Content uploaded by Toru Seo
Author content
All content in this area was uploaded by Toru Seo on Mar 27, 2018
Content may be subject to copyright.
Traffic State Estimation with the Advanced Probe
Vehicles using Data Assimilation
Toru Seo, Takahiko Kusakabe, and Yasuo Asakura
Abstract—This paper proposes a method for estimating traffic
state from data collected by the advanced probe vehicles, namely,
probe vehicles with spacing measurement equipment. The probe
vehicle data are assumed to include spacing information, in
addition to conventional position information. The spacing in-
formation is collected as secondary products from advanced
vehicle technologies, such as automated vehicles. Traffic states
and a fundamental diagram are derived from the probe vehicle
data. Then, a traffic state estimator based on a data assimilation
technique and a traffic flow model is formulated. This procedure
is intended to mitigate negative effects in traffic state estimation
caused by high fluctuations in microscopic vehicular traffic. The
validation results with a simulation experiment suggested that the
proposed method works reasonably; for example, the proposed
method was able to estimate precise traffic state compared with
the previous methods. Therefore, we expect that the proposed
method can estimate precise traffic states in wide area where the
advanced probe vehicles are penetrated, without depending on
fixed sensor infrastructures nor careful parameter calibration.
I. INTRODUCTION
Road traffic monitoring is one of the most essential role of
intelligent transport systems. In order to achieve efficient traffic
management, a road administrator need to understand the
traffic situation. In past half a century, various methods have
been investigated and implemented to realize more effective
monitoring of road traffic. Especially, recent achievements in
information and communications technology have enabled us
to obtain various types of information by employing a wide
variety of data collection methods. Traffic situation is often
defined as traffic state, which is a set of the flow, density and
speed in a specific spatiotemporal point or area in a road net-
work. Since completely continuous and long-term observation
in the entire road network is practically impossible, states in
unobserved area are often estimated from partially observed
traffic data. This is referred to traffic state estimation (TSE) in
which numerous studies have investigated, for example, [1]–
[8]. Especially, probe vehicle-based TSE methods [4]–[8] have
received attention due to probe vehicles’ wider data collection
range compared with conventional roadside detectors.
Vehicle automation technologies have begun to spread in
the practical uses. Currently, these technologies are mainly
utilized for microscopic-scale traffic controls, such as advanced
driver assistance, autonomous driving, and vehicle-to-vehicle
cooperation [9]. Meanwhile, data collected by these technolo-
gies, namely, surrounding environment of the vehicles [10], is
also expected to be utilized for macroscopic traffic condition
T. Seo, T. Kusakabe and Y. Asakura are with the Tokyo Institute of
Technology. Email: t.seo@plan.cv.titech.ac.jp (T. Seo)
Part of this work was financially supported by the Research Fellow (DC2)
program of the Japan Society for the Promotion of Science (KAKENHI Grant-
in-Aid for JSPS Fellows #26010218).
monitoring and macroscopic controls. For example, [7], [8]
proposed TSE methods supposing that the spacing data from
advanced vehicles are available for inputs of TSE—they re-
ferred such utilization of advanced vehicles as probes to “probe
vehicle with spacing measurement equipment (PVSME).” In-
vehicle data collection systems for similar purposes were de-
veloped by [11]–[14]. Meanwhile, the TSE methods proposed
by [7], [8] do not rely on traffic flow models strongly. It implies
both advantage and disadvantage: the methods can be applied
for any traffic situation without information about characteris-
tics of traffic, though the methods’ estimation precision can be
relatively low, especially in high resolution. One of the most
significant reasons for this disadvantage is high fluctuation
in microscopic vehicular traffic, such as vehicle platoons and
lane-changing.
TSE methods were incorporated with traffic flow models
by existing studies [1]–[6] in order to interpolate traffic states
in unobserved area and/or reduce observation noises in ob-
served area. Especially, frameworks of data assimilation (e.g.,
Kalman filtering-like techniques) are often utilized for TSE
with traffic flow models [1]. The LWR model [15], [16] is
the typical traffic flow model that represents theoretical aspect
of macroscopic traffic dynamics. It represents traffic based on
two important principles: the fundamental diagram (FD) and
the conservation law (CL). An FD (also known as flow–density
relation) determines relation among flow, density and speed.
Therefore, it plays significant role in the LWR model (and
many other traffic flow models), although its functional form
and parameters are uncertain in actual traffic. It made existing
GPS-equipped probe vehicle-based TSE methods, such as
[4]–[6], applicability low; because they require careful FD’s
parameter calibration based on exogenous information, such as
detector data (a detector-based TSE method can endogenously
estimate FD’s parameters [2], [3]).
