Article

A simple nonparametric car-following model driven by field data

Abstract

Car-following models are always of great interest of traffic engineers and researchers. In the age of mass data, this paper proposes a nonparametric car-following model driven by field data. Different from most of the existing car-following models, neither driver's behaviour parameters nor fundamental diagrams are assumed in the data-driven model. The model is proposed based on the simple k-nearest neighbour, which outputs the average of the most similar cases, i.e., the most likely driving behaviour under the current circumstance. The inputs and outputs are selected, and the determination of the only parameter k is introduced. Three simulation scenarios are conducted to test the model. The first scenario is to simulate platoons following real leaders, where traffic waves with constant speed and the detailed trajectories are observed to be consistent with the empirical data. Driver's rubbernecking behaviour and driving errors are simulated in the second and third scenarios, respectively. The time-space diagrams of the simulated trajectories are presented and explicitly analysed. It is demonstrated that the model is able to well replicate periodic traffic oscillations from the precursor stage to the decay stage. Without making any assumption, the fundamental diagrams for the simulated scenario coincide with the empirical fundamental diagrams. These all validate that the model can well reproduce the traffic characteristics contained by the field data. The nonparametric car-following model exhibits traffic dynamics in a simple and parsimonious manner.
A simple nonparametric car-following model driven by field
data
Zhengbing He
a
, Liang Zheng
b
, Wei Guan
a,
a
MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing, China
b
School of Traffic and Transportation Engineering, Central South University, Changsha, China
article info
Article history:
Received 26 November 2014
Received in revised form 14 July 2015
Accepted 15 July 2015
Keywords:
Car-following model
Data-driven approach
k-nearest neighbour
Traffic oscillation
Fundamental diagram
abstract
Car-following models are always of great interest of traffic engineers and researchers. In
the age of mass data, this paper proposes a nonparametric car-following model driven
by field data. Different from most of the existing car-following models, neither driver’s
behaviour parameters nor fundamental diagrams are assumed in the data-driven model.
The model is proposed based on the simple k-nearest neighbour, which outputs the
average of the most similar cases, i.e., the most likely driving behaviour under the current
circumstance. The inputs and outputs are selected, and the determination of the only
parameter kis introduced. Three simulation scenarios are conducted to test the model.
The first scenario is to simulate platoons following real leaders, where traffic waves with
constant speed and the detailed trajectories are observed to be consistent with the empir-
ical data. Driver’s rubbernecking behaviour and driving errors are simulated in the second
and third scenarios, respectively. The time–space diagrams of the simulated trajectories are
presented and explicitly analysed. It is demonstrated that the model is able to well repli-
cate periodic traffic oscillations from the precursor stage to the decay stage. Without mak-
ing any assumption, the fundamental diagrams for the simulated scenario coincide with
the empirical fundamental diagrams. These all validate that the model can well reproduce
the traffic characteristics contained by the field data. The nonparametric car-following
model exhibits traffic dynamics in a simple and parsimonious manner.
Ó2015 Elsevier Ltd. All rights reserved.
1. Introduction
Car-following models are always of great interests of traffic engineers and researchers. As one of the most important traf-
fic analysis tools, the first car-following models (Pipes, 1953; Chandler et al., 1958) were proposed as early as sixty years ago.
From then on, a large number of car-following models were addressed. For example, the optimal velocity model (Bando et al.,
1995) and the full velocity difference model (Jiang et al., 2001) were proposed based on the intuitive speed changing of vehi-
cles, and the well-known Gipps model (Gipps, 1981) and the intelligent driver model (Treiber et al., 2000) were proposed in
the perspective of driver’s acceleration and deceleration strategies. More recently, to study the trigger and formation of the
stop-and-go traffic oscillations, Laval and Leclercq (2010) and Chen et al. (2012b) incorporated non-equilibrium (timid and
aggressive) driver behaviour into Newell’s simplified car-following model (Newell, 2002). Laval et al. (2014) captured traffic
oscillations by using a desire acceleration model with a white noise term. It showed that small driver errors can result in the
http://dx.doi.org/10.1016/j.trb.2015.07.010
0191-2615/Ó2015 Elsevier Ltd. All rights reserved.
Corresponding author.
E-mail addresses: he.zb@hotmail.com (Z. He), zhengliang@csu.edu.cn (L. Zheng), weig@bjtu.edu.cn (W. Guan).
Transportation Research Part B 80 (2015) 185–201
Contents lists available at ScienceDirect
Transportation Research Part B
journal homepage: www.elsevier.com/locate/trb
stop-and-go oscillation. Detailed introduction about more car-following models can be found in the review articles (see e.g.
Brackstone and Mcdonald, 1999; Hoogendoorn, 2001; Helbing, 2001; Saifuzzaman and Zheng, 2014). Performance
comparison amongst important car-following models can be found in the recently published book (Treiber and Kesting,
2013b). In the work, the simulated time–space diagram and the fundamental diagrams were used as core elements to
evaluate performance in a figure so-called ‘‘fact sheet’’.
A process prior to analysing or simulating real traffic dynamics is calibration, including the fundamental diagrams, dri-
ver’s behaviour parameters, etc. Only calibrated models are able to approximate real traffic dynamics and conditions.
Calibration, however, is a challenging topic needed to be deeply investigated (see e.g. Brockfeld et al., 2004; Kesting and
Treiber, 2008; Punzo et al., 2012; Treiber and Kesting, 2013a). Even the shapes of the fundamental diagrams in assumptions
are still controversial (see e.g. Kerner, 1998; Schönhof and Helbing, 2009; Treiber et al., 2010).
Recently arising high-fidelity traffic data and data-driven modelling approaches provide an opportunity to stride over the
modelling and calibrating, and to extract traffic dynamics and driver’s behaviour directly from mass field data. The ‘‘manual’’
errors produced in modelling and calibrating are not a concern of these approaches. Artificial neural networks (ANN),
support vector regression (SVR), nonparametric regression, etc. are all prevailing data-driven modelling approaches. There
approaches have been widely applied and investigated in the field of short-term traffic flow forecasting; see e.g. Zhang
and Ge (2013), Wu et al. (2004) and Zheng and Su (2014) and a recent review article (Vlahogianni et al., 2014). In recent
years more high-fidelity traffic data, such as the trajectory data provided by Next Generation Simulation Project (NGSIM,
2006), are collected and released. This makes it possible to model car-following behaviour directly from a large number
of field data.
