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Traffic State Estimation with the Advanced Probe Vehicles Using Data Assimilation



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 information 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.
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.
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)
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
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
- desired data resolution T,X
Probe vehicle data:
- spacing am(AT,X )
- position tm(AT,X )
- functional form F
- constant parameter
region: B
Fundamental diagram:
- parameters
Traffic state:
- density ˆ
Observation vector:
Data assimilation:
- Ensemble Kalman filter
-ft: Cell transmission model
Deviations of noise terms:
Conservation law,
1st order traffic
flow model
Filter ensemble of state vector:
- density and FD parameters
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.
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
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
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-
Step 1 Generate an ensemble of the initial states {xn
Let t1.
Step 2 Prediction step:
Step 2.1 Generate an ensemble of the system noises
Step 2.2 Calculate xn
each n.
Step 3 Filtering step:
Step 3.1 Generate an ensemble of the observation
noises {ωn
Step 3.2 Obtain the filter ensemble {xn
n=1 for
each n, by calculating eq (3) based on
yt, Ht, Rt,{xn
Step 4 Increment the timestep, tt+ 1. Go back to Step 2
until t=tmax.
i(t), kc
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:
In this study, following mean vector of the filter ensemble
is defined as an “estimation result” at timestep t:
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),
1(t), kc
2(t), . . . , kc
i(t), . . . kc
κ1(t), κ2(t), . . . , κi(t), . . . κM(t)(6)
where, ki(t)is a density, ui(t)is a free flow speed, kc
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
region AT,X for the observation of a traffic state
spacing: am(AT,X )
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
i(t) = εk
i(t) min li1ki1(t),
i(t)(liκi(t)liki(t)) ,(7a)
ki(t+ 1) = ki(t) + Qin
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
(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
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
i(t+ 1) N (kc
i(t), σ2
κi(t+ 1) N (κi(t), σ 2
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]:
k(AT,X ) = mP(AT,X )tm(AT ,X )
mP(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)|XxX+ ∆X, T tT+ ∆T}
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)
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:
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=βt, (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 Ttt < (t+ 1)∆tT+ ∆T
and Xxi< xi+lX+ ∆X
where xiis a space coordinate of the upstream edge of cell i.
Therefore, following relation holds true:
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) =
|P(Aa(t))|,if a=band aM
u,if a=band M < a 2M
kc,if a=band 2M < a 3M
κ,if a=band 3M < a 4M
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 aMand |P(Aa(t))|>0
1,if a=band a > M
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 )mP(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
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.
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.
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
with (P, T , X) = (5%,1min,300 m), and (c) represents
estimated value kwith 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
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), where pis a precision index, namely RMSE or
MAPE. The value of PoI represents improvement of estimated
density kcompared 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.
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
Scenario parameters Precision indices Percentage of improvement
PT(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].
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... Sequential training aims to first calibrate parameters of the PICG (i.e., parameter calibration) and then encode the known physics into the PUNN for training. Parameter calibration has been extensively studied in TSE using nonlinear programming [19], genetic algorithm [20], least-squares fitting [21][22][23]71], and kernel smoothing [24]. The physicsbased parameters include ρ max , u max , and other nominal parameters. ...
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For its robust predictive power (compared to pure physics-based models) and sample-efficient training (compared to pure deep learning models), physics-informed deep learning (PIDL), a paradigm hybridizing physics-based models and deep neural networks (DNNs), has been booming in science and engineering fields. One key challenge of applying PIDL to various domains and problems lies in the design of a computational graph that integrates physics and DNNs. In other words, how the physics is encoded into DNNs and how the physics and data components are represented. In this paper, we offer an overview of a variety of architecture designs of PIDL computational graphs and how these structures are customized to traffic state estimation (TSE), a central problem in transportation engineering. When observation data, problem type, and goal vary, we demonstrate potential architectures of PIDL computational graphs and compare these variants using the same real-world dataset.
... Besides, streaming-data-driven approaches are regarded as more robust against uncertainties while requiring a large amount of streaming data to perform the prediction. Some research contributes to the streaming-datadriven models, such as [31] and [32]. One of the advantages of data-driven models is that they require less prior info on traffic dynamics and can be more accurate than model-based methods. ...
