Supplementary Technical Details and Results

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1 Supplementary Technical Details and Results April 6, Introduction This document provides additional details to augment the paper Efficient Calibration Techniques for Large-scale Traffic Simulators. In what follows, we provide sections that include additional technical details on the implementation of experiments, specifications on the simulation model, and additional analysis on the case studies. Section 2 provides additional technical details on implementation of the analytical traffic model, the simulation model, and the Simulation-based Optimization (SO) algorithm. Section 3 provides additional analysis for the toy and Berlin network case studies. Section 4 provides a detailed description of the route choice model used in the simulator, MATSim. 2 Implementation Details 2.1 Estimation of the exogenous parameters of the analytical traffic model We solve the analytical traffic model to get the analytical approximation of link flows λ i (θ). In order to solve the analytical traffic model, we first need to estimate the exogenous parameters. The exogenous parameters of the system of nonlinear differentiable equations are as follows. d s expected travel demand for OD pair s; µ i service rate of queue i; l i space capacity of queue i; l i length of link i (road length); v i maximum speed of link i; S set of OD pairs; Q set of queues; R set of routes; R s set of routes of OD pair s; G ij set of routes that consecutively go through queues i and j; H i set of routes that go through queue i; T i set of routes that start with queue i; Ψ r set of links of route r. In general, the exogenous parameters of the analytical traffic model are inferred mainly based on the input to the simulation counterpart, MATSim [2]. The input to the simulation model consists of: (1) configuration. (2) network which describes the simulation network infrastructure 1

2 in terms of links and nodes. In the network, the attributes associated with links and nodes are specified such as link length, maximum speed, and flow capacity, etc. (3) plans/population which describes the demand to the simulation model in terms of a list of day plans. A day plan is a chained activities and trips which a traveler intend to go through for a whole day. Each trip is a sequence of links which has its first link as origin and the last link as destination. A time stamp is attached to the trip as well. The multiple realizations of trips between a given OD pair make the route choice set. For link i, exogenous parameters such as link length l i, maximum speed v i, service rate µ i as the flow capacity can be obtained directly from the network input to the simulation model. The space capacity l i of queue i is approximated by the following equation: l i = l i /d, (1) where d is the average distance between the front bumpers of two successive vehicles which is set to 7.5 meters. This value is adopted from the effective cell size defined in the simulator, MATSim. The fraction is rounded up to the nearest integer. The expected travel demand d s for OD pair s S is estimated based on the day plans of all travelers of the simulation model. We first extract all trips being selected (there is only one trip in use out of the multiple alternative routes in every assignment iteration) which depart within a given period (i.e., the time step we are interested in, for example, 8-9 am), then obtain the OD pairs for all these trips, and at last summarize the frequencies of all unique OD pairs. The frequency of a specific OD pair s represents the number of travelers traveled during the given period which is the expected travel demand d s in the unit of vehicles per time period (e.g., vehicles/hour). The set S consists of all unique OD pairs. For the simulation model, the set of route alternatives for a given OD pair is endogenous and the number of alternative routes for a given OD pair is fixed (i.e., 10). For the purpose of tractability, a route choice set of fixed size is synthesized for the analytical traffic model. Specifically, we synthesize a route choice set R s of fixed size (i.e., 10) for any given OD pair s S. This resulting route choice set is exogenous to the analytical traffic model. For details regarding the generation of route choice set, please refer to Section 2.3. The pool of all route choice sets for all OD pair s S constitutes the set of routes R. For each route r R, the links on this route constitutes the set of links Ψ r of route r. All links involved in these trips consists the set of queues Q. The set of routes T i that start with queue i is obtained by find the subset of routes from the set of routes R which have link i as their first link. For all routes in R which depart during the given period, we filter out a subset of routes which contain link i. This subset of routes constitutes set H i which is the set of routes go through queue i. Similarly, G ij which is the set of routes that consecutively go through links i and j can be formed by finding out the subset of routes which have link i and link j consecutively. 2.2 Feasible region of behavioral parameter θ The behavioral parameter θ represents the travelers sensitivity to travel time. The behavioral parameter is negative and upper bounded by zero since additional travel time decreases the 2

