# Leader-Follower Identification with Vehicle-Following Calibration for Non-
Lane-Based Traffic

Mihir Mandar Kulkarni111These authors have contributed equally to this work
Ankit Anil Chaudhari222These authors have contributed equally to this work
<EMAIL_ADDRESS>Karthik K. Srinivasan  Bhargava Rama
Chilukuri  Martin Treiber  Ostap Okhrin

###### Abstract

Most car-following models were originally developed for lane-based traffic.
Over the past two decades, efforts have been made to calibrate car-following
models for non-lane-based traffic. However, traffic conditions with varying
vehicle dimensions, intermittent following, and multiple leaders often occur
and make subjective Leader-Follower (LF) pair identification challenging. In
this study, we analyze Vehicle Following (VF) behavior in traffic with a lack
of lane discipline using high-resolution microscopic trajectory data collected
in Chennai, India. The paper’s main contributions are threefold. Firstly,
three criteria are used to identify LF pairs from the driver’s perspective,
taking into account the intermittent following, lack of lane discipline due to
consideration of lateral separation, and the presence of in-between vehicles.
Second, the psycho-physical concept of the regime in the Wiedemann-99 model is
leveraged to determine the traffic-dependent ”influence zone” for LF
identification. Third, a joint and consistent framework is proposed for
identifying LF pairs and estimating VF parameters. The proposed methodology
outperforms other heuristic-based LF identification methods from the
literature in terms of quantitative and qualitative performance measures. The
proposed approach can enable robust and more realistic LF identification and
VF parameter calibration with practical applications such as LOS analysis,
capacity, and travel time estimation.

###### keywords:

Traffic microsimulation , Naturalistic trajectory data , Heterogeneous and
non-lane-based traffic , Leader-follower identification , Joint and consistent
calibration , Vehicle-following models

††journal: arXiv.org

[inst1]organization=Zachry Department of Civil & Environmental Engineering,
Texas A&M University, postcode=3136, addressline=TAMU, city=College Station,
state=TX, country=USA

[inst2]organization=Chair of Econometrics and Statistics esp. in the Transport
Sector, Institute of Transport and Economics,addressline=Faculty of Transport
and Traffic Sciences, Technische Universität Dresden, postcode=01187,
city=Dresden, country=Germany

[inst3]organization=Department of Civil Engineering,addressline=Indian
Institute of Technology Madras, city=Chennai, postcode=600036, country=India
[inst4]organization=Center for Scalable Data Analytics and Artificial
Intelligence (ScaDS.AI),city=Dresden/Leipzig, country=Germany

##  1 Introduction

Microscopic traffic simulation models are widely used to predict traffic
performance measures like delays (Tian et al., 2002), level of service
(Geistefeldt et al., 2014), or vehicular emissions (Abou-Senna et al., 2013).
Traffic simulations are based on several models, including car-following and
lane-changing models, in order to represent the lateral and longitudinal
movement and placement of vehicles in the traffic stream. Vehicle following
behavior refers to how a subject vehicle reacts to the motion of its leader
(the vehicle immediately ahead) when they are sufficiently close to each other
such that the leader has a considerable influence on the follower.

Traffic conditions can be classified based on the presence or absence of lane
discipline into the following two categories: Lane-Based (LB) flow and Non-
Lane-Based (NLB) flow (both categories may include heterogeneous traffic,
i.e., several types of vehicles). Compared to LB traffic, which has received
the bulk of the attention in existing literature (Gazis et al., 1961;
Wiedemann, 1974; Gipps, 1981; Bando et al., 1998; Treiber et al., 2000),
analyzing car-following behavior and identification of LF pairs is more
challenging in NLB traffic flow for a number of reasons. The interaction among
different vehicle types is complex, and the influence of surrounding vehicles
beyond the assumed leader and follower pair must be considered. Moreover, the
movements of smaller vehicles like two-wheelers and their partial presence in
the influence region between the leader and follower pair can make the car-
following process intermittent rather than a continuous one. Furthermore, the
lack of lane discipline and varying vehicle sizes necessitates considering the
role of lateral overlap in car-following behavior. For these reasons, methods
for identifying LF pairs based on lane-based traffic are unlikely to be
effective for NLB streams. To address these challenges, this paper develops a
joint and consistent methodology for LF identification and car-following model
parameter calibration for such NLB traffic flows.

The methodology in the NLB context is illustrated in this paper using
trajectory data from a divided arterial in Chennai City, India. Trajectory
data offers several advantages for the calibration of car-following models.

First, it can provide information about the complete set of driving regimes,
enabling a thorough understanding of driving behavior (Sharma et al., 2019a).
As a result, it is shown that trajectory data can be used for more accurate
and unbiased calibration (Chaudhari et al., 2022). Further, it provides richer
and more extensive information than cross-sectional and probe-based data
(Sekuła et al., 2018). Moreover, it provides complete information on all
trajectories in a certain spatiotemporal region, which is a prerequisite for
determining the relevant vehicles in the local neighborhood. Finally, the
variety and quantity of such data enable the development of data-driven and
machine learning-based models (Papathanasopoulou and Antoniou, 2015), which
may be more accurate for forecasting purposes. For these reasons, there is a
growing use of such data in developing and calibrating (Kesting and Treiber,
2008) different car-following models such as the Wiedemann-99 model
(Wiedemann, 1974), the IDM (Treiber et al., 2000), the OVM (Bando et al.,
1998) and the Gipps car-following model (Gipps, 1981). As this study is
focused on different types of leaders and followers, we are using the more
general term vehicle following (VF) instead of car following. An important
step in VF model parameter estimation in many models is the identification of
leader-follower pairs. For this purpose, different studies (Zhu et al., 2018;
Chong et al., 2013; LeBlanc et al., 2013; Higgs and Abbas, 2013; Durrani et
al., 2016) use a variety of heuristics in the literature which are based on
some strong assumptions and restrictions discussed in the next section.

This research is motivated by the following considerations. First, the methods
for LF identification vary widely across the literature and are based on
heuristics involving trial and error, subjectivity, or the use of arbitrary
thresholds without calibration, e.g., for the longitudinal clearance or the
headway. As a result, it is difficult to transfer and compare parameters
across different studies. For instance, Brockfeld et al. (2004) used all
vehicle pairs in calibrating model parameters, which can lead to an
unrepresentative sample of the following behavior. Recognizing this
limitation, Anand et al. (2019) use simple heuristics for LF identification
but are based solely on spatial or temporal proximity (e.g., distance or
headway thresholds). Two main limitations of the heuristic approaches for LF
identification include: 1. choice of thresholds is arbitrary, and 2. trial and
error procedures may yield erroneous LF pairs, affecting and potentially
invalidating the subsequent estimation of the VF parameters.

The second main motivation is that there are several reasons that mere
physical proximity (or its absence) may not always be an adequate indicator of
the influence of the leader’s motion on a subject vehicle’s behavior. Due to
differences in the vehicles (age, dimensions, etc.), the relative speed
between leader and follower, or driver characteristics (aggressiveness, skill,
etc.), the response to the same physical or temporal gap may be different and
thus inadequate to demarcate following vs. non-following regimes. Furthermore,
proximity-based measures do not recognize the possibility of different driving
regimes, including some non-following episodes within the same gap or time
headway. Besides, the following process may be discontinuous due to
psychophysical constraints such as visual angle or its rate of change.

Third, existing methods for LF identification do not adequately account for
non-lane-based traffic features. For instance, the criteria of being in the
same lane is not properly defined because of staggered following or simply
because there are no lanes. Further, it is not easy to identify whether
intervening vehicles are present between a potential LF pair due to these
varying spatial configurations and their partial presence in the influence
area, as illustrated in Section 3. The presence and varying influence of other
surrounding vehicles between the subject vehicle and the staggered leader can
also lead to intermittent following in the NLB traffic context.

To address these gaps and considerations, the following objectives are pursued
in this paper:

  1. (a)

Demonstrate the limitations of existing heuristics for LF identification in
mixed traffic.

  2. (b)

Develop a new methodology for LF identification in mixed traffic that
incorporates

    1. i.

joint and consistent procedure for LF identification and vehicle-following
parameters estimation,

    2. ii.

information about psychophysical driving regimes,

    3. iii.

and more accurate representation of heterogeneous traffic such as the presence
of potential surrounding intervening vehicles and lateral separation between
the leader and follower.

