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Calibration of fundamental diagram using trajectories of probe

vehicles: Basic formulation and heuristic algorithm∗

Toru Seo†‡ Takahiko Kusakabe§Yasuo Asakura‡

August 9, 2016

Abstract

A fundamental diagram (FD), also known as ﬂow–density relation, is one of the most important

principles in traﬃc ﬂow theory. It would be valuable if an FD could be calibrated by GPS-equipped probe

vehicles; since they can continuously collect data from wide spatiotemporal area, compared to traditional

ﬁxed sensors. This paper proposes methods for calibrating an FD from trajectories of sampled vehicles.

We formulate a method that identiﬁes values of a free-ﬂow speed and a critical density of a triangular FD,

while it relies on exogenous assumptions on FD’s functional form and a value of its jam density. Then,

a heuristic algorithm for FD calibration in actual traﬃc environment is developed based on the proposed

method. It was validated using traﬃc data generated by microscopic traﬃc simulator. The results suggested

that the proposed methods can calibrate an FD precisely and robustly. It implies that FDs in road sections

on which congestion happens frequently can be calibrated using probe vehicles, if probe vehicle data were

collected for a long period. Therefore, the proposed methods would contribute to signiﬁcant improvement

of applicability of probe vehicle-based traﬃc management methods.

Keywords: traﬃc ﬂow theory, fundamental diagram, GPS-equipped probe vehicle, trajectory

1 Introduction

Fundamental diagram (FD), also known as ﬂow–density relation, is literally one of the most fundamental

concepts in the traﬃc ﬂow theory. An FD describes relation between ﬂow and density1in stationary traﬃc.

Stationary traﬃc is deﬁned as traﬃc in which all the vehicles have the same constant speed and spacing

(Daganzo, 1997). In theory, an FD itself contains useful information on traﬃc features, such as value of

free-ﬂow speed and ﬂow capacity, and distinction between free-ﬂow and congested states. Empirical studies

also showed that clear relation can be found between ﬂow and density in actual traﬃc at near-stationary states

(Cassidy, 1998). In addition, macroscopic traﬃc ﬂow dynamics can be modeled by combining an FD and other

principles: the most known example is the Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham,

1955; Richards, 1956). Moreover, FDs can describe microscopic vehicle behavior to some extent (Newell,

2002). Therefore, FDs are applied to various academic and practical purposes in traﬃc and transportation

engineering, such as macroscopic traﬃc simulation (e.g., Daganzo, 1994), traﬃc control (e.g., Papageorgiou

et al., 2003), and traﬃc state estimation (e.g., Deng et al., 2013).

∗Manuscript presented at International Symposium of Transport Simulation (ISTS) and the International Workshop on Traﬃc Data

Collection and its Standardisation (IWTDCS) 2016; to appear in Transportation Research Procedia

†corresponding author. Email: t.seo@plan.cv.titech.ac.jp, Tel.: +81 3 5734 2575.

‡Transport Studies Unit, Tokyo Institute of Technology, 2-12-1-M1-20, O-okayama, Meguro, Tokyo 152-8552, Japan

§Center for Spatial Information Science, the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan

1Technically, relation between two variables among ﬂow or headway, density or spacing, and speed or pace (Laval and Leclercq,

2013).

1

To calibrate parameters of an FD in actual traﬃc, one has to collect data from traﬃc, to assume its FD’s

functional form, and then to calibrate the FD’s parameters by ﬁtting the FD to the data. Those data are

commonly collected using ﬁxed sensors (e.g., cameras, detectors) from the era of Greenshields (1935); for

example, see Chiabaut et al. (2009), Dervisoglu et al. (2009), Coifman (2014), and references therein. This is

a straightforward way because usual ﬁxed sensors can measure traﬃc count and occupancy, which are closely

related to ﬂow and density, respectively, at their location. The limitation of the ﬁxed sensor-based calibration

is obvious: it cannot calibrate FDs where sensors are not installed. Therefore, FDs on roads without sensors,

such as most of arterial roads, are diﬃcult to be known. Identiﬁcation of bottlenecks’ exact locations and

characteristics is also diﬃcult, even on freeways with some sensors.

Probe vehicles received high attention in these days.2The advantage of probe vehicles is their signiﬁcantly

wider data collection range (in spatiotemporal domain) compared with ﬁxed sensors (Herrera et al., 2010).

