1

A Learning-Based Approach for Lane Departure
Warning Systems with a Personalized
Driver Model

arXiv:1702.01228v1 [cs.LG] 4 Feb 2017

Wenshuo Wang, Ding Zhao, Junqiang Xi, and Wei Han

Abstract—Misunderstanding of driver correction behaviors
(DCB) is the primary reason for false warnings of lane-departureprediction systems. We propose a learning-based approach to
predicting unintended lane-departure behaviors (LDB) and the
chance for drivers to bring the vehicle back to the lane.
First, in this approach, a personalized driver model for lanedeparture and lane-keeping behavior is established by combining
the Gaussian mixture model and the hidden Markov model.
Second, based on this model, we develop an online model-based
prediction algorithm to predict the forthcoming vehicle trajectory
and judge whether the driver will demonstrate an LDB or a DCB.
We also develop a warning strategy based on the model-based
prediction algorithm that allows the lane-departure warning
system to be acceptable for drivers according to the predicted
trajectory. In addition, the naturalistic driving data of 10 drivers
is collected through the University of Michigan Safety Pilot Model
Deployment program to train the personalized driver model and
validate this approach. We compare the proposed method with a
basic time-to-lane-crossing (TLC) method and a TLC-directional
sequence of piecewise lateral slopes (TLC-DSPLS) method. The
results show that the proposed approach can reduce the falsewarning rate to 3.07%.
Index Terms—Learning-based approach, lane departure warning system, Gaussian mixture model, hidden Markov model,
personalized driver model

I. I NTRODUCTION

1. Will the driver steer back
from the side by him/herself?

2. Should I send a
warning to the driver?
Negative alarm

Positive alarm

Fig. 1: Negative alarm and positive alarm for a lane departure
scenario. If the future vehicle trajectory is the dash line, the
warning is a positive alarm; if the future vehicle trajectory is
the solid line, the warning is a negative alarm.

is to reduce the false alarm rate (FAR), which occurs when
drivers become aware of the lane departure and plan to correct
the maneuvers in the next moment [8]. A high-rate of false
alarms may reduce the driver’s trust in the LDW system or
cause the driver to become annoyed. As shown in Fig. 1, the
LDW system needs to accurately predict the driver’s intention
and provides warnings only when needed. A successful LDW
design needs to understand driving style of a specific person
and offer a personalized assistance.

A. Motivation

L

ANE departure is an unintentional drifting towards the
boundary of the driving lane, which usually occurs when
the driver is drowsy or fatigued [1]. In the United States,
37.4% of fatal crashes are due to single-vehicle lane departure
[2], which makes it the leading cause of fatalities [3]. Lanedeparture warning (LDW) systems aim to alert the driver when
the lane departure begins and has great potential for vehicle
safety [4]. LDW systems can detect or predict lane-departure
events and give a warning in an auditory, haptic, or visual
form [5]–[12]. The challenge to designing a successful LDW
The first two authors, W. Wang and D. Zhao, have made equal contributions
to this work.
W. Wang is with the School of Mechanical Engineering, Beijing Institute
of Technology (BIT), Beijing, China, 100081, and is studying in Vehicle Dynamics & Control Lab, Department of Mechanical Engineering, University of
California at Berkeley (UCB), CA, 94706 USA. e-mail: wwsbit@gmail.com.
wwsvdc2015@berkeley.edu
D. Zhao is with the University of Michigan Transportation Research
Institute, Ann Arbor, MI 48109-2150, USA. e-mail: zhaoding@umich.edu.
J. Xi is with the School of Mechanical Engineering, Beijing Institute of
Technology (BIT), Beijing, China, 100081. e-mail: xijunqiang@bit.edu.cn.
W. Han is with the Department of Computer Science, Tsinghua University,
Beijing, China, 100084. e-mail: hanwei1949@gmail.com.

B. Related Research
Most LDW systems use time to lane crossing (TLC) to
determine whether to activate the warning. TLC is defined as
the time duration available for the driver before lane-boundary
crossing. However, the TLC-based method has been criticized
for having a high FAR [8], [13] because of its inability to
predict driver’s intention. The TLC-based warning could be
triggered when the TLC reaches the predefined critical value
(usually more than 0.9 s). Yet drivers can usually keep the
vehicle close to the lane boundary and then bring the vehicle
back to the center of the lane without the warning in the
second step, as the solid blue line shows in Fig. 1, which
illustrates driver correction behavior (DCB). A warning given
when the driver can presumably keep the vehicle in the lane
is likely to cause annoyance. Therefore, it is crucial to infer
the driver’s upcoming behavior and judge whether the driver
could bring the vehicle back to the lane center and then decide
when the warning should be given to the driver, thus reducing
the annoyance level of excessive false warning and avoiding
crashes.

2

Several approaches have been proposed to reduce the FAR.
Angkitrakul et al. [8] proposed a stochastic driver model to
predict the vehicle trajectory. Experimental results showed
that the approach had a 17% FAR in detecting an intentional
correction in the subsequent 0.5 s. When the prediction time
was lengthened to 1.5 s, the FAR exceeded 100%. Another
indicator of whether a warning will be given to the driver is
based on whether the driver will bring the car back to the lane
center after the vehicle approaches the boundary. If the driver
plans to bring the vehicle back with some minor departure
(i.e., the solid blue line in Fig. 1), it is not necessary to send a
warning to the driver; if the driver is not aware of the situation
and the departure may exceed a certain threshold (i.e., the
dashed blue line in Fig. 1), a warning should be sent to the
driver to avoid a crash. Saito et al. [10] applied the idea in
Fig. 1 with a dual control scheme in driver drowsiness. In
their study [10], the steering angle was used to estimate the
driver’s state and then decide whether an assist control should
be implemented. Though this approach can improve safety
by identifying driver drowsiness, it can not predict the future
trajectory of the vehicle and the driver’s future operations.
C. Contributions
We propose an online model-based prediction algorithm of
vehicle lateral trajectory to prevent false warnings by inferring
whether the driver’s forthcoming behavior can bring the vehicle back to the driving lane or cross the lane boundary and potentially cause a crash. The goal is to employ the personalized
driver model (PDM) to infer the upcoming lateral trajectory
of a vehicle by using a model-based prediction algorithm. We
apply a PDM in our research because the driving styles of lane
departure and lane keeping behaviors for drivers differ greatly,
and the PDM can discern the characteristics of individual’s
driving style. The proposed PDM describes the driver’s lanekeeping behavior and lane-departure behavior by using a jointprobability density distribution of Gaussian mixture model
(GMM) between vehicle speed, relative yaw angle, relative
yaw rate, lateral displacement, and road curvature. Then, a
hidden Markov model (HMM) is used to estimate the vehicle’s
lateral displacement based on the trained GMM. An online
model-based prediction algorithm is developed to predict the
vehicle’s lateral displacement. We can then predict the future
trajectory of the vehicle and judge whether the driver will
approach the lane boundary and then bring the vehicle back,
keeping the vehicle in the driving lane. Our main contributions
are (1) the personalized driver model (2) and the model-based
prediction algorithm.
D. Paper Organization
The contents of this paper are organized as follows. Section
II describes the key concept of the lane-departure predictions
and warnings. Section III presents the structure of the proposed
approach and model-based prediction algorithm. Section IV
describes the experiments and data collection. Section V and
VI include a further analysis and a discussion of the results.
Section VII gives the conclusions.

Road edge
𝑦

Trajectory
𝑥

Δ𝑦

𝑌

𝜓
𝑋

Fig. 2: Illustration of a lane departure event on a road.
II. L ANE D EPARTURE P REDICTION
Lane departure prediction (LDP) aims to estimate whether a
vehicle will depart from the lane, thus allowing a longer time
for a driver to take effective action to avoid a crash. The LDP
algorithms in the literature can be roughly classified into three
groups: TLC-based prediction [8], [14], [15], vehicle-variablebased vehicle-position estimation [9], [16]–[18], and detection
of the lane boundary using real-time road images [19]–[21].
Some of the methods in the first and second categories share
a common feature, namely, they all use real-time images. The
vision- or vehicle-variable-based method can improve TLCbased prediction, but have a hard time predicting DCB, these
methods are difficult to deal with it. In this paper, we focus on
predicting vehicle trajectory and deciding whether to send a
warning, that is , developing a more effective warning system,
instead of lane-detection techniques.
A. Time to Lane Crossing (TLC)
The TLC-based algorithm has been widely reported in the
literature and used on production vehicles. These systems
estimate the lane states [9], [17] (i.e., lane markers, lane width,
and lane curvature, etc.) and vehicle states based on visionbased equipment, and then calculate the TLC online using a
variety of algorithms [14]. When the TLC reaches a threshold,
the LDW sends alerts the driver. The computation methods of
TLC differ with regard to various road geometries and vehicle
types. The most common method for calculating the TLC is to
predict the road boundary and the vehicle trajectory and then
calculate the time when they intersect. When road curvature
is small, the TLC can be computed as the ratio of lateral
distance to lateral velocity or the ratio of the distance to the
line crossing [14]:
∆y − (D/2 − lf · tan ψ)
(1)
v sin ψ
where ∆y is the lateral distance from the vehicle’s center of
gravity (CoG) to the line that would be crossed, ψ is the
relative yaw angle between the longitudinal coordinate of the
vehicle and the road direction as shown in Fig. 2, v is the
vehicle speed, D is the width of vehicle, and lf is the distance
between the front axle and the CoG of the vehicle.
tT LC =

B. Excessive False Warning
Many studies have observed that the TLC-based methods
tend to have a higher FAR when the ego vehicle drives close

3

Trajectory
Mode i

GMM

(a) Unintentional lane-departure behavior.

Mode k

Mode j

Trajectory

(b) Intentional driver correction behavior.

Fig. 3: Examples of an unintentional lane-crossing event and
intentional driver correction behavior. The red circle represents
the TLC-based warning methods.

to the lane boundary [8], [13], [14]. This is primarily due
to using an oversimplified model to reduce computational
complexity, viz. neglecting drivers’ steering characteristics
and vehicle dynamics. Another reason for the high FAR is
that some drivers will control the vehicle back to the center
of the lane without the help of a warning with DCB. The
problem, however, is that most LDW systems can not predict
the forthcoming driver behaviors or vehicle trajectories. Fig.
3 shows two cases consisting of unintentional LDB and
intentional DCB. In the case shown in Fig. 3(b), an LDW is
not desired because the driver can guide the vehicle back after
being close to the lane boundary. This kind of false warning
is difficult to reduce by improving the accuracy of sensor data
(e.g., road curvature) or a TLC-based calculation method (e.g.,
considering the attributes of vehicle dynamics). Therefore, to
reduce this kind of false warning, we need to estimate the
driver’s forthcoming behavior by asking, “Will the driver bring
the vehicle back to the center of the driving lane within a short
span of time?” or “Does the driver correctly understand the
driving situation?”.
III. P ROPOSED M ETHOD
HMM has been widely used to model driver dynamic
behaviors due to its powerful ability to describe the dynamic
process and infer unobserved (hidden) states [22]. In the
HMM, we apply the component of GMM to representing the
hidden modes (Fig. 4), which makes it easier to determine
the Markov chain modes. We use the GMM to model the
dependent relationship between variables that could describe
driver behaviors because the GMM method has been applied to
model driving tasks and has shown its effectiveness [8], [23],
thus more off-the-shelf techniques for estimating the model
parameters can be directly used. Based on the GMM, we use
the observations to infer the hidden states and design a modelbased prediction algorithm to infer driver upcoming behaviors.
A. Feature Parameter Selection
For the LDP, we focus on the driver’s lateral-control behaviors. Drivers predict the future trajectory of a vehicle based on
their “internal model” [24] and the driving situation as well as

Data
sequence

HMM

Mode i

Mode i

i

𝛼","

Mode j

Mode j

𝑟'()

i

𝛼",$

j

𝛼$ ,$

Mode k

Mode k

𝑟'(*

𝑟'(+

j

𝛼$ ,%

k

𝛼% ,%

k

Hidden
states

Observations

Fig. 4: The illustration of the proposed method by combining
GMM and HMM, and r(∗) ∈ N+ is the duration for mode i, j,
and k.
vehicle states, and then take a lateral-control action to keep the
vehicle in a safe and comfortable region. We describe drivers’
lane-keeping or lane-departure behaviors using the following
variables:
• Vehicle Speed (v): Speed selections have a great influence
on drivers’ lateral control [25] and drivers’ risk perception
[26]. Drivers will usually compensate for steering errors
by adjusting the vehicle speed [27], keeping the TLC
constant. Also, according to Mammar [14], the speed is
one key to influencing the TLC. With the same relative
yaw angle of ψ 6= 0, higher speed leads to a smaller TLC.
• Relative Yaw Angle (ψ): According to May and Baldwin
[1], the TLC is also influenced by the relative yaw
angle (Fig. 2). For example, a larger relative yaw angle
generates a shorter TLC and has a greater potential for
crashes. Therefore, the relative yaw angle is selected as
one variable for modeling driver behaviors.
• Relative Yaw Rate (ψ̇): The relative yaw rate can indirectly show a driver’s intentions. For example, when
the vehicle is approaching the road boundary, an inverse
relative yaw rate can slow down the approaching speed
towards the road boundary and avoid a crash.
• Road Curvature (ρ): Estimation of road profiles is a
big challenge for calculating TLC due to the difficulty
of accurately estimating road geometry. In general, to
simplify the model and reduce computation cost, road
curvature is assumed to be constant or affine varying [14]
over a short span of time. The means of computing TLC
differ with respect to a straight road and a curved road.
In this paper, we mainly consider the scenarios where the
road has a small curvature ρ ≤ 10−4 m−1 .
• Lateral Displacement (∆y): Drivers differ in terms of
their preferences for maintaining a vehicle’s lateral position with respect to the road boundary, related to many
factors such as the driver’s ability to perceive risk [26]

4

and driver states [28]. Drivers can laterally displace the
vehicle according to their own physical and psychological
states. Therefore, lateral displacement is also selected to
describe driver behaviors.
In sum, five variables are employed to model drivers’ lanekeeping and lane-departure characteristics. We can describe
driver behavior using a driving-data sequence,

•

Pkl+1 (ξ t )

B. GMM

b l+1
µ
k

1) Structure of GMM: GMM is used to establish the
dependent relationships among the five variables in vector ξ.
The joint probability density function of ξ is in a form of the
multivariate Gaussian distribution (MGD) function’s weighted
sum:

b l+1
Σ
k

•

Update log-likelihood: At the end of each iteration, we
bl+1 ) by
compute and update the log-likelihood L(θ
b
L(θ

1
=
ωk
d/2 |Σ |1/2
(2π)
k
k=1


1
× exp − (ξ t − µk )> Σk (ξ t − µk )
2

L(θ) =

n
X

log (p(ξ t ; θ))

•

(3)

θ ∗ = arg max L(θ)
θ

(4)

To achieve (4), we denote the EM estimate of θ at step l be
l
b
bl to θ
bl+1 is achieved by the following
θ . The iteration from θ
E-Step and M-Step.

)=

n
X

l

b)
L(θ

(7)

Convergence Condition: Repeat the iteration (5) - (7) until
bl+1 ) − L(θ
bl ) < ε,
the following condition is valid: L(θ
where ε is a very small positive value. In this work, we
set ε = 10−10 .

C. HMM
Based on the trained GMM, a connected HMM representation of driver lane-keeping behavior can be built. Each
component of GMM is treated as a state in the HMM. Since
the goal is to infer the driver’s upcoming behavior based on
the driving situation, we define the following variables for the
HMM:
•

•

t=1

Because of the nonlinearity of (3), it is difficult to directly
derive (3) with respect to parameters θ and get an optimal
solution. Therefore, the iterative version of the ExpectationMaximization (EM) algorithm is employed, which can guarantee a monotonic increase in the likelihood value of model
(2) at each step of the iteration, with the objective of searching
an optimal parameter θ ∗

l+1

t=1

(2)

K
where θ = {θk }K
k=1 = {ωk , µk , Σk }k=1 are the parameters
of model (2); N (ξ t ; µk , Σk ) is the MGD of ξ, with the mean
center µk ∈ Rd×1 and covariance matrix Σk ∈ Rd×d ; K is
the number of GMM components, which can be determined
using Bayesian information criterion (BIC) [29];P
ωk ∈ (0, 1] is
K
the weight of the k th Gaussian component and k=1 ωk = 1.
2) Parameter Identification: Given a data sequence ξ 1:n
and model configuration (2), the model parameter (θ) can
be estimated using the maximum-likelihood (ML) method.
The goal of the ML method is to find the parameter θ that
maximizes the likelihood of the GMM function (2):

(5)

1 X l+1
P (ξ t )
(6a)
n t=1 k
Pn
(ξ t · Pkl+1 (ξ t ))
= t=1
(6b)
Pn
Pkl+1 (ξ t )
t=1


Pn
l+1
l+1 >
l+1
b
b
P
(ξ
)(ξ
−
µ
)(ξ
−
µ
)
t
t
t
k
k
t=1
k
=
Pn
l+1
t=1 Pk (ξ t )
(6c)

ωk N (ξ t ; µk , Σk )

k=1
K
X

l

b )
b lj , Σ
ω
bjl · N (ξ t ; µ
j

n

ω
bkl+1 =

K
X

=P
K

M-Step: Then, update the model parameters by

where ξ t = {vt , ψt , ρt , ∆yt , ψ̇t } is the data point at time t
and n is the number of data point.

p(ξ t ; θ) =

bl )
b lk , Σ
ω
bkl · N (ξ t ; µ
k

j=1

•

ξ 1:n = {ξ 1 , ξ 2 , · · · , ξ t , · · · , ξ n },

E-Step: For each iteration, we compute the posterior
probability for each component k by using the GMM
bl :
parameter from the previous iteration θ

•

•

Hidden Modes: mt ∈ M = {1, 2, · · · , K} is the hidden
mode at time t, with K as the number of possible hidden
modes;
Observable States: O = {ζ t }nt=1 is the set of observable
variable states, where ζ t = {vt , ψt , ρt , ∆yt } ∈ R4×1 is
the observable state at time t;
Hidden States: H = {ψ̇t }nt=1 is the set of hidden states,
where ψ̇t is the hidden state that needs to be estimated
at time t;
Transition Matrix: T = {αi,j } ∈ RK×K is the transition
matrix, where αi,j is the transition probabilities from the
ith to j th hidden modes and i, j ∈ M.

The transition matrix T can be estimated from the training
data. See Appendix A.
In the training phase of HMM, the observation ξ t = [ζ t , ψ̇t ]
consists of the observable states and hidden states. The joint
distribution between the hidden modes and the observations is
presented by

5

p(m0:t , ξ 1:t ) = p(m0 )

t
Y

[p(mi |mi−1 ) · p(ξ i |mi )]

(8)

assumptions made above, the vehicle speed and road curvature
are constant during the prediction steps in Algorithm 1. See
(14).

i=1

Thus, we obtain an MGD, p(ξ i |mi ), with mode mi and model
parameter θi . The model parameter θi can be estimated using
the EM algorithm as shown above. In the phase of inferring
the hidden states, we will estimate the hidden states (i.e.,
relative yaw rate) at time t from the consecutive values of
ḃ
the driving situation using GMM–HMM, i.e., ψ is estimated
as the conditional expectation of ψ̇ given the sequence ζ 1:t
[11]:
ḃ
ψ t = E[ψ̇t |ζ 1 , ζ 2 , · · · , ζ t ]
K
h
i
X
ψ̇ζ
ζζ −1
=
βk,t µψ̇
(ζ t − µζk )
k + Σk (Σk )

Algorithm 1 Model-based Prediction algorithm for q-step
prediction.
1: Based on the observation ξ t at time t, to predict the
relative lateral position ∆b
yt+(q+1)∆t at time t+(q +1)∆t,
q ∈ N;
2: Set i = 0 and define ∆ψt = ψ̇t ;
3: while i ≤ q & i ∈ N do
4:
Compute the relative yaw angle ψt+(i+1)∆t at time
t + (i + 1)∆t by
ψt+(i+1)∆t = ψt+i·∆t + ∆ψt+i·∆t · ∆t

(9)

5:

k=1

(11)

Compute the lateral displacement ∆yt+(i+1)∆t at time
t + (i + 1)∆t by
∆yt+(i+1)∆t = ∆yt+i·∆t + vt+i·∆t · sin ψt+i·∆t · ∆t (12)

where

"
µk =

#

"

µζk
Σζζ
k
ψ̇ , Σk =
µk
Σψ̇ζ
k

Σζkψ̇
ψ̇
Σψ̇
k

#

6:

Estimate the relative yaw rate ψ̇t+(i+1)∆t at time t +
(i + 1)∆t according to (9) by
(9) ḃ
ζ t+(i+1)∆t −−→ ψ t+(i+1)∆t
ḃ
ψ̇t+(i+1)∆t ≈ ψ t+(i+1)∆t

and βk,t is the mixing weight for mode k at time t, computed
as the probability of being in mode k and observing the
sequence ζ 1:t . The computation of βk,t is given by

(13)

∆ψt+(i+1)∆t = ψ̇t+(i+1)∆t
P

K
j=1





ζ t ; µζk , Σζζ
k



βk,t−1 · αj,k · N
hP
i


βk,t = P
K
K
ζ
ζζ
β
·
α
·
N
ζ
;
µ
,
Σ
r
k,t−1
j,r
t
r
r=1
j=1
(10)
with initialized value
βk,1

ωk · N (ζ 1 ; µζk , Σζζ
k )
h
i
=P
K
ζ
ζζ
j=1 ωj · N (ζ 1 ; µj , Σj )

D. Iterative Algorithm for Prediction
The proposed method must be able to predict the future trajectory based on the historical information ζ 1:t . This prediction
is computed by iteratively applying the driver model defined
in (9) and propagating the driving situation. The prediction
algorithm is based on a kinematic point mass model for the
ego vehicle and some assumptions as follows:
• Over a short period of time, vehicle-speed changes are
small and can be treated as a constant for the lanedeparture behavior, i.e., the vehicle speed is constant
during the iteration process of prediction. The emergent
brake behaviors, such as collision avoidance, are not
considered.
• Road curvature is continuous and can be differentiated. It
changes slowly during the iteration process of prediction.
To reduce calculation complexity, we consider driving
scenarios on a road with a small curvature (ρ < 10−4
m−1 ). Therefore, road curvature can be treated as a
constant during prediction.
Based on the above assumptions, the prediction algorithm is
shown in Algorithm 1, in which ∆t is the discretization time
of the driver model and set ∆t = 0.1 s. According to the

7:

Update speed and curvature value at time t+(i+1)∆t
vt+(i+1)∆t = vt+i·∆t
ρt+(i+1)∆t = ρt+i·∆t

(14)

Assign ∆b
yt+(i+1)∆t ⇐ ∆yt+(i+1)∆t
Update i = i + 1;
end while
11: Return the estimated relative lateral position ∆b
yt+(q+1)∆t
at time t + (q + 1)∆t.
8:
9:
10:

IV. E XPERIMENTS AND DATA C OLLECTION
A. Data Collection
All the data come from naturalistic driving and are collected from the Safety Pilot Model Deployment program [30].
Volunteers were recruited from Ann Arbor city with their
own vehicles mounted with Mobileye, DSRC, and a data
acquisition system (DAS) (Fig. 5). The vehicle-based variables
such as vehicle speed and the positions of the gas/brake pedal
were obtained from the CAN-Bus and the road-based variables
are collected from the Mobileye systems.
Data recorded from ten drivers were used in the experiments. Drivers had an opportunity to become accustomed to
the equipped vehicles. They performed casual daily trips for
several months without any restrictions on or requirements for
their trips, the duration of the trips, or their driving style. While
the vehicle was running, the onboard PC recorded driving data
with a frequency of 10 Hz. The data process and recording
equipment were hidden from the drivers, thus avoiding the
influence of recorded data on driver behavior.

6

×106

-4

-3.1
-3.2

(a)

-3.3

(b) (c)

-2.5

-4.3
-4.4

2 4 6 8 10 12
×10

-4.2

-4.5

BIC
-1.5
-1.55

The data sets for training a PDM were extracted from the
entire set of data. The training data consisted of many events
that included the DCB and/or LDB. The following rules were
considered to determine the beginning points and endpoints
for each event of interest:
•

•

•
•
•

•

Detect the case data points ξ in which the vehicle lateral
position are close to the road boundary with ∆y ≤ 0.5
m or crossing the road boundary [10].
The data points that were behind the case data points
with 15 s and before the case data point with 15 s were
extracted.
The data points with road curvature ρ > 10−4 m−1 were
deleted.
Only the data with road width of about 3.7 m was
considered.
We deleted the event with turning lights on, avoiding the
lane change behaviors. Also, we checked the event where
the lateral displacement changes from one lane to the
center of the adjacent lane was deleted. This can avoid
some lane change behaviors without turning lights on.
The event with a duration of less than 15 s was deleted.

