Flexible car–following models for mixed traffic and weak lane–discipline conditions
Abstract
Heterogeneous mixture of vehicle types and lack of lane discipline are common characteristics of cities in the developing countries. These conditions lead to driving manoeuvres that combine both longitudinal and lateral movements. Modeling this driving behavior tends to be complex and cumbersome, as various phenomena, such as multiple–leader following, should be addressed. This research attempts to simplify mixed traffic modeling by developing a methodology, which is based on data–driven models. The methodology is applied on mixed traffic, weak lane–discipline trajectory data, which have been collected in India. A well–known car–following model, Gipps’ model, is also applied on the same data and is used as a reference benchmark. Regarding the lateral manoeuvres, the focus is given on identification of significant lateral changes, which could indicate a lane–changing situation. Methods that allow monitoring structural changes in regression models could be used for this purpose. The ability of capturing lane changes is explored. A typical example is illustrated and further discussion is motivated.
Keywords
Mixed traffic Weak lane discipline Datadriven models Machinelearning Virtual lanes1 Introduction
Traffic simulation models have been typically formulated for lane–based conditions and European traffic. However, simulation of mixed traffic flow in weak lane based heterogeneous conditions poses additional challenges. In recent years, there has been an increasing interest in modeling driving behavior in developing countries where conditions, such as nonlane discipline and heterogeneity in vehicle types, prevail. Wong et al. [1] have explored traffic characteristics of mixed traffic flows in urban arterials and have focused on motorcycles, the proportion of which is high in Asian countries. Traffic flow in the developing countries is very complex in nature and safety issues arise.
Due to the lack of lane discipline, it is difficult to identify leader–follower pairs and to decide if a car–following or a lane changing model should be applied. This research aims to provide some more input into this ongoing active research field. Car–following and lane–changing models describe the longitudinal and lateral movements of drivers. However, these two behavioral models may not be able to describe the integrated driving behavior independently [2]. Lateral interactions take place along with longitudinal interactions in mixed traffic conditions. There have been many attempts to model these behaviors separately. Some of these attempts found in the literature are described in the next section.
This research was motivated by several considerations. Car–following models, which replicate the behavior of a driver following another vehicle, are widely used in the deployment of traffic simulation models. However, only fewer studies have focused on mixed traffic conditions. Due to the complex driver behavior and vehicular interactions and manoeuvers it is difficult to model the traffic flow through analytical methods [3]. Modeling driving behavior in mixed traffic streams is still a challenge. Heterogeneous mixture of vehicle types and violation of lane regulations are common characteristics in cities in developing countries. These characteristics are difficult to be simulated using conventional microscopic models. In cases of carfollowing situations, there is difficulty in the determination of leader–follower pairs due to multiple–leader following. Furthermore, in cases of lane changing situations there is difficulty in the determination of lanes, as drivers do not obey the real lane marks. To overcome some of the associated limitations, in this research a methodology is proposed using temporary virtual lanes in order to capture heterogeneity in vehicle width and speed.
The existing approaches for mixed traffic conditions do not adapt dynamically to the current conditions. Lanes, strips or cells with a predefined width [2, 4], which are used to simulate mixed traffic, do not ensure that the appropriate width has been selected, as half vehicle or two vehicles may fit into this width. Heterogeneity in vehicle types lead to various widths of virtual lanes and various speeds. On the other hand, temporary virtual lanes allow only one vehicle to fit in each lane.

to explore the feasibility of modeling mixed traffic conditions using data–driven models

to compare the performance of data–driven models versus conventional models, in particular Gipps model

to estimate model efficiency considering the difference in following behavior across different vehicle pairs

to introduce the concept of temporary virtual lanes based on identification of significant lateral changes.
An integrated methodology is developed for modeling mixed traffic conditions. The focus is given on data–driven car–following models and the identification of significant lateral positions that may be indicative of the traffic situation of a vehicle (carfollowing, lanechanging or free flow). In the case study, a data–driven model has been used for speed estimation using mixed traffic trajectory data from India. Then, lateral manoeuvres are investigated using an algorithm for identification of structural changes in data. Finally, issues for further analysis and future prospects are discussed.
2 Literature review
Asaithambi et al. [5] have reviewed driver behavior models under mixed traffic conditions and have pointed out limitations of current models, arguing that the main limitation is that they do not explicitly consider the wider range of situations that drivers in mixed traffic face. Munigety and Mathew [2] have identified that due to weak lane discipline, drivers maneuvering in mixed traffic streams exhibit some peculiar patterns such as maintaining shorter headways, swerving, and filtering. They have also proposed that the lane should be divided into small strips in order to handle virtual lane movements. Li et al. [6] have proposed a car–following model that considers the effect of two–sided lateral gaps and they have shown that their model has larger stable region compared to a car–following model that captures the impacts from the lateral gap on only one side. In addition, Parsuvanathan [7] used proxy lanes between the main lanes.
It is assumed that free space is perceived as lanes by small vehicles. However, distribution and types of vehicles could affect the width of the lanes. A grid–based modeling approach akin to cellular automata [8] and a stripbased modelling method [4] have also been proposed. Mathew et al. [4] have based their idea on portions of traffic queues instead of regular main lane queues. Kanagaraj et al. [3] have evaluated the performance of different car following models under mixed traffic conditions. However, they have not taken into account the fact that a vehicle may not be exactly in line with its leading vehicle due to weak lane discipline in mixed traffic. Metkari et al. [9] have modified an existing car–following model in order to take into account lateral movements and include mixed traffic conditions. Choudhury and Islam [10] have developed a latent leader acceleration model.
Maurya [11] developed comprehensive driver behavior model which considers concurrently both longitudinal and lateral interaction with roadway and traffic features. Chunchu et al. [12] analyzed vehicle composition, lateral distribution of vehicles, lateral gaps and longitudinal gaps, in mixed traffic stream. Lan and Chang [13] used General Motors model to simulate the motorcycle’s following behaviors in two cases: (1) only one leading vehicle in front; (2) two or more leading vehicles in front and neighboring–front (including leftfront, right–front or both). The present research attempts to cover more cases. The problem of dealing with non–lane discipline conditions has been treated either by splitting lanes into small strips [4] or small cells using cellular automata model [14]. Furthermore, a veering angle and a path selection have been used to update the lateral position [15, 16]. Social Force models and friction forces [17] have also been proposed. Considering the existing literature review, the concept of temporary virtual lanes is innovative and flexible enough to adapt to mixed traffic conditions.
3 Methodology
In this section, a methodology is developed for simulation of mixed traffic conditions. Mixed traffic flow is considered when speed differential among different types of vehicles is quite substantial and the desired number of overtaking increases with limited opportunities to overtake [18]. Mehar et al. [19] define as mixed traffic conditions when there are several categories of vehicles sharing and moving on same carriageway width without any physical segregation between motorized and nonmotorized vehicles, and without proper lane discipline. Due to the wide variations in physical dimensions and speeds of various vehicles, it is difficult to impose lane discipline. The vehicles occupy any available lateral position on the road space, while the small vehicles, such as motor cycles often utilize gaps between larger vehicles in the traffic stream. In mixed traffic flow, there are different combinations of vehicles for leaderfollower pairs [3].
This study focuses on heterogeneity of vehicle width by considering temporary virtual lanes with different width. On the other hand, heterogeneity of vehicle speed is also critical and is considered by using data–driven models that use as input speeds. For example, three wheeler passenger vehicles and goods vehicles are having almost similar width in shape, but their speeds and driver behavior could be completely different.
3.1 Virtual lanes and leader–follower pair identification
3.1.1 Determination of virtual lanes
where x_{t} is the position of the center of the vehicle, measured from the left–most side of the roadway for each time instant t+i and w_{v} is the width of the vehicle.
The estimation procedure of virtual lane width takes place between two consecutive breakpoints. The same procedure could be applied in all types of urban carriageways. However, the sensitivity of the algorithm, which identifies major changes in data sequence, should be set to adapt conditions of the respective road network. For instance, on a highway, larger lateral movements are expected to imply a lane change manoeuvre.
3.1.2 Identification of leader–follower vehicles
The probability of a given front vehicle to be the governing leader depends on the type of the lead vehicle and the extent of lateral overlap with the following vehicle [10].
where i: 0,1,2,n vehicle index \(x_{c_{i}}\): lateral coordinate of the front center of vehicle i, \(x_{l_{i}}\): lateral coordinate of the front left bound of vehicle i, \(x_{r_{i}}\): lateral coordinate of the front right bound of vehicle i, w_{i}: width of vehicle i s_{i}: a lateral safety distance for vehicle i.
A scenario with two leaders and one follower case is also possible. For instance, a bus could be the follower and a part of its front side may overlap with two leaders such as two motorcycles or a small vehicle and a motorcycle. In this case the closest vehicle according to the direction of movement is chosen as the most critical leader [20]. If no vehicles are identified as leaders, then the driving situation of the vehicle is free flow.
3.1.3 Operationalization process
3.2 Data–driven modeling
Estimation has been achieved without assuming any predefined functional form; instead a flexible regression method. Various machine learning techniques could be used in this context. Other data–driven methods, including neural networks [21], Gaussian processes [22] and Kernel methods offering similar capabilities, have also been used in applications [23]. In this research locally weighted regression has been used, as it comprises much of the simplicity of linear least squares regression with the flexibility of nonlinear regression.
Locally weighted regression (loess) could be considered as a generalization of the k–nearest neighbor method [24]. It was firstly introduced by Cleveland [25] and the following analysis is based on [26].
Locally weighted regression y_{i}=g(x_{i}) + ε_{i}, where i=1,…, n index of observations, g is the regression function and ε_{i} are residual errors, provides an estimate g(x) of each regression surface at any value x in the ddimensional space of the independent variables. Correlations between observations of the response variable y_{i} and the vector with the observations dtuples x_{i} of d predictor variables are identified. Local regression provides an estimation of function g(x) near x=x_{0} according to its value in a particular parametric class. This estimation could be achieved by adapting a regression surface to the data points within a neighborhood of the point x_{0}, which is bounded by a smoothing parameter: span. The span determines the percentage of data that are considered for each local fit and hence the smoothness of the estimated surface is influenced [27]. The span ranges from 0 (wavy curve) to 1 (smooth curve). Each local regression uses either a first or a second degree polynomial that it is specified by the value of the “degree” parameter of the method (degree =1 or degree =2).
3.3 Evaluation
4 Case study set–up
4.1 Data collection
In order to evaluate the feasibility of the methodological framework on mixed traffic trajectory data, data collected in India were used [30]. The video data were collected on a sixlane separated urban arterial road at the Maraimalai Adigalar Bridge in Saidapet, Chennai, India. The section was on a bridge, which ensured that the road geometry was uniform and that there were no nearby intersections, bus stops, parked vehicles, or other side factors that could affect drivers’ behavior. Furthermore, there was no interaction between the vehicle traffic and pedestrians, because the pedestrian walkway is segregated by a barrier. A detailed description of the data could be found in [30]. The data are presented in two parts two excel files for the data collected in the periods 2:45–3:00 PM and 3:00–3:15 PM, on February 13, 2014. Each excel sheet contains columns of variables, such as time, vehicle type, length and width, longitudinal position, speed, acceleration and lateral position, speed, acceleration. Longitudinal position is the position of the front of the vehicle, measured from the upstream end of the section, while lateral position is the position of the center of the vehicle, measured from the leftmost side of the roadway. The trajectory data are available at the address http://toledo.net.technion.ac.il/downloads/.
4.2 Data processing
where v_{i}: resultant speed of vehicle i, \(v_{long_{i}}\): longitudinal speed of vehicle i and \(v_{lat_{i}}\): lateral speed of vehicle i.
In addition, a new column is added which includes the observed speed for the next time instant, namely the speed that should be predicted for each observation. Actually this is the speed that corresponds to time t + 0.5 s and to the same vehicle ID. If there is no observation for this vehicle and for the next time instant, NA is given. Afterwards, rows with NA in this column are omitted, as there is no observed speed to compare with the estimated one by the proposed methodology.
Due to the nature of mixed traffic data, the next step was to define the car–following sequence, namely which vehicle is in front of the other. [30] have identified that in 45% of the observations the overlap between the leader and the follower is less than half the follower width. The methodology described in section was adopted for the identification of the front vehicle. Observations that correspond to vehicles with no leading vehicle were excluded. As lateral safety distance, s=0.20 m is considered for each vehicle on both sides. As distance L in Eq. 4, L=200 m is considered. If no vehicles are identified as leaders, then these observations are omitted, as they do not correspond to carfollowing state.The same procedure was also used with the validation on dataset data300. Finally, dataset “data245” includes 47036 observations corresponding to 1511 vehicle pairs and dataset “data300” 45982 observations corresponding to 1488 vehicle pairs.
4.3 Estimation of conventional models
There are several traffic microsimulation packages, such as AIMSUN, PARAMICS, TransModeler and VISSIM, that could be used as a reference benchmark in terms of conventional models. AIMSUN utilizes a safety distance carfollowing model, the Gipps model, while PARAMICS uses the Fritzsche car–following model [31] and VISSIM is based on a psychophysical model. Mehar et al. [19] found that the VISSIM in its original form is not able to simulate mixed traffic conditions that prevail on Indian highways and proposed a method for model calibration appropriate for mixed traffic. A few modifications to the default behavioral parameters of VISSIM are required to effectively simulate Indian mixed traffic conditions [32]. Several studies have also demonstrated the use of VISSIM in simulating mixed traffic in different countries [33, 34]. On the other hand, VISSIM model contains the largest number of parameters which are also not easily interpreted to familiar driving factors such as the desired speed. The Fritzsche model of Paramics is similar to VISSIM model and includes the same number of parameters. However, AIMSUN is the model with the smallest number of parameters and the most interpretable ones, allowing the best possible results with less calibration work [35]. In addition, Kanagaraj et al. [3] evaluated four different car following models, in particular Gipps Model, Intelligent Driver Model (IDM), Krauss Model and Das and Asundi Model, under mixed traffic conditions and have shown that Gipps model is able to replicate the field conditions better than other models in nonsteady state. Among the aforementioned models, Gipps’ model [36], which is used in AIMSUN, is considered for this case study. More traffic simulation models should be also tested as future prospect.
Observations per vehicle type used for calibration of each data sample
a/a  Vehicle type  Sample 1  Sample 2  Sample 3 

1  Motorcycle  2665  2701  2626 
2  Car  1347  1292  1347 
3  Bus  145  156  156 
4  Truck  41  29  15 
5  Light commercial vehicle  56  59  78 
6  Auto–rickshaw  746  763  778 
Initial values, bounds and optimization results for Gipps’ parameter values
Parameters  Initial values  Constraints  Opt. values  Mean values  

Min  Max  Sample 1  Sample 2  Sample 3  
α  0.8  0.8  2.6  0.81  0.82  0.82  0.82 
b  3.2  5.2  1.6  5.18  5.17  5.08  5.14 
V  14.4  10.4  29.6  10.45  10.44  10.44  10.44 
s  5.9  5.6  7.5  5.62  5.60  5.60  5.61 
b  3.1  4.5  3.0  3.01  3.01  3.00  3.01 
RMSN        0.21  0.22  0.21   
5 Application of data–driven models
In this research the explanatory variables per each time instant t have been considered as independent predictor variables for the estimation of the response variable (for instance speed) for the next time instant (t+ τ), where τ is the apparent reaction time. Estimation is achieved without assuming any predefined functional form; instead a flexible regression method can be used. The next step is the fitting of the proposed methodology for car–following situations using data–driven models. The problem to be addressed is the speed estimation of each vehicle, when the available data include its speed, the speed of the preceding vehicle and the distance between the two vehicles (in the previous time instant). Locally weighted regression could be used for the application. In the training step the flexible carfollowing model is fitted or calibrated on the surveillance data and validated on the other dataset.
5.1 Exploration of data–driven car–following models
In Fig. 6, the proposed method outperforms Gipps’ model and produces a more reliable speed prediction. The estimated RMSN for dataset “data300” is 0.19 using the Gipps’ model and 0.12 using the loess model. The flexible model outperforms the conventional model and produces a more reliable speed prediction.
In Figs. 7 and 8, an analysis of the results per vehicle type is attempted. Figure 7 shows the Empirical Cumulative Distribution Function (ECDF) of RMSN per vehicle type. The best performance of loess method is achieved for cars and light commercial vehicles, while higher RMSN are observed for other vehicle types, especially for trucks and auto–rickshaws. In Fig. 8 ECDF of RMSN are outlined per vehicle type of the leader when the follower is a car. Vehicles pairs car– car and motorcycle–car (leader– follower) have almost 80% of RMSN values lower than 0.1. The curve of vehicle pair truck–car corresponds to higher RMSN than the other vehicle pairs. It is evident that vehicle type plays a significant role in driving behavior.
Finally, in Fig. 9 observed speeds are plotted versus predicted speeds per vehicle type. Linearity is evident for all vehicle types.
5.2 Identification of virtual lane changes
Models developed for lanebased traffic conditions may not be appropriate to simulate traffic situations in developing countries, where weak lane discipline is often observed. Traffic in the developing world is so heterogeneous that often lanebased models cannot be realistic. To overcome some of the associated limitations, in this research a methodology is proposed using temporary virtual lanes. An algorithm for the identification of significant lateral changes has been applied and the feasibility of the method has been explored.
5.2.1 Breakpoints
In order to identify structural changes in sequence of lateral positions, ’strucchange’ package [42] was used in R statistical software [43]. This package is appropriate for testing, monitoring and dating structural changes in regression models. Breakpoints are marked in positions with significant lateral changes.
5.2.2 Results
6 Conclusions and future prospects
Models developed for lane–based traffic conditions may not be appropriate to simulate traffic situations in developing countries, where weak lane discipline is often observed. Traffic in such conditions is so heterogeneous that often lane–based models cannot be realistic. To overcome these limitations, in this research a methodology is proposed based on data–driven models and using temporary virtual lanes. An algorithm for the identification of significant lateral changes has been applied and the feasibility of the method has been explored. In this research, the algorithm has identified all the breakpoints on the available data without constraints. However, the sensitivity of the algorithm could be further explored by setting a minimal segment size either given as fraction relative to the sample size or as an integer giving the minimal number of observations in each segment. The use of other algorithms, such as ’segmented’ package [44] and ’changepoint’ [45], should be also checked for the same purpose. A method for estimation of virtual lane width has been also described.
Data driven approaches could be a promising tool for modeling mixed traffic. They lead to flexible carfollowing models and thus to more robust and reliable representation of driving behavior. This simple methodological approach outperforms the reference (Gipps’) model for the available data. For the available data, speed prediction with RMSN 12% is achieved using loess method, while 19% using Gipps model. Datadriven estimation techniques are designed to address cases in which the traditional approaches do not perform well or cannot be effectively applied without including undue labor. Furthermore, the findings have interesting implications for the role of vehicle type. More specifically, vehicles pairs car– car and motorcycle–car (leader– follower) have almost 80% of RMSN values lower than 0.1, while the curve of vehicle pair truck–car corresponds to higher RMSN. Regarding the identification of lane–changing manoeuvres, breakpoints are marked in positions with significant lateral changes for few trajectories and seem to correspond to lane changing manoeuvres. However, further experimental analysis is required.
This research has highlighted the difficulties in modeling mixed traffic conditions and has explored the feasibility of data–driven models versus Gipps model in this context. Different vehicle pairs resulted in different model efficiency, showing the need for vehicle–dependent models. Finally, this research contributed to the introduction of an alternative method for setting temporary virtual lanes under mixed traffic conditions. As the proposed methodology is data–driven, its transferability is feasible to any another section/ corridor or city in India and other developing countries. Furthermore, the proposed methodology is also useful for urban road networks without strict compliance to road traffic lanes, observed mainly in South European countries. More specifically, in Europe motorcycles and sometimes bicycles share the same road space with cars and tend to move through the lateral gaps. However, the appropriate input data should be used to fit the model for each case. It is suggested that training data come from similar network and traffic conditions with the explanatory data.
As future prospects, swarm–like models and crowd simulation models could also be considered for modeling mixed traffic and weak–lane discipline conditions. In addition, the proposed methodology allows incorporation of further variables moving towards an integrated solution for the simulation of mixed traffic. For instance, vehicle–dependent models need to be developed in case of heterogeneous traffic, as the drivers of vehicles with unequal dimensions tend to have different driving behaviors; furthermore, different vehicle types are characterized by varying vehicle kinematics. Thus, it is foreseen that further exploration into this could open up opportunities to understand and simulate driving behavior in non–lane discipline conditions with heterogeneity of vehicle types.
Notes
Acknowledgments
The authors would like to thank Prof. Tomer Toledo from Technion  Israel Institute of Technology for making the data from India freely available.
Funding
There was no funding for this research. Not applicable.
Availability of data and materials
Data used in this research include weak lanediscipline trajectory data, which have been collected in India and are available at http://toledo.net.technion.ac.il/downloads. A detailed description of the data could be found in [30].
Authors’ contributions
CA and VP developed the proposed methodology for modeling mixed traffic conditions. CA created the outline of the paper. VP validated the methodology using a case study and drafted the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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