論文 / 著書情報
Article / Book Information
題目(和文)

周辺移動体を観測する移動体観測による交通流の推定

Title(English)

Traffic Estimation with Vehicles Observing Other Vehicles

著者(和文)

瀬尾亨

Author(English)

Toru Seo

出典(和文)

学位:博士（工学）,学位授与機関:東京工業大学,
報告番号:甲第9982号,
授与年月日:2015年9月25日,
学位の種別:課程博士,
審査員:朝倉 康夫,屋井 鉄雄,福田
料 隆雅

大輔,室町

泰徳,花岡

Citation(English)

Degree:,Conferring organization: Tokyo Institute of Technology,
Report number:甲第9982号,
Conferred date:2015/9/25,
Degree Type:Course doctor,
Examiner:,,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Tokyo Institute Research Repository)伸也,井Traﬃc Estimation with Vehicles Observing Other Vehicles
byToru Seo

Submitted to the
Department of Civil Engineeringin partial fulfillment of the requirements for the degree ofDoctor of Engineering
at theTokyo Institute of Technology

August, 2015

Abstract
This dissertation is entitled “Traﬃc estimation with vehicles observing other vehicles” by
Toru Seo, submitted to the Department of Civil Engineering in partial fulfillment of therequirements for the degree of Doctor of Engineering at the Tokyo Institute of Technology,
August, 2015. The dissertation supervisor is Professor Yasuo Asakura, Department ofCivil Engineering, Tokyo Institute of Technology. The examiners are Professor Yasuo
Asakura, Professor Tetsuo Yai, Associate Professor Yasunori Muromachi, Associate
Professor Shinya Hanaoka, Associate Professor Daisuke Fukuda (Tokyo Institute ofTechnology) and Professor Takamasa Iryo (Kobe University).
This dissertation summarizes methodology of estimating traﬃc using probe vehicles
which observe other nearby vehicles. It consists of seven chapters.
In chapter 1, the background and objective are summarized. Automobile traﬃc takes
significant roles in today’s civilization. In order to achieve high eﬃciency of traﬃc,
traﬃc managements are required. The managements require knowledge on current traﬃcsituation. However, knowing traﬃc situation in anytime and anywhere is not yet possible
due to theoretical and practical reasons. The aim of this dissertation is to propose
methodology for traﬃc state estimation that is theoretically and practically applicable forvast time–space domain. To achieve it, this dissertation formulates and validates traﬃcstate estimation method utilizing probe vehicle with spacing measurement equipment.
In chapter 2, related studies are reviewed and summarized. It includes traﬃc flow
theory, traﬃc data collection methods, traﬃc state estimation methods and advanced
vehicle technology. Traﬃc flow theory can be summarized by employing the concept of
cumulative flow. Then current traﬃc data collection methods and traﬃc state estimation
methods are reviewed; and their characteristics are described based on the concept of
cumulative flow. As a result, a challenging task remained in traﬃc engineering filed isoutlined: traﬃc state estimation methods that can applicable for vast time–space domain
without theoretical discrepancies. Recent developments in advanced vehicle technology
include vehicle automation, by which vehicle can observe other nearby vehicles. Spacing,inamely, distance from a vehicle’s head to its leading vehicle’s one, is typical informationcollected by such automated vehicles. As a result, a possible approach for the traﬃc state
estimation is pointed out: methods utilizing probe vehicles with spacing measurement
equipment.
In chapters 3 and 4, methodology of estimating traﬃc state utilizing probe vehicles’
position and spacing information is formulated. An estimation method are formulated
as requiring less exogenous assumptions as possible and consistent with traditional
definition in traﬃc flow theory. These characteristics enable the methods to be appliedin various situations without theoretical discrepancies. The method’s characteristics,
such as accuracy and precision, are investigated analytically. Then, some well-accepted
assumptions on traﬃc flow are integrated to the methods in order to improve theproposed methodology’s reasonability in common traﬃc situations. One of the method
estimates traﬃc state based on the concept of cumulative flow by assuming existence of aconservation law in a link. The other method estimates traﬃc state by assuming existenceof a conservation law and a fundamental diagram whose parameters are endogenously
estimated from the probe vehicle data.
In chapters 5 and 6, the proposed methodology is empirically validated. The datafor validation are actual probe vehicle data collected via a field experiment conductedin the real world, as well as synthetic data generated by simulation experiment. A
simulation experiment-based validation can clarify the methods’ characteristics underidealized situation, while a field experiment-based validation can evaluate the method’s
characteristics under actual traﬃc environment.

As a result, relation among theestimation methods’ precision and accuracy, time–space resolution, penetration rate ofprobe vehicles and assumptions is clarified. Their consistency with analytical results is
confirmed. Therefore, the methods’ applicability in actual traﬃc environment is shown.
In chapter 7, achievements and future directions are summarized. The achievements
are formulation and validation of methodology of traﬃc state estimation utilizing
probe vehicle with spacing measurement equipment.

The future directions include

sophistication of the methodology, such as further integration of traﬃc flow theory.
ii

Acknowledgements
The traﬃc detector data was generously provided by Metropolitan Expressway Co., Ltd.
I would like to appreciate Prof. Yasuo Asakura and Dr. Takahiko Kusakabe forsupervising this research. I also would like to appreciate Prof. Carlos F. Daganzo (University of California, Berkeley), Prof. Masao Kuwahara (Tohoku University), Prof. Toshio
Yoshii (Ehime University), Prof. Takamasa Iryo (Kobe University), Dr. Nobuhiro Uno(Kyoto University), Dr. Daisuke Fukuda (Tokyo Institute of Technology), Dr. Kentaro
Wada (University of Tokyo), Dr. Chong Wei (Beijing Jiaotong University), anonymous
reviewers, Dr. Hiroshi Warita (Metropolitan Expressway Co., Ltd.), Dr. Yasuhiro Nonaka,
Mr. Takashi Ishida (Highway Planning Co., Ltd.), Mr. Jun Tanabe (Regional Futures
Research Center Co., Ltd.), Metropolitan Expressway Co., Ltd., staﬀs of Survey ResearchCenter Co., Ltd. who assisted our experiment, Dr. Van Hong Tan, Dr. Long Xuan
Nguyen (Ho Chi Minh City University of Technology), students of Ho Chi Minh
City University of Technology who assisted our experiment, Ms. Kiko Yamada-Kawai,Prof. Tetsuo Yai, Dr. Yasunori Muromachi, Dr. Shinya Hanaoka, and current and former
members of Asakura Lab. (Tokyo Institute of Technology) from its beginning to 2015,
who directly contributed to this research by their constructive comments, insightful
suggestions and dedicated assistance during seminars, conferences, workshops, symposia,
dojos, reviewing, surveys, experiments, proofreading and examining, as well as private
communication. I further would like to appreciate numerous people who kindly supported
and warmly encouraged me during this research.
I am indebt to following financial supports to parts of this research: the Research
Fellow (DC2) program of the Japan Society for the Promotion of Science entitled“Road network traﬃc estimation using information from on-vehicle cameras of probevehicles” (KAKENHI Grant-in-Aid for JSPS Fellows #26010218/#14J10218), and thecollaborative research project of the Japan Society of Traﬃc Engineers entitled “Traﬃcmonitoring at Asian cities without sensor infrastructures”.
Without all the above help, this research would not have been possible.

iii

Contents
1

2

Introduction

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Literature Review

6

2.1

Traﬃc flow theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.1

Traﬃc state . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.2

Fundamental diagram . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.3

Traﬃc flow model . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.4

Area-wide traﬃc: Macroscopic fundamental diagram . . . . . . .

16

Traﬃc data collection and traﬃc state estimation . . . . . . . . . . . . .

18

2.2.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2.2

Traﬃc data collection methods . . . . . . . . . . . . . . . . . . .

23

2.2.3

TSE methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Advanced vehicle technologies . . . . . . . . . . . . . . . . . . . . . . .

32

2.3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.3.2

Utilization for TSE-like purposes . . . . . . . . . . . . . . . . .

33

2.2

2.3

3

Basis: Method for Estimating Traﬃc State

34

3.1

Assumptions and concepts . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.2

Collectable data by PVSMEs . . . . . . . . . . . . . . . . . . . . . . . .

36

3.3

Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.4

Analytical characteristics of estimators . . . . . . . . . . . . . . . . . . .

38

3.4.1

Derivation of the approximated bias . . . . . . . . . . . . . . . .

40

3.4.2

Derivation of the approximated precision . . . . . . . . . . . . .

41

3.5

Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.6

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

iv

4

Integrating Traﬃc Dynamics into Estimation Methods

49

4.1

Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.2

Method with CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2.1

Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2.2

Estimation method for cumulative flow at probe vehicle’s trajectory 51

4.2.3

Estimation method for traﬃc state . . . . . . . . . . . . . . . . .

54

4.2.4

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Method with FD and CL . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.3.1

Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.3.2

Ensemble Kalman filter . . . . . . . . . . . . . . . . . . . . . . .

60

4.3.3

TSE method formulation . . . . . . . . . . . . . . . . . . . . . .

61

4.3.4

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Comparison among three TSE methods . . . . . . . . . . . . . . . . . .

68

4.3

4.4
5

Simulation Experiments-based Validation

71

5.1

The basic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.1.1

Simulation Scenario . . . . . . . . . . . . . . . . . . . . . . . .

71

5.1.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

The method with CL . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5.2.1

Simulation scenario . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.2.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.2.3

Eﬀect of incorporating CL . . . . . . . . . . . . . . . . . . . . .

79

The method with FD and CL . . . . . . . . . . . . . . . . . . . . . . . .

80

5.3.1

Simulation scenario . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.3.3

Eﬀect of incorporating FD and CL . . . . . . . . . . . . . . . . .

84

5.2

5.3

6

Field Experiment-based Validation

86

6.1

Field experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

6.1.1

87

Inner Circular Route . . . . . . . . . . . . . . . . . . . . . . . .
v

6.1.2
6.2

6.3

Probe vehicle data . . . . . . . . . . . . . . . . . . . . . . . . .

89

The basic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

6.2.1

Visualizations of estimation results . . . . . . . . . . . . . . . .

95

6.2.2

Eﬀect of time–space resolution and PR . . . . . . . . . . . . . .

97

6.2.3

Eﬀect of traﬃc situation . . . . . . . . . . . . . . . . . . . . . . 100

The method with CL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3.1

Ground truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3.3

Comparison

betweensimulation

experiment-

and

field

experiment-based validations . . . . . . . . . . . . . . . . . . . . 105
7

Conclusions

107

7.1

Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2

Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2.1

The proposed TSE methods themselves . . . . . . . . . . . . . . 110

7.2.2

Transportation system estimation with travelers observing othertravelers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A Expansion to Network: OD Flow

113

A.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.2 Simulation experiment-based validation . . . . . . . . . . . . . . . . . . 115
A.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B Field Experiment at Arterial Road with Mixed Traﬃc117

B.1 Field experiment environment . . . . . . . . . . . . . . . . . . . . . . . 117
B.2 Results of TSE on cars . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
References

121

vi

List of Figures
1.1

Transportation system information collection approaches. . . . . . . . . .

4

1.2

Structure of dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1

definition of traﬃc state. . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Three-dimensional representation of traﬃc flow using the cumulative flow8

2.3

Example of triangular FD in four coordinate systems . . . . . . . . . . .

10

2.4

Example of realistic FD. . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.5

Example of well-defined MFD. . . . . . . . . . . . . . . . . . . . . . . .

17

2.6

Examples of various traﬃc data. . . . . . . . . . . . . . . . . . . . . . .

20

3.1

Time–space region A and probe vehicle data. . . . . . . . . . . . . . . .

36

3.2

Numerical examples of error of the flow estimator with high resolution. .

44

3.3

Numerical examples of error of the flow estimator with low resolution. . .

45

3.4

Examples of A shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.1

Three-dimensional representation of traﬃc flow. . . . . . . . . . . . . . .

52

4.2

Method conceptualized as time–space diagram. . . . . . . . . . . . . . .

52

4.3

Illustrated examples of estimation method on cumulative curves. . . . . .

55

4.4

Illustrated concept of the procedure of the proposed TSE method. . . . .

59

4.5

Triangular FD and its parameters in cell i at time t. . . . . . . . . . . . .

62

4.6

Coordinates for the method with FD and CL and the probe vehicle data. .

63

4.7

Illustrated comparison of three TSE methods. . . . . . . . . . . . . . . .

69

5.1

Results of the basic method visualized as time–space diagrams. . . . . . .

75

5.2

Time–space diagrams of traﬃc state and cumulative flow. . . . . . . . . .

78

5.3

Scatter diagrams of estimated variables vs. ground truth variables. . . . .

79

5.4

Typical estimation results visualized as time–space diagrams. . . . . . . .

82

5.5

Observed FD, disaggregated headway–spacing relation and aggregatedflow–density relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

83

6.1

Map of the Inner Circular Route. . . . . . . . . . . . . . . . . . . . . . .

88

6.2

Schematic of the Inner Circular Route (counterclockwise direction). . . .

88

6.3

Time–space diagram of density in C1 as observed by the detectors. . . . .

88

6.4

Preparation of probe vehicles and equipments. . . . . . . . . . . . . . . .

89

6.5

Trajectories of probe vehicles. . . . . . . . . . . . . . . . . . . . . . . .

90

6.6

Examples of images taken by the cameras on the probe vehicles. . . . . .

92

6.7

Part of the probe vehicle data. . . . . . . . . . . . . . . . . . . . . . . .

93

6.8

Spot speed distributions in probe vehicle data and detector data. . . . . .

94

6.9

Spacing distributions in probe vehicle data and detector data. . . . . . . .

94

6.10 Visualization of estimated flow, density, and speed. . . . . . . . . . . . .

96

6.11 Visualization of estimated density with lower PR and lower resolution. . .

97

6.12 Scatter plots of estimated variables versus the corresponding values
observed by the detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.13 Time–space diagrams of traﬃc state and cumulative flow in field
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.14 Scatter diagrams of estimated variables vs. detectors’ observed variables
in field experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.1 Example of probe vehicle data in network . . . . . . . . . . . . . . . . . 114
A.2 Network for simulation experiment. . . . . . . . . . . . . . . . . . . . . 115
A.3 Comparison between estimated flow and ground truth. . . . . . . . . . . 116
B.1 Experiment at Pasteur Str. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.2 Cumulative curves at Pasteur Str. . . . . . . . . . . . . . . . . . . . . . . 119

viii

List of Tables
3.1

Parameters for numerical examples . . . . . . . . . . . . . . . . . . . . .

43

4.1

Summary of three TSE methods . . . . . . . . . . . . . . . . . . . . . .

70

5.1

Parameters of the model in simulation . . . . . . . . . . . . . . . . . . .

72

5.2

Precision and bias indices over estimation scenarios. . . . . . . . . . . .

74

5.3

Analytically expected precision and bias. . . . . . . . . . . . . . . . . .

74

5.4

Vehicle behavior parameter setting in simulation. . . . . . . . . . . . . .

76

5.5

Error indices for scenarios with proposed method. . . . . . . . . . . . . .

80

5.6

Comparison between the method with CL and the basic method. . . . . .

80

5.7

Summary on the estimation performance . . . . . . . . . . . . . . . . . .

84

6.1

Spot speed and spacing statistics of probe vehicle data and detector data. .

95

6.2

Precision and bias indices over estimation scenarios. . . . . . . . . . . .

99

6.3

Error indices for scenarios with proposed method in field experiment. . . 104

6.4

Comparison between results of field experiment with the method with CL
and the basic method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.5

Diﬀerence between validations based on simulation experiment and field
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

A.1 OD matrix for simulation experiment. . . . . . . . . . . . . . . . . . . . 115
B.1 Flow estimation results at Pasteur Str. . . . . . . . . . . . . . . . . . . . 120

ix

List of Notations
Following notations are used throughout in this dissertation unless otherwise specified.

qflow (veh/h)kdensity (veh/km)v

speed (km/h)ttime (h)xspace (km)N(t, x)

cumulative flow at time t and space x (veh)nindex for a vehiclemindex for a probe vehicleˆmark for an estimated valueA

time–space region|A|area of time–space region A (km h)an (A)

time–space region between vehicle n and n − 1 in region A

N(A)set of all the vehicles in region AP(A)set of all the probe vehicles in region A

tn (A)

time spent by vehicle n in region A

dn (A)

distance traveled by vehicle n in region Ah̄n (A)

mean headway experienced by vehicle n in region As̄n (A)

mean spacing experienced by vehicle n in region Aufree-flow speed (km/h)

kc , k c

critical density (veh/km)

κ

jam density (veh/km)

µ

mean

σ

standard deviationxList of Abbreviations
Following abbreviations are used throughout in this dissertation.

TSE

Traﬃc State Estimation

GPS

Global Positioning System

AVT

Advanced Vehicle Technologies

PVSME

Probe Vehicle with Spacing Measurement Equipment

LWR

Lighthill–Whitham–Richards

FD

Fundamental Diagram

CL

Conservation Law

FIFOFirst-In First-Out

OD

Origin–Destination

UE

User Equilibrium

SO

System Optimum

CTM

Cell Transmission Model

MFD

Macroscopic Fundamental Diagram

KFT

Kalman Filtering(-like) Techniques

EnKF

Ensemble Kalman Filter

PR

Penetration Rate of probe vehicles in entire vehicles

MAE

Mean Absolute Error

MAPE

Mean Absolute Percentage Error

RMSE

Root Mean Square Error

RMSPE

Root Mean Square Percentage Error

PoI

Percentage of Improvement

C1

Inner Circler RouteJCT

JunCTionID

IDentificationxi

Chapter 1 Introduction
1.1

Background

Transportation—“a system for carrying people or goods from one place to another using
vehicles, roads, etc.” (Hornby et al., 2010)—is one of the most essential elements incivilization. Eﬃciency of a transportation system must be kept as high as possible. A
transportation system consists of• people who travel within the system based on their decision,• modes/infrastructures by/on which people travel, and• information by which people decide,In general, civil engineers can keep the eﬃciency high by controlling and regulating
modes/infrastructures based on information, and providing information to people. To
do so, civil engineers have to answer following questions:• how many people are there and how fast they are?
• how people decide?
In fact, these are not easy questions. The first point can be answered by collecting
information from the transportation system; however, collecting information from theentire system is not always possible due to technological and financial limitations. The
second point may be more diﬃcult. If complete rational travelers with perfect information
are assumed, it can be answered partially (Sheﬃ, 1985); however, as mentioned above,
the perfect information condition is not always satisfied (and perhaps people are notcomplete rational). Under such circumstances, people’s decision-making should bemodeled theoretically and be validated empirically—this requires information, again. In
short conclusion, information on transportation system is required as• basis for coding controls, and1

Chapter 1

Introduction• input data for controls.
Traﬃc is a special mode in a transportation system, where automobiles travel throughone-dimensional roads (Daganzo, 1997). In the modern era, automobiles take significant
role in transportation, because of their high flexibility, high speed, low cost and good
capability of carrying.

It has the same information problem mentioned in above.

Ironically, mainly due to their high flexibility, automobiles often cause traﬃc jams andnegative externalities such as time loss, safety issues and environmental issues—user
equilibrium (UE) vs social optimum (SO) problem in traﬃc network (Wardrop, 1952;
Braess, 1968). To mitigate these externalities, carefully planed and precisely performed
traﬃc control measures are required. As a consequence, it can be said that traﬃc datacollection can be one of the most essential elements in civilization. From emergenceof traﬃc engineering (Greenshields, 1935), several methodologies have been developed
to collect traﬃc data. Their approaches can be roughly categorized twofold: Eulerianand Lagrangian sensing. Eulerian sensing, also known as fixed observation, collects
traﬃc data at fixed-point on a road; the typical examples are road side traﬃc detectors
and manual counting. They can acquire suﬃcient traﬃc data at installed location;
however, spatiotemporally wide ranging data collection is diﬃcult due to their installationand maintenance cost. Lagrangian sensing, also known as floating, mobile and probe
observation, collects traﬃc data from a vehicle that moves along the traﬃc; the typicalexamples are Global Positioning System (GPS)-equipped probe vehicles. They can
only acquire position and speed data; however, spatiotemporally wide ranging datacollection is possible due to their nature and low running cost. Still, there are no
feasible methodologies that are capable of collecting suﬃcient traﬃc information for
spatiotemporally wide range. This makes diﬃcult to understand and control traﬃc.
Recently, advanced vehicle technologies (AVT), such as adaptive cruise control,
collision avoidance, and autonomous vehicles, have been developed practically forthe driving safety and eﬃciency purposes (c.f.,National Highway Traﬃc Safety

Administration, 2013). These systems automate some of the driving functions of vehicles
by recognizing the surrounding environment of a vehicle. Indeed, these technologies
2Chapter 1

Introduction

are valuable as they improve eﬃciency of an individual vehicle’s driving; however, it
does not always guarantee optimal improvement at social level, just like the UE vs SO
problem. Meanwhile, they can be more valuable if they are utilized from social-scale
view point. The information collected by them, especially spacing (i.e., vehicle-to-vehicle
distance), is valuable from the point of view of traﬃc data collection—spacing is closely
related to traﬃc state as well as vehicle’s decision-making behavior. The information is
impossible to be acquired with conventional data collection methods, such as road side
traﬃc detectors and GPS-equipped probe vehicles. Therefore, such advanced vehicles are
expected to be utilized as a source of new traﬃc data and contribute to transportation fromsocial-scale. Fig 1.1 illustrates traﬃc data collection approaches; a gray arrow represents
a vehicle trajectory, orange area represents observable area, and a red dot/line represents
observable information.

1.2

Objective

The primal objective of this dissertation is to propose methodology for probe vehiclebased traﬃc state estimation (TSE) that can collect suﬃcient traﬃc data where probevehicles observe their surrounding vehicles with onboard equipments. In particular, thisdissertation proposes methods for estimating the flow, density, and speed based on thespacing and position information collected by the probe vehicles. This dissertation names
such probe vehicles “probe vehicles with spacing measurement equipment (PVSME)”.
The primal objective is achieved by accomplishing following secondary objectives:
review (chapter 2), formulation (chapters 3 and 4) and validation (chapters 5 and 6). In thereview chapter, background and challenging tasks for this dissertation are clarified. Then,in the formulation chapters, TSE methods using PVSME are formulated considering thereview results. In the validation chapters, empirical characteristics of the formulated
methods are investigated; and their performance is shown.
Specifically, this dissertation is organized as follows. Chapter 2 reviews existingstudies on related topics to this dissertation, that is, traﬃc flow theory, traﬃc data3

Chapter 1

Introduction

Time

Time

Space

Fixed sensor

Fixed sensor(a) Eulerian sensing (e.g., detectors).

Space

Space

Space

Fixed (b) Conventional Lagrangian sensing (e.g.,sensors (orange pillars) collect information

GPS-equipped probe vehicles). Some of theof travelers (gray arrows).

travelers (red arrows) report information of
their own.

Time

Space

Space
(c) Proposed

Lagrangiansensing(i.e.,PVSME). Some of the travelers (red arrows
and orange belts) collect information of
nearby travelers.
Figure 1.1: Transportation system information collection approaches.

4

Chapter 1

Introduction
Chapter 1
Introduction

Chapter 2
Literature Review

Formulation
Chapter 3
Basis: Method for Estimating Traffic State
Chapter 4Integrating Traffic Dynamics into Estimation Methods

Chapter 5
Simulation Experiment-based Validation

Chapter 6
Field Experiment-based Validation

Empirical Validation
Chapter 7
Conclusions

Figure 1.2: Structure of dissertation.

collection, TSE and advanced vehicle technologies. Chapter 3 formulates the basic
method for estimating traﬃc state from the assumed probe vehicle data. It is formulated
as much as simple and consistent with traﬃc flow theory, in order to utilize it as basisof more advanced methods. In addition, the basic method is analytically investigatedin order to obtain general insight.

Then chapter 4 develops more advanced TSE

methods considering other factors in traﬃc flow theory, especially on those describe
traﬃc dynamics. Chapters 5 and 6 validate the proposed methods in chapters 3 and 4,
by using simulation experiment datasets and an actual traﬃc dataset we collected via
a field experiment, receptively. Through the validation, performance of the proposed
methodology is shown empirically. Finally, chapter 7 concludes achievements and future
direction of the proposed methodology. Fig 1.2 illustrates the structure of this dissertation.

