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1 A NEW GRADIENT APPROXIMATION METHOD FOR DYNAMIC ORIGIN-DESTINATION MATRIX 2 ESTIMATION ON CONGESTED NETWORKS 3 4 Rodric Frederix (Corresponding author) 5 Department of Mechanical Engineering 6 Section Traffic and Infrastructure 7 Katholieke Universiteit Leuven 8 Celestijnenlaan 300A - PO Box 2422 9 3001 Heverlee, Belgium 10 Tel. +32 16 329614 11 Fax. +32 16 322986 12 [email protected] 13 14 Francesco Viti 15 Department of Mechanical Engineering 16 Section Traffic and Infrastructure 17 Katholieke Universiteit Leuven 18 Celestijnenlaan 300A - PO Box 2422 19 3001 Heverlee, Belgium 20 Tel. +32 16 321673 21 Fax. +32 16 322986 22 [email protected] 23 24 Ruben Corthout 25 Department of Mechanical Engineering 26 Section Traffic and Infrastructure 27 Katholieke Universiteit Leuven 28 Celestijnenlaan 300A - PO Box 2422 29 3001 Heverlee, Belgium 30 Tel. +32 16 321669 31 Fax. +32 16 322986 32 [email protected] 33 34 Chris M.J. Tampère 35 Department of Mechanical Engineering 36 Section Traffic and Infrastructure 37 Katholieke Universiteit Leuven 38 Celestijnenlaan 300A - PO Box 2422 39 3001 Heverlee, Belgium 40 Tel. +32 16 321673 41 Fax. +32 16 322986 42 [email protected] 43 44 Word count: 4919 + 6 Tables/Figures = 6419 45 Submission date: November 9, 2010 46 47 Submitted for presentation at the 90 th meeting of the Transportation Research Board, January 23-27 48 2011 and for publication in Transportation Research Record R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 2

1 [1] ABSTRACT 2 In origin-destination (OD) estimation methods the relationship between the link flows and OD flows is 3 typically approximated by a linear function described by the assignment matrix corresponding with the 4 current estimate of the OD flows. However, this relationship implicitly assumes the link flows to be 5 separable, which leads to biased results in congested networks. We suggest the use of a different linear 6 approximation of the relationship between OD flows and link flows that takes into account that link 7 flows are non-separable. However, deriving this relationship is cumbersome in terms of computation 8 time. In the present paper, we propose to use Marginal Computation (MaC), a computationally efficient 9 method that performs a perturbation analysis using Kinematic Wave Theory principles, to derive this 10 relationship. The use of MaC for dynamic OD estimation is tested on a study network and on a real 11 network. In both cases the proposed methodology performs better than traditional OD estimation 12 approaches, indicating its merit. 13 R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 3

