Tra State Estimation with
Multi-Agent Simulations
vorgelegt von
Diplom-Ingenieur
Gunnar Flötterö d
aus Bielefeld
Von der Fakultät V Verkehrs- und Mashinensysteme
der Tehnishen Universität Berlin
zur Erlangung des akademishen Grades
Doktor der Ingenieurwissenshaften
Dr.Ing.
genehmigte Dissertation
Promotionsausshuÿ:
Vorsitzender: Prof. Dr. rer. nat. Volker Shindler
Gutahter: Prof. Dr. rer. nat. Kai Nagel
Gutahter: Prof. Mihel Bierlaire, PhD in Mathematis
Gutahter: Dr. rer. nat. Peter Wagner
Tag der wissenshaftlihen Aussprahe: 23. April 2008
Berlin 2008
D 83
Aknowledgments
I am indebted to Claudia, my family, and my friends, who were patient with me
throughout these years and ontinuously oered enouragement and distration.
I would like to express my gratitude to the members of my thesis ommittee,
Prof. Kai Nagel, Prof. Mihel Bierlaire, and Dr. Peter Wagner, for their interest
and guidane.
Thanks to Johannes for programming muh of the prototype simulator.
Prof. Tim Nattkemp er and the Center for Biotehnology at the University of
Bielefeld kindly made available their omputing failities at short notie. TU
Berlin's mathematial faulty also provided omputing time.
This researh was partially funded by the German researh so iety DFG under
the grant State estimation for tra simulations as oarse grained systems.
2
Zusammenfassung
Die vorliegende Dissertation b eshreibt ein neuartiges Verfahren zur gänz-
lih disaggregierten Nahführung des Mobilitätsverhaltens von Autofahrern auf
Grundlage aggregierter Messungen von Verkehrsüssen, -dihten o der -geshwin-
digkeiten, welhe durh eine b egrenzte Anzahl von Sensoren im Netzwerk auf-
genommen werden. Das Problem wird mittels eines bayesshen Ansatzes gelöst,
wob ei das gegeb ene a priori Wissen über die Auswahlverteilung der Verhal-
tensalternativen eines jeden Individuums mit der Likelihoo d-Funktion der ver-
fügbaren Messungen in eine geshätzte a p osteriori Verhaltensverteilung kom-
biniert wird. Der Ansatz ist insofern simulationsbasiert, als daÿ (i) allein ein
Simulationssystem zur Repräsentation der a priori Verhaltensannahmen b enö-
tigt wird und (ii) das Verfahren ausshlieÿlih Ziehungen aus der a p osteriori
Verhaltensverteilung generiert.
Das Verfahren b ehandelt den Simulator des a priori Verhaltens soweit wie mög-
lih als eine Blak Box. Die nahführbaren Verhaltensdimensionen reihen von
einfaher Routenwahl bis hin zur Auswahl von Plänen für einen ganzen Tag.
Eine gleihgewihtsbasierte Mo dellierungsannahme ist eb enso zulässig wie ein
Telematikmo dell unvollständig informierter Fahrer.
Die Verwendung aggregierter Sensordaten zur disaggregierten Verhaltensshät-
zung wird durh eine kombinierte mikroskopishe/makroskopishe Mobilitätssi-
mulation ermögliht, welhe individuelle Fahrzeuge auf Grundlage eines makro-
skopishen Mo dells der Verkehrsussdynamik bewegt. Das Modell erlaubt eine
lineare Vorhersage des Eektes von individuellem Verhalten auf den aggregierten
Verkehrszustand und ermögliht auf diese Weise eine lineare Approximation der
logarithmierten Likeliho o d-Funktion der Sensordaten in Abhängigkeit von dem
Verhalten der Fahrerp opulation. Diese Linearisierung wird von zwei op erativen
bayesshen Shätzern genutzt.
Der
aept/rejet estimator
maht keine weitergehenden Annahmen üb er die a
priori Verhaltensverteilung. Er zieht eine Anzahl von Realisierungen aus dieser
Verteilung und behält nur eine Teilmenge dieser Ziehungen b ei. Diese Teilmen-
ge wird unter Berüksihtigung der Likelihoo d-Funktion der Messungen derar-
tig ausgewählt, daÿ sie näherungsweise äquivalent zu einer Stihprob e aus der
a p osteriori Verhaltensverteilung ist. Der
utility-modiation estimator
addiert
einen Korrekturterm zu der Nutzenfunktion einer jeden Verhaltensalternative,
die ein simulierter Reisender vor einer Entsheidung auswertet. Diese Korrektur
ist eb enfalls durh die Likeliho o d-Funktion der Messungen b estimmt. Für eine
b estimmte Form der a priori Verhaltensverteilung ist das resultierende Verhalten
3
näherungsweise äquivalent zu einer Ziehung aus der a p osteriori Verhaltensver-
teilung.