The statistical results of the driving data for 10 drivers are
listed in Table I. The number of events ranges from 330 to

×10

-3.5

-4.9

2 4 6 8 10 12

-3.3

6

-3.7

2 4 6 8 1012

K

×106

×10

-3.6

-4.7
-4.8

-3.8

2 4 6 8 10 12

K

×106

-3.4

BIC

-4.2
-4.3

-4.5

2 4 6 8 10 12

K

6

K
-4.1

-2.6

2 4 6 8 1012

-4.6

-4.4

B. Data Preprocessing

BIC

6

BIC

Average time [s]
21.37
22.66
20.93
22.75
21.16
22.20
21.52
21.20
21.07
22.00
21.60

-2.5

K

-1.45

BIC

Total time [min]
247.80
181.69
759.52
705.76
1244.43
2238.01
118.70
1698.36
1504.66
1617.04
-

6

BIC

-1.4

# of events
696
481
2177
1861
3529
6048
331
4806
4285
4411
-

-4.8

2 4 6 8 10 12

×10

-2.4

K

TABLE I: Statistical Results of the Extracted Driving Data for
10 Drivers
Driver
1
2
3
4
5
6
7
8
9
10
Average

-2.3

-4.6

×10

2 4 6 8 10 12

K

6

-4.4

-4.5
-4.6

×10

BIC

BIC

-4.3

Fig. 5: Examples of the experiment vehicle with datacollection equipment. (a) Example of an experiment vehicle;
(b) Mobileye; and (c) DAS.

-2.6

2 4 6 8 1012

K

6

-4.4

×106

-2.4

-4.2

K
-4.2

-2.3

-4.1

BIC

BIC

-3

×106

BIC

-2.9

-3.5
-3.6

2 4 6 8 10 12

K

-3.7

2 4 6 8 1012

K

Fig. 6: BICs with respect to different numbers of GMM
component for 10 drivers (black line with triangle) and the
average value of BIC (red line with triangle).

6000 and the driving-time duration for each event averages
about 22 s.

C. Training Model
A cross-validation (CV) method was selected to train the
PDM. Driving data for each driver was evenly divided into
ten folds. Nine folds were used to train the driver model, and
the remaining one was used to assess the performance. The
CV method guarantees that the training data and test data are
not joint.
We used BIC to determine the number of GMM components
by finding the “elbow point”. Fig. 6 presents the experiment
results of BIC for different numbers of GMM component in
10 drivers. As the BIC is convergent at about K = 10, we
select K = 10.

7

0.7
q=5

q = 10

q = 15

q = 20

q = 25

q = 30

Prediction performance

0.6
0.5
0.4
0.3
0.2
0.1
0

1

2

3

4

5

Driver

6

7

8

9

10

Fig. 7: The errors and stand deviations for 10 drivers with different prediction q-steps.
0.35

Prediction Error

0.3
0.25
0.2
0.15
0.1
0.05

0

5

10

15

20

Prediction step q

25

30

35

Fig. 8: Prediction errors on average of 10 drivers with respect
to different steps q.

[t : t + q∆t] is less than a value γ1 as well as the vehicle
trajectory at future time [t + q∆t] is less than γ2 , is a warning
sent. For example, for γ1 = −0.1, γ2 = 0.1 and q = 30,
even though the condition (15a) was valid, if the upcoming
trajectories [∆b
yt:t+3 ] are always larger than −0.1 m ( i.e., the
driver will not cross the road boundary a lot) or the future
position ∆b
yt+3 is larger than 0.1 m (i.e., the driver will bring
the vehicle back to the center of driving lane), a warning will
not be sent. Conditions (15b) and (15c) give drivers more time
to bring the vehicle back and keep the vehicle in the driving
lane, instead of giving them a warning and being a nuisance.
Parameter τ is a threshold value of TLC. In [10], the authors
agreed that the time margin should be larger than 0.9 s (i.e.,
τ > 0.9 s), thus allowing enough time for a response and
applying a correct reaction for drivers. Thus, in this paper, we
set τ = 1.0 s.

D. TLC-Based LDP with PDM
The approach proposed in this paper is named as a TLCPDM method because it is based on a PDM. To reduce the
FAR, we designed a personalized warning strategy based on
the predicted vehicle trajectory. Assuming that we can estimate
the relative lateral position of a vehicle at the upcoming time
t + q∆t by the Algorithm 1, we can also know whether
drivers will bring the vehicle back from road boundaries in the
upcoming behaviors, keeping vehicles in the driving lane. Only
when the TLC reaches a predefined criterion, is a warning sent,
i.e.,
tT LC < τ

(15a)

min{∆b
yt:t+q∆t } < γ1

(15b)

∆b
yt+q∆t < γ2

(15c)

If the driving conditions (15a) – (15c) are valid, a warning
is sent to drivers for the TLC-PDM method. Equation (15a)
checks the TLC condition, (15b) detects the upcoming vehicle
trajectory in time [t : t + q∆t] and (15c) judges whether
drivers will bring the vehicle back to the center of the lane
at future time t + q∆t. Thus, only when the TLC is valid
and the minimum relative lateral position during future time

E. Comparison with Other Approaches
We compare the TLC-PDM approach with two recent methods that have been proven to demonstrate good performance in
LDW applications. One is the basic TLC-based method, which
considers no human factors or features of vehicle dynamics.
The second one is the directional sequence of the piecewise
lateral slopes (DSPLS) method [8], called the TLC-DSPLS
method.
1) Basic TLC: An alarm signal is given to drivers when
the following condition (16) is valid.
tT LC < τ

(16)

This method is very simple but relatively efficient in a simple
driving scenario.
2) TLC-DSPLS: TLC-DSPLS method [8] is also based
on a probabilistic driver model, in which the trajectory of
driving signals is described by the DSPLS which then infers
drivers’ upcoming episodes of vehicle trajectory. The Bayes
rule is used to compute the probabilities of p(LCB|state)
and p(DCB|state). The decision of the occurrence of LCB
is given by