5

Chapter 2 Literature Review
This chapter reviews existing studies that are related to this dissertation. Section 2.1
summarizes theoretical aspects of traﬃc flow, from how traﬃc states are defined to how
traﬃc dynamics are modeled. Section 2.2 classifies existing TSE methods, as well as
traﬃc data collection methods, according to approaches, techniques and technologies
employed by them. Finally, section 2.3 reviews recent advancement in the vehiculartechnologies from a point of view of traﬃc and transportation engineering.

2.1

Traﬃc flow theory

This section briefly*1 reviews how traﬃc has been theoretically represented and modeledin existing studies.
In order to simplify discussion, following considers suﬃciently long length street withsingle lane without vehicle’s overtaking nor merging/diverging sections (i.e., a first-in
first-out (FIFO) condition and a conservation law (CL) is satisfied), unless otherwise
specified. The space coordinate x increases as downstream direction; the time coordinatet increases as future direction; the vehicle count n increases from leading vehicles tofollowing vehicles.

2.1.1

Traﬃc state

A traﬃc state can be defined as a set of flow q, density k and average speed v, inmacroscopic scale which focuses on aggregated vehicles and does not always distinguisheach vehicles. The flow, also known as flow rate and traﬃc volume, is a number of*1

See following literatures for the detail on traﬃc flow theory. Daganzo (1997) provided systematic anddetailed introduction to traﬃc flow theory. Helbing (2001) summarized comprehensive review on traﬃcflow from various perspective. Wageningen-Kessels et al. (2014) described taxonomy of traﬃc flow models
with excellent artwork.

6

Chapter 2

Literature Review

Space xSpace xsmall flow, large density, small speed

small flow

small speed

large density

large speed
small density
large flow
large flow, small density, large speed

Time tTime t(a) Specified definition often utilized in practical situ-(b) Edie’s generalized definition

ation
Figure 2.1: definition of traﬃc state.

vehicles that pass per unit time; density is a number of vehicles that exist per unit space;
and the average speed is a mean of instantaneous speeds among vehicles. In practicalsituation, point flow in space, space density at a time point and space mean speed areoften utilized; they are shown in Fig 2.1a. Edie (1963) proposed generalized definition of
traﬃc state; the traﬃc state in a time–space region A can be defined as follows:
d(A)
,
|A|
t(A)
k(A) =,|A|
d(A)v(A) =,t(A)q(A) =

(2.1)
(2.2)
(2.3)

where d(A) is the total distance traveled by all of the vehicles in the region A (veh km),t(A) is the total time spent by all of the vehicles in the region A (veh h), and |A| is thetime–space area of the region A (km h). It is shown in Fig 2.1b. These definition include
specific definitions mentioned before. Note that the relation of q = kv must be satisfied.
They are often utilized for representing macroscopic traﬃc. Usually, a set of flow, density
and speed with high time–space resolution (e.g., minutes- and meters-scale) is referred to7

Chapter 2

Literature Review

Figure 2.2: Three-dimensional representation of traﬃc flow using the cumulative flowa traﬃc state.
The concept of a cumulative flow can be introduced to generalize above. A cumulative
flow N(t, x) is defined as a number of vehicles that passed position x by time t. A
cumulative curve or count, which is a cumulative flow at a certain position (i.e., N(t, x =given)), is well-known for analyzing queues. A continuous cumulative flow Ñ(t, x) shapes
a continuous and smooth “surface” in vehicle–time–space (n–t–x) domain, in which all
the traﬃc state mentioned above are included (Moskowitz, 1965; Makigami et al., 1971).
For example, a contour line of the same Ñ is a vehicle trajectory X(n, t) and T (n, x), a
slope of Ñ in the t direction is the flow q, and the slope of Ñ in the −x direction is thedensity k. Therefore, following equations hold true:∂N(t, x),∂t
∂N(t, x)k(t, x) =,∂xq(t, x).
v(t, x) =k(t, x)q(t, x) =

(2.4)
(2.5)
(2.6)

Fig 2.2 shows an example of a cumulative flow.
In general, a traﬃc state can be defined as representation of traﬃc situation in a certain
8

Chapter 2

Literature Review

time–space domain. In microscopic scale, which focuses on disaggregated vehicles anddistinguishes each vehicles, a state of traﬃc can be defined as position of vehicle n at timet, that is, X(n, t). Its first order derivative, namely, dX(n, t)/dt = Ẋ(n, t), is instantaneous
speed; and its second order derivative Ẍ(n, t) is acceleration, respectively, of vehicle n attime t. A spacing of vehicle n, namely, distance to its leading vehicle, is X(n−1, t)−X(n, t).
They are often utilized for modeling vehicle driving behaviors. In fact, they are not often
referred to “traﬃc state”.

2.1.2

Fundamental diagram

A fundamental diagram (FD), also known as flow–density relation, found by Greenshields(1935), is a relation among the traﬃc state (i.e., flow, density and speed) in equilibrium
state*2 (i.e., no vehicles have incentive to change their speed and spacing) at a time–spacepoint. An FD can be represented asq(t, x) = Q(k(t, x), t, x),(2.7)

where Q(k, t, x) is the FD function that returns flow where density is k at time t andposition x. Usually (in single-regime model), an FD in a flow–density plane is a convex
curve that has a peak. An FD in a speed–density plane is a weakly monotonic decrease
curve. An equilibrium traﬃc state is a stationary, that is, all the vehicles have the same
speed and spacing, if the FD in speed–density plane is strongly monotonic decreasing.
Fig 2.3 shows an example of an FD. A traﬃc state with a density higher than a criticaldensity kc is considered as congested state; and that lower than or equal to kc is considered
as free-flow state. One of the most widely-used functional form for an FD is triangular*2

It should be noted that, a flow–density plot, namely, scatter plot that drawn with observed traﬃc state by

traﬃc detector with fixed time interval, looks like an FD but is not an FD. It is because such flow–density
plot can include non-equilibrium states (e.g., the observation duration includes both free-flow state andcongested state). The wide scatter often found in flow–density plot is mainly because of this issue; if such
non-equilibrium states are removed, an actual data-based flow–density plot will be well-defined bivariately(Cassidy, 1998; Coifman, 2014). Nevertheless, actual FD may have scatter; it is mentioned in later.

9

Chapter 2

Literature Review(kc , qc )Flow qFlow q(u, qc )u(κ, 0)(u, 0)
Speed v

(1/kc , u)(0, u)(kc , u)Speed v

Speed v

Density k(κ, 0)

u/(1/kc − 1/κ)
(1/κ, 0)Density k

Spacing s

Figure 2.3: Example of triangular FD in four coordinate systems

function (Newell, 1993a), which simplify an FD as piecewise linear function with twolines in flow–density plane, as follows:




 uk,
q=


 − ukc k +κ−kc

0 ≤ k < kc ,u,1/kc −1/κkc ≤ k ≤ κ,

(2.8)

where qc is maximum flow, kc is critical density, and κ is jam density. For simplicity
purpose, backward wave ukc /(κ − kc ) is sometimes noted using w, and u/(1/kc − 1/κ) isnoted using 1/τ. Fig 2.3 illustrates a triangular FD with four coordinate systems, namely,
q–k, k–v, q–v and s–v (they are equivalent). The blue lines represent free-flow state andthe red lines represent congested state.
Although several studies have investigated FDs characteristics from theoretical,
analytical and empirical perspective, there are no complete theory and explanation for
10Chapter 2

Literature Review

q
qc ?

κ?

kc ?

k

Figure 2.4: Example of realistic FD.

determining an FD; there are only approximations*3 .

This is due to that an FD

depends on, ultimately, behavior of drivers whom the traﬃc consists of. For example,
Fig 2.4 illustrates an FD often found in empirical data (Edie, 1961; Treiterer and Myers,1974). The scatter in congested area is referred to hysteresis phenomenon, which is aphenomenon where trajectories of a transition from lower density to higher and that fromhigher to lower draw diﬀerent curves. Its reason is considered to heterogeneity in vehicle’s
driving style for acceleration and deceleration (Newell, 1962; Treiterer and Myers, 1974;
Laval, 2011). The variety in capacity is referred to capacity drop, which is a phenomenon
where capacity decreases when free-flow state turns into congested state. Its reason isconsidered to eﬀect from upstream (Edie, 1961), lane-changing (Daganzo, 2002) andeﬀect of a platoon (Shiomi et al., 2011). Heterogeneity in vehicle length also aﬀects
to above phenomena (Coifman, 2015).

2.1.3

Traﬃc flow model

Representing and modeling dynamics of traﬃc is important. The modeling approaches
can be roughly divided by two: microscopic and macroscopic. In microscopic models,
individual vehicle’s behavior is disaggregately and explicitly modeled. On the other hand,in macroscopic models, vehicles are approximated as continuous fluid; and behavior of*3

By employing an approximated functional form, an FD can be estimated empirically based on

suﬃciently high resolution traﬃc data (e.g., Chiabaut et al., 2009; Qu et al., 2015).

11

Chapter 2

Literature Review

the fluid is aggregately and explicitly modeled.
(1)

Microscopic approach

It is natural that traﬃc is described from a point of view of a vehicle individually. Carfollowing models (Pipes, 1953; Kometani and Sasaki, 1958, 1961; Gazis et al., 1961)
describe one-dimensional (i.e., longitudinal) movement of individual vehicle behavior
by considering interaction among vehicles. In typical car-following models, a vehicle’s
speed are determined by considering distance to the leading vehicle and speed of it; forexample, the car-following model of Kometani and Sasaki (1958) is described asβẌ(n, t) = Ẋ(n − 1, t − tr ) − Ẋ(n, t − tr )(2.9)

where β is constant coeﬃcient, and tr is a reaction time (of vehicle n). Newell (2002)
connected this model and macroscopic traﬃc flow model, such as capacity, which is
described in later.
Lane-changing models (Gipps, 1986) describe lane-changing (i.e., lateral) movementof individual vehicle behavior by considering interaction among vehicles. In typical lanechanging models, a vehicle determines its lane by considering its desired speed, safetyand destination.
Microscopic vehicle behavior (i.e., combination of car-following and lane-changing
behavior) has been modeled by using diverse manners, such as gap acceptance decision(Ashton, 1971), behavioral models (Gipps, 1981, 1986), game theory (Kita, 1999) andMarkov chained (i.e., dynamic) latent class models (Choudhury and Ben-Akiva, 2013;
Yanagihara et al., 2015); because it should be determined considering a lot of objectives
and constrains that relate with human behavior, vehicle performance and road alignment.
These models are mainly utilized for investigating traﬃc safety (Kometani and Sasaki,1958), traﬃc at merging/diverging section and intersection (Sarvi and Kuwahara, 2007),
capacity drop due to platoon (Shiomi et al., 2011), and eﬀects of introducing new typesof vehicles such as ADAS (Kikuchi et al., 2003; Talebpour and Mahmassani, 2015)—inother words, where heterogeneity among drivers significantly aﬀects the results and is the12Chapter 2

Literature Review

main interest. Their reproductivity in macroscopic scale is not always guaranteed (is notalways required), mainly due to their too many parameters, diﬃculty to be calibrated andmathematical intractability.
(2)

Macroscopic approach

From macroscopic point of view, traﬃc can be approximated as continuous fluid whose
flow, density and speed correspond to the original traﬃc state.
The Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham, 1955;
Richards, 1956), also known as first order traﬃc flow model and kinematic wave theory,is the simplest and yet suﬃcient model that describes traﬃc dynamics. The LWR model
employs two principles: an FD and a CL. In the LWR model, a traﬃc state is assumed toalways equilibrates*4 . The LWR model can be described as following equations:q(t, x) = Q(k(t, x), t, x),(2.10)∂q(t, x) ∂k(t, x)
+
= 0,
∂x
∂t

(2.11)

where eq (2.10) represents an FD, eq (2.11) represents a CL, and Q(k, t, x) is afundamental diagram at time–space point (t, x).
One of the objectives of studying macroscopic traﬃc flow models is to determine thecumulative flow in the time–space area of interest based on given boundary conditions.
Therefore, several numerical computation methods for solving the LWR model have been
proposed. Numerical computation with time–space discretization (i.e., t–x coordinate
system) is a typical approach for solving the LWR model (Daganzo, 1994; Lebacque,1996). In this approach, the initial condition is the density at a certain time in the roadsections and boundary condition is in/outflow to/from the road sections. Other studies*4

It implies that the vehicle’s speed can change instantaneously. It can be flaw in representing realistic

traﬃc phenomena, such as hysteresis phenomena and capacity drop. Some studies developed more complex
traﬃc flow models, such as the second order model (Payne, 1971). However, they may produces some unpreferable phenomena as traﬃc (Daganzo, 1995); and their tractability may be low. Other studies addressed
this issue by modifying the LWR model (Leclercq et al., 2011; Srivastava et al., 2015; Delpiano et al., 2015).

13

Chapter 2

Literature Review

introduced Lagrangian coordinates (i.e., n–t and n–x coordinate system) for solving the
LWR model (Leclercq et al., 2007; Laval and Leclercq, 2013). Lagrangian coordinateshave following advantages and disadvantages: they can explicitly use vehicle trajectories;
however, special treatments are required for dealing non-FIFO situation.
The LWR model can be simplified by assuming that the triangular and constant FD
shown in eq (2.8), which have three variables u, kc and κ (and two auxiliary variables
w and τ for concise notation). With this simplification, solving the LWR model basedon boundary conditions concludes a minimization problem of two quantities that can becalculated from the boundary cumulative flow, the free-flow speed and the wave speed(Newell, 1993a,b,c). It is referred as Newell’s simplified kinematic wave model andNewell’s three detector problem. It can be represented as follows:
)
(
)
}
{ (
xd − xm
xm − xu, xu , N t −, xd + κ(xd − xm ) ,N(t, xm ) = min N t −uw(2.12)

where xm is a position of a detector whose cumulative flow is to be calculated, xu is aposition of an upstream detector, and xd is a position of a downstream detector. If the firstterm in min{·} in eq (2.12) is smaller than or equal to the second one, the traﬃc between
xu and xm is in free-flow state. Otherwise, it is partially congested and therefore the traﬃc
between xd and xm is in congested state.
The cell transmission model (CTM; Daganzo, 1994) is a well-known computation
method for the simplified LWR model based on time–space discretization (i.e., theGodunov scheme; Lebacque, 1996). It is described as follows:
fiin (t) = min{}w
Ni−1 (t), (lκ − Ni (t)) ,
u

in
(t),
Ni (t + ∆t) = Ni (t) + fiin (t) − fi+1

(2.13a)(2.13b)where fiin (t) is number of vehicles that enter cell i during timestep t, Ni (t) is number ofvehicles in cell i at timestep t (it can be equal to N(t, il)), ∆t is width of a timestep, andl is length of a cell which must be equal to u∆t in order to avoid numerical errors. Eq(2.13a) determines flow based on state of cell i and its upstream cell i − 1. If the first termin min{·} in eq (2.13a) is smaller than or equal to the second one, cell i during timestep
14Chapter 2

Literature Review

t is in free-flow state; and all the vehicles move with the free-flow speed. if not, it is incongested state. Eq (2.13b) represents transmission of vehicles between adjacent cells.
A car-following model that is equivalent to the simplified LWR model was
also proposed (Newell’s simplified car-following model;

Newell, 2002; Daganzo,2006)—obviously, it can be utilized for computing the model from a microscopic pointof view. It can be represented as follows:


1


 X(n − 1, t − τ) − κ , if n is following n − 1
X(n, t) =  X(n, t − ∆t) + u∆t, otherwise{}
1= min X(n − 1, t − τ) − , X(n, t − ∆t) + u∆t .
κ

(2.14a)(2.14b)In eq (2.14a), “n is following n − 1” means that traﬃc is congested or saturated. As
connection with the microscopic models, it showed that relation 0 ≤ tr ≤ τ must be
satisfied between an FD and car-following models’ reaction time; and tr = τ/2 wassuggested for usual conditions.
The variational theory (Daganzo, 2005a,b) is eﬃcient computation methodology
using the variational method, based on the idea of Newell’s three detectors problem,with rigorous mathematical proofs. Then above mentioned calculation methods weregeneralized for the three coordinate systems including Lagrangian ones (i.e., t–x, n–t and
n–x coordinate systems), by applying the theory of Hamilton–Jacobi partial diﬀerential
equations and the idea of the variational theory (Laval and Leclercq, 2013). It can besummarized as follows:
N-model calculates quantity N at point (t, x):
{
}
(
x )
N(t, x) = min min{N(0, y) + ukc t − kc (x − y)}, N t − , 0
y∈B
u
{
}
θ−i
≃ min N(t − ∆t, x − i∆x) + ∆x,i=−1,0,...,θ
θ+1(2.15a)(2.15b)where ∆x = ∆n/κ and ∆t = τ∆n,
T-model calculates quantity T at point (n, x):
{
(
}
x
n)
T (n, x) = max T (n, 0) + , T 0, x +
+ τn
u
κ
15

(2.16a)Chapter 2

Literature Review≃ max{T (n, x − ∆x) + ∆t, T (n − ∆n, x + ∆x) + θ∆t},(2.16b)where ∆t = ∆x/u and ∆x = ∆n/κ,
X-model calculates quantity X at point (n, t):
}
{
{
}
n
n−y
, X (0, t − τn) −X(n, t) = min min X(y, 0) + ut −y∈Bkcκ≃ min{X(n, t − ∆t) + θ∆x, X(n − ∆n, t − ∆t) − ∆x},(2.17a)(2.17b)where ∆n = κ∆x and ∆t = ∆x/w,
where B is a boundary (for initial condition) curve of the area of interest, and θ = u/w
is required to be an integer. In eqs (2.15)–(2.17) sub-equations (a) represent generalsolution; and sub-equations (b) represent corresponding discretized solution (i.e., cellular
automata). The Newell’s three detector problem and CTM can be regarded as special
cases of N-model. The variational theory can be regarded as a special N-model thatcan eﬃciently calculate N at arbitrarily (t, x); therefore, it has capability of dealing
vehicle trajectories. The Newell’s simplified car-following model and some of the existing
Lagrangian models can be regarded as special cases of X-model.
More advanced traﬃc flow models such as that dealing with multi-lane (Daganzo,
2002; Laval and Daganzo, 2006; Zheng, 2014; Shiomi et al., 2015), multi-class (Daganzo,
2002; Wageningen-Kessels et al., 2010), stochastic phenomena (Sumalee et al., 2011) andcapacity drop (Leclercq et al., 2011) have been developed.

2.1.4

Area-wide traﬃc: Macroscopic fundamental diagram

Recently, the concept of an FD is expanded to an area-wide region that includes
non-stationary states (Geroliminis and Daganzo, 2008); it is referred to macroscopic
fundamental diagram (MFD). An MFD is relation between area-flow and area-density
which are defined using Edie’s generalized definition. It is conjectured that, under “certain
conditions”, following features exist in area-wide traﬃc:• area-wide traﬃc is steady,16

Chapter 2

Literature Review

Area-flow(kc , qc )FDu(κ, 0)
Area-density
Figure 2.5: Example of well-defined MFD.

• relation between area-flow and area-density is clearly bivariate (except hysteresis
phenomena mentioned by below), namely, a convex curve with a peak just like anFD,• hysteresis phenomena exist just like an FD, and• flow at an area’s perimeter correlates to its area-flow.
It is referred to “well-defined MFD”. These characteristics are expected to enable traﬃccontrol measures that require less information and operation from theoretical and practicalperspectives (Daganzo, 2007; Daganzo et al., 2011, 2012). Fig 2.5 illustrates an example
of well-defined MFD (solid curve) along with an FD (gray dashed curve). The red curve
represents gridlock state, where eﬀect of queue spillover is significant. Note that relation
between the MFD and FD in Fig 2.5 is just a conjecture.
The flaw is that the “certain conditions” for a well-defined MFD are not clarified yet,although a lot of applications that assumed existence of well-defined MFDs are proposed.
Some of the “certain conditions” were analytically proven, such as homogeneous road17

Chapter 2

Literature Review

with signals without in/outflow (Daganzo and Geroliminis, 2008), that with stochastic(heterogeneous) schematics and controls (Laval et al., 2015) and two link networkwith route choice (Leclercq and Geroliminis, 2013; Jin et al., 2013). Other conditions
were empirically guessed, such as some-sort-of-homogeneous network (Geroliminis andDaganzo, 2008; Geroliminis and Sun, 2011) and one-to-many network under dynamic
user equilibrium (和田・佐津川, 2014). Meanwhile, existence of steady network traﬃcand therefore steady area-wide traﬃc were proven under steady traﬃc demands, supplies
and route choice principles (Jin, 2015).

2.2

Traﬃc data collection and traﬃc state estimation

2.2.1

Definition

As a preparation for reviewing existing studies, definitions of traﬃc data, traﬃc datacollection and traﬃc state estimation in this dissertation and their basic characteristics aredescribed.
(1)

Traﬃc dataTraﬃc data are defined as measured cumulative flow in a certain time–space regions withappropriate origins in this dissertation. If measurement errors were ignored, the definition
can be represented as{}
D(A) = {N(t, x) − N(t0 (A), x0 (A)) | (t, x) ∈ A} | A ∈ A(2.18)

where D(A) is a traﬃc dataset in time–space regions A, A is a set of time–space points
included in A, N is a global cumulative flow and (t0 (A), x0 (A)) is a time–space point
representing an origin of A (i.e., reference point for local cumulative flow in A). Notethat A can be closed time–space region, multiple closed region and discrete points. Thisdefinition includes common traﬃc information such as: vehicle flow counts at a fixed
position, traﬃc volume at a fixed position in a time interval, a traﬃc state with a time–
space interval, a vehicle trajectory, and actual travel time between two positions with
18

Chapter 2

Literature Review

starting time. Fig 2.6 visualizes traﬃc data on time–space diagram, where red lines/dots
represent measured integer N, red filled areas represent measuring range of A (where Ñ
is measured), red dotted lines represent connection among regions in one A (they cannot
measure N), and dashed lines represent unobserved vehicles. The details on Fig 2.6 will
be described in later.
Traﬃc data with measurement errors can be represented as{
D′ (A) = {N ′ (t′ , x′ ) − N ′ (t0′ (A), x0′ (A)) |
(x)
(N)
′
′
t′ = t + ε(t)t,x,A , x = x + εt,x,A , N (t, x) = N(t, x) + εt,x,A ,(x)′t0′ (A) = t0 (A) + ε(t)t0 ,x0 ,A , x0 (A) = x0 (A) + εt0 ,x0 ,A , (t, x) ∈ A} | A ∈ A}(2.19)

where




ε(t)

t,x,A ∼ pt (t, x, A),



 (x)
εt,x,A ∼ p x (t, x, A),







 ε(N) ∼ pN (t, x, A),t,x,Aand pt (t, x, A), p x (t, x, A) and pN (t, x, A) are probabilistic distributions of measurement
errors on t, x and N, respectively, at (t, x) and A.
Traﬃc data are required for understating, planning and managing traﬃc andtransportation systems. In general, the suﬃcient characteristics of a traﬃc data collection
method and a TSE method can be described as follows:
1. any two of the traﬃc state’s variables can be simultaneously acquired,
2. a spatially and/or temporally wide-ranging area can be covered,
3. spatial and/or temporal detailed information can be acquired, and
4. accurate information can be acquired.
Using expressions in eq (2.18), above characteristics can be represented as follows:
1. the following is satisfied:• A covers multiple N values (i.e., is not a point nor a line with constant N
value), and19

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Space xSpace xSpace xTime t(a) Ground truth

A2

A2

A1

A1

Time tTime t(b) Dual detectors/fixed cameras (c) Aggregated data from dual
detectors/fixed cameras

Space xSpace xSpace x

A1

A2
A1

A1
A2

A4

A3
Time t(d) Signal timing

Time tTime t(e) Probe vehicles trajectories (f) Anonymized probe vehicle
dataSpace xSpace x

A1

Space x

A1

A1

A2

A2

Time t(g) AVI

Time t(h) A moving observer

Time t(i) Probe vehicles with spacing
measurement equipment

Figure 2.6: Examples of various traﬃc data.