In the studies utilising the data-driven approaches to model driver’s behaviour, Panwai and Dia (2007) proposed a
car-following model by using ANN with the back-propagation and fuzzy ARTMAP architectures. The model classified field
data into five driving modes. Before moving, each vehicle was classified into a mode first. Space headway and leader’s speed
were the inputs of ANN and the follower’s speed were the output. The dataset for training was limited, which was collected
by an instrumented following vehicle for only 300 s. Simulation results showed that the model outperformed the Gipps and
psychophysical models. Colombaroni and Fusco (2014) also modelled car-following behaviour by using ANN. Car-following
data were collected by using several global-positioning-system-equipped vehicles that followed each other on urban roads.
More variables were added into the training inputs in order to incorporate driver’s reaction delay. Nevertheless, the analyses
and comparisons based on several immediate followers might be insufficient in the above two literature.
Forward-propagating traffic waves triggered by leader’s deceleration could be observed in Fig. 13 in Colombaroni and
Fusco (2014), which might be unrealistic. In Khodayari et al. (2012), four inputs, i.e., reaction delay, relative speed, relative
distance and follower’s speed, were chosen to train ANN, and the follower’s acceleration was output. Although the proposed
model was studied and compared, the study was only limited in leader–follower pairs. Zheng et al. (2013) further presented
an explicit reaction delay model based on ANN, and tested the model in a nine-vehicle platoon following a given leader. By
utilising SVM, Wei and Liu (2013) investigated the asymmetric characteristic in car-following behaviour and its impact on
traffic flow evolution. Space headway, follower’s speed, and relative speed were taken as the inputs, and follower’s speed was
output. Similarly, in the model test only an immediate follower was simulated and compared with the real immediate one.
In the existing literature of data-driven modelling, most of the inputs and outputs, such as reaction delay and accelera-
tion, are continuous variables, which are also indirect to computer simulation; only a few followers are simulated to test the
model, which is insufficient; ANN and SVR are complex in architecture. These are expected to be improved, in particular
when apply them in a simulation scenario. To this end, the paper proposes a simple nonparametric car-following model dri-
ven by field data. In this model, all inputs and outputs are based on vehicle positions, which are straightforward to reproduce
traffic dynamics; thousands of vehicles are simulated to test and validate the model; k-nearest neighbour, one of the most
simplest data-driven approach, is employed, which is simple and parsimonious. It is demonstrated that without making any
assumption or calibration, the model is able to well replicate important traffic characteristics contained by the field data,
such as the traffic oscillations from the precursor stage to the decay stage and the fundamental diagrams.
The remainder of the paper is organised as follows: The field data driving the nonparametric model are introduced in
Section 2; the model is proposed in Section 3; a simulation scenario of platoons following real vehicles is presented in
Section 4in order to primarily validate the model; Section 5simulates a 1.25 km roadway with rubbernecking, and the
results are thoroughly analysed and compared; Section 6introduces driver errors into the model, and the response of the
model to small perturbations is demonstrated; conclusions are made at last.
2. Site description and the data
The nonparametric car-following model proposed in the paper is driven by a large number of field data. The well-known
NGSIM providing high-fidelity trajectory dataset satisfies the demand. We propose and analyse the nonparametric
car-following model based on the trajectory dataset collected on a 6-lane segment in the vicinity of Lankershim Avenue
on southbound US-101 freeway in Los Angeles, California. The time period of data collection ranges from 7:35 a.m. to
8:35 a.m. on June 15, 2005. We adopt the trajectory dataset collected on Lane 1 (median lane), 2, and 3, where lane changes
are relatively few. Most of the speeds are less than 60 km h
1
(except a few faster vehicles leaving the segment), which indi-
cates that the segment is congested during the study period. Stop-and-go oscillations originate and propagate upstream at a
186 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
constant wave speed of approximately 16 km h
1
. The original dataset provides a data sample every 0.1 s. This paper simply
averages these data sample every 1 s (arithmetic mean) to smooth out the detection errors (for the feature of the errors, see
e.g. Thiemann et al. (2008); Punzo et al. (2011); Montanino and Punzo (2013)). It is also consistent with the selected simu-
lation time step in the upcoming simulation scenarios.
3. Nonparametric car-following model
3.1. k-nearest neighbour approach
k-nearest neighbour (kNN) is a nonparametric approach that is very simple to understand but works incredibly well in
practice. The approach selects the most similar historical cases, and takes the average of their outputs as the estimate of this
time. Specifically, the approach estimates y
0
in focal ðx
0
;y
0
Þas follows.
^
y
0
¼P
k
i¼1
y
i
kð1Þ
where x
i
with respect to y
i
is one of the kmost similar samples to x
0
. Distance between points or vectors in space is usually
used to measure the similarity. A basic assumption of kNN is that history are repeating. It is reasonable to believe that drivers
are repeating their behaviour in the similar circumstances including traffic and geography conditions. Thus, it is suitable for
kNN to model car-following behaviour.
There are still several more advanced nonparametric approaches, such as kernel estimation and loess (Cleveland, 1979).
However, more parameters or functions are introduced in these approaches. To keep parsimony of the model, we adopt kNN,
where only kis needed to be specified.
3.2. Inputs and outputs of the model
It is known that when a vehicle follows a leader, the movement of the vehicle is closely related to the speed and accel-
eration of its leader, as well as its speed, acceleration, and space headway. In the model we directly input vehicle positions,
and output vehicle positions. The inputs and outputs of positions avoid numerically solving differential equations that are
required by many traditional parametric car-following models (Chapter 10.2 in Treiber and Kesting (2013b)), and can be
directly used in simulation.