For example, one of the most typical and useful utilization of the advantage is traﬃc state estimation, which

estimates ﬂow, density, and speed of traﬃc using partial measurement data (c.f., Deng et al., 2013, and

references within). However, existing studies on such application assume their FDs exogenously, in order to

relate ﬂow and density to speed measured by probe vehicles. This can be signiﬁcant limitation for application of

probe vehicles, since FD calibration is not a trivial task. Especially, FD calibration using probe vehicle data has

been not well studied. Appendix in Herrera et al. (2010) mentioned a manual inference method for FD using

probe vehicle data under special conditions. However, it lacks rigorous formulation and is not computable.

Considering these days high availability of probe vehicle data, systematic and computational approaches for

FD calibration would be desirable.3

The aim of this paper is to propose a method of calibrating FDs using probe vehicle data. In order to enable

the calibration, we allow the method to rely minimum exogenous assumptions, such as FD’s functional form

and values of some of the FD’s parameters (e.g., jam density which can be inferred from external knowledge).

In addition, to show empirical validity of the proposed method, a computable method for FD calibration is

developed and validated using microscopic traﬃc simulation data.

The rest of this paper is organized follows. Section 2 formulates a method of identifying FD parameters

using sampled vehicle trajectories under idealized conditions. Section 3 describes a heuristic algorithm for

actual traﬃc environment based on the proposed method. Section 4 validates the proposed heuristic algorithm

using noisy traﬃc data generated by microscopic traﬃc simulation. Section 5 concludes this paper.

2 Formulation

This section describes a method of calibrating FD using sampled vehicle trajectories under simpliﬁed

conditions.

Concept of the proposed method is as follows. The subjects of estimation are values of free-ﬂow speed

and critical density of a triangular FD. The jam density is assumed exogenously, as it can be easily inferred by

external knowledge such as average vehicle body length. Suppose that two probe vehicles are driving the same

road with homogeneous FD; and c−1vehicles exist between the probe vehicles. If traﬃc in a time–space

region between two probe vehicles is stationary (c.f., stationary traﬃc is deﬁned as traﬃc in which all the

vehicles have the same constant speed and spacing (Daganzo, 1997)), a traﬃc state (i.e., ﬂow, density, and

speed) in the region follows the FD by the deﬁnition. If the region is stationary and the distance traveled by the

probe vehicles are the same, a traﬃc state of probe vehicles only in the region (i.e., probe-traﬃc-state) follows

2In this paper, unless otherwise speciﬁed, the term “probe vehicle” refers to be the Global Positioning System (GPS)-equipped

probe vehicle, which continuously measure and report its position and time.

3If probe vehicles can measure spacing to their leading vehicles using advanced technologies (c.f., Huber et al., 1999; Seo et al.,

2015a), FDs can be directly derived from such probe vehicle data in detail, for example, continuously (Kotani and Iwasaki, 1999),

individually (Duret et al., 2008), stochastically (Jabari et al., 2014), and jointly with traﬃc state (Seo et al., 2015b). However, these

technologies are not commonly available for large-scale data collection at this moment. Therefore, application of such advanced

technologies is out of scope of this study.

2

a curve which is similar to the FD with 1/c scale (i.e., probe-FD). If the probe vehicles trajectories satisﬁes

certain conditions (which are likely to be satisﬁed in many situations), the probe-FD can be identiﬁed. Finally,

since the probe-FD is similar to the FD with 1/c scale and the value of the FD’s jam density is already known,

the free-ﬂow speed and the critical density can be identiﬁed.

2.1 Assumptions and preconditions

Following two factors are assumed:

(a) a functional form of an FD is triangular, and

(b) value of jam density.

Note that parameters of a triangular FD are three: free-ﬂow speed, critical density, and jam density. Therefore,

parameters to be calibrated are free-ﬂow speed and critical density.

Input data for the proposed method are:

•Multiple vehicles trajectories in a time–space region whose FD is to be calibrated.

These data are considered to be continuously collected by probe vehicles without error. Hereafter, the

time–space region whose FD is to be calibrated is referred as target region.

The required conditions for calibrating FD’s parameter are as follows:

(i) The FD is constant with the assumed functional form and jam density in the target region.

(ii) A ﬁrst-in ﬁrst-out (FIFO) principle and a conservation law are satisﬁed in the target region.

