Fascicle II.3 _ Rec. E.506 3 ANNEX B (to Recommendation E.506) Example using weighted least squares method B.1 Telex data The telex traffic between the following countries has been analyzed: _ Germany (D) _ Denmark (DNK) _ USA (USA) _ Finland (FIN) _ Norway (NOR) _ Sweden (S) The data consists of yearly observations from 1973 to 1984 [19]. B.2 Forecasting Before using the weighted least squares method, separate forecasts for the traffic matrix have to be made. In this example a simple ARIMA (0,2,1) model with logarithmic transformed observations without explanatory variables is used for forecasting. It may be possible to develop better forecasting models for the telex traffic between the various countries. However the main point in this example only is to illustrate the use of the weighted least squares technique. Forecasts for 1984 based on observations from 1973 to 1983 are given in Table B_1/E.506. TABLE B_1/E.506 Forecasts for telex traffic between Germany(D), Denmark(DNK), USA(USA), Finland(FIN), Norway(NOR) and Sweden(S) in 1984 From D DNK USA FIN NOR S Sum Forecas ted To sum D _ 486 12 287 239 28 005 27 788 9 630 9 7 523 0 DNK _ 127 10 831 10 805 519 165 751 0 195 6 5 9 USA 11 131 _ 165 17 193 17 009 103 3 719 7 240 1 FIN _ 6496 6458 265 715 741 489 189 5 6 NOR 125 _ 7580 7597 241 5 182 541 154 5 1 8 S 182 179 133 _ 12 063 12 053 482 1 228 8 3 8 3 Sum 26 997 19 668 714 13 197 3 130 8 6 034 Forecast 26 996 19 665 711 12 ed sum 097 7 353 9 0 914 It should be noticed that there is no consistency between row and column sum forecasts and forecasts of the elements in the traffic matrix. For instance, the sum of forecasted outgoing telex traffic from Germany is 28 005, while the forecasted row sum is 27 788. To adjust the forecasts to get consistency and to utilize both row/column forecasts and forecasts of the traffic elements the weighted least squares method is used. B.3 Adjustment of the traffic matrix forecasts To be able to use the weighted least squares method, the weights and the separate forecasts are needed as input. The separate forecasts are found in Table B_2/E.506, while the weights are based on the mean squared one step ahead forecasting errors. Let yt be the traffic at time t. The ARIMA (0,2,1) model with logarithmic transformed data is given by: zt = (1 _ B)2 ln yt = (1 _ qB) at or zt = at _ qat_1 where zt = ln yt _ 2 ln yt_1 + ln yt_2 at is white noise, q is a parameter, B is the backwards shift operator. The mean squared one step ahead forecasting error of zt is: MSQ = S (zt _ z›t_1(1))2 where t_1(1) is the one step ahead forecast. The results of using the weighted least squares method is found in Table B_3/E.506 and show that the factors in Table B_1/E.506 have been adjusted. In this example only minor changes have been performed because of the high conformity in the forecasts of row/column sums and traffic elements. TABLE B_2/E.506 Inverse weights as mean as squared one step ahead forecasting errors of telex traffic (100_4) between Germany(D), Denmark(DNK), USA(USA), Finland(FIN), Norway(NOR) and Sweden(S) in 1984 From D DNK USA FIN NOR S Sum To D _ 7.77 28.72 13.18 11.40 8.29 44.61 DNK _ 10.61 5.91 43.14 18.28 39.99 18.40 USA _ 21.27 23.76 39.19 42.07 50.72 51.55 FIN _ 17.46 23.05 12.15 99.08 34.41 19.96 NOR _ 20.56 21.47 40.16 132.5 24.64 17.15 7 S _ 6.48 6.38 12.95 28.60 28.08 8.76 Sum 6.15 3.85 14.27 9.55 12.94 8.53 TABLE B_3/E.506 Adjusted telex forecasts using the weighted least squares method From D DNK USA FIN NOR S Sum To D _ 4850 12 2858 2383 27 865 684 5090 DNK _ 750 1257 10 825 5185 1674 1959 USA 11 1321 _ 717 1644 17 090 001 2407 FIN 715 _ 487 6471 2633 745 1891 NOR 1258 540 _ 7617 2402 1870 1547 S 1817 1788 1331 _ 12 066 4823 2307 Sum 26 9961 19 6653 7102 12 044 280 894