1 Bayesian Calibration of the Intelligent Driver Model Chengyuan Zhang Graduate Student Member IEEE and Lijun Sun Senior Member IEEE

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Bayesian Calibration of the Intelligent Driver Model

Chengyuan Zhang, Graduate Student Member, IEEE, and Lijun Sun, Senior Member, IEEE

Abstract—Accurate calibration of car-following models is essen-

tial for understanding human driving behaviors and implementing

high-ﬁdelity microscopic simulations. This work proposes a

memory-augmented Bayesian calibration technique to capture

both uncertainty in the model parameters and the temporally

correlated behavior discrepancy between model predictions

and observed data. Speciﬁcally, we characterize the parameter

uncertainty using a hierarchical Bayesian framework and model

the temporally correlated errors using Gaussian processes. We

apply the Bayesian calibration technique to the intelligent driver

model (IDM) and develop a novel stochastic car-following model

named memory-augmented IDM (MA-IDM). To evaluate the

effectiveness of MA-IDM, we compare the proposed MA-IDM

with Bayesian IDM in which errors are assumed to be i.i.d.,

and our simulation results based on the HighD dataset show

that MA-IDM can generate more realistic driving behaviors and

provide better uncertainty quantiﬁcation than Bayesian IDM.

By analyzing the lengthscale parameter of the Gaussian process,

we also show that taking the driving actions from the past ﬁve

seconds into account can be helpful in modeling and simulating

the human driver’s car-following behaviors.

Index Terms—car-following model, serial correlation, Bayesian

inference, hierarchical model, Gaussian processes

I. INTRODUCTION

Microscopic car-following models are powerful tools to

study and simulate human driving behaviors in trafﬁc ﬂows at

the trajectory level. They reveal the mechanisms of complex

interactions between a subject vehicle and its leading vehicle.

These interactions are the essential factors that affect the

dynamics of trafﬁc ﬂow and create diverse macroscopic trafﬁc

phenomena [1]. Although experiences indicate that we cannot

calibrate a zero error model that perfectly ﬁts the data [2],

some car-following models are still valid with which one

can adopt various calibration (or parameter identiﬁcation)

methods to reproduce reality to different extents. Given that the

performance and ﬁdelity of microscopic car-following models

are heavily dependent on accurate calibration, improving the

quality of calibration methods is a critical research question.

In recent studies, probabilistic calibration methods have

become an emerging and promising approach with a solid

statistical foundation. There are essentially two probabilistic

approaches for model calibration. The ﬁrst is maximum likeli-

hood estimation (MLE): we often solve MLE as an optimization

problem and ﬁnally obtain a point estimation for model

(Corresponding author: Lijun Sun)

C. Zhang and L. Sun are with the Department of Civil Engineering, McGill

University, Montreal, QC H3A 0C3, Canada. E-mail: enzozcy@gmail.com (C.

Zhang), lijun.sun@mcgill.ca (L. Sun).

The authors would like to thank the McGill Engineering Doctoral Awards

(MEDA), the Institute for Data Valorisation (IVADO), the Interuniversity

Research Centre on Enterprise Networks, Logistics and Transportation

(CIRRELT), Fonds de recherche du Qu

ebec – Nature et technologies (FRQNT),

and the Natural Sciences and Engineering Research Council (NSERC) of

Canada for providing scholarships and funding to support this study.

parameters; see, e.g., [3]. The second is Bayesian inference,

which allows us to exploit the full posterior distribution for

model parameters through Markov chain Monte Carlo (MCMC)

or variational inference (VI); see, e.g., [4], [5]. In general, the

Bayesian inference approach is advantageous in two aspects:

(1) we can perform uncertainty quantiﬁcation based on the

full posterior distribution. This is particularly important for a

simulation model as we are often interested in the distribution

and uncertainty of the simulation results; (2) Bayesian inference

offers a hierarchical modeling scheme that allows us to learn

parameter distributions at both population and individual

levels, where the population distribution prevents overﬁtting

by imposing certain dependencies on the parameters. This

is particularly important for calibrating car-following models

since samples are often short trajectories collected from a large

number of drivers with diverse vehicle conﬁgurations/dynamics.

