Particle Filter Based Traffic Data Assimilation with Sensor Informed Proposal Distribution

Transcription

1 Particle Filter Based Traffic Data Assimilation with Sensor Informed Proposal Distribution Peisheng Wu Dept. of Computer Science Georgia State University Atlanta, Georgia, USA Haidong Xue Dept. of Computer Science Georgia State University Atlanta, Georgia, USA Xiaolin Hu Dept. of Computer Science Georgia State University Atlanta, Georgia, USA ABSTRACT This paper presents a particle filter with sensor informed proposal distribution for traffic state estimation. An agent-based traffic simulator is employed to simulate the traffic network and vehicle behaviors with the help of the sensor and accident component. The proposed framework estimates the average speed of a traffic network along with the location of any traffic accidents. This framework uses observed data to construct the system state by proposing local sensor proposal distributions. The performance is validated and evaluated by means of identical twin experiments carried out on both Bootstrap and Sensor Informed Particle Filter (SIPF). The result shows that the proposed data assimilation framework outperforms Bootstrap Particle Filter with respect to accuracy and efficiency. Author Keywords traffic simulation, particle filter,data assimilation, proposal distribution INTRODUCTION Freeway traffic congestion in urban areas has become a large problem that results in increasing number of accidents and higher emissions. According to [7], the wasted time and fuel caused by traffic congestion were worth approximately 10 billion dollars in the urban area of New York and Los Angeles in 2006 and With effective traffic control systems, the congestion can be significantly reduced, as well as the economic loss caused by it. However, the traffic control system will not help unless real time estimation of the traffic states can be accurately obtained, which leads to the question of how to learn and determine the traffic state. Traffic simulation has been used as a vital technique to study traffic systems. It can be grouped into macro and micro level models on different levels of abstraction. The macro model, known as traffic flow model, describes traffic as the continuum of vehicles [3] and attempts to study the average behavior of the vehicles, such as spatial density and average velocity. On the contrary, micro models simulate the behavior of each individual vehicle and construct the simulation system by the composition of all vehicles. Compared to traffic flow model, micro model studies more details such as driver s decision to change lanes and decelerate to avoid accidents. This makes it suitable for complex traffic problems such as traffic ANSS 2015 April 12-15, 2015, Alexandria, VA Copyright c 2015 Society for Modeling & Simulation International (SCS) intersection, congestion and accidents. However, the complexity of the micro model involves interaction with other drivers, road conditions and even with the drivers physical condition. Therefore, it is really difficult to obtain a perfect micro model. Typically, the model will only take some of the factors into consideration. For example, the advanced Intelligent Driver Model (IDM) [6] effectively models the behaviors of vehicles on freeways by setting intuitive parameters, such as desired velocity, comfortable deceleration and maximum acceleration. But the road conditions are ignored. For example, vehicles on clear lanes will keep accelerating unless other cars are close ahead to them even when they are passing congested areas. For better estimation and prediction of the traffic without struggling for a perfect and probably complex model, data assimilations methods are applied to existing traffic models. These methods estimate the current system state based on previous states and observation information, and then iteratively update the estimation at each time step as the simulation advances. In previous work, we developed a data assimilation framework for complex simulation models such as wildfire simulation [19] and agent-based crowd behavior simulation. The developed framework is based on Particle Filter framework with an effective proposal distribution making use of knowledge from both simulation model and sensor data. In this paper, the developed framework is applied to a microscopic traffic simulator MovSim 1. This work differs from previous work in two aspects: An agent-based micro simulation model is employed to carry out the data assimilation, while existing data assimilate works on traffic system operates on higher level models. the proposed data assimilation framework develops a sensor informed proposal instead of system transition which improve the performance of particle filter. To compare the performance, we design identical-twin experiments [19] for both the bootstrap and sensor informed methods. As will be shown in the results, with same number of particles the proposed sensor informed proposal method is much more effective compared to the conventional bootstrap 1 Multi-model open-source vehicular-traffic Simulator: A java based open-source traffic simulator, initiated by Arne Kesting in June 2011 and further contributed by Martin Budden, Ralph Germ and Martin Treiber, 264

