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