20

Chapter 2

Literature Review• ∂x/∂t at N is available,
2. diﬀerence between minimum t (and/or x) and maximum t (and/or x) in all A in A
is large,
3. the following is satisfied:• A is large region, or• a distance between regions (or points, lines) in A in A is small,and4. values of N in D are correct.
However, these characteristics are diﬃcult to achieve simultaneously because of cost andtechnological limitations. Purposes of utilizing data determine what levels of above
characteristics are suﬃcient. Thus, diﬀerent data collection and/or TSE methods areapplied to specific purposes based on their feasibility and information required.
(2)

Traﬃc data collection

A traﬃc data collection method is defined as a method that collects traﬃc data without
significant assumptions in this dissertation. Assumptions can be regarded as significant
if they cannot be derived from knowledge other than traﬃc flow theory. A traﬃc datacollection method can be represented as a method that collects D′ (A) directly. Therefore,characteristics of a traﬃc data collection methods depend on definition of collecting range
A, measurement errors pt , p x , pN , and operational costs.
(3)

Traﬃc state estimationTraﬃc data in unobserved area can be estimated from collected data in observed area—
it is referred to TSE. TSE can be defined as estimation of traﬃc states at unobserved
area and observed area based on data collected at observed area and, if any, exogenous
assumptions. This can be represented as follows:D̂(B) = F (D′ (A), θ)
21(2.20)

Chapter 2

Literature Review

where B is a set of time–space regions whose traﬃc states are estimated, D̂(B) is
estimated traﬃc data in B, F is a function representing a TSE method, θ is a set of
exogenous parameters, and D′ (A) is collected traﬃc dataset in area A. F and θ canbe regarded as exogenous assumptions for the TSE method. Usually, A ⊂ B is satisfied,meaning that traﬃc states in unobserved area are estimated. If D̂(A) = D′ (A) is satisfied,
the observed data are considered to be perfect; otherwise, the observed data may beconsidered as containing errors and corrected by the TSE method. Operational costs (i.e.,
computational complexity) of TSE methods can be important if their purposes are realtime estimation; otherwise, the costs can be ignored as they are relatively small comparedto that of data collection methods.
(4)Remarks

In order to simplify the discussion of this dissertation, following topics are not discussed
in detail.
• Traﬃc data collection can be regarded as a special case of TSE. This is becausethat data collection procedures require exogenous assumptions to obtain traﬃc statefrom sensors’ raw signal, even for the most straightforward traﬃc data collection
method (e.g., Coifman et al., 1998). The assumptions are distinguished dependingon their significance in this dissertation; however, it is diﬃcult to be quantified.
• Short-term future traﬃc prediction. It can be regarded as TSE whose B contains
future. It requires demand prediction in addition to standard procedures of TSE
methods. For this reason, some studies employed statistical or machine learning
approaches which are based on historical data (c.f., Vlahogianni et al., 2014).
• Travel time estimation/prediction. It is studied by numerous studies, since travel
time is important information for road user’s route choice. It can be regarded as a
special case of TSE, because travel time is corresponding to a vehicle trajectory:
actual travel time from x0 to x1 is defined as T (n, x1 ) − T (n, x0 ). Meanwhile, it canbe regarded as an application of TSE, since travel time can be estimated/predicted

22

Chapter 2

Literature Review

based on TSE results. For these reasons, some studies employed approaches similar
to TSE, such as traﬃc simulation and statistical/machine learning methods (c.f., Oh
et al., 2015; Mori et al., 2015).
• Supplemental information regarding TSE such as a confidence interval is sometimes derived in addition to D̂(B).

2.2.2

Traﬃc data collection methods

For the eﬃcient collection of traﬃc data, various methods have been proposed to date.
Especially, recent achievements in information and communications technology (ICT)have enabled various information to be acquired by a wide variety of data collection
methods.
(1)

Eulerian data

Eulerian data are based on t–x coordinate system and collected by sensors fixed in space.
In general, fixed data collection methods can simultaneously acquire all the traﬃc statevariables at their installed points.
Roadside traﬃc detectors are a typical equipment used for fixed data collection
method (c.f., Klein et al., 2006). A detector’s sensor detects that whether a vehicle
occupies the detector’s position or not by using electromagnetic eﬀect of a metal-madevehicle (i.e., inductive loop detector), ultrasonic sensor, etc.; therefore, it can measure
flow and time occupancy directly at sensor’s position. A detector with single sensor (i.e.,single detector) can derive density and speed by assuming a value of the vehicle length.
A detector that has two sensor with short space interval (i.e., dual detector) can derivedensity and speed by assuming steady speed in space between the sensors. Disadvantagesof detectors are operation cost and accuracy. Wide-ranging traﬃc data collection with
detectors is diﬃcult due to high cost of installing and maintaining detectors. Accuracyof detector can be low depending on situation—for example, loop detectors often miss or
double-count vehicles and ultrasonic detector’s sensitivity is aﬀected by air temperature.

23

Chapter 2

Literature Review

This issue requires careful maintenance of detectors and therefore increases the operation
cost.
Image data collected by (video) cameras can also be utilized as fixed data collection(Hilbert et al., 1980; Coifman et al., 1998), and provide spatially detailed informationnear the installed points. Image-based detectors do not require assumptions employedby the loop/ultrasonic detectors; however, they require well trained/calibrated patternrecognition machines. An extreme example of utilizing image data is use of aircrafts,
unmanned aerial vehicles (Coifman et al., 2006) and satellites (McCord et al., 2003);
however, they may not be feasible in ordinary civil uses due to their high operating (fuel)cost per observation duration.
Signal phase information can be regarded as Eulerian traﬃc data, since cumulative
flow at signal’s location must be constant during its red phases.
In general, using the expression in eq (2.18), the characteristics of Eulerian data canbe represented as follows. Time coverage of A is large; but space coverage of A is small.
If the data were disaggregated, A is defined as a line that is parallel with the time axis
or an Eulerian rectangle (i.e., two edges are parallel with the time axis and the other twoedges are parallel with the space axis; c.f., Fig 2.6b) whose space length is shorter thanthat of time. Sometimes the data are aggregated due to limitation of data handling ability;
in such cases, A is defined as points on a line that is parallel with the time axis or vertexesof Eulerian rectangles (c.f., Fig 2.6c). Signal phase information can be represented as aline whose position is the signal and the time is during red sign (Fig 2.6d), Space interval
between multiple A can be large. Deviations of pt and p x are neglectable. Deviation ofpN may or may not be small.
In conclusion, the fixed data collection methods can provide temporally detailed
information at their installed points. However, information that is spatially detailed andwide-ranging is diﬃcult to be acquired simultaneously; because the observable area ofthe methods is limited to the vicinity of the sensors’ installed positions.

24

Chapter 2(2)

Literature Review

Lagrangian data

Lagrangian data are based on n–t or n–x coordinate system and collected by sensors
moving along with vehicles.
A probe vehicle is defined as having an onboard data collection devices that collects
traﬃc data while the vehicle is in the traﬃc (Zito et al., 1995; Sanwal and Walrand,
1995; Murakami and Wagner, 1999). Probe vehicle data can be analyzed to acquirewide-ranging and spatiotemporally detailed information; because they are often driven
by travelers who range over the entire road network. Today, the most common probevehicles are the ones equipped with a Global Positioning System (GPS) device to identify
their position and speed. From the GPS-equipped probe vehicle data, we can identify
the probe vehicles’ origins, destinations and routes by overlaid on a road map. Oncethese information are identified, the probe vehicle data directly provide the speed inroad sections that the probe vehicles drove. Mobile communication devices have also
been utilized as probes for specific purposes, such as individual tracking (Asakura andHato, 2004; Asakura and Iryo, 2007). Usually, probe vehicles consists of ordinary
road users who are using position identification systems for their own purposes, suchas route navigation and fleet management (Liu et al., 2009). Therefore, additional cost
regarding the data collection is relatively low compared with the fixed data collection;
it only requires data transmission and storage costs. In addition, probe vehicles cancollect data while severe disasters such that fixed data collection cannot operate due toblackouts (Hara and Kuwahara, 2015). Probe vehicle data have become popular withadvances in ICT, and many studies have utilized them. With penetration rate of few
percentage on a highway, they can collect spatiotemporally detailed data compared withusual traﬃc detectors (Herrera et al., 2010). However, GPS-equipped probe vehicle data
do not provide the flow and density information, unless otherwise a PR is 100% or puttingstrong assumptions such as an FD and a PR. Unfortunately, a 100% PR may be diﬃcult
to be achieved due to privacy issues (Antoniou and Polydoropoulou, 2015).
Wardrop and Charlesworth (1954) proposed other type of probe vehicle, that is, a25

Chapter 2

Literature Review

moving observer method that estimates the speed and flow of predetermined routes byusing a single probe vehicle. The original method relied on manually counting the numberof vehicles that the probe vehicle overtakes, that overtake the probe vehicle, and thatpass in the opposite direction, along with the journey time of the probe vehicle. Special
equipments such as laser imaging detection and ranging (LIDAR) can automate thismethod (Redmill et al., 2011; Coifman and Hall, 2014). However, unfortunately, evenwith recent technological advancement, this method of estimating flow may be diﬃcult
to be automated and often miss-counts vehicles using common equipment, such as video
cameras; this may be because of diﬃculty of using background subtraction against traﬃcin other lanes where background also moves*5 .
Matching a vehicle (AVI: automatic vehicle (re-)identification) that observed at
multiple fixed sensors is sometimes possible. By using the AVI, not only traﬃc state
at sensors position but also OD pair, travel route and average travel speed of some ofthe vehicles can be acquired. Note that, an AVI system cannot always matches all thevehicles. For example, electromagnetic signature observed by loop detectors (Coifman,1998), license numbers and visual features observed by cameras (Asakura et al., 2000;
Sumalee et al., 2012) can be utilized. Fare collection system can match vehicles based on
the user IDs (Ohba et al., 1999; 上田ら, 2014).
In general, using the expression in eq (2.18), the characteristics of GPS-equippedprobe vehicle data can be represented as follows. Time and space coverage of A dependon probe vehicles’ movement. Shape of A is defined as a line that follows probe vehicle’s
trajectory (i.e., N(t, x) = const, (t, x) ∈ A; c.f., Fig 2.6e). In practical, probe vehicle ID
may be removed from the probe vehicle data due to privacy concerns; it can be represented
as Fig 2.6f. In addition, data from multiple probe vehicles are sometimes aggregatedinto single average speed datum due to limitation in data handling ability and further
privacy concerns; this aggregated data are no longer considered as traﬃc data defined in*5

Currently, the idea of the moving observer is employed as analogy for explaining derivation of traﬃcflow models (Lighthill and Whitham, 1955) and calculation of them (Daganzo, 2005a,b) apart from its
original motivation.

26

Chapter 2

Literature Review

this dissertation. Time and space interval between multiple A depend on probe vehicles’
penetration rate (PR). Deviation of pt is neglectable but that in p x may be not neglectable.
An error on N is zero if the road is single-lane (i.e., N ′ (t′ , x′ ) = N ′ (t0′ (A), x0′ (A)) is always
satisfied). An AVI can be visualized as Fig 2.6g and the moving observer method can bevisualized as Fig 2.6h.
In conclusion, the probe vehicle data can provide spatiotemporally wide-ranging anddetailed information if the PR is high enough. However, the widely used probe vehicle
data, namely, GPS-equipped ones, cannot acquire quantity-related information, namely,
flow and density.

2.2.3

TSE methods

Following summarizes some of the existing TSE studies. First, direct utilizations of thetraﬃc flow theory are mentioned. Then, TSE methods considering practical problems,
namely traﬃc flow theory-based and statistical/machine learning-based method, aredescribed.
(1)

Deterministic theory-based methods

According to the definition of TSE, solving the LWR model based on given initial/boundary conditions can be TSE. Such TSE methods can be regarded as deterministic
theory-based; because they assume their theories and collected data are perfect andwell-posed. In other words, using expressions in eq (2.20), deterministic theory-based
methods can be represented as follows: F represents a computation method for a traﬃcflow model, θ represents the model parameters, D′ (A) represents boundary condition forthe model, and B represents area inside A.
Following methods described in section 2.1.3 can be regarded as F of deterministic
theory-based TSE methods, although they did not limit their motives to TSE: Newell’s
simplified kinematic wave model (referred to Newell’s three detector problem; Newell,
1993a,b,c), Eulerian coordinates-based (Godunov scheme) calculation methods of the27

Chapter 2

Literature Review

LWR model (Daganzo, 1994; Lebacque, 1996), the variational theory (Daganzo,
2005a,b), and Lagrangian coordinates-based calculation methods of the LWR model(Newell, 2002; Laval and Leclercq, 2013). θ includes parameters of an FD for above
methods.
Deterministic theory-based TSE methods may have limitations.

Theories only

represent abstracted features of actual phenomena, meaning that F and θ do not always
hold true in actual phenomena. Especially in traﬃc flow theory, attributes of an FD
are uncertain for general traﬃc, as mentioned in section 2.1.2. Even if the theories
were perfect, following issues remain: F may not be perfect (c.f., numerical errors,
non-closed-form expressions); and D(A) contain measurement errors.
(2)

Practical traﬃc flow theory-based methods

Some TSE methods are based on traﬃc flow theory as well as collected data and considers
discrepancies between theories and actual traﬃc data appropriately. Basically, they
introduce procedures that can handle discrepancies between theories and actual traﬃcdata, by introducing such as stochastic factors (then apply maximum likelihood method)
and residual minimization procedures. In these methods, definitions of terms in eq (2.20)
are varied. Following describes features of existing these TSE methods regarding to theirdefinitions of D̂(B), D′ (A), F and θ.
On D̂(B), namely, variables to be estimated and areas of interest of TSE. Usually,
detailed aggregated data, such as minutes–hundreds meters scale traﬃc states, areestimated (e.g., Wang and Papageorgiou, 2005). Disaggregated data are sometimes
estimated (e.g., Yuan et al., 2012). Network-wide TSE methods and area-wide TSE
methods are also developed, such as missing link complementation based on CLs (Castillo
et al., 2013) and area-flow estimation assuming that a well-defined MFD exists (Gayahand Dixit, 2013).
On D′ (A), namely, input data for TSE. Eulerian data are often utilized for the input
data (e.g., Wang and Papageorgiou, 2005). Lagrangian data are also utilized for newer
TSE methods, with recent GPS-equipped probe vehicles’ emergence (e.g., Nanthawichit
28

Chapter 2

Literature Review

et al., 2003); however, it should be noted that most of them still uses Eulerian data (e.g.,Nanthawichit et al., 2003; Herrera and Bayen, 2010; Mehran et al., 2012; Yuan et al.,2012) as boundary conditions. At signalized intersections, signal timing information canbe utilized for input data, since flow at red signal is known to be zero (Ban et al., 2011;
Mehran et al., 2012; Comert, 2013; Hao et al., 2013; Sun and Ban, 2013; Lee et al., 2015;
Wada et al., 2015). If an estimation method is based on Eulerian coordinate system,
aggregated data are utilized (Nanthawichit et al., 2003; Wang and Papageorgiou, 2005;
Herrera and Bayen, 2010). Disaggregated data are also utilized in order to reproduce
detailed traﬃc phenomena from vehicles trajectories using Lagrangian coordinates (Yuanet al., 2012; Mehran et al., 2012; Wada et al., 2015). Utilizing disaggregated data may
require special treatment on non-FIFO conditions (e.g., a probe vehicle overtaking otherprobe vehicle will violate the coordinate system; Mehran et al., 2012). For example, Yuan
et al. (2012) avoided this issue by reproducing disaggregate data from data aggregatedfrom raw disaggregated data; Mehran et al. (2012) and Wada et al. (2015) assumed that
overtaking never happens.
On F and θ, namely, methodology of TSE. Basically, the methodology introduces
stochastic factors based on traﬃc flow theory and then applies maximum likelihood
method or residual minimization procedures to estimate traﬃc states.
The most typical methodology is Kalman filtering-like techniques (KFT), such as
Kalman filter (Kalman and Bucy, 1961), extended Kalman filter, ensemble Kalman filter,
particle filter, etc. It may be due to their capability to integrate traﬃc flow theory and fuseheterogeneous traﬃc data within a TSE method. General state-space model for KFT canbe represented as follows:
D(Bt ) = St (D(Bt−1 ), Vt ),(2.21)

D′ (At ) = Ht (D(Bt ), Wt ),(2.22)

where eq (2.21) is a system equation, eq (2.22) is an observation equation, Bt is a partof B those belong to time t, At is a part of A those belong to time t, St is a systemmodel, Ht is an observation model, Vt is a system noise, and Wt is an observation
29

Chapter 2

Literature Review

noise, at time t, respectively. The objective of KFT-based TSE is to estimate D(B)based on D′ (A), St , Ht , Vt and Wt ; the estimator is D̂(B). St corresponds to a
traﬃc flow model; and D′ (At ), Ht and Wt correspond to a data collection method. The
disadvantage are such as numerical errors, computation costs and calibration diﬃculties
due to linear approximation, numerical diﬀerential, Monte Caro simulation and/or ad
hoc parameters. Various combinations of observation and system models have been
investigated by existing studies. They can be summarized as follows: aggregated detector
data and random walks (Gazis and Knapp, 1971), aggregated speed data of probe vehicles,
detector data and second order traﬃc flow model (Nanthawichit et al., 2003), freeway
detector data and second order traﬃc flow model (Wang and Papageorgiou, 2005), stateand parameters estimation using detector data and first order traﬃc flow model (Lintet al., 2008), state and parameters estimation using detector data and second order traﬃcflow model (Wang et al., 2009), various heterogeneous data from detector, probe vehicle
and AVI (van Lint and Hoogendoorn, 2010), aggregated speed data of probe vehicles,
boundary detector data and speed-based traﬃc flow model (Herrera and Bayen, 2010),
and disaggregated probe vehicle data, boundary detector data and Lagrangian-coordinatedfirst order traﬃc flow model (Yuan et al., 2012). van Lint and Hoogendoorn (2010);
El Faouzi et al. (2011) provided reviews including KFT-based TSE methods.
Other methodologies are also employed. For example, the deterministic theory-based
methods were modified into practical TSE methods that can handle discrepancies between
theories and actual traﬃc data (Muñoz et al., 2003; Sumalee et al., 2011; Mehran et al.,2012; Laval et al., 2012; Deng et al., 2013; Lei and Zhou, 2014; Wada et al., 2015;
Sun et al., 2015), using maximum likelihood method, residual minimization, etc. Naı̈ve
extrapolation method with less assumption on traﬃc flow theory is also useful in somecases (Coifman, 2002).
The concept of an FD (or MFD sometimes) has a significant role in F (as well as St in
KFT) and therefore the TSE method. Most existing TSE methods assume conditions on
an FD (e.g., its functional form and parameters) exogenously*6 . However, as mentioned
in section 2.1.2, an FD is a complicated phenomenon that involves various factors and30

Chapter 2

Literature Review

cannot be described or predicted completely. Therefore, they usually requires carefulparameter calibration prior to TSE. Furthermore, in principle, this calibration must require
knowledge on traﬃc states in B—it can be a vicious circle.
(3)

Statistical/machine learning-based methods

Statistical methods and machine learning methods are also utilized for TSE. Basically,
they estimate traﬃc state based on collected current traﬃc data and some relation (e.g.,
spatial and/or temporal correlation among data) abstracted from stored historical traﬃcdata. Therefore, F represents a method for abstracting relation or a traﬃc state calculation
method based on abstracted relation; θ represents parameters of an abstracting method orabstracted relation. Methods for abstracting the relation are varied. Data required for the
methods are also varied, as long as the statistical methods and machine learning methods
allow, including both of Eulerian data and Lagrangian data.
These methods are often utilized for short-term future prediction; because theabstracted relation may include changes in demand in short-term future without explicit
traveler behavior model (c.f., Karlaftis and Vlahogianni, 2011; Vlahogianni et al., 2014).
In short-term future prediction, A represents current and past at a certain space; and B
represents future at the same space.
They are also utilized for missing link complement problems (花岡ら, 2013; Kataoka
et al., 2015). For example, probe vehicles cannot always collect data from every links
in a network; therefore, traﬃc states of such links are required to be estimated. In these
problems, A represents links with probe vehicles at a time; and B represents links without
probe vehicles at the same time.
The shortcoming is that they require long term historical data for abstracting therelation and assume the relation always holds true. It implies that these methods are valid
only under ordinal conditions; because extraordinary data, in which the ordinal relation
does not hold true, are diﬃcult to be collected*7 . In addition, they can be black box and*6

Under special conditions (e.g., detector observes flow and density), the parameters can be estimatedalong with traﬃc states by using special procedures (Lint et al., 2008; Wang et al., 2009; Sun et al., 2015).

31

Chapter 2

Literature Review

case-specific which sometimes contradict traﬃc flow theory.

2.3

Advanced vehicle technologies

2.3.1

Overview

Recently, advanced vehicle technologies (AVT) have began spread in the practical uses.
The AVT includes automated vehicles and connected vehicles.
The automated vehicles includes functions, such as adaptive cruise control, collision
avoidance and autonomous vehicles (Bengler et al., 2014). Several technologies for the
automated vehicles have been developed practically for the driving safety and eﬃciency
purposes (National Highway Traﬃc Safety Administration, 2013; Diakaki et al., 2015).
To realize automated vehicles, the surrounding environment of a vehicle must berecognized using on-vehicle equipments with high accuracy and frequency (Rajamaniand Shladover, 2001; Sivaraman and Trivedi, 2013). The typical and essential factorof the surrounding environment of a vehicle is a spacing, namely, distance between
a vehicle and its leading vehicle. In fact, spacing measuring technology is not very
new development; it have been studied from mid 1990s (Saneyoshi, 1996; Bertozzi and
Broggi, 1998) and achieves high performance even with commercialized equipments.
For example, Saneyoshi (1996), which utilized stereo camera and was commercialized
recently, reported that the precision is within ±0.01 m and ±1 m at distances of 5 m and100 m, respectively; and the maximum detectable range is 130 m.
The connected vehicles establishes bidirectional information connection among
nearby vehicles (i.e., VANET: vehicular ad hoc network) using on-vehicle equipment
and/or road-based infrastructure (Hartenstein and Laberteaux, 2008).

Basically, it

intends to enable eﬃcient information exchange so that driving can be eﬃcient, suchas information provision and cooperative vehicle control combined with the automated*7

The characteristics can be utilized for detecting extraordinary conditions themselves by comparing

predicted traﬃc state and revealed actual traﬃc state (成岡ら, 2015).

32

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Literature Review

technologies.
Eﬀects of AVT for traﬃc and transportation are investigated and optimistic results
are often obtained, such as improvements on safety and capacity in both of microscopic
and macroscopic scales (Kikuchi et al., 2003; Talebpour and Mahmassani, 2015; Zhang,2015; Diakaki et al., 2015).

2.3.2

Utilization for TSE-like purposes

Some of the data collected by AVT satisfy the definition of traﬃc data in this dissertation.
For example, spacing data collected by automated vehicles are represented as traﬃc data
whose A containing two values of N, namely, the automated vehicle itself and its leading
vehicle. This is the probe vehicle with spacing measurement equipment (PVSME). It canbe visualized as Fig 2.6i.
In fact, several studies investigated utilization of such AVT data to TSE-like purposes.
For example, capacity of road sections can be measured continuously using a vehicle
with spacing measurement equipment (小谷ら, 2003). Image recognition techniques thatmeasure spacing in order to quantity-related variable estimation were developed (Rabieet al., 2002; Panmungmee et al., 2012; Yokoi et al., 2013); however, their estimation
outputs as well as estimation methods are not specified. Some studies (Messelodi et al.,2009; Diaz et al., 2012) called data collected by spacing measurement and visible vehicle
counting functions “extended Floating Car Data (xFCD)”. An estimation method for
occupancy was developed based on statistical relation between the xFCD and detector
data (Wilby et al., 2014). Data from connected vehicles can be utilized for traﬃc density
estimation with some assumptions (Goodall et al., 2014; Darwish and Bakar, 2015).

33

Chapter 3 Basis: Method for Estimating Traﬃc
State
This chapter*1 describes the basic method for estimating traﬃc state from probe vehicle
data including position and spacing. The method estimates the flow, density, and speed
at a predetermined time and space resolution for a corridor that the probe vehicles drive
through. In order to keep it simple and easy to expansion, the method is designed as notrelying on significant exogenous assumptions.
Sections 3.1 and 3.2 describe premises for the estimation method.
formulates traﬃc state estimators based on the premises.

Section 3.3

Section 3.4 investigatesanalytical characteristics of the estimators by using approximation. Section 3.5 shows
numerical examples of the estimators. Section 3.6 summarizes the chapter and discusses
characteristics of the estimators.

3.1

Assumptions and concepts

The proposed estimation method is for traﬃc state in a corridor. The corridor is assumed
to be one-way traﬃc that can include multi-lane, merging, and diverging sections. Edie’s
generalized definitions (Edie, 1963) were employed for the traﬃc flow variables; thesedefine the traﬃc flow variables (i.e., flow, density, and speed) in a time–space region
based on the trajectories of all vehicles in the region and the area of the region. The
definitions indicate generalized forms of the well-known traﬃc variable definitions, suchas the spot flow rate, space density, and space mean speed.
The probe vehicle data are used as the inputs for the estimation. The data consist of
the following continuously measured variables:*1

This chapter is mainly based on joint research with Dr. Takahiko Kusakabe and Prof. Yasuo Asakura,

published in Transportation Research Part C (Seo et al., 2015a).

34

Chapter 3

Basis: Method for Estimating Traﬃc State• spacing of the probe vehicle (i.e., space headway between the probe vehicle and its
leading vehicle in the same lane) and• position of the probe vehicle.
To simplify the situation, the following conditions are assumed:• The measurements have no error.
• The route that the probe vehicle drives is identified without error.
• The probe vehicles are randomly distributed in the traﬃc; the probe vehicle
penetration rate (PR) is unknown to analysts.
• The driving behavior of the probe vehicles is the same as that of other vehicles.
The first and second assumptions indicate an error free assumption. The third and fourthassumptions indicate a random sampling assumption.
The traﬃc state estimation procedure is described as follows.
Step 1 Time–space region subject to the traﬃc state estimation is divided into multiple
discrete time–space regions.
Step 2 The traﬃc flow variables of every time–space region are estimated independently
based on data collected by the probe vehicles in each region.
Fig 3.1 shows a time–space diagram and time–space region, where the horizontally
hatched region A is a discrete time–space region, the thick solid lines are trajectories ofthe probe vehicles, the dashed lines are those of non-probe vehicles, the vertically hatchedregions sn are time–space regions between the n-th probe vehicle and its leading vehicle,
and the 5th, 8th and 12th vehicles are probes.
Any rule can be used to divide the time–space region of the traﬃc flow (step 1). The
simplest rules is employed in this method, where the traﬃc flow is divided into identicalsize of Eulerian rectangles. This is a familiar coordination in today’s traﬃc flow data,
where fixed-point detectors are installed at a certain time and space resolution or interval.
This coordination can be represented as follows:
Aij = {(x, t) | xi ≤ x ≤ xi+1 , t j ≤ t ≤ t j+1 }35

i ≥ 0, j ≥ 0,

(3.1)

Chapter 3

Basis: Method for Estimating Traﬃc StateSpace x

n=5

a 5(

A)

Region A
a 8(

n=8

A)

n = 12Time tFigure 3.1: Time–space region A and probe vehicle data.

xi+1 = xi + ∆x,(3.2)

t j+1 = t j + ∆t,(3.3)

where i and j are non-negative indices for space and time, respectively, (x0 , t0 ) is thecoordinates of a predetermined origin, (xi , t j ) is the coordinates of the lower-left cornerof the region Aij , ∆x is a predetermined space resolution, and ∆t is a predetermined time
resolution. Note that Aij is simply written as A in the rest of this dissertation for the
purpose of conciseness.
The estimation method of the step 2 is described in the next subsection.

3.2

Collectable data by PVSMEs

The PVSMEs are supposed to collect their position and spacing as mentioned. In thissection, they are defined in mathematical form.
A position is a coordinate in time–space plane. A vehicle trajectory consists ofcontinuous positions of a vehicle. Trajectory of vehicle m can be defined as X(m, t)

36

Chapter 3

Basis: Method for Estimating Traﬃc Stateand T (m, x), where X(m, t) represents position that vehicle m exists at time t and T (m, x)
represents time that vehicle m passes position x. Obviously, these two of the same mshould be equivalent to each other, and following relation holds true: x = X(m, T (m, x)).
A spacing is space distance between a position of a vehicle at a time and that of its
leading vehicle at the same time. Then, a spacing of vehicle m at time t can be defined as
X(l(m, t), t) − X(m, t), where l(m, t) represents a leading vehicle of vehicle m at time t. If
FIFO is satisfied, it can be simplified as X(m − 1, t) − X(m, t).
We define dm (A): distance traveled by vehicle m in region A, tm (A): time spent by
vehicle m in region A, and am (A): total spacing of vehicle m in region A, as follows:dm (A) = max{X(m, t) | (t, X(m, t)) ∈ A} − min{X(m, t) | (t, X(m, t)) ∈ A},(3.4)

tm (A) = max{T (m, x) | (T (m, x), x) ∈ A} − min{T (m, x) | (T (m, x), x) ∈ A},(3.5)

am (A) = {(t, x) ∈ A | X(m, t) < x < X(l(m, t), t)},(3.6)

If FIFO is satisfied, am (A) can be simplified as
am (A) = {(t, x) ∈ A | X(m, t) < x < X(m − 1, t)}.

(3.7)

In practical, data collected by a probe vehicle are discrete, e.g., GPS log with 1 stime interval. Therefore, to obtain dm (A), tm (A) and am (A), probe vehicle data must be
interpolated somehow. Note that precision of am (A) is significantly aﬀected by quality ofinterpolation, while those of dm (A) and tm (A) are not.

3.3

Estimators

The traﬃc flow variables defined by Edie (1963), namely, the flow q(A), density k(A),and speed v(A), in a time–space region A are as follows:
d(A)
,
|A|
t(A)
,
k(A) =|A|
d(A)v(A) =.
t(A)q(A) =

37

(3.8)
(3.9)
(3.10)

Chapter 3

Basis: Method for Estimating Traﬃc State

These definitions are easily transformed into eqs (3.11)–(3.13) by summing the spacingsand trajectories of each vehicle in the region A:∑n∈N(A) dn (A)q(A) = ∑,(3.11)n∈N(A) |an (A)|∑n∈N(A) tn (A)k(A) = ∑,(3.12)n∈N(A) |an (A)|∑n∈N(A) dn (A)v(A) = ∑(3.13),n∈N(A) tn (A)∑if n∈N(A) |an (A)| > 0, where an (A) is a time–space region between vehicle n and its
leading vehicle in the region A (an (A) = A ∩ sn , cross-hatched regions in Fig 3.1), N(A)is a set of all vehicles in the region A, dn (A) is the distance traveled by vehicle n in theregion A, and tn (A) is the time spent by vehicle n in the region A.
By replacing N(A) in eqs (3.11)–(3.13) with the set of all probe vehicles in region A,
namely, P(A), estimators based on the probe vehicle data can be defined as follows:∑n∈P(A) dn (A)q̂(A) = ∑,(3.14)n∈P(A) |an (A)|∑n∈P(A) tn (A)k̂(A) = ∑,(3.15)n∈P(A) |an (A)|∑n∈P(A) dn (A)v̂(A) = ∑.
(3.16)n∈P(A) tn (A)These estimators denote the flow, density, and speed of probe vehicles that works informal
moving blockade. Note that eqs (3.14)–(3.16) are exactly the same as eqs (3.11)–(3.13) if
all of the vehicles were the probe vehicles (P(A) = N(A)).

3.4

Analytical characteristics of estimators

It is valuable to investigate characteristics of the estimators in analytical way prior
to applied them to empirical data; because it gives general insights. The estimation
procedures can be described as random process of selecting probe vehicles with a certain
probability. For example, flow estimator (3.14) can be represented as∑n∈N(A) bn dn (A),q̂(A) = ∑n∈N(A) bn an (A)38(3.17)

Chapter 3

Basis: Method for Estimating Traﬃc Statebn ∼ B(P),(3.18)

where B(P) is Bernoulli distribution which is equal to 1 with probability P and 0
otherwise, and P is PR of probe vehicle. In general, indices for accuracy (i.e., bias)
and precision are defined as[]
Bias(q̂(A)) = E q̂(A) − q(A) ,√[]RMSE(q̂(A)) = E (q̂(A) − q(A))2 .

(3.19)(3.20)

where Bias(θ̂) is the bias, which is an accuracy index of an estimator θ̂, where RMSE(θ̂) is
the root mean square error, which is a precision index of an estimator θ̂. Unfortunately, eqs(3.17)–(3.20) are diﬃcult to be expressed in closed form. Therefore, following discusses
the bias and precision by using approximations that hold true in special cases, and Taylor
approximations.
The estimators represented in eqs (3.14) and (3.15) are not unbiased in general. With
third-order Taylor expansion, the bias in the flow and density estimators shown in eqs(3.14) and (3.15) can be approximated as:[]
1 Var h̄n (A)Bias(q̂(A)) ≃,|P(A)| E [h̄n (A)]3[]
1 Var s̄n (A),Bias(k̂(A)) ≃|P(A)| E [ s̄n (A)]3(3.21)(3.22)

where h̄n (A) and s̄n (A) are the mean time and space headway, respectively, of vehicle nin the region A (see section 3.4.1 for their derivation). According to these equations, aninverse correlation exists between the bias and number of probe vehicles in the region.
Therefore the bias tends to be mitigated as the size of the region is increased (i.e.,time/space resolution is lowered) or the PR is increased. The bias also correlates to thevariance of all of the vehicles’ mean time/space headway, which depends on the traﬃcflow characteristics. There is no bias in the speed estimator.
In a similar way, the precision of the flow and density estimators shown in eqs (3.14)
and (3.15) can be approximated as
RMSE(q̂(A)) ≃

√
1

[
]2
E h̄n (A)39[]
Var h̄n (A),|P(A)|(3.23)

Chapter 3

Basis: Method for Estimating Traﬃc State√
RMSE(k̂(A)) ≃1[]2
E s̄n (A)[]
Var s̄n (A),|P(A)|(3.24)(see section 3.4.2 for their derivation). The tendencies in the precision are quite similar
to those in the bias; namely, the precision of the flow increases as the number of probevehicles in a region increases and the variance in the mean time headway decreases.

3.4.1

Derivation of the approximated bias

The approximation of |an (A)| ≃ dn (A)h̄n (A) can be appropriate for most situations, such as
when the size of region A is suﬃciently larger than the spacing. With this approximation,
the true flow in eq (3.11) can be transformed to[]
E dn (A)q(A) ≃ [].
E dn (A)h̄n (A)(3.25)

We assumed that dn (A) and h̄n (A) are independent from each other. Therefore,q(A) ≃

1[].
E h̄n (A)

(3.26)In the same way, the flow estimator in eq (3.14) can be transformed toq̂(A) ≃

1[],
EP(A) h̄n (A)(3.27)[ ]where EP(A) θn is the mean of θn for each vehicle in P(A). The expected value of theestimator is[]E q̂(A) ≃ E

[

]
1
[
] .
EP(A) h̄n (A)(3.28)[ ]The third-order Taylor expansion of f (X) = 1/X around E X is derived as
[ ])
[ ])2
[ ])3
1
1 (
1
1 (
1 (
≃ [ ] − [ ]2 X − E X + [ ]3 X − E X − [ ]4 X − E X .
X E X
E X
E X
E X
We can then obtain
[
]
[
[
]]
Var EP(A) h̄n (A)
1
1
E
[
] ≃ [
[
]] + [[]]3 ,EP(A) h̄n (A)E EP(A) h̄n (A)E EP(A) h̄n (A)40(3.29)(3.30)

Chapter 3

Basis: Method for Estimating Traﬃc State[]by taking the expectations and substituting EP(A) h̄n (A) for X. If the probe vehicles
[
[
]]
[
]
are randomly sampled from all of the vehicles, E EP(A) h̄n (A) = E h̄n (A) and
[
[
]]
[
]
Var EP(A) h̄n (A) ≃ Var h̄n (A) /|P(A)| are satisfied. Therefore, eq (3.30) can betransformed to
[

]
[
]
1
1
1 Var h̄n (A)
E
.
[
] ≃ [
]+
|P(A)| E [h̄n (A)]3
EP(A) h̄n (A)E h̄n (A)

(3.31)

From eqs (3.26), (3.28), and (3.31), the relation between flow and its estimator can beapproximated as[]
E q̂(A) ≃ q(A) +

3.4.2

[
]
1 Var h̄n (A).
|P(A)| E [h̄n (A)]3(3.32)

Derivation of the approximated precision

The index for the precision, that is, root mean square error (RMSE), is represented asfollows:√[]
E (q(A) − q̂(A))2
√ (
)2 


1
1
≃ E  []−[] E h̄n (A)EP(A) h̄n (A)
v
t 



[
]




2
1
1


−
E
+
E
= E  [

[
]
[
]
]2 
[
]2 
E h̄n (A) EP(A) h̄n (A)E h̄n (A)EP(A) h̄n (A)
v
t


[
]


1
2
1
1

=
−
E
+
E

[
]
[
]
[
]2
[
]2 

E
h̄
(A)
E
h̄
(A)
E h̄n (A)EP(A) h̄n (A)n
P(A) n
v
t



[
]



2
1 Var h̄n (A) 
1
1
1



≃
+
E
]  []+
[]2 − [[][]
2
|P(A)| E h̄n (A) 3
E h̄n (A) E h̄n (A)E h̄n (A)EP(A) h̄n (A)
v
t 

[
]


2 Var h̄n (A)11= E 
(3.33)[]2  − []2 − |P(A)| []4 .
EP(A) h̄n (A)E h̄n (A)E h̄n (A)RMSE(q̂(A)) =[ ]The third-order Taylor expansion of f (X) = 1/X 2 around E X is derived as
[ ])
[ ])2
[ ])3
2 (
1
1
3 (
4 (
−
≃
X
−
E
X
+
X
−
E
X
−
X
−
E
X . (3.34)
[
]
[
]
[
]
[
]
4
5
X2 E X 2 E X 3
E X
E X
41

Chapter 3

Basis: Method for Estimating Traﬃc State

We can then obtain


[
[
]]


3Var EP(A) h̄n (A)
1
1
E 
(3.35)[]2  ≃ [
[
]]2 +
[
[
]]4 ,
EP(A) h̄n (A)E EP(A) h̄n (A)E EP(A) h̄n (A)[]by taking the expectations and substituting EP(A) h̄n (A) for X. If the probe vehicles are
randomly sampled from all of the vehicles, this can be transformed to


[
]


1
3 Var h̄n (A)
1


E 
[
]2  ≃ []2 + |P(A)| []4 .
EP(A) h̄n (A)E h̄n (A)E h̄n (A)From eqs (3.33) and (3.36), the precision can be approximated as√[]
Var h̄n (A)1
RMSE(q̂(A)) ≃ [.
]2|P(A)|
E h̄n (A)

3.5

(3.36)(3.37)

Numerical examples

In order to investigate general behavior of the estimators, and validity of the approximations for their accuracy/precision, numerical examples on the flow estimator is shown. In
the numerical examples, traﬃc is assumed to be steady as follows.
Distribution of mean time headway of a vehicle, h̄n (A), is assumed to follow theGamma distribution, which is often employed for representing the headway distribution(c.f., Katakura, 1971; Zhang et al., 2007), with parameters k = kh and θ = θh . Variable
dn (A) is also assumed to follow the Gamma distribution with parameters k = kd andθ = θd , and is independent of h̄n (A). Note that dn (A) is actually almost independent ofh̄n (A) where A has long time length comparing to the space length and the vehicle speed,
such that all the vehicles in A travel distance of the space length of A.
The probability density function of the Gamma distribution isf (x) = xk−1e−x/θ,Γ(k)θk

(3.38)

where k and θ are parameters of the distribution, and Γ is the Gamma function. The mean
and variance are kθ and kθ2 , respectively.
The parameters of the distributions (mean µ(·) and standard deviation σ(·)), the total
number of vehicles N(A), and the number of probe vehicles P(A) are given for numerical
42

Chapter 3

Basis: Method for Estimating Traﬃc State

Table 3.1: Parameters for numerical examplesµ(h̄n (A)) (s)σ(h̄n (A)) (s)µ(dn (A)) (m)σ(dn (A)) (m)

(a)

1.5

0.5

100

5

(b)

3

2

100

5

(c)

3

2

100

20

(d)

8

9

100

20

examples. If the mean of h̄n (A) was large, it means that the flow is low. If the varianceof h̄n (A) was large, it implies that the flow is in free-flow state. If the variance of dn (A)was large, it implies that the variance of speed is large. If N(A) was large, it implies thatsize of A is large, namely, the estimation resolution is low. If P(A) was large compared to
N(A), it means that PR is high.
The parameters of time headway distribution were adopted from field data describedby Katakura (1971), while that of speed were assumed. Three combination of the
distribution parameters values were employed as shown in Table 3.1, where (a) represents
near-saturated state, (b) represents free-flow state, (c) represents free-flow state with larger
speed variance and (d) represents sparse free-flow state. The value of N(A) is either 100
veh or 5000 veh, which represents high and low resolution, respectively. The P(A) is
selected from various values. Note that congested states are not validated in this section,
since mean headway variance in congested states can be significantly smaller than that offree-flow states so that the estimation accuracy/precision may be high.
Errors defined by eqs (3.17)–(3.20) are numerically computed using Monte Carlosimulation (i.e., simulated random sampling of bn where dn (A) and an (A) are given for
all n). Then, they are compared with the analytically approximated ones (3.21) and(3.23). Figs 3.2 and 3.3 show the results. Each subfigure shows relation between errors
and PR under specific state and resolution. Errors include numerical computed bias
and RMSE (blue lines), and analytically approximated ones (red lines). According to
the figure, the numerical computed errors decrease as PR increases, resolution lowers,
and variances in headway and speed lowers. The eﬀect of speed variance is relatively

43

Chapter 3

Basis: Method for Estimating Traﬃc State(a) Near-saturated state (2400 veh/h)(b) Free-flow state (1200 veh/h)(c) Free-flow state (1200 veh/h) with larger speed(d) Sparse free-flow state (450 veh/h)variance
Figure 3.2: Numerical examples of error of the flow estimator with high resolution.

insignificant. The tendencies of the numerical computed errors are the same to theanalytical approximated ones. In addition, the analytical approximated errors are fairly
close to the numerical computed ones under some conditions, such as in low resolution,
in dense state, for bias, etc. Therefore, they can be valid and useful for predicting theaccuracy/precision of the TSE method in prior to applications under such conditions.
On the other hand, the approximation is not appropriate under high penetration rate orsparse traﬃc flow conditions. It may be due to third order Taylor approximation and
[
[
]]
[
]
approximation Var EP(A) h̄n (A) ≃ Var h̄n (A) /|P(A)|.

44

Chapter 3

Basis: Method for Estimating Traﬃc State(a) Near-saturated state (2400 veh/h)(b) Free-flow state (1200 veh/h)(c) Free-flow state (1200 veh/h) with larger speed(d) Sparse free-flow state (450 veh/h)variance
Figure 3.3: Numerical examples of error of the flow estimator with low resolution.

3.6

Discussions

With the method we can estimate the flow, density, and speed depending only on probevehicle data. This characteristic is valuable for a TSE method because it makes wideranging state estimation easy, as noted in chapter 2. The method does not presume an
FD, which is often used to estimate the traﬃc state in existing studies (e.g., Nanthawichit
et al., 2003; Yuan et al., 2012; Mehran et al., 2012). Because of this independency froman FD, the results obtained with this method are robust against unpredictable or uncertain
factors of traﬃc flow, such as traﬃc incidents. In addition, the results can be utilized to45

Chapter 3

Basis: Method for Estimating Traﬃc StateSpace xSpace xSpace xTime t(a) Higher resolution

Time tTime t(b) Lower resolution(c) Area-wide resolution

Figure 3.4: Examples of A shapes.

estimate or calibrate traﬃc flow models, including FDs.
The resolution of the estimation (∆t and ∆x) can be adjusted to the requirements by theplanners and analysts. As shown in eqs (3.21)–(3.24) and section 3.5, there is a tradeoﬀ
between the estimation accuracy and precision and resolution, as well as a positive relation
between the accuracy and precision and PR. Therefore, analysts can choose a desired
resolution depending on the estimation purpose and the number of probe vehicles inorder to maintain the required accuracy and precision. Fig 3.4 illustrates examples ofthe shapes of A where the dashed lines represent each boundaries of As. If the numberof probe vehicles is suﬃciently large, a high resolution can be set—detailed and dynamicchanges in traﬃc states can be recognized (Fig 3.4a). In contrast, even if the number ofprobe vehicles is very small, analysts can still obtain some knowledge on the traﬃc bysetting a low resolution, such as an hourly traﬃc volume (Fig 3.4b) and area-flow/density(Fig 3.4c).
Section 3.5 shows numerical examples of the accuracy/precision depending on
traﬃc situation, estimation resolution and PR by assuming simple spacing and speed
distributions; and suggested naı̈ve benchmarks on the capability of the proposed method
and validity of the analytically approximated errors.

In order to investigate thesecharacteristics under more realistic traﬃc situation, in which speed and spacing are

46

Chapter 3

Basis: Method for Estimating Traﬃc Statedescribed by nonlinear traﬃc phenomena, they are quantitatively analyzed based on datacollected via simulation experiments and a field experiment in chapters 5 and 6
The proposed method relies on assumptions that may not always satisfied in thereal world, namely, the error free and random sampling assumptions. The error freeassumption can be reasonable except in situations with very sparse traﬃc where leading
vehicles are not often detectable. The reasons are as follows. Small errors in position
measurement*2 do not significantly aﬀect accuracy and precision of the proposed method.
If the size of the time–space region A is larger enough than expected error in position
measurement, errors in dn (A), tn (A) and an (A) can be neglectable. On the other hand,errors in spacing measurement directly aﬀect an (A) and therefore accuracy and precisionof the proposed method. However, errors in spacing measurement should be small
because practically implemented spacing measurement technologies have to provide good
features in order to guarantee the safety of the ADAS. Although the spacing measurement
requires assumption on vehicle length, this can be only a weak assumption; because
variation of vehicle length is expected to be relatively smaller comparing to that of thespacing.
Leading vehicles are not always detectable due to technical problems, such as
excessive distance. This “missing spacing” issue always exist, and is more significant atsparse traﬃc. This issue may arise over estimation if such data were discarded; therefore,
it cannot be neglected. For example, ratio of missing spacing under traﬃc condition
described in section 3.5, 80 km/h average speed, and maximum detectable range 130 m(Saneyoshi, 1996) is calculated as follows: (a) 0.0%, (b) 9.2%, (c) 9.6% and (d) 44.4%.
Therefore, for practical purposes, imputation methods for missing spacing are desirable*3 .

The random sampling assumption may not always satisfied in the real world. This
*2
*3

Usual GPS-based map-matching techniques’ accuracy is around 10 m (Quddus et al., 2007).
The amount of flow over estimation caused by discarding missing spacing data in the example are as

follows: (a) 0.0, (b) 213.6, (c) 226.3 and (d) 1110.1 veh/h—they are far more than neglectable. If themissing spacing data are simply substituted by the maximum detectable range, flow over estimation can be
mitigated to (a) 0.0, (b) 64.9, (c) 77.6 and (d) 482.9 veh/h.

47

Chapter 3

Basis: Method for Estimating Traﬃc State

means that the probe vehicles may have diﬀerent driving behaviors compared with othervehicles. In general, the estimation biases caused by biases in probe vehicles’ driving
behavior can be explained as follows. If average speed of probe vehicles is larger thanthat of all the vehicles, the flow and speed will be proportionally overestimated. If average
spacing of probe vehicles is larger than that of all the vehicles, the flow and density will
be proportionally overestimated. Note that the biases are diﬃcult to be specified in prior
due to the nonlinear relation between speed and spacing.

48

Chapter 4 Integrating Traﬃc Dynamics into
Estimation Methods
This chapter*1 describes how the basic method can be integrated with dynamics oftraﬃc. Traﬃc is dynamic phenomena that propagates over time and space. If conditions
describing such dynamics were integrated into TSE methods, their accuracy and precision
can be improved while keeping them consistent with traﬃc flow theory. For example,
the negative eﬀect by local fluctuation in microscopic vehicular traﬃc, such as vehicle
platoons and lane-changing, can be mitigated. This chapter develops two TSE methods
integrated with traﬃc dynamics based on the basic method formulated at chapter 3.
Section 4.1 describes some directions of integrating dynamics of traﬃc into TSE
methods. Sections 4.2 and 4.3 describe TSE methods integrated with a CL and withboth of an FD and a CL, respectively. Both sections consist of the method’s concept,
its formulation and its summary. Finally, section 4.4 provides qualitative comparison
among the three TSE methods, namely, the basic method (chapter 3), the method with CL(section 4.2) and the method with FD and CL (section 4.3).

4.1

Directions

Basically, traﬃc dynamics can be described with a fundamental diagram (FD) and aconservation law (CL) (c.f., the LWR model, chapter 2). An FD represents relation among
flow, density and speed in equilibrium traﬃc. A CL means that vehicles do not warp,
appear and disappear without specific sinks and sources.
The following develops two TSE methods. One of them utilizes the condition of a*1This chapter is mainly based on joint research with Dr. Takahiko Kusakabe and Prof. Yasuo Asakura,

published in Journal of JSCE (瀬尾ら, 2013), ISTTT21, Transportation Research Part C (Seo and Kusakabe,
2015, in press) and IEEE ITSC 2015 (Seo et al., 2015b, accepted).

49

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

CL where a CL must be satisfied (i.e., road sections that do not have merging/diverging
sections). The other utilizes the conditions of an FD and a CL; a functional form of an FD
is given exogenously while its parameters are endogenously estimated from probe vehicle
data. Note that methods with an FD but without a CL is not considered here; because it istheoretically pointless for TSE purposes in a corridor*2 .

4.2

Method with CL

This section develops a method that estimates the traﬃc state based on the collectedspacing and position data of probe vehicles by considering a CL to represent the traﬃcdynamics. On the other hand, it does not use an FD in order to eliminate exogenous
factors.

4.2.1

Concept

The proposed method estimates traﬃc states in a road section where some of the vehicles
in the flow are probes that measure the geographic position and spacing of the vehicle
ahead without any errors. The road section’s schematics are assumed to be known toanalysts. The probe vehicles’ driving behavior is assumed to be the same as that of nonprobe vehicles, i.e., the probes are randomly sampled from all of the vehicles. In addition,
the traﬃc flow is assumed to be single-lane traﬃc that satisfies the FIFO condition in order
to simplify the situation.
The spacing observed by a vehicle at a point depends on microscopic vehicular
phenomena that depend on macroscopic traﬃc flow phenomena. For example, the spacingof a vehicle takes a volatile value that is mainly determined by whether the vehicle or notis the leader of a platoon; meanwhile, the development of the platoon is determined by the
global traﬃc state. Therefore, aggregation is needed in order to estimate the traﬃc statefrom the observed vehicular variables. In this method, the observed vehicular variables*2

Methods with an FD without a CL may be interesting in terms of area-wide traﬃc state estimation,

namely, methods with MFD.

50

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

are aggregated based on the CL. Specifically, the number of vehicles between two specificprobe vehicles is a constant along a section where flow discontinuity points (e.g., a node
in a road network) do not exist. This number of vehicles is estimated by aggregating
the data observed by the two boundary probe vehicles. The estimation procedure is asfollows:
Step 1 The number of vehicles between two consecutive probe vehicles and twoconsecutive discontinuity points is estimated based on the observed data of the twoprobe vehicles.
Step 2 The cumulative flow at the probe vehicle trajectories is calculated from theestimated number of vehicles.
Step 3 The continuous cumulative flow over the entire time–space region is estimated byinterpolating the cumulative flow at the probe vehicle trajectories.
Step 4 The traﬃc state (i.e., flow, density and speed) is derived by partially diﬀerentiating
the continuous cumulative flow.
Fig 4.1 visualizes the relation among the cumulative flow, vehicle trajectory, and traﬃcstate (for details, see Makigami et al., 1971); the height is the continuous cumulative flowÑ, the contour line of Ñ is a vehicle’s trajectory, the slope of Ñ in the t direction (∂Ñ/∂t)is the flow q, and the slope of Ñ in the −x direction (−∂Ñ/∂x) is the density k. Thismethod can be regarded as based on rectangles on the n–x coordinate plane, which issimilar to the n–t Lagrangian coordinates of Leclercq et al. (2007); Wageningen-Kessels
et al. (2013) and is identical to coordinates of T-model by Laval and Leclercq (2013),
because the primary subject of estimation is the number of vehicles in the n–x rectangles.

4.2.2

Estimation method for cumulative flow at probe vehicle’s
trajectory

Consider the time–space region Amj (c.f., Fig 4.2) surrounded by the• trajectory of the m-th probe vehicle,
51

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

Figure 4.1: Three-dimensional representation of traﬃc flow.

Space xprobe m − 1probe mj

( j + 1)-th discontinuity point, x j+1

region am−1 (Am )jregion am (Am )jregion Am

j-th discontinuity point, x jjnumber of vehicles |N(Am )| − 2Time tFigure 4.2: Method conceptualized as time–space diagram.

• trajectory of the leading vehicle of the (m − 1)-th probe vehicle,• position of the j-th discontinuity point, and• position of the ( j + 1)-th discontinuity point.
Because of this definition and the FIFO assumption, the number of vehicles in the region
Amj is always identical to the number of vehicles between the m-th probe vehicle and theleading vehicle of the (m − 1)-th probe vehicle. The following notations are employed to52

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

represent the time and space coordinates: xmt is the position of the m-th probe vehicle atthe time point t, tmx is the time point of the m-th probe vehicle at the position x, and x j is
the position of the j-th discontinuity point.
The cumulative flow N(t, x) at a probe vehicle’s trajectory—with the 0th probevehicle*3 as a reference point—is described by using the number of vehicles between
two probe vehicles as follows:
N(t0x , x) = 0,

(4.1a)x, x) + |N(Amj )| − 2N(tmx , x) = N(tm−1(m ≥ 1, x j < x < x j+1 ),(4.1b)

where N(A) is the set of all vehicles in the time–space region A and |N(A)| is the numberof vehicles of N(A). Eq (4.1) means that the number of vehicles between the m-th andx(m − 1)-th probe vehicles at the position x (i.e., N(tmx , x) − N(tm−1, x)) is identical to thenumber of vehicles in the region N(Amj ) minus two*4 (i.e., |N(Amj )| − 2) if the position x is
between the j-th and ( j + 1)-th discontinuity points.
The only unknown variable in eq (4.1) is N(Amj ). This can be estimated from the probevehicle data as follows:|Amj |) ,|N̂(Amj )| = (|am−1 (Amj )| + |am (Amj )| /2(4.2)

where am (A) is the time–space region between the m-th probe vehicle and its leading
vehicle in region A, which should be observed by the probe vehicle, and |A| is the areaof the region A. Eq (4.2) means that all vehicles in time–space region Amj are distributedwith a time–space interval of (|am−1 (Amj )| + |am (Amj )|)/2, which is the average of the twoprobe vehicle’s intervals. This is consistent with the generalized definitions of Edie
(1963). It also implies that the number of vehicles between two vehicles is determinedby the two vehicles’ spacing at a road section without discontinuity points. By using theestimated value, the cumulative flow at a probe vehicle’s trajectory (4.1) can be calculated
as follows:N̂(t0x , x) = 0,*3
*4

(4.3a)

This vehicle is to be specified by an analyst
the (m − 1)-th probe vehicle and its leading vehicle

53

Chapter 4Integrating Traﬃc Dynamics into Estimation MethodsxN̂(tmx , x) = N̂(tm−1, x) + |N̂(Amj )| − 2

4.2.3(m ≥ 1, x j < x < x j+1 ).

(4.3b)

Estimation method for traﬃc state

The flow, which is defined as q = ∂Ñ/∂t, can be estimated asq̂(t, x) =xN̂(tmx , x) − N̂(tm−1, x)xx
tm − tm−1x(tm−1< t ≤ tmx ).

The continuous cumulative flow Ñ is estimated from∫ tˆq̂(τ, x)dτ.
Ñ (t, x) =

(4.4)

(4.5)

t0x

In fact, eqs (4.4) and (4.5) mean that Ñˆ is linearly interpolated from the discrete N̂ of eq(4.3) over the t-axis (Fig 4.3a) where uniform distribution of the vehicles between tmx andxtm−1
is assumed.

Once the continuous cumulative flow is obtained, the density and speed can be
estimated as∂Ñˆ (t, x),∂xq̂(t, x)v̂(t, x) =,k̂(t, x)k̂(t, x) = −

(4.6)
(4.7)

based on their definitions, namely, k = −∂Ñ/∂x and v = q/k, except at the discontinuity
points.
The estimation procedure can also be visualized as Fig 4.3. First, the number ofvehicles between two convective probe vehicles |N̂(Amj )| − 2 is estimated based on thexobserved data from probe vehicles. Then, the cumulative flow N̂(tm−1, x), N̂(tmx , x) andcontinuous cumulative flow Ñˆ (t, x) are estimated along with the cumulative flow’s slopexon the n–t plane, which means the flow q̂(x, t) between tm−1
and tmx , as shown in Fig 4.3a.
By using the continuous cumulative flow Ñˆ (t, x), the density k̂(x, t) can be estimated as

the slope of Ñˆ (t, x) on the n–x plane, as shown in Fig 4.3b and c.
54

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods
Cumulative count N

probe mN(tmx , x)N(t, x)Ñˆ (t, x)N̂(tmx , x)j

|N(Am )| − 2j|N̂(Am )| − 2
x , x)
N(tm−1x , x)
N̂(tm−1q̂(x, t)probe m − 1

x
tm−1

tmx

Time t(a) n–t cumulative curve at certain point x in space(x j , x j+1 ).
Cumulative count Nt )N(t, xm

probe mj

|N(Am )| − 2jt )N̂(t, xm|N̂(Am )| − 2−k̂(x, t)probe m − 1tN(t, xm−1)

N(t, x)Ñˆ (t, x)tN̂(t, xm−1)

x j xt

t
xm−1

m

x j+1

Space x(b) n–x cumulative curve at certain time point t wherett
x j ≤ xm
< xm−1≤ x j+1 is satisfied.

Cumulative count N

probe m − 1probe mt )N(t, xmt )N̂(t, xm−k̂(x, t)N(t, x)Ñˆ (t, x)xj

x j+1 xt Space x
m−1

t
xm

(c) n–x cumulative curve at certain time point t wherett
x j ≤ xm
< x j+1 < xm−1
is satisfied.

Figure 4.3: Illustrated examples of estimation method on cumulative curves.

55

Chapter 4

4.2.4

Integrating Traﬃc Dynamics into Estimation Methods

Discussions

The proposed method can derive the flow, density, and speed without an explicit FD,
meaning that no exogenous assumptions on an FD. Therefore, the proposed method caneasily be applied to various traﬃc conditions, especially when an FD is unknown or
unpredictable. Its independence from an FD also means that the proposed method canbe utilized to estimate the FD itself. In fact, estimating the cumulative flow based on
eq (4.2) is identical to estimating the area-flow (c.f., MFD; Geroliminis and Daganzo,
2008; Daganzo and Geroliminis, 2008) between two merging/diverging positions andtwo consecutive probe vehicles. It implies that if a well-defined MFD exist in theregion (Daganzo and Geroliminis (2008); Laval et al. (2015) prove this fact under certain
conditions such as a link with constant FD), the proposed method could be accurate.
The proposed method utilizes the CL in order to estimate the traﬃc state fromvehicles’ spacing. This means that the traﬃc state of a certain time point and positionis estimated based on spacings observed by the same vehicle over a wider area of timeand space. Therefore, the precision of the estimated traﬃc state is relatively robust against
fluctuations in microscopic vehicular phenomena (e.g., vehicle platoons, lane-changing)comparing to the basic method proposed of chapter 3, which does not consider the CL.
As noted earlier, equation (4.2) is identical to estimating the area-flow from the probevehicle data based on Edie’s generalized definition. Therefore, the accuracy and precision
of |N̂(Amj )| can be analytically approximated based on eqs (3.21) and (3.23) of the basic
method, as follows:()2|Amj | σ h̄n (A)E |N̂(Amj )| − |N(Amj )| ≃()3 ,2(x j+1 − x j )µ h̄n (A)
√ (
)2
j
)
(
σ
h̄
(A)
n
|A
|
m
,
RMSE |N̂(Amj )| ≃()22
(x − x )µ h̄ (A)
[

]

j+1

j

(4.8)

(4.9)

n

where µ() represents the mean, σ()2 represents the variance, h̄n (A) represents the mean
headway of a vehicle n in the region Amj , and RMSE is root mean square error and taken
as an index for precision. Eqs (3.21) and (3.23) mean that the accuracy and precision of56

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

the estimated cumulative flow N are improved as mean of the vehicle’s mean headways
increases and the variance of the vehicle’s mean headways decreases. Therefore, theaccuracy and precision of the estimated traﬃc state have similar tendencies.
The basic method estimates a traﬃc state in a short road segment using data collectedfrom the segment itself; therefore, estimation error can be large because vehicles may
keep a platoon formed in the segment (i.e., σ(h̄n (A))2 may be large). In contrast, theproposed method in this section estimates a traﬃc state in a position using data collectedfrom the entire link; therefore, estimation error can be smaller because platoons may be
newly formed and/or declined in the link (i.e., σ(h̄n (A))2 may be smaller).
The deficiencies of the proposed method are as follows. First, single-lane traﬃc and
FIFO condition were assumed in order to estimate the number of vehicles between twoprobe vehicles; such assumptions are not always satisfied in the real world. However,the proposed method can be applied to traﬃc where these conditions are not satisfied
if the probe vehicle’s driving behavior is the same as other vehicles. This is because
the expected number of vehicles that overtake a probe vehicle and number that are
overtaken by a probe vehicle is equal to each other. In addition, estimation error may
sometimes decrease under non-FIFO condition; because a platoon, by which mean
headway variances increase, will be formed and/or declined frequently under suchcondition. Note that if the road has multiple lanes, the cumulative number (4.3) needs tobe multiplied by the number of lanes. Second, the linear interpolation of the cumulative
flow over time is limited in its ability to reproduce shockwaves. Another deficiency is that
real-time states cannot be estimated. Therefore, short-term future prediction is requiredin order to obtain the real-time states*5 .

*5This prediction can be based on the estimated traﬃc state and an FD that can be estimated with theproposed method.

57

Chapter 4

4.3

Integrating Traﬃc Dynamics into Estimation Methods

Method with FD and CL

This section develops a method that estimates the traﬃc state based on the observed
spacing and position data of probe vehicles by considering an FD and CL to represent
the traﬃc dynamics.

4.3.1

Concept

The following conditions are assumed for traﬃc that is subject of TSE in this method. 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 probevehicle and its leading vehicle ahead) and position without error.
The procedure of the proposed TSE method can be described as follows. First, time–
space resolution for TSE, time–space resolution for FD’s parameter estimation, and FD’sfunctional 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 FDparameters, estimated traﬃc states in the previous timestep, and a traﬃc flow model.
Specifically, the basic TSE method is utilized for observing traﬃc states.

FDparameters are derived by regressing spacing and headway (spacing divided by speed)
relation in the probe vehicle data. The cell transmission model (CTM; Daganzo, 1994),
a numerical computation method for solving the LWR model, is applied for the traﬃcflow model. As a data assimilation technique, Ensemble Kalman Filter (EnKF; Evensen,
1994) is employed due to its capability for dealing nonlinear phenomena in traﬃc (thesystem 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.
Fig 4.2 summarizes the procedures of the proposed method.

58

59

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 parameterregion: B

Traffic state:- density k̂i (t)Fundamental diagram:- parametersûi (t), kˆc i (t), κ̂i (t)Observation vector:- ytflow model

1st order traffic

Conservation law,

Estimation result:- density k ∗ (AT,X )- FD parameters

Filter ensemble of state vector:- density and FD parameters{xnt|t }Nn=1

Data assimilation:- Ensemble Kalman filter· xt = ft (xt , νt )· yt = Ht xt + ωt
- ft : Cell transmission model

Deviations of noise terms:- σk , σu , σk c , σκ ,ξk , ξu , ξkc , ξκ

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

Figure 4.4: Illustrated concept of the procedure of the proposed TSE method.

Chapter 4

4.3.2

Integrating Traﬃc Dynamics into Estimation Methods

Ensemble Kalman filter

This part briefly describes EnKF by employing the expressions from 樋口 (2011).
A state–space model for EnKF can be described as follows
xt = ft (xt , νt ),(4.10)

yt = Ht xt + ωt ,(4.11)

where eq (4.10) is a system equation, eq (4.11) is an observation equation, xt is a statevector, ft is a system model, νt is a system noise vector, yt is an observation vector, Ht isan observation matrix, and ωt is an observation noise vector, at timestep t respectively.
The observation noise vector ωt follows normal distribution whose average is 0 andvariance-covariance matrix is Rt , namely, ωt ∼ N(0, Rt ).
The general procedure of EnKF can be described as follows:N. Let t ← 1.
Step 1 Generate an ensemble of the initial states {xn0|0 }n=1

Step 2 Prediction step:N
Step 2.1 Generate an ensemble of the system noises {νtn }n=1.

Step 2.2 Calculate xnt|t−1 = ft (xnt−1|t−1 , νtn ) for each n.
Step 3 Filtering step:N
Step 3.1 Generate an ensemble of the observation noises {ωtn }n=1.
N
Step 3.2 Obtain the filter ensemble {xnt|t }n=1
for each n, by calculating eq (4.12)
NN
based on yt , Ht , Rt , {xnt|t−1 }n=1
, {ωtn }n=1.

Step 4 Increment the timestep, t ← t + 1. Go back to Step 2 until t = tmax .
The filter ensemble in Step 3.2 can be derived as follows:)(
xnt|t = xnt|t−1 + K̂t yt + ω̆tn − Ht xnt|t−1 ,(4.12))−1(K̂t = V̂t|t−1 Ht′ Ht V̂t|t−1 Ht′ + Rt ,(4.13a)where,

60

Chapter 4Integrating Traﬃc Dynamics into Estimation MethodsN
1 ∑ jV̂t|t−1 =x̆ (x̆ j )′ ,N − 1 j=1 t|t−1 t|t−1

1∑ j
x ,
N j=1 t|t−1(4.13b)Nx̆nt|t−1 = xnt|t−1 −1ω̆tn = ωtn −

N
∑

N j=1

ωtj .

(4.13c)(4.13d)In this method, following mean vector of the filter ensemble is defined as an“estimation result” at timestep t:1∑ nx ,N n=1 t|tNx̄t|t =

4.3.3

∀t.

(4.14)

TSE method formulation

In this subsection, the TSE method is formulated by specifying the terms and the function
in eqs (4.10) and (4.11).
(1)

System model

A nonlinear system equation that represents traﬃc dynamics is formulated.
The state vector xt is defined as follows:(
xt = k1 (t), k2 (t), . . . , ki (t), . . . k M (t),u1 (t), u2 (t), . . . , ui (t), . . . u M (t),k1c (t), k2c (t), . . . , kic (t), . . . kcM (t),)
κ1 (t), κ2 (t), . . . , κi (t), . . . κ M (t)(4.15)

where, ki (t) is a density, ui (t) is a free-flow speed, kic (t) is a critical density, and κi (t) isa jam density, of cell i at timestep t respectively. Note that ki (t) is the traﬃc state andui (t), kic (t), κi (t) are the FD parameters. The number of elements in vector xt is 4M, where
M is total number of discretized spaces (cells) in the system model.
The system model ft consists of two parts: on traﬃc state dynamics and on FDparameters. The first part of the system model ft (on traﬃc state dynamics) is the CTM
61

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

Flow q(kic (t), kic (t)ui (t))

ui (t)(κi (t), 0)
Density k
Figure 4.5: Triangular FD and its parameters in cell i at time t.

with noise term. It is described as follows:








 li−1 ki−1 (t),inkQi (t) = εi (t) min 
,
c (t)


k
 i (l κ (t) − l k (t)) κi (t)kic (t)i i(4.16a)i iinki (t + 1) = ki (t) + Qin
i (t)/li − Qi+1 (t)/li ,(4.16b)where Qini (t) is a number of vehicles that enters cell i at timestep t, li is the length of celli, and εki (t) is a system noise of cell i at timestep t that is expressed as εki (t) ∼ N(1, σ2k )(truncated to non-negative values). Triangular FD is employed by eq (4.16); the relation
between triangular FD and its parameters is shown in Fig 4.5. The original CTM
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 (4.16) without the noise term, εki (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 later, namely, eq (4.23) and Fig 4.6; because it has significant relation
with the observation.
The other part of the system model ft (on FD parameters) is assumed to be random
62

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

region B is larger than AT,X and cell i∆Tregion AT,X for the observation of a traffic state

Space x∆X

spacing: am (AT,X )cell i for the system model (CTM)li∆tTime tFigure 4.6: Coordinates for the method with FD and CL and the probe vehicle data.

walks. They can be represented asui (t + 1) ∼ N(ui (t), σ2u ),(4.17a)kic (t + 1) ∼ N(kic (t), σ2kc ),(4.17b)κi (t + 1) ∼ N(κi (t), σ2κ ),(4.17c)where σu , σkc and σκ are deviations for the random walks. The size of the deviations are
given by the analysts.
The system noise vector νt is assumed to be consists of σk , σu , σkc and σκ withouttime, space nor flow dependency.
(2)

Observation

A linear observation equation can be formulated based on the basic method.
The observation vector yt is defined to have the same property with the state vector xt
as()
yt = . . . , k̂i (t), . . . , ûi (t), . . . , kˆc i (t), . . . , κ̂i (t), . . .
63(4.18)

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

where the ˆ mark indicates that the variable is directly derived (i.e., observed) from theprobe vehicle data. The number of elements in vector yt is also 4M.
From the probe vehicle data, a density in a time–space region can be derived asfollows:∑k̂(AT,X ) = ∑m∈P(AT,X ) tm (AT,X )m∈P(AT,X ) |am (AT,X )|.

(4.19)

This is equivalent to eq (3.15). This is the observed state. The definition of AT,X is
AT,X = {(x, t) | X ≤ x ≤ X + ∆X, T ≤ t ≤ T + ∆T }(4.20)

where ∆T and ∆X are predetermined resolutions for time and space, respectively. Eq(4.20) means that region AT,X is a rectangle in time–space plane with time length ∆T andspace length ∆X (c.f., Fig 4.6).
The FD parameters are derived from regression of headway–spacing relation in
the probe vehicle data. Values of ûi (t), kˆc i (t), κ̂i (t) are assumed to be constant in a
pre-determinded time–space region B (i.e., the observed FD parameters are constant in
B). They can be determined asûi (t) = u(B),∀(t, i) ∈ B(4.21a)kˆc i (t) = w(B)κ(B)/(u(B) + w(B)),∀(t, i) ∈ B(4.21b)κ̂i (t) = κ(B),∀(t, i) ∈ B,

(4.21c)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 solutionof the following optimization problem:
argmin∑()
D hm,τ , sm,τ , u(B), w(B), κ(B) 2 ,u(B),w(B),κ(B) (m,τ)∈P(B)s.t. u(B) ≥ 0, w(B) ≥ 0, κ(B) ≥ 0,

(4.22)

where hm,τ and sm,τ are stationary headway and spacing, respectively, of vehicle m at timeτ, D is 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 F is a function representing a triangular64

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

FD. Therefore, problem (4.22) finds FD parameter values that minimizes total distance
between observed stationary headway–spacing points and the FD curve. Stationary meansthat 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 Cassidy (1998).
The relation between region AT,X for the observation and cell i at timestep t in thesystem model are as follows. The resolution of the observation and that of the systemmodel 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 method, the size of region AT,X is set to includes one or multiple cell(s) andtimestep(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, ∆t and li are determined as follows:
li = αûi (t)∆t,∀(t, i) ∈ B(4.23a)∆T = β∆t,(4.23b)∆X = γl,(4.23c)where α is a coeﬃcient larger than or equal to 1 (equal to 1 is desirable), and β andγ are natural numbers. Fig 4.6 shows an example with α = 1, β = 3 and γ = 2. In
the figure, solid curves represent vehicle trajectories, dotted grid represents regions forthe observation, and dashed gird represents cells for the system model. The analysts
can select appropriate values for ∆t, α, β and γ according to their interests. Following
definitions are introduced in order to represent the relation between region AT,X and cell i
at timestep t:
Ai (t) = AT,X ,if T ≤ t∆t < (t + 1)∆t ≤ T + ∆T
and X ≤ xi < xi + l ≤ X + ∆X

(4.24)

where xi is a space coordinate of the upstream edge of cell i. Therefore, following relation
65

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

holds true:k̂i (t) = k̂(Ai (t)) = k̂(AT,X ).

(4.25)The variance of observation noise ωt , namely, Rt , is determined as follows. Section
3.4 showed that the precision of the estimator (4.19) 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 Rt is defined as

ξk2



, if a = b and a ≤ M
|P(Aa (t))|







ξu2 ,
if a = b and M < a ≤ 2M




 2
Rt (a, b) = (4.26)
ξk c ,
if a = b and 2M < a ≤ 3M







ξκ2 ,
if a = b and 3M < a ≤ 4M







 0,
otherwise,
where ξk is deviation for observing traﬃc state, and ξu , ξkc and ξκ are deviations forobserving respective FD parameters. The size of the deviations are given by the analysts.
An element (i, i) of the observation matrix Ht represents whether the cell i is observed
or not. Therefore, element (a, b) of Ht is defined as1, if a = b and a ≤ M and |P(Aa (t))| > 0





Ht (a, b) = 
1, if a = b and a > M






 0, otherwise,(4.27)

according to the presence of probe vehicles.

4.3.4

Discussions

As results of the filtering steps, the filter ensemble’s mean vector x̄t|t can be obtained. The
density elements (1st to M-th elements) in the mean vectors x̄t|t are the final output of theTSE procedure. This estimated density is declared as k∗ (AT,X ) with resolution of ∆T and∆X.
66

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

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 isassumed to be triangular. On the other hand, the parameters of the FD are not required
to be assumed; they are endogenously estimated from probe vehicle data. Note thatthe 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, A for state observation,
B for 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 theother variables of a traﬃc state, namely, flow and speed, can be easily derived from theestimated 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 middleof the link. Traﬃc in a network with multiple links can be treated by implementing a nodeconservation law and a merging/diverging model in the proposed approach.
For large-scale real world application, the method may be not very costly in termsof computation and data-handling. EnKF requires less computation costs compared withother techniques like the particle filter. Data required for the method are position andspacing, which are calculated by distributed on-vehicle systems for their own purposes;
therefore, not heavy data analyses, transmission and storage are required.
The method may be easy to be incorporated with other traﬃc data collection methods(e.g., detectors, GPS-equipped probe vehicles), since the method is based on data
assimilation. It can be achieved by following procedures: incorporating the collected data
to yt , and defining appropriate Ht and Rt considering characteristics of the data collection
67

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

methods.

4.4

Comparison among three TSE methods

Chapter 3 and this chapter proposed three methods for TSE, namely, the basic method,
the method with CL, and the method with FD and CL. This section qualitatively compares
the three methods and discusses their expected characteristics.
The basic method estimates traﬃc state in an arbitrary region using data collectedfrom the region. Since it only assumes random sampling condition, it can be applicablewithout theoretical discrepancies regardless of traﬃc condition. However, the resolutionof estimated traﬃc state is identical to the data collection area. Therefore, amount of data
will be limited if an analyst estimates high resolution traﬃc state. It implies that the basic
method is not robust against local fluctuations in microscopic traﬃc, such as platoon andlane-changing.
The method with CL estimates traﬃc state in an arbitrary point in a certain region.
The region is product of predetermined space, which is assumed to be satisfy a CL,
and area between two probe vehicles. Therefore, amount of data for TSE is relatively
large compared with the basic method; it may improve precision of estimating highresolution traﬃc state. On the other hand, a CL is not completely satisfied under non-FIFO
condition, such as in multi-lane and multi-class traﬃc.
The method with FD and CL estimates traﬃc state in an arbitrary region. Traﬃcdynamics, which means that traﬃc propagation between adjacent regions, is explicitly
considered by assuming an FD and a CL. The FD’s parameters are endogenously
estimated from the probe vehicle data. Therefore, amount of data for TSE is relatively
large compared with the method with CL as well as the basic method. The method
employs traﬃc flow model with an Eulerian coordinate system. It means both advantage
and disadvantage: the method cannot utilize information of probe vehicle trajectories,
while the method is not severely aﬀected by the discrepancies under non-FIFO condition.
Assuming existence of constant FD means that the method is not robust against
68Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

Space xSpace xinput data (reso. of A)

Space xinput data (reso. of A)

input data (reso. of A)

output state (reso. of B)
output state (reso. of B)

output state (reso. of B)

Time t(a) The basic method

Time t(b) The method with CLTime t(c) The method with FD and CL

Figure 4.7: Illustrated comparison of three TSE methods.

unpredicted traﬃc phenomena, such as incident.
Fig 4.7 and Table 4.1 summarize above mentioned characteristics of the three TSE
methods. In Fig 4.7, resolution of input data for TSE (i.e., size of an element in the set A
in eq (2.20)) and resolution of estimated traﬃc state (i.e., that of B) are illustrated alongwith the probe vehicle data.

69

predetermined space and an area between two probe vehicles
Region B that is identical to thepredetermined area with constant FD• Satisfaction of a CL in predeter-

mined time–space area

• Random sampling• Satisfaction of a CL in predeter-

70• Noise terms (θ)• Functional form of the FDs

termined time–space areas• Existence of constant FDs in prede-mined time–space area

Region A that is a product of the• Random sampling

The method with CL

The method with FD and CL

Predetermined arbitrary time–space• Random sampling

The basic method
region A

Data collection range (an element of A)

Assumption (F )Proposed TSE method

Table 4.1: Summary of three TSE methods

Predetermined arbitrary region A in B

Point in A

Identical to A

(an element of B)

Resolution of estimated traﬃc state

Chapter 4Integrating Traﬃc Dynamics into Estimation Methods

Chapter 5 Simulation Experiments-based
Validation
This chapter*1 describes an empirical analysis of the proposed methods based onsynthetic traﬃc data generated by a microscopic traﬃc simulator.

The simulation

experiments-based validation have some advantages compared with the field experimentbased validation that is described by chapter 6. For example, measurement errors andbiases in probe vehicles driving behavior can be eliminated in simulation. Therefore, thisvalidation complements the limitations of the field experiment-based validation of chapter
6. However, simulation experiments have limitations in the model and scenario setting.
The simulation model describing vehicle behavior is car-following model, which controls
the input variable of the proposed TSE methods directory. In addition, scenario setting ofsimulation, namely, demand and supply to road sections, is given ad hoc.
Sections 5.1, 5.2 and 5.3 describe validation results of the basic estimation method,the estimation method with CL, and that with FD and CL, respectively, based on synthetic
traﬃc data generated by a microscopic traﬃc simulator.

5.1

The basic method

5.1.1

Simulation Scenario

An experiment environment is prepared as follows.

To generate traﬃc conditions

investigated in a validation analysis, Aimsun (TSS-Transport Simulation Systems),
microscopic traﬃc simulator based on a car-following of Gipps (1981) and lane-changing
model of Gipps (1986), is employed. The parameters of the car-following model are*1

This chapter is mainly based on joint research with Dr. Takahiko Kusakabe and Prof. Yasuo Asakura,

published in ISTTT21, Transportation Research Part C (Seo and Kusakabe, 2015, in press) and IEEE ITSC
2015 (Seo et al., 2015b, accepted).

71

Chapter 5

Simulation Experiments-based Validation

Table 5.1: Parameters of the model in simulation
Parameter name

Mean

Deviation

60

10.0

Max acceleration (m/s )3

0.2

Normal deceleration (m/s2 )4

0.5

Max deceleration (m/s2 )6

0.5

Min spacing

1

0.3

Length (m)40.5Desired speed (km/h)
2

shown in Table 5.4. The road is a homogeneous freeway that has a bottleneck at theend 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. The average value of ground truth
traﬃc states are 2162.8 veh/h, 128.5 veh/km and 32.6 km/h for flow, density and speed,
respectively.
The bias and the root mean square error (RMSE) are employed as accuracy and√precision indices, where Bias(θ̂) = E[θ̂ − θ] and RMSPE(θ̂) = E[((θ̂ − θ))2 ]. Variableθ denotes a ground truth traﬃc flow variable (one of q, k and v) at a certain time andspace calculated based on Edie’s definition; and variable θ̂ is one of those estimated fromthe probe vehicle data. The parameters set for scenarios were PR P (%), time resolution∆t (min), and space resolution ∆x (km). The PR P is selected from 10.0%, 5.0% and0.1%. The time resolution ∆t is selected from 1 min or 60 min; and the space resolution∆x is selected from 0.1 km or 3 km. The error indices are calculated from the resultsof 20 estimation iterations. In each iteration, probe vehicle data for the estimation werecomposed of diﬀerent combinations of probe vehicles whose PR was expected to be thescenario parameter P. Traﬃc state in a region Aij where probe vehicles do not exist is
substituted by that of its neighborhood region, Aij−1 .

72

Chapter 5

5.1.2

Simulation Experiments-based Validation

Results

Fig 5.1 visualized estimated traﬃc state by the basic method and corresponding groundtruth as time–space diagrams, where plot color represents value of traﬃc state.*2 The
scenario parameters are (P, ∆t, ∆x) = (10, 0.1, 1). The error indices are RMSE(q) = 719.2,
Bias(q) = 158.1, RMSE(k) = 23.6, Bias(k) = 6.2, RMSE(v) = 4.8, and RMSE(v) =−0.1. According to the figure, estimation errors are relatively large at free-flow state andrelatively small in congested state. These phenomena can be explained by the headway
and spacing distribution (c.f., sections 3.4 and 3.6); it is known that they tend to be large
in free-flow state in general.
Table 5.2 summarizes the results of the estimation. The precision and bias often
improves as P, ∆t and ∆x increased; these are reasonable features and as analytically
expected (c.f., sections 3.4 and 3.6). The biases are positive in most of the scenarios.
However, negative biases are also found, such as (P, ∆t, ∆x) = (0.1, 0.1, 1) scenario. Thismay be caused by following two reasons: random error due to small number of samples(simulation iterations), and the interpolation method for regions without probe vehicles.
In order to validate the analytically expected bias and precision shown in eqs (3.21)–(3.24), they are calculated from all the vehicle data and compared with actual errors. Thed k̂) and RMSE([ k̂) represent the analyticallyresults on density are shown in Table 5.3. Bias(
expected bias and precision, respectively, of estimated density. The analytically expected
errors and the actual ones seems to be corresponded well in P = 5.0, 10.0 (%) scenarios.
This implies that the analytical approximations discussed at section 3.4 may be reliablein such situations, which are much generalized conditions than that of required for strict
satisfaction of eqs (3.21)–(3.24).

*2

The figure (and most of other similar figures in this dissertation) employs the color scheme developedby Kusakabe et al. (2010), which aims appropriate visualization of traﬃc data, in both of printing andelectrical presentation.

73

Chapter 5

Simulation Experiments-based Validation

Table 5.2: Precision and bias indices over estimation scenarios.
Scenario parameters

Flow (veh/h)Density (veh/km)

Speed (km/h)

P (%)∆t (min)∆x (km)RMSE(q)Bias(q)RMSE(q)Bias(k)RMSE(q)Bias(v)

0.1

1

0.1

1102.3−622.8

112.1

−70.9

18.2

5.1

0.1

60

0.1

849.8

323.6

58.7

12.1

13.9

5.5

0.1

1

3.0

1140.9−684.9

76.3

−10.3

12.3

−2.6

0.1

60

3.0

522.1

120.3

46.4

12.8

3.9

0.3

5.0

1

0.1

809.5

117.4

33.4

5.2

5.9

−0.3

5.0

60

0.1

202.3

12.8

11.9

1.2

1.2

0.1

5.0

1

3.0

540.7

126.4

38.0

11.8

3.8

−0.2

5.0

60

3.0

147.5

50.1

10.9

2.6

0.6

0.1

10.0

1

0.1

719.2

158.1

23.6

6.2

4.8

−0.1

10.0

60

0.1

121.9

15.4

7.4

1.7

0.8

0.0

10.0

1

3.0

414.9

96.2

26.9

6.8

2.5

0.0

10.0

60

3.0

82.8

−23.0

7.0

−1.5

0.5

0.0

Table 5.3: Analytically expected precision and bias.
Scenario parameters

Precision (veh/km)

Bias (veh/km)

RMSE(k̂)

d k̂)
Bias(

Bias(k̂)P (%)∆t (min)∆x (km)

[ k̂)
RMSE(

0.1

1

0.1

112.1

159.8

585.9

−70.9

0.1

60

0.1

58.7

81.4

88.8

12.1

0.1

1

3.0

76.3

261.9

639.0

−10.3

0.1

60

3.0

46.4

89.8

87.6

12.8

5.0

1

0.1

33.4

26.6

16.0

5.2

5.0

60

0.1

11.9

11.6

1.8

1.2

5.0

1

3.0

38.0

33.6

11.7

11.8

5.0

60

3.0

10.9

12.7

1.8

2.6

10.0

1

0.1

23.6

18.8

8.0

6.2

10.0

60

0.1

7.4

8.2

0.9

1.7

10.0

1

3.0

26.9

23.7

5.8

6.8

10.0

60

3.0

7.0

9.0

0.9

−1.5

74

Chapter 5

Simulation Experiments-based Validation(a) Estimated flow(b) Ground truth flow(c) Estimated density(d) Ground truth density(e) Estimated speed(f) Ground truth speed

Figure 5.1: Results of the basic method visualized as time–space diagrams.

5.2

The method with CL

This section presents the validation of the proposed TSE method with CL though asimulation experiment.

75

Chapter 5

Simulation Experiments-based Validation

Table 5.4: Vehicle behavior parameter setting in simulation.
Parameter name

Mean

Deviation

80

10.0

Max acceleration (m/s )3

0.2

Normal deceleration (m/s2 )

4

0.25

Max deceleration (m/s2 )6

0.5

Min spacing (m)10.3

Vehicle length (m)40.5Desired speed (km/h)
2

5.2.1

Simulation scenario

A microscopic traﬃc flow simulator called AIMSUN (TSS-Transport Simulation Systems), which is based on the car-following model of Gipps (1981) and the lane-changing
model of Gipps (1986), was employed for the simulation. The road section in thesimulation had a single-lane and length of 5 km with homogeneous geometry except fora bottleneck at the end of the section. Note that this road setting always satisfies theassumptions of the proposed method*3 . The vehicles in the simulation had heterogeneous
driving behaviors, such as the desired speed. Table 5.4 summarizes the parameters. The
probe vehicles were randomly sampled from all the vehicles to simulate a given PR. The
position and spacing of the probe vehicles with 1 s interval were used for estimating a
traﬃc state.
We generated a traﬃc situation with a queue by setting the demand from upstreamof the section and the capacity at the bottleneck at the end of the section. A ground truth
state was available because this was a simulation experiment. Fig 5.2a–d show the groundtruth state (flow q, density k, speed v) and cumulative flow Ñ as time–space diagrams. The
contour lines of Ñ were drawn at intervals of 100 veh. By applying the proposed method
to the generated traﬃc situation, the characteristics of the proposed method in a various
traﬃc situations (e.g., free-flow, congested flow, flow with backward wave and flow with

*3

Except the first vehicle enters the road; because it does not have its leader. We removed it fromcandidates for the probe vehicles.

76

Chapter 5

Simulation Experiments-based Validation

forward wave) can be investigated.

5.2.2

Results

The estimated variables were the traﬃc state at certain time–space resolutions (i.e.,
the intervals for the central diﬀerences) and the cumulative flow in the section. The
cumulative flow was calculated at 1 min time intervals and 100 m space intervals
according to eq (4.5). The density k̂ was calculated based on the central diﬀerences at
100 m intervals according to eq (4.6).
Fig 5.2e–h show the estimated traﬃc state (q̂, k̂, v̂) and cumulative flow Ñˆ as atime–space diagram where the probe vehicle penetration is 3.5%. The contour lines ofÑˆ were drawn at intervals of 100 veh. Through comparison with the ground truth shownin Fig 5.2a–d, the estimation can be used to acquire tendencies of the ground truth, suchas the queue propagation and dissolution. More detailed phenomena such as stop-and-go
waves were partially captured, but with some errors. This is a deficiency of the methodof interpolating the continuous cumulative flow in the proposed methods, namely, linerinterpolation over the time.
Fig 5.3 compares the estimated variables and ground truth as scatter diagrams. The
estimated variables well corresponded at most of true states although the values wereslightly overestimated, as suggested by the analytically expected bias in eq (3.23). In
Fig 5.3d, a notable bias can be found around Ñ ∈ [800, 1000]. According to Fig 5.2d,
this value of Ñ is located at the area with rapidly diminishing queue, namely, around t ∈[30, 40]. It implies that, at the area, a probe vehicle may experience much diﬀerent traﬃcstate from its preceding probe vehicle’s experience. Therefore, the headway variance may
increases, so that the bias becomes large.
Table 5.5 summarizes the error indices of the traﬃc state estimation for diﬀerent
combinations of the PR P. The RMSPE and bias were employed as error indices, which√
are defined as RMSPE(θ̂) =E[((θ̂ − θ)/θ)2 ] and Bias(θ̂) = E[θ̂ − θ], respectively.
The error indices of each scenario were calculated from 100 iterations under diﬀerent

77Chapter 5

Simulation Experiments-based Validation(a) Ground truth flow q.

(b) Ground truth density k.

(c) Ground truth speed v.

(d) Ground truth cumulative flow Ñ.

(e) Flow q̂ estimated with proposed method.

(f) Density k̂ estimated with proposed method.

(g) Speed v̂ estimated with proposed method.

(h) Cumulative flow Ñˆ estimated with proposed
method.

Figure 5.2: Time–space diagrams of traﬃc state and cumulative flow.

78

Chapter 5

Simulation Experiments-based Validation(a) Flow q.

(b) Density k.

(c) Speed v.

(d) Cumulative flow Ñ.

Figure 5.3: Scatter diagrams of estimated variables vs. ground truth variables.

samplings of probe vehicles. The results showed clear improvements in the estimation
accuracy and precision as the PR was increased. Almost all of the scenarios overestimatedthe states similar to Fig 5.3d and for the same reasons.

5.2.3

Eﬀect of incorporating CL

In order to investigate the eﬀect of incorporating the CL, the flow was estimated using
the basic method which does not consider the CL but uses the same probe vehicle data.
This is defined as q̂withoutCL in this section. The time and space resolution for calculatingq̂withoutCL were identical to the intervals of the central diﬀerence in the proposed method:79

Chapter 5

Simulation Experiments-based Validation

Table 5.5: Error indices for scenarios with proposed method.
PR
P

Flow (veh/h)Density (veh/km)

Speed (km/h)

RMSPE(q̂)

Bias(q̂)

RMSPE(k̂)

Bias(k̂)

RMSPE(v̂)

Bias(v̂)

0.2%

59%

151.0

107%

−1.6

134%

1.1

1.0%

55%

134.8

87%

2.2

61%

−2.1

3.5%

42%

83.6

52%

2.1

36%

−0.7

5.0%

38%

69.6

48%

2.4

36%

−0.2

10.0%

33%

63.8

45%

2.4

38%

0.7

Table 5.6: Comparison between the method with CL and the basic method.
P

RMSPE(q̂)

RMSPE(q̂withoutCL )

PoI

0.2%

59%

66%

12%

1.0%

55%

63%

7%

3.5%

42%

59%

27%

5.0%

38%

53%

38%

10.0%

33%

52%

43%

1 min and 100 m. Table 5.6 presents the results of the comparison. The percentage ofimprovement (PoI) is defined as (RMSPE(q̂withoutCL ) − RMSPE(q̂))/RMSPE(q̂) and is anindex that represents the positive eﬀect of incorporating the CL. According to the results,
incorporating the CL clearly improved the estimation precision.

5.3

The method with FD and CL

In this section, numerical characteristics of the proposed TSE method with FD and CL
are investigated.

5.3.1

Simulation scenario

Road schematics and vehicle behavior models for this experiment environment are
identical to that in section 5.1.1.
The PR P was selected from 5%, 1% and 0.5%. Two types of time–space resolution
for observation were selected, namely, (∆T , ∆X) ∈ {(1 min, 300 m), (10 min, 1000 m)}.
80

Chapter 5

Simulation Experiments-based Validation

The observation error (i.e., precision of the basic method) is expected to be smaller as
the observation resolution increased as shown in chapter 3 and section 5.1. In order
to simplify the discussion, region B is set to be the entire time–space region—the FDparameters are assumed to be constant throughout the simulation.
A timestep width in the system model, ∆t, was set to 0.111 min. The other variableson the resolution of the system model, namely, α, β, and γ, were set to certain values suchthat the cell length li is identical to 100 m; their exact values vary in estimation iterations
depending on observed free-flow speed u and given observation resolution ∆T and ∆X.
With these setting, the value of α is 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 section 4.3.3, eq (4.23).
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 thatgeneral 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 rateduring 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 N was set to 200. In general, larger N value makes accuracy of
EnKF’s Monte Carlo simulation better at the cost of computation.

5.3.2

Results

Fig 5.4 shows traﬃc density as time–space diagrams, whose horizontal axes represent
time, vertical axes represent space, and color represents density*4 .

In Fig 5.4, arepresents the ground truth value k, b represents observed value k̂ with (P, ∆T, ∆X) =(5%, 1 min, 300 m), and c represents estimated value k∗ with the identical resolution.
According to Fig 5.4, the noises in the observation were reduced in the filtering result,81

Chapter 5

Simulation Experiments-based Validation(a) ground truth traﬃc state k(b) observed traﬃc state k̂(c) estimated traﬃc state k∗
Figure 5.4: Typical estimation results visualized as time–space diagrams.

82

Chapter 5

Simulation Experiments-based Validation

Figure 5.5: Observed FD, disaggregated headway–spacing relation and aggregated flow–density relation.

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.
Fig 5.5 shows observed FD by probe vehicles (red line), disaggregated headway–spacing relation of vehicles (gray dots) and aggregated flow–density relation intime–space regions (blue crosses), with (P, ∆T, ∆X) = (5%, 1 min, 300 m). The observed
FD’s parameter values were û = 55.4 (km/h), kˆc = 73.7 (veh/km) and κ̂ = 388.5(veh/km). The disaggregated relation is widely scattered due to the high fluctuation andheterogeneity in microscopic vehicle behaviors. On the other hand, observed FD is closed
to the aggregated relation which is clearly bivariate.
Table 5.7 summarizes the estimation performance. The root mean square error(RMSE) and mean absolute percentage error (MAPE) were employed as precisionindices on estimated density. Their definitions were as follows: 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 S is 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.
*4

In addition, theIn order to emphasize the diﬀerence between results, color scheme for this figure is diﬀerent to theother figures.

83

Chapter 5

Simulation Experiments-based Validation

Table 5.7: Summary on the estimation performance
Scenario parameters

Precision indices

Percentage of improvement∆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%

5

500

45.6

34.6%

8.3%

19.0%

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%

5

500

32.3

22.9%

6.5%

24.4%

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%

5

500

13.6

9.4%

−6.8%

16.7%

5.0%

10

1000

7.4

3.8%

5.8%

15.3%

P

percentage of improvement (PoI) was employed for evaluate the eﬀect of the data
assimilation. It was defined as PoI(p) = (p(k̂) − p(k∗ ))/p(k̂), where p is a precision index,
namely RMSE or MAPE. The value of PoI represents improvement of estimated densityk∗ compared to observed density k̂; if the value is positive, the data assimilation worked
positively for precision. According to Table 5.7, distinguishable trends can be foundfor precision. The precision is increased as probe vehicle penetration rate increases andtime–space resolution lowers. In addition, according to PoI value, the data assimilation
almost always reduced estimation error, especially on MAPE.