2 particle filter; Even with less particles, the estimated state of the traffic including the location of the obstacle is commensurately accurate. The rest of this paper is organized as follows. The related work of data assimilation on traffic simulation is discussed in the next section. Then the details about the sensor informed proposal distribution method will be elaborately illustrated a- long with the microscopic simulator. The performance of this method is then evaluated by comparison with bootstrap particle filter afterwards. Conclusions and future research issues are highlighted in the last section. RELATED WORK Ever since the sensor information became available for traffic simulation, researchers have been using data assimilation for better estimation of traffic systems. Most existing literature dealing with traffic state estimation employ variants of the Kalman Filter algorithm based on macroscopic model. Extended Kalman filter (EKF), for instance, applied in [13, 14, 16, 17]. [14] presents traffic states estimation and prediction based on a cell transmission model transcribed in a closed analytical state-space form. In [13, 16, 17] a different stochastic macro traffic flow model, Lighthill-Whitham- Richards (LWR) partial differential equation (PDE) model [8, 11], is implemented to estimate freeway traffic states. Unscented Kalman Filter (UKF), another deterministic filter algorithm has also been applied for traffic state estimation based on a compositional traffic model [4]. Different from EKF, UKF uses weighted samples to represent the target distribution deterministically. All of the above use modified Kalman filters which were originally designed for linearization of the state, whereas the studied traffic flow model, L- WR model for instance, is well known for its non-linear [3] non-gaussian [1] behavior. Particle filter, which is inherently suitable for nonlinear models, is then adopted to deal with the nonlinearities. [9] presents particle filter framework based on a hybrid stochastic model and compares the estimation results between PF and UKF with respect to accuracy and complexity. In [5], particle filter is applied to conduct a muti-step speed prediction using speed measurements based on a combination model of both the LWR model and the Van Aerde traffic stream model. Similar method is developed in [2], but with a different second-order macroscopic traffic flow model. [10] employs cellphone handoff data as a measurement of the estimated models to conduct the PF framework. All these literature base on macro models and concern for aggregated state of the traffic system (density and average velocity). The microscopic model is seldom found in existing work on data assimilation for traffic simulation. However, particle filter has been approved as a promising tool for micro level models in [20, 21]. In [21], information from vehicles equipped with positioning system is employed to estimate the number, position and speed of vehicles without the equipment. And in [20], the dynamic data driven event reconstruction for traffic flow simulation is presented, in which PF assimilating real time sensor data into the simulation model is applied to estimate the traffic state, such as slow vehicles or accidents. PARTICLE FILTER WITH SENSOR INFORMED PROPOSAL DISTRIBUTION As previously discussed, to improve the accuracy of the micro model this paper assimilates sensor data in complex scenarios such as accidents and car crash. The following subsection will briefly introduce the MovSim simulator and the necessary modification made for data assimilation framework. The data assimilation framework for MovSim simulator will be discussed in details in the second subsection. MovSim Simulator MovSim is a microscopic lane-based traffic simulation [15], which implements various car-following models of different model categories such as time-continuous models, iterated maps and cellular automata. To utilize the data assimilation framework, we define a traffic system as a seven-tuple termed as sensor monitored spatialtemporal [18], which is < A, S,f,M, C, s, c >. A is the system area, denoting the two dimensional area on which system states are defined; S is the system state that consists of all possible system information; f is the system transition function that describes the advance of a system state: s t = f(s t 1, u t ), in which u t denotes other inputs containing random variables. M is the measurement function which determines the true sensor reading given a system state and a sensor location. C is the sensor set that consists of sensor locations, detection range and the sensor reading. s defines the distance function of two system states. Similarly, c defines the distance function of two sensor reading. Besides, the data assimilation framework also has the following assumptions. System state can be broken into several local states and the local states can reconstruct the system states, if the union of these local states is the same as system state. The analytical forms of transition and measurement function are unknown, but can be used as black-boxes. Based on the sensor reading and measurement model, the system