Specifically, we propose the input vector of kNN (refer to Fig. 1)as
x
n
ðtþ
s
Þ¼ðd
n1
ðtþ
s
Þ;d
n1
ðtÞ;s
n
ðtÞ;s
n
ðt
s
ÞÞ:ð2Þ
where
s
is the simulation time step; ðn1Þis the leader of vehicle n;d
n1
ðtÞis the moving distance of the leader between
time ðt
s
Þand tgiven as:
d
n1
ðtÞ¼l
n1
ðtÞl
n1
ðt
s
Þ ð3Þ
where l
n
ðtÞis the position of vehicle nat time t;s
n
ðtÞis the space headway of vehicle nat time t:
s
n
ðtÞ¼l
n1
ðtÞl
n
ðtÞ ð4Þ
In this paper, we set
s
to be 1 s, and the data samples in the database are averaged every 1 s accordingly. Note that the mov-
ing distance corresponds to the speed of a vehicle, i.e., moving distance over a simulation step, and two successive moving
distances imply the acceleration of a vehicle. Through the space headway, the speed and acceleration of both the leader and
follower are included in the inputs.
The output is the moving distance of vehicle n, i.e.,
y
n
ðtþ
s
Þ¼d
n
ðtþ
s
Þ ð5Þ
Theoretically, the inputs and outputs will not introduce back-forward movement. The estimate of kNN is an average of
selected historical cases. No negative moving distance exists in field data, and taking average of non-negative numbers will
not result in a negative number.
In addition, the computational efficiency is usually concerned when we apply or train a data-driven model. In particular,
the input of this model is four-dimensional, which further increases the burden. Reducing search time is a hot topic in com-
puter science. A k-dimensional tree is a useful method, which can reduce the time complexity of kNN. Making use of a hash
table can put the searching of four-dimensional samples off-line, and then desirable simulation speed would be achieved.
These are all effective speed-up methods for a data-driven model in computer science.
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 187
3.3. Elimination of autocorrelation
Because of strong autocorrelation of time-series trajectory data, it is easy for all ksamples coming from a leader–follower
pair, in particular when the leader and follower move in a constant speed. Such dominance could reduce the reliability of the
model. To overcome this issue, we make all ksamples selected from different leader–follower pairs.
3.4. Distance between two data samples
This paper adopts the ‘‘ordinary’’ and simple scaled Euclidean distance, i.e., adjusting the input x
ji
2x
i
by its mean
x
j
and
standard deviation S
j
before calculating Euclidean distance in a metric space, where jindicates an element in an input vector.
The formula reads
x
i
;x
0
Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
J
j¼1
ðz
ji
z
j0
Þ
2
v
u
u
tð6Þ
where
z
ji
¼x
ji
x
j
S
j
;ð7Þ
and Jis the total number of all elements in an input vector.
3.5. Determination of k and similarity
The parameter kin kNN indicates the number of the historical cases that are considered to be similar to the estimated
case. It apparently relates to the underlying database. Usually, kis estimated by experience. In the paper, we determine k
by comparing estimation errors under different k-values. To measure the similarity of the kselected samples to the esti-
mated case, we employ the scaled Euclidean distance of kth nearest sample to the estimated case, which is the longest dis-
tance in all ksamples. Denote by D
k
the distance.
Before introducing the determination, we first build the following three databases with different sizes by using the data-
sets collected on Lane 2 and 3. (i) Database 1: the small-size database containing 22,202 input–output samples, which is
built by using the dataset collected on Lane 2 during the first 15 min; (ii) Database 2: the medium-size database containing
78,683 samples, which is built by using the dataset collected on Lane 2 during the all study 45 min; (iii) Database 3: the
large-size database containing 152,637 samples, which is built by using the dataset collected on Lane 2 and 3 during the
all study 45 min. No data from Lane 1 is included in these three databases. Such databases are employed to estimate the
movement of the followers on Lane 1. The purpose is to avoid involving the empirical followers to be estimated into the
database.
Now we estimate the followers of two typical vehicles (i.e., Vehicle 422 and 1989) who traverse stop-and-go oscillations
on Lane 1, and present the detail in Fig. 2. In each estimation, the inputs are all real field data, and the output is estimated
moving distance of the follower. Database 3 is used in this test, and the absolute error in the figure is the absolute difference
between the estimated and real values. From the upper plots of Fig. 2, it can be seen that smaller kbasically makes smaller
Fig. 1. Inputs and outputs of the proposed nonparametric car-following model.
188 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
errors, but the errors may fluctuate more. One can compare the green and blue lines in the figure, for example. It is easily
understood that the closer historical samples better reflect estimated case, but averaging a small number of samples results
in instability. This is a general tendency but not always true, because we can also see a few good estimations with long dis-
tances (for example, the points with absolute errors of around zero in the blue line), which relates to the quality of database
and the number of similar historical samples. The lower plots of Fig. 2 present the distance D
k
of each estimation. The figure
shows that for Database 3, when k¼10, the estimations are usually good with the distance smaller than 0.2, i.e. D
k
<0:2.
To make a more general conclusion, we estimate the movement of more followers. We select automobile leader–follower
pairs who move over 200 m on Lane 1 during the first 15 min. These vehicles are considered to be stable on Lane 1, i.e., hav-
ing little impact from lane-changing. To be representative, a part of these pairs (50 pairs, here) are randomly selected from
total 371 pairs on Lane 1. Database 3 is used, and total 2918 estimations are made. Note that using a random part of vehicles
instead of all is to show that a randomly selected part of samples may be sufficient to determine a proper k. Besides, as the
traffic on Lane 1 is similar to those on Lane 2 and 3, Lane 1 may be able to reflect the driving behaviour contained by the
database. Similar tendency of k-values is observed when we increase the size of the part.
