.00 5322.00 23532.00 0.00 0.00 17 Mar-90 3 25725 26708.77 5043.08 28854.00 -3129.00 12.16 18 Apr-90 4 19080 23064.07 3913.52 31751.85 -12671.85 66.41 19 May-90 5 25392 25890.51 3772.18 26977.59 -1585.59 6.24 20 Jun-90 6 19604 22766.48 2875.55 29662.69 -10058.69 51.31 21 Jul-90 7 19645 21530.49 2340.97 25642.03 -5997.03 30.53 A B C D E F G H I 108 Oct-97 94 67866 65874.08 1743.87 61530.43 6335.57 9.34 109 Nov-97 95 60359 62641.24 1096.81 67617.95 -7258.95 12.03 110 Dec-97 96 64486 64250.84 1163.48 63738.05 747.95 1.16 111 Jan-98 97 48230 65414.32 -17184.32 35.63 112 Feb-98 98 56090 66577.81 -10487.81 18.70 113 Mar-98 99 63467 67741.29 -4274.29 6.73 Again, very poor MSE and MAPE for the test set 39  Accuracy Measures While the MSE and MAPE for the initialization set improve with optimization, they become very poor for the test set. Clearly Holt’s linear method is not forecasting future values as well as it does so for known values – This is also evident from the time plots Purchase Value of Housing from all Lenders in Victoria 140000 120000 100000 80000 Value (Yt) Forecast(Ft) 60000 40000 20000 Sep-91 May-93 Sep-96 May-98 Sep-01 Nov-90 Nov-95 Nov-00 Mar-94 Mar-99 Jan-90 Jan-95 Jan-00 Jul-92 Jul-97 0 40  Appendix: Strategy for evaluating any forecasting methodology Stage 1 – The time series of interest is divided into two parts: initialization and test sets, so that an evaluation of a forecasting method can be conducted Stage 2 – A forecasting is chosen from a list of possible methods Stage 3 – The initialization set is used to get the forecasting method started – Estimates of any trend components, seasonal components and parameter values are made at this stage 41  Stage 4 – The method is applied to the test set to see how well it does on data that were not used in estimating the components of the model – After each forecast, the forecasting error is determined over the complete data set and accuracy measures such as MSE, MAPE are determined 42