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Accurately monitoring road traffic state and speed is crucial for various applications, including travel time prediction, traffic control, and traffic safety. However, the lack of sensors often results in incomplete traffic state data, making it challenging to obtain reliable information for decision-making. This paper proposes a novel method for imputing traffic state data using Gaussian processes (GP) to address this issue. We propose a kernel rotation re-parametrization scheme that transforms a standard isotropic GP kernel into an anisotropic kernel, which can better model the propagation of traffic waves in traffic flow data. This method can be applied to impute traffic state data from fixed sensors or probe vehicles. Moreover, the rotated GP method provides statistical uncertainty quantification for the imputed traffic state, making it more reliable. We also extend our approach to a multi-output GP, which allows for simultaneously estimating the traffic state for multiple lanes. We evaluate our method using real-world traffic data from the Next Generation simulation (NGSIM) and HighD programs. Considering current and future mixed traffic of connected vehicles (CVs) and human-driven vehicles (HVs), we experiment with the traffic state estimation scheme from 5% to 50% available trajectories, mimicking different CV penetration rates in a mixed traffic environment. Results show that our method outperforms state-of-the-art methods in terms of estimation accuracy, efficiency, and robustness.
... 1) Sequential training Sequential training aims to first calibrate parameters of the PICG (i.e., parameter calibration) and then encode the known physics into the PUNN for training. Parameter calibration has been extensively studied in TSE using non-linear programming [58], genetic algorithm [59], least-squares fitting [60]- [63], and kernel smoothing [64]. The physics based parameters include ρ max , u max and other nominal parameters. ...
Full-text available
For its robust predictive power (compared to pure physics-based models) and sample-efficient training (compared to pure deep learning models), physics-informed deep learning (PIDL), a paradigm hybridizing physics-based models and deep neural networks (DNN), has been booming in science and engineering fields. One key challenge of applying PIDL to various domains and problems lies in the design of a computational graph that integrates physics and DNNs. In other words, how physics are encoded into DNNs and how the physics and data components are represented. In this paper, we provide a variety of architecture designs of PIDL computational graphs and how these structures are customized to traffic state estimation (TSE), a central problem in transportation engineering. When observation data, problem type, and goal vary, we demonstrate potential architectures of PIDL computational graphs and compare these variants using the same real-world dataset.
Traffic state estimation is a challenging task due to the collection of sparse and noisy measurements from specific points of the traffic network. The emergence of Connected and Automated Vehicles (CAVs) provides new capabilities for traffic state estimation using extended floating car data such as position, speed and spacing information. In this work we propose a Bayesian Traffic State Estimation (BTSE) methodology for estimating the traffic density based on extended floating car data. BTSE utilizes the Bayesian paradigm to express any prior information to derive probability distributions of the traffic density of different road segments of the traffic network. Two variations of the BTSE methodology are developed to handle the offline and online estimation problem. The BTSE methodology is evaluated both using realistic SUMO micro-simulations for M25 Highway, London, U.K., and a real-life vehicle-trajectory dataset from German highways, extracted from videos recorded by drones. The efficiency and accuracy of the BTSE methodology is compared to an existing methodology in the literature. We present results for the estimation performance of the methods showing that the Bayesian methodology consistently results in lower mean absolute percentage error than the compared literature method. The BTSE methodology yields high-quality estimation results even for a low penetration rate of CAVs (e.g. 5%).
Real-time and accurate traffic state identification can provide reference for urban traffic control and guidance. Due to the randomness and complexity of traffic flow, it is difficult to identify the traffic state accurately. The existing researches mainly adopted traffic state feature vectors and machine learning to identify the traffic state. Few studies attempted to use the feature map and deep learning for traffic state recognition. Therefore, this paper proposes a traffic state recognition model based on the traffic state feature map and deep learning. The feature map is a chromatogram form of digital traffic state feature vector. And the deep learning has strong predictive performance in image identification. In the model, the road traffic state feature vector is extracted from the vehicle trajectory data, and the Gram Angle Field (GAF) is adopted to transform the feature vector into feature map. Then, the deep learning algorithm is utilized in traffic state offline classification and online recognition. In the offline training phase, the historical vehicle trajectory data and the DeepCluster algorithm are used to establish the mapping relationship between the feature maps and the traffic states. In the online identification stage, the real-time vehicle trajectory data and CoATNet algorithm are utilized to identify the current traffic state. In experiments, the proposed model is compared with the Convolutional Neural Network (CNN), Support Vector Machine (SVM), and K-Nearest Neighbor (KNN) models using the open vehicle trajectory data of Shanghai North Cross Channel. The results showed that the proposed model achieved the best performance with the accuracy of 92.06% and could provide support for road traffic control and guidance.
Observability and controllability are two critical requirements for a partially observable transportation system. This paper proposes a stepwise signal optimization framework with connected vehicle (CV) data as input to solve both challenges. First, a Bayesian deduction method based on low‐penetration CV data is established to estimate the traffic volume. Thereafter, an offline signal optimization model is constructed to simultaneously optimize the flexible lane settings and signal timings, which are set as the prior information for the third step. In the third step, an online deep recurrent Q‐learning (DRQN) signal optimization model dynamically adjusts signal settings based on real‐time traffic information. Numerical experiments demonstrate that the model outperforms the actuated control, the online DQRN model without offline filter, and the back‐pressure model by 9%–66% and 7%–29% in two networks. This study innovatively combines traffic state estimation and traffic signal control as an integrated process. It contributes to an improved understanding of traffic control in a CV environment and lays a solid foundation for future traffic control strategies.