Für die exp erimentellen Untersuhungen dient ein erweitertes
el l-transmission
model
als Mobilitätssimulation und ein randomisierter Kurzwegalgorithmus als
Platzhalter für eine vollständige Verhaltenssimulation. Die Exp erimente werden
unter synthetishen Bedingungen durhgeführt, wob ei die Sensordaten durh
eine externe Mo dellinstanz erzeugt werden. Der Testfall umfasst ein Netzwerk
von
2 459
Kanten und eine mikroskopishe Population von
206 353
Fahrern. Die
exp erimentellen Ergebnisse zeigen, daÿ das implementierte Verfahren die fol-
genden Eigenshaften aufweist: (i) Es nutzt in ezienter Weise eine b egrenzte
Menge verfügbarer Verkehrszählungen, um das individuelle Routenwahlverhal-
ten in der Population derartig nahzuführen, daÿ eine deutlih realistishere
globale Verkehrslage resultiert. (ii) Es ist sowohl auf ein gleihgewihtsbasiertes
Simulationssystem als auh auf einen Simulator ohne Gleihgewihtsannahme
anwendbar. (iii) Wenngleih der verfügbare Testfall etwas zu groÿ ist, um in
Ehtzeit nahgeführt zu werden, sind in dieser Hinsiht realisierbare Szenarien
niht um Gröÿenordnungen kleiner.
4
Abstrat
This dissertation desrib es a novel method for the fully disaggregate estimation
of motorist b ehavior from aggregate measurements of ows, densities or velo-
ities that are obtained at a limited set of network loations. The problem is
solved in a Bayesian setting, where the prior assumption ab out an individual's
hoie distribution is ombined with the available measurements' likeliho o d into
an estimated p osterior hoie distribution. The approah is simulation-based in
that it (i) only requires a simulation system to represent the behavioral prior
distribution and (ii) only generates realizations from the b ehavioral posterior
distribution.
The estimator treats the b ehavioral simulation system as a blak box to the
greatest p ossible extent. The p ossibly estimated b ehavioral aspets range from
single route hoie to the seletion of full-day plans, and an equilibrium-based
mo deling assumption is just as feasible as a telematis mo del of imp erfetly
informed drivers.
The inorp oration of aggregate sensor data into this b ehaviorally disaggregate
estimation proedure is enabled by a mixed miro/maro mobility simulation
that moves individual drivers through a marosopi model of tra ow dy-
namis. This mo del allows to linearly predit the eet of individual b ehavior
on aggregate tra onditions, and through this it provides a linear approxima-
tion of the sensor data's log-likeliho o d given a partiular b ehavioral pattern in
the driver p opulation. This linearization is utilized by two op erational Bayesian
estimators.
The aept/rejet estimator funtions without further assumptions about the
b ehavioral prior distribution. Its takes a numb er of draws from this prior and
retains only a subset of these draws. This subset is hosen in onsideration of
the measurements' likeliho o d suh that it is equivalent to a sample from the b e-
havioral posterior. The utility-mo diation estimator adds a orretion term to
the utility of every b ehavioral alternative a simulated traveler evaluates b efore
making a hoie. This orretion also is a funtion of the measurements' likeli-
ho o d. Given a partiular form of the b ehavioral prior, the resulting b ehavior is
equivalent to a draw from the b ehavioral p osterior.
For exp erimental investigations, an extended ell-transmission mo del is imple-
mented as the mobility simulation, and a randomized best-path routing logi
serves as a plaeholder for a full b ehavioral simulator. The exp eriments are
onduted in a syntheti setting, where the sensor data is generated by an ex-
ternal mo del instane. The test ase omprises a network of
2 459
links and a
5
mirosopi p opulation of
206 353
drivers. The exp erimental results show that
the implemented estimator has the following properties: (i) It eiently utilizes
limited tra ounts to adjust the p opulation's individual-level route hoie
suh that a signiantly more realisti global tra situation results. (ii) It
is equally appliable to an equilibrium-based and to a non-equilibrium-based
simulation system. (iii) While the available test ase is somewhat to o large to
b e monitored in real-time, a feasible senario for an online appliation of the
estimator is not smaller by orders of magnitude.