state sample can be drawn easily. Thanks to the open source feature, all the models even the simulator can be modified without much efforts. In addition, although originally designed for basic traffic scenarios, complex scenarios can easily be built since the road network is constructed based on OpenDRIVE format 2. To fulfill the requirements of sensor monitored spatialtemporal system, two necessary components are added to the simulator. Sensor deployment component: Sensors are essential for data assimilation since measurement of the traffic state is based on the observation from the sensors. With evenly distributed sensors, the measurement can be calculated at 2 OpenDRIVE is an open file format for the logical description of road networks that was developed and is being maintained by a team of simulation professionals with large support from the simulation industry. More details can be found at org/ 265

3 each time step. Moreover, measurement function based on sensor reading is also implemented for framework development. Obstacle component: MovSim loads the complete setting of the simulation through xml files, which means nothing can be changed after initialization. The obstacle component allows users to dynamically place obstacles to anywhere at any time step so that the time of car crash can be controlled. This component also contributes for sensor informed proposal distribution since the local sensor proposal will need the ability of proposing new obstacles. With the help of these 2 components, MovSim is capable of building a sensor monitored spatial-temporal system. Particularly, the traffic system contains several road segments with evenly distributed sensors. The state space is defined as all possible states represented by vehicles on the road, such as number, position and velocity of vehicles. Transition function f is constructed by the revised MovSim. Measurement function M returns the number, average velocity and acceleration of vehicles within the detection range of sensors. Except for the modification of the simulator, vehicle behavior models such as Intelligent Driver Model and lane-change model remain unmodified. The parameters for these models keep the same for all experiments. SMC Framework The proposed data assimilation framework is based on Sequential Monte Carlo (SMC) methods. These methods employ a collection of system state samples to approach the system state posterior distributions conditional on observed data. As the system time advances, the sequence of the estimated system states can be built up by iteratively evolving samples and updating their importance weights at every step. The system to be estimated is considered as discrete state model with properties as follows: s t = f(s t 1, u t ) (1) m t = g(s t, v t ) (2) f in equation 1 denotes the system transition function. s t and s t 1 are system states at time step t and t 1. Both u t and v t denotes the vector of inputs that may include random variables. g in equation 2 denotes the observation function. It is known from equation 1, that the system state can be considered as a first order Markov process since s t depends only on s t 1. Furthermore, with the random variable added to transition function, s t can be regarded as random variable that follows the probability distribution of p(s t s t 1 ). Similarly, m t, the observation of s t, is also conditionally dependent on s t, of which the probability distribution is p(m t s t ). However, it s always difficult to draw samples directly from the distribution p(s t s t 1 ) for the reason of complexity. SM- C methods are designed to estimate the posterior distribution without knowledge about it. More specifically, the Importance Sampling (IS) step draws samples from proposal distribution q according to equation 3. E p(x) (h(x)) = E q(x) ( h(x)p(x) ) 1 q(x) N N i=1 p(x i ) q(x i ) h(x i) (3) Furthermore, since p(x) and W (x) can be approximated by p(x) W (x) N i=1 δ(x x w i) and W (x) (x i), N i=1 w (x i) E p(x) (h(x)) is then estimated as E p(x) (h(x)) 1 N N i=1 w (x i ) N i=1 w (x i ) h(x i)δ(x x i ) (4) where δ(x x i ) is the Dirac-delta function. With Importance Sampling from proposal distribution and resampling step after weight updating, the posterior distribution of system state can be approximated by the drawn particles. The procedure of SMC methods is illustrated in algorithm 1. Data: Proposal distribution q; Result: Set of particles Initialization at time step t = 0; Draw N samples from proposal distribution; Weight updating for each particle; Resample to obtain N equally weighted particles; Iterative estimation at Time step t 2; while t T do Draw N samples from the updated proposal distribution on the basis of previous state; Weight updating for each particle based on observation; Resampling according to weights; end return N Samples from the estimated distribution Algorithm 1: SMC procedure Specially, when updating the importance weight for the samples, the proposal distribution is always the critical problem s- ince choosing an optimal proposal distribution is not straightforward. For instance, Bootstrap particle filter choose the system transition distribution as the proposal to achieve simplicity of weight updating, at the loss of performance. The samples drawn from the transition distribution may fail on the estimation when the support of transition distribution is much larger than the likelihood function or the majority of the likelihood function locates in the tail of the transition distribution. Correspondingly, the presented SIPF utilizes the most recent observation to better estimate the system state. However, difficulty arises for analytical forms of the transition distribution and proposal distribution. The proposed framework introduces a 3-step sampling algorithm to generate the samples taking advantage of both sensor readings and simulation model. The weight updating strategy is modified accordingly to estimate the system transition and proposal distribution by 266