(iii) Two vehicles’ trajectories traveled through following three time–space regions in the target region:

– a stationary region4whose density is less than the critical density,

– a stationary region whose density is greater than or equal to the critical density, and

– a stationary region whose density is greater than the critical density and is not equal to those

of the above regions.

The conditions (i) and (ii) are assumptions on traﬃc ﬂow characteristics. They means the traﬃc is described

by the LWR model. The condition (iii) is an assumption on appearance of probe vehicles. Practical meaning of

these assumptions and conditions are discussed later (see section 2.3.2).

The output estimated by the proposed method are:

•values of free-ﬂow speed and critical density, and

•number of vehicles exist between the two vehicles.

The objective of the proposed method is to obtain the former values.

2.2 Calibration method

The calibration method can be described by following general formula:

ﬁnd u, kc,{ci,j }∀i,j ,(1a)

s.t. qi(a) = Q(ki(a), u, kc/ci,j , κ/ci,j ),(1b)

∀a∈A0(i, j, A),(1c)

∀i, j ∈P(A),(1d)

where

4In this paper, the term “stationary region” is deﬁned as a time–space region in which traﬃc is stationary.

3

ufree-ﬂow speed,

kccritical density,

κjam density,

Athe target region which satisﬁes conditions (i) and (ii),

Q(k, u, kc, κ)FD function which represents ﬂow under density k, free-ﬂow speed u, critical

density kc, and jam density κ,

P(A)set of all the probe vehicles in region A,

ci,j total number of vehicles exist between probe vehicles iand j,

qi(a), ki(a), vi(a)ﬂow, density, and speed with respect to probe vehicle ionly in region a.

Hereafter, they are refereed to be probe-traﬃc-state, Their deﬁnition is provided

by eqs (4)–(6),

Si(a)probe-traﬃc-state vector, namely, (qi(a), ki(a))>,

A0(i, j, A)set of time–space regions which satisﬁes following conditions:

– it is in region A,

– its edges are trajectories of probe vehicles iand jin region A,

– its traﬃc is stationary

Eqs (1a) and (1b) mean that values of FD parameters are determined such that traﬃc state estimated from probe

vehicle data follows the FD, considering conditions (i) and (ii). Eqs (1c) and (1d) mean that probe vehicle data

which satisfy condition (iii) are utilized for calibration. Following traﬃc state is deﬁned:

q(a), k(a)ﬂow and density with respect to all the vehicle in region a

S(a)traﬃc state vector, namely, (q(a), k(a))>

The traﬃc state (considering all the vehicles) and probe-traﬃc-state (which is traﬃc state considering a probe

vehicle only) are deﬁned based on the Edie (1963)’s generalized deﬁnition as follows:

q(a) = Pn∈N(a)dn(a)

|a|,(2)

k(a) = Pn∈N(a)tn(a)

|a|,(3)

qi(a) = di(a)

|a|,(4)

ki(a) = ti(a)

|a|,(5)

vi(a) = di(a)

ti(a),(6)

where

N(a)set of all the vehicle in region a,

dn(a)distance traveled by vehicle nin region a,

tn(a)time spent by vehicle nin region a,

|a|area of region a.

Following explains the logic of the FD calibration. For the sake of brevity, it explains a special case of the

method where an element in A0(i, j, A)is deﬁned as a time–space region between trajectories of probe vehicles

iand jin space [x, x + ∆x), where ∆xis arbitrary distance. Such element is denoted as a(x)

i,j . Under this

speciﬁcation, the calibration method can be illustrated as Figs. 1 and 2. Fig 1 is a time–space diagram showing

free-ﬂow traﬃc and congested traﬃc divided by a shockwave. Fig 2 is a ﬂow–density plane corresponding to

4

Fig 1. In Fig 2, a dot indicates traﬃc state or probe-traﬃc-state; a solid line indicates transition of traﬃc states;

and a dashed line indicates a FD. The blue areas (in Fig 1) and dots (in Fig 2) indicate stationary regions with

free-ﬂow traﬃc; the green ones indicate non-stationary regions with free-ﬂow and congested traﬃc; and the red

ones indicate stationary regions with congested traﬃc.

