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A NEW GRADIENT APPROXIMATION METHOD FOR DYNAMIC ORIGIN-DESTINATION MATRIX
ESTIMATION ON CONGESTED NETWORKS
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Submitted for presentation at the 90 th meeting of the Transportation Research Board, January 23 -27
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2011 and for publication in Transportation Research Record
Rodric Frederix (Corresponding author)
Department of Mechanical Engineering
Section Traffic and Infrastructure
Katholieke Universiteit Leuven
Celestijnenlaan 300A - PO Box 2422
3001 Heverlee, Belgium
Tel. +32 16 329614
Fax. +32 16 322986
[email protected]
Francesco Viti
Department of Mechanical Engineering
Section Traffic and Infrastructure
Katholieke Universiteit Leuven
Celestijnenlaan 300A - PO Box 2422
3001 Heverlee, Belgium
Tel. +32 16 321673
Fax. +32 16 322986
[email protected]
Ruben Corthout
Department of Mechanical Engineering
Section Traffic and Infrastructure
Katholieke Universiteit Leuven
Celestijnenlaan 300A - PO Box 2422
3001 Heverlee, Belgium
Tel. +32 16 321669
Fax. +32 16 322986
[email protected]
Chris M.J. Tampère
Department of Mechanical Engineering
Section Traffic and Infrastructure
Katholieke Universiteit Leuven
Celestijnenlaan 300A - PO Box 2422
3001 Heverlee, Belgium
Tel. +32 16 321673
Fax. +32 16 322986
[email protected]
Word count: 4919 + 6 Tables/Figures = 6419
Submission date: November 9, 2010
R.F REDERIX , F. V ITI , R. C ORTHOUT , C.M.J. T AMPÈRE
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ABSTRACT
In origin-destination (OD) estimation methods the relationship between the link flows and OD flows is
typically approximated by a linear function described by the assignment matrix corresponding with the
current estimate of the OD flows. However, this relationship implicitly assumes the link flows to be
separable, which leads to biased results in congested networks. We suggest the use of a different linear
approximation of the relationship between OD flows and link flows that takes into account that link
flows are non-separable. However, deriving this relationship is cumbersome in terms of computation
time. In the present paper, we propose to use Marginal Computation (MaC), a computationally efficient
method that performs a perturbation analysis using Kinematic Wave Theory principles, to derive this
relationship. The use of MaC for dynamic OD estimation is tested on a study network and on a real
network. In both cases the proposed methodology performs better than traditional OD estimation
approaches, indicating its merit.
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2
1. INTRODUCTION
Daily congestion dynamics on motorway networks originate from variable traffic patterns, which are
generated from specific combinations of origin-destination (OD) flows distributed along the most
convenient route alternatives. The problem is that there can be man y combinations of demand patterns
that result in the same link flow values, thus the problem of estimating the OD matrix from traffic
counts is typically underdetermined and the set of possible solutions usually grows with the size of the
network in consideration, and the travel alternatives available for each OD pair, while it usually
reduces by adding more information sources. Numerous studies on network OD estimation focus on the
underdeterminedness problem and how to solve it. However, in congested networks and for a withinday dynamic context, the underdeterminedness is not the only cause of discrepancy between estimated
and real OD matrices.
Congestion causes the relationship between the link flows and the OD flows to become highly
non-linear in time, mainly because of spatial and temporal dynamics of queues and delays, spillback
and rerouting effects, and corresponding changes in split proportions at the nodes. This non -linearity
makes the problem highly non-convex, and it is possible that the estimated OD matrix converges to a
local minimum that is far from representing the real demand patterns. Notably, errors in this procedure
are carried over when reproducing the observed traffic data, revealing an incorrect estimation and