The aim of this paper is to propose a PVSME-based
TSE method that is robust against the high fluctuation in
microscopic vehicular traffic. To mitigate the fluctuation, the
proposed TSE method is incorporated with macroscopic traffic
flow model with an FD and a CL (i.e., the LWR model)
by using data assimilation. Since an FD is equivalent to a
headway–spacing relation which is included in the PVSME
data, it can be estimated endogenously. By comparing to the
conventional GPS-equipped probe vehicle-based TSE methods,
the proposed method can be almost self-sufficient; because it
is not required to predetermine FD parameters. This makes the
proposed method’s applicable range wider, if the advanced ve-
hicles have penetrated in the real world. The remainder of this
paper is organized as follows. Part II describes formulation of
the TSE method. Part III investigate numerical characteristics
of the TSE method through a simulation experiment.
Real world:
- road schematics
- advanced probe vehicle
(PVSME)
- desired data resolution ∆T,∆X
Probe vehicle data:
- spacing am(AT,X )
- position tm(AT,X )
FD:
- functional form F
- constant parameter
region: B
Fundamental diagram:
- parameters
ˆui(t),ˆ
kci(t),ˆκi(t)
Traffic state:
- density ˆ
ki(t)
Observation vector:
-yt
Data assimilation:
- Ensemble Kalman filter
·xt=ft(xt,νt)
·yt=Htxt+ωt
-ft: Cell transmission model
Deviations of noise terms:
-σk,σu,σkc,σκ,
ξk,ξu,ξkc,ξκ
Conservation law,
1st order traffic
flow model
Filter ensemble of state vector:
- density and FD parameters
{xn
t|t}N
n=1
Estimation result:
- density k∗(AT,X )
- FD parameters
Fig. 1. Illustrated concept of the procedure of the proposed TSE method. The notation is defined at parts II-B and II-C.
II. ME TH OD
This part describes formulation of the proposed TSE
method within a framework of data assimilation.
A. Concept
The following conditions are assumed for traffic that is
subject of TSE in this study. The road schematics (e.g.,
position and length of links, link connectivity, position of
nodes, number of lanes) are pre-given information for the
proposed method. Probe vehicles are randomly distributed in
traffic with unknown penetration rate. The probe vehicles are
PVSMEs, which continuously collect data including spacing
(i.e., distance between probe vehicle and its leading vehicle
ahead) and position without error. Note that to measure spacing
by probe vehicles, vehicle length is needed to be derived
somehow.
The procedure of the proposed TSE method can be de-
scribed as follows. First, time–space resolution for TSE, time–
space resolution for FD’s parameter estimation, and FD’s
functional form are set by analysts. Then, traffic states and
FD parameters are respectively derived (i.e., observed) from
the probe vehicle data. Finally, by using a data assimilation
technique, traffic states are updated from the observed traffic
states, the observed FD parameters, estimated traffic states in
the previous timestep, and a traffic flow model.
Specifically, the TSE method proposed by [7] is utilized for
observing traffic states. FD parameters are derived by regress-
ing spacing and headway (spacing divided by speed) relation
in the probe vehicle data. The cell transmission model [17],
a numerical computation method for solving the LWR model,
is applied for the traffic flow model. As a data assimilation
technique, Ensemble Kalman Filter (EnKF) [18] is employed
due to its capability for dealing nonlinear phenomena in traffic
(the system model). Note that the observation model of the
proposed TSE method can be represented by a linear system;
therefore, EnKF is an appropriate way for this problem. Figure
1 summarizes the procedures of the proposed method.
B. Ensemble Kalman Filter
This part briefly describes EnKF by employing the expres-
sions from [19].
A state–space model for EnKF can be described as follows
xt=ft(xt,νt),(1)
yt=Htxt+ωt,(2)
where eq (1) is a system equation, eq (2) is an observation
equation, xtis a state vector, ftis a system model, νtis
a system noise vector, ytis an observation vector, Htis an
observation matrix, and ωtis an observation noise vector,
at timestep trespectively. The observation noise vector ωt
follows normal distribution whose average is 0 and variance-
covariance matrix is Rt, namely, ωt∼ N (0, Rt).
The general procedure of EnKF can be described as fol-
lows:
Step 1 Generate an ensemble of the initial states {xn
0|0}N
n=1.
Let t←1.
Step 2 Prediction step:
Step 2.1 Generate an ensemble of the system noises
{νn
t}N
n=1.
Step 2.2 Calculate xn
t|t−1=ft(xn
t−1|t−1,νn
t)for
each n.
Step 3 Filtering step:
Step 3.1 Generate an ensemble of the observation
noises {ωn
t}N
n=1.
Step 3.2 Obtain the filter ensemble {xn
t|t}N
n=1 for
each n, by calculating eq (3) based on
yt, Ht, Rt,{xn
t|t−1}N
n=1,{ωn
t}N
n=1.
Step 4 Increment the timestep, t←t+ 1. Go back to Step 2
until t=tmax.
(kc
i(t), kc
i(t)ui(t))
(κi(t),0)
ui(t)
Density k
Flow q
Fig. 2. Triangular FD and its parameters in cell iat time t.
The filter ensemble in Step 3.2 can be derived as follows:
xn
t|t=xn
t|t−1+ˆ
Ktyt+˘
ωn
t−Htxn
t|t−1,(3)
where,
ˆ
Kt=ˆ
Vt|t−1H′
tHtˆ
Vt|t−1H′
t+Rt−1,(4a)
ˆ
Vt|t−1=1
N−1
N
j=1
˘
xj
t|t−1(˘
xj
t|t−1)′,(4b)
˘
xn
t|t−1=xn
t|t−1−1
N
N
j=1
xj
t|t−1,(4c)
˘
ωn
t=ωn
t−1
N
N
j=1
ωj
t.(4d)
In this study, following mean vector of the filter ensemble
is defined as an “estimation result” at timestep t:
¯
xt|t=1
N
N
n=1
xn
t|t,∀t. (5)
C. TSE Method Formulation
In this part, the TSE method is formulated by specifying
the terms and the function in eqs (1) and (2).
1) System Equation: A nonlinear system equation that
represents traffic dynamics is formulated.
The state vector xtis defined as follows:
xt=k1(t), k2(t), . . . , ki(t), . . . kM(t),
u1(t), u2(t), . . . , ui(t), . . . uM(t),
kc
1(t), kc
2(t), . . . , kc
i(t), . . . kc
M(t),
κ1(t), κ2(t), . . . , κi(t), . . . κM(t)(6)
where, ki(t)is a density, ui(t)is a free flow speed, kc
i(t)
is a critical density, and κi(t)is a jam density, of cell i
at timestep trespectively. Note that ki(t)is the traffic state
and ui(t), kc
i(t), κi(t)are the FD parameters. The number of
elements in vector xtis 4M, where Mis total number of
discretized spaces (cells) in the system model.
Time t
Space x
∆T
∆X
region AT,X for the observation of a traffic state
spacing: am(AT,X )
∆t
licell ifor the system model (CTM)
region Bis larger than AT,X and cell i
Fig. 3. Coordinates for the proposed TSE method and the probe vehicle data:
solid curves represent vehicle trajectories; dotted grid represents regions for
the observation; dashed gird represents cells for the system model.
The system model ftconsists of two parts: on traffic state
dynamics and on FD parameters.
The first part of the system model ft(on traffic state
dynamics) is the CTM with noise term. It is described as
follows:
Qin
i(t) = εk
i(t) min li−1ki−1(t),
kc
i(t)
κi(t)kc
i(t)(liκi(t)−liki(t)) ,(7a)
ki(t+ 1) = ki(t) + Qin
i(t)/li−Qin
i+1(t)/li,(7b)
where Qin
i(t)is a number of vehicles that enters cell iat
timestep t,liis the length of cell i, and εk
i(t)is a system noise
of cell iat timestep tthat is expressed as εk
i(t)∼ N (1, σ2
k)
(truncated to non-negative values). Triangular FD is employed
by eq (7); the relation between triangular FD and its parameters
is shown in figure 2. The original CTM [17] represents traffic
dynamics by solving the LWR model (partial differential
equation system of continuous fluid approximated traffic) using
the Godunov scheme; it can be represented as eq (7) without
the noise term, εk
i(t).
The CTM requires time and space discretization. The size
of the CTM’s timestep is ∆t. The cell length should be equal
to the timestep size multiplied by the free flow speed in the
cell (i.e., li= ∆tui) in order to keep consistency with the
LWR model; and should be larger than the value in order to
satisfy the CFL condition. The detailed discretization method
is described in the next part II-C2, eq (14) and figure 3, because
it has significant relation with the observation.
The other part of the system model ft(on FD parameters)
is assumed to be random walks. They can be represented as
ui(t+ 1) ∼ N (ui(t), σ 2
u),(8a)
kc
i(t+ 1) ∼ N (kc
i(t), σ2
kc),(8b)
κi(t+ 1) ∼ N (κi(t), σ 2
κ),(8c)
where σu,σkcand σκare deviations for the random walks.
The size of the deviations are given by the analysts.
The system noise vector νtis assumed to be consists of
σk,σu,σkcand σκwithout time, space nor flow dependency.
2) Observation Equation: A linear observation equation
can be formulated based on the previous study [7].
The observation vector ytis defined to have the same
property with the state vector xtas
yt=. . . , ˆ
ki(t), . . . , ˆui(t), . . . , ˆ
kci(t),...,ˆκi(t), . . . (9)
where theˆmark indicates that the variable is directly derived
(i.e., observed) from the probe vehicle data. The number of
elements in vector ytis also 4M.
From the probe vehicle data, a density in a time–space
region can be derived as follows [7]:
ˆ
k(AT,X ) = m∈P(AT,X )tm(AT ,X )
m∈P(AT,X )|am(AT ,X )|,(10)
where, AT,X is a time–space region, ˆ
k(AT,X )is an estimated
generalized density in AT,X ,P(AT ,X )is a set of all the
probe vehicles in AT,X ,tm(AT ,X )is a total time spent by
vehicle min AT,X , and |am(AT ,X )|is an area of a time–
space region am(AT,X ), which is a time–space region between
vehicle mand its leading one in AT,X . This is the observed
state. The region am(AT,X )is the key information collected
by the PVSMEs. The definition of AT,X is
AT,X ={(x, t)|X≤x≤X+ ∆X, T ≤t≤T+ ∆T}
(11)
where ∆Tand ∆Xare predetermined resolutions for time and
space, respectively. Equation (11) means that region AT,X is a
rectangle in time–space plane with time length ∆Tand space
length ∆X(c.f., figure 3).
The FD parameters are derived from regression of
headway–spacing relation in the probe vehicle data. Values
of ˆui(t),ˆ
kci(t),ˆκi(t)are assumed to be constant in a pre-
determinded time–space region B(i.e., the observed FD pa-
rameters are constant in B). They can be determined as
ˆui(t) = u(B),∀(t, i)∈B(12a)
ˆ
kci(t) = w(B)κ(B)/(u(B) + w(B)),∀(t, i)∈B(12b)
ˆκi(t) = κ(B),∀(t, i)∈B,(12c)
where u(B),w(B)and κ(B)are the free flow speed, the wave
speed and the jam density, respectively, in time–space region
B. The values of u(B),w(B)and κ(B)are the solution of
the following optimization problem:
argmin
u(B),w(B),κ(B)
(m,τ)∈P(B)
D(hm,τ , sm,τ , u(B), w(B), κ(B))2,
s.t. u(B)≥0, w(B)≥0, κ(B)≥0,(13)
where hm,τ and sm,τ are stationary headway and spacing,
respectively, of vehicle mat time τ,Dis a function that returns
the minimum distance from a point (q, k) = (1/hm,τ ,1/sm,τ )
to a curve q=F(k, u(B), w(B), κ(B)), and Fis a function
representing a triangular FD. Therefore, problem (13) finds
FD parameter values that minimizes total distance between
observed stationary headway–spacing points and the FD curve.
Stationary means that the change rates of these variables in
small time duration ∆τ(e.g., (hm,τ −hm,τ −∆τ)/hm,τ ) are
small enough. This procedure is required because a theoretical
FD is defined under stationary condition and an empirical FD
can be clear triangular relation under stationary condition [20].
The relation between region AT,X for the observation and
cell iat timestep tin the system model are as follows. The
resolution of the observation and that of the system model can
be different; because that of the observation depends on the
analysts purposes, while that of the system model depends on
the timestep and free flow speed in the CTM. In this study, the
size of region AT,X is set to includes one or multiple cell(s)
and timestep(s) completely (i.e., without extending a cell and
a timestep over multiple regions for the observation) in order
to make the coordinate system concise and to avoid causing
numerical errors. Specifically, the value of ∆T,∆x,∆tand
liare determined as follows:
li=αˆui(t)∆t, ∀(t, i)∈B(14a)
∆T=β∆t, (14b)
∆X=γl, (14c)
where αis a coefficient larger than or equal to 1 (equal to
1 is desirable), and βand γare natural numbers. Figure 3
shows an example with α= 1,β= 3 and γ= 2. The analysts
can select appropriate values for ∆t,α,βand γaccording
to their interests. Following definitions are introduced in order
to represent the relation between region AT,X and cell iat
timestep t:
Ai(t) =AT,X ,(15)
if T≤t∆t < (t+ 1)∆t≤T+ ∆T
and X≤xi< xi+l≤X+ ∆X
where xiis a space coordinate of the upstream edge of cell i.
Therefore, following relation holds true:
ˆ
ki(t) = ˆ
k(Ai(t)) = ˆ
k(AT,X ).(16)
The variance of observation noise ωt, namely, Rt, is
determined as follows. The previous study [7] showed that
the precision of the estimator (10) in a region AT,X is
approximately in inverse proportion to the number of the probe
vehicles in AT,X , namely, |P(AT,X )|. Therefore, deviations
of traffic state observation noises are assumed to be inversely
proportional to |P(AT,X )|. As a result, element (a, b)of the
matrix Rtis defined as
Rt(a, b) =
ξ2
k
|P(Aa(t))|,if a=band a≤M
ξ2
u,if a=band M < a ≤2M
ξ2
kc,if a=band 2M < a ≤3M
ξ2
κ,if a=band 3M < a ≤4M
0,otherwise,
(17)
where ξkis deviation for observing traffic state, and ξu,ξkc
and ξκare deviations for observing respective FD parameters.
The size of the deviations are given by the analysts.
An element (i, i)of the observation matrix Htrepresents
whether the cell iis observed or not. Therefore, element (a, b)
of Htis defined as
Ht(a, b) = 1,if a=band a≤Mand |P(Aa(t))|>0
1,if a=band a > M
0,otherwise,
(18)
according to the presence of probe vehicles.
D. Summary of the proposed TSE Method
As results of the filtering steps, the filter ensemble’s mean
vector ¯
xt|tcan be obtained. The density elements (1st to M-
th elements) in the mean vectors ¯
xt|tare the final output
of the TSE procedure. This estimated density is declared as
k∗(AT,X )with resolution of ∆Tand ∆X.
The proposed TSE method estimates traffic density
k∗(AT,X )from the probe vehicle data. The probe vehicle data
consist of continuous position and spacing of probe vehicles,
namely, tm(AT,X )and am(AT,X )∀m∈P(AT ,X )∀T , X.
The framework of EnKF with the CTM is employed for
the estimation procedure. The functional form of the FD is
assumed to be triangular. On the other hand, the parame-
ters of the FD are not required to be assumed; they are
endogenously estimated from probe vehicle data. Note that the
proposed approach possibly employs FDs with other functional
forms; and use time and/or space varying FD parameters by
defining appropriate B. The parameters regarding to time–
space resolution (i.e., cell and timestep size for CTM, Afor
state observation, Bfor FD parameters observation) of the
estimation are given by the analysts according to their interest.
The size of the noise terms are required to be given. Note that
the other variables of a traffic state, namely, flow and speed,
can be easily derived from the estimated density and FD.
The method assumes that probe vehicles are representing
whole traffic without biases, so that the FD parameters and
traffic states can be estimated without biases. However, in
mixed traffic condition with traditional vehicles and automated
ones, probe vehicles may have biased driving behavior and
therefore the proposed TSE method will be biased.
The method treats traffic in a link—no merging/diverging
sections exist in the middle of the link. Traffic in a network
with multiple links can be treated by implementing a node con-
servation law and a merging/diverging model in the proposed
approach.
For large-scale real world application, the method may be
not very costly in terms of computation and data-handling.
EnKF requires less computation costs compared with other
techniques like the particle filter. Data required for the method
are position and spacing, which are calculated by on-vehicle
systems for their own purposes; therefore, not heavy data
transmission and storage are required.
III. VALIDATI ON
In this part, numerical characteristics of the proposed TSE
method are investigated through a simulation experiment.
A. Experiment Environment
An experiment environment is prepared as follows. To
generate traffic conditions investigated in a validation analysis,
Aimsun [21], microscopic traffic simulator based on a car-
following and lane-changing model, is employed. The param-
eters of the car-following model are shown in table I. The road
is an almost homogeneous freeway that has a bottleneck at the
end of the section. It has two lanes and 3 km length. Based
on the above setting, traffic condition for 1 hour with a queue
was generated.
TABLE I. PARA MET ER S OF TH E SI MUL ATIO N MOD EL
Parameter name Mean Deviation
Desired speed (km/h) 60 10.0
Max acceleration (m/s2) 3 0.2
Normal deceleration (m/s2) 4 0.5
Max deceleration (m/s2) 6 0.5
Min spacing (m) 1 0.3
Vehicle length (m) 4 0.5
The probe vehicle penetration rate Pwas selected from 5%,
1% and 0.5%. Two types of time–space resolution for obser-
vation were selected, namely, (∆T,∆X)∈ {(1 min, 300 m),
(10 min, 1000 m)}. The observation error is expected to be
smaller as the observation resolution increased as shown in
previous study [7]. In order to simplify the discussion, region
Bis set to be the entire time–space region—the FD parameters
are assumed to be constant throughout the simulation.
A timestep width in the system model, ∆t, was set to 0.111
min. The other variables on the resolution of the system model,
namely, α,β, and γ, were set to certain values such that the
cell length liis identical to 100 m; their exact values vary in
estimation iterations, depending on observed free flow speed
uand given observation resolution ∆Tand ∆X. With these
setting, the value of αis almost always slightly larger than
1; therefore, the CFL condition is expected to be satisfied.
Above mentioned parameters are set to satisfy the conditions
mentioned in part II-C2, eq (14).
The size of the noise terms were set ad hoc as follows:
σk= 0.1,σu= 0.5(m/s), σkc= 0.002 (veh/m), σκ= 0.01
(veh/m), ξk= 0.01 (veh/m), ξu= 5.0(m/s), ξkc= 0.1(veh/m)
and ξκ= 0.2(veh/m). The reason for such larger size of the
observation noises was that general traffic does not always
follow the triangular FD, especially at a microscopic scale.
Criteria for stationary headway–spacing determination is set
as less than 10% change rate during 5 s (= ∆τ). In general,
if this criteria is strict, precision of FD observation will be
increased. However, if the criteria is too strict compared with
the total amount of data, amount of data that passes the criteria
becomes few so that the precision can be low. The number of
ensembles Nwas set to 200. In general, larger Nvalue makes
accuracy of EnKF’s Monte Carlo simulation better at the cost
of computation.
B. Results
Figure 4 shows traffic density as time–space diagrams,
whose horizontal axes represent time, vertical axes represent
space, and color represents density. In figure 4, (a) represents
the ground truth value k, (b) represents observed value ˆ
k
with (P, ∆T , ∆X) = (5%,1min,300 m), and (c) represents
estimated value k∗with the identical resolution. According
to figure 4, the noises in the observation were reduced in
the filtering result, especially at free flow areas, such as
traffic before 10 min. On the other hand, difference between
observed and estimated states in a congested area was not so
remarkable—both were close to the ground truth states.
Figure 5 shows observed FD by probe vehicles (red line),
disaggregated headway–spacing relation of vehicles (gray dots)
and aggregated flow–density relation in time–space regions
(blue crosses), with (P, ∆T , ∆X) = (5%,1min,300 m). The
observed FD’s parameter values were ˆu= 55.4(km/h), ˆ
kc=
(a)
(b)
(c)
Fig. 4. Typical estimation results visualized as time–space diagrams: (a)
ground truth traffic state k, (b) observed traffic state ˆ
k, (c) estimated traffic
state k∗, RMSE(k∗)=21.7 (veh/km), and MAPE(k∗)=20.1%.
73.7(veh/km) and ˆκ= 388.5(veh/km). The disaggregated
relation is widely scattered due to the high fluctuation and
heterogeneity in microscopic vehicle behaviors. On the other
hand, observed FD is closed to the aggregated relation which
is clearly bivariate.
Table II summarizes the estimation performance. The
root mean square error (RMSE) and mean absolute per-
centage error (MAPE) were employed as precision in-
dices on estimated density. Their definitions were as fol-
Fig. 5. Observed FD, disaggregated headway–spacing relation and aggregated
flow–density relation.
lows: RMSE(k∗) = 1/S ∀(T,X )(k(AT,X )−k∗(AT ,X ))2,
MAPE(k∗) = 1/S ∀(T,X )|k(AT,X )−k∗(AT ,X )|/k(AT ,X ),
where Sis total size of the results (i.e., combination of
(T, X)). The simulation were performed for 50 times in order
to get the average performance. In addition, the percentage
of improvement (PoI) was employed for evaluate the effect
of the data assimilation. It was defined as PoI(p)=(p(ˆ
k)−
p(k∗))/p(ˆ
k), where pis a precision index, namely RMSE or
MAPE. The value of PoI represents improvement of estimated
density k∗compared to observed density ˆ
k; if the value is
positive, the data assimilation worked positively for precision.
According to table II, distinguishable trends can be found
for precision. The precision is increased as probe vehicle
penetration rate increases and time–space resolution lowers. In
addition, according to PoI value, the data assimilation almost
always reduced estimation error, especially on MAPE.
C. Discussion
The results showed that the estimation precision improved
due to the data assimilation in the most cases. According to
figure 5, FD parameters were precisely observed by PVSMEs;
therefore, the prediction steps by the traffic flow model are
expected to work well. Then, according to figure 4 and table
II, improvements of the estimation precision due to the filtering
steps were confirmed. Especially, they were remarkable at low
density regime; for example, free flow area shown in figure 4
and PoI(MAPE) in table II. This is a preferable feature for a
TSE method with the PVSMEs; because the precision of the
previous study’s method [7] was relatively low at low density
traffic. Contrary, the precision in high density (congested)
regime was not improved significantly. This may be due to
that the vehicular fluctuation are low in congested regime,
since vehicle movements are restricted by the jam. These
implied that estimation errors caused by high fluctuations in
microscopic vehicular traffic were reduced as intended.
IV. CONCLUSIONS
This study proposed TSE method whose inputs are position
and spacing data collected by PVSMEs (i.e., the advanced
probe vehicles), which are possible secondary utilization of
automated vehicles. The proposed method estimates traffic
states and FD parameters jointly and endogenously from probe
TABLE II. SUMMARY ON THE ESTIMATION PERFORMANCE
Scenario parameters Precision indices Percentage of improvement
P∆T(min) ∆X(m) RMSE (veh/km) MAPE PoI(RMSE) PoI(MAPE)
0.5% 1 300 65.0 68.5% 8.4% 16.2%
0.5% 10 1000 31.0 12.7% –4.3% 13.5%
1.0% 1 300 49.7 58.7% 12.6% 19.0%
1.0% 10 1000 11.6 7.5% 15.1% 21.2%
5.0% 1 300 20.3 23.0% 20.0% 29.6%
5.0% 10 1000 7.4 3.8% 5.8% 15.3%
vehicle data. In order to reduce noises, traffic states are
updated using a traffic flow model-based data assimilation.
As a consequence, the proposed method can be applied in
anytime and anywhere regardless traffic conditions, compared
with existing GPS-based TSE methods.
The results of the simulation experiment showed that the
data assimilation successfully mitigated the noises in the esti-
mation result caused by fluctuation in the microscopic vehicle
behavior. Therefore, the estimation precision in higher resolu-
tion and lower probe vehicle penetration rate was increased. It
means that the proposed method is robust against microscopic
fluctuations, compared with the previous PVSMEs studies [7],
[8], This might help precise traffic control measures.
Following future works are considerable to improve the
proposed TSE approach. The first is to expand the proposed
TSE approach for road network, as mentioned in part II-D. It
is valuable for probe vehicle-based traffic monitoring; because
it can interpolate traffic state in unobserved links where probe
vehicles have not traveled, by using information from adjacent
links. The second is development of methods that can correct
the bias caused by probe vehicles’ behavioral biases, as men-
tioned in part II-D, too. Possibly useful information to tackle
this problem is the PVSME data themselves, by which vehicle
behavior of other vehicles and probe vehicles can be estimated.
The third is elimination of remained exogenous factors from
the TSE method. Deviations of noise terms and a functional of
an FD form are given exogenously in this study. It also includes
endogenous identification method for time–space regions with
constant FD. If they are determined endogenously from probe
vehicle data, the applicability of the TSE method will be
significantly increased. Alternatively, sensitivity analyses on
the factors are required. The forth is to employ traffic flow
models based on Lagrangian coordinates as a system model
of the proposed TSE approach. Lagrangian traffic flow models
can utilize information of trajectories of probe vehicles [5],
[6], [8] so that estimation performance can be increased. The
last is validation based on the real world dataset [7].
REF ER EN CE S
[1] Y. Wang and M. Papageorgiou, “Real-time freeway traffic state esti-
mation based on extended Kalman filter: a general approach,” Trans-
portation Research Part B: Methodological, vol. 39, no. 2, pp. 141–167,
2005.
[2] A. Hegyi, D. Girimonte, R. Babuska, and B. De Schutter, “A comparison
of filter configurations for freeway traffic state estimation,” in Intelligent
Transportation Systems Conference, IEEE, 2006, pp. 1029–1034.