Real-time traffic monitoring represents a key component for transportation management. The increasing penetration rate of connected vehicles with positioning devices encourages the utilization of trajectory data for real-time traffic monitoring. The use of commercial fleet trajectory data could be seen as the first step towards mobile sensing networks. The main objective of this research is to estimate space occupancy of a single road segment with partially observed trajectories (commercial fleet trajectories in our case). We first formulate the trajectory-based traffic estimation as a video computing problem. Then, we reconstruct trajectory series into video-like data by performing spatial discretization. Following this, video input is embedded using a tubelet embedding strategy. Finally, a Revised Video Vision Transformer (RViViT) is proposed to estimate traffic state from video embeddings. The proposed RViViT is tested on a public dataset of naturalistic vehicle trajectories collected from German highways around Cologne during 2017 and 2018. The results witness the effectiveness of the proposed method in traffic estimation with partially observed trajectories.
This work proposes an estimator for the vehicle density of every road section of a large urban traffic network using a data-based approach. We assume that a limited number of flow and turning ratio sensors can be installed and that aggregate Floating Car Data are available. To validate the estimator, real data obtained in the city of Grenoble in France is used. The results show that the trajectories of the real and estimated flows and density are very close, showing the good performance of the proposed methods.
The management of vehicle traffic congestion in public road networks is becoming a key factor for economic growth. Indeed, road traffic congestion hampers economic growth: it profoundly impacts travel time, causing delays that result in a significant financial loss of billions of dollars spent on extra hours of travel and fuel consumption. Traffic congestion, which can result in traffic accidents, lost time, hampered mobility, traffic delays, and even severe safety problems, is often difficult to avoid. Accordingly, understanding and modeling traffic dynamics is crucial to enhance traffic management, transportation safety, and cost savings. Firstly, in this chapter, we present an overview of traffic modeling approaches, including microscopic, mesoscopic, and macroscopic modeling. Next, we review the macroscopic traffic modeling with its last enhanced version, the piecewise switched linear traffic model (PWSL). Then we provide more detail about PWSL parameters calibration with an illustrative example. Finally, we shed some light on potential future research lines.
This paper addresses real-time joint traffic state and model parameter estimation on freeways using data from fixed sensors and connected vehicles. It investigates how the combined usage of both types of sensing data improves the performance of traffic state estimation (TSE) and what role the online model parameter estimation (OMPE) plays therein. The paper first presents a state-of-the-art overview for freeway TSE with mixed sensing, focusing on a few critical issues such as filtering methods, Eulerian and Lagrangian formulation for traffic flow modeling/sensing/estimation, OMPE, and fusion of disparate sensing data, to determine the strengths and weaknesses of various technical paths, and figure out a viable roadmap for future studies. Three representative approaches to the design of freeway traffic state estimators using mixed sensing data are then investigated, which are based on a first-order, a second-order traffic flow model, and a speed-uniformity assumption, respectively. The paper intends to check if the gradual richness of mobile sensing data (in the era of connected vehicles) would compensate the deficiency of first-order models (as compared to second-order models) for TSE; if OMPE would still be essential for TSE in the mixed sensing case compared to the fixed sensing case; if the increasing usage of mobile sensing data would reduce the necessity of OMPE for TSE? The designed traffic state estimators have been evaluated thoroughly using NGSIM data, with the above questions answered.
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This paper proposes a method of estimating a traffic state based on probe vehicle data that contain spacing and position of probe vehicles. The probe vehicles were assumed to observe spacing by utilizing an advanced driver assistance system, that has been implemented in practice and is expected to spread in the near future. The proposed method relies on the conservation law of the traffic flow but is independent of a fundamental diagram. The conservation law is utilized for reasonable aggregation of the spacing data to acquire the traffic state, i.e., a flow, density and speed. Its independence from a fundamental diagram means that the proposed method does not require predetermined nor exogenous assumptions with regard to the traffic flow model parameters. The proposed method was validated through a simulation experiment under ideal conditions and a field experiment conducted under actual traffic conditions; and empirical characteristics of the proposed method were investigated.
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Probe data provide rich information of vehicle trajectories which include driving modes (e.g., acceleration/deceleration) and reflect traffic conditions. However, most of the applications only use probe data to measure travel times. This paper presents two alternative applications of vehicle trajectories on signalized arterials: one is a traffic signal timing estimation and the other is a traffic volume estimation. Both applications are based on a simple methodology that combines vehicle trajectories and traffic engineering concept shock wave. By testing the methodology using real world data, we demonstrate that probe data have more potential for traffic measurements.