Considering substantial differences in personal driving styles

and vehicle conﬁgurations/dynamics, it is critical to learn

the population distribution that can generalize the parameter

distribution from short trajectory data. In addition, the learned

population posterior distribution can help create diverse driving

behaviors to create more realistic simulations with driver

heterogeneity. Note that in the following of this paper, we

refer to “driver behavior” as the characteristics resulting from

both the driver and the vehicle.

A fundamental question in performing probabilistic cali-

bration is how to deﬁne the probabilistic model and data

generation process. Most existing car-following models seek

parsimonious structures by simply taking observations from

the most recent (i.e., only one) step as input to generate

acceleration/speed as output for the current step. However,

given physical inertia, delay in reaction, and missing important

covariates (e.g., car follower models, in general, do not take

the status of the breaking light of the leading vehicle as

input variable), we should expect unexplained behavior (ie,

the discrepancy between predicted acceleration/speed and

observed acceleration/speed) to be temporally correlated [6],

[7]. For instance, Wang et al. [7] show that the best-performed

deep learning model essentially takes states/observations from

the most recent

∼

10 sec as input. However, most existing

probabilistic calibration methods are developed based on a

simple assumption—the errors are independent and identically

distributed (i.i.d.), and as a result, ignoring the autocorrelation

in the residuals will lead to biased calibration [8], [9]. For

example, Fig. 1 shows the residual process in acceleration

(

), speed (

), and gap (

) when calibrating an IDM model

with i.i.d. noise, and we can see that the residuals have strong

serial correlation (i.e., autocorrelation). Essentially, two classes

of methods are developed in the literature to model serial

correlation. One approach is to directly process the time series

data to eliminate serial correlations. For instance, Hoogendoorn

arXiv:2210.03571v2 [stat.AP] 12 Jan 2024

0 10 20 30 40 50 60

time (s)

-1.5

-1.0

-0.5

0.0

0.5

a (m/s2)

IDM

Human

0 10 20 30 40 50 60

time (s)

-0.4

-0.2

0.0

0.2

0.4

¢a (m/s2)

Data

Residuals

Fig. 1. The ﬁrst panel shows the predicted acceleration from a well-calibrated

IDM and the real-world observation from a human driver. The second panel

shows the corresponding discrepancy. Note that the literature usually assumes

aHuman =aIDM +error

, where the error is assumed i.i.d. However, we can

clearly see strong serial correlations in the residual. Ignoring such serial

correlations will undoubtedly lead to biased modeling ﬁtting.

and Hoogendoorn [3] adopted a difference transformation (see

[10]) to eliminate the serial correlation. However, they used

empirical correlation coefﬁcients to perform the transformation

instead of jointly learning the IDM parameters and the

correlations. Another approach is to explicitly model the serial

correlations (e.g., by stochastic processes); for example, Treiber

and Kesting [11] introduced the Wiener process to model the

temporally correlated error process.

In this paper, we propose a novel probabilistic calibration ap-

proach that takes advantage of Bayesian methods by capturing

both uncertainty in the model parameters and the temporally

correlated behavior discrepancy. We apply this approach to

calibrate the IDM and develop an MA-IDM, which models

the serially correlated errors as zero-mean Gaussian processes

(GP). Taking advantage of the Bayesian methods, we can jointly

learn the model parameters and the GP hyperparameters by

MCMC. In this work, the MA-IDM is presented in three forms,

i.e., pooled model, hierarchical model, and unpooled model.

We conducted numerical experiments on the Highway Drone

(HighD) data set [12], and the results demonstrate that our

hierarchical model not only learns consistent driving styles at

the population level but also depicts heterogeneity of driving

behavior at the individual level. Additionally, a stochastic

simulation method is developed to obtain the posterior motion

states in terms of acceleration, speed, and location.

The overall contributions of this work are threefold:

We develop a novel Bayesian calibration approach to

learn unbiased parameters and their full posterior distribu-

tion, in which GP is introduced to model autocorrelated

errors. This approach is applied to calibrate IDM. The

autocorrelation in the residuals reveals that incorporating

observations from the past

∼5

seconds improves the

modeling of car-following decisions.

With a hierarchical MA-IDM, heterogeneous but consis-

tent driving behaviors can be learned at the individual

level. Therefore, we can generate enormous drivers with

heterogeneous driving behaviors/styles governed by the

same population distribution. In such a way, our model

can help create simulations with driver/car heterogeneity.