3 utility of traveling. The lower bound of the behavioral parameter depends on the network and is calculated using Equation (2) which was proposed in [1]. θ L = ln2 (1 ξ)ν sp, (2) where ξ is a constant which is greater than 1. ν sp is the shortest route cost among all OD pairs. Equation (2) defines a scale parameter θ L such that the probability of choosing a route of cost ξν sp is half the probability that the shortest path being chosen. By applying this method with ξ = 1.042, the lower bound for the behavioral parameter θ for the toy network and the Berlin network is calculated to be 60 for travel time in the unit of hours. Therefore, we obtain the feasible region Θ for the behavioral parameter θ as [ 60, 0]. The resulting feasible region for the behavioral parameter not only contains the pre-calibrated behavioral parameter value (i.e., -6) defined in the simulator, MATSim, but also retains a relatively large region to take possible local variance of the behavioral parameter into consideration. 2.3 Route choice set of the analytical traffic model The simulator, MATSim, makes route choice from a choice set of fixed size (i.e., 10). Each route in a choice set is assigned the score (i.e., utility) which it was realized last time. In each assignment iteration of the simulation, most of the travelers (i.e., 83.3%) make route choice according to the Multinomial Logit (MNL) model based on the scores of routes, and a small portion of travelers (i.e., 16.7%) generates new routes and eliminates one of the current routes with worst score. The travelers who generate new route are randomly sampled in each assignment iteration. Therefore, the route choice set is endogenous which varies with behavioral parameter θ and across assignment iterations. As a matter of convenience, we consider an exogenous route choice set for the analytical traffic model. To keep consistent with the simulator, we derive a route choice set of fixed size (i.e., 10) for each OD pair. The route choice set for the analytical traffic model does not vary with behavioral parameter θ and across iterations. However, a desired route choice set should take into account the variability of route choice sets in the simulator. We propose an algorithm for generating the route choice set for the analytical traffic model. By running simulation until convergence with multiple different behavioral parameters, we obtain a route choice set from the last assignment iteration of each of these experiments. These route choice sets are different from each other due to different behavioral parameters. We put the route choice sets associated with different behavioral parameters for the same OD pair s together. The route choice set R s for the analytical traffic model for a given OD pair s is derived by selecting the most representative routes of the fixed size (i.e., 10) from the pool of simulated route choice sets associated with different behavioral parameters. The most representative route choice set for a given OD pair s is the one which has the maximum overlap with any of the simulated route choice sets for the OD pair s. The overlap of two route choice sets are measured by the proportion of total distance of common unique links of these two route choice sets. The resulting route choice set R for all OD pairs is provided as exogenous input for the analytical 3

4 traffic model. 2.4 Initialization of the endogenous variables of the analytical traffic model The system of nonlinear equations of the analytical traffic model is solved iteratively. For each iteration of the solution process, we first solve the assignment model given fixed network conditions, and then update the network conditions given a fixed assignment. Therefore, we need to estimate the expected link travel time t i of link i at the very beginning of the iterative solution process. The expected link travel time t i is initialized as the free flow travel time on link i which is defined as: t i = l i v i. (3) With the initial expected link travel time estimated, we can proceed to solve for the whole system of equations. For details regarding the iterative process of solving the analytical traffic model, please refer to Section Numerical evaluation of the analytical traffic model With the route choice set generated, exogenous parameters estimated, and endogenous variables initialized, we proceed to solve the analytical traffic model to get the analytical approximation of link flows. In this section, we state technical details of how the system of equations of the analytical traffic model is numerically evaluated. At first, we recap the formulation of the analytical traffic model used in our SO calibration algorithm. f r = s S d s p sr r R (4) p sr = e θtr j R s e θt j s S, r R s (5) t r = i Ψ r t i r R (6) t i = l i v i + ñi λ i i Q (7) 4