6
Contents
1 Intro dution 16
1.1 Denition of Problem Domain . . . . . . . . . . . . . . . . . . . . 16
1.1.1 Maro- and Mirosimulation . . . . . . . . . . . . . . . . . 17
1.1.2 Behavioral and Physial Simulation . . . . . . . . . . . . 17
1.1.3 Transp ortation Planning and Telematis . . . . . . . . . . 18
1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 Estimation Without Behavioral Mo deling . . . . . . . . . 20
1.2.2 Estimation With Behavioral Mo deling . . . . . . . . . . . 22
1.2.2.1 Stati Tra Assignment . . . . . . . . . . . . . 22
1.2.2.2 Dynami Tra Assignment . . . . . . . . . . . 23
1.2.2.3 Multi-Agent Tra Simulation . . . . . . . . . . 25
1.3 Thesis Contribution and Outline . . . . . . . . . . . . . . . . . . 26
1.3.1 Coneptual Outline . . . . . . . . . . . . . . . . . . . . . . 26
1.3.2 Metho dologial Contribution . . . . . . . . . . . . . . . . 29
1.3.3 Struture of Thesis . . . . . . . . . . . . . . . . . . . . . . 30
2 Marosopi Mobility Simulation 31
2.1 Design Choies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 The Kinemati Wave Mo del . . . . . . . . . . . . . . . . . . . . . 32
2.3 Intersetion Flow Calulation Sheme . . . . . . . . . . . . . . . 33
2.4 Intersetion Sp eiation . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Straight Connetions . . . . . . . . . . . . . . . . . . . . . 36
2.4.2 Merges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.3 Diverges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.4 General Connetions . . . . . . . . . . . . . . . . . . . . . 39
2.5 Simulation Logi . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7
2.5.1 Cell Boundaries . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.2 Connetor Flow Rate Update . . . . . . . . . . . . . . . . 42
2.5.3 Cell State Up date . . . . . . . . . . . . . . . . . . . . . . 42
2.5.4 Exp erimental Investigation of Simulation Preision . . . . 43
2.6 Network Disretization . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.1 Sp eiation . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.2 Berlin Test Case . . . . . . . . . . . . . . . . . . . . . . . 45
2.7 State Spae Notation . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Mirosopi Behavioral Simulation 49
3.1 Coupling Miro- and Marosimulation . . . . . . . . . . . . . . . 49
3.1.1 Representation of Behavioral Heterogeneity . . . . . . . . 50
3.1.2 Partile Movement . . . . . . . . . . . . . . . . . . . . . . 50
3.1.2.1 Sp eiation . . . . . . . . . . . . . . . . . . . . 50
3.1.2.2 Simulation on Variable Time Sales . . . . . . . 51
3.1.3 Partile Route Choie . . . . . . . . . . . . . . . . . . . . 52
3.1.3.1 Sp eiation . . . . . . . . . . . . . . . . . . . . 52
3.1.3.2 Simulation on Variable Time Sales . . . . . . . 54
3.1.4 Computational Mo del Investigation . . . . . . . . . . . . 56
3.1.4.1 Preision of Miro/Maro Coupling . . . . . . . 56
3.1.4.2 Computational Performane . . . . . . . . . . . 60
3.2 Simulation of Drivers' Choies . . . . . . . . . . . . . . . . . . . . 62
3.2.1 Choie Formalism . . . . . . . . . . . . . . . . . . . . . . 62
3.2.1.1 Denition of the Choie Problem . . . . . . . . . 62
3.2.1.2 Generation of Alternatives . . . . . . . . . . . . 64
3.2.1.3 Evaluation of Attributes of Alternatives . . . . . 65
3.2.1.4 Choie . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1.5 Implementation . . . . . . . . . . . . . . . . . . 66
3.2.2 Sp ei Mo deling Assumptions . . . . . . . . . . . . . . . 67
3.2.2.1 Random Utility Mo dels . . . . . . . . . . . . . . 67
3.2.2.2 Mo dels of Route Choie . . . . . . . . . . . . . . 68
3.2.2.3 Mo dels of Plan Choie . . . . . . . . . . . . . . . 70
8
4 Estimation 72
4.1 Steering Agent Behavior . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 Mo died Utility Pereption . . . . . . . . . . . . . . . . . 74
4.1.2 Linearization of Global Ob jetive Funtion . . . . . . . . 74
4.1.3 Consistent Linearization for Many Agents . . . . . . . . . 77
4.1.4 Behavioral Justiation . . . . . . . . . . . . . . . . . . . 78
4.2 Heuristi Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Mo deling of Aggregate Tra Measurements . . . . . . . 80
4.2.2 Steering Agents Towards the Measurements . . . . . . . . 81
4.3 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 General Formulation of Estimator . . . . . . . . . . . . . 82
4.3.2 Op erational Aept/Rejet Estimator . . . . . . . . . . . 84
4.3.3 Op erational Utility-Mo diation Estimator . . . . . . . . 87
4.3.4 Appliability of Heuristi Estimator . . . . . . . . . . . . 88
4.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.1 Senario Desription . . . . . . . . . . . . . . . . . . . . . 89