We introduce an unbiased stochastic simulator, which is

inspired by the corresponding generative process of our

Bayesian calibration approach. As a result, the simulator

can produce more realistic results and better uncertainty

quantiﬁcation than those with homogeneous parameters

or random parameters.

The structure of the remaining contents is organized as

follows. First, Section II introduces related literature on cali-

brating car-following models and modeling serial correlations.

Section III emphasizes some preliminaries and formulates the

car-following problem based on IDM. Then in Section IV,

we propose a novel Bayesian calibration method and develop

several novel car-following models, i.e., the Bayesian-IDM (B-

IDM) and the MA-IDM, with different hierarchies. Next, we

conduct extensive experiments and simulations, then thoroughly

analyze the results in Section V and Section VI. Finally,

Section VII wraps up this work and discusses several potential

directions worth further exploration.

II. RELATED WORK

A. Car-Following Model Calibration

Proper choice of key parameters in a car-following model can

help to depict and reproduce complex driving behaviors. As in

most cases, we will never know the values of these parameters.

Typically, one could collect observable measurements from the

ﬁeld car-following data in diverse scenarios and ﬁt the model

to the observed data by adjusting the parameters to optimize

certain objective functions. The overall process is known as

model calibration [13]. We refer readers to [2] for a review of

the calibration of car-following models.

Various calibration methods have been studied in the litera-

ture. The genetic algorithm (GA) is one of the most typical and

traditional tools as a heuristic evolutionary algorithm [2], [14].

For example, Punzo et al. [2] conducted extensive experiments

to study

goodness-of-ﬁt functions (GoF) with combinations

of three measures of performance (MoP) and provided sys-

tematic guidelines on GA-based calibration. The combinations

of GoF could be selected according to the speciﬁc problem

and trafﬁc scenarios, leading to the multi-objective calibration

approach [15], [16]. However, as an optimization tool, GA

presents some limitations because it is not only computationally

expensive but also sensitive to different choices/combinations

of GoF and MoP. Besides, the best combination of one dataset

may not be suitable for another. Maximum likelihood (MLE) is

a powerful approach. Hoogendoorn et al. [3], [17] ﬁrst proposed

to calibrate car-following models based on MLE. Treiber and

Kesting [8] thoroughly investigated the MLE approach on

the calibration and validation of car-following models. MLE

seeks to estimate a single “best” value of an unknown quantity

based on optimization algorithms. The Bayesian method, as

an alternative approach, aims to estimate the distribution of an

unknown variable (i.e., posterior). Rahman et al. [4] calibrated

leading vehicle

following vehicle

gap (m)

speed (m/s) speed (m/s)

Fig. 2. Physical settings of a car-following scenario.

car-following models based on a Bayesian approach. The

experiments showed that the Bayesian approach provided much

better results than the deterministic optimization algorithm.

With the Bayesian approach, Abodo et al. [5] further developed

a hierarchical model to calibrate the IDM, which has also

achieved promising results.

B. Serial Correlation Modeling

Regression plays a vital role in model calibration. In the

literature, many regression models are developed heavily based

on an assumption—the errors are i.i.d. random variables.

However, due to the omission of relative covariates and model

misspeciﬁcation, we often see autocorrelated errors when

estimating regression models on time series data. In this case,

assuming the errors to be i.i.d. will lead to suboptimal parameter

identiﬁcation. Generally, as mentioned in [13], the observations

are usually regarded as the true process with an i.i.d. observation

error; while in calibration methods, observations are modeled

as the combination of a calibrated model and an independent

model inadequacy function with an i.i.d. observation error. In

calibrations involving time series data, the model inadequacy

function is mainly used to capture the autocorrelations.

In general, there are two ways to perform model estimation

in the presence of serial correlations: (1) by directly processing

the nonstationary data and eliminating serial correlations (e.g.,

performing the differencing operation), so that one can safely

ignore the model inadequacy function and obtain stationary

time series; or (2) by explicitly modeling the serial correlations

based on speciﬁc model inadequacy functions. For instance,

Hoogendoorn [3] performed a differencing transformation

to eliminate serial correlations, which then did not show

signiﬁcant differences between autocorrelation coefﬁcients and

zeros in the previously mentioned Durbin–Watson test [18].