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During the last decade, there has been an enormous interdisciplinary effort by the automobile industry and numerous research institutions worldwide towards the development, testing and employment of a variety of Vehicle Automation and Communication Systems (VACS) with the main aims to improve road safety and driver convenience. Some VACS, however, have a direct impact on road efficiency as well and could therefore be exploited to relieve road networks from the significant congestion problems and their negative consequences for travel times, safety, fuel consumption, the environment and the quality of life in general. In other words, some of the available VACS could also be used as novel or innovative sensors, actuators and tools towards a new era of traffic management. This paper provides an overview of proposed and available VACS and discusses their perspectives from the motorway traffic management point of view. Classifications of the different systems in this respect are also provided, while SWOT (Strengths–Weaknesses–Opportunities–Threats) analyses are used to identify specific exploitation ways. Current trends and future perspectives of VACS within a motorway traffic management context are finally summarised. An extended version of the overview and analysis is provided in the "Technical Report: Overview and Analysis of Vehicle Automation and Communication Systems from a Motorway Traffic Management Perspective".
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Probe vehicles provide some of the most useful data for road traffic monitoring because they can acquire wide-ranging and spatiotemporally detailed information at a relatively low cost compared with traditional fixed-point observation. However, current GPS-equipped probe vehicles cannot directly provide us volume-related variables such as flow and density. In this paper, we propose a new probe vehicle-based estimation method for obtaining volume-related variables by assuming that a probe vehicle can measure the spacing to its leading one. This assumption can be realized by utilizing key technologies in advanced driver assistance systems that are expected to spread in the near future. We developed a method of estimating the flow, density, and speed from the probe vehicle data without exogenous assumptions on traffic flow characteristics, such as a fundamental diagram. In order to quantify the characteristics of the method, we performed a field experiment at a real-world urban expressway by employing prototypes of the probe vehicles with spacing measurement equipment. The result showed that the proposed method could accurately estimate the 5 min and hourly traffic volumes with probe vehicle penetration rate of 3.5% and 0.2%, respectively.
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Traffic information from probe vehicles has great potential for improving the estimation accuracy of traffic situations, especially where no traffic detector is installed. A method for dealing with probe data along with conventional detector data to estimate traffic states is proposed in this paper. The probe data was integrated into the observation equation of the Kalman filter, in which state equations are represented by a macroscopic traffic flow model. Estimated states were updated with information from both stationary detectors and probe vehicles. The method was tested under several traffic conditions based on hypothetical data, giving considerably improved estimation results compared to those estimated without probe data. Finally, the application of the proposed method was extended to the estimation and short-term prediction of travel time. Travel times were obtained indirectly through the conversion of speeds estimated/predicted by the proposed method. Experimental results show that the performance of travel time estimation/prediction is comparable to those of some existing methods.
Conference Paper
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We present a comparison for several filter configurations for freeway traffic state estimation. Since the environmental conditions on a freeway may change over time (e.g., changing weather conditions), parameter estimation is also considered. We compare the performance of the extended Kalman filter and the unscented Kalman filter for state estimation, parameter estimation, joint estimation and dual estimation. Furthermore, the performance is evaluated for different detector configurations. The main conclusions from the simulations are that (1) the performance of the extended Kalman filter and the unscented Kalman filter is comparable, (2) joint filtering performs significantly better than dual filtering, and (3) a larger number of detectors results in better state estimation, but has no significant influence on the parameter estimation error
In this paper, we propose a camera-based probe car system for traffic condition estimation. The probe car, which mounts a stereo camera and a GPS sensor, measures the traffic density using a car distance map from the stereo camera and receives the velocity and position information of the car from the GPS sensor. A traffic control center creates a traffic condition map according to the probe car data and uses it for traffic control, car navigation, and bus/taxi scheduling. We implement the camera-based probe car system as an in-vehicle device using an image recognition processor and show its enough performance and speed for practical use.
This paper provides a review of the literature in on-road vision-based vehicle detection, tracking, and behavior understanding. Over the past decade, vision-based surround perception has progressed from its infancy into maturity. We provide a survey of recent works in the literature, placing vision-based vehicle detection in the context of sensor-based on-road surround analysis. We detail advances in vehicle detection, discussing monocular, stereo vision, and active sensor-vision fusion for on-road vehicle detection. We discuss vision-based vehicle tracking in the monocular and stereo-vision domains, analyzing filtering, estimation, and dynamical models. We discuss the nascent branch of intelligent vehicles research concerned with utilizing spatiotemporal measurements, trajectories, and various features to characterize on-road behavior. We provide a discussion on the state of the art, detail common performance metrics and benchmarks, and provide perspective on future research directions in the field.
A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well-known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble.