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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].

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