The statistical results are illustrated in Fig. 3. From the upper plot in Fig. 3, it can be seen that there indeed exists an opti-
mal k-value for minimum absolute errors. For Database 3, the value is around 10. When k-value is smaller or bigger than the
optimal value, the mean and standard deviation of the absolute errors become larger simultaneously. From the lower plot in
Fig. 3, it can be seen that the distance is positively correlated to k-values. Moreover, for the optimal k¼10, the distance D
k
(a
mean plus a standard deviation) is smaller than 0.2. It implies that if we set k¼10, an estimation is usually good when
D
k
<0:2. It is noticed that the estimation is not very sensitive to the k-value for the database. Seeing Fig. 3, specifying
k¼9ork¼11 for Database 3 may be also able to result in satisfied results.
The size of the database also affects the estimation; see Fig. 4. In the figure, kis fixed to be 10, and Database 1, 2, and 3 are
used to estimate, respectively. It is clear that the larger Database 3 results in better estimations (i.e., lower mean and stan-
dard deviation of the absolute errors), and further shortens the distance D
k
.
Therefore, we specify k¼10 for the database built on the US-101 dataset, and an estimation is considered to be satisfied if
D
k
<0:2. In practice, one can determine kby comparing the estimation results with different k-values as this subsection did.
This determination is a premise to apply kNN, because the optimal k-value is highly related to the underlying database.
In a simulation scenario, any situation may occur, and we cannot guarantee that there are historical cases that are close to
every estimated case. It thus requires us to monitor D
k
in order to judge if an estimation is within the scope of the underlying
database. A time–space diagram of D
k
is designed to play the role in Section 5. Large distance means the simulation result
may not be reasonable, and substitution is needed. One may add more field data in the database, or simulate this case using a
traditional parametric car-following model.
3.6. Analysis on avoiding collisions
The kNN approach is based on the assumption that history repeats. Generally speaking, averaging similar collision-free
historical cases would not lead to a collision. As a data-driven approach, however, the collision-free may not be mathemat-
ically proved. Instead, we can show the collision-free in statistic. To this end, we calculate the relative error of the estimated
space headway as follows:
10 20 30 40 50 60
−4
−2
0
2
Absolute error (m)
k=200 k=100 k=50 k=20 k=10
10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
Time (sec)
Distance
200th 100th 50th 20th 10th 1st
(
a
)
Following Vehicle 422 on Lane 1
10 20 30 40 50 60
−2
0
2
Absolute error (m)
k=200 k=100 k=50 k=20 k=10
10 20 30 40 50 60
0
0.2
0.4
0.6
Time (sec)
Distance
200th 100th 50th 20th 10th 1st
(
b
)
Following Vehicle 1989 on Lane 1
Fig. 2. Estimation errors under different k-values, and the distance D
k
between estimated and historical cases. The leaders are Vehicle 422 and 1989 on
Lane 1 on the US-101 segment, respectively. Database 3 is used.
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 189
RE ¼s
sim
n
ðtÞs
data
n
ðtÞ
s
data
n
ðtÞð8Þ
where s
sim
n
ðtÞand s
data
n
ðtÞare the estimated and ground-truth data, respectively. All automobile leader–follower pairs (total
371 pairs) moving over 200 m on lane 1 is estimated by using Database 3. In the results, 19,035 out of 20,733 estimations
(91.8%) have the distances D
k
smaller than 0.2, and the results are presented in Fig. 5. It shows that the relative errors are
around zero and the maximum negative error is about 0.3. For a minimum space headway of around 6 m, the magnitude of
the relative errors implies collision-free in a single estimation.
1
There is still an extreme situation needed to be treated carefully. When both the leader and follower are in a standstill at
time t(i.e., d
n1
ðtÞ¼0 and s
n
ðtÞ¼s
n
ðt
s
Þ) and the leader has not moved yet at time (tþ
s
) (i.e., d
n1
ðtþ
s
Þ¼0), the move-
ment of the follower at time (tþ
s
) can be predetermined to be zero, and no estimation is needed.
2
Whilst in the situation,
kNN may make the standstill follower begin to approach its standstill leader, because kNN averages similar samples but not
exactly same samples, and any selected historical sample outputting a (even very small) positive movement of the follower will
result in the unrealistic movement. Thus, when dn1ðtÞ¼0;snðtÞ¼snðtsÞand dn1ðtþsÞ¼0, we should directly set the
output dnðtþsÞ¼0.
110 20 30 40 50 100 200
0.5
0.7
0.9
1.1
Absolute error (m)
mean
standard deviation
6 7 8 9 10 11 12 13 14 15 16
0.5
0.52
0.54
110 20 30 40 50 100 200
0
0.1
0.2
0.3
0.4
0.5
k−value
distance
mean
standard deviation
6 7 8 9 10 11 12 13 14 15 16
0.09
0.1
0.11
0.12
Fig. 3. Estimation results of following 50 randomly selected leaders on Lane 1 by using Database 3 and various k-values.
1 2 3
0.4
0.5
0.6
0.7
Absolute error (m)
mean
standard deviation
1 2 3
0.05
0.1
0.15
0.2
Database
distance
mean
standard deviation
Fig. 4. Estimation results of following 50 randomly-selected leaders on Lane 1 by using Database 1–3 and k= 10.
1
Although no collision has been observed in our analysis, it may occur when the model is applied to other data due to the data-driven nature of kNN.
2
In practice, it is uneasy to capture that snðtÞ¼snðtsÞdue to the discrete nature of compute simul ation. One could approximate it with
snðtÞ;snðtsÞ6smin.
190 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
3.7. Transferability of the model and the database
Considering the basic assumption of kNN, i.e., drivers repeat their behaviour in similar circumstances, an underlying data-
base could be well transferred to any site with similar circumstances including driving habits, roadway geometry, etc. To
show the transferability of the model and its database, we estimate driving behaviour from other site by using the
US-101 database. Another well-known trajectory dataset collected from I-80 by NGSIM is ideal to do this. The I-80 segment
is located in Emeryville, California. The length is about 500 m, and there are six lanes. From the left (median lane) to the right,
we label them as Lane 1–6. Note that Lane 1 is a High Occupancy Vehicle lane. The trajectory data were collected for 45 min
(4:00–4:15 p.m. and 5:00–5:30 p.m.) on April 13, 2005. We may consider the driving behaviour on the I-80 and US-101 seg-
ments are similar, because both of them are typical freeways in U.S.