  3. (c)

Evaluate the proposed approach in comparison with the existing LF
identification methods.

Using our new methodology, we show that it outperforms existing heuristics in
terms of the accuracy of predicted trajectories.

The structure of the paper is as follows. Section 2 summarizes various
criteria used in the literature to identify LF pairs in homogeneous and
heterogeneous traffic and their associated advantages and disadvantages.
Section 3 outlines the data utilized in the study and preliminary analysis to
illustrate the limitations of existing LF identification methods. Section 4
provides the rationale and an elaborate explanation of the proposed
methodology. In Section 5, the salient findings and results from this study
are presented, including an evaluation of the proposed method in relation to
other LF identification methods. Section 6 presents some concluding remarks
along with directions for future work.

##  2 Literature Review

A comprehensive review of car-following literature is provided in Olstam and
Tapani (2004). To analyze vehicle-following behavior using trajectory data, it
is crucial to identify the points of the _influence zone_ for the assumed
leader-follower pair where the follower’s motion is influenced by the leader.
Various methods have been used to identify influence zones, ranging from
observations of traffic videos to measuring lagged acceleration correlations
between an assumed leader and an assumed follower. Heuristic thresholds on
longitudinal clearance, relative speed, lateral overlap, and time headways
have been widely applied to identify the influence zone. These thresholds
depend on the type of traffic and method of data collection. For example,
verifying that the assumed leader and follower are traveling in the same lane
may be sufficient for LB traffic but not for NLB traffic because of the lack
of lane discipline. If the data is collected by a range sensor on the
following vehicle detecting only vehicles straight ahead, it may be sufficient
to ensure that the leader remains in the same lane (Higgs and Abbas, 2013). If
the data, however, contains all trajectories (e.g., the NGSIM (US Department
of Transportation, 2008) or Chennai data (Kanagaraj et al., 2015)), then it
becomes crucial to identify periods where a given leader-follower pair is not
interrupted by other surrounding vehicles.

There is a growing interest in developing vehicle-following models for NLB
traffic in the last decade (Mathew and Ravishankar, 2011; Kanagaraj et al.,
2013; Das et al., 2019; Adavikottu et al., 2023; Raveendran et al., 2024).
Some studies have tried to include lateral gaps and movement in the analysis
(Gaddam and Rao, 2020; Kanagaraj and Treiber, 2018; Sharath and Velaga, 2020;
Nirmale et al., 2024). A variety of VF models have been used, but only a few
have included psycho-physical driving regime information (Raju et al., 2021;
Chaudhari et al., 2022) in the model.

Many of these studies have not explicitly dealt with LF pair identification
for various reasons. Some studies have used synthetic data (Ravishankar and
Mathew, 2011; Das et al., 2019) or simulated data (Mathew and Radhakrishnan,
2010) instead of naturalistic data. Other studies on NLB have used aggregated
macroscopic measures to calibrate the vehicle-following parameters (Mathew and
Radhakrishnan, 2010; Asaithambi et al., 2018), and hence, LF identification
was not examined sufficiently. A few researchers have used heuristics such as
aggregate hysteresis for LF identification (Raju et al., 2021; Das et al.,
2019). However, the comprehensive treatment of LF and VF simultaneously is
underexplored.

Table 1: Criteria and thresholds for the influence zone from existing
literature. If several criteria from the same source are listed, all need to
be true for a valid influence zone.

Literature |  Criteria |  Threshold value |  Remarks
---|---|---|---
Zhu et al. (2018) |  Longitudinal clearance |  $<120\,\mathrm{m}$ |  Expressway in China
|  Lateral distance between center of follower and radar target |  $<2.5\,\mathrm{m}$
Sun et al. (2021) |  Longitudinal clearance |  $>7\,\mathrm{m}$ |  NGSIM data
|  Longitudinal clearance |  $<120\,\mathrm{m}$ |
|  Relative speed |  $<2.5\,\mathrm{m/s}$ |
|  Speed of follower |  $>5\,\mathrm{m/s}$ |
|  Correlation between acceleration of follower and speed difference |  $>0.6$ |
Anand et al. (2019) |  Longitudinal clearance |  $<30\,\mathrm{m}$ |  NLB Traffic
|  Lateral displacement with leader |  $<3\,\mathrm{m}$
|  Headway |  $<2\,\mathrm{s}$
|  Continuous following |  $>5\,\mathrm{s}$
Fernandez (2011) |  Longitudinal clearance |  $<100\,\mathrm{m}$ |  Urban arterial roads
|  Length of following section |  $>100\,\mathrm{m}$
Chong et al. (2013) |  Lateral clearance |  $<1.9\,\mathrm{m}$ |
Raju et al. (2021) |  Lateral clearance |  $<1.5\,\mathrm{m}$ |  NLB traffic
Papathanasopoulou and Antoniou (2018) |  Lateral safety distance of both sides |  $<0.2\,\mathrm{m}$ |
Nagahama et al. (2021) |  Edge to edge lateral clearance |  $<0.3\,\mathrm{m}$ |
LeBlanc et al. (2013) |  Relative speed |  $<\pm 2\,\mathrm{m/s}$ |
|  Follower’s Speed |  $>11.2\,\mathrm{m/s}$ |
|  Continuous following |  $>15\,\mathrm{s}$ |
Higgs and Abbas (2013) |  Follower’s Speed |  $>5.55\,\mathrm{m/s}$ |  To minimise effect of traffic jam
|  Length of vehicle following period |  $>30\,\mathrm{s}$ |  For trucks
|  Longitudinal clearance |  $>61\,\mathrm{m}$
|  Longitudinal clearance |  $<120\,\mathrm{m}$

Table 1 gives an overview of different LF identification criteria used in
literature, which are explained in detail as follows. Longitudinal clearance
between the assumed leader and follower has been widely used to determine the
influence zone by (Zhu et al., 2018; Sun et al., 2021; Anand et al., 2019;
Higgs and Abbas, 2013; Fernandez, 2011). Lateral clearance or lateral
overlapping width between leader and follower is another criterion to identify
the influence zone (Zhu et al., 2018; Chong et al., 2013; Raju et al., 2021;
Papathanasopoulou and Antoniou, 2018; Nagahama et al., 2021). In lane-based
traffic, it could be sufficient to ensure that the assumed leader and follower
are in the same lane (LeBlanc et al., 2013; Sun et al., 2021). Variables like
relative speed (follower speed minus leader speed, i.e., approaching rate)
(LeBlanc et al., 2013; Sun et al., 2021), speed of the follower (LeBlanc et
al., 2013; Higgs and Abbas, 2013; Sun et al., 2021), and time headway (Anand
et al., 2019) have also been used to determine the influence zone.

After classifying individual trajectory points as falling within or outside
the influence zone, the pair should be classified as an LF or non-LF pair. The
following duration (Anand et al., 2019; LeBlanc et al., 2013; Higgs and Abbas,
2013) and length of the following segment (Fernandez, 2011) are used to
identify whether the pair can be recognized as LF or non-LF. Sun et al. (2021)
used an additional condition based on the Pearson correlation. Raju et al.
(2021); Das et al. (2019) used hysteresis plots to determine leader-follower
pairs. To identify LF pairs in non-lane-based traffic, it is crucial to
consider the effect of other surrounding vehicles between the assumed leader
and follower. This situation may not arise in lane-based traffic but can
frequently happen in non-lane-based traffic due to the varying dimensions of
vehicles. Raju et al. (2021) used space-time plots to identify other
surrounding vehicles that could influence the interaction between the assumed
leader-follower pair. Unfortunately, visual observation of space-time plots is
not scalable if the data is large. Moreover, if an intermediate vehicle is
intermittently present for more time between potential LF pair, it may not be
practical to consider the whole potential pair for calibration. Nagahama et
al. (2021) used Voronoi-triangulation-based logic to detect the influence of
surrounding vehicles. According to them, if an edge of a triangle exists
between the leader and follower, it serves as one of the influencing
conditions. Sharma et al. (2019b) generated different trajectory groups and
studied the impact of trajectory .Chen et al. (2020) incorporate human factors
in driving modeling by identifying long-term and short-term driving
characteristics and calibrating the vehicle following models for those
characteristics; these driving characteristics identification is similar to
influence points identification as proposed in Section 4.1.

