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Recommendation E.506
FORECASTING INTERNATIONAL TRAFFIC1)
1 Introduction
This Recommendation is the first in a series of three Recommendations that
cover international telecommunications forecasting.
In the operation and administration of the international telephone
network, proper and successful development depends to a large degree upon
estimates for the future. Accordingly, for the planning of equipment and circuit
provision and of telephone plant investments, it is necessary that
Administrations forecast the traffic which the network will carry. In view of the
heavy capital investments in the international network, the economic importance
of the most reliable forecast is evident.
The purpose of this Recommendation is to give guidance on some of the
prerequisites for forecasting international telecommunications traffic. Base
data, not only traffic and call data but also economic, social and demographic
data, are of vital importance for forecasting. These data series may be
incomplete; strategies are recommended for dealing with missing data. Different
forecasting approaches are presented including direct and composite methods,
matrix forecasting, and top down and bottom up procedures.
Recommendation E.507 provides guidelines for building forecasting models
and contains an overview of various forecasting techniques. Recommendation E.508
covers the forecasting of new international telecommunications services.
2 Base data for forecasting
An output of the international traffic forecasting process is the
estimated number of circuits required for each period in the forecast horizon. To
obtain these values, traffic engineering techniques are applied to forecast
Erlangs, a measure of traffic. Figure 1/E.506 outlines two different approaches
for determining forecasted Erlangs.
The two different strategies for forecasting are the direct strategy and
the composite strategy. The first step in either process is to collect raw data.
These raw data, perhaps adjusted, will be the base data used to generate the
traffic forecasts. Base data may be hourly, daily, monthly, quarterly, or annual.
Most Administrations use monthly accounting data for forecasting purposes.
With the direct strategy, the traffic carried in Erlangs, or measured
usage, for each relation would be regarded as the base data in forecasting
traffic growth. These data may be adjusted to account for such occurrences as
regeneration (see Recommendation E.500).
Figure 1/E.506 - T0200800-87
In both strategies (direct and composite) it is necessary to convert the
carried traffic into offered traffic Erlangs. The conversion formula can be found
in Recommendation E.501 for the direct strategy and in this Recommendation for
the composite strategy.
Composite forecasting uses historical international accounting data of
monthly paid minute traffic as the base data. The data may be adjusted by a
number of factors, either before or after the forecasting process, that are used
for converting paid minutes on the basis of the accounting data into busy-hour
Erlang forecasts.
As seen in Figure 1/E.506, the forecasting process is common to both the
direct and composite strategy. However, the actual methods or models used in the
process vary. Forecasts can be generated, for example, using traffic matrix
methods (see S 4), econometric models or autoregressive models (see S 3,
Recommendation E.507). There are various other data that are input to the
forecasting process. Examples of these are explanatory variables, market
segmentation information and price elasticities.
Wherever possible, both the direct and composite forecasting strategies
should be used and compared. This comparison may reveal irregularities not
evident from the use of only one method. Where these are significant, in
particular in the case of the busy hour, the causes for the differences should be
identified before the resulting forecast is adopted.
1) The old Recommendation E.506 which appeared in the Red Book was split into two
Recommendations, revised E.506 and new E.507 and considerable new material was added to
both.
Fascicle II.3 - Rec. E.506 PAGE1
In econometric modelling especially, explanatory variables are used to
forecast international traffic. Some of the most important of these variables are
the following:
- exports,
- imports,
- degree of automation,
- quality of service,
- time differences between countries,
- tariffs,
- consumer price index, and
- gross national product.
Other explanatory variables, such as foreign business travellers and
nationals living in other countries, may also be important to consider. It is
recommended that data bases for explanatory variables should be as comprehensive
as possible to provide more information to the forecasting process.
Forecasts may be based on market segmentation. Base data may be segmented,
for example, along regional lines, by business, non-business, or by type of
service. Price elasticities should also be examined, if possible, to quantify the
impact of tariffs on the forecasting data.
3 Composite strategy - Conversion method
The monthly paid-minutes traffic is converted to busy-hour Erlangs for
dimensioning purposes by the application of a number of traffic related
conversion factors for each service category. The conversion is carried out in
accordance with the formula:
A = Mdh/60e (3-1)
where
A is the estimated mean traffic in the busy hour,
M is the monthly paid-minutes,
d is day-to-month ratio,
h is the busy hour-to-day ratio, and
e is the efficiency factor.
The formula is described in detail in Annex A.
4 Procedures for traffic matrix forecasting
4.1 Introduction
To use traffic matrix or point-to-point forecasts the following procedures
may be used:
- Direct point-to-point forecasts,
- Kruithof's method,
- Extension of Kruithof's method,
- Weighted least squares method.
It is also possible to develop a Kalman Filter model for point-to-point
traffic taking into account the aggregated forecasts. Tu and Pack describe such a
model in [16].
The forecasting procedures can be used to produce forecasts of internal
traffic within groups of countries, for example, the Nordic countries. Another
application is to produce forecasts for national traffic on various levels.
4.2 Direct point-to-point forecasts
It is possible to produce better forecasts for accumulated traffic than
forecast of traffic on a lower level.
Hence, forecasts of outgoing traffic (row sum) or incoming traffic (column
sum) between one country and a group of countries will give a relatively higher
precision than the separate forecasts between countries.
In this situation it is possible to adjust the individual forecasts by
taking into account the aggregated forecasts.
On the other hand, if the forecasts of the different elements in the
traffic matrix turn out to be as good as the accumulated forecasts, then it is
not necessary to adjust the forecasts.
Evaluation of the relative precision of forecasts may be carried out by
comparing the ratios s¢(X)/X where X is the forecast and eq \o(\s\up4(^),s)(X)
the estimated forecast error.
4.3 Kruithof's method
Kruithof's method [11] is well known. The method uses the last known
traffic matrix and forecasts of the row and column sum to make forecasts of the
traffic matrix. This is carried out by an efficient iteration procedure.
Kruithof's method does not take into account the change over time in the
PAGE4 Fascicle II.3 - Rec. E.506
point-to-point traffic. Because Kruithof's method only uses the last known
traffic matrix, information on the previous traffic matrices does not contribute
to the forecasts. This would be disadvantageous. Especially when the growth of
the distinct point-to-point traffic varies. Also when the traffic matrices
reflect seasonal data, Kruithof's method may give poor forecasts.
4.4 Extension of Kruithof's method
The traditional Kruithof's method is a projection of the traffic based on
the last known traffic matrix and forecasts of the row and column sums.
It is possible to extend Kruithof's method by taking into account not only
forecasts of the row and column but also forecasts of point-to-point traffic.
Kruithof's method is then used to adjust the point-to-point traffic forecasts to
obtain consistency with the forecasts of row and column sums.
The extended Kruithof's method is superior to the traditional Kruithof's
method and is therefore recommended.
4.5 Weighted least squares method
Weighted least squares method is again an extension of the last method.
Let {Cij}, {Ci.} and {C.j} be forecasts of point-to-point traffic, row sums and
column sums respectively.
The extended Kruithof's method assumes that the row and column sums are
"true" and adjust {Cij} to obtain consistency.
The weighted least squares method [2] is based on the assumption that both
the point-to-point forecasts and the row and column sum forecasts are uncertain.
A reasonable way to solve the problem is to give the various forecasts different
weights.
Let the weighted least squares forecasts be {Dij}. The square sum Q is
defined by:
Q = eq \i\su(ij, ,aij)(Cij - Dij)2+eq \i\su(i, , ) bi(Ci. - Di.)2+eq
\i\su(j, , ) cj(C.j - D.j)2 (4-1)
where {aij}, {bi }, {cj} are chosen constants or weights.
The weighted least squares forecast is found by:
MinQ(Dij)
Dij
Fascicle II.3 - Rec. E.506 PAGE1
subject to
Di. = eq \i\su(j, , ) Dij i = 1, 2, . . .
(4-2)
and
D.j = eq \i\su(i, , ) Dij j = 1, 2, . . .
A natural choice of weights is the inverse of the variance of the
forecasts. One way to find an estimate of the standard deviation of the forecasts
is to perform ex-post forecasting and then calculate the root mean square error.
The properties of this method are analyzed in [14].
5 Top down and bottom up methods
5.1 Choice of model
The object is to produce forecasts for the traffic between countries. For
this to be a sensible procedure, it is necessary that the traffic between the
countries should not be too small, so that the forecasts may be accurate. A
method of this type is usually denoted as "bottom up".
Alternatively, when there is a small amount of traffic between the
countries in question, it is better to start out with forecasting the traffic for
a larger group of countries. These forecasts are often used as a basis for
forecasts for the traffic to each country. This is done by a correction procedure
to be described in more detail below. Methods of this type are called "top down".
The following comments concern the preference of one method over another.
Let sT2 be the variance of the aggregated forecast, and si2 be the variance
of the local forecast No. i and gij be the covariance of the local forecast No. i
and j. If the following inequality is true:
eq \o(\s\up4(^),s)\s(2,T) < eq \i\su(i, , ) eq \o(\s\up
4(^),s)\s(2,i) + eq \i\su(i ¹, , )\I\su( j, , )gij (5-1)
then, in general, it is not recommended to use the bottom up method, but to use
the top down method.
In many situations it is possible to use a more advanced forecasting model
on the aggregated level. Also, the data on an aggregated level may be more
consistent and less influenced by stochastic changes compared to data on a lower
level. Hence, in most cases the inequality stated above will be satisfied for
small countries.
5.2 Bottom up method
As outlined in S 5.1 the bottom up method is defined as a procedure for
making separate forecasts of the traffic between different countries directly. If
the inequality given in S 5.1 is not satisfied, which may be the case for large
countries, it is sufficient to use the bottom up method. Hence, one of the
forecasting models mentioned in Recommendation E.507 can be used to produce
traffic forecasts for different countries.
5.3 Top down procedure
In most cases the top down procedure is recommended for producing
forecasts of international traffic for a small country. In Annex D a detailed
example of such a forecasting procedure is given.
The first step in the procedure is to find a forecasting model on the
aggregated level, which may be a rather sophisticated model. Let XT be the
traffic forecasts on the aggregated level and sT the estimated standard deviation
of the forecasts.
The next step is to develop separate forecasting models of traffic to
different countries. Let Xi be the traffic forecast to the ith country and s¢i
the standard deviation. Now, the separate forecasts [Xi] have to be corrected by
taking into account the aggregated forecasts XT. We know that in general
XT ¹ eq \i\su(i, , ) Xi (5-2)
Let the corrections of [Xi] be [X`i], and the corrected aggregated
forecast then be X`T = S X`i.
The procedure for finding [X`i] is described in Annex C.
6 Forecasting methods when observations are missing
6.1 Introduction
Most forecasting models are based on equally spaced time series. If one
observation or a set of observations are missing, it is necessary either to use
an estimate of missing observations and then use the forecasting model or to
modify the forecasting model.
All smoothing models are applied on equally spaced observations. Also
autoregressive integrated moving average (ARIMA)-models operate on equally spaced
time series, while regression models work on irregularly spaced observations
PAGE4 Fascicle II.3 - Rec. E.506
without modifications.
In the literature it is shown that most forecasting methods can be
formulated as dynamic linear models (DLM). The Kalman Filter is a linear method
to estimate states in a time series which is modelled as a dynamic linear model.
The Kalman Filter introduces a recursive procedure to calculate the forecasts in
a DLM which is optimal in the sense of minimizing the mean squared one step ahead
forecast error. The Kalman Filter also gives an optimal solution in the case of
missing data.
6.2 Adjustment procedure based on comparable observations
In situations when some observations are missing, it may be possible to
use related data for estimating the missing observations. For instance, if
measurements are carried out on a set of trunk groups in the same area, then the
traffic measurements on various trunk groups are correlated, which means that
traffic measurements on a given trunk group to a certain degree explain traffic
measurements on other trunk groups.
When there is high correlation between two time series of traffic
measurements, the relative change in level and trend will be of the same size.
Suppose that a time series xt of equidistant observations from 1 to n has
an inside gap . xt is, for instance, the yearly increase. The gap consists of k
missing observations between r and r + k + 1.
A procedure for estimating the missing observations is given by the
following steps:
i) Examine similar time series to the series with missing observations and
calculate the cross correlation.
ii) Identify time series with high cross correlation at lag zero.
iii) Calculate the growth factor Dr+i between r and r + k of the similar
time series yt:
Dr+i = eq \f( yr+i - yr, yr+k+1 - yr) i = 1, 2, . . . k (6-1)
iv) Estimates of the missing observations are then given by:
eq \o(\s\up4(^,x))!Unexpected End of Expression.r+i = xr + Dr+i
(xr+k+1 - xr) i = 1, 2, . . . k (6-2)
Fascicle II.3 - Rec. E.506 PAGE1
Example
Suppose we want to forecast the time series xt. The series is observed
from 1 to 10, but the observations at time 6, 7 and 8 are missing. However a
related time series yt is measured. The measurements are given in Table 1/506.
TABLE 1/E.506
Measurements of two related time series; one with missing observations
t 1 2 3 4 5 6 7 8 9 10
xt 100 112 125 140 152 - - - 206 221
yt 300 338 380 422 460 496 532 574
PAGE4 Fascicle II.3 - Rec. E.506
622 670
The last known observation of xt before the gap at time 5 is 152, while
the first known observation after the gap at time 9 is 206.
Hence r = 5 and k = 3. The calculation gives:
D6 = eq \f( 496 - 460, 622 - 460) = eq \f( 36, 162)
D7 = eq \f( 532 - 460, 622 - 460) = eq \f( 72, 162)
D8 = eq \f( 574 - 460, 622 - 460) = eq \f( 114, 162)
eq \o(\s\up4(^),x)6 = 152 +eq \f( 36, 162) (206 - 152) = 164
eq \o(\s\up4(^),x)7 = 152 +eq \f( 72, 162) (206 - 152) = 176
eq \o(\s\up4(^),x)8 = 152 +eq \f( 114, 162) (206 - 152) = 190
6.3 Modification of forecasting models
The other possibility for handling missing observations is to extend the
forecasting models with specific procedures. When observations are missing, a
modified procedure, instead of the ordinary forecasting model, is used to
estimate the traffic.
Fascicle II.3 - Rec. E.506 PAGE1
To illustrate such a procedure we look at simple exponential smoothing.
The simple exponential smoothing model is expressed by:
eq \o(\s\up4(^),m)t = (1 - a) yt + aeq \o(\s\up4(^),m)t-1 (6-3)
where
yt is the measured traffic at time t
eq \o(\s\up4(^),m)t is the estimated level at time t
a is the discount factor [and (1 - a) is the smoothing parameter].
Equation (6-3) is a recursive formula. The recursion starts at time 1 and
ends at n if no observation is missing. Then a one step ahead forecast is given
by:
eq \o(\s\up4(^),y)t (1) = eq \o(\s\up4(^),m)t (6-4)
If some observations lying in between 1 and n are missing, then it is
necessary to modify the recursion procedure. Suppose now that y1, y2, . . ., yr,
yr+k +1, yr+k +2, . . ., yn are known and yr+1, yr+2, . . ., yr+k are unknown. n
the time series has a gap consisting of k missing observations.
The following modified forecasting model for simple exponential smoothing
is proposed in Aldrin [2].
(1 - a) yt + a eq \o(\s\up4(^),m)t-1 t = 1, 2, . . . , r
eq \o(\s\up4(^),m)t = (1 - ak) yt + akeq \o(\s\up4(^),m)t t =
r+k+1 (6-5)
(1 - a) yt + a eq \o(\s\up4(^),m)t-1 t = r+k+2, . . . , n
where
ak = eq \f( a,1 + k(1-a)2) (6-6)
By using the (6-5) and (6-6) it is possible to skip the recursive
procedure in the gap between r and
r + k + 1.
In Aldrin [2] similar procedures are proposed for the following
forecasting models:
- Holt's method,
- Double exponential smoothing,
- Discounted least squares method with level and trend,
- Holt-Winters seasonal methods.
Wright [17] and [18] also suggests specific procedures to modify the
smoothing models when observations are missing.
As mentioned in the first paragraph, regression models are invariant of
missing observations. When using the least squares method, all observations are
given the same weight. Hence, missing observations do not affect the estimation
procedure and forecast are made in the usual way.
PAGE4 Fascicle II.3 - Rec. E.506
On the other hand it is necessary to modify ARIMA models when observations
are missing. In the literature several procedures are suggested in the presence
of missing data. The basic idea is to formulate the ARIMA model as a dynamic
linear model. Then the likelihood function is easy to obtain and the parameters
in the model can be estimated recursively. References to work on this field are
Jones [9] and [10], Harvey and Pierse [8], Ansley and Kohn [3] and Aldrin [2].
State space models or dynamic linear models and the Kalman Filter are a
large class of models. Smoothing models, ARIMA models and regression models may
be formulated as dynamic linear models. This is shown, for instance, in Abraham
and Ledolter [1]. Using dynamic linear models and the Kalman Filter the
parameters in the model are estimated in a recursive way. The description is
given, for instance, in Harrison and Stevens [7], Pack and Whitaker [13],
Moreland [12], Szelag [15] and Chemouil and Garnier [6].
In Jones [9] and [10], Barham and Dunstan [4], Harvey and Pierse [8],
Aldrin [2] and Bílviken [5] it is shown how the dynamic linear models and the
Kalman Filter handle missing observations.
Fascicle II.3 - Rec. E.506 PAGE1