5 ñ i = ρ i (l i + 1) ρ li+1 i 1 ρ i 1 ρ l i+1 i i Q (8) ρ i = λ i µ i i Q (9) λ i = γ i + j Q p ji λ j i Q (10) p ij = r G ij f r k H i f k i Q, j Q (11) γ i = r T i f r i Q (12) To apply the iterative heuristic, we first discompose the above system of equations into two separate systems of equations. The first system of equations consists of Equations 4, 5, 6, 11, and 12. The second system of equations consists of Equations 7 through 10. A typical iteration of the heuristic consists of two main steps. At step one, the first system of equations is solved for turning probabilities p ij given the expected link travel time t i (free flow travel time for the very first iteration). Then, the second system of equations is solved for the updated network conditions (i.e.,expected link travel time t i ) with the turning probabilities p ij fixed. We use stopping (or convergence) criteria that stop the iterative process when a pre-defined condition is satisfied. Commonly used stopping criteria are based on the change of solution achieved during one iteration. For instance, given a small ϵ > 0, the iterative process is stopped at iteration a when the following condition is met: x a x a 1 < ϵ, (13) where we denote the solution of iteration a of the iterative process by x a. maximum acceptable number of iterations is usually specified. Additionally, a Specifically, we keep track of the change of variables t i and p ij of the system of equations of two consecutive iterations a 1 and a of the iterative heuristic until it is less than a threshold ϵ as shown in (14). i Q ( t a i t a 1 ) 2 i + ( ) p a ij p a 1 2 ij < ϵ, (14) t p i,j Q 5

6 where t and p are the maxima of t i and p ij respectively and defined as: which are used to normalize these two variables t = max t i, i Q (15) p = max i,j Q p ij. (16) The threshold ϵ is set to 10 3 and the maximum number of iteration is set to 100. The iterative solution process terminates either when the threshold ϵ is satisfied or the number of iteration reaches the maximum iteration. 2.6 Estimation of f(θ) Due to stochasticity, the stochastic user equilibrium flows are calculated as the expectation over multiple simulation replications. Each simulation replication is independent and depended on a different random seed. And all simulation replications are simulated afresh in parallel. Since each simulation is computational expensive, we choose a total number of simulation replications U for evaluating the expectation as a tradeoff between the expensive simulation cost and the stochasticity of the simulator. Before the simulation starts, we use Matlab function randi() to generate U pseudo-random integers uniformly between 1 and the maximum integer as random seeds with which to start each simulation replication. The expected stochastic user equilibrium flow for link i which is denoted by E[F i (θ; x)] for the behavioral parameter θ is calculated as the average of the simulated flows over all U independent simulation replications for link i as: E[F i (θ; x)] = U 1 U F j i (θ; x), (17) j=1 where F j i (θ; x) is the stochastic user equilibrium link flows for a given behavioral parameter θ of simulation replication j where j [1, U]. Assuming the total number of assignment iterations for a given simulation replication is M, the replication-specific stochastic user equilibrium link flow F j i (θ; x) for link i for a given behavioral parameter θ of simulation replication j where j [1, U] are calculated as the average of flows from the last P assignment iterations for link i: M F j i (θ; x) = k=m P +1 1 P F jk i (θ; x), (18) where F jk i (θ; x) is the simulated flow on link i of the kth assignment iteration of the jth replication for a given behavioral parameter θ. The motivation of using Equation (18) is to smoothen out the simulation observations within a given simulation replication. 6

7 By applying Equations 17 and 18 together, we are able to achieve an evaluation of the stochastic user equilibrium flows with relatively low variance. 2.7 SO algorithm The proposed calibration algorithm uses the Simulation-based Optimization (SO) algorithm in [3]. In this section, we state the changes made to the traditional SO algorithm. In general, the traffic measurement we obtain by solving the analytical traffic model can be a good approximation to the simulation-based traffic measurement, and is usually better than the initial guess θ 0. Different from the traditional SO algorithm, we firstly solve the calibration problem with only the analytical traffic model. To be specific, we solve a similar optimization problem to the metamodel optimization problem using a metamodel with only the analytical problem-specific approximation λ i (θ). The optimization problem solved in the first iteration is formulated as follows: min θ (y i m i (θ)) 2 i I (19) θ L θ θ U h 2 (θ, ṽ; q) = 0. (20) (21) And the metamodel for link i used here takes the following mathematical form: m i (θ) = λ i (θ). (22) In addition, the optimization problem is solved despite the trust region constraint so as to be able to search over the full feasible region Θ. For solving the system of nonlinear equations of the analytical traffic model, we use the iterative method as described in Section 2.5 instead of the Matlab routine fsolve. Similar to the traditional SO algorithm, we use the Matlab routine fmincon for solving the metamodel optimization problem, however, the implementation is different. We implement the metamodel optimization problem with only the behavioral parameter θ as the decision variable. We don t explicitly formulate the analytical traffic model as constraints and only relate it to the objective function. Thus, all endogenous variables of the analytical traffic model do not appear to be decision variables in this problem. The number of decision variables will always be the same regardless of the scale of the network. This makes our calibration algorithm applicable regardless of the scalability issues. The optimization objective function is formulated as a function of the analytical model so that for any behavioral parameter θ proposed by the solver the analytical traffic model yields the corresponding link flows λ i (θ), and hence the corresponding value of the objective function is calculated. 7

8 Toy network Berlin network max min γ inc γ dec θ L θ U 0 0 η τ µ n max Table 1: Algorithmic parameters initialization A list of the algorithmic parameters used for the SO calibration algorithm is displayed below in Table 1. In this table, max, min, and 0 are respectively the maximum, minimum and initial value of trust region radius. γ inc and γ dec are the factors for increasing and decreasing the trust region radius. η is the threshold for accepting new trial points. τ is the threshold for sampling new points. µ is the threshold on the number of consecutive rejected trial points required to reduce the trust region radius. And n max is the maximum number of simulation runs (i.e., computational budget). 2.8 Simulator settings and APIs The Application Program Interface (API) of the traffic simulator, MATSim, is used for interacting with the simulator such as starting simulation externally from Matlab, configure the simulation settings, and extracting simulation observations. In this section, we briefly describe the simulator settings and how we control the simulation model in terms of APIs. In order to obtain the user equilibrium flows for a given behavioral parameter θ, we need to modify the corresponding simulation model parameter by configuring the settings and specify the random seed for different simulation replication. This is achieved by modifying the parameters traveling car and random seed in the configuration file. Then, we start the simulation by supplying the network infrastructure (network file) and demand (plans file). During the simulation, we obtain the stochastic link traffic measurement by counting the number of vehicles on a link during a time period. This is achieved by using the MATSim API LinkLeaveEventHandler to keep track of all vehicles leaving this specific link during this time period. To get the simulation observations, we listen to the completion of an assignment iteration by using the MATSim API EventListener and then calculate the simulated link flows for each link in I for this assignment iteration. We also listen to the end of simulation (i.e., shutdown ) and write the collected simulated link flows for all assignment iterations into the output file. This output file is used for analyzing the objective function and fitting metamodel parameters. The strategy modules used for routing in the simulation are SelectExpBeta and ReRoute. SelectExpBeta doesn t create new plan and selects one of the existing plans of the traveler based on the Multinomial Logit (MNL) model. For details regarding the route choice model in 8

9 MATSim, please refer to Section 4. And ReRoute uses the routing algorithm (i.e., Dijkstra) to calculate new least cost routes using travel times from the previous iteration and create new plans. In other words, ReRoute module is used to take care of the cases when the existing choice set is not suitable for a given behavioral parameter θ. The probability of using SelectExpBeta module is set to 1.0 and using ReRoute module is set to 0.2. In the simulation, each traveler chooses exactly one module per iteration, and this setting defines the probability that a module will be chosen by a traveler. Hence, a traveler has a probability of 5/6 to choose SelectExpBeta module and 1/6 to choose ReRoute module. The selection of these numerical values are to achieve the goal of route choice which updates the choice set slowly and following the multinomial logit model. These settings are also helpful for stabilizing the simulation process. These two modules are in effect until the last assignment iteration for each simulation. 3 Additional results and analysis In this section, we augment the analyses in the paper by providing additional tables and discussions for both the toy network and Berlin network case studies. In order to further analyze and compare the algorithm performance in terms of convergence of model Am and model Aϕ, we consider the convergence statistics listed as follows. Stat 1: Number of convergences. This is the total number of cases which converge for a method and a true value. Stat 2: Number of enter/exit/re-enter-converge. The number of cases which enter, leave, and re-enter the green region (once or for multiple times) but finally stay within the green region. Stat 3: Number of enter/exit/do-not-converge. The number of cases which enter and leave the green region (once or for multiple times) and finally stay off the green region. Stat 4: Number of never-enter/do-not-converge. The number of cases which never enter the green region and stay off the green region all the time. For the toy network, Table 2 displays the results for each method and a pair of true value and initial value (i.e., each experiment). Each row of this table corresponds to one of the figures of algorithmic solutions versus iterations in the paper. Table 3 summarizes the results for each method and a true value. And the values in the table are written as percentage of all cases in a certain category. It is worth noting that the total number of cases for a method and a true value is 9 for the toy network in Table 3. Based on Table 3, we can see that all cases for method Am converge, while there are 3 cases for method Aϕ do not converge. For the general-purpose method Aϕ, there are cases which they first enter the green region, then left the green region, and end up with a un-converged solution. However, this never happens for method Am. 9

10 stat 1 stat 2 stat 3 stat 4 Method Am Aϕ Am Aϕ Am Aϕ Am Aϕ θ 0 = θ = 5 θ 0 = θ 0 = θ 0 = θ = 20 θ 0 = θ 0 = θ 0 = θ = 55 θ 0 = θ 0 = Table 2: Case specific convergence statistics for the toy network stat 1 stat 2 stat 3 stat 4 Method Am Aϕ Am Aϕ Am Aϕ Am Aϕ θ = 5 100% 100% 11.1% 22.2% 0% 0% 0% 0% θ = % 77.8% 33.3% 0% 0% 22.2% 0% 0% θ = % 88.9% 11.1% 0% 0% 11.1% 0% 0% Table 3: Convergence statistics for the toy network On the whole, both method Am and method Aϕ are satisfying with similar performance for the toy network. However, method Am outperforms method Aϕ since it identifies good solutions within less number of SO iterations and always converges. We further analyze and compare the algorithm performance in terms of the convergence statistics for the Berlin network. We first present the convergence statistics for each case with a different initial value of θ in Table 4. Rows in this table correspond to Figures respectively. We then present the convergence statistics for a given initial value in Table 5 as percentages. The total number of cases for each category is 3 in Table 5. Based on Table 5, we can see that all cases for method Am converges, while there are only 2 cases for method Aϕ converge. Different from the toy network, there are 3 cases of method Aϕ which never enter the green region and eventually do not converge. On the whole, method Am outperforms method Aϕ in terms of convergence. In the paper, we carried out a more detailed analysis within the initial green region [ 6, 1]. The stat 1 stat 2 stat 3 stat 4 Method Am Aϕ Am Aϕ Am Aϕ Am Aϕ θ 0 = θ 0 = θ 0 = Table 4: Case specific convergence statistics for the Berlin network 10