This test for transferability estimates the movement of the followers on Lane 2 and 3 in the I-80 segment by using
Database 3 built on the US-101 database. Similarly to the above approach, we obtain 347 and 206 automobile leader–fol-
lower pairs who move over 200 m on the lanes during the first 15 min. For Lane 2, we make 17,447 out of 18,186 (95.9%)
estimations with D
k
<0:2, and for lane 3 it is 8905 out of 9359 (95.1%) estimations. Fig. 6 presents the histogram plots
for the relative errors of the estimated space headway (see Eq. (8)). It can be seen that most of the relative errors are around
0 and between 0.1 and 0.1, and the maximum deviations are about 0.2 and 0.2 (except really few estimation with large
deviation). It indicates that the US-101 database could be transferred to the case of I-80.
4. A simulation scenario of a platoon following a real vehicle
To test this proposed nonparametric car-following model, this section simulates platoons following real vehicles. Two
typical vehicles who traverse stop-and-go oscillations on Lane 1, i.e., Vehicle 422 and 1989, are employed as the empirical
leaders. Database 3 without the data coming from Lane 1 is used.
4.1. A platoon with real boundary conditions
This subsection estimates platoons following the real leaders with real entry boundary conditions. The real entry bound-
ary conditions mean that the entry speed and time intervals of simulated followers are the same to those of the real follow-
ers. At each simulation step, we input the current situation of the estimated vehicle into the database, search for the k
nearest samples, and average their output as an estimate of its movement.
Total 23 and 31 followers are simulated in the platoons following the two vehicles, respectively. The time–space diagrams
of the real and simulated trajectories are plotted and compared in Fig. 7. We mark the estimated cases with the distance D
k
in order to monitor the similarity between the estimated case and the database. In the simulation scenario most of D
k
are
smaller than 0.1, which means that the database is proper to underlie the simulation scenario.
From Fig. 7, it can be seen that the time–space shapes of simulated platoons are close to the real ones. The wave speeds
revealed in the simulated and real platoons are all about 16 km h
1
. It means that the congestion propagates in similar speed.
To compare more clearly, we additionally plot the last vehicles of the real platoons in the simulated platoons (the black
curves in Fig. 7(b) and (d)). In the comparison of the last vehicles, Fig. 7(d) shows satisfied accuracy, whilst a relatively large
deviation occurs in Fig. 7(b). The deviation is resulted from a vehicle who fails to follow its leader along the wave not as most
of the vehicles usually do; see the trajectory pointed by an arrow in Fig. 7(a). This random behaviour diminishes the wave,
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
0
200
400
600
800
1000
1200
1400
1600
1800
Relative error
Frequency
Fig. 5. A histogram plot for the relative errors of the estimated space headways when D
k
<0:2. Total 371 vehicles from Lane 1 are estimated using Database
3.
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 191
whilst it is missed by the nonparametric model that reflects averaging behaviour. Seeing the vehicle in front of the pointed
vehicle, e.g., 10th vehicle, the estimation deviation is much smaller and satisfied. It should not expect that the same wave is
reproduced, because various random factors and lane changes widely exist in the real world. This can also be seen from the
distribution of vehicles in the time–space diagrams. The simulated vehicles distribute more uniformly, whilst the gaps
between real vehicles are quite different.
4.2. A platoon with different boundary conditions
In this subsection, the following platoons of the real leaders are estimated given different boundary conditions. A new
vehicle is sent on the roadway when the gap between the entrance and the last vehicle on the roadway exceeds a given
value. 30 m, 40 m, and 50 m for the gap are selected to test. It is clear that the smaller the gap is, the higher the demand
is. The initial moving distance is set to be 15 m (i.e., 54 km h
1
for the initial speed).
Fig. 8 presents the time–space diagrams of the trajectories of the simulated following platoon. It can be seen that the
waves originated from the real leaders propagate with a speed of 16 km h
1
, and the shape of the waves are consistent with
the real ones in the US-101 dataset. The impact of sudden deceleration of the leaders is relatively small when the arrival
demand is low (see Fig. 8(c) and (f)), and the waves thus fail to stably propagate. It also demonstrates that the model can
well reflect the impact of realistic speed drop. Meanwhile, a few monitored distances D
k
are within [0.2,0.3], but it seems
that the limited cases with relatively large distance do not make large impact on the simulation results.
4.3. Necessity of each input
Along with this simulation scenario, we demonstrate the necessity of each input proposed in Eq. (2). To this end, we
remove one input from the total four inputs, and conduct the same simulation presented in Fig. 7(d). See Fig. 9 for the results.
Comparing with Fig. 7(d) resulted by all four inputs in the same scenario, Fig. 9 well demonstrates the necessity of all four
inputs, because the traffic waves become unrealistic or even collisions occur without inputing any one of the four.
5. A rubbernecking simulation scenario
In Chen et al. (2012b, 2014), rubbernecking caused by the clean-up work is reported on Lane 1 between 7:50 a.m. and
8:05 a.m. on the study segment of US-101. In order to test the nonparametric car-following model, this section simulates
the rubbernecking scenario by using the nonparametric car-following model.
Before presenting the scenario, we briefly introduce four distinct stages of traffic oscillations addressed in Chen et al.
(2014). Our simulation results will be analysed based on the stages. The four stages are (i) precursor: the speeds of the decel-
eration and acceleration waves are close to zero, indicating localised slow-and-go driving motions; (ii) growth: the waves
propagate backward in space at the speed of 16–24 km h
1
, and the minimum speed of vehicles decreases significantly as
the waves propagate; (iii) stable: the amplitude remains relatively constant as waves propagate backward in space; (iv)
decay: the amplitude diminishes as waves propagate.