1 2 1. INTRODUCTION 3 Daily congestion dynamics on motorway networks originate from variable traffic patterns, which are 4 generated from specific combinations of origin-destination (OD) flows distributed along the most 5 convenient route alternatives. The problem is that there can be many combinations of demand patterns 6 that result in the same link flow values, thus the problem of estimating the OD matrix from traffic 7 counts is typically underdetermined and the set of possible solutions usually grows with the size of the 8 network in consideration, and the travel alternatives available for each OD pair, while it usually reduces 9 by adding more information sources. Numerous studies on network OD estimation focus on the 10 underdeterminedness problem and how to solve it. However, in congested networks and for a within- 11 day dynamic context, the underdeterminedness is not the only cause of discrepancy between estimated 12 and real OD matrices. 13 Congestion causes the relationship between the link flows and the OD flows to become highly 14 non-linear in time, mainly because of spatial and temporal dynamics of queues and delays, spillback 15 and rerouting effects, and corresponding changes in split proportions at the nodes. This non-linearity 16 makes the problem highly non-convex, and it is possible that the estimated OD matrix converges to a 17 local minimum that is far from representing the real demand patterns. Notably, errors in this procedure 18 are carried over when reproducing the observed traffic data, revealing an incorrect estimation and 19 prediction of the actual traffic states in a network. To avoid local minima, many studies stress the 20 importance of having a good initial matrix that should not deviate too much from the actual OD matrix. 21 Very often this is referred to as the OD adjustment process. However, only in rare cases the available 22 initial OD matrix yields to a reliable adjusted matrix. Moreover, even having a good initial matrix 23 might not be a sufficient condition for obtaining a good adjusted matrix (see Frederix et al.( 1)). 24 There is a need for a practical OD estimation methodology that could be applicable to heavily 25 congested networks and that does not necessarily rely on the quality of the initial matrix. Recently, 26 Frederix et al. (2) conducted a theoretical analysis of dynamic OD estimation in congested networks. 27 They identified two conditions for unbiased OD estimation. A first condition is that the initial OD 28 matrix needs to produce traffic patterns with the same congestion state as observed in reality. The 29 second condition deals with the relationship between link flows and OD flows. There is no closed form 30 of this relationship, and therefore this relationship is approximated. The second condition states that the 31 approximation of this relationship needs to account for the sensitivity of the assignment matrix to the 32 OD flows. The proposed methodology of Frederix et al. (2) provides a general framework for dynamic 33 OD estimation in congested networks, but it does not address the problem of high computation inherent 34 to calculating the sensitivity of the link flows to the OD flows. While the advancement of computer 35 technologies and simulation techniques enables the development of very complex models that can 36 simulate the spatial and temporal properties of queues on networks with increasingly high accuracy, 37 adopting these more complex and slower models in OD estimation, is at the expense of simplifications 38 in the sensitivity of the link flows to the OD flows, in order to maintain computation times to a 39 reasonable extent. Efficient techniques to calculate the sensitivity of the link flows the OD flows are 40 necessary to allow the applicability of the proposed methodology to other than small, theoretical 41 networks. The scope of this paper is to apply the methodology described in Frederix et al. ( 2) on a real 42 case study. For this aim we propose the method of Marginal Computation (MaC) which enables one to 43 calculate the sensitivity of the link flows the OD flows in an efficient manner, still allowing the 44 adoption of rather complex traffic models. 45 This paper is structured as follows. Section 2 provides an overview of the within-day dynamic 46 OD estimation methods and the relevant literature on this topic. We summarize a number of 47 fundamental requirements from Frederix et al. (2) to obtain reliable OD estimates that reproduce the 48 observed congestion dynamics. The method of MaC that allows one to account for the non-linear 49 relationship between link and OD flows is presented. Next we apply this new method to a real dataset in 50 section 3. Finally section 4 provides recommendations, conclusions and describes the future steps of 51 this research stream. 52

53 2. METHODOLOGY 54 R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 4

1 2.1 Problem statement 2 For the purpose of our study, reference is made exclusively to past research on demand estimation 3 methods that use local sensors as input, i.e. we do not consider the adoption of moving sensors such as 4 floating cars. The common ground is the relationship between any origin-destination flow distributed on 5 each (used) route alternative and each link flow in the network. We introduce the following notation: 6 7 8 Mathematically speaking, OD estimators can be formulated in a general way as follows: 9 10 (1) 11 12 where z1 and z2 are distance functions with z1 measuring the similarity between the estimated OD matrix 13 and the elements of the initial matrix , and z2 measuring the similarity between the estimated and 14 observed link counts, respectively and . Different forms of the optimization problem (1) can be found 15 in literature. Popular functions are the Maximum Likelihood estimator (Nguyen ( 3)) and the 16 Generalized Least Squares (GLS) method (e.g., Cascetta (4)). 17 By definition, each element of the vector of link flows satisfies the generic relationship: 18 19 (2)