4.4.2 Aept/Rejet Estimator . . . . . . . . . . . . . . . . . . 91
4.4.3 Utility-Mo diation Estimator . . . . . . . . . . . . . . . 92
5 Test Case 96
5.1 Exp erimental Overall Setting . . . . . . . . . . . . . . . . . . . . 96
5.1.1 Senario Desription . . . . . . . . . . . . . . . . . . . . . 96
5.1.1.1 Invariable Settings . . . . . . . . . . . . . . . . . 96
5.1.1.2 Variable Settings . . . . . . . . . . . . . . . . . . 97
5.1.2 Simulation and Estimation Logi . . . . . . . . . . . . . . 98
5.1.2.1 Simulation . . . . . . . . . . . . . . . . . . . . . 98
5.1.2.2 Estimation . . . . . . . . . . . . . . . . . . . . . 99
5.1.3 Sensor and Validation Data . . . . . . . . . . . . . . . . . 102
5.1.3.1 Sensor Data . . . . . . . . . . . . . . . . . . . . 102
5.1.3.2 Validation Data . . . . . . . . . . . . . . . . . . 103
5.1.3.3 Quantitative Error Measures . . . . . . . . . . . 103
5.2 Planning Exp eriments (Equilibrium Situation) . . . . . . . . . . 104
5.2.1 Senario Generation . . . . . . . . . . . . . . . . . . . . . 105
5.2.1.1 Investigation of Senario Stability . . . . . . . . 105
9
5.2.1.2 Measurement and Validation Data Generation . 107
5.2.1.3 Comparison of Senarios . . . . . . . . . . . . . 108
5.2.2 Exp erimental Results . . . . . . . . . . . . . . . . . . . . 110
5.2.2.1 Desription of Results . . . . . . . . . . . . . . . 111
5.2.2.2 Disussion of Results . . . . . . . . . . . . . . . 113
5.2.2.3 Estimation Dynamis . . . . . . . . . . . . . . . 116
5.3 Telematis Exp eriments (Non-Equilibrium Situation) . . . . . . . 118
5.3.1 Rolling Horizon Estimation . . . . . . . . . . . . . . . . . 118
5.3.2 Senario Generation . . . . . . . . . . . . . . . . . . . . . 120
5.3.2.1 Simulation of Imp erfetly Informed Drivers . . . 120
5.3.2.2 Investigation of Senario Stability . . . . . . . . 121
5.3.2.3 Measurement and Validation Data Generation . 122
5.3.2.4 Comparison of Senarios . . . . . . . . . . . . . 122
5.3.3 Exp erimental Results . . . . . . . . . . . . . . . . . . . . 124
5.3.3.1 Oine Estimation . . . . . . . . . . . . . . . . . 124
5.3.3.2 Online Estimation in Rolling Horizon Mo de . . . 125
5.3.3.3 Computational Performane . . . . . . . . . . . 128
5.4 Further Disussion . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Summary and Outlo ok 136
6.1 Reapitulation of Work . . . . . . . . . . . . . . . . . . . . . . . 136
6.2 Researh Contributions . . . . . . . . . . . . . . . . . . . . . . . 137
6.3 Classiation of Results . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Further Researh Topis . . . . . . . . . . . . . . . . . . . . . . . 140
6.4.1 Towards a Real-World Appliation . . . . . . . . . . . . . 140
6.4.1.1 Mo del Calibration and Validation . . . . . . . . 140
6.4.1.2 Measurement Soures and Sensor Typ es . . . . . 141
6.4.1.3 Performane Tuning . . . . . . . . . . . . . . . . 141
6.4.2 Combined Behavioral and Physial Estimation . . . . . . 142
6.4.3 Combined Telematis and Planning Estimation . . . . . . 143
6.4.3.1 Fusion of
Λ
Co eients . . . . . . . . . . . . . . 143
6.4.3.2 Choie Set Mo diations . . . . . . . . . . . . . 143
6.4.4 Behavioral Parameter Estimation . . . . . . . . . . . . . . 143
6.4.4.1 Estimation of Population Parameters . . . . . . 144
10
6.4.4.2 Estimation of RUM Parameters . . . . . . . . . 145
6.4.5 Integration with MATSim . . . . . . . . . . . . . . . . . . 146
6.4.5.1 Coneptual Asp ets . . . . . . . . . . . . . . . . 146
6.4.5.2 Tehnial Asp ets . . . . . . . . . . . . . . . . . 147
6.4.6 Strutural Mo del Renements . . . . . . . . . . . . . . . . 149
6.4.6.1 Physial Simulation . . . . . . . . . . . . . . . . 149
6.4.6.2 Behavioral Simulation . . . . . . . . . . . . . . . 150
A Implementation of GPRC Integer Sets 151
B Sensitivity Analysis for the GPRC 154
B.1 Initialization of Sensitivities . . . . . . . . . . . . . . . . . . . . . 155
B.2 Calulation of
∂ξ(m+1
/2)/∂ξ(0)
and
∂ξ(m+1
/2)/∂β
. . . . . . . . . 155
B.3 Calulation of
∂ξ(m+1)/∂ξ(0)
. . . . . . . . . . . . . . . . . . . . 157
B.4 Calulation of
∂ξ(m+1)/∂β
. . . . . . . . . . . . . . . . . . . . . 160
B.5 Completition of Sensitivities . . . . . . . . . . . . . . . . . . . . . 161
C Calulation of Cell Velo ities 163
D Gridlo k Resolution 164
E Stationary Limit of Turning Counter Variane 166
11
List of Figures
1.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1 Lo al supply and demand omprise a fundamental diagram . . . 34
2.2 A p oint-like intersetion with
I
ingoing and
J
outgoing links . . 34
2.3 A straight onnetion . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 A merge with
I
ingoing links . . . . . . . . . . . . . . . . . . . . 37
2.5 A diverge with
J
outgoing links . . . . . . . . . . . . . . . . . . . 38
2.6 A general onnetion with
I
ingoing and
J
outgoing links . . . . 39
2.7 Spae-time plots with variable spatiotemp oral disretizations . . 44
2.8 Ma jor road network of Greater Berlin . . . . . . . . . . . . . . . 46
2.9 Eet of network time onstant on ell ount . . . . . . . . . . . 47
2.10 Time step duration histogram . . . . . . . . . . . . . . . . . . . . 47
3.1 Partile movement aross ell b oundaries . . . . . . . . . . . . . 51
3.2 Partile movement on variable time sales . . . . . . . . . . . . . 52
3.3 Turning ounter dynamis . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Simulated Berlin morning p eak . . . . . . . . . . . . . . . . . . . 57
3.5 Preision of miro/maro mo del synhronization . . . . . . . . . 58
3.6 Mean normalized bias and error tra jetories . . . . . . . . . . . . 59
3.7 Mirosopi and marosopi omputation times . . . . . . . . . 60
3.8 Real time ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.9 Route hoie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.10 Generalized path hoie . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 Three routes example . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Fixed p oint of utility orretions . . . . . . . . . . . . . . . . . . 78
12
4.2 Three routes example, rep eated . . . . . . . . . . . . . . . . . . . 89
4.3 Measurement t . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4 Estimated path sizes . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 Inner-urban part of Berlin . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Exemplary sensor lo ations . . . . . . . . . . . . . . . . . . . . . 102
5.3 RMS
x
and RMSA
x
[6 EUR/h VOT simulation℄ . . . . . . . . . 105
5.4 RMS
x
and RMSA
x
[12 EUR/h VOT simulation℄ . . . . . . . . . 106
5.5 RMS
x
and RMSA
x
[18 EUR/h VOT simulation℄ . . . . . . . . . 106
5.6 RMS
x
and RMSA
x
[no-toll simulation℄ . . . . . . . . . . . . . . . 107
5.7 Satterplots for omparison of planning referene simulations . . 109
5.8 Result overview for planning exp eriments . . . . . . . . . . . . . 112
5.9 Comparison of true and estimated ows (planning) . . . . . . . . 114
5.10 Comparison of true and estimated o upanies (planning) . . . . 115
5.11 RMS
x
and RMSA
x
[6 EUR/h VOT estimation℄ . . . . . . . . . . 116
5.12 RMS
x
and RMSA
x
[18 EUR/h VOT estimation℄ . . . . . . . . . 117
5.13 RMS
x
and RMSA
x
[no-toll estimation℄ . . . . . . . . . . . . . . . 117
5.14 RMS(A)
x
[no-toll planning/telematis simulation℄ . . . . . . . . . 121
5.15 Satterplots for omparison of telematis referene simulations . 123
5.16 Result overview for telematis oine exp eriments . . . . . . . . 124
5.17 Comparison of true and estimated ows/o upanies (telematis) 126
5.18 RMS
x
[30 min. rolling horizon estimation℄ . . . . . . . . . . . . . 127
5.19 RMS
x
[0-30 min. rolling horizon predition℄ . . . . . . . . . . . . 129
5.20 RMS
x
[5-10 min. rolling horizon predition℄ . . . . . . . . . . . . 130
5.21 RMS
x
[15-20 min. rolling horizon predition℄ . . . . . . . . . . . 131
5.22 RMS
x
[25-30 min. rolling horizon predition℄ . . . . . . . . . . . 132
6.1 Estimated quantities . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.1 Resoure variations for rst half of GPRC sensitivity alulation 156
B.2 Resoure variations for seond half of GPRC sensitivity alulation 159
D.1 Mo died fundamental diagram . . . . . . . . . . . . . . . . . . . 165
13
List of Algorithms
1 General pro ess of resoure onsumption . . . . . . . . . . . . . . 35
2 Steering a p opulation of agents . . . . . . . . . . . . . . . . . . . 79
3 Aept/rejet estimator . . . . . . . . . . . . . . . . . . . . . . . 86
4 Utility-mo diation estimator . . . . . . . . . . . . . . . . . . . . 88
5 GPRC sensitivity alulation logi . . . . . . . . . . . . . . . . . 155
6 First half of GPRC sensitivity alulation . . . . . . . . . . . . . 158
7 Seond half of GPRC sensitivity alulation . . . . . . . . . . . . 162
15
Chapter 1
Intro dution
The 2007 world limate report emphasizes the signiant inuene of fossil fuel
burning on the urrent and future limate hange [81, 82℄, whereas a large share
of the global greenhouse gas pro dution stems from present transp ortation sys-
tems [105℄. Mobility is an essential go o d that justies a ertain environmental
prie. However, its neessity as well as the very prie it entails make it highly
desirable to op erate transp ortation systems at working p oints of greatest e-
ieny and to optimally exploit the available infrastruture. This goal needs to
b e pursued b oth in long-term planning onsiderations and in short-term tra
management eorts.
From an engineering p ersp etive, a p owerful to ol to ahieve suh ob jetives are
algorithms for mo del-based predition and ontrol. They allow to evaluate the
p erformane of a tra system in various settings before hoosing the most
promising measure. Pivotal to the suess of these approahes is the availability
of a realisti mo del. Usually, this is ahieved by building a struturally orret
mo del whih is alibrated based on omparisons of its outputs and available
measurements. Numerous metho ds have b een developed to more or less auto-
matially solve the latter task.
This thesis ontributes to that eld. It desrib es a metho d to estimate the travel
b ehavior of individual motorists from measurements of aggregate tra features
suh as ows, densities or velo ities that are obtained at a limited set of network
lo ations. Knowing what trips p eople will make allows to predit and p ossibly
redue ongestion. But no matter if this information is used to hoose ontrol
measures, for driver information servies or to ollet long-term data: It always
provides a valuable basis for prosp erous deision making.
1.1 Denition of Problem Domain
Tra state estimation is a broad eld, whih neessitates the preliminaries
given in this setion. Their purp ose is to outline this dissertation's work sop e
and to introdue some terminology.
16
A mo del-based estimation approah is pursued. Blind mo deling tehniques
that provide general-purpose mappings of a system's inputs to its outputs with-
out an underlying problem-sp ei mo del struture are exluded from onsid-
eration. For example, a neural network that maps lo al tra volumes on
network-wide travel times do es not ontain a strutural model and thus is not
in the sop e of this thesis.
The notion of state estimation is introdued informally as the measurement-
based adjustment of a strutural mo del's time-dep endent prop erties. This ter-
minology is made inreasingly preise as the onsidered lass of mo dels is spe-
ied throughout Chapters 2 and 3. This order of presentation aompanies the
overall omp osition of this work, whih is geared by the transp ortation spei
asp ets of the estimation problem.
1.1.1 Maro- and Mirosimulation
Marosopi tra mo dels treat a p opulation of travelers as a ontinuous quan-
tity and express mobility in terms of equally marosopi tra streams. Real
travelers are disrete entities. This requires their aggregation into suiently
large homogeneous groups for this approah to work. While b eing partiularly
amenable to a mathematial treatment, marosopi models are unable to repre-
sent highly heterogeneous traveler p opulations. The p ossibilities to marosop-
ially represent b ehavioral onstraints, whih often are of a rule-based nature
and might greatly vary aross a p opulation, are limited as well.
Mirosimulations apture travelers and their b ehavior individually. This gives
them a greater expressive p ower. Still, sine their p opulation mo del an only b e
a sample of the real p opulation, it is inherently sto hasti. The inreased realism
of a struturally detailed mapping of the real world on a mirosopi simulation
system also introdues the real world's mathematial intratabilities into the
mo del. This op ens a gap b etween the ease of implementing a mirosopi mo del
and the diulties in understanding the resulting mo del dynamis.
This work adopts a mirosimulation approah to the estimation of individual
b ehavior. Mirosimulation greatly simplies the mo deling and likewise ompli-
ates the estimation task. Consequently, every property of the mo del that is to
b e estimated has to b e arefully mathed by a formal representation that allows
for a mathematial treatment. The formal requirements set up in this thesis
aim to apture a wide variety of mirosopi aspets while ensuring tratability
of the mathematial estimation problem.
1.1.2 Behavioral and Physial Simulation
Mirosimulations of vehiular tra usually onsist of at least two sub-mo dels,
one of tra ow dynamis and one of travel b ehavior:
•
Tra ow dynamis
desrib e the physial laws of the tra system
under onsideration. They determine how well a road network serves a
traveler's need of driving most onveniently along a route to a destina-
tion in a p otentially ongested tra situation. To serve the purp ose of
17
this thesis, driver b ehavior in terms of breaking, aeleration, and lane
hanging is subsumed in the physial representation of tra ow.
•
Travel b ehavior
results from the demand for mobility aross a network.
Various asp ets suh as route, destination, and departure time hoie an
b e mo deled one a representation for the traveler p opulation itself is found
[71, 128℄. If only motorists are onsidered, mode hoie winds down to the
deision if a ar trip is made or not. Long-term deisions suh as ar
ownership and residential hoie are b eyond the time sales onsidered in
this thesis.
This work is restrited to the estimation of b ehavioral asp ets. That is, the
present approah assumes the tra ow dynamis to b e mo deled without error.
A p ossible augmentation towards the onurrent estimation of b ehavior and
physis is outlined as a sub jet of future researh.
Given the fo us on b ehavioral estimation, no dierentiation b etween freeway and
intra-urban tra is neessary in priniple sine their ma jor dierene onsists
in their tra dynamis. Only the granularity of the physial mo deling has a
limiting eet on the proposed metho d's appliability.
1.1.3 Transp ortation Planning and Telematis
Mirosimulation an b e applied b oth in transp ortation planning and transp orta-
tion telematis, and the prop osed estimation metho d is appliable in b oth elds
as well.
At rst glane, this is not surprising sine planning and telematis onstitute
two dierent views of the same system. Planning metho ds have evolved over
many deades, while telematis app eared quite reently as an ospring of trans-
p ortation planning and adopted many metho ds from this eld. Still, there are
systemati dierenes that must b e aounted for:
•
Planning
mo dels usually assume that travelers obtain global knowledge
of average system states through many days of exploration and that the re-
sulting b ehavioral patterns resemble some kind of equilibrium. Typially,
suh mo dels work at the granularity of average within-day tra jetories
but do not repro due within-day utuations of the system states as they
o ur in reality due to the sto hasti nature of tra [35℄.
•
Telematis
mo dels expliitly deal with utuations within a day. They
neither assume global driver knowledge nor do they assume an equilibrium.
The b ehavioral mo del omp onent in suh a system may represent driver
reations to new and p ossibly unforeseeable tra situations, to provided
information, and to guidane [24, 25℄. Without these utuations, there
would b e little use in guiding the system in one way or another sine
under normal onditions travelers have already found go o d travel options
via day-to-day exp erimentation [74℄.
18
This distintion arries over to the temp oral onstraints for a tra state es-
timation algorithm. In a planning appliation, there is at least one night to
adjust a mo del to reently olleted measurements. This is onsidered as an
oine
estimation problem. In a telematis appliation, usually just a few min-
utes are available to inorp orate the most reent measurements into the urrent
estimate. The adjustment takes plae while the mo del progresses through (real)
time, onstituting an
online
estimation problem. However, a telematis esti-
mator may also b e used in oine mo de for the ex p ost analysis of a partiular
day's tra situation.
While the ab ove distintion is lear, that of appliable estimation metho ds is
not. Coneptually, it do es not make a dierene to a reursive algorithm if it
is used for inremental over-night adjustment of a planning mo del or on a 5-
minutes time sale in a real-time ontext. However, the p ortability of traditional
planning to ols to telematis appliations is limited. The need for substantial
researh in this eld has been reognized about two deades ago, e.g., [160℄, and
has spawned ongoing investigation eorts b oth nationally, e.g., [123, 169℄, and
internationally, e.g., [42, 58, 143℄. Still, many metho dologial p otentials are yet
to b e explored [139℄.
But there are not only limitations. Mutual b enets of dierent estimation ap-
proahes naturally result from their ommon ob jet of investigation. Online
tra monitoring systems usually rely on some kind of a priori knowledge about
the average system b ehavior as provided by a planning simulation. Vie versa,
the daily generation of high-resolution state estimates provides valuable data
for the ontinuous alibration of a planning mo del.
The prop osed estimator is ompatible with b oth a planning and a telematis
mo deling assumption. However, its immediate b enets are greatest in online
tra monitoring, and further pro essing of its outputs is likely to be neessary
for typial planning purp oses. The following literature review therefore fo uses
on the online tra state estimation problem and gives referenes to more tra-
ditional planning methods only where their interplay with the online problem
is of relevane.
1.2 State of the Art
Many approahes to the online tra state estimation problem draw from trans-
p ortation planning's established metho ds and enhane them by a dynamial
omp onent. Arguably, the most frequently adopted metho ds are those of stati
origin-destination (OD) matrix estimation. An OD matrix mo dels the demand
of a given time interval in terms of the number of trips from every origin to
every destination of a tra system. The originally onsidered problem was to
estimate suh a matrix from observed link volumes given a linear assignment
mapping of demand on link ows (assignment matrix). Various methods suh
as entropy maximization and information minimization [168℄, Bayesian estima-
tion [113℄, generalized least squares [12, 34℄, and maximum likelihoo d estimation
[162℄ have b een prop osed to solve this task. Early overviews on this sub jet an
b e found in [37, 180℄. Nonlinear assignment mappings an b e inorp orated by a
bilevel-approah that iterates b etween a nonlinear assignment and a linearized
19
estimation problem [114, 181, 182℄ until a xed point of this mutual mapping is
reahed [39℄. The ombined estimation of OD matries at subsequent time slies
was demonstrated in [36℄, and many originally stati metho ds have b een applied
to dynamial problems in this vein, e.g., [111, 158℄ and the referenes in Setion
1.2.2.2. Beyond the dierent mo deling requirements, temp oral onstraints are
most ritial to the online deployment of these approahes.
Many advaned online appliations employ systems engineering metho dologies
to a suitably formalized tra mo del. The most prominent of these metho ds
is without doubt the Kalman lter in one of its many guises. Assuming a
sto hasti disturbane up on an originally linear dynamial system [90℄, it has
evolved to an estimator for systems with smo oth, nonlinear dynamis [161℄ as
well as for systems with a merely simulation-based representation [88, 89℄. More
generally appliable partile lters even trak multimodal state distributions [6℄.