By the deﬁnition, traﬃc state S(a(x)

i,j )follows FD if region a(x)

i,j is stationary. Probe-traﬃc-state Si(a(x)

i,j )

is equal to S(a(x)

i,j )/ci,j if region a(x)

i,j is stationary and distance traveled by each vehicles in a(x)

i,j is equal to

the others. It means that Si(a(x)

i,j )follows a probe-FD which is a curve whose scale is 1/ci,j of the true FD

under the aforementioned conditions. The probe-FD can be also represented as q=Q(k, u, kc/ci,j , κ/ci,j ). In

Fig 1, regions a(0,1,4,5)

i,j are stationary; therefore, they are on the probe-FD which is represented as thick dashed

lines in Fig 2. On the other hand, regions a(2,3)

i,j are non-stationary; therefore, they are not on the probe-FD and

discarded (not utilized by calibration) (eq (1c)).

Stationarity of region a(x)

i,j can be inferred using trajectories of probe vehicles iand j. For example,

vi(a(x)

i,j ) = vj(a(x)

i,j )is a necessary condition for stationarity.

Finally, if the probe vehicles iand jhave traveled other congested region a(o)

i,j (the gray dot in Fig 2; it

may locates at downstream outside of Fig 1), A0(i, j, A)consists of a(0)

i,j ,a(1)

i,j ,a(4)

i,j ,a(5)

i,j ,a(o)

i,j . Then, the FD

can be uniquely determined by choosing values of u,kc, and ci,j such that all the points in {Si(a(x)

i,j )|a(x)

i,j ∈

A0(i, j, A)}are on the probe-FD q=Q(k, u, kc/ci,j , κ/ci,j )(eqs (1a) and (1b)).

2.3 Discussion

2.3.1 Summary of method

The proposed method can be summarized as follows.

Suppose that there is traﬃc described by the LWR model; two probe vehicles, namely, iand j, travel

through a free-ﬂow region and some congested regions in the traﬃc; and ci,j −1non-probe vehicles are

traveling between the two probe vehicles (Fig 1). Let q(a)be ﬂow in time-space region a;k(a)be density in

time-space region a;Q(k, u, kc, κ)be an FD function, which determine ﬂow under density k, free-ﬂow speed

u, critical density kc, and jam density κ.

Now the time–space region between the two probe vehicles is divided to multiple subregions (ain Fig

1) according to a speciﬁc rule. If traﬃc in a subregion is stationary, its traﬃc state (S(a)=(q(a), k(a))

in Fig 2) will follow the true FD (q(a) = Q(k(a), u, kc, κ)) by the deﬁnition. If traﬃc in a subregion is

stationary and distance traveled in the subregion by each vehicle is equal to the others, its probe-traﬃc-state,

Si(a)=(qi(a), ki(a)), which is ﬂow and density of probe vehicles only, will be 1/ci,j times the true traﬃc

state in the subregion (Si(a) = S(a)/ci,j ). Such probe-traﬃc-states follow probe-FD of which scale is 1/ci,j

of the true FD. It can be represented as qi(a) = Q(ki(a), u, kc/ci,j , κ/ci,j ).

If probe vehicles traveled through regions that satisfy speciﬁc conditions, the probe-FD can be identiﬁed

from the probe vehicle data only. For example, in Fig 2, a triangle with its vertex is at point (0,0), its edge is on

line q= 0, and its edges pass blue point, red point, and gray point can be determined uniquely. In other words,

values of u,kc/ci,j, and κ/ci,j can be determined from the probe vehicle data by assuming triangular shape for

the FD. Finally, since the value of jam density κis given, values of the rest of variables—namely, number of

vehicles between probe vehicles ci,j, free-ﬂow speed u, and critical density kc—can be determined.

2.3.2 Practical meaning of the assumptions and preconditions

The proposed method assumes a triangular FD with known jam density (assumptions (a) and (b)). This

assumption can be regarded as useful and easy to be assumed. Triangular FD has several desirable

characteristics in theory (Newell, 1993), is often employed by existing studies (c.f., Daganzo, 1997), and can

5

Time t

Space xProbe vehicle jProbe vehicle i

Non-probe

vehicles

(ci,j −1veh)

Shockwave

Free-ﬂow area F,S(F)=(q(F), k(F))