20 21 where are elements of the assignment matrix A, which controls the fraction of flows from any OD pair 22 j which uses link i (see also Cascetta (5) for a more extensive discussion on this problem formulation). 23 The dynamic OD estimation problem is usually expressed as a bi-level problem. It is solved by 24 iterating between the lower level, in which a relationship between the OD flows and the link flows is 25 determined with the help of a Dynamic Network Loading model (DNL) or a Dynamic Traffic 26 Assignment (DTA) model, and the upper level in which we use this relationship to find OD flows that 27 better resemble reality when assigned onto the network. 28 An alternative to the bi-level formulation is to define the OD estimation problem as a 29 mathematical program with equilibrium constraints (MPEC). In this approach the relationship between 30 link flows and OD flows is implicitly accounted for by the constraints. Such methods become quite 31 complex if the incorporated DNL model needs to capture congestion spillback, complex node 32 interactions, etc. In that case approximations are required when solving the MPEC (see Waller et al. 33 (6)), and these can have an influence on the modeling of spillback. In Frederix et al. ( 1) the necessity of 34 proper spillback modeling for dynamic OD estimation in congested networks is discussed. 35 Note that it is also possible to derive the relationship between the OD flows and the link flows from 36 detailed travel time measurements (see Ashok (7)), but in many networks no such data is available in 37 sufficient size and reliability. 38 In the remainder of the paper we focus only on the extensively used OD estimation methods 39 that use a DNL or DTA model for deriving the relationship between OD flows and link flows. 40 Most OD estimation methods use a linear approximation of the non-linear relationship between 41 OD flows and link flows in equation (2). Two very popular examples are GLS estimators (e.g. Cascetta 42 & Postorino (8); Yang et al. (9)) and state-space models (e.g. Okutani (10); Ashok & Ben-Akiva (11)). 43 The linear relationship assumed in these approaches is described by the assignment matrix 44 corresponding with the current OD matrix , and can be written down as follows: 45 46 (3) 47 48 Since the assignment fraction is zero for all OD flows that do not pass link i in time interval k when is

49 assigned, a summation is made over Ji instead of J in equation (3). This relationship therefore assumes 50 that the flow on link i during time interval k cannot be changed by changing one of the OD flows that 51 does not pass the link in time interval k when is assigned. In other words, this type of linear

R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 5

1 relationship assumes that the link flows are separable. This assumption is incompatible with some

2 typical phenomena in congested networks, such as spillback of congestion, time lags due to congestion 3 effects, and interdependencies between crossing (or opposing) flows through intersections. In these 4 cases it is very likely that increasing another OD flow (one that does not pass that time-space interval) 5 will cause delays somewhere else in the network, hereby altering the amount of flow passing the link in 6 the considered time interval. The above described error has already been clearly addressed in past 7 studies (e.g., Yang (12); Lindveld (13); Tavana (14)). In a more recent study of Frederix et al. ( 15), the 8 authors of this paper have shown, through synthetic examples, the importance of including a second 9 term in the calculation of the descent direction in the upper level (as is discussed immediately 10 hereafter), otherwise the OD estimation is likely to fail in capturing the actual congestion dynamics. 11 A common manner to specify a linear relationship at a certain point is using a Taylor 12 approximation. We will now make a comparison between the traditional linear relationship (3) that is 13 used in the optimization step of OD estimation methods, and the Taylor approximation. The first-order 14 Taylor approximation of the link flows at point has the following form: 15 16 (4) 17 18 The difference between equation (3) and (4) is the second term in equation (4). This term incorporates 19 the sensitivity of the assignment matrix to changes of the OD flows. In this second term a summation is 20 made over J, thereby allowing link flows to have non-separable behavior: the flow on link i during time 21 interval k now also depends on OD flows that do not pass link i when is assigned. Note that the