5.3.3

Eﬀect of incorporating FD and CL

The results showed that the estimation precision improved due to the data assimilationin the most cases. According to Fig 5.5, FD parameters were precisely observed by
PVSMEs; therefore, the prediction steps by the traﬃc flow model are expected to work
well. Then, according to Fig 5.4 and Table 5.7, 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 Fig 5.4 and PoI(MAPE) in Table 5.7. Thisis a preferable feature for a TSE method with the PVSMEs; because the precision of84

Chapter 5

Simulation Experiments-based Validation

the basic method was relatively low at low density traﬃc. Contrary, the precision inhigh density (congested) regime was not improved significantly. This may be due tothat 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.

85

Chapter 6 Field Experiment-based Validation
This chapter*1 describes an empirical analysis of the proposed methods based on actual
traﬃc data collected via a field experiment. The advantage of a field experiment-based
validation is singular: it can be realistic. This is the most preferable feature for the
proposed TSE methods; because not all factors in the microscopic vehicle behavior, which
determines spacing of vehicle in traﬃc, can be represented by existing traﬃc flow models,
as described in chapter 2. In addition, traﬃc condition, namely, demand and supply of theroute, is also realistic.
Section 6.1 describes the details of a field experiment we conducted at an urban
expressway in Tokyo. Sections 6.2 and 6.3 describes the validation of the basic method ofchapter 3 and the method with CL of section 4.2, respectively, based on the data collected
via the experiment. Note that the method with FD and CL of section 4.3 is not validated
here because of technical limitations of the field experiment—the resolution of the data
was not suﬃcient for it.

6.1

Field experiment

The field experiment was conducted on Tuesday, September 24, 2013, from 15:00 to
16:00. The location was the counterclockwise direction in the Inner Circular Route of theMetropolitan Expressway, which is an urban expressway in Tokyo, Japan. The weather
was cloudy with occasional light rain. Twenty probe vehicles equipped with spacing andposition measurement devices were employed.

*1This chapter is mainly based on joint research with Dr. Takahiko Kusakabe and Prof. Yasuo Asakura,

published in the 34th JSTE Conference (瀬尾ら, 2014), ISTTT21 (Seo and Kusakabe, 2015, in press) andTransportation Research Part C (Seo et al., 2015a; Seo and Kusakabe, 2015, in press).

86Chapter 6

6.1.1

Field Experiment-based Validation

Inner Circular RouteThe Inner Circular Route (i.e., C1) is a ring-shaped urban expressway located in central
Tokyo. In the experiment, the probe vehicles ran in the counterclockwise direction.
The loop length is 14.2 km, and most sections have two lanes for cruising and passing.
The speed limit is 50 km/h. Figures 6.1 and 6.2 show a map and a schematic of C1,
respectively. The space coordinates denoted in Fig 6.2 are used throughout this chapter.
In Fig 6.2, a junction is abbreviated as JCT, arrows on the upper side represent the merging
from or diverging to other highway routes, and arrows on the lower side represent on/oﬀ
ramps. Note that the survey section was limited to an 11 km length section from the 0 km
point to the 11 km point, because a series of tunnels at the 11–14.2 km disturbed the GPSfunctions.
A large number of dual supersonic traﬃc detectors are installed in C1. They enable us
to obtain highly detailed and accurate data on the traﬃc flow variables*2 . The traﬃc
detectors observed the flow, spot speed, and spot occupancy directly, and these wereconverted to density and speed for validation of the proposed method. The observation
with the traﬃc detectors is performed at a time resolution of 1 min and average spaceresolution of 250 meters per lane. In this experiment, these detector observed data areused for comparison purposes. Fig 6.3 shows the density observed by the detectors in thecruising lane as a time–space diagram; where the horizontal axis indicates the time andfuture is to the right, the vertical axis indicate space and downstream is in the upward
direction. Some remarkable events were occurred during the experiment. For example,
Edobashi JCT at the 8 km point become a merging bottleneck and caused a 3 km queue
was formed at 15:50. Queue spillovers from other highway routes were observed via
Takebashi JCT at the 10 km point and Hamazakibashi JCT at the 3.8 km point.

*2

白石ら (2012) reported that measurement error in 5 min flow of a dual supersonic traﬃc detector wasusually less than 10%.

87

Chapter 6

Field Experiment-based Validation

Figure 6.1: Map of the Inner Circular Route.
Tanimachi JCT
Hamazakibashi JCT
Ichinohashi JCT

Takebashi JCT
Miyakezaka JCT
Edobashi JCT

loop
0km

2km

4km

6km

8km

10km 11km 12km

Figure 6.2: Schematic of the Inner Circular Route (counterclockwise direction).

Figure 6.3: Time–space diagram of density in C1 as observed by the detectors.

88

14km

Chapter 6

Field Experiment-based Validation

Figure 6.4: Preparation of probe vehicles and equipments.

6.1.2

Probe vehicle data

Twenty standard-sized passenger vehicles with non-professional drivers were employed
as the probe vehicles. The vehicles were equipped with mono-eye cameras and GPS
loggers on their dashboards to identify and record the spacing and position.

The

survey section was limited to the cruising lane of the 0–11 km section because of law
restrictions*3 and the GPS function. Most of the probe vehicles drove three laps on C1,
and a total of 59 laps were performed during the experiment period. This corresponded to42.1 veh/h/lane. The flow of all vehicles in the survey section of C1 was 1255.2 veh/h/laneaccording to the detector data. Therefore, the PR was roughly calculated to 3.5%. Probevehicle data, including positions and spacing, were collected at intervals of 15 s. Thus,
an average number of 38.8 data points was recorded per lap. Fig 6.4 shows preparation ofprobe vehicles and data collection equipments.
(1)

Position data

Vehicle positions were identified by the GPS data and simple map-matching procedure.
The procedure is described as follows:
1. a digital map consists of the experiment site of links and nodes was acquired froman external source,*3

According to Japan’s Road Traﬃc Act, vehicles should drive in the cruising lane unless changing

routes, overtaking within the speed limit, or avoiding danger.

89Chapter 6

Field Experiment-based Validation(a) Raw data of trajectories plotted on a (b) Map-matched trajectories on time–space diagram of speed
map
Figure 6.5: Trajectories of probe vehicles.

2. vehicle coordinates identified by GPS were matched to their closest point on thelinks,
3. data points located at 11–14.2 km section where the series of tunnels exist were
removed.
Fig 6.5 shows the trajectories of the probe vehicles. Fig 6.5a shows the raw data plottedon a map of C1 where the red solid line indicates the shape of C1 and small circles
indicate the vehicle coordinates. According to the figure, the measurement errors were notsignificant excluding tunnels. Fig 6.5b shows the map-matched data on the time–space
diagram with speed; the trajectories seemed to be reasonable. For example, the probevehicles tended to be slow (i.e., the slopes of trajectories decreased) when they went
through the bottleneck at the 8 km point just like other vehicles (i.e., the plot color wasdark). Although small errors were found in the GPS measurement and map-matching, theproposed traﬃc state estimation method is not significantly aﬀected by such small errors
as mentioned in section 3.6.

90

Chapter 6(2)

Field Experiment-based Validation

Spacing data

This experiment employs a spacing measurement method using a mono-eye camera
equipped on the dashboard of the probe vehicles. Fig 6.6 shows some images taken by
the cameras on the probe vehicles during the experiment; typical road environments andvehicle characteristics at the C1 are shown. The procedure of the spacing measurement
method is described as follows:
1. the leading vehicles of the probe vehicles in the image were identified,
2. the apparent size of the leading vehicles’ body width were measured,
3. the spacing were calculated based on the apparent size, its assumed actual size,
angle of view of camera, etc.
The reason for the simple method is because implementation is relatively easy and theysatisfy the requirement for this experiment. That is, the method is• capable of detecting the leading vehicle at relatively long distance that often appears
in highways,• capable of determining the lane driven by the probe vehicle and the leading vehicle,
and• based on a clear logic (i.e., not a black box).
Note that any methods of measuring spacing are considerable for the proposed traﬃcstate estimation methodology as long as the measurement method satisfies the required
capability.
Specifically, the spacing s was estimated from images taken by the probe vehicles
based on the following pinhole camera model assuming that the road is not curved.
s=

2wp0
+ l,
p tan(θ0 /2)(6.1)

where w is the actual body width of the leading vehicle, p0 is the width of the image, p is
the apparent size of the body width of the leading vehicle in the image, θ0 is the horizontal
angle of view of the camera, and l is the actual body length of the leading vehicle. Here,91

Chapter 6

Field Experiment-based Validation

Figure 6.6: Examples of images taken by the cameras on the probe vehicles.

the variables p0 and θ0 were known. The actual body width w was assumed to be 1.92 m.
This assumption comes from prior knowledge on the average width of a standard vehicle(1.8 m), large vehicle (2.4 m), and the average ratio of large vehicles on the route (20%).
The actual body length l was assumed to be the same as that of the probe vehicles (5 m).
Variable p was manually measured from the images. If these variables have errors (i.e.,
wrong assumptions on w and l, measurement error in p), the estimated spacing may bebiased. The errors aﬀect biases almost proportionally: therefore, the biases are not verysignificant. If the spacing was not measurable (i.e., the leading vehicle was not visible),a value of 100 m was assumed for the value. The most frequently appeared reason of the
failure in spacing measurement was too large distance to leading vehicles. Therefore, thelarge value that is close to the maximum detectable range was substituted for the missing
spacing data. Such cases occurred for 1.86% of all the data.
Fig 6.7 shows a part of the probe vehicle data as a time–space diagram. A dot
represents a position datum collected by a probe vehicle. Data between two data points arelinearly interpolated. Red areas represent areas between probe vehicles and their leading
vehicles.

92Chapter 6

Field Experiment-based Validation

Figure 6.7: Part of the probe vehicle data.

If the assumed or measured variables have errors, the estimated spacing will haveerrors. We cannot determine the amount of the errors directly; because the ground truth
data for the vehicle size during the experiment do not exist. However, the method may
not be biased significantly and practicable for the experimental measurement because theassumed variables were based on common knowledge on statistics and regulations anddetector data. In addition to that, the measurement was based on high resolution images.
(3)

Basic validation

In order to validate the probe vehicle data, we compared the data with the spacing and spot
speed of all of the vehicles at four specific locations as calculated by the detectors’ pulse
data*4 . Figures 6.8 and 6.9 show the speed and spacing distributions. Their statistics√(mean µ, standard deviation σ, and Kolmogorov–Smirnov (K–S) test value nDn ) aresummarized in Table 6.1. If the K–S test value is larger than 1.36, the diﬀerence between
probe vehicle data and detector data is statistically significant at the 5% level. According
to these comparisons, there considered to be diﬀerences between the two data at some*4

The detectors’ pulse data include every individual vehicle’s spot speed, body length, and time headway

at the detector location.

93

Chapter 6

Field Experiment-based Validation(a) At 0.390 km point.

(b) At 2.980 km point.

(c) At 7.140 km point.

(d) At 8.640 km point.

Figure 6.8: Spot speed distributions in probe vehicle data and detector data.

(a) At 0.390 km point.

(b) At 2.980 km point.

(c) At 7.140 km point.

(d) At 8.640 km point.

Figure 6.9: Spacing distributions in probe vehicle data and detector data.

locations. The probe vehicles had slightly faster speed and larger spacing at the threepoints (0.390 km, 2.980 km and 7.140 km points), and the reverse occurred at the8.640 km point. The probe vehicles’ routes were assigned to maintain cruising lane,
therefore, their driving behavior may have slightly diﬀered from that of other vehicles ofwhich paths were going to diverge or had just merged. For example, the 8.640 km point
was located downstream of the merging at Edobashi JCT (c.f., Fig 6.2). At this location,
vehicles from C1 are often required to slow down to merge because of its road alignment.
Therefore, the probe vehicles may have tended to be slower than other vehicles. Becauseof these reasons, the probe vehicle data may contain location-specific biases. This meansthat the random sampling assumption was not rigorously satisfied in the experiment.

6.2

The basic method

We quantitatively analyzed the traﬃc state estimation results of the basic method
presented in chapter 3.

Section 6.2.1 visualize examples of estimation results as
94

Chapter 6

Field Experiment-based Validation

Table 6.1: Spot speed and spacing statistics of probe vehicle data and detector data.
0.390 km point

Speed

Spacing

2.980 km point

7.140 km point

8.640 km point

Detector data (µ, σ) [km/h](24.2, 10.7)(25.8, 10.5)(26.3, 11.4)(29.7, 09.3)

Probe vehicle data (µ, σ) [km/h]√
K–S test value nDn(27.7, 11.8)(28.4, 06.8)(29.5, 10.2)(26.2, 11.0)

0.684

1.248

0.763

1.568

Detector data (µ, σ) [m](19.4, 11.2)(19.2, 11.7)(20.7, 12.3)(21.5, 11.8)

Probe vehicle data (µ, σ) [m]√K–S test value nDn(21.8, 11.5)(22.4, 09.9)(24.5, 11.4)(19.7, 08.5)

1.008

1.784

1.503

0.518

time–space diagrams in order to investigate how well the traﬃc phenomena can be
reproduced by the proposed method. Section 6.2.2 summarizes indices for accuracy andprecision of the proposed method for various settings of the PR and the resolution. Section
6.2.3 describes relation between accuracy and precision of the proposed method and thetraﬃc state.

6.2.1

Visualizations of estimation results

Fig 6.10 visualizes traﬃc states estimated by the proposed method (left side: a, c, e),as well as ones observed by the detectors (right side: b, d, f). The time and spaceresolution of the estimation were set to 5 min and 500 m, respectively, that is close to
average distance between detectors and detector’s data aggregation time interval in urban
expressways in Japan. The PR was 3.5%. At this resolution and PR, the expected numberof probe vehicles per region was 3.83; therefore, bias in the estimator is expected to besmall (c.f., section 3.3). Traﬃc phenomena that were observed in the traﬃc detector data
could be recognized in this estimation result. For example, Fig 6.10c shows a significant
queue whose head was at the 8 km point and that propagated to a length of around 3 km
at 15:50. There were other congested areas whose heads were at around the 3.5 km and
10 km points. These characteristics were almost identical to those described in chapter 3
and Fig 6.10d by the detector data.
The resolution of the estimation can be set flexibly corresponding to the objectiveof the analysis and PR (c.f., section 3.6). Lower PR scenarios were generated from95

Chapter 6

Field Experiment-based Validation(a) Flow estimated by the proposed method.

(b) Flow observed by the detectors.

(c) Density estimated by the proposed method.

(d) Density observed by the detectors.

(e) Speed estimated by the proposed method.

(f) Speed observed by the detectors.

Figure 6.10: Visualization of estimated flow, density, and speed.

random sampling of the probe vehicles. Fig 6.11a shows the density estimation resultat a resolution of 30 min and 4 km with a 1% PR; Fig 6.11b shows the density observedby the detectors aggregated with the corresponding resolution. The estimation results
showed the tendencies of the traﬃc flow at lower resolutions. For example, the 0–4 km
section had a relatively dense traﬃc flow, while the 4–8 km section tended to have sparse
traﬃc at 15:00–15:30 and dense traﬃc at 15:30–16:00.
96

Chapter 6

Field Experiment-based Validation(a) Density estimated by the proposed(b) Density observed by the detectors

methodFigure 6.11: Visualization of estimated density with lower PR and lower resolution.

6.2.2

Eﬀect of time–space resolution and PR

To quantify the relations among accuracy and precision of the estimation, time andspace resolution, and PR, we compared accuracy and precision indices for the estimation
scenarios. The bias and the root mean square percentage error (RMSPE) are employedas accuracy and precision indices, where Bias(θ̂) = E[θ̂ − θ] and RMSPE(θ̂) =√E[((θ̂ − θ)/θ)2 ]. Variable θ denotes a traﬃc flow variable (one of q, k and v) at a certain
time and space observed by the detectors; and variable θ̂ is one of those estimated fromthe probe vehicle data. In addition to that, analytically expected bias and precision shownd θ̂) and RMSPE([ θ̂), were estimated for every scenario.
in eq (3.21)–(3.24), that is, Bias(
The estimation was based on eqs (3.21)–(3.24) by estimating expectation of mean andvariance spacing and time headway from the probe vehicle data. The parameters set for
scenarios were PR P (%), time resolution ∆t (min), and space resolution ∆x (km). The
PR P was 3.5% or 0.2%. In scenarios where P = 3.5%, data from all of the probevehicles were used for the estimation. In scenarios where P = 0.2%, data from two
randomly sampled laps were used on average (i.e., the expected flow of probe vehicles
is 2 veh/h). The indices for the lower PR scenarios were calculated from the results of
20 estimation iterations. In each iteration, probe vehicle data for the estimation were97

Chapter 6

Field Experiment-based Validation

composed of diﬀerent combinations of probe vehicles’ laps whose PR was expected
to be the scenario parameter P. The time resolution ∆t was 5 min or 60 min, whichcorresponded to the resolutions used for Japanese highway traﬃc management and traﬃc
census data, respectively. The space resolution ∆x was 0.5 km or 11 km, which roughly
corresponded to the usual detector interval and entire survey section length, respectively.
Table 6.2 summarizes the results of above estimation. Note that the mean values ofthe flow, density, and speed observed by the detectors were 1255.2 veh/h, 40.7 veh/km
and 30.8 km/h, respectively.
The biases were aﬀected by three diﬀerent factors: the analytical characteristics of theestimators (c.f., sections 3.3 and 5.1), biases in the probe vehicles’ driving behavior (c.f.,sections 3.6 and 6.1.2), and biases in the spacing measurement method (c.f., section 6.1.2).
d θ̂), were almost
According to Table 6.2, amounts of the analytical expected biases, Bias(
negligible compared with the amounts of states for most of scenarios. It is also confirmedby the simulation-based validation shown in section 5.1. However, the actual biases on theflow and density calculated from the detector data were not identical to the analytically
expected ones. For example, most of the actual biases were negative while the analytical
expected one is positive. This can be explained by the probe vehicles’ driving behavior
biases as follows. In the (3.5%, 60 min, 11 km) scenario, whose analytically expected bias
is small, slightly underestimated the flow, density, and speed. This suggests that the probevehicles’ driving behavior was slightly biased compared with the rest of traﬃc (section
3.6): the probe vehicles had 10% slower speed than other vehicles on average, and they
had almost the same spacing with other vehicles on average. The amount of bias causedby the spacing measurement method can be expected to be small as mentioned in section
6.1.2. As consequence, the results implied that the amount of bias decreases as P, ∆tand ∆x increased, as the analytical results suggested. However, in the field experiment,
the probe vehicles’ driving behavior was slightly biased; therefore, the amount of bias issignificantly smaller than the analytical result. The results of simulation-based validation
shown in section 5.1 support this interpretation.
The tendencies in the precision RMSPE can be found as analytically expected; the98Chapter 6

Field Experiment-based Validation

Table 6.2: Precision and bias indices over estimation scenarios.
(a) Flow.
Scenario parameters

Errors against the detector data

Analytically expected errors

P (%)∆t (min)∆x (km)RMSPE(q̂) (%)

Bias(q̂) (veh/h)[ q̂) (%)
RMSPE(d q̂) (veh/h)
Bias(

0.2

5

0.5

43

21.1

67

111.5

0.2

5

11

33

−48.7

25

26.6

0.2

60

0.5

25

−59.9

27

37.3

0.2

60

11

16

−159.0

13

29.6

3.5

5

0.5

26

−82.4

17

42.6

3.5

5

11

14

−151.9

7

5.2

3.5

60

0.5

14

−147.1

6

4.5

3.5

60

11

13

−161.43

1.2(b) Density.
Scenario parameters

Errors against the detector data

Analytically expected errors

P (%)∆t (min)∆x (km)RMSPE(k̂) (%)

Bias(k̂) (veh/km)[ k̂) (%)
RMSPE(d k̂) (veh/km)
Bias(

0.2

5

0.5

49

−0.1

69

4.1

0.2

5

11

45

2.1

27

3.3

0.2

60

0.5

28

−2.9

44

1.3

0.2

60

11

13

−1.1

17

1.1

3.5

5

0.5

28

−2.6

17

1.2

3.5

5

11

15

−0.6

11

0.5

3.5

60

0.5

14

−4.2

7

0.2

3.5

60

11

3

−1.4

4

0.1

(c) Speed.
Scenario parameters

Errors against the detector dataP (%)∆t (min)∆x (km)RMSPE(v̂) (%)

0.2

5

0.5

43

0.4

0.2

5

11

31

−0.3

0.2

60

0.5

23

0.7

0.2

60

11

11

−3.1

3.5

5

0.5

18

−0.7

3.5

5

11

13

−3.0

3.5

60

0.5

12

−0.6

3.5

60

11

10

−3.0

Bias(v̂) (km/h)

99Chapter 6

Field Experiment-based Validation

precision improves as P, ∆t and ∆x increased. However, the actual precision was lower
than the analytically expected one in most of the scenarios. This is also able to be
explained by the probe vehicles’ driving behavior biases. Therefore, if biases in the probevehicle data did not exist (i.e., if the random sampling assumption were satisfied), RMSPE
is expected to improve.

6.2.3

Eﬀect of traﬃc situation

Fig 6.12 compares the estimated variables and the corresponding values observed by thedetectors based on the results of the (3.5%, 5 min, 0.5 km) scenario. The colors represent
the number of cases. The estimated variables correlated well with the corresponding
values observed by the detectors in most cases. However, the relative error for the density
tended to be larger when the density was low. This may be due to the variances in the mean
headway. The variances tended to be larger at lower densities than at higher densities,
and the accuracy and precision may have suﬀered, as suggested by eqs (3.21)–(3.24) insection 3.3. This means that the estimation accuracy and precision depend on the traﬃcstate, although the amount of error was not significant among traﬃc states from free-flow
to congested state.
Note that the data do not contain situations where the flow was low, such as sparse
traﬃc, extremely congested traﬃc, and signalized traﬃc. Therefore, the capabilities ofthe method in such situations were not validated with this data. According to section 3.3,
the accuracy and precision may decrease for sparse traﬃc and signalized traﬃc becauseof the larger variance in the mean headway distribution.

6.3

The method with CL

In order to investigate the applicability of the proposed method to the real world, theproposed method was validated by using data collected from a field experiment at an
actual road.

100

Chapter 6

Field Experiment-based Validation(a) Flow.

(b) Density.

(c) Speed.
Figure 6.12: Scatter plots of estimated variables versus the corresponding values observed by the detectors.

6.3.1

Ground truth

For a comparison purpose, Fig 6.13a–d show the state (flow q, density k, speed v) observedby the detectors (basically, they means the same state with Fig 6.10b, d, f) and cumulative
flow Ñ as time–space diagrams. The contour lines of Ñ were drawn at intervals of 200
veh. The cumulative flow was calculated from the flow observed by detectors with aprobe vehicle’s trajectory as a reference point. Discontinuities in the contour lines present
junctions (merging/diverging sections from/to other highway routes) in C1; these arerepresented as arrows on the upper side in Fig 6.2. Note that the cumulative flow Ñ did
not always increase as t increased and x decreased when there were no discontinuities.
This is because of the on/oﬀ-ramps, lane-change by vehicles, and systematic errors in themeasurements of the detectors.

101

Chapter 6

6.3.2

Field Experiment-based Validation

Results

The estimation targets were the traﬃc state at certain time and space resolutions and thecumulative flow in the survey section. The cumulative flow was calculated at 1 min timeintervals and 100 m space intervals according to eq (4.5). The density k̂ was calculated
based on central diﬀerences at 100 m intervals according to eq (4.6). In order to compare
the estimated states with the detector data, the estimated states were averaged at a time
resolution of 5 min and space resolution of 500 m. The proposed method requires the
positions of the discontinuity points to be specified. In this estimation, only junctions
were considered to be discontinuity points. Thus, on/oﬀ-ramps and lane-change wereignored.
Fig 6.13e–h show the estimated traﬃc state (q̂, k̂, v̂) with a 5 min × 500 m resolution
and cumulative flow Ñˆ as a time–space diagram. The contour lines of Ñˆ were drawn
at intervals of 200 veh. Through a comparison with Fig 6.13a–d, the estimation canbe used to acquire the tendencies of the detectors’ observed variables (e.g., a queue
propagation from the 8 km point). Fig 6.14 compares the estimated variables anddetectors’ observed variables as scatter diagrams. According to the figures, the estimatedvariables and detectors’ observed variables correlated well. However, the values tended to
be underestimated. One of the reasons was a bias in the probe vehicles’ driving behavior(i.e., their driving behaviors tended to be safer, which meant a slower speed and larger
spacing than for other vehicles).
Table 6.3 summarizes the error indices of the traﬃc state estimation for diﬀerent
PRs. The 0.2% PR was reproduced by randomly sampling all of the probe vehicles.
This corresponded to roughly 2 probe veh/h. The error indices of scenarios with a 0.2%
PR were calculated from 25 iterations under diﬀerent sampling conditions of the probevehicles. According to the results, the estimation precision clearly increased with the PR.
Note that most of the scenarios underestimated the states, similar to Fig 6.14 and for thesame reasons. Table 6.4 compares results of the proposed method and the method without
considering the CL; the precision was improved with the proposed method.

102

Chapter 6

Field Experiment-based Validation(a) Flow q observed by detectors.

(b) Density k observed by detectors.

(c) Speed v observed by detectors.

(d) Cumulative flow Ñ observed by the detectors.

(e) Flow q̂ estimated with proposed method.

(f) Density k̂ estimated with proposed method.

(g) Speed v̂ estimated with proposed method.

(h) Cumulative flow Ñˆ estimated with proposed
method.

Figure 6.13: Time–space diagrams of traﬃc state and cumulative flow in field experiment.

103

Chapter 6

Field Experiment-based Validation(a) Flow q.

(b) Density k.

(c) Speed v.

(d) Cumulative flow Ñ.

Figure 6.14: Scatter diagrams of estimated variables vs. detectors’ observed variables in field experiment.

Table 6.3: Error indices for scenarios with proposed method in field experiment.
PR
P

Flow (veh/h)Density (veh/km)

Speed (km/h)

RMSPE(q̂)

Bias(q̂)

RMSPE(k̂)

Bias(k̂)

RMSPE(v̂)

Bias(v̂)

0.2%

30%

−158.9

39%

−4.9

60%

0.9

3.5%

23%

−133.9

30%

−2.3

24%

−2.1

104

Chapter 6

Field Experiment-based Validation

Table 6.4: Comparison between results of field experiment with the method with CL and the basic method.
P

RMSPE(q̂)

RMSPE(q̂withoutCL )

PoI

0.2%

30%

43%

30%

3.5%

23%

26%

13%

Table 6.5: Diﬀerence between validations based on simulation experiment and field experiment.
The simulation experiment

The field experiment

FIFO condition

Always satisfied

Not always satisfied

CL

Always satisfied

Not always satisfied

Traﬃc flow model

Car-following model of Gipps (1981)

Actual traﬃc

Traﬃc condition

Synthetic scenario

Actual traﬃc

Driving behavior of probe vehicles

Not biased

Slightly biased

Ground truth

Available

Detectors

RMSPE(q̂) at P=0.2%

59%

30%

RMSPE(q̂) at P=3.5%

42%

23%

Bias(q̂) at P=0.2%

151.0 veh/h−158.9 veh/hBias(q̂) at P=3.5%

83.6 veh/h−133.9 veh/h

PoI due to incorporating CL at P=0.2%

12%

30%

PoI due to incorporating CL at P=3.5%

27%

13%

6.3.3

Comparison between simulation experiment- and field
experiment-based validations

Table 6.5 summarizes the diﬀerence between the simulation experiment (section 5.2) andthis field experiment. The simulation experiment was conducted under traﬃc conditions
where the assumptions of the proposed method were always satisfied. On the other hand,the field experiment-based validation had some limitations, such as slight violations of theassumption. However, it allowed the proposed method’s characteristics to be validated
under actual traﬃc conditions. This advantage is important for validating the proposed
method’s practical characteristics because of the proposed method’s strong dependence
on the spacing, which was determined by a synthetic model (i.e., car-following model) in
the simulation experiment.
105

Chapter 6

Field Experiment-based Validation

The validation results suggest that the proposed method works well in an actual
traﬃc environment, although the assumptions of the proposed method (i.e., the CL) were
not exactly satisfied because of the on/oﬀ-ramps and lane-changing. For example, theprecision increased with the PR in both the simulation (Table 5.5) and field experiment(Table 6.3). In fact, the proposed method showed better precision in the field experimentthan in the simulation experiment.

Ironically, one of the reasons is existence oflane-changing. They decrease the variance in the mean headway; then the estimation
precision of the cumulative flow improved (c.f., eq (3.23)).
The notable diﬀerence between the two validation is a sign of the biases. The
simulation-based validation (Table 5.5) and analytical approximation eq (3.21) showed
positive biases, while the field experiment-based validation showed negative biases. Thismay be due to biases in the probe vehicles’ driving behaviors. The probe vehicles tended
to be slower than other vehicles, according comparison between the probe vehicle’s speed
and traﬃc speed observed by detectors.
Incorporating the CL improved the precision in both validations (Tables 5.6 and 6.4).
However, in the field experiment scenario, the PoI decreased as the PR was increased;
this tendency was the inverse of that in the simulation-based validation. This may be due
to violations of the assumptions. If a large amount of probe vehicle data is available (i.e.,
the traﬃc state can be estimated precisely even without the CL being incorporated) the
non-exact assumptions of the CL will negatively aﬀect the precision.

106

Chapter 7 Conclusions
Traﬃc data are crucial for today’s civilization as they are required for understanding
behavior of traﬃc, developing traﬃc control measures, and executing the measures.
However, current situation of traﬃc data collection is far from perfect, due to practicaland theoretical limitations. Current Eulerian data collection method, namely, detectors,
only work at its installed point, while the number of installed points are limited. Current
Lagrangian data collection method, namely, GPS-equipped probe vehicles, cannot collectquantity-related information. To address this issue, this dissertation proposed traﬃc state
estimation methods utilizing probe vehicle with spacing measurement equipment, which
can be applicable in the near future due to technological advancement.

7.1

Achievements

This dissertation proposed traﬃc state estimation methods utilizing probe vehicle withspacing measurement equipment. First, our literature survey outlined that capability ofexisting traﬃc data collection/state estimation methods by defining traﬃc data generally.
Second, methods of estimating traﬃc states and/or traﬃc flow model parameters fromthe collected data were formulated and analytically investigated. Third, the formulated
methods are empirically validated based on data obtained via a field experiment in thereal world, as well as synthetic simulation experiments. As a result, methodology thatenables us to estimate traﬃc state for vast time–space domain, is established. It may
contribute to monitoring traﬃc in everywhere, enhancing its eﬃciency, and understanding
wide-ranging traﬃc dynamics in the near future.
In chapters 1 and 2, from both perspectives of academic and practical, a challenging
task remained in traﬃc and transportation field is clarified: traﬃc state estimation
with probe vehicles. Traﬃc state estimation—estimating flow, density and speed frompartially observed and/or partially unreliable data—is essential to understand and manage
107

Chapter 7

Conclusions

automobile-based movements of people and goods (i.e., traﬃc) in modern civilization.
Academic research on traﬃc and transportation engineering is identical to understanding
traﬃc; and managing traﬃc is important in terms of economical and environmental
eﬃciency in practical. However, estimating traﬃc state in anytime and anywhere is notan easy task due to observation cost, required theoretical assumptions and privacy issues.
Fixed observation such as detectors cannot collect data from wide range with feasible
cost; and conventional probe vehicles require strong exogenous assumptions or violate
privacy to estimate traﬃc state. As possible solution for this problem is found in recent
advancement in vehicle technologies: on-vehicle spacing measurement equipment.
In chapters 3 and 4, methodology of estimating traﬃc state using probe vehicles
with spacing measurement equipment is formulated. The basic method in chapter 3 isformulated as requires less exogenous assumptions as possible. It is consistent withtraditional definition in traﬃc flow theory. These characteristics enable the method
possible to be applied in various situations without theoretical discrepancies. Qualitative
characteristics of the method is analytically investigated in detail. The method’s accuracy
and precision are expected to improves as time–space resolution of estimation lowered
and probe vehicles penetration rate increased.
The methods in chapter 4 are formulated based on the basic method considering
additional assumptions that are often accepted by traditional traﬃc flow theory.
These assumptions are integrated to the methods in order to improve the proposed
methodology’s reasonability in common traﬃc situations, as well as to improve accuracy
and precision of its estimation. Namely, one of the methods considers a conservation
law of traﬃc. A conservation law is incorporated with the estimation method by utilizing
information of probe vehicles trajectories, which is ignored by the basic method. The
other method in chapter 4 considers a fundamental diagram of traﬃc as well: a functional
form of a fundamental diagram is given exogenously but the parameters are estimatedendogenously. A fundamental diagram is a key concept describing traﬃc dynamics (i.e.,
time–space transition of traﬃc state) and vehicle behavior. The method estimates traﬃcstates and fundamental diagrams simultaneously; then they are integrated by using a data108

Chapter 7

Conclusions

assimilation framework.
In chapters 5 and 6, the proposed methodology is empirically validated; and practical
performance of the proposed methodology is clarified. In chapter 6, actual probe vehicle
data were collected via a field experiment conducted in an urban expressway in thereal world. The actual data are preferable for validating the proposed methodology
because spacing calculated by synthetic models is not always realistic. On the other
hand, the actual data contain unpreferable factors such as probe vehicle’s bias andmeasurement errors. These unpreferable factors are complemented by validating the
proposed methodology using synthetic data generated by simulation experiment inchapter 5. As a result of the validation, relation among the estimation methods’ precision
and accuracy, time–space resolution and penetration rate of probe vehicles is clarified
quantitatively. It is confirmed that the basic method’s characteristics are fairly consistent
with the analytical investigation. The eﬀect of incorporating traﬃc dynamics into thebasic method is also confirmed. For example, the methods’ precision improves as
resolution lowered and penetration rate increased, as analytical results suggested. In
partially congested urban expressway, the basic method can estimate 5 min traﬃc volume
fairly well with 3.5% penetration rate; and it can estimate 1 hour traﬃc volume precisely
with 0.2% penetration rate (i.e., 2 probe vehicle/h/lane). The integration of a conservation
law and a fundamental could improve precision by 10–30%. As results, the methods’
applicability in actual traﬃc environment is shown*1 .
Summary of the proposed traﬃc state estimation methods is as follows. The methods
estimate traﬃc state based on data collected by probe vehicles with spacing measurement
equipment. The basic method relies only on random sampling and error free assumptions.
It means that the method can be applicable to general traﬃc without theoretical
discrepancies. The other two methods employ existence of conservation law and/or afundamental diagram as additional assumption(s), in order to incorporate traﬃc dynamics.
Therefore, they can get reasonable results (i.e., accurate and precise estimation results) if*1

Another field experiment in an arterial road with mixed traﬃc also supports this. See appendix B for

its summary.

109

Chapter 7

Conclusions

the assumptions are actually satisfied. The validation result suggested that the proposed
methodology can estimate precise traﬃc state, which can be applicable for traﬃc control,with few percentages of probe vehicle penetration rate in partially congested highway.
On the other hand, the proposed methods can estimate lower resolution traﬃc state likehourly traﬃc volume, which can be applicable for transportation planning, with few probevehicles per hour.

7.2

Future directions

The future works to this dissertation are twofold: to sophisticate the proposed TSE
methods and to extend the proposed methodology for other general transportation fields.

7.2.1

The proposed TSE methods themselves

Following improvements are considerable to the proposed TSE methods.
Less biased estimators. The basic estimators (3.14) and (3.15) are biased.

The

third-order term in the Taylor series may be valuable in formulating less biased
estimators, since the approximation was shown to be reasonable (c.f., sections 3.5,
5.1 and 6.2).
Unbiased estimations method that can be applied to biased probe vehicle data. It
will relax the random sampling assumption. It is an important topic because theadvanced technologies may cause biases in the probe vehicles’ driving behavior.
Use of information on the surrounding vehicles of the probe one, which are
expected to be not biased, may be valuable to obtain the unbiased estimation.
Adaptive estimation of FD parameters in the method with FD and CL. It will relax
the assumption of constant FD and improve estimation capability. To achieve this,two approaches are considerable: automatic clusterization of region B based onprobe vehicle data and adaptive parameter estimation through data assimilation(c.f., 樋口, 2011).
110Chapter 7

Conclusions

Incorporation of Lagrangian traﬃc flow model. It will improve reasonability andperformance of a TSE method, since the Lagrangian coordinate system can utilize
information of vehicle trajectories (c.f., Yuan et al., 2012; Laval and Leclercq,
2013; Wada et al., 2015). However, it requires explicit consideration of non-FIFO(multi-lane) and multi-class traﬃc, which is ignored by the existing studies.
Incorporation of node conservation law. It is required to make the method applicablein networks; and requires merging/diverging models. Since probe vehicle data can
be available throughout a network (and some links may not have probe vehicles
at a moment), considering a node conservation law may improve reasonability andperformance of estimation (c.f., Castillo et al., 2013).
Consideration of practical problems. In a real world application, several problems
that are almost ignored by the proposed TSE methods may arise. For example,
first, leading vehicles cannot be always recognized because of excessive distances,
curves, etc., as described at section 3.6. Therefore, imputation methods for such
missing spacing are required. Second, exact vehicle length is not known. This
problem can be solved with on-line estimation of vehicle length distribution bymeasuring it in neighbor lanes. Third, the proposed TSE methods may have low
precision at surface streets which have intersections, signals, parking, etc. Current
methods require lowering time–space resolution in order to get high precision undersuch situations. However, low resolution information is sometimes useful, such ashourly traﬃc volume and annual average daily traﬃc for planning purposes and
MFD for possible area-wide traﬃc managements. Nevertheless, if the signal timing
information is known, the information can be incorporated with the method with FD
and CL (c.f., Mehran et al., 2012; Wada et al., 2015).

111

Chapter 7

7.2.2

Conclusions

Transportation system estimation with travelers observing
other travelers

In chapter 1, it was mentioned that information is a key to enhance eﬃciency of atransportation system; and the challenging problem is to collect information from theentire system so that the system becomes understandable and controllable. As a solution
to the problem, this dissertation proposed methodology for estimating transportation
system’s states and traveler’s model parameters using travelers who observe othertravelers in the system.
This dissertation formulated and validated a special case of the problem, namely,
traﬃc state estimation, where travelers’ movement is one-dimensional (in spatial). Thismethodology can be generalized for network-wide movement, two-dimensional movement and day-to-day movement, such as traﬃc network, pedestrian space, multi-mode
urban transportation, and day-to-day learning/adaptation, at least in theoretical*2 . It may
be worth to be investigated, though it will arise (have arose) a lot of diﬃcult problems intechnological and ethical.

*2

Simple examples exist for traﬃc networks and mixed traﬃc. See appendices A and B for theirsummaries.

112Appendix A Expansion to Network: OD Flow
This appendix*1 describes how the proposed method can be expanded for estimation of
network traﬃc. Specifically, OD flow estimation method using PVSME data is developed
and validated using simulation experiment.
OD flow, demand and matrix are valuable information for transportation planning.
Typical approach in existing studies is to estimate current (dynamic) OD matrix using
current partial collected traﬃc data and prior information on OD matrix, assuming UE
conditions (Bell, 1991; Yang et al., 1992; Wei and Asakura, 2013). Yamamoto et al.
(2009) applied GPS-equipped probe vehicle data for the approach by assuming FDs.
Other approach estimates dynamic OD flow using AVI data without prior information
on an OD matrix (Asakura et al., 2000).
In the spirit of this dissertation, the method should be containing assumptions as lessas possible (e.g., prior knowledge on OD matrix and route choice principles must be
eliminated). In fact, it is possible by simply expanding the basic method described inchapter 3.

A.1

Formulation

Probe vehicles are assumed• to measure position and spacing without errors,• to be randomly sampled from all the vehicles, and• to report their OD pairs.
The third condition is added compared to the basic method. Network schematics are
assumed to be known. An example of probe vehicle data under above conditions are*1

This appendix is mainly based on joint research with Prof. Yasuo Asakura, published in the 49th

Conference of JSCE Infrastructure Planning and Management (瀬尾・朝倉, 2014).

113

Appendix A

Expansion to Network: OD Flow

Space
Dest.

3

Node 32

Node 2

1

Node 1

23

Orig./Dest.

12

Orig.

Time

Figure A.1: Example of probe vehicle data in network

shown in Fig A.1. The network is consists of three nodes and two links. Node 1 is a
producing node, node 2 is a producing and attracting node, and node 3 is an attracting
node, of traﬃc; therefore, OD pairs are 1 → 2, 1 → 3 and 2 → 3. Red thick lines
represent probe vehicles’ trajectories; red filled areas represent observed time–space
regions by probe vehicles; black lines represent non-probe vehicles’ trajectories; solidlines represent vehicles’ trajectories whose OD pair is 1 → 3; and dashed lines represent
vehicles’ trajectories whose OD pair is 1 → 2 or 2 → 3.
The objective is to estimate link flow who has an particular OD pair in a network. By
expanding the basic estimator of flow (3.14), estimator for flow per OD pair can be easily
defined as∑m∈P(uw) (A) dm (A)q̂(uw) (A) = ∑,m∈P(A) |am (A)|(A.1)

where q(uw) (A) is a flow in a time–space region A whose OD pair is uw, and P(uw) (A)is a set of probe vehicles in A whose OD pair is uw. By substituting A with A(t)i j , that
is, a time–space region that covers link i j and time t in eq (A.1), a link OD flow can be
estimated—in the following part, the flow is represented as quw
i j if the time can be ignored.

114

Appendix A

Expansion to Network: OD Flow
Orig.
13, Highway

Dest.
3

1
Bottleneck
2

24, Arterial

4
Dest.

Orig.

Figure A.2: Network for simulation experiment.
Table A.1: OD matrix for simulation experiment.
3

4

1

600

300

2

300

600

Note that following relation is satisfied by the definition:q̂(A) =∑q̂(uw) (A).

(A.2)

uw

A.2

Simulation experiment-based validation

Numerical characteristics of the proposed method are validated using simulation
experiment.
Aimsun (TSS-Transport Simulation Systems) is employed as a simulator; where
vehicle behavior is described by the car-following model of Gipps (1981), route choice
behavior is described by quasi dynamic user optimum, and vehicle departure is described
as Poisson arrival. The network schematics and OD matrix are shown in Fig A.2 andTable A.1 that simulate a highway with a bottleneck and a parallel arterial road. The time
width of A is set to one hour.
An estimation result with 5% of PR is shown in Fig A.3; the estimated variables arewell corresponded to the ground truth. The RMSE of flow calculated from 10 iterations
is 36.6 veh/h.

115Appendix A

Expansion to Network: OD Flow

Figure A.3: Comparison between estimated flow and ground truth.

A.3

Remarks

This appendix demonstrated expansion of the basic method of chapter 3 to network flow
estimation. Only additional assumption is availability of information on probe vehicles’
OD pair. The validation results with small toy network suggested that the proposed
method works well. Even if number of combination of OD pairs are large, the method
may work by aggregating OD pairs appropriately.
Network traﬃc information such as OD flow is often employed for transportation
planning purposes, which require relatively low resolution data, such as hourly traﬃcvolume of a link and annual average daily traﬃc volume. As shown in section 6.2, datafrom PVSMEs with quite low PR can be utilized for precise estimation of low resolution
traﬃc information. Therefore, this application of PVSME data may be feasible in the very
near future, compared with traﬃc control-motivated TSE.
As a future direction, the method with FD and CL can be expanded to network flow
estimation. To do so, a conservation law at a node should be considered so that traﬃcstates in unobserved links can be interpolated.

116Appendix B Field Experiment at Arterial Road
with Mixed Traﬃc
In this appendix*1 , we focus on mixed traﬃc flow formed by car (i.e., four-wheeledvehicle) and motorcycle which is very popular in many Asian cities of developing
countries. Under the mixed traﬃc condition, motorcycles in small sizes which do not need
to follow the lane disciplines, could travel alongside with lane-based driving cars, follow
obliquely a lead cars with short headways (Robertson, 2002), swerve away to overtake or
to avoid collision with the preceding cars (Lee et al., 2009; Nguyen et al., 2012, 2014).
Therefore, the occurrence of motorcycles may weaken the lane discipline of the cars andcould make the traﬃc flow diﬃcult to be observed.

B.1

Field experiment environment

Pasteur Str. located in central area of Ho Chi Minh City, Vietnam was chosen for thesurvey location, as shown in Fig B.1a. The road is a one-way road with 3 lanes: two outerlanes for cars and one inner lane for motorcycles. However, cars and motorcycles could
share their lanes each other. The experimental road segment is approximately 1.5 km longwith 7 signalized intersections. Four fixed cameras were set at two diﬀerent places on theboth sidewalk of the road, as illustrated by circles in Fig B.1a to obtain validation dataof car flow rate. Probe vehicles were asked to run on the road segment and return to thestarting point for the next round by using Nam Ky Khoi Nghia Str. parallel to this road.
Taxis were employed as probe vehicle, equipped with a GPS logger and a video
camera. A GPS logger is put inside the taxi to log GPS coordinates every 15 seconds toa file. Then a logger application opens the file to provide data of position and speed. The*1This appendix is mainly based on joint research with Dr. Long Xuan Nguyen, Dr. Takahiko Kusakabe,Dr. Hong Tan Van and Prof. Yasuo Asakura, published in 19th HKSTS Conference (Nguyen et al., 2014).

117

Appendix B

Field Experiment at Arterial Road with Mixed Traﬃc(a) Map of Pasteur Str.

(b) View from probe vehicle(c) Part of probe vehicle data
Figure B.1: Experiment at Pasteur Str.

video camera is fixed on the dashboard of the front passenger seat to record movies. The
images extracted from the movies contain information utilized for measuring distances
between the probe vehicle and other vehicles in its surroundings. An example of theimages and part of the probe vehicle data are shown in Fig B.1b and c, respectively.
The survey was conducted on March, 7th 2014 for 1.5 hours from 4:00 pm to 5:30
pm in the dry weather. Ten taxis equipped with video cameras and GPS loggers gatheredon the Tan Cang parking area near the survey location. At 4:00 pm, the probe vehicles
departed one-by-one every five minutes and ran on the course. In addition, four fixedcameras on the sidewalk were used to observe the real traﬃc state of the traﬃc flow. At

118

Appendix B

Field Experiment at Arterial Road with Mixed Traﬃc(a) 800 m point(b) 1350 m point

Figure B.2: Cumulative curves at Pasteur Str.

5:30 pm, the probe vehicles left the course at the same time and returned to the parking
area. Regarding the behaviors of taxi drivers, they were asked to run normally as desiring.
It means that each driver could run with their desired speed when the traﬃc flow is free,or overtake the preceding vehicles to attain a higher speed when traﬃc become congested.
The flow of all the cars was about 600 veh/h. The flow of all the motorcycles was about
4500 veh/h.

B.2

Results of TSE on cars

The traﬃc state of cars was estimated using the basic method.
Fig B.2 shows cumulative curves at two locations of Pasteur Str. Dashed lines
represent ground truth obtained by road side observer, and blue lines represent estimatedvalues by all the probe vehicle with 5 min time resolution. According to the figure, therough tendency of estimated cumulative curves are close to observed ones. However,errors in high resolution state such as 5 min flow are relatively large. This characteristic
is similar to the results of sections 5.1 and 6.2.
Table B.1 summarizes estimation performance at Pasteur Str. The same tendency tosections 5.1 and 6.2 can be found: accuracy and precision improves as P, ∆t increases.
119Appendix B

Field Experiment at Arterial Road with Mixed Traﬃc

Table B.1: Flow estimation results at Pasteur Str.
P∆t (min)

RMSPE(q̂)

Bias(q̂) (veh/h)

0.2%

5

57.9%

63.8

0.2%

60

42.0%

91.0

1.0%

5

52.7%

52.9

1.0%

60

18.4%

−11.4

2.5%

5

55.4%

14.3

2.5%

60

7.9%

−20.5

However, in quantitatively, the precision is lower than the field experiment in the urban
expressway. It may be due to larger variance in headway due to the higher fluctuation of
the mixed traﬃc and signals.

120

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