Congested area C,S(C)=(q(C), k(C))

a(0)

i,j

a(1)

i,j

a(2)

i,j

a(3)

i,j

a(4)

i,j

a(5)

i,j

Figure 1: Vehicle trajectories on time–space diagram

Density k

Flow q

κ

Actual FD: q=Q(k, u, kc, κ)

kc

u

κ/ci,j

kc/ci,j

probe-FD: q=Q(k, u, kc/ci,j , κ/ci,j )

S(F),S(a(0,1)

i,j )

q(F)

k(F)

S(C),S(a(4,5)

i,j )

q(C)

k(C)

S(a(o)

i,j )

Si(a(0,1)

i,j )

q(F)/ci,j

k(F)/ci,j

Si(a(4,5)

i,j )

q(C)/ci,j

k(C)/ci,j

Si(a(o)

i,j )

Si(a(2)

i,j )

Si(a(3)

i,j )

Shockwave

Figure 2: Traﬃc states transition on ﬂow–density plane

6

be found in empirical data (Cassidy, 1998). As for the parameter assumption, we can expect that reasonable

value can be assumed without strong dependency on time, space, nor ﬂow characteristic; because value of jam

density mainly depends on average length of vehicle bodies.

The method requires that the FD is constant in the target region (condition (i)). This assumption can be

the method’s limitation to some extent, especially on application to freeways; because the constant FD means

there are no bottlenecks.5Therefore, in freeway cases, the method has to be applied to near-homogeneous road

section only. Note that bottlenecks at the boundaries of the section are acceptable for the method. For example,

the method can be applied to a homogeneous section whose downstream end is an bottleneck. It should be

noted that, although the FD of the bottleneck cannot be calibrated by the method in such situation, its capacity

can be inferred; because ﬂow during congested state in the section can be derived based on the calibrated FD

and the probe vehicle data. On the other hand, the condition (i) will not be signiﬁcant limitation of the method

on application to arterial roads, where traﬃc is governed by signals rather than bottlenecks. In fact, the method

would be useful in such arterial roads, since the condition (iii) is likely to be satisﬁed near signals (see below).

The method requires certain properties for presence of probe vehicles (condition (iii)). This condition is

likely to be satisﬁed in the real-world. For example, it will be satisﬁed at upstream congestion of a merging

section, queue at signal, stop-and-go wave, and congestion due to a moving bottleneck. Furthermore, it will be

satisﬁed if amount of probe vehicle data is large enough (e.g., data collection time period) and it is randomly

sampled, even if the penetration rate is low.

3 Heuristic algorithm for actual traﬃc

The method presented in section 2 can determine an FD theoretically, if all the preconditions (i)–(iii) were

satisﬁed (and appropriate computation procedure was implemented). However, actual traﬃc may not follow

the LWR model (e.g., ﬁnite acceleration, heterogeneity among vehicles and lanes, measurement error). It

means that the preconditions may not be satisﬁed in actual traﬃc. In order to derive FD in such actual traﬃc,

this section describes a heuristic (and computational) algorithm for actual traﬃc based on the method of section

2.

The concept of the proposed algorithm is as follows. Exact solution of the problem (1) may not exist since

the preconditions are not exactly satisﬁed in actual traﬃc. Therefore, an algorithm determines FD parameters by

solving a residual minimization problem which relaxes eq (1) is proposed. Because of the residual minimization

problem may not be convex, the proposed algorithm is considered as heuristic.

3.1 Algorithm

The algorithm can be described as follows:

Step 1 Deﬁne region A.

Step 2 Select probe vehicle ifrom set P(A). Go to Step 5 if all of the probe vehicles are already

selected.

Step 3 Execute the following procedure:

Step 3.1 Select probe vehicle jfrom P(A)such that i6=j. Go back to Step 2 if all of the

combination (i, j)are already selected. Deﬁne empty set A0(i, j, A).

Step 3.2 Deﬁne region a(x)

i,j . Go back to step 3.1 if all of the combination (i, j, x)are

already selected.

Step 3.3 Determine whether region a(x)

i,j is stationary or not based on the trajectories of i

and j. Go back to Step 3.2 if it is not stationary.

5Lane-drops and sags are the typical bottlenecks in this LWR context. Contrary, merging sections and signals are not bottlenecks in

the context, as FDs around those sections are commonly considered as constant.

7

Step 3.4 Add probe-traﬃc-state Si(a(x)

i,j )to set A0(i, j, A).