22 sensitivity of the assignment matrix to changes of the OD flows consists of two elements: a sensitivity 23 of the flow propagation, and a sensitivity of the route choice. The former sensitivity is caused by travel 24 time delay and congestion spillback, while the latter is caused by rerouting effects. For more details 25 about this twofold sensitivity, we refer to Frederix et al. (2). Although we acknowledge the importance 26 of route choice, in the present paper we focus exclusively on the sensitivity of the flow propagation, 27 and therefore only consider networks without route choice. The effect of route choice will be the 28 subject of future research. 29 It is clear that using equation (4) results in a correct calculation of the gradient of goal function 30 (1), while equation (3) results in a biased approximation of the gradient. Although using equation (4) 31 instead of equation (3) might be theoretically more sound, the feasibility of calculating this second term 32 can be questioned. It is possible through finite differences to calculate the sensitivity of the link flows 33 to the OD flows, but if we have J OD pairs and N departing intervals for these OD pairs, it would 34 require to run a DNL/DTA model J×N times. Since this is usually not feasible in terms of computation 35 time, many researchers prefer the use of equation (3) to (4). The combination of the difficulty to derive 36 an exact calculation of equation (4) and the implicit assumption of separable behavior of equation (3) 37 has also attracted attention towards the use of gradient approximation methods, among which the 38 Simultaneous Perturbation Stochastic Approximation (SPSA, Spall (16)) has been used extensively in 39 dynamic OD estimation problems (e.g. Balakrishna & Koutsopoulos (17); Cipriani et al.(18)) since it 40 allows one to identify a descent direction with significantly lower computational resources than through 41 explicit calculation. However, because SPSA makes use of a very approximate gradient the convergence 42 process requires a large number of iterations, even on small-sized networks (see Cipriani et al. (18)). 43 Therefore if it is possible to calculate the gradient in an efficient manner, traditional gradient-based 44 methods should be preferred. 45 46 2.2 Solution method 47 In the present paper we calculate the sensitivity (Jacobian matrix) of the link flows to the OD flows 48 through finite differences with MaC simulations. MaC performs approximate DNL to reduce 49 computation time. The basic idea is that a small increase of a single variable (here: OD flows per 50 departure time interval) has a rather local effect. For example, if an OD flow changes, link flows in the 51 opposite direction are hardly influenced; the same holds for all simulated OD flow that departed from 52 the origin prior to the time slice that is currently varied. First a base simulation is performed with a 53 standard DNL model. The MaC algorithm takes the outcome of this base simulation and only 54 recalculates the links flows on (and close to) the affected routes while leaving the other variables R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 6

1 unchanged. The procedure is further detailed in Corthout et al. (19). Note that even though MaC 2 substantially reduces computation time for the Jacobian, the computation effort remains considerable 3 and may exceed the requirements for online OD estimation. The proposed method is therefore only 4 suitable for off-line OD estimation purposes. 5 It is important to realize that the linear relationship, whether given by equation (3) or (4), is a 6 local (approximated) relationship. By using this relationship in a gradient-based optimization step we 7 hope to get closer to the solution. Therefore we need to make sure that the local relationship points in 8 the correct direction, else we might converge to a local optimum that deviates strongly from the actual 9 solution. When dealing with congestion, it is therefore important to have an initial OD matrix that 10 produces traffic patterns with the same congestion state as observed at the measurement site. For more 11 details about the importance of starting with a correct congestion pattern, we refer to Frederix et al. 12 (2).To obtain the correct traffic patterns, extra information might be necessary, for instance one can use 13 information on speeds or link densities to identify the state of the network unambiguously. This 14 information can then be used to adjust the OD flows in such a way that the same congestion pattern as 15 observed in reality occurs. For a discussion of this point one can find details in Frederix et al. ( 20). 16 The essence of the proposed methodology can be summarized as follows. First of all we 17 propose the use of a full linear relationship (4) between OD flows and link flows rather than just an 18 assignment matrix (equation (3)). Secondly we suggest to use an initial OD matrix that produces the 19 same congestion pattern as is observed in reality. The solution algorithm that will be used in the next 20 section can be outlined as follows: 21 Step 1: Adjust the initial OD matrix such that it produces the same congestion pattern as 22 observed in reality (for more details, see Frederix et al. (20)). 23 Step 2: Assign the current OD matrix to the network to obtain the base simulation outcome. 24 Step 3: For every OD flow, use MaC to calculate the sensitivity of all link flows to that OD 25 flow. 26 Step 4: Determine the gradient of problem (1) using equation (4). 27 Step 5: Perform a line search along the direction of the gradient to determine the new estimate 28 of the OD matrix. 29 Step 6: If the convergence criterion is met, stop; otherwise go to step 2. 30

31 3. CASE STUDY 32 In this section the methodology described in section 2 is applied using the MaC method to 33 approximately calculate the second term of equation (4). Firstly it is tested on a study network. As a 34 proof of concept, a comparison is made with traditional OD estimation that make use of equation (3) to 35 approximate the link-OD flow relationship on one hand, and an OD estimation that uses explicit 36 simulation to calculate the second term of equation (4) on the other hand. The Link Transmission Model 37 (Yperman et al. (21)) is used as DNL model in both OD estimation methods, as well as for determining 38 the outcome of the base simulation that is used in MaC. 39 Next we compare the traditional OD estimation with the described methodology on a real network to 40 analyze the feasibility of the approach for practitioners. As mentioned in the previous section we focus 41 on networks without route choice. 42 43 3.1 Study network 44 The study network is depicted in figure 1(a). It is a simple merge network. Each branch of the merge is 45 subdivided in four links. Each link has a length of 1 km, a free flow speed of 90 km/h, a capacity of 46 1800 veh/h, and a jam density of 140 veh/km. There are two OD pairs, and the real OD flows are 47 depicted in figure 1(b). 48 49 50 (a) (b) 51 FIGURE 1 Study network (a) and its OD flows (b). 52 53 When the OD flows increase after 15 minutes a bottleneck activates at the merging point, and a queue 54 spills back onto the left branch. Note that there is no queue on the right branch. The maximum length of R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 7

1 the queue is indicated in figure 1(a). When both OD flows decrease again after 95 minutes, the queue 2 starts decreasing, but also the flow in the queue increases because of reduced “competition” of OD2 for 3 the bottleneck capacity. For the OD estimation synthetic flow measurements from a DNL run with the 4 correct OD matrix are generated for all nodes indicated with an ‘x’. No detectors are assumed available 5 on the right branch. However the flow on this branch can easily be deduced using flow conservation 6 equations. 7 The effect of a different linear model between the OD flows and the link flows is investigated: an 8 incorrect linear model (equation (3)) that only uses the assignment matrix for expressing the 9 relationship (a), and a correct linear model (equation (4)) that also takes the sensitivity of the 10 assignment matrix into account. This sensitivity can be calculated through explicit simulation (b) or by 11 using the MaC model (c). In total three different OD estimations are performed: 12 a) Incorrect linear model 13 b) Correct linear model, explicit simulation 14 c) Correct linear model, MaC model 15 A gradient-based optimization method is used: the gradient is calculated at the current point, and we 16 take a step in this direction. Next the gradient at this new point is calculated, and this process repeats 17 until convergence. The initial OD matrix can be seen in figure 2. This OD matrix produces congestion 18 at the correct place at the correct moment in time, though with incorrect demand values. The real OD 19 matrix is also depicted in dashed lines as a point of reference. 20 21 22 FIGURE 2 Initial OD matrix. 23 24 The estimated OD matrices are shown in figure 3. It can be seen that in case (a) the OD estimation 25 process converges to an OD matrix that deviates strongly from the correct OD matrix . Note that in case 26 b) the exact solution is not fully reached because the problem is underdetermined for OD flows from 2 27 to 3 in the time period between the start of the congestion spillback and the moment that this queue 28 spills back over the first detector on the left branch. Apart from this temporary effect the estimation is 29 quite accurate. The difference between the estimated OD matrix using MaC is very small. These 30 differences are due to approximation errors in the MaC method. This result suggests that the MaC 31 method is capable of correctly deriving the sensitivity of the link flows to the OD flows. 32

33 34 (a) (b) 35 R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 8

1 2 (c) 3 FIGURE 3 Estimated OD matrices for case (a), (b) and (c). 4 5 3.2 Inner ringway of Antwerp 6 This network covers the inner ring way around Antwerp, Belgium (see figure 4). Though this is still a 7 small-sized network (56 links and 39 nodes), it is an interesting example because of the large amount of 8 congestion that is present in the peak periods. In this study we consider a typical morning peak period 9 between 05.30h and 10.30h. Both flow and speed measurements from loop detectors are available at the 10 on- and off-ramps and on some intermediate sections. This data is available on a 5-minute interval.

11 12 (a) (b) 13 FIGURE 4 (a) Ring way around Antwerp and (b) xt plot of measured speeds (in km/h). 14 15 We will use the flow measurements to calibrate the OD matrix, and verify the goodness of fit. Even 16 though speed measurements are not used in the calibration, they are used for validation. Two different 17 measures of performance are used to quantify the deviation between measured and simulated flows and 18 speeds, namely the Root Mean Square Error (RMSE) and the Root Mean Square Normalized (RMSN): 19 20 (5) 21 (6) 22 R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 9

1 The dynamic OD flows are estimated with a 15-minute departure interval. The initial OD matrix is an 2 existing static OD matrix superimposed by a time profile. Moreover, the original structure was 3 deliberately altered: all OD flows were halved, and then a selection of OD flows was increased in such 4 a way that the initial OD matrix produces a congestion pattern similar to reality. The optimization 5 method of the OD estimation method is again a gradient-based optimization method. An OD estimation 6 is performed using both the incorrect linear relationship of equation (3) (case a) and the correct linear 7 relationship of equation (4) (case b). The results are summarized in table 1 and in figure 5. 8 9 TABLE 1 Results for the two cases. Initial OD OD estimate of case a OD estimate of case b RMSE flows (veh/h) 1391 662 440 RMSN flows (%) 43 21 14 RMSE speed (km/h) 23 27 20 RMSN speed (%) 33 39 29 10

11 12 (a) (b) 13 FIGURE 5 xt plot of measured speeds for case a and b. 14 15 The OD estimate obtained by using a correct linear relationship (case b) performs better than the other 16 one (case a) in terms of fit to both the flow and speed measurements. The results in figure 5 suggest 17 that for specific routes travel time prediction in case(a) will deviate strongly from actual travel times. It 18 should be pointed out that the RMSE and RMSN values of case (b) seem rather high. This can be 19 caused by a number of factors, for instance: 20  a simplified first-order traffic model has been used (LTM), which uses a fixed speed in free 21 flow, so independent of the flow. This explains the high deviation from the speed measurements, 22 even when the congestion pattern of the model is quite similar to reality. 23  the OD flows are fixed for 15 minutes, while measurements are available every 5 minutes. 24 Therefore it is inevitable that certain variations cannot be captured. 25  the approximation errors in the MaC model are transferred to errors in the gradient calculation, 26 which can cause convergence to a local optimum. 27

28 4. CONCLUSIONS 29 This paper aims to provide a practical OD estimation methodology that could be applicable to heavily 30 congested networks. Traditional OD estimation methods implicitly assume that link flows are separable 31 with respect to OD flows. In congested networks this assumption leads to biased OD estimates. In 32 Frederix et al. (2) a theoretical methodology for dynamic OD estimation is presented that is applicable 33 in congested networks. However, this methodology requires the calculation of the sensitivity of the link 34 flows to the OD flows, which is computationally troublesome. To ensure the methodology to be 35 applicable on real-sized networks more efficient techniques to calculate this sensitivity are necessary. R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 10