In the rubbernecking simulation scenario, a 1.25 km one-lane roadway is simulated for 1 h. A new vehicle enters the road-
way with an initial speed of 54 km h
1
, when its leader has left the entrance 30 m away. The rubbernecking zone is located at
section [1,1.05] km. When vehicles enter the zone, they have a probability rto rubberneck and then slow down by a
−0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
200
400
600
800
1000
1200
Relative error
Frequency
Frequency
(
a
)
Lane2,I-80
−0.3 −0.2 −0.1 0 0.1 0.2 0.3
0
200
400
600
800
1000
1200
Relative error
(
b
)
Lane3,I-80
Fig. 6. Histogram plots for the relative errors of the estimated space headways when D
k
<0:2. The estimated vehicles are from the I-80 segment, and the
database for kNN is from the US-101 segment (Database 3).
192 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
percentage of ð1pÞ. The output, i.e., follower’s moving distance in Eq. (5), is multiplied by pin order to describe the influ-
ence of the speed drop in the model. If rubbernecking occurs, it will occur at most once. The influence of rubbernecking lasts
for hsimulation steps, i.e., hsec in the scenario. The database contains all data collected from Lane 1, 2, and 3 during all
45 min. Neither the fundamental diagrams nor parameters about driver behaviour is needed to be specified or calibrated.
Note that only congested traffic could be well simulated here, because the underlying database only contains congested traf-
fic data.
The simulation results are demonstrated in the time-diagrams of trajectories, which are considered as very important
tools to validate models (see e.g. Laval and Leclercq (2010) and Chen et al. (2012b), and Chapter 10–13 in Treiber and
Kesting (2013b)). See Fig. 10 for the diagrams, in which r¼0:05;p¼0:8, and h¼5. In the figure, Fig. 10(a) is a regular
time–space diagram of trajectories, in which different speeds are highlighted. Whilst Fig. 10(b) is a time–space diagram
of the distance D
k
, and the colours represent different distances of the estimations. We employ this type of diagram to mon-
itor the similarity between the database and the estimation cases. From Fig. 10(b), we can see that almost all distances of the
estimations are less than 0.2. It means that all the estimated cases in the simulation scenario could find similar historical
cases in the database, and the results highly likely have a high quality as stated in Section 3.5.
We first focus on important and representative traffic characteristics in Fig. 10. It can be seen from the figure that the
simulated oscillations transform from localised disturbances to well-developed ones. The wave speed in the stable stage
is around 16 km h
1
, which is consistent with the wave speed of the US-101 dataset. The period and amplitude of oscilla-
tions at the location of 400 m are around 5 min and 58 km h
1
. The observations are all consistent with the existing findings
030 60 90 120
0
100
200
300
400
500
600
−16 km/h
10th follower
time (sec)
space (m)
speed (km/h)
0
20
40
60
80
(a) Real platoon following Vehicle 422
030 60 90 120
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
10th follower
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(b) Simulated platoon following Vehicle 422
030 60 90 120 150
0
100
200
300
400
500
600
−16 km/h
time (sec)
space (m)
speed (km/h)
0
20
40
60
80
(
c
)
Real platoon following Vehicle 1989
030 60 90 120 150
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(
d
)
Simulated platoon following Vehicle 1989
Fig. 7. Comparisons of the real and simulated platoons following Vehicle 422 and 1989 on Lane 1 of the US-101 segment. The time and initial speed of the
simulated followers entering the road are the same to the real platoons. Database 3 is used. The black curves are the last vehicles in the real platoons, and
the blue curve is the 10th follower in the real platoon following Vehicle 422. (For interpretation of the references to colour in this figure legend, the reader is
referred to the web version of this article.)
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 193
and empirical observations, such as wave speed empirically ranging from 10 to 20 km h
1
and the period ranging from 2
to 15 min (Mauch and Cassidy, 2002; Ahn and Cassidy, 2007; Schönhof and Helbing, 2007; Laval and Leclercq, 2010; Zheng
et al., 2011a; Zheng et al., 2011b; Chen et al., 2012a,b, 2014; Jiang et al., 2014).
To see more detail, some regions in Fig. 10(a) are zoomed in and analysed. Fig. 10(c)–(e) present three patterns that a
stable wave forms, which could be also observed in the US-101 dataset. In Fig. 10(c), several slowdowns of two vehicles make
the traffic in the precursor stage grow. In Fig. 10(d), a slowdown primarily triggers slow traffic, whilst it is amplified by a
series of slowdowns afterwards at the same location. In Fig. 10(e), several waves join together and finally form a stable wave
propagating backward in spaces. Fig. 10(f) presents an oscillation in the decay stage. Few empirical study focuses on the traf-
fic in this stage probably due to lack of empirical data. Considering the good performance of the nonparametric model, it may
be able to be studied with the aid of the model.
We further present the fundamental diagrams to show the relations amongst the fundamental traffic parameters.
Suppose that virtual detectors are installed in the roadside, and the traffic flow q, density
q
, and speed
v
within a time period
Tare measured as follows.
q¼N
T;
q
¼P
N
n¼11
v
n
T;and
v
¼q
q
¼N
P
N
n¼11
v
n
ð9Þ
where Nis the count of the vehicles passing the detection location within the time period T, and
v
n
is the passing speed of a
detected vehicle. Fig. 11 compares the fundamental diagrams for the empirical and simulated traffic, in which the aggrega-
tion time period is set to be 60 s. It is observed that the shapes of the simulated fundamental diagrams are quite similar to
the empirical ones for the US-101 segment. It means that the simulated traffic has the similar relations amongst flow, den-
sity, and speed contained by US-101 dataset. We can thus believe that the simulation scenario is conducted based on the
fundamental diagrams similar to the real ones. It demonstrates that the simulated traffic evolves similarly to the real traffic,
and the nonparametric car-following model is able to well replicate the fundamental diagrams.
Meanwhile, hysteresis is observed in the fundamental diagrams for the simulation scenario. Particularly see the
flow-density relations in Fig. 11(a). The lower branch is produced by the upstream traffic far away from the rubbernecking
location (i.e., 200 m, 400 m, and 600 m). At these locations, the oscillations have been in the stable stage, during which lower
flow-density relations are generated. This is consistent with the empirical findings given in Chen et al. (2014), in which the
hysteresis during the stable stage of an oscillation is demonstrated in a perspective of the trajectories of vehicle platoons.
