ANNEX C
(to Recommendation E.506)
Description of a top down procedure
Let
XT be the traffic forecast on an aggregated level,
Xi be the traffic forecast to country i,
eq \o(\s\up4(^),s)T the estimated standard deviation of the aggregated
forecast,
eq \o(\s\up4(^),s)i the estimated standard deviation of the forecast to
country i.
Usually
XT eq \i\su(i, , )!Unexpected End of Expression. Xi, (C-1)
so that it is necessary to find a correction
[X`i] of [Xi] and [X`T] of [XT]
by minimizing the expression
Q = a0(XT - XT`)2 + eq \i\su(i, , )!Unexpected End of Expression.
ai(Xi - Xi`)2 (C-2)
subject to
XT` = i Xi` (C-3)
where a and [ai] are chosen to be
a0 = eq \f(1,\o(\s\up4(^),s)\s(2,T)) and ai = eq \f(1,\
o(\s\up4(^),s)\s(2,i)) i = 1, 2, . . . (C-4)
The solution of the optimization problem gives the values [X`i]:
Xi` = Xi - eq \o(\s\up4(^),s)\s(2,T) \f(\i\su( ,i, ) Xi - XT,\i\su(
,i, ) \o(\s\up4(^),s) + \o(\s\up4(^),s))(C-5)
A closer inspection of the data base may result in other expressions for
the coefficients [ai], i = 0, 1, . . . On some occasions, it will also be
reasonable to use other criteria for finding the corrected forecasting values
[X`i]. This is shown in the top down example in Annex D.
If, on the other hand, the variance of the top forecast XT is fairly
small, the following procedure may be chosen:
The corrections [Xi] are found by minimizing the expression
Q` = eq \i\su(i, , )!Unexpected End of Expression. ai (Xi - Xi`)2 (C-
6)
subject to
XT = eq \i\su(i, , )!Unexpected End of Expression. Xi` (C-7)
If ai, i = 1, 2, . . . is chosen to be the inverse of the estimated
variances, the solution of the optimization problem is given by
Xi` = Xi - eq \o(\s\up4(^),s)\s(2,i) eq \f(\i\su( , , ) Xi -
XT,\i\su( , , )\o(\s\up4(^),s)\s(2,i)) (C-8)
Fascicle II.3 - Rec. E.506 PAGE1