1 This paper introduces the use of Marginal Computation (MaC) for dynamic OD estimation. MaC 2 performs approximate DNL to reduce computation time. These approximate DNL simulations are used 3 to calculate the sensitivity through finite differences. The methodology is successfully tested both on a 4 study network and on a real network. However, computation time still remains an issue. Future research 5 will focus on further extensions of the MaC method to increase the computational efficiency. Other 6 interesting research directions include the use of heuristics in the optimization of the goal function to 7 ensure faster convergence. 8 9 References 10 (1) Frederix R., Tampère C.M.J., Viti F., Immers, L.H. (2010). The effect of Dynamic Network Loading models on 11 DTA-based OD estimation. In Tampère C.M.J., Viti F., Immers L.H. (eds): New Developments in Transport 12 Planning: Advances in Dynamic Traffic Assignment. Edward Elgar, Cheltenham UK and Northampton, MA, 13 USA. 14 (2) Frederix R., Tampère C.M.J., Viti F. (2010). Dynamic origin-destination estimation in congested networks. In 15 Proceedings of the 3rd Dynamic Traffic Assignment Symposium, July 2010, Takayama, Japan. 16 (3) Nguyen S. (1977). Estimating an OD Matrix from network data: a network equilibrium approach. Publication 17 No. 60, Centre de Recherche sur les Transports, Universite de Montreal, Quebec. 18 (4) Cascetta E. (1984). Estimation of trip matrices from traffic counts and survey data: a generalized least squares 19 approach. Transportation Research Part B, 18, 289-299. 20 (5) Cascetta E. (2001). Transportation systems engineering: theory and methods, Kluwer Academic Publishers, the 21 Netherlands. 22 (6) Waller S.T., Kumar R., Nezamuddin N. (2010). A Graph Theoretical Combinatorial Algorithm and Dual 23 Approximation Scheme for Large-Scale Dynamic Traffic Assignment Calibration Problems. In Proceedings of 24 the 3rd Dynamic Traffic Assignment Symposium, July 2010, Takayama, Japan. 25 (7) Ashok K. (1996). Estimation and prediction of time-dependent origin-destination flows. Ph.D. dissertation, 26 Center for Transportation Studies, Massachusetts Institute of Technology,Cambridge, MA. 27 (8) Cascetta, E & Postorino, MN (2001). ‘Fixed point approaches to the estimation of O/D matrices 28 using traffic counts on congested networks’, Transportation Science, vol. 35, no. 1, pp. 134-147. 29 (9) Yang, H, Meng, Q, & Bell MGH (2001). ‘Simultaneous estimation of the origin–destination 30 matrices and travel-cost coefficient for congested network in a stochastic user equilibrium’, 31 Transportation Science vol. 35, pp 107–123. 32 (10) Okutani, I (1987). ‘The Kalman filtering approach in some transportation and traffic problems’, 33 N. H. Gartner and NHM Wilson, eds. Transportation and Traffic Theory. Elsevier, New York, pp. 34 397–416. 35 (11) Ashok, K, & Ben-Akiva, ME (2002). ‘Estimation and prediction of time-dependent origin– 36 destination flows with a stochastic mapping to path flows and link flows’, Transportation Science 37 vol. 36, pp 184–198. 38 (12) Yang H., (1995). Heuristic algorithms for the bilevel origin–destination matrix estimation problem, 39 Transportation Research B 29, 231–242. 40 (13) Lindveld K., (2003). Dynamic O-D Matrix estimation: a behavioral approach. Ph.D. thesis, Delft University of 41 Technology. 42 (14) Tavana H., (2001). Internally-consistent estimation of dynamic network origin–destination flows from 43 intelligent transportation systems data using bi-level optimization’, Ph.D. thesis, Department of Civil 44 Engineering, University of Texas at Austin, Austin, Texas. 45 (15) Frederix R., Viti F., Tampère C.M.J., (2010). How important is capturing congestion dynamics in dynamic OD 46 estimation? Presented at the 2nd Traffic Flow Theory Committee Summer Meeting, July 2010, Annecy, France. 47 (16) Spall J.C., (1998). An Overview of the simultaneous perturbation method for efficient optimization. Johns 48 Hopkins University APL Technical Digest 19(4), 482-492. 49 (17) Balakrishna, R., Koutsopoulos H.N., (2008). Incorporating within-day transitions in the simultaneous off-line 50 estimation of dynamic origin-destination flows without assignment matrices. Proceedings of the 87th TRB 51 Meeting, Washington DC. 52 (18) Cipriani E., Florian M., Mahut M. and Nigro M., (2010). Investigating the efficiency of a gradient 53 approximation approach for solution of dynamic demand estimation problem. In Tampère C.M.J., Viti F., 54 Immers L.H. (eds): New Developments in Transport Planning: Advances in Dynamic Traffic Assignment, 55 Edward Elgar, Cheltenham UK and Northampton, MA, USA. R.FREDERIX, F. VITI, R. CORTHOUT, C.M.J. TAMPÈRE 11

1 (19) Corthout, R, Tampère CMJ, Frederix R, Immers LH (2011). ‘Marginal Dynamic Network 2 Loading for Large-scale Simulation-based Applications’. Submitted to the 90th annual meeting of 3 the Transportation Research Board, Washington, USA. 4 (20) Frederix R., Viti F., Tampère C.M.J. (2010). A density-based dynamic OD estimation method that reproduces 5 within-day congestion dynamics. Paper submitted for the IEEE-ITSC 2010 conference, September 19-21, 6 Madeira Island, Portugal. 7 (21) Yperman, I, Logghe, S, Tampere, CMJ & Immers, LH (2006). ‘The multi-commodity link 8 transmission model for dynamic network loading’, presented at the 85th annual meeting of the 9 Transportation Research Board, Washington, USA.