The above literature on the identification of LF pairs reveals several gaps:
1) Using a fixed threshold for longitudinal clearance may result in inaccurate
identification of the influence zone. 2) Lateral clear gap or overlap
thresholds may not be the same for all vehicle types. For instance, a 0.2 m
overlap might influence a two-wheeler following a vehicle but not necessarily
a truck. 3) Existing literature relies on heuristic-based thresholds for the
duration of the continuous following, with limited consideration for
variations in traffic conditions; long following durations can often be
observed in LB traffic because of the lane discipline, but for NLB traffic,
intermittent following is frequent, (Chaudhari et al., 2022), also as shown in
Section 3.2. 4) Hysteresis plots are not easily scalable, and they are
subjective. 5) Previous works do not comprehensively address the
identification of other potentially influencing vehicles at every point in the
trajectory.

The next section explains the data used for the study and some preliminary
analysis of the existing LF identification methods.

##  3 Data Description and Preliminary Analysis

###  3.1 Data Description

This study is based on video data with two frames per second collected from a
six-lane divided urban arterial in Chennai, India, as detailed in Kanagaraj et
al. (2015). The selected study section, depicted in Figure 1, is situated on a
bridge, ensuring a uniform and consistent roadway segment for analysis. In the
vicinity, there are no immediate intersections, bus stops, parked vehicles, or
any other factors that could potentially influence the drivers’ behavior.
Additionally, there is no interaction between vehicle traffic and pedestrians.
The width of the section under study is 11.2 m and its length 245 m. The video
data was recorded from 10:00 AM to 3:30 PM. The coordinates, dimensions, and
class of all vehicles in the video sequences during 30 min between 2:45 PM and
3:15 PM were obtained using a trajectory extractor. The data consists of 3016
vehicles with six different vehicle classes and 130,137 data points. The
vehicle population consist of 1703 (57$\%$) motorized two-wheelers, 802 (27%)
cars, 367 (12%) three-wheelers, 95 (3%) buses, 40 (1%) light trucks, and 9
(0.29%) heavy trucks. The study area has three nominal lanes, which are
largely ignored.

Figure 1: Study Section in Chennai, India (Kanagaraj et al., 2015)

From the trajectory data, we selected 2078 potential LF pairs using the
influence area method proposed by Anand et al. (2019) (cf. Table 1). These
pairs consist of 1099 motorized two-wheelers, 623 cars, 249 three-wheelers, 59
buses, 42 light trucks, and 6 heavy trucks as follower vehicles. This work
focuses on the 623 potential LF pairs with a car following. For each pair, we
extracted the longest continuous duration where the longitudinal clearance
between the assumed leader and the assumed follower remained positive. These
extracted parts of trajectories are used for further analysis to ensure that
the potential vehicle-following periods are not discontinuous.

###  3.2 Intervening surrounding vehicles

Figure 2: Trajectory of a potential LF pair with surrounding vehicles and
intermittent following.

Figure 2 shows an example trajectory of a potential LF pair in the data. At
$t$ = 0 s, the red surrounding vehicle (SV1) is present between the potential
leader (Green) and follower (Blue), but it may not be influencing the LF
interaction. The SV1 moves away, and at $t$ = 2.5 s, there is no surrounding
vehicle between the leader and follower. But in the next few seconds, another
vehicle (SV2) cuts in. From $t$ = 5 s to $t$ = 7.5 s, we can say that the SV2
(purple) influences the leader-follower interaction. Intervening surrounding
vehicles are often observed in the NLB traffic. However, no methods in the
literature look at the potential intervening vehicles at each point of the
trajectory.

One of the goals of this work is to evaluate existing LF identification
methods. Limitations of the existing methods are pointed out in the following
section.

###  3.3 Limitation of existing LF identification methods

Existing LF identification methods are evaluated based on the procedure of
pair identification and the performance of identified LF pairs for calibration
of the Wiedemann-99 model. Existing methods are as follows:

  1. (a)

LF identification based on heuristic thresholds of longitudinal clearance and
time headway (Fernandez, 2011; Zhu et al., 2018; Sun et al., 2021; Anand et
al., 2019)

  2. (b)

LF identification based on hysteresis plots (Raju et al., 2021)

####  3.3.1 Limitations on heuristic thresholds based methods

(a)

(b)

Figure 3: Identification of the influence zone for one LF pair using a)
Longitudinal clearance threshold b) Headway threshold.

It is crucial to critically evaluate existing LF identification methods based
on physical thresholds, as changes in thresholds can have significant
implications for influence point determination. Figure 3 shows the effect of
changing thresholds for two of the heuristic methods. In the first method,
longitudinal clearance between leader and follower is used to classify points
into influencing or non-influencing. However, when the longitudinal clearance
threshold is changed from 20 s to 30 m, a considerable number of points in
blue got reclassified as influence points (Fig. 3(a)). We call the
reclassified points a mixed points. The same ambiguity can be observed in the
second method, where a time gap threshold is used to classify points into
influencing or non-influencing (Fig.3(b)). If the threshold is changed from 2
s to 2.5 s, a considerable number of points shown in blue are reclassified as
influence points. A mere change of 0.5 s in the headway threshold led to a
notable difference in the proportion of influence points. This is a drawback
of using heuristic-based methods to identify influence points.

####  3.3.2 Limitations of Hysteresis plots based LF identification

The core idea of the Wiedemann (1974) psycho-physical model revolves around
how drivers of faster-moving vehicles respond when approaching slower
vehicles. When these drivers reach their perception threshold, they initiate a
deceleration process. Nevertheless, due to the inherent challenges in
accurately estimating the speed of the lead vehicle, the driver’s velocity
might temporarily dip below that of the lead vehicle. Consequently, after
crossing another threshold, the driver may engage in a minor acceleration
phase. This cyclic pattern of acceleration and deceleration occurs
iteratively, giving rise to a phenomenon known as hysteresis, which can be
attributed to the imperfections in drivers’ ability to precisely gauge the
exact speeds of the lead vehicles.

(a)

(b)

(c)

(d)

Figure 4: Hysteresis plots for LF Identification

Figure 4 shows the hysteresis plots of a sample of LF pairs for the adapted
method of Raju et al. (2021) to identify LF pairs. Numbers associated with
points in the figure show time progression. Plot (a) shows clear hysteresis
behavior as, for the most part, the longitudinal clearance is almost constant;
hence, the follower is trying to maintain the same gap with the leader, and
the cyclic pattern is observed at relative speed zero. Thus, the pair can be
classified as an LF pair. In plot (b), the leader and follower are clearly not
interacting since the leader is significantly faster than the follower, so
there is no need for the follower to respond. Plot (c) is interesting since it
represents two episodes of the following phase with an approaching phase
between points 7 and 14. Similarly, in plot (d), the following phase is only
between points 1 to 12, and stimulus-response can be observed only in some
parts of the trajectories. Hence, the conclusions could be very subjective by
using this method of LF identification.

###  3.4 Performance of LF pairs for calibration of Wiedemann-99 model

In this subsection, LF pairs are identified using the existing methods given
in the literature, such as heuristics and hysteresis plots, as explained
above. These methods are evaluated by calibration of the Wiedemann-99 model by
identified LF pairs. The considered methods are shown below:

  * •

M1: All pairs are used.

  * •

M2: Clearance $<$ 30 m, % lateral overlap $>$ 0%, continuous following
duration $>$ 5 s Anand et al. (2019).

  * •

M3: Headway $<$ 2 s % lateral overlap $>$ 0%, continuous following duration
$>$ 5 s Anand et al. (2019).

  * •

M4: Clearance $<$ 30 m, % lateral overlap $>$ 50%, Total following duration
$>$ 5 s Chaudhari et al. (2022).

  * •

M5: Visual Observation of hysteresis plot Raju et al. (2021).