[3] Y. Wang, M. Papageorgiou, A. Messmer, P. Coppola, A. Tzimitsi, and
A. Nuzzolo, “An adaptive freeway traffic state estimator,” Automatica,
vol. 45, no. 1, pp. 10–24, 2009.
[4] C. Nanthawichit, T. Nakatsuji, and H. Suzuki, “Application of probe-
vehicle data for real-time traffic-state estimation and short-term travel-
time prediction on a freeway,” Transportation Research Record: Journal
of the Transportation Research Board, vol. 1855, no. 1, pp. 49–59, 2003.
[5] Y. Yuan, J. W. C. van Lint, R. E. Wilson, F. van Wageningen-Kessels,
and S. P. Hoogendoorn, “Real-time Lagrangian traffic state estimator
for freeways,” Intelligent Transportation Systems, IEEE Transactions
on, vol. 13, no. 1, pp. 59–70, 2012.
[6] K. Wada, T. Ohata, K. Kobayashi, and M. Kuwahara, “Traffic measure-
ments on signalized arterials from vehicle trajectories,” Interdisciplinary
Information Sciences, vol. 21, no. 1, pp. 77–85, 2015.
[7] T. Seo, T. Kusakabe, and Y. Asakura, “Estimation of flow and density
using probe vehicles with spacing measurement equipment,” Trans-
portation Research Part C: Emerging Technologies, vol. 53, pp. 134–
150, 2015.
[8] T. Seo and T. Kusakabe, “Probe vehicle-based traffic state estima-
tion method with spacing information and conservation law,” Trans-
portation Research Part C: Emerging Technologies, 2015, in press,
doi:10.1016/j.trc.2015.05.019.
[9] C. Diakaki, M. Papageorgiou, I. Papamichail, and I. Nikolos, “Overview
and analysis of vehicle automation and communication systems from
a motorway traffic management perspective,” Transportation Research
Part A: Policy and Practice, vol. 75, pp. 147–165, 2015.
[10] S. Sivaraman and M. M. Trivedi, “Looking at vehicles on the road:
A survey of vision-based vehicle detection, tracking, and behavior
analysis,” Intelligent Transportation Systems, IEEE Transactions on,
vol. 14, no. 4, pp. 1773–1795, 2013.
[11] S. Messelodi, C. M. Modena, M. Zanin, F. G. D. Natale, F. Granelli,
E. Betterle, and A. Guarise, “Intelligent extended floating car data
collection,” Expert Systems with Applications, vol. 36, no. 3, Part 1,
pp. 4213–4227, 2009.
[12] J. J. V. Diaz, D. F. Llorca, A. B. R. Gonzalez, R. Q. Minguez, A. L.
Llamazares, and M. A. Sotelo, “Extended floating car data system:
Experimental results and application for a hybrid route level of service,”
Intelligent Transportation Systems, IEEE Transactions on, vol. 13, no. 1,
pp. 25–35, 2012.
[13] K. Yokoi, Y. Suzuki, T. Sato, T. Abe, H. Toda, and N. Ozaki, “A camera-
based probe car system for traffic condition estimation,” in Proceedings
of 20th ITS World Congress, 2013.
[14] B. Coifman and H. Hall, “An overview of the on-going OSU instru-
mented probe vehicle research,” The Ohio State University, Tech. Rep.,
2014.
[15] M. J. Lighthill and G. B. Whitham, “On kinematic waves. II. a theory
of traffic flow on long crowded roads,” Proceedings of the Royal Society
of London. Series A. Mathematical and Physical Sciences, vol. 229, no.
1178, pp. 317–345, 1955.
[16] P. I. Richards, “Shock waves on the highway,” Operations Research,
vol. 4, no. 1, pp. 42–51, 1956.
[17] C. F. Daganzo, “The cell transmission model: A dynamic representation
of highway traffic consistent with the hydrodynamic theory,” Trans-
portation Research Part B: Methodological, vol. 28, no. 4, pp. 269–287,
1994.
[18] G. Evensen, “Sequential data assimilation with a nonlinear quasi-
geostrophic model using Monte Carlo methods to forecast error statis-
tics,” Journal of Geophysical Research: Oceans, vol. 99, no. C5, pp.
10 143–10 162, 1994.
[19] T. Higuchi, Ed., Introduction to Data Assimilation. Asakura Publishing,
2011, (in Japanese).
[20] M. J. Cassidy, “Bivariate relations in nearly stationary highway traffic,”
Transportation Research Part B: Methodological, vol. 32, no. 1, pp.
49–59, 1998.
[21] TSS-Transport Simulation Systems, “Aimsun,” www.aimsun.com.