8

5

2
Predicted
Real

3

1
50

2

55

60

65

1

3

1
50

2

55

60

1
0

50

100

150

0

200

5

0

5
Predicted
Real

50

100

1

2

55

60

150

65

1
50

100

0

200

0

5

2
Predicted
Real

1

3

50

2

55

60

65

1
0

50

2

55

60

150

200

0

65

50

100

150

200

Time[s]
(c) q = 15

4 1.5

∆ y[m]

∆ y[m]

4 1.5

50

3 1

Time[s]
(b) q = 10

2

3

Predicted
Real

1

Time[s]
(a) q = 5

0

65

2

4 1.5

2
Predicted
Real

4 1.5

∆ y[m]

0

Predicted
Real

4 1.5

∆ y[m]

∆ y[m]

4 1.5

5

2

∆ y[m]

5

3 1
50

2

55

60

65

1
0

Time[s]
(d) q = 20

50

100

150

200

0

0

50

Time[s]
(e) q = 25

100

150

200

Time[s]
(f) q = 30

Fig. 9: One example of the prediction results of the relative lateral position ∆y for driver #10 with different prediction steps
using the TLC-PDM method.

p(DCB|state)
< γ0
p(LCB|state)

(17)

where γ0 is a predefined threshold. In this paper, we just
reproduce and re-examine this method as was done in [8] to
show the benefits of our proposed method. We will not further
discuss TLC-DSPLS. More details can be found in [8]. Thus,
warning if both (16) and (17) are valid.

B. Performances for LDW system

V. R ESULTS AND A NALYSIS
We discuss and analyze the experimental results in terms of
the prediction performance of lateral positions and the FAR
of the LDW system for DCB.
A. Prediction of Lateral Positions
Being able to predict the performance of the vehicle’s
upcoming trajectories is crucial for designing a personalized
LDW system and reducing the FAR. The performance criterion
is computed using the errors of predicted and experimental
vehicle trajectory:
t+q

e∆yt =

1X
|∆b
yt:t+q∆t − ∆yt:t+q∆t |
q t

prediction step q for 10 drivers. Fig. 9 gives an example of
predicted lateral position with respect to different prediction
steps. We note that a larger prediction time (i.e., q = 30) leads
to a larger prediction error. This error could be influenced
by the assumption that during iteration of each step in the
prediction horizon in Algorithm 1, the road curvature and
vehicle speed are treated as constant.

(18)

Fig. 7 presents the prediction results of the vehicle trajectory. We know that the proposed method can predict the
vehicle lateral trajectory precisely across different prediction
steps. For q = 5 (i.e., prediction time q∆t = 0.5 s), the
prediction error ranges from 0.063 (driver #8) to 0.1696 (driver
#9). For q = 30 (i.e., prediction time q∆t = 3.0 s), the
prediction error ranges from 0.2090 (driver #8) to 0.5138
(driver #9). For the same driver, we note that the prediction
performance decreases as the prediction step increases. Fig. 8
shows the average value of prediction errors with respect to

Fig. 10 presents an example comparing different LDW
algorithms using the basic TLC method and the TLC-PDM
method for driver #10. We note that the basic TLC method
is unable to predict or infer whether the driver will bring the
vehicle back to the center of the driving lane. Thus, the LDW
system will send a warning once condition (15a) is satisfied,
even if the driver intends to bring the vehicle back before
or after the warning. Furthermore, we can note that a larger
critical value (τ ) of TLC leads to a higher FAR, which tends
to annoy drivers.
For the TLC-PDM method, we note that in region A in
Fig. 10, the driver does not receive a warning from the
LDW system, even though the vehicle is laterally crossing
the boundary a little (i.e., less than 0.05 m) because the TLCPDM method can predict that the driver can steer the vehicle
back to the center of the driving lane by him/herself in a short
time. Region B is the same case as Region A. Green cross
symbols in Fig. 10 indicate that the TLC-PDM criteria (i.e.,
(15a) - (15c)) are valid and a warning is triggered, because
Algorithm 1 can predict the vehicle lateral trajectory in the
upcoming 1s and estimate that (1) the vehicle will obviously
cross the lane boundary (i.e., min{∆b
yt:t+q∆t } < γ1 ), and (2)
the driver will not be able to bring the vehicle back to the
current lane in a short time (i.e., ∆b
yt+q∆t < γ2 ). Therefore,
the TLC-PDM method gives the driver a warning, avoiding a

9

4
0.2

3

0.2

q = 5

0.05

0

0

70

80

90

100

γ2

γ2

γ1

(a) q = 5

0.01

γ1

(b) q = 10

0.01
0.005

A

-0.01

0.2

0

B

0.1

60

70

1.3

80

84

90

86

0.05

0.05

0

0

100

γ2

1
0.9

1.5

0.8

A

1

(c) q = 15
84

(d) q = 20
0.2

0.2

0.15

100

0.1

0.05

0

0

NT LC−P DM
(19a)
N
NT LC
η2 =
(19b)
N
NT LC−DSP LS
η3 =
(19c)
N
where NT LC−P DM and NT LC are the number of warning
points using the TLC-PDM method and the basic TLC method,
η1 =

γ1

γ2

6
0. .5
0 4
0. .3
0 .2
0 1
0.
0 .1
-0 2
.
-0 .3
-0 .4
-0

γ2

(e) q = 25

crash and reducing the FAR. The TLC-PDM method considers
drivers’ upcoming behaviors and provides a more acceptable
warning decision (i.e., criteria (15a) – (15c)). Furthermore,
for the TLC-PDM method, parameters q, γ1 and γ2 in (15a) –
(15c) play a major role in determining when the LDW system
will send a warning.
1) Influences of parameters q, γ1 , γ2 : From (15a) – (15c),
we know that the TLC-PDM warning strategy is developed
from the basic TLC warning condition. Different values of
parameters q, γ1 , and γ2 lead to different warning results.
To show the influences of parameters q, γ1 , and γ2 on the
warning performance, we use a warning frequency to describe
the difference, given by:

0.1

0.05

6
0. .5
0
4
0. .3
0 .2
0
1
0.
0 .1
-0 2
.
-0 .3
-0 .4
-0

Fig. 10: Example of LDWS using the basic TLC algorithm
with τ = 1 s and TLC-PDM method with γ1 = −0.05, γ2 =
0.1, q = 10 for driver #10. The red circle represents that (16) is
valid for the basic TLC algorithm. The green cross represents
that the TLC-PDM criteria (15a) – (15c) are valid.