Moreover, we conduct the experiment with various settings and present the periods and amplitudes in Fig. 12. Notice that
the periods stay around 2–4 min when r>0:08;ris negatively correlated with the period, and pis positively correlated with
020 40 60 80 100
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(a) Vehicle 422, entry gap 30 m
050 100 150
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(b) Vehicle 422, entry gap 40 m
050 100 150
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(c) Vehicle 422, entry gap 50 m
050 100 150
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(
d
)
Vehicle 1989, entry gap 30 m
050 100 150
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(
e
)
Vehicle 1989, entry gap 40 m
050 100 150
0
100
200
300
400
500
600
time (sec)
space (m)
−16 km/h
0.1<distance 0.2
0.2<distance 0.3
0.3<distance
speed (km/h)
0
20
40
60
80
(
f
)
Vehicle 1989, entry gap 50 m
Fig. 8. Simulated platoons following real vehicles on the US-101 segment. The ‘‘distance’’ in the legend is the distance from the kth (i.e., 10th) sample to the
estimated case, i.e., D
k
.
194 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
the period. The relationship between a period and rappear to be convex and rapidly approaching a stable value. Although the
period become stable, the amplitudes continue to drop. These observations are quite consistent with the outcome of the vari-
ants of Newell’s model (Newell, 2002) proposed in Laval and Leclercq (2010) and Chen et al. (2012b).
6. A simulation scenario with driver errors
In the study of the trigger of traffic oscillations in the absence of lane changes, it has been shown that even a small per-
turbation, e.g., driver errors, may result in traffic instability, i.e., the oscillation (see e.g. Treiber et al., 1999; Igarashi et al.,
2001; Wilson, 2008; Orosz et al., 2009; Laval et al., 2014). To test the response of the nonparametric model to perturbations,
this section introduces driver errors to the model and simulate in a condition without external disturbance. By conducting
the experiments with various driver errors, we observe a critical error intensity to trigger traffic instability.
We model the driver errors in a form of a white Gaussian noise with diffusion coefficient
r
2
(Laval et al., 2014). Thus, the
moving distance of a follower with driver errors is written as follows:
~
d
n
ðtþ
s
Þ¼d
n
ðtþ
s
ÞþWð
e
Þ ð10Þ
where
Wð
e
Þ¼
1
r
ffiffiffiffi
2
p
p
e
e
2
2
r
2
;d
jam
<d
n
ðtþ
s
Þ<d
free
0;otherwise
(ð11Þ
030 60 90 120 150
0
100
200
300
400
500
600
−16 km/h
time (sec)
space (m)
speed (km/h)
0
20
40
60
80
(a) Inputs without dn1(tτ)
030 60 90 120 150
0
100
200
300
400
500
600
−16 km/h
time (sec)
space (m)
speed (km/h)
0
20
40
60
80
(b) Inputs without dn1(t)
030 60 90 120 150
0
100
200
300
400
500
600
−16 km/h
time (sec)
space (m)
speed (km/h)
0
20
40
60
80
(c) Inputs without sn(tτ)
030 60 90 120 150
0
100
200
300
400
500
600
−16 km/h
time (sec)
space (m)
speed (km/h)
0
20
40
60
80
(d) Inputs without sn(t)
Fig. 9. Simulated platoons taking three inputs amongst the total four inputs. The leader is Vehicle 1989 on Lane 1 of the segment of US-101.
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 195
where d
jam
and d
free
are the moving distance/speed around jam density and free-flow conditions, respectively. Limiting the
perturbation only occurring when d
n
ðtþ
s
Þ2ðd
jam
;d
free
Þroughly reflects the fact that human errors (i.e., speed variability) is
maximal at the critical density, and is minimal at the free-flow condition and jam density (Jia et al., 2011; Laval et al., 2014).
(a) Time-space diagram coloured by speed
(b) Time-space diagram coloured by distance
660 680 700 720 740 760 780 800 820
700
800
900
1000
1100
1200
−16 km/h
time (sec)
space (m)
(c) Region1
100 150 200 250 300
600
700
800
900
1000
1100
1200
−16 km/h
time (sec)
space (m)
(d) Region2
850 900 950 1000 1050
500
600
700
800
900
1000
1100
1200
−16 km/h
time (sec)
space (m)
(
e
)
Region3
550 600 650 700
700
800
900
1000
1100
1200
−16 km/h
time (sec)
space (m)
(
f
)
Region4
Fig. 10. Time–space diagrams of trajectories in the rubbernecking scenario (r¼0:05;p¼0:8, and h¼5).
196 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
020 40 60 80 100 120
0
500
1000
1500
2000
2500
3000
density (veh/km)
flow (veh/h)
600 m
500 m
400 m
300 m
200 m
100 m
(a) flow-density diagram for empirical traffic
020 40 60 80 100 120
0
500
1000
1500
2000
2500
3000
density (veh/km)
flow (veh/h)
hysteresis
1200 m
1000 m
800 m
600 m
400 m
200 m
(b) flow-density diagram for simulated traffic
0 500 1000 1500 2000 2500 3000
0
10
20
30
40
50
60
70
80
90
100
flow (veh/h)
speed (km/h)
600 m
500 m
400 m
300 m
200 m
100 m
(c) speed-flow diagram for empirical traffic
0 500 1000 1500 2000 2500 3000
0
10
20
30
40
50
60
70
80
90
100
flow (veh/h)
speed (km/h)
1200 m
1000 m
800 m
600 m
400 m
200 m
(d) speed-flow diagram for simulated traffic
020 40 60 80 100 120
0
10
20
30
40
50
60
70
80
90
100
density (veh/km)
speed (km/h)
600 m
500 m
400 m
300 m
200 m
100 m
(
e
)
speed-density diagram for empirical traffic
020 40 60 80 100 120
0
10
20
30
40
50
60
70
80
90
100
density (veh/km)
speed (km/h)
1200 m
1000 m
800 m
600 m
400 m
200 m
(
f
)
speed-density diagram for simulated traffic
Fig. 11. The fundamental diagrams for the rubbernecking scenario and for the empirical US-101 segment. The empirical fundamental diagrams are drawn
using the NGSIM dataset between 7:50 a.m. and 8:35 a.m., and the data are aggregated every 60 s.