Figure 5: (a) Calibration Performance of existing LF Identification Methods
(b)% of LF Pairs by existing methods

Figure 5(a) shows the calibration performance of the Wiedemann-99 model using
existing methods for LF identification. The results show that the prediction
performance (RMSE) varies across the methods, and it depends on the various
arbitrary threshold values, which are subjective. Also, Figure 5 (b) shows the
percentage of LF pairs identified by different methods varies substantially,
indicating that these methods are not robust. Therefore, there is a need to
calculate threshold values tailored to specific traffic conditions which, in
turn, influence the calibration results. In the following, we propose a method
for joint and consistent LF identification and VF model calibration.

##  4 Methodology

This section presents an overview of the proposed optimization-based framework
consisting of two tasks:

  1. 1.

LF pair identification process given the set of VF parameters

  2. 2.

Calibration of VF parameters is carried out conditional on the set of
identified LF pairs from task 1.

Figure 6: Overview of Proposed Methodology for Joint and Consistent LF
Identification and VF parameter estimation

Due to their interdependence, the two tasks are carried out iteratively until
convergence. Figure 6 provides a visual summary of the proposed methodology. A
brief outline of the two tasks is given below, and a detailed discussion is
presented in the following subsections. The LF identification process invokes
three necessary conditions: strong interaction between the subject vehicle and
potential leader based on the driving regime, significant lateral overlap
between the pair, and the absence of any (substantially) intervening vehicle
between them.

We specify the condition of strong interaction in terms of driving regimes of
the Wiedemann-99 psycho-physical model depending on some VF parameters of this
model (Sect. 4.1.1). The VF parameters, in turn, are calibrated for identified
LF pairs. This mutual dependence between identifying LF pairs and calibrating
the VF parameters calls for consistency in estimating the two sets of
parameters.

The remaining criteria, clear gap between the pair (Sect. 4.1.2) and absence
of other intervening vehicles (Sect. 4.1.3), require the calibration of three
threshold parameters, namely 1. the gap threshold $c_{0}$ (negative, if an
overlap is required), 2\. the minimum duration $t_{\rm cont}$ of continuous
influence as defined by the interaction criterion above for the given
potential LF pair, and 3. the minimum fraction $f_{\rm min}$ of (possibly
discontinuous) influence between this pair. The lateral gap criterion is
applied at each time instance, whereas the remaining two criteria relate to
the entire trajectory of the subject vehicle, which is required to have a
minimum total duration trajectory of 5 s (which is not subject to
calibration). A potential LF pair that satisfies the three necessary
conditions (either continuously or with intermittent breaks) is deemed to be
an actual LF pair.

In the actual simultaneous calibration process for the VF and LF
threshold/identification parameters, we use as initial guesses the standard
Wiedemann-99 parameters (Vortisch, 2015), and $c_{0}=0.116\,\mathrm{m}$,
$t_{\rm cont}=5\,\mathrm{s}$, and $f_{\rm min}=0.36\,\mathrm{}$ as the initial
VF parameters. Initial guesses are identified by using 1000 randomly generated
sets of values, and set for parameters for which RMSE of the position is
minimum is considered as an initial guess for optimization. With these initial
values for both parameter sets, an initial set of LF pairs is identified, and
the Wiedemann-99 CF model is estimated by calibration on these LF pairs,
minimizing the RMSE of the positions, thus providing the first update of the
CF parameters.

Next, LF identification parameters are optimized using a nested optimization
approach as described in detail in Sect. 4.4. These updated VF and LF
parameters are fed back into the LF identification process, and the LF
identification process is iterated until convergence of both sets of LF pairs
and associated VF parameters, i.e., they are now mutually consistent. Once all
LF pairs are determined with calibrated parameters, the results as obtained
through different LF identification methods discussed in Sect. 3.4 are
validated using performance metrics based on Wiedemann-99 regimes
classification, estimated VF parameters, trajectory prediction accuracy, and
statistical validation via bootstrapping.

###  4.1 Determination of influence points

At a particular instant of time $t$, it is assumed that a subject vehicle is
influenced by another vehicle that is immediately ahead of it under the
following conditions:

####  4.1.1 Influencing regime criterion

The influencing regime criterion assesses whether, for a candidate LF pair,
the leader’s motion exerts an influence on the follower’s response as
evaluated by the Wiedemann-99 model (Vortisch, 2015). Specifically, the
subject vehicle (follower) is considered to be influenced by the leader if it
is in either the following, closing, or emergency braking regimes of this
model and satisfies the corresponding relative speed and spacing thresholds
(see Figure 7 below).

Figure 7: Influencing Region Based on Wiedemann-99 Model (Wiedemann, 1974)

In order to obtain a numerical criterion, we observe that there is _no_
influence if one or more of the following conditions are satisfied: the speed
of the vehicle exceeds its free flow speed, the relative spacing exceeds SDX
(or the minimum of SDX and SDV if the latter is defined), or the relative
speed is smaller than the corresponding OPDV threshold. If none of these three
conditions are satisfied, then the point belongs to either following, closing
or emergency braking and is thus considered an influencing point. It is
important to note that these thresholds depend on the vehicle following
parameters (CC0, CC1, CC2, CC3, CC4, CC5, and CC6), as discussed in Appendix
A. Thus, one of the inputs for the influence regime criterion consists of a
set of VF parameters necessitating feedback between the VF estimation and LF
pair identification procedures.

####  4.1.2 Lateral gap criterion

The lateral gap criterion examines if the potential LF pair is in sufficiently
close lateral proximity. This condition assumes that the leader can influence
the follower when the lateral clear gap is less than the threshold $c_{0}$, as
shown in Figure 8.

Figure 8: Lateral conditions to identify influence points showing ‘lateral
clear gap’ and ‘lateral overlap’ criteria.

The proposed lateral clear gap criterion is compared with two other lateral
criteria in the literature.

a. Percentage (%) lateral overlap (Kanagaraj et al., 2015),

$W_{\rm o}/W_{\rm f}>O_{\rm lat}.$ (1)

As shown in Figure 8, $W_{\rm o}$ is the overlapping width, $W_{\rm f}$ is the
follower’s width, and $O_{\rm lat}$ is a calibrated threshold. According to
this condition, the leader influences the follower if the lateral overlap
relative to the follower’s width is greater than the threshold. We emphasize
that this criterion is ineffective in the case of a two-wheeler as a leader
because it considers width $W_{\rm l}<O_{\rm lat}*W_{\rm f}$ that is exactly
centered ahead of the follower as non-interacting.

b. Absolute lateral overlap, $W_{\rm o}$$>$$O_{\rm abs}$ The percentage
overlap condition also depends on the width of the follower, which is not
considered by the absolute overlap condition. Both criteria assume a minimal
required positive overlap, unlike the proposed clear gap criterion.

####  4.1.3 No substantial intermediate vehicle effect criterion

This criterion checks whether there is any substantially influencing
intermediate vehicle between the subject vehicle and its potential leader. If
such a vehicle is present, the interaction between the subject vehicle and the
potential leader will be significantly weaker. Hence, the associated pair is
treated as a non-LF pair. For evaluating this criterion, a rectangular
intermediate zone (RIZ) is defined as the area enclosed by the horizontal
lines representing the rear of the potential leader and front of the subject
vehicle and vertical lines representing the left-most and right-most edges of
these two vehicles. Notice that this criterion is only evaluated, and an RIZ
is only defined if a candidate LF pair interacts according to the Wiedemann
regimes and satisfies the lateral gap criterion because all criteria for a
valid LF pair must be satisfied simultaneously.

Figure 9: Rectangular intermediate zone (RIZ) using the proposed method.

The rectangular intermediate zone is defined as shown in Figure 9 by dashed
lines, where,

  * •

Left boundary = $min(x_{\rm leader}^{\rm left},x_{\rm follower}^{\rm left})$;

  * •

Right boundary = $max(x_{\rm leader}^{\rm right},x_{\rm follower}^{\rm
right})$ ;

  * •

Back boundary = $y_{\rm follower}^{\rm front}$;

  * •

Front boundary = $y_{\rm leader}^{\rm back}$.