η1

90

η1

80

Time[s]

0
0 .6
0 .5
0. .4
0 3
0. .2
1
- 0
-0 0.1
- .2
-0 0.3
.4

70

q = 30

q = 25

0.15

60

γ1

86

B
50

γ2

γ1

6
0. 5
0. .4
0 .3
0 2
0. .1
0
0 .1
-0 .2
-0 .3
-0 .4
-0

1.1

6
0. 5
0. .4
0 .3
0 .2
0
1
0.
0 .1
-0 2
.
-0 .3
-0 .4
-0

1.2

0
0 .6
0 .5
0. .4
3
0
0. .2
1
- 0
-0 0.1
-0 .2
-0 .3
.4

-0.015

50

2

∆ y[m]

0.1

-0.01

2.5

0.5

q = 20

0.15

0
0 .6
0 .5
0. .4
0 3
0. .2
1
-0 0
- .1
-0 0.2
-0 .3
.4

-0.03

η1

-0.005

-0.02

0.2

q = 15

0.15

η1

0

0
0 .6
0 .5
0. .4
0 3
0. .2
1
- 0
-0 0.1
- .2
-0 0.3
.4

Relative Yaw[rad]

0.02

6
0. .5
0 4
0. .3
0 .2
0 1
0.
0 .1
-0 2
.
-0 .3
-0 .4
-0

60

6
0. .5
0
4
0. .3
0 .2
0
1
0.
0 .1
-0 2
.
-0 .3
-0 .4
-0

50

0.1

0.05

B

A

0.1

0
0 .6
0 .5
0. .4
3
0
0. .2
1
- 0
-0 0.1
-0 .2
-0 .3
.4

0

η1

η1

1

q = 10

0.15

0
0 .6
0 .5
0. .4
3
0
0. .2
1
- 0
-0 0.1
-0 .2
-0 .3
.4

TLC[s]

0.15

2

γ1

(f) q = 30

Fig. 11: Example of the influences of parameters γ1 and γ2
on the warning over different steps q by using the TLC-PDM
method for driver #10.

respectively; N is the total number of driving data. A larger
value of η1 , η2 or η3 indicates the corresponding method
obtains a higher warning frequency.
According to criteria (15a) – (15c), we know that
NT LC−P DM could not be larger than NT LC (i.e., η1 ≤ η2 )
for the same driver, because the number of warning data points
is always less than the total number of driving data. Fig. 11
gives an example of the influences of parameters q, γ1 , and γ2
on the warning performance η1 for driver #10 and the warning
performance with the basic TLC is η2 = 0.2034. The number
of warning points increases with values of γ1 or γ2 increasing,
but is always less than NT LC . For example, in Fig. 11, we
can note that when (γ1 , γ2 ) = (0.6, 0.6), η1 approaches the
maximum value but is always less than η2 = 0.2034 for all q.
For a fixed q, we know that a larger value of parameters γ1
or/and γ2 generates a higher warning frequency. Parameters γ1
and γ2 in the warning strategy (15a) – (15c) allow designers to

10

TABLE II: Warning Frequency for Different Methods with
γ0 = 1, γ1 = −0.05, γ2 = 0.1, q = 10 and Prediction Time
is 1 s.

Mean and std. of ∆ y

2.5
2
1.5
1
0.5

1

2

3

4

5

6

7

8

9

10

(a)
0.5
Basic TLC

TLC-DSPLS

TLC-PDM

0.4

Driver
1
2
3
4
5
6
7
8
9
10
Average

NT LC−P DM (η1 )
0.0936
0.1711
0.2361
0.2823
0.0124
0.0143
0.1916
0.1660
0.1660
0.0703
0.1404

NT LC (η2 )
0.2034
0.4263
0.4079
0.3818
0.0622
0.0534
0.4081
0.3273
0.2735
0.2034
0.2747

NT LC−DSP LS (η3 )
0.1590
0.3355
0.3414
0.3146
0.0456
0.0369
0.3221
0.3552
0.2489
0.1673
0.2336

η

0.3
0.2

6

0.1

5

0.8
1

2

3

4

5

6

Driver
(b)

7

8

9

10

Fig. 12: Relationship between warning frequency and driving
styles.

False rate

0

Basic TLC
TLC-DSPLS
TLC-PDM

1

4

0.6

3

0.4

2

0.2
0

1

set a preferred parameter for drivers. For instance, by setting
a larger value of γ1 , the warning frequency will be larger, but
a larger γ1 can guarantee a higher safety level. Parameter γ1
has the same influence as parameter γ2 on warning frequency.
For the same value of parameters q, γ1 and γ2 , the warning
frequency also differs between drivers. Table II lists the
experimental results of warning frequency for 10 drivers with
parameters γ0 = 1, γ1 = −0.05, γ2 = 0.1, and q = 10. For
driver #5 and driver #6 the TLC-PDM obtains a very lower
warning frequency at about 0.013. Driver #1 and driver #10
can reach a middle level of warning frequency at about 0.08.
For other drivers, however, the value of warning frequency
is greater than 0.16. The difference across drivers is due to
their differing driving styles. Drivers who prefer to drive a car
in the middle of the road will naturally get a lower warning
frequency; but some drivers, however, prefer to drive close to
the road boundary. Fig. 12 presents the relationship between
driving styles and warning frequency. We know that driver #5
and driver #6, who usually drive a vehicle in the center of
the road (i.e., ∆y ≈ 1.75 m, which is half the width of the
driving lane), can get a lower warning frequency. Conversely,
driver #3 and driver #4, who drive close to the boundary of
the driving lane, get a higher warning frequency.
2) False-Alarm Rate: We assess the performance of the
LDW system using an FAR. The false warning event is
recorded as a warning is triggered when the DCB occurs
during the upcoming prediction interval. The FAR is computed
by
λ=

# of false warning event
# of all warning event

(20)

For example, when the prediction time is 1 s, if a warning is
triggered at time t and the DCB will occur during [t, t+1], then
the warning is a false warning. For the TLC-DSPLS method,

0

5

10
10

20

30
15

20

25

30

Time[s]× 10 -1

Fig. 13: Comparison of the FAR between the basic TLC
method, TLC-DSPLS method, and TLC-PDM method. For
the TLC-DSPLS method, we set γ0 = 1. For the TLC-PDM
method, we set γ1 = −0.05 and γ2 = 0.1.

the DCB behavior is defined by the slope of a piecewise lateral
displacement. See [8]. For the TLC-PDM method, the DCB
is determined by parameter γ2 .
To further show the benefits of the TLC-PDM method,
we defined that the DCB occurred when the future lateral
displacement ∆yt+q∆t was larger than 0.1. Fig. 13 shows the
experimental results of FAR for three different methods, where
γ1 = −0.05. The average and standard variance values of FAR
for driver #10 are recorded. The FAR for LDWS with the basic
TLC method was from 18.8 % and 492.2 %. The FAR for the
LDW system with the TLC-DSPLS method was from 6.44%
and 198.1%. The FAR for the proposed TLC-PDM method
was between 1.13 % and 74.51 %.
For the basic TLC method, the FAR will reach over 100 %
when the prediction time is 1.5 s. For the TLC-DSPLS method,
the FAR nearly reaches 100 % when the prediction time is 2.0
s because the average error of LCB and DCB classification
increases fast with the increase in prediction time [8]. For the
TLC-PDM method, the largest FAR is about 75 %, as the
proposed prediction algorithm can predict the future lateral
trajectory and its tendency.
VI. F URTHER D ISCUSSION AND F UTURE W ORK
We proposed a TLC-PDM method for the LDW system,
reducing the FAR caused by driver correction behavior. We

11

also developed a model-based prediction algorithm for predicting vehicle trajectory and as a warning strategy for an LDW
system. The experimental results show good performance for
predicting the lateral trajectories of a vehicle. In this paper, we
mainly focus on the framework of methods and the experiment
validation in a driving scenario with small road curvature.
A. Parameter Selection
In this research, the parameters (γ1 , γ2 , τ , and q) in the
warning strategy were defined subjectively by the researchers.
We should note that these parameters can be tuned and set
according to different design requirements. For example, if
the designers wish to pay more attention to the safety level
than to the warning frequency, they can set a larger value
of γ1 or/and γ2 . Furthermore, from Fig. 12, we know that
driving styles are highly related to warning performance. In
future work, driver characteristics will be considered based on
the TLC-PDM method to design a more driver-friendly LDW
system.
B. Vehicle Dynamics and Road Curvature
In the model-based prediction algorithm (Algorithm 1),
the attributes of vehicle dynamics are not considered. Lateral
trajectory prediction using vehicle dynamics (e.g., steering
angle, brake/gas pedal position, acceleration) can improve the
prediction accuracy [14]. In our work, we treated vehicle speed
and road curvature as a constant during the iteration process
and did not consider the effects of steering angle. Furthermore,
road profiles (such as straight road or curved road with or
without constant road curvature) greatly impact warning functions, see [14]. We use a simple TLC-based risk function to
determine whether a warning should be given. In future work,
we will take vehicle dynamics and more complicated road
profiles into consideration in order to improve the performance
of the TLC-PDM method.
C. Driver State and Intention
We do not classify or recognize the drivers’ physical and
physiological states [31], such as the level of drowsiness or
fatigue [10], [32], drunkenness, or, aggression [33]. Recognizing drivers’ intentions, such as lane changing, is also beyond
the scope in this paper. To design a more acceptable LDW
system for drivers, drivers’ intentions should be considered.
Therefore, in future applications of the TLC-PDM method,
drivers’ lane change behaviors [34], [35] must be adequately
recognized and considered.
VII. C ONCLUSION
In this paper, a new method (TLC-PDM method) for reducing the false-warning rate of lane-departure warning systems is
proposed. First, we model driver behaviors using five variables,
including vehicle speed, relative yaw angle, the change rate of
the relative yaw angle, road curvature, and relative lateral displacement. Then a personalized driver model is developed by
combining the Gaussian mixture model and the hidden Markov
model. The model parameters are determined by learning from

the naturalistic driving data collected from 10 drivers. Based
on the personalized driver model, a model-based prediction
algorithm for predicting the upcoming lateral vehicle trajectory
is proposed and validated. Second, based on personalized
driver model and the proposed prediction algorithm, a warning
strategy is also developed. We also discuss the influences
of different parameters of warning strategy on the warning
performance. Last, to show the advantages of the TLC-PDM
method, we compare it with the basic TLC method and the
TLC-DSPLS method. The experimental results show that the
TLC-PDM method can predict the upcoming lateral trajectory
of a vehicle and obtain the lowest false-warning rate of 3 %
with 1 s prediction time.
A PPENDIX A
In this Appendix, the calculation method of transfer matrix,
T , in Section III, C. Hidden Markov Model is presented. Given
the training data set with n data points ξ t :
ST rain = {ξ 1 , ξ 2 , · · · , ξ t , · · · , ξ n }
For each point ξ t , we define ξ t is subject to mode mk ∈ M =
{1, 2, · · · , K} if
mk =

max
k∈{1,2,··· ,K}

N (ξ t ; µk , Σk )

(21)

Therefore, each point ξ t has a mode mt ∈ {1, 2, · · · , K},
and we have a mode sequence with the same number of
training data points
{mt }nt=1 ⇐⇒ ST rain = {ξ t }nt=1

(22)

Based on (22), we estimate the transfer probability between
mode i and mode j by
αi,j =

Fi,j
, i, j = 1, 2, · · · , K
ni

(23)

where Fi,j is the count of transferring from mode mi to mj ,
and ni is the P
total number of training data points at mode i.
K
In this paper, i=1 ni = n > 5 × 105 . Finally, we can get the
transfer matrix T as


α1,1
 α2,1

T = .
 ..

α1,2
α2,2
..
.

···
···
..
.

α1,K−1
α2,K−1
..
.


α1,K
α2,K 

.. 
. 

αK,1

αK,2

···

αK,K−1

αK,K

(24)

K×K

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