However, the information conveyed by the serial correlations

is directly discarded, which prevents us from modeling the

generative processes of observations. Another way is to

explicitly develop the formation of serial correlations based on

further assumptions. For example, dynamic regression models

leverage linear regression and autoregressive integrated moving

average (ARIMA) into a single regression model to forecast

time series data [19]. For the main scope of this paper, i.e.,

calibration and simulation of car-following models, we use

GP [20] to model serially correlated errors (i.e., for the model

inadequacy part). GP provide a solid statistical solution to

learn the autocorrelation structure, and more importantly, it

allows us to understand the temporal effect in driving behavior

through the lengthscale parameter

, which partially explains

the memory effect (see [21]) of human driving behaviors.

III. PRELIMINARIES AND PROBLEM FORMULATION

A. IDM and Probabilistic IDM

The traditional IDM [22] is a continuous nonlinear function

f:R37→ R

which maps the gap, the speed, and the speed

difference (approach rate) to acceleration

aIDM

at a certain

time point. Here, we denote

as the gap between the following

vehicle and the leading vehicle,

as the speed of the following

vehicle, and

∆v=−ds/dt =v−vl

(

denotes the speed

of the leading vehicle) as the speed difference. The physical

meaning of these notation is illustrated in Fig. 2. With these

notations, IDM is deﬁned as

aIDM =f(s, v, ∆v)≜α 1−v

v0δ

−s∗(v, ∆v)

s2!,

(1)

s∗(v, ∆v) = s0+s1rv

+v T +v∆v

2√α β ,(2)

where,

v0, s0, T, α, β,

and

are model parameters with the

following meaning: the desired speed

is the free-ﬂow speed;

the jam spacing

denotes a minimum gap distance from

the leading vehicle; the safe time headway

represents the

minimum time interval between the following vehicle and the

leading vehicle; the acceleration

and the comfortable braking

deceleration

are the maximum vehicle acceleration and the

desired deceleration to keep safe, respectively; the exponential

coefﬁcient

is a constant, usually set as

at default [22]. In

IDM, the deceleration is controlled by the desired minimum

gap

s∗

in Eq.

(2)

, in which we set

s1= 0

following [22] to

obtain a model with interpretable parameters.

To make the notation concise and compact, we denote

the IDM parameters by

θ= [v0, s0, T, α, β]∈R5

. For

a certain vehicle

, we have the IDM actions

a(t)

IDM,d =

f(s(t)

d, v(t)

d,∆v(t)

d;θd)

, where the subscript

represents the

index for each driver, and the superscript

(t)

indicates the

timestamp. Compactly, we denote the inputs at time

as a

vector

h(t)

d= [s(t)

d, v(t)

d,∆v(t)

d],∀t∈ {t0, . . . , t0+ (T−1)∆t}

Here, we adopt the scheme in [8], [23] with a step of

∆t

update vehicle speed and location as following:

v(t+∆t)=v(t)+a(t)∆t, (3a)

x(t+∆t)=x(t)+v(t)∆t+1

2a(t)∆t2.(3b)

However, real-world driving actions cannot be fully modeled

by IDM and thus it is inevitable to observe some discrepancies,

as stated by Treiber and Kesting (see Section 3.2 of Chapter

12 in [11]). It can be modeled by adding some acceleration

noise of standard deviation

σϵ

aIDM

. In such a setting, we

consider

aIDM

as a rational behavior model, while the random

term as the imperfect driving behaviors that cannot be modeled

by IDM. By taking the random term with i.i.d assumption into

consideration, one can develop a stochastic version of IDM,

i.e., the probabilistic IDM [24], [25], given by

a(t)

d|h(t)

d,θd∼

i.i.d. N(a(t)

IDM,d, σ2

ϵ),(4)

where

a(t)

is the true acceleration, and

N(µ, σ2)

denotes a

Gaussian distribution with

as the mean and

σ2

as the variance.

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1BayesianCalibrationoftheIntelligentDriverModelChengyuanZhang,GraduateStudentMember,IEEE,andLijunSun,SeniorMember,IEEEAbstract—Accuratecalibrationofcar-followingmodelsisessen-tialforunderstandinghumandrivingbehaviorsandimplementinghigh-fidelitymicroscopicsimulations.Thisworkproposesamemory-augmented...

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1 Bayesian Calibration of the Intelligent Driver Model Chengyuan Zhang Graduate Student Member IEEE and Lijun Sun Senior Member IEEE

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