11 stat 1 stat 2 stat 3 stat 4 Method Am Aϕ Am Aϕ Am Aϕ Am Aϕ θ 0 = 0 100% 33.3% 33.3% 0% 0% 0% 0% 66.7% θ 0 = % 33.3% 33.3% 0% 0% 66.7% 0% 0% θ 0 = % 0% 0% 0% 0% 0% 0% 100% Table 5: Convergence statistics for the Berlin network green region based on the more detailed estimation of the simulation-based objective function is re-defined as: [ 5, 4.5], [ 3.75, 2.75], and [ 2.25, 2]. The convergence in terms of the newly estimated green region for each case of the Berlin network is summarized in Table 6. The percentage in the table is calculated over all experiments (i.e., 3 experiments) for a specific method and initial value θ 0. In the last row of the table, we also display the overall convergence performance for a method regarding all experiments. Method Am Aϕ θ 0 = % 0% θ 0 = % 33.3% θ 0 = % 0% Overall 55.6% 11.1% Table 6: The convergence percentages for each method and each initial value for the Berlin network In terms of the more detailed analysis, the convergence percentages decrease for both methods. However, method Am still outperforms method Aϕ with a much higher chance of convergence. 4 Route choice model in the simulation model We specify the route choice model used in MATSim in this section as a reference for readers based on the MATSim book [2]. As we mentioned previously, the SelectExpBeta module in MATSim makes route choice between plans according the Multinomial Logit (MNL) model. The probability for a traveler to choose plan i among all alternative plans is formulated mathematically as: p i = ebeta brain score i j ebeta brain score j (23) where score i is the score of plan i from the previous assignment iteration. The scores are taken as utilities. The parameter beta brain is treated as a scale parameter in this model. In our case, we fix beta brain at 1. The score/utility of plan is computed as the sum of all activity utilities V act,w plus the sum of all travel utilities V trav,mode(w). Mathematically, it is formulated as: 11

12 W 1 score i = w=0 W 1 Vact,w i + w=0 V i trav,mode(w) (24) where W is the number of activities associated with plan i. Trip w is the trip that follows activity w. For scoring, the last activity is merged with the first activity to produce an equal number of trips and activities. Specifically, the score/utility of an activity w is calculated as: V act,w = V dur,w + V wait,w + V late.ar,w + V early.dp,w + V short.dur,w (25) where V dur,w is the utility of performing activity w, V wait,w is the utility of waiting for performing activity w, V late.ar,w and V early.dp,w specifies the late arrival penalty and early departure respectively, and V short.dur,w is the penalty for a too short activity. The score/utility for traveling on leg w is calculated as follows: V trav,w = C mode(w) + α trav,mode(w) t trav,w + α m m w + (α d,mode(w) + α m γ d,mode(w) )d trav,w + α transfer x transfer,w (26) where C mode(w) is a mode-specific constant, α trav,mode(w) is the direct marginal utility of time spent traveling by mode, t trav,w is the travel time between activities w and w + 1, α m is the marginal utility of money, m w is the change in monetary budget caused by fares/tolls, α d,mode(w) is the marginal utility of distance, γ d,mode(w) is the mode-specific monetary distance rate, d trav,w is the distance traveled between activities w and w + 1, α transfer is the public transit transfer penalty, and x transfer,w is the binary variable indicating whether a transfer is involved in the current leg. In our model, only the car mode is considered. References [1] M. Bierlaire and G. Flötteröd. Metropolis-Hastings sampling of alternatives for route choice models. In Proceedings of the International Choice Modelling Conference, Leeds, England, July [2] A. Horni, K. Nagel, and K. W. Axhausen. The multi-agent transport simulation MATSim. Zurich and Berlin, [3] C. Osorio and M. Bierlaire. A simulation-based optimization framework for urban transportation problems. Operations Research, 61(6): ,