Based on the presence or absence of an intermediate vehicle in this zone
(RIZ), as well as the extent of the presence of the intermediate vehicle
(corner, edge, centroid presence) and overlap with potential leader and
follower, six possible cases are delineated as shown in Figure 10. During the
analysis of all leader-follower (LF) pairs, each data point is scrutinized to
determine the influence of other surrounding vehicles on the considered LF
pair. It is based on the factors such as vehicle sizes, positions, and lateral
overlaps. The various scenarios of intermediate vehicle effects are shown in
Figure 10 and described below:

Figure 10: Different categories of identification of external vehicle’s
influence on assumed leader and follower.

  1. (A)

No part of a vehicle is present in RIZ: This is a clear case that the
intermediate vehicle plays no role between the potential LF pair. This case
indicates the absence of external influence and, consequently, the likelihood
of vehicle-following between leader and follower provided other conditions
such as driving regime and lateral overlap are satisfied. In the empirical
data, 39829 time instances of this category were observed, where no other
vehicle is partially or fully detected within the rectangular boundary.

  2. (B)

Centroid (as shown in Figure 9) of at least one vehicle is present in RIZ: The
presence of centroid indicates a strong external influence of this intervening
vehicle, which disrupts vehicle-following between the subject vehicle and its
potential leader. In this dataset, 12805 time instances of this category are
observed. During these instances, due to this intermediate vehicle, the
interaction between the subject vehicle and the potential leader is assumed to
be non-following.

  3. (C)

Intermediate vehicle’s edges(s) is (are) detected in the RIZ ahead of the
follower and has lateral overlap with it: Although only the corner of the
intermediate vehicle is present, the considerable lateral overlap with leader
suggests a notable influence of the intervening vehicle. In the 2341 time
instances of this category, it is assumed that due to the effect of the
vehicle in-between, there is no following between the potential leader and the
subject vehicle. (If a time instance satisfies both (B) and (C), it is counted
for Criterion (B). Generally, a time instance is counted for the first
criterion that applies.)

  4. (D)

Intermediate Vehicle’s corner is detected in RIZ whose back edge is behind the
follower’s front edge, and has greater lateral overlap with the leader than
the lateral overlap between potential leader and follower: This is another
instance where despite the small presence of an intermediate vehicle in the
RIZ, it has a significant effect on disrupting vehicle following behavior due
to the considerable overlap with the leader. Hence, in the 974 time instances
of this category, the intermediate vehicle has a stronger effect than (F) and
is assumed to result in non-following.

  5. (E)

Intermediate vehicle’s corner(s) is (are) detected in the RIZ ahead of the
follower but without any lateral overlap: In this case, there is the marginal
presence of intermediate vehicle, but unlike case C it is much weaker due to
the lack of overlap with subject vehicle. Among the 1643 instances of this
category in the sample data, it is assumed that the intermediate vehicle is
staggered away from the LF pair and does not interfere with the vehicle-
following process.

  6. (F)

Intermediate Vehicle’s corner is detected in RIZ whose back edge is behind the
follower’s front edge and has smaller lateral overlap with the leader than the
lateral overlap between potential leader and follower: Due to the lower
marginal presence of an intermediate vehicle and lesser overlap with the
leader, in the 599 instances of this category, the leader-follower interaction
is not affected much by the intermediate vehicle. Hence vehicle-following
behavior is plausible for the LF pair.

For each case, the intermediate vehicle’s potential presence or absence of
influence on the potential LF pair is also analyzed. From Figure 10, it is
clear that cases B, C, and D correspond to the potential influence of the
intermediate vehicle; in the case of C and D, the intermediate vehicle’s
centroid doesn’t need to be present in RIZ., whereas A, E, and F represent the
relatively little impact of the intermediate vehicle. Thus, A, E, and F could
represent influencing points where the subject vehicle in blue is directly
influenced by the leader ahead in green, whereas, in the remaining cases, the
subject vehicle will not be considered to be influenced by the leader due to
the intervening vehicle effect. The instant of time, when all these conditions
are satisfied for a specific pair of follower and leading vehicles, is
considered an influence point for this pair. For a given potential LF pair,
the subject vehicle is under a given lead vehicle’s influence at some time
instants but not at others.

###  4.2 Determination of LF pairs from influence points

The presence of an influence point is suggestive but not confirmatory of a
following behavior between the leading and following vehicles. For instance,
the two vehicles may come into mutual influence even during lane change or
lateral movements, which may be momentary or temporary. Thus, influence points
are necessary but not sufficient to determine whether a given pair of vehicles
is indeed a leader-follower pair. To resolve this ambiguity, it is postulated
that for a subject vehicle that is following a lead vehicle, either the
duration of continuous influence and/or the fraction of total influence points
will be substantial. Accordingly, a given pair of vehicles is said to be in
leader-follower mode if one (or both) of the following conditions is
satisfied:

  1. 1.

Duration of continuous influence points ($t$) exceeds a minimum threshold
($t_{\rm cont}$) or

  2. 2.

The percentage of influence points in the subject vehicle’s trajectory is
greater than a given threshold $f_{\rm min}$.

The first condition implies that if the continuous duration when the subject
vehicle is likely to be influenced by the vehicle ahead is substantial (say
more than 1 minute), this is consistent with the subject vehicle following the
leader. The second condition implies that even if the duration of influence is
not continuous, but the subject vehicle was in close and direct proximity of
the vehicle ahead for a substantial fraction of its trip (say 50%), the
subject vehicle may be in the following mode.

###  4.3 VF Parameter Estimation

Algorithm 1 Calibration Methodology, adopted from Chaudhari et al. (2022)

1:Initialisation:

2:Initial Wiedemann-99 (CC) parameters

3:

4:for $follower(i)=1,2,....,N$ do $\triangleright$ $N$ is number of followers

5: $a^{\rm sim}_{i}(t=1)=a^{\rm obs}_{i}(t=1)$

6: $v^{\rm sim}_{i}(t=1)=v^{\rm obs}_{i}(t=1)$

7: $x^{\rm sim}_{i}(t=1)=x^{\rm obs}_{i}(t=1)$

8:

9: for $iteration(t)=1,2,\ldots,(T_{i}-1)$ do $\triangleright$ $t=1$ is the
begin of the leader-follower episode and $T_{i}$ is the duration of leader-
follower episode $i$ 1. Calculation of Wiedemann-99 thresholds AX, ABX, SDX,
CLDV, OPDV, and SDV 2. Identification of Regimes (Closing, Following, Free-
flow, Emergency Braking, and Opening) 3. Prediction of acceleration $a^{\rm
sim}_{i}(t+\tau)$ 4. Prediction of Speed $v^{\rm sim}_{i}(t+\tau)$ and
Position $x^{\rm sim}_{i}(t+\tau)$ using numerical integration method

10: end for

11: Predicted trajectory for follower $i$

12: $RMSE(x)_{i}=\sqrt{\frac{1}{T}_{i}\Sigma_{t=1}^{{T}_{i}}{[x^{\rm
obs}_{i}(t)-x^{\rm sim}_{i}(t)]^{2}}}$

13:end for

14:$Z(CC0,...,CC7)=\frac{\Sigma_{i=1}^{N}RMSE(x)_{i}}{N}$

15:$Minimize$ $Z$ to get optimized Wiedemann-99 (CC) Parameters

Once the LF pairs are identified by the above procedure (Section 4.2) for an
assumed initial set of VF parameters, the VF parameter estimation is performed
by calibration, as shown in Algorithm 1 and briefly outlined below. The leader
trajectory is assumed to be fixed. Based on initial VF parameters, regimes are
identified for each subject vehicle (follower) and each time $t$. For the
corresponding regime, the modified Wiedemann-99 model equations (as modified
in Chaudhari et al. (2022)) are used to compute the acceleration response of
the subject vehicle at time $t$\+ $\tau$ where $\tau$ represents the timestep.
Next, numerical integration is used to determine the speed at the next time
interval $t$\+ $\tau$, which is then integrated to compute the position at the
next time step. For this updated time step, the longitudinal gap and speed
difference with the leader are estimated, and the time step is incremented.
This process is continued until all points of the subject vehicle and all
subject vehicles are processed. The resultant output is the predicted
trajectory (position, speed, and acceleration profile) of the subject vehicle.
The RMSE in position between the predicted and actual trajectory of the
subject vehicle is computed and minimized to obtain the desired VF parameters.
Punzo et al. (2021) concluded that calibration on spacing is to be favored
over speed as minimization of spacing errors is equivalent to simultaneous
minimization and optimal time allocation of speed errors. Algorithm 1 gives an
overview of the VF calibration procedure.