Step 3.5 Go back to Step 3.1.

Step 4 Go back to Step 2.

Step 5 Determine free-ﬂow speed u, critical density kc, vehicle numbers {ci,j }by using residual

minimization with set A0(i, j, A).

Note that Steps 2, 3, and 4 correspond to eqs (1c) and (1d); and Step 5 corresponds to eqs (1a) and (1b). Among

the procedure, Step 5 is considered as heuristic while the other steps are not.

In Step 3.3, a region is determined to be stationary if relative diﬀerence of speed of probe vehicles iand j

is less than θ, namely,

|vi(a(x)

i,j )−vj(a(x)

i,j )|

vi(a(x)

i,j )≤θ, (7)

where θis a constant which should be given appropriately. If values of θand ∆xare too small, a(x)

i,j will not be

regarded as stationary due to small-scale noises in traﬃc. On the other hand, if they are too large, a(x)

i,j will be

regarded as stationary even if it includes a shockwave and therefore should not be stationary.

The residual minimization in Step 5 is described as follows:

min

u,kc,{ci,j }∀i,j

X

a∈A0(i,j,A)

∀i,j∈P(A)

diﬀ(ci,j Sm(a), u, kc, κ)2,(8)

s.t. u≥0,

0≤kc≤κ,

∀ci,j ≥0, i, j ∈P(A),

where diﬀrepresents distance between a point (q, k)and a curve q=Q(k, u, kc, κ). Note that

diﬀ(ci,j Sm(a), u, kc, κ)and ci,j diﬀ(Sm(a), u, kc/ci,j , κ/ci,j )are equivalent to each other except ci,j = 0

case. The optimization can be executed by quasi-Newton method with multiple initial points; because

optimization problem (8) is not necessarily convex.

3.2 Discussion

By using the proposed algorithm, values of the FD parameters can be calibrated. The result can be considered as

an “average FD” where heterogeneity over time and space are averaged out. The heterogeneity among vehicles

can be also averaged out if the data contains large number of probe vehicles.

The proposed algorithm can be robust against measurement error. The variables di(a),ti(a), and |a|are

derived from a vehicle trajectory over a distance of ∆x. Therefore, measurement error will be negligible if ∆x

is suﬃciently larger than GPS measurement error (typically around 10 meters).

The optimization problem (8) is not necessarily convex as mentioned. In addition, an obvious global

optimal is ci,j = 0 (∀i, j ), which is physically meaningless for FD calibration. However, it can be expected that

local optima exist near the actual FD parameters. Therefore, FD parameters would be estimated properly by

using a local optimization method (e.g., quasi-Newton method) with certain initial points. Validness of applying

quasi-Newton method with various initial points to this problem (i.e., robustness of the proposed algorithm) is

presented in section 4.

4 Validation

This section describes validation of the calibration algorithm of section 3 by applying it to synthetic traﬃc data

generated by microscopic traﬃc simulation.

8

Figure 3: Density kin the simulation Figure 4: An example of probe vehicle trajectories in

the simulation

4.1 Simulation environment

Traﬃc data was generated by a microscopic traﬃc ﬂow simulator Aimsun (TSS-Transport Simulation Systems,

2011), which is based on Gipps (1981)’s car-following model and Gipps (1986)’s lane-changing model. The

car-following model’s parameters are shown by Table 1. The road was conﬁgured to have two lanes corridor

with a bottleneck at its downstream end. Probe vehicles were randomly sampled from the entire vehicles with

5% probability. As results, the synthetic traﬃc does not satisfy the precondition (i) and (ii) exactly and does

not satisfy (iii) at all depending on a combination of probe vehicles—these are similar to actual traﬃc.

Fig 3 shows the traﬃc density generated by the simulation. According to the ﬁgure, several traﬃc features,

such as queue extension, queue diminishing, and stop-and-go waves in the queue can be found. Fig 4 shows an

example of probe vehicle trajectories. The curves represent trajectories of probe vehicles and the colored

dots represent non-stationarity deﬁned by the left hand side of eq (7), where thicker color shows larger

non-stationarity.

Values of the parameter of the algorithm were set as follows: κ= 200 (veh.km), θ= 0.05, and ∆x= 100

(m). The initial values for the optimization problem (8) were given by certain i.i.d. uniform distributions such

that u∈[70,100] (km/h), kc∈[10,40] (veh/km), and ci,j ∈[20,100] (veh).