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 197
0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.3
0
2
4
6
8
10
12
14
probability to rubberneck (r)
period (min)
p=0.6
p=0.7
p=0.8
(a)
Period (h=4)
0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.3
30
35
40
45
50
55
60
65
70
probability to rubberneck (r)
amplitude (km/h)
p=0.6
p=0.7
p=0.8
(b)
Amplitude (h=4)
0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.3
0
2
4
6
8
10
12
14
probability to rubberneck (r)
period (min)
p=0.6
p=0.7
p=0.8
(c)
Period (h=5)
0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.3
30
35
40
45
50
55
60
65
70
probability to rubberneck (r)
amplitude (km/h)
p=0.6
p=0.7
p=0.8
(d)
Amplitude (h=5)
0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.3
0
2
4
6
8
10
12
14
probability to rubberneck (r)
period (min)
p=0.6
p=0.7
p=0.8
e
Period
h=6
0.04 0.06 0.08 0.1 0.12 0.15 0.2 0.3
30
35
40
45
50
55
60
65
70
probability to rubberneck (r)
amplitude (km/h)
p=0.6
p=0.7
p=0.8
f
Amplitude
h=6
Fig. 12. The periods and amplitudes of traffic oscillations at the location of 400 m.
198 Z. He et al./ Transportation Research Part B 80 (2015) 185–201
Although driver errors are directly added to the outputted moving distance, collisions will not be introduced if the diffusion
coefficient is small.
In the simulation scenario, a 900 m one-lane roadway is simulated for 3600 s. The white Gaussian noise is only added
when a vehicle is moving in the section between the locations of 300 m and 600 m. This is analogous to an uphill section.
It is set that d
free
=54 km h
1
;d
jam
=15kmh
1
, and the entry speed and gap are 54 km h
1
and 20 m, respectively.
We first test the model with different values of the diffusion coefficient. The results
3
show that the stop-and-go oscillation
is only triggered by the noise greater than r¼0:2. No oscillation occurs in the experiment when r¼0 and 0.1. Thus, there is a
threshold of the error leading to traffic instability. It also shows that even without external disturbance, driver errors can also
result in traffic oscillations, and the required error may be quite small. It is consistent with the existing findings in physics and
transportation studies.
Fig. 13 presents a part of the simulation results. In Fig. 13(a) and (b) the periodic waves and their speeds are all consistent
with the real oscillation. The precursor stages of oscillations in Fig. 13(c) and (d) have realistic shapes. It is also noticed in
Fig. 13(a) and (d) that stable oscillations form in a section without driver errors, i.e., in the upstream of the section with
(a) Time-space diagram of trajectories (σ=0.2)
(b) Time-space diagram of trajectories (σ=0.5)
250 300 350 400 450
100
200
300
400
500
600
−16 km/h
time (sec)
space (m)
(
c
)
Region 1
1900 1950 2000 2050 2100
50
100
150
200
250
300
350
400
450
−16 km/h
time (sec)
space (m)
(
d
)
Region 2
Fig. 13. Time–space diagrams of trajectories in the simulation scenario with driver errors.
3
All Dkare within satisfied ranges, i.e., smaller than about 0.2. It is not shown to save space.
Z. He et al./ Transportation Research Part B 80 (2015) 185–201 199
driver errors. It implies a situation that the traffic slowly moves in the uphill section whilst stop-and-go waves occur in its
upstream without a clear trigger.
7. Conclusions
Based on the truth that drivers repeat their behaviour in similar circumstances, this paper proposes a data-driven
car-following model by using the simple kNN. The approach takes the average of the most similar cases as its output, which
implies the most likely driving behaviour under the current circumstance. Four inputs are selected, i.e., leader’s moving dis-
tances and follower’s space headways in the latest two time steps, and the output is follower’s moving distance. The only
parameter kis determined through analysing estimation errors with different k-values. A time–space diagram of the distance
of kth sample is designed to monitor the simulation scope. Three simulation scenarios are conducted, i.e., following real lead-
ers, rubbernecking, and driver errors. Time–space diagrams containing thousands of vehicle trajectories, fundamental dia-
grams, and periods and amplitudes of oscillations are presented. Explicit analysis demonstrates that the model is able to
reproduce realistic stop-and-go oscillations. We conclude the advantages of the proposed nonparametric car-following
model as follows:
(i) Neither mathematical equation nor calibration is needed to be concerned in the model.
(ii) Neither the fundamental diagrams nor driver’s behaviour parameters is assumed.
(iii) The model is simple and parsimonious particularly in the conceptual point of view, and the only parameter is k.
(iv) All inputs and outputs are based on vehicle positions, which are straightforward to reproduce traffic dynamics in com-
puter simulations.
(v) The model is able to well reproduce traffic characteristics contained by the underlying database, such as all stages of
stop-and-go oscillations, fundamental diagrams, periods and amplitudes of oscillations.
Based on field data, the nonparametric car-following model can be rapidly deployed and replicate reliable traffic charac-
teristics without model calibration. However, the simplicity, in return, may also limit the freedom to set simulation scenarios
due to lack of mechanism description.
Due to no mathematical formula, it may be easy to incorporate various factors for specific purposes or extend to
multi-lane situations. One could just concentrate on the inputs and outputs, and does not need to worry about the mathe-
matical expression. These are future work directions.
Acknowledgements
The authors are grateful to three anonymous referees whose comments greatly improved the quality of this paper, and to
Prof. Rui Jiang and Prof. Ning Jia for their helpful suggestions. This research has been funded by NFSC (71471014) and 973
Program of China (2012CB725403).
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