prediction of the actual traffic states in a network. To avoid local minima , many studies stress the
importance of having a good initial matrix that should not deviate too much from the actual OD matrix.
Very often this is referred to as the OD adjustment process. However, only in rare cases the available
initial OD matrix yields to a reliable adjusted matrix. Moreover, even having a good initial matrix
might not be a sufficient condition for obtaining a good adjusted matrix (see Frederix et al.(1)).
There is a need for a practical OD estimation methodology that could be applicable to heavily
congested networks and that does not necessarily rely on the quality of the initial matrix. Recently,
Frederix et al. (2) conducted a theoretical analysis of dynamic OD estimation in congested networks.
They identified two conditions for unbiased OD estimation. A first condition is that the initial OD
matrix needs to produce traffic patterns with the same congestion state as observed in reality. The
second condition deals with the relationship between link flows and OD flows. There is no closed form
of this relationship, and therefore this relationship is approximated. The second condition states that the
approximation of this relationship needs to account for the s ensitivity of the assignment matrix to the
OD flows. The proposed methodology of Frederix et al. (2) provides a general framework for dynamic
OD estimation in congested networks, but it does not address the problem of high computation inherent
to calculating the sensitivity of the link flows to the OD flows. While the advancement of computer
technologies and simulation techniques enable s the development of very complex models that can
simulate the spatial and temporal properties of queues on netwo rks with increasingly high accuracy,
adopting these more complex and slower models in OD estimation, is at the expense of simplifications
in the sensitivity of the link flows to the OD flows, in order to maintain computation times to a
reasonable extent. Efficient techniques to calculate the sensitivity of the link flows the OD flows are
necessary to allow the applicability of the proposed methodology to other than small, theoretical
networks. The scope of this paper is to apply the methodology described in Frederix et al. (2) on a real
case study. For this aim we propose the method of Marginal Computation (MaC) which enables one to
calculate the sensitivity of the link flows the OD flows in an efficient manner, still allowing the
adoption of rather complex traffic models.
This paper is structured as follows. Section 2 provides an overview of the within-day dynamic
OD estimation methods and the relevant literature on this topic. We summarize a number of
fundamental requirements from Frederix et al. (2) to obtain reliable OD estimates that reproduce the
observed congestion dynamics. The method of MaC that allows one to account for the non-linear
relationship between link and OD flows is presented. Next we apply this new method to a real dataset
in section 3. Finally section 4 provides recommendations, conclusions and describes the future steps of
this research stream.
2. METHODOLOGY
2.1 Problem statement
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For the purpose of our study, reference is made exclusively to past research on demand estimation
methods that use local sensors as input, i.e. we do not consider the adoption of moving sensors such as
floating cars. The common ground is the relationship between any origin -destination flow distributed
on each (used) route alternative and each link flow in the network. We introduce the following notation:
yik is the link flow at link i during time period k
x tj is the flow between OD pair j departing in time period t
X is a vector containing all OD flows x tj
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aik, ,jt is the proportion of xtj that passes link i during time period k
J is the set containing all OD pairs
I is the set containing all links
J i is the set of OD pairs passing link i
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Mathematically speaking, OD estimators can be formulated in a general way as follows:



(1)
where z 1 and z 2 are distance functions with z 1 measuring the similarity between the estimated OD
matrix X * and the elements of the initial matrix X̂ , and z 2 measuring the similarity between the
estimated and observed link counts, respectively Y and Ŷ . Different forms of the optimization
problem (1) can be found in literature. Popular functions are the Maximum Likelihood estimator
(Nguyen (3)) and the Generalized Least Squares (GLS) method (e.g., Cascetta (4)).
By definition, each element yik of the vector of link flows Y satisfies the generic relationship:
k
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
ˆ  z Y(X), Y
ˆ 
X*  arg min  z1 X, X
2


X
Ji
yik    aik, ,jt  X  xtj
t 1 j 1
(2)
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where aik, ,jt are elements of the assignment matrix A, which controls the fraction of flows from any OD
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pair j which uses link i (see also Cascetta (5) for a more extensive discussion on this problem
formulation).
The dynamic OD estimation problem is usually expressed as a bi-level problem. It is solved by
iterating between the lower level, in which a relationship between the OD flows and the link flows is
determined with the help of a Dynamic Network Loading model (DNL) or a Dynamic Traffic
Assignment (DTA) model, and the upper level in which we use this relationship to find OD flows that
better resemble reality when assigned onto the network.
An alternative to the bi-level formulation is to define the OD estimation problem as a
mathematical program with equilibrium con straints (MPEC). In this approach the relationship between
link flows and OD flows is implicitly accounted for by the con straints. Such methods become quite
complex if the incorporated DNL model needs to capture congestion spillback, complex node
interactions, etc. In that case approximations are required when solving the MPEC (see Waller et al.
(6)), and these can have an influence on the modeling of spillback . In Frederix et al. (1) the necessity of
proper spillback modeling for dynamic OD estimation in congested networks is discussed.
Note that it is also possible to derive the relationship between the OD flows and the link flows from
detailed travel time measurements (see Ashok (7)), but in many networks no such data is available in
sufficient size and reliability.
In the remainder of the paper we focus only on the extensively used OD estimation methods
that use a DNL or DTA model for deriving the relationship between OD flows and link flows .
Most OD estimation methods use a linear approximation of the non-linear relationship between
OD flows and link flows in equation (2). Two very popular examples are GLS estimators (e.g. Cascetta
R.F REDERIX , F. V ITI , R. C ORTHOUT , C.M.J. T AMPÈRE
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& Postorino (8); Yang et al. (9)) and state-space models (e.g. Okutani (10); Ashok & Ben-Akiva (11)).
The linear relationship assumed in these approaches is described by the assignment matrix
corresponding with the current OD matrix X * , and can be written down as follows:
yik    aik, ,jt  X*  xtj
k
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Ji
(3)
t 1 j 1
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Since the assignment fraction aik, ,jt is zero for all OD flows that do not pass link i in time interval k
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when X * is assigned, a summation is made over J i instead of J in equation (3). This relationship
therefore assumes that the flow on link i during time interval k cannot be changed by changing one of
the OD flows xtj that does not pass the link in time interval k when X * is assigned. In other words, this
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type of linear relationship assumes that the link flows are separable. This assumption is incompatible
with some typical phenomena in congested networks, such as spillback of congestion, time lags due to
congestion effects, and interdependencies between crossing (or opposing) flows through intersections.
In these cases it is very likely that increasing another OD flow (one that does not pass that time -space
interval) will cause delays somewhere else in the network, hereby altering the amount of flow passing
the link in the considered time interval. The above described error has already been clearly addressed in
past studies (e.g., Yang (12); Lindveld (13); Tavana (14)). In a more recent study of Frederix et al.
(15), the authors of this paper have shown, through synthetic examples, the importance of including a
second term in the calculation of the descent direction in the upper level (as is discussed immediately
hereafter), otherwise the OD estimation is likely to fail in capturing the actual congestion dynamics.
A common manner to specify a linear relationship at a certain point X * is using a Taylor
approximation. We will now make a comparison between the trad itional linear relationship (3) that is
used in the optimization step of OD estimation methods, and the Taylor approximation. The first-order
Taylor approximation of the link flows at point X * has the following form:
Ji
k
y   a
k
i
j 1 t 1
k ,t
i, j
X  x
*
*t
j
J
k

dyik  X* 
dxtj
j 1 t 1
x
t
j
 xtj * 
 k

d
aik, ,jt''  X  x tj'' 
 t'1 j '
k
J k

J
i

   aik, ,jt  X*  x*tj    
j 1 t 1
j 1 t 1
dx tj
x
Ji
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t
j
 x tj * 
(4)
X*
k

d  aik, ,jt''  X  
*t '

   a X  x    x  x   
x j' 
j 1 t 1
j 1 t 1
dxtj
t '1 j 'Ji

X*


Ji
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k
k ,t
i, j
*
t
j
J
k
t
j
t*
j
The difference between equation (3) and (4) is the second term in equation (4). This term incorporates
the sensitivity of the assignment ma trix to changes of the OD flows. In this second term a summation is
made over J, thereby allowing link flows to have non -separable behavior: the flow on link i during time
interval k now also depends on OD flows that do not pass link i when X * is assigned. Note that the
sensitivity of the assignment matrix to c hanges of the OD flows consists of two elements: a sensitivity
of the flow propagation, and a sensitivity of the route choice. The former sensitivity is caused by travel
time delay and congestion spillback, while the latter is caused by rerouting effects. For more details
about this twofold sensitivity, we refer to Frederix et al. (2). Although we acknowledge the importance
of route choice, in the present paper we focus exclusively on the sensitivity of the flow propagation,
and therefore only consider networks without route choice. The effect of route choice will be the
subject of future research.
It is clear that using equation (4) results in a correct calculation of the gradient of goal function
(1), while equation (3) results in a biased approximatio n of the gradient. Although using equation (4)
instead of equation (3) might be theoretically more sound, the feasibility of calculating this second term
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can be questioned. It is possible through finite differences to calculate the sensitivity of the link flows
to the OD flows, but if we have J OD pairs and N departing intervals for these OD pairs, it would
require to run a DNL/DTA model J×N times. Since this is usually not feasible in terms of computation
time, many researchers prefer the use of equation ( 3) to (4). The combination of the difficulty to derive
an exact calculation of equation (4) and the implicit assumption of separable behavior of equation (3)
has also attracted attention towards the use of gradient approximation methods, among which the
Simultaneous Perturbation Stochastic Approximation (SPSA, Spall (16)) has been used extensively in
dynamic OD estimation problems (e.g. Balakrishna & Koutsopoulos (17); Cipriani et al.(18)) since it
allows one to identify a descent direction with significantly lower computational resources than through
explicit calculation. However, because SPSA makes use of a very approximate gradient the
convergence process requires a large number of iterations, eve n on small-sized networks (see Cipriani
et al. (18)). Therefore if it is possible to calculate the gradient in an efficient manner, traditional
gradient-based methods should be preferred.
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3. CASE STUDY
2.2 Solution method
In the present paper we calculate the sensitivity (Jacobian matrix) of the link flows to the OD flows
through finite differences with MaC simulations. MaC performs approximate DNL to reduce
computation time. The basic idea is that a small increase of a single variable (here: OD flows per
departure time interval) has a rather local effect. For example, if an OD flow changes, link flows in the
opposite direction are hardly influenced; the same holds for all simulated OD flow that departed from
the origin prior to the time slice that is currently varied. First a base simulation is performed with a
standard DNL model. The MaC algorithm takes the outcome of this base simulation and only
recalculates the links flows on (and close to) the affected routes while leaving the other variables
unchanged. The procedure is further detailed in Corthout et al. (19). Note that even though MaC
substantially reduces computation time for the Jacobian, the computation effort remains considerable
and may exceed the requirements for online OD estimation. The proposed method is therefore only
suitable for off-line OD estimation purposes.
It is important to realize that the linear relationship, whether given by equation (3) or (4), is a
local (approximated) relationship. By using this relationship in a gradient-based optimization step we
hope to get closer to the solution. Therefore we need to make sure that the local relationship points in
the correct direction, else we might converge to a local optimum that deviates strongly from the actual
solution. When dealing with congestion, it is therefore important to have an initial OD matrix that
produces traffic patterns with the same congestion state as observed at the measurement site. For more
details about the importance of starting with a correct con gestion pattern, we refer to Frederix et al.
(2).To obtain the correct traffic patterns, extra information might be necessary, for instance one can use
information on speeds or link densities to identify the state of the network unambiguously. This
information can then be used to adjust the OD flows in such a way that the same congestion pattern as
observed in reality occurs. For a discussion of this point one can find details in Frederix et al. (20).
The essence of the proposed methodology can be summarized as follows. First of all we
propose the use of a full linear relationship (4) between OD flows and link flows rather than just an
assignment matrix (equation (3)). Secondly we suggest to use an initial OD matrix that produces the
same congestion pattern as is observed in reality. The solution algorithm that will be used in the next
section can be outlined as follows:
Step 1: Adjust the initial OD matrix such that it produces the same congestion pattern as
observed in reality (for more details, see Frederix et al. (20)).
Step 2: Assign the current OD matrix to the network to obtain the base simulation outcome.
Step 3: For every OD flow, use MaC to calculate the sensitivity of all link flows to that OD
flow.
Step 4: Determine the gradient of problem (1) using equation (4).