4.2 Calibration results

Fig 5 shows a calibration result where the red lines represent a calibrated FD and blue dots represent traﬃc

states which are not necessarily stationary. The estimated FD parameter values are ˆu= 74.2(km/h) and

ˆ

kc= 28.8(veh/km). The calibrated FD seems to describe the traﬃc well. Fig 6 compares estimated ˆci,j and

its truth. According to the ﬁgure, the results of ci,j estimation can be categorized into two: almost proper

estimation and estimated as zero. The latter is caused by indeﬁnite ci,j , which is because of non-satisfaction of

the precondition (iii), and the feature of the optimization problem discussed in section 3.2.

Table 1: Parameters of the car-following model

mean std.div.

Vehicle length (m) 4 0.5

Minimum spacing (m) 1 0.3

Desired speed (km/h) 80 10.0

Maximum acceleration rate for acceleration (m/s2) 3 0.2

Standard acceleration rate for deceleration (m/s2) 4 0.25

Maximum acceleration rate for deceleration (m/s2) 6 0.5

9

Figure 5: An example of estimated FD Figure 6: An example of estimated {ci,j }

Figure 7: Distribution of free-ﬂow

speed ˆu

Figure 8: Distribution of critical

density ˆ

kc

Figure 9: Distribution of cov. of

ˆci,j

Robustness of the algorithm is validated by performing multiple calibration iterations for 500 times with

diﬀerent initial value to the non-convex optimization problem (8). Fig 7 and 8 shows distributions of estimated

free-ﬂow speed ˆuand critical density ˆ

kc, respectively. According to the ﬁgure, the calibration is almost stable,

although slight variation can be found. Fig 9 shows distribution of the coeﬃcient of variation (cov.) of each

ˆci,j . According to the ﬁgure, this result is not stable, since ˆci,j can be zero as evidenced by Fig 6. Table 2

summarizes the calibration results. The results of ˆuand ˆ

kcwere fairly stable. It means that although derivation

of ˆci,j was not necessarily stable due to the issue mentioned in section 3.2, the algorithm can estimate values

of the FD parameters robustly. In addition, the mean of all the iterations ﬁts well to the actual traﬃc; therefore,

obtaining a ﬁnal result by taking mean of multiple iterations would be precise.

In summary, the results suggested that the proposed algorithm can be precise and robust for traﬃc in which

the preconditions are not exactly satisﬁed, although the optimization problem has some theoretical issues (e.g.,

non-convexity). The algorithm can be considered as precise and accurate as the calibration results are close to

the actual phenomena, and can be considered as robust as the results did not strongly depend on initial points

of the algorithm.

Table 2: Statistics of distributions of calibration results

mean std.div. cov.

free-ﬂow speed u(km/h) 74.2 2.1 0.028

critical density kc(veh/km) 28.7 1.4 0.048

capacity kcu(veh/h) 2126.1 106.7 0.050

cov. of ci,j 0.997 – –

10

5 Conclusion

This paper proposed methods of calibrating a fundamental diagram based on probe vehicle data. First, we

formulated a method which identiﬁes value of free-ﬂow speed and critical density under idealized conditions,

while it relies on exogenous assumptions about functional form of FDs and value of jam density. Then, a

heuristic algorithm for calibrating FDs under actual traﬃc conditions is developed based on the method. The

results of simulation-based validation suggested that the algorithm can calibrate an FD precisely and robustly.

Therefore, it can be expected that the proposed method calibrate FDs in road sections on which congestion

happens frequently, if probe vehicle data were collected for a long period. It means that applicability of probe

vehicle-based traﬃc management methods can be signiﬁcantly increased.

Several future works are considerable and being investigated by the authors. The ﬁrst is to develop

endogenous identiﬁcation method for time–space regions with constant FDs. This is essentially important

to improve the methods’ applicability, since the current methods themselves do not identify a bottleneck.

The second is to develop theoretically sophisticated algorithm (e.g., avoiding meaningless global optimal,

developing convex optimization-based algorithm) for more rigorous calibration. The third is to validate the

methods under more general conditions (e.g., using real world datasets). The forth is to adopt more precise

identiﬁcation method for stationary states (e.g., Yan et al., 2016) and to perform related sensitivity analyses.

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