###  4.4 Joint and Consistent Estimation Procedure

The proposed methodology aims to determine LF pairs and calibrate the
Wiedemann-99 car-following model jointly and consistently.

Overall, the combined LF identification and vehicle following model have a
total of 10 parameters, 7 out of 10 CC parameters for Wiedemann-99 models, and
3 for identification of LF pairs. Notice that the parameters CC0, CC6, and CC9
are not considered for calibration. According to Chaudhari et al. (2022),
CC0=0.65 m is a constant for a given vehicle type (here, cars), CC6 is assumed
as the default value, and CC9 is not used in the modified acceleration
equations. The LF identification thresholds include parameters on lateral
clearance (or other lateral criteria), duration of continuous following, and
fraction of influential points.

The joint and consistent estimation is accomplished by creating feedback
between the LF identification procedure, which takes VF parameters as input
and determines driving regimes, which in turn form the basis of classification
into influencing and non-influencing points. The resultant (tentative) LF
pairs are then fed into the VF model calibration procedure, and the VF
parameters are obtained. The updated VF parameters are then used in the next
iteration of LF identification, and the process continues until the
convergence of both LF identification and VF model parameters, which are
mutually interdependent. One key advantage of this procedure is that it
provides an optimization method to determine important thresholds for LF pair
identification in lieu of trial and error methods currently used in
literature. The objective function used in the above calibration is the
deviation between observed and computed longitudinal positions. Thus,
calibration is formulated as an optimization problem to determine the best set
of model parameter values that minimizes this RMS error. The objective
function is given by

${Z(\vec{\beta}_{\rm w},\vec{\beta}_{\rm
LF})}=\frac{\Sigma_{i=1}^{N}\sqrt{\frac{1}{T_{i}}\Sigma_{t=1}^{T_{i}}[x^{\rm
obs}(i,t)-x^{\rm sim}(i,t,\vec{\beta}_{\rm w})]^{2}}}{N}{\
\stackrel{{\scriptstyle!}}{{=}}\textrm{min}}$ (2)

where, Wiedemann parameters $\vec{\beta}_{\rm w}=(CC0,...,CC8)^{\prime}$, LF
parameters $\vec{\beta}_{\rm LF}=(\tau_{\rm cont},f_{\rm
min},c_{0})^{\prime}$, $T_{i}=T_{i}(\vec{\beta}_{\rm wr},\vec{\beta}_{\rm
LF})$, N=N$(\vec{\beta}_{\rm wr},\vec{\beta}_{\rm LF})$ is the number of LF
pairs that are used for model calibration. $x^{\rm obs}(i,t)-x^{\rm
sim}(i,t,\vec{\beta}_{\rm w})$ is the difference between observed and
simulated longitudinal coordinates of the follower for pair $i$, $x^{\rm
sim}(i,t)$ is obtained from the numerical integration of velocity which is
obtained from the numerical integration of acceleration. Notice that, because
of initially setting the simulated position to the data, the objective
function (2) is equivalent to minimizing the RMS of the gap deviations.

Figure 11: Convergence of Objective Function with the Nested Calibration

A nested calibration approach is used for this calibration, wherein,
initially, the LF parameters $\vec{\beta}_{\rm LF}$ are fixed, and the
Wiedemann parameters $\vec{\beta}_{\rm w}$ are estimated. In the next
iteration, $\vec{\beta}_{\rm w}$ is fixed, and $\vec{\beta}_{\rm LF}$ is
estimated, and so on. The process is repeated until both CC and LF (indicated
as LF identification parameters in Figure 11) parameters converge. Figure 11
shows the convergence pattern of the objective function. The nested
optimization process is more stable than a direct simultaneous estimation of
all the 10 parameters of ($\vec{\beta}_{\rm LF},\vec{\beta}_{\rm w})$ because
the number of LF pairs depends on $(\vec{\beta}_{\rm LF})$ resulting in
discontinuities in the objective function. Using the nested process restricts
the discontinuities to the calibration of the three LF parameters increasing
the stability and performance of the whole optimization process.

To summarize, a new joint and consistent framework is proposed to estimate LF
identification and vehicle following parameters simultaneously. This procedure
takes as inputs trajectory data and initial VF and LF starting values. By
optimizing the RMSE of position, the procedure produces the following outputs:
RMSE of position, number, and set of leader-follower pairs, converged VF
parameters, and LF identification thresholds. The performance of the proposed
procedure is evaluated with respect to other LF identification methods from
the literature in the following results section.

##  5 Results and Findings

This section presents a comparison between existing LF identification methods
and the proposed approaches. The comparison is conducted based on several
criteria, including $RMSE$ of position, the number of LF pairs, regime
distributions of LF and non-LF pairs, and performance on the validation set.
Additionally, a statistical comparison is made between different VF parameters
using bootstrapping. Table 2 provides a detailed description of the various LF
identification methods (existing and proposed) utilized in this study,
calibrated (non-calibrated for some methods) values of LF identification
parameters, additional thresholds required by each method for identifying LF
pairs, Goodness of Fit $(GOF)$ which is $RMSE$ of position for each method and
total number of LF pairs identified by each method (cf. Table 2).

  1. (a)

M1-M5 represent existing methods without modifying heuristic thresholds, as
explained in Section 3.3.

  2. (b)

M6-M7 existing methods (M2 and M3 respectively) have been adapted with the
proposed approach of joint and consistent calibration. In these cases,
thresholds are calibrated rather than relying on heuristic values.

  3. (c)

M8 is our proposed method that takes into account and calibrates the clear gap
threshold according to Algorithm 1 M9 is Method M8 without calibration of LF
identification parameters.

  4. (d)

M10-M11 are methods from the literature that involve calibrating LF
identification parameters. M10 is based on visual angle.(Yousif and Al-Obaedi,
2011). M11 uses Delaunay triangulation similar to (Nagahama et al., 2021) to
identify intermediate vehicle.

  5. (e)

M12-M13 are modified versions of the proposed methods (M8) considering
different heterogeneous features such as Absolute Lateral Overlap and %
lateral overlap instead of the lateral clear gap as proposed.

[b]

Method |  Description |  O_lat |  $t_{\rm cont}$ (Sec) |  $f_{\rm min}$ |  Additional Thresholds |  GoF (m) |  Number of LF pairs
---|---|---|---|---|---|---|---
M1 |  All Pairs |  – |  – |  – |  – |  7.06 |  623
M2 |  Longitudinal Clearance Heuristic (Neither Joint nor Consistent) |  0.0%* |  5* |  – |  Longitudinal Clearance = 30 m* |  6.40 |  274
M3 |  Headway Heuristic (Neither Joint nor Consistent) |  0.0%* |  5* |  – |  Headway = 2 s* |  5.77 |  241
M4 |  Influence Area Heuristic (Neither Joint nor Consistent) |  50.0%* |  – |  – |  Longitudinal Clearance = 30 m*, Total Duration of Following = 5 s* |  7.44 |  159
M5 |  Hysteresis Plots-based LF Identification by Raju et al. (2021) (Neither Joint nor Consistent) |  – |  – |  – |  – |  5.89 |  403
M6 |  Longitudinal Clearance Heuristic (Joint and Consistent) |  33.0% |  3.5 |  0.66 |  Longitudinal Clearance = 30 m* |  6.25 |  199
M7 |  Headway Heuristic (Joint and Consistent) |  63.0% |  6.5 |  0.60 |  Headway = 2 s* |  3.65 |  44
M8 |  Lateral Clear Gap (Joint and Consistent) |  – |  7 |  0.54 |  Lateral Clear Gap Threshold: 0.157m |  3.51 |  100
M9 |  Lateral Clear Gap (Neither Joint nor Consistent) |  – |  5* |  0.35* |  Lateral Clear Gap Threshold: 0.116m |  4.03 |  143
M10 |  Visual Angle (Joint and Consistent) |  -1.0%# |  7 |  0.65 |  Visual Angle Threshold = 0.0029 |  6.40 |  253
M11 |  Voronoi Triangulation (Joint and Consistent) |  – |  6.5 |  0.57 |  Clear Gap Threshold = 0.3m* |  3.62 |  116
M12 |  Absolute Lateral Overlap (Joint and Consistent) |  – |  5 |  0.45 |  0.007 m |  3.83 |  107
M13 |  Lateral % Overlap (Joint and Consistent) |  – |  4.5 |  0.55 |  % Lateral Overlap Threshold = 0.06% |  3.89 |  108

  * –

Threshold is not applicable

  * *

Uncalibrated or initial values

  * #

-1% indicates that influence would be there even for no overlap

  * •

Calibrated CC parameters of M8: CC0: 0.65 m; CC1: 0.30 s; CC2: 7.00 m; CC3:
-3.90 s; CC4: -2.34 m/s; CC5: 2.41 m/s; CC6: 11.44 1/ms; CC7: 0.61 m/$s^{2}$;
CC8: 3.09 m/$s^{2}$

Table 2: LF identification parameters for different methods

###  5.1 Benefits of Joint and Consistent Calibration

Figure 12(a) shows the benefit of joint and consistent calibration. For the
longitudinal clearance heuristic (M2), headway heuristic (M3), and lateral
clear gap model (M9), we can see an improvement in RMSE for the corresponding
‘joint and consistent’ calibration methods M6, M7, and M8 respectively. As
shown in Figure 12(b), the number of LF pairs identified by the joint and
consistent calibration are lower than that of the corresponding uncalibrated
models. For M7, the number of LF pairs drops down considerably because the
calibrated value of the minimum overlap $O_{lat}$ for joint and consistent
calibration is 63%, which implies a strict following condition. Based on the
performance metric, the proposed model M8 with joint and consistent
calibration shows the best results with the lowest GoF measure at a reasonable
number of LF pairs.

Figure 12: Benefits of joint and consistent calibration

###  5.2 Advantage of psychophysical regime based LF identification

Figure 13(a) points out the advantage of psychophysical regime-based LF
identification. Only the proposed model (M8) uses a psychophysical concept for
LF identification. For more details of the visual angle model (M10), the
reader is referred to Yousif and Al-Obaedi (2011). We calibrate the angular
velocity value for the visual angle model in the suggested range. In terms of
RMSE, the proposed model performs the best. The performance of the joint and
consistent headway heuristic (M7) is closer to that of the proposed model. In
Figure 13(b), a wide variation can be observed in the number of LF pairs
identified by each of the models. This shows the difference in selectivity for
LF pairs across methods.

Figure 13: Advantage of psychophysical regime based LF identification.

###  5.3 Regime Distribution of LF and Non-LF pairs

(a)

(b)

Figure 14: Wiedemann-99 regime distribution for LF pairs (a) and non-LF pairs
(b) using different methods of LF identification

To demonstrate the advantage of the psychophysical concept in LF
identification, we compared the regime distribution in LF and non-LF pairs for
5 methods as shown in Figure 14. The fraction of points in the following
regime of LF pairs (Figure 14 (a)) is the highest (67.20%) for the proposed
model (M8) which is an indicator of accurate LF identification. The second-
best following regime fraction for LF pairs is for the headway model (M7),
while other methods such as (M4) identified almost half of the points of LF
pairs in the emergency braking regime, which leads to inaccurate prediction of
trajectories as depicted with high GoF (Table 2). When regime distributions
for non-LF pairs are compared (Figure 14 (b)), the proposed model (M8) has the
highest fraction of free-flow regime (74.50%), which ascertains non-following
behavior in non-LF pairs. The free-flow regime fraction in non-LF pairs of the
headway model (M7) is considerably less than that of the proposed model (M8).
A relatively higher fraction following regime in non-LF pairs of the headway
model (M7) and a lesser number of LF pairs (17.20%) suggest that some of the
pairs which are LF pairs, are identified as non-LF pairs by the headway model.
Performance metrics and accurate LF identification results prove the superior
performance of the proposed model (M8) compared to all the other methods.

###  5.4 Comparison of Methods Using Two-Dimensional Criteria

While Methods M1-M7 only use longitudinal criteria, Methods M8-M13 make full
use of the two-dimensionality of non-lane-based mixed traffic by defining
lateral criteria and criteria for identifying intermediate vehicles.

Figure 15(a) shows the comparison of our method M8 with Delaunay triangulation
method M11 similar to (Nagahama et al., 2021) to identify an intermediate
vehicle. If an edge of the triangulation appears between the assumed leader
and follower, we say that the leader is influencing the follower at that
instant. For M11, we consider the value of lateral clear gap threshold $c_{0}$
= 0.3 m as considered in Nagahama et al. (2021), and with two variants M12 and
M13 of our method using absolute and relative overlaps (negative gaps),
respectively, instead of positive gaps. As shown in Figure 15(b), a lesser
variation is observed in the number of identified LF pairs. Consideration of
lateral gap and intervening surrounding vehicles brings two important features
of NLB traffic in LF identification and vehicle-following calibration. This
result suggests that consideration of two-dimensional traffic features
significantly improves model calibration. The proposed model, which does joint
and consistent calibration, performs the best. Amongst the different lateral
conditions, the superior performance of the lateral clear gap model suggests
that when a car is a follower, positive overlap with the leader may not be
required to have influence. Influence can be there even if there is a clear
gap. The calibrated value of the threshold for the lateral clear gap is 0.157
m. Hence, by comparison between M8 and M12-M13 we can say that gaps are better
suited to define thresholds than overlaps

Figure 15: Comparison methods using two-dimensional criteria

###  5.5 Statistical comparison of parameters using bootstrapping

In this section, we conduct a comparative analysis of VF parameters obtained
from bootstrapping using an unpaired t-test. As an example, we compare the
parameters obtained from the proposed method (M8) with those from the headway
method (M7). This procedure can be extended to all the methods. The aim of
bootstrapping is to estimate the variance of each CC parameter and to
determine if the parameter values obtained from the two methods can be
considered to be equal.

During the bootstrapping process, 100 pairs are selected with repetition from
the LF pairs identified by the proposed method. Similarly, 44 pairs are
selected from the headway method (M7). Notice that while the number of pairs
in the sets is the same as the available number of LF pairs for the respective
method, the sets are still different from the complete collection because, due
to the selection with repetition, some pairs are drawn several times and some
not at all. A hundred such sets are prepared for both methods. CC parameters
are then calibrated for each set, and sample standard deviation $s_{1}$ and
$s_{2}$ are reported in Table 3. The pooled standard deviation $s_{p}$ is
calculated using (3) and the standard error is calculated using (4). The
t-statistic for each parameter is calculated using (5). In this case, $n_{1}$
= $n_{2}$= 100 since there are 100 sets for each method.

$s_{\rm
p}=\sqrt{\frac{(n_{1}-1){s_{1}}^{2}+(n_{2}-1){s_{2}}^{2}}{n_{1}+n_{2}+2}}$ (3)

$SE(\bar{x_{1}}-\bar{x_{2}})=s_{p}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}$ (4)

$T=\frac{\bar{x_{1}}-\bar{x_{2}}}{SE(\bar{x_{1}}-\bar{x_{2}})}$ (5)

The null hypothesis $H_{0}$ states that the investigated parameter has the
same value for both methods. For CC1, CC2, CC3, CC4, CC5, and CC8, we can
reject $H_{0}$ at error probabilities $p<5\,\mathrm{\%}$ while, for CC7
($p=5.8\%$) there is not sufficient evidence for a rejection. Notice that,
during the bootstrapping investigation, we keep the LF parameters at the
values of Table 2.

Table 3: Statistical comparison of CC parameters using bootstrapping

Method |  | CC1 | CC2 | CC3 | CC4 | CC5 | CC7 | CC8
---|---|---|---|---|---|---|---|---
Proposed (M8) | Sample mean ($\bar{x}_{1}$) | 0.303 | 6.799 | -3.707 | -2.439 | 2.408 | 0.665 | 3.062
Sample Std. dev. ($s_{1}$) | 0.074 | 1.818 | 2.385 | 0.467 | 0.449 | 0.176 | 0.400
Headway (M7) | Sample mean ($\bar{x}_{2}$) | 0.559 | 9.090 | -7.559 | -1.721 | 1.916 | 0.603 | 2.843
Sample Std. dev. ($s_{2}$) | 0.136 | 1.929 | 2.360 | 0.459 | 0.472 | 0.213 | 0.412
| Pooled Std. dev. ($s_{p}$) | 0.122 | 2.273 | 2.911 | 0.569 | 0.560 | 0.232 | 0.495
| Standard error ($SE$) | 0.017 | 0.321 | 0.412 | 0.080 | 0.079 | 0.033 | 0.070
| t-statistic ($T$) | -14.907 | -7.126 | 9.357 | -8.927 | 6.223 | 1.903 | 3.134
| Statistical significance | Different | Different | Different | Different | Different | Same | Different

###  5.6 Performance of validation set

To validate the proposed methodology, it is necessary to assess its
performance on unseen data. The available data consists of 623 potential LF
pairs which are randomly divided into 70% for estimation and the remaining 30%
for holdout validation. The estimation set contains 436 pairs, and the holdout
set has 187 pairs. We perform the validation on all joint and consistent
methods from Table 2 (Figure 16). The proposed lateral clear gap (M8) method
clearly shows the best results with the lowest RMSE position.

Figure 16: Performance of different methods on the Validation set

The results show that the proposed joint and consistent framework performs
better for conventional heuristics as well as the proposed method. While, on
calibration, the methods M11-M13 nearly have the same fit quality as M8, there
is a distinct difference in validation, showing the robustness of the proposed
method.

##  6 Conclusion and Future Work

In this paper, a novel method is proposed for jointly and consistently
identifying LF pairs and calibrating vehicle-following model parameters in
non-lane-based traffic flow. Existing LF identification methods are compared
with the proposed method using empirical trajectory data from Chennai City,
India. The proposed methodology leverages the concept of the regime in the
psycho-physical Wiedemann-99 model to determine ”influence zones”. A logic to
identify potential intervening vehicles between leader and follower was
developed and used as one of the influencing conditions. Three LF
identification parameters are introduced and calibrated, namely ‘minimum
duration of the continuous following’ ($t_{\rm cont}$), ‘minimum fraction of
influence points’ ($f_{\rm min}$) and ‘lateral gap threshold’ $c_{0}$. Nested
calibration was employed which iteratively calibrates CC parameters and LF
identification parameters. The objective of the calibration was to minimize
the mean of RMSE of position for LF pairs.

The results, particularly the validation, demonstrate the superiority of the
proposed joint and consistent calibration procedure compared to models that do
not perform LF identification at all as well as those based on proximity or
hysteresis-based heuristics.

Overall, this study provides valuable insights into LF identification methods
and their impact on vehicle-following behavior. The proposed methodology can
be used to extract LF pairs from vehicle trajectory data under varying traffic
conditions as well as using alternative vehicle-following models. In similar
traffic conditions to our study area, the calibrated Wiedemann-99 car-
following parameters and LF identification parameters like lateral clear gap,
can be directly adopted to obtain LF pairs. Lateral thresholds and continuous
following duration thresholds could be used to further investigate lateral
(lane-changing) behavior of heterogeneous and non-lane-based traffic which is
an insufficiently explored research area.

This research has implications for traffic engineering, as the proposed
approach can enable robust and more realistic LF identification and VF
parameter calibration. The incorporation of these techniques within a
microsimulation framework can yield more accurate traffic fundamental diagrams
(FDs) for non-lane-disciplined traffic, which has a wide range of applications
including LOS, capacity, and travel time analysis. With a better understanding
of vehicle-following behavior, highway capacities can be estimated accurately
and FDs could be used to determine reliable travel time estimates. In our next
research thread, we aim to develop FDs using the traffic simulation packages.

There are several directions in which this work could be extended. Currently,
we only analyzed the behavior of the cars as a follower, but two-wheelers are
also significant contributors to the traffic volume in the study area.
Investigating the influence conditions for two-wheelers would improve traffic
simulation. Moreover, our proposed model focuses on vehicle-following behavior
irrespective of the type of leader, but, the following behavior could be
impacted by the vehicle type of the leader. The proposed methodology could be
utilized to understand the following behavior of various LF types. The
evaluation of alternative vehicle-following models within this framework is
yet another promising avenue of research. Also, we have only considered the
longitudinal aspects (VF), and future work includes developing a trajectory-
based calibration methodology for the fully 2D behavior, e.g., for the
Intelligent-Agent Model (Treiber and Chaudhari, 2023).

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

The work is funded by a scholarship from the Ministry of Education, Government
of India. DAAD (The German Academic Exchange Service), KOSPIE fellowship. And
DFG (Deutsche Forschungsgemeinschaft) under project grant no. 456691906,
”Enhancing Traffic Flow Understanding by Two-Dimensional Microscopic Models -
ETF2D” at TU Dresden. The authors also acknowledge Kanagaraj et al. (2015) for
sharing the trajectory dataset.

## CRediT authorship contributions statement

Mihir Mandar Kulkarni: Conceptualization, Methodology, Investigation, Writing
– original draft, Writing – review & editing. Ankit Anil Chaudhari:
Conceptualization, Methodology, Investigation, Writing – original draft,
Writing – review & editing. Karthik K Srinivasan: Conceptualization,
Methodology, Writing – review & editing. Bhargava Rama Chilukuri:
Conceptualization, Methodology, Writing – review & editing. Martin Treiber:
Conceptualization, Methodology, Writing review & editing. Ostap Okhrin:
Conceptualization, Methodology, Writing – review & editing. All authors
reviewed the results and approved the final version of the manuscript.

## Declaration of generative AI and AI-assisted technologies in the writing
process

During the preparation of this work, the author(s) used ChatGPT in order to do
grammar checks and language corrections. After using this tool/service, the
author(s) reviewed and edited the content as needed and take(s) full
responsibility for the content of the publication.

##  Appendix A Wiedemann-99 Model Regime Classification

The equations for calculating Wiedemann-99 regimes based on vehicle-following
parameters are as follows, (Chaudhari et al., 2022):

$\displaystyle AX$ $\displaystyle=CC0,$ $\displaystyle ABX$
$\displaystyle=CC0+CC1\cdot V_{\text{slow}},$ $\displaystyle SDX$
$\displaystyle=ABX+CC2,$ $\displaystyle CLDV$
$\displaystyle=CC5+\frac{CC6}{17000}\cdot DX^{2},$ $\displaystyle OPDV$
$\displaystyle=CC4-\frac{CC6}{17000}\cdot DX^{2},$ $\displaystyle SDV$
$\displaystyle=CC5+\frac{DX-SDX}{CC3},$

where

  * •

AX: the desired distance between two vehicles in a stopped condition;

  * •

ABX: the desired minimum safe following distance in moving state, as a lower
limit of the following regime;

  * •

SDX: the maximum following distance as the upper limit of the following
regime;

  * •

CLDV: the points at short distances (less than SDX) where drivers perceive
that their speeds are higher than their lead vehicle speeds;

  * •

OPDV: the points at short distances (less than SDX) where drivers perceive
that they are traveling slower than their leader;

  * •

SDV: the points at long distances (more than SDX) where drivers perceive that
they are approaching slower vehicles.

Wiedemann-99 car following parameters are explained as follows:

  * •

CC$0$ [0.65 m]: The desired gap between two vehicles in a stopped condition;

  * •

CC$1$ [0.9 s]: Time gap the following driver keeps for a safety-in-moving
state;

  * •

CC$2$ [4 m]: Range of the gap between vehicles in the following regime;

  * •

CC$3$ [-8 s]: The time between the beginning of deceleration after perceiving
a slow-moving leader and starting the unconscious following behavior;

  * •

CC$4$ [-0.35 m/s]: Speed difference during the following process— CC$4$
controls speed differences during the opening process (negative relative
speed);

  * •

CC$5$ [0.35 m/s]: Speed difference during the following process— CC$5$
controls speed differences in the closing process (positive relative speed);

  * •

CC$6$ [11.44 1/ms]: Influence of distance on speed oscillation during the
following condition;

  * •

CC$7$ [0.25 m/$s^{2}$]: Actual acceleration during oscillation in the
unconscious following regime;

  * •

CC$8$ [3.5 m/$s^{2}$]: Desired acceleration when the vehicle starts from the
standing condition.

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