4 from equation 5; Figure 1. The setting of traffic data assimilation system meas of Kernel Method. The corresponding proposal distribution is named as SenSim proposal and the framework as SenSim framework. SENSIM DATA ASSIMILATION FRAMEWORK As previously mentioned, to draw desirable samples of high density value and high likelihood the SenSim framework follows a 3-step sampling algorithm. The first step is Simulation Model Generated States, which employs system transition function f including the input vector u i t to generate a system state s i t from the previous state s i t 1. Same as Bootstrap particle filter, samples are drawn from the system transition distribution. Fortunately, the next 2 steps will generate the samples with a high density value and a high likelihood. Step 2, Sensor Reading Generated Local States, local s- tate samples will be drawn from local sensor proposal p(s At T, s ). Supposing T is a sensor cluster covering A T, s is the system state generated by system model, the probability density of local state s At is conditional on the sensor reading T and the simulated system state s. Apparently, the union of the local states will construct a sample with high likelihoods compared to simulated sample. However, with low transition density value the local sensor proposal may fail if there is less uncertainty in simulation model. Therefore, the sampled system state, both from sensor reading and simulation model, will then be drawn in the next step. Step 3, Sampling Local States, confidence levels for simulation model and sensor reading are defined as c Ai sen and c Ai sim for each local area A j. Then a local state sample can be drawn p(s t,aj s i t,a j, s i t,a j ) = + c Ai sen c Ai sen + c Ai sim c Ai sim c Ai sen + c Ai sim δ(s t,aj s i t,a j ) δ(s t,aj s i t,a j ) (5) in which s denotes the sample from sensor reading, and s denotes the sample from simulation model. The final system state will be constructed by combining all the local states subsequently. However, even the samples can be drawn conveniently the analytical form of p(s t s t 1 ) and q(s t s t1, m t ) are still unknown. Therefore, the kernel method [12] is employed estimate the values of system transition and proposal probability density functions. Kernel method is a non-parametric method for density function estimation, defined as equation 6. p(x) 1 nh nx n i=1 K( x x i ) (6) h where K is a symmetric probability function, termed as Kernel function, n x is the number of dimensions of x, and h > 0 is the bandwidth. A perfect pair of K and h can minimize the error between the true density function and the estimated. To utilize the power of kernel function, corresponding samples need to be drawn from the estimated distribution, which in this case the proposal distribution q(s t s t 1, m t ) and system transition distribution p(s t s t 1 ). As mentioned in the sampling method, the samples from transition distribution can be drawn by the simulation model and the samples from proposal distribution can be drawn by means of the 3-step sampling algorithm. Hence, the estimates of p(s i t s i t 1) and q(s i t s i t 1, m t ) is obtained accordingly. After sampling and weight updating, the same resampling s- trategy is carried out to avoid degeneracy of the particles. So far, the SenSim data assimilation framework is consequently established. PERFORMANCE EVALUATION The modified MovSim is employed to carried out the identical twin experiment for both Bootstrap and SenSim framework. To show the advantage of the proposed data assimilation framework, a traffic accident is modelled at the beginning of the simulation and both methods will try to estimate the average density and the location of the accident. The detailed experiment setting of the sensor monitored spacial-temporal system is defined as follows: Experiment Setting System Area A is defined as a ring road of 2 lanes, the length of which is 1000m. System state S is defined as the combination of vehicle velocities, numbers, accelerations and positions. Transition function f is controlled by MovSim. Accidents and random moves are manually added. Specifically, an accident is randomly placed to the current state with a low probability at each simulation step. Similarly, the random 267

5 move, that all vehicles move to a random close position, happens at each step as well. Measurement function M returns the vehicle number, average velocity and average acceleration within the sensor s detection area. In this experiment, 4 sensors, whose detection range is 100m, are evenly distributed along the 4 road segments. Distance function of system states s is defined as the sum of the vehicle density difference over all 4 road segments. Distance function of sensor reading c is defined as the weighted average of all 3 variables. The weight assigned to each variable is based on the importance: 0.5 for numbers of vehicles, 0.3 for velocities and 0.2 for accelerations. The traffic data assimilation system is then constructed, as shown in figure 1. The local sensor proposal is designed intuitively based on the observed average velocity of the sensor. If the average velocity is faster than 20m/s, the road is considered to be clear and if an accident is placed on that road segment it will be removed. On the contrary, if the average velocity is slower than 8m/s, the road is more likely obstructed. In this case, if no accident is placed on that road segment, an accident will be randomly placed in the local area with the probability of 40%. The local area starts from the left detection bound of the current sensor to the left bound of the next sensor. As an identical twin experiment, a pair of MovSim simulations will run simultaneously: one is considered as the real system and the other as simulated system. Both simulations will have 15 time steps of 15 seconds per step, totally 225 seconds simulation time. Data assimilation framework obtains sensor reading from real system at each simulation step, and samples of particles will be drawn and resampled based on the sensor information. The real system will be artificially placed an accident at 15 seconds as the rare event of this system so that no chance of the simulated system will approach the same system state as the simulation proceeds. The simulation results after 225 seconds of both simulations are shown in figure 2. As figure 2(a) depicts, the Real System has the congestion location exactly at where the accident is placed, while in figure 2(b) the simulated system doesn t have an accident placed. The data assimilation framework will benefit from sensor reading and propose better estimation. Experiment Results Both Bootstrap and SenSim framework are carried out with different numbers of particles. Considering simulated system as the baseline, Bootstrap and SenSim framework are then compared to each other, regarding the estimation of vehicle number on each road segment and the location of the accident. Weighted average errors in each step of the particles are calculated and displayed in figure 4. As is shown in both Figure 3(a) and 4(a), the weighted error starts almost the same as simulated system, and as the simulation proceeds the error of simulated system, without the knowledge of the accident, continues growing. On the other hand, with the help of data assimilation framework and the sensor information, the errors decrease over time and drop into the range of [0, 0.05] after 15 steps in most cases. The only exception is the Boot- (a) Real System (b) Simulated System Figure 2. The setting of traffic data assimilation system 268

6 PARTICLE NUMBER PARTICLE NUMBER PARTICLE NUMBER Boorstrap Sensim Boorstrap Sensim Boorstrap Sensim (a) 10 Particles (b) 20 Particles (c) 30 Particles Figure 5. Comparison (a) Bootstrap Figure 3. Estimate for number of vehicles (a) SenSim Figure 4. Estimate for number of vehicles strap case with 10 particles resulting in the range [0.05, 0.1] because the number of proposed accidents are not sufficient enough to approach the true system state. By comparison of the Bootstrap and SenSim framework with different number of particles in figure 5, it is obvious that the SenSim proposal outperforms Bootstrap. In figure 5(a), without enough particles Bootstrap filter is left far behind by SenSim. The local sampling proposes more realistic local system state with the sensor readings so that the estimation is more close to true system state. As the number of particles increases Bootstrap filter will obtain comparable accuracy with SenSim though taking more time, as shown in 5(b) and 5(c). In other words, efficiency is the advantage of SenSim framework. Estimating the location of the accident in the real system is the other goal of employing data assimilation framework. In order to show clearly how the particles pursue the location of accidents, an obstacle graph is designed for better illustration. In the obstacle graph, no vehicles will be shown except the accident (an crashed vehicle). All particles that contains an accident will be included in the graph. Therefore, for each time step, an obstacle graph can be generated to show how close the estimated location is to the real location. Specifically, the estimated location will be denoted as red dot, while the real location as blue dot as is shown in Figure 6. Figure 6(a) and Figure 6(b) depicts the 11th step of Bootstrap and SenSim with 20 particles respectively. As the simulation advances, particles with better estimation are chosen so that in both figures the red dots are located on the road segment 3, where the real accident locates. Furthermore, the red dots of Bootstrap are distributed all along the road segment, while the red dots of SenSim are centered around the real accident which implies better performance of SenSim proposal. The numerical results of the accident location estimation are also provided in Figure 7. It is clear that SenSim framework speed up the converge compared to Bootstrap as all lines of SenSim drops under 0.2 for only 5 simulation steps while it takes 8 steps for the Bootstrap with 30 particles. In terms of accuracy, the error of all 3 SenSim experiments ends around 0.02 while it takes no less than 30 particles for Bootstrap framework to achieve. In such a manner, SenSim framework is proved more efficient than Bootstrap and commensurately accurate. CONCLUSION In this paper, a particle filter based data assimilation framework is applied on a micro level agent- based simulator. Different from existing work of data assimilation, this framework 269

7 Figure 7. Results for Accident Estimation proposes sensor informed Particle Filter for better estimation taking advantage of both sensor information and simulation model. In order to validate and evaluate the performance, i- dentical twin experiments have been carried out on both Bootstrap and sensor informed proposal particle filter. As is shown in the experiment results, SenSim framework outperforms Bootstrap in both efficiency and accuracy. Specifically, with same number of particles SenSim reduce the estimated error faster; and to achieve same accuracy SenSim framework needs less particles. (a) Bootstrap at step 11 REFERENCES 1. Antoniou, C., Ben-Akiva, M., and Koutsopoulos, H. N. Nonlinear Kalman filtering algorithms for on-line calibration of dynamic traffic assignment models. Intelligent Transportation Systems, IEEE Transactions on 8, 4 (2007), Bi, J., Chang, C., and Fan, Y. Particle Filter for Estimating Freeway Traffic State in Beijing. Mathematical Problems in Engineering 2013 (2013). 3. Blandin, S., Couque, A., Bayen, A., and Work, D. On sequential data assimilation for scalar macroscopic traffic flow models. Physica D: Nonlinear Phenomena 241, 17 (2012), Boel, R., and Mihaylova, L. Modelling freeway networks by hybrid stochastic models. In Intelligent Vehicles Symposium, 2004 IEEE, IEEE (2004), Chen, H., Rakha, H., and Sadek, S. Real-time freeway traffic state prediction: A particle filter approach. In Intelligent Transportation Systems (ITSC), th International IEEE Conference on (Oct 2011), (b) SenSim at step 11 Figure 6. Obstacle Graph 6. Kesting, A., Treiber, M., and Helbing, D. Enhanced intelligent driver model to access the impact of driving strategies on traffic capacity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, 1928 (2010), Lewis, D. America s Traffic Congestion Problem: Toward a Framework for National Reform. The Brookings 51 (2008). 270

8 8. Lighthill, M. J., and Whitham, G. B. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 229, 1178 (1955), Mihaylova, L., Boel, R., and Hegyi, A. Freeway traffic estimation within particle filtering framework. Automatica 43, 2 (2007), Nanthawichit, C., Nakatsuji, T., and Suzuki, H. Application of probe-vehicle data for real-time traffic-state estimation and short-term travel-time prediction on a freeway. Transportation Research Record: Journal of the Transportation Research Board 1855, 1 (2003), Richards, P. I. Shock waves on the highway. Operations research 4, 1 (1956), Rosenblatt, M., et al. Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics 27, 3 (1956), Schreiter, T., Van Hinsbergen, C., Zuurbier, F., Van Lint, J., and Hoogendoorn, S. Data-model synchronization in extended Kalman filters for accurate online traffic state estimation. In TFTC Summer Meeting 2010, no. EPFL-CONF (2010). 14. Tampère, C. M., and Immers, L. An extended Kalman filter application for traffic state estimation using CTM with implicit mode switching and dynamic parameters. In Intelligent Transportation Systems Conference, ITSC IEEE, IEEE (2007), Treiber, M., and Kesting, A. An open-source microscopic traffic simulator. Intelligent Transportation Systems Magazine, IEEE 2, 3 (2010), Wang, Y., and Papageorgiou, M. Real-time freeway traffic state estimation based on extended Kalman filter: a general approach. Transportation Research Part B: Methodological 39, 2 (2005), Wang, Y., Papageorgiou, M., and Messmer, A. Real-time freeway traffic state estimation based on extended Kalman filter: A case study. Transportation Science 41, 2 (2007), Xue, H. Data Assimilation Based on Sequential Monte Carlo Methods for Dynamic Data Driven Simulation. PhD thesis, Georgia State University, Atlanta, Georgia, US, August Xue, H., Gu, F., and Hu, X. Data Assimilation Using Sequential Monte Carlo Methods in Wildfire Spread Simulation. ACM Trans. Model. Comput. Simul. 22, 4 (Nov. 2012), 23:1 23: Yan, X., Gu, F., Hu, X., and Engstrom, C. Dynamic data driven event reconstruction for traffic simulation using sequential Monte Carlo methods. In Proceedings of the 2013 Winter Simulation Conference: Simulation: Making Decisions in a Complex World, IEEE Press (2013), Zhou, Y., Yang, X., and Mi, C. State Estimation of Unequipped Vehicles Utilizing Microscopic Traffic Model and Principle of Particle Filter. CMES: Computer Modeling in Engineering & Sciences 89, 6 (2012),