Step 5: Perform a line search along the direction of the gradient to determine the new estimate
of the OD matrix.
Step 6: If the convergence criterion is met, stop; otherwise go to step 2.
R.F REDERIX , F. V ITI , R. C ORTHOUT , C.M.J. T AMPÈRE
In this section the methodology described in section 2 is applied using the MaC method to
approximately calculate the second term of equation (4). Firstly it is tested on a study network. As a
proof of concept, a comparison is made with traditional OD estimation that make use of equation (3) to
approximate the link-OD flow relationship on one hand, and an OD estimation that uses explicit
simulation to calculate the second term of equation (4) on the other hand. The Link Transmission
Model (Yperman et al. (21)) is used as DNL model in both OD estimation methods, as well as for
determining the outcome of the base simulation that is used in MaC .
Next we compare the traditional OD estimation with the described methodology on a real network to
analyze the feasibility of the approach for practitioners. As mentioned in the previous section we focus
on networks without route choice.
3.1 Study network
The study network is depicted in figure 1(a). It is a simple merge network. Each branch of the merge is
subdivided in four links. Each link has a length of 1 km, a free flow speed of 9 0 km/h, a capacity of
1800 veh/h, and a jam density of 140 veh/km. There are two OD pairs, and the real OD flows are
depicted in figure 1(b).
X
3
X
13
X
12
X
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OD1
Flow (veh./h)
OD2
10
1200
8
4
X
800
X
X
X
1
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T (h)
1.58
2.5
1.58
2.5
8
800
500
7
OD 2
T (h)
5
1
4
0.25
6
2
5
X
OD 1
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3
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2
(a)
FIGURE 1 Study network (a) and its OD flows (b) .
0.25
(b)
When the OD flows increase after 15 minutes a bottleneck activates at the merging point, and a queue
spills back onto the left branch. Note that there is no queue on the right branch. The maximum length of
the queue is indicated in figure 1(a). When both OD flows decrease again after 95 minutes, the queue
starts decreasing, but also the flow in the queue increases because of reduced “competition” of OD2 for
the bottleneck capacity. For the OD estimation synthetic flow measurements from a DNL run with the
correct OD matrix are generated for all nodes indicated with an ‘x’. No detectors are assumed available
on the right branch. However the flow on this branch can easily be deduced using flow conservation
equations.
The effect of a different linear model between the OD flows and the link flows is investigated: an
incorrect linear model (equation (3)) that only uses the assignment matrix for expressing the
relationship (a), and a correct linear model (equation (4)) that also takes the sensitivity of the
assignment matrix into account. This sensitivity can be calculated through explicit simulation (b) or by
using the MaC model (c). In total three different OD estimations are performed:
a) Incorrect linear model
b) Correct linear model, explicit simulation
c) Correct linear model, MaC model
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A gradient-based optimization method is used: the gradient is calculated at the current point, and we
take a step in this direction. Next the gradient at this new point is calculated, and this process repeats
until convergence. The initial OD matrix can be seen in figure 2. This OD matrix produces congestion
at the correct place at the correct moment in time , though with incorrect demand values. The real OD
matrix is also depicted in dashed line s as a point of reference.
Flow (veh./h)
Start OD 1-3
1770
1200
OD 1-3
800
200
0.25
T (h)
1.58
800
2.5
Start OD 2-3
500
OD 2-3
T (h)
40
0.25
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1.58
2.5
FIGURE 2 Initial OD matrix.
The estimated OD matrices are shown in figure 3. It can be seen that in case (a) the OD estimation
process converges to an OD matrix that deviates strongly from the correct OD matrix . Note that in case
b) the exact solution is not fully reached because the problem is underdetermined for OD flows from 2
to 3 in the time period between the start of the congestion spillback a nd the moment that this queue
spills back over the first detector on the left branch. Apart from this temporary effect the estimation is
quite accurate. The difference between the estimated OD matrix using MaC is very small. These
differences are due to approximation errors in the MaC method. This result suggests that the MaC
method is capable of correctly deriving the sensitivity of the link flows to the OD flows.
(a)
(b)
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(c)
FIGURE 3 Estimated OD matrices for case (a), (b) and (c).
3.2 Inner ringway of Antwerp
This network covers the inner ring way around Antwerp, Belgium (see figure 4). Though this is still a
small-sized network (56 links and 39 nodes), it is an interesting example because of the large amount of
congestion that is present in the peak periods. In this study we consider a typical morning peak period
between 05.30h and 10.30h. Both flow and speed measurements from loop detectors are available at the
on- and off-ramps and on some intermediate sections. This data is available on a 5 -minute interval.
(a)
(b)
FIGURE 4 (a) Ring way around Antwerp and (b) xt plot of measured speeds (in km/h) .
We will use the flow measurements to calibrate the OD matrix, and verify the goodness of fit. Even
though speed measurements are not used in the calibration, they are used for validation. Two different
measures of performance are used to quantify the deviation between measured and simulated flows and
speeds, namely the Root Mean Square Error (RMSE) and the Root Mean Square Normalized (RMSN):
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^
n  y  y( x) 
RMSE 
n
RMSN 
9
2
^

n  y  y ( x) 

n 
(5)
2
(6)
^
y
n
3
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The dynamic OD flows are estimated with a 15-minute departure interval. The initial OD matrix is an
existing static OD matrix superimposed by a time profile. Moreover, the original structure was
deliberately altered: all OD flows were halved, and then a selection of OD flows was increased in such
a way that the initial OD matrix produces a congestion pattern similar to reality. The optimization
method of the OD estimation method is again a gradient -based optimization method. An OD estimation
is performed using both the incorrect linear relationship of equation (3) (case a) and the correct linear
relationship of equation (4) (case b). The results are summarized in table 1 and in figure 5.
TABLE 1 Results for the two cases.
Initial OD
RMSE flows (veh/h)
1391
RMSN flows (%)
43
RMSE speed (km/h)
23
RMSN speed (%)
33
OD estimate of case a
662
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FIGURE 5 xt plot of measured speeds for case a and b.
(b)
The OD estimate obtained by using a correct linear relationship (case b) performs better than the other
one (case a) in terms of fit to both the flow and speed measurements. The results in figure 5 suggest
that for specific routes travel time prediction in case(a) will deviate strongly from actual travel times. It
should be pointed out that the RMSE and RMSN values of case (b) seem rather high. This can be
caused by a number of factors, for instance:
 a simplified first-order traffic model has been used (LTM), which uses a fixed speed in free
flow, so independent of the flow. This explains the high deviation from the speed measurements,
even when the congestion pattern of the model is quite similar to reality .
 the OD flows are fixed for 15 minutes, while measurements are available every 5 minutes.
Therefore it is inevitable that certain variations cannot be captured.
R.F REDERIX , F. V ITI , R. C ORTHOUT , C.M.J. T AMPÈRE
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the approximation errors in the MaC model are transferred to errors in the gradient calculation,
which can cause convergence to a local optimum.
4. CONCLUSIONS
This paper aims to provide a practical OD estimation methodology that could be applicable to heavily
congested networks. Traditional OD estimation methods implicitly assume that link flows are separable
with respect to OD flows. In congested networks this assumption leads to biased OD estimates. In
Frederix et al. (2) a theoretical methodology for dynamic OD estimation is pr esented that is applicable
in congested networks. However, this methodology requires the calculation of the sensitivity of the link
flows to the OD flows, which is computationally troublesome. To ensure the methodology to be
applicable on real-sized networks more efficient techniques to calculate this sensitivity are necessary.
This paper introduces the use of Marginal Computation (MaC) for dynamic OD estimation. MaC
performs approximate DNL to reduce computation time. These approximate DNL simulations are used
to calculate the sensitivity through finite differences. The methodology is successfully tested both on a
study network and on a real network. However, computation time still remains an issue. Future research
will focus on further extensions of the MaC method to increase the computational efficiency. Other
interesting research directions include the use of heuristics in the optimization of the goal function to
ensure faster convergence.
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