These developments have made Kalman ltering inreasingly appliable to the
high omplexity of tra systems. However, with these apabilities omes a
growing omputational burden that renders the real-time observation of truly
large-sale systems still imp ossible. Beause of its equivalene with a reursive
least squares estimator, the Kalman lter an also b e reformulated as a problem
of mathematial programming, whih broadens the eld of p otentially appliable
algorithms [23℄.
The following presentation is organized with resp et to the underlying mo del.
It dierentiates b etween estimation metho ds that use a b ehavioral mo del and
those that do not. At the limit of this lassiation are approahes that rely on
spatially non-orrelated probabilities of turning move o urrenes at interse-
tions. These metho ds represent route hoie merely as a sequene of indep endent
turning deisions and thus are not onsidered to b e based on a b ehavioral mo del.
1.2.1 Estimation Without Behavioral Mo deling
No strutural mo deling at all is required if general-purp ose system representa-
tions are used. Auto-regressive moving average mo dels and artiial neural net-
works learn a regression-typ e relation b etween urrent measurements and ur-
rent or future tra states. Pattern mathing tehniques suh as nonparametri
regression or lustering metho ds ompare previously olleted tra state tra-
jetories to urrently available information and provide most similar historial
data for estimation and foreast. Laking a strutural mo del, these approahes
are mentioned only for ompleteness. A omprehensive overview of data-driven
metho ds in tra estimation and predition is given in [46℄.
A linear road do es not allow for the type of b ehavioral deisions onsidered in
thesis but is amenable to the mo deling of tra ow dynamis. Sine tra
ow is a dynamially rather restrited system, this yields useful additional in-
formation. Mo dels for ow on a link have gone from the fundamental diagram
(where density and velo ity are uniquely related, and ow is a funtion of either
density or velo ity [72℄) via the Lighthill-Whitham-Rihards theory of kinemati
waves (where the fundamental diagram is inserted into an equation of ontinuity
[108, 151℄) to seond-order mo dels (where a seond equation introdues inertia
[144℄).
20
Various approahes based on Kalman lters (and, more reently, partile lters)
have b een prop osed to estimate parameters and/or states of tra ow mo d-
els from loal measurements in a variety of settings, e.g., [75, 112, 122, 165℄.
As a typial example of these, the RENAISSANCE approah is desrib ed fur-
ther b elow. ASDA and FOTO (Automatishe Staudynamikanalyse: Automati
Traing of Moving Tra Jams and Foreasting of Tra Ob jets) onstitute
a pattern-based monitoring and predition system that traks tra jams along
a freeway [95, 96℄. The adaptive smoothing metho d uses a nonlinear lter
that aounts for the dierent diretions of disturbane propagation in free and
ongested tra onditions to interp olate and extrap olate stationary detetor
data on freeways [167℄.
If network tra is onsidered, turning deisions at intersetions need to b e
mo deled. If no suh mo del is at hand, a simple approah is to dene turning
probabilities. The simulation of individual vehiles by this metho d results in pa-
rameterized random walks through the network. In a marosopi mo del, ows
aross an intersetion diverge at ingoing links aording to turning frations that
equal these probabilities and additively merge at outgoing links. For the result-
ing linear model, (reursive) least squares and Kalman ltering an b e applied
to trak the turning frations from link volume measurements [13, 50, 107, 135℄.
The inorp oration of signal timing information was prop osed in [93, 117℄, and
the provision of estimated turning ows as supplementary measurements to a
network-wide OD matrix estimator was found to signiantly inrease the over-
all estimation quality in [68, 118℄.
The Urban Tra Analyzer UTA uses a marosopi queuing mo del of inner-
urban tra ow to predit network-wide ows and travel times. However, it
requires that likewise network-wide measurements of urrent ows and turn-
ing frations are available, and no data fusion b eyond a temp oral averaging of
measurements is desrib ed [94, 95℄.
A system that is in ontinuous op eration in Germany is OLSIM (Online Traf-
Simulation) [45, 137, 174℄. It uses a mirosopi tra mo del. Additional
vehiles are inserted where sensors reord more vehiles than the mo del, and
vehiles are removed where sensors reord fewer vehiles than the mo del [92℄.
Measurements are extrap olated by having the vehiles move forward along links
aording to realisti driving rules and having them turn at intersetions aord-
ing to historial or diretly measured turning probabilities [116℄. The system
predits network onditions based on a pre-lustering of typial measurement
tra jetories: At a given p oint in time, the measurements themselves are pre-
dited as a weighted average of the most reent observations and representa-
tive historial tra jetories. Based on this predition, the aforementioned sensor
adaptation pro edure is ontinued into the future [46℄.
Reently, the RENAISSANCE (Real-Time Freeway Network Tra Surveil-
lane To ol) tra monitoring and predition system has been op erationalized
[178℄. Its estimation mo dule onsists of an extended Kalman lter [175, 176,
177℄, whih is applied to the marosopi tra ow mo del METANET [101℄. A
random walk assumption is imp osed on mo del parameters suh as road apa-
ities, free ow veloities, and turning frations, whih allows to estimate these
parameters together with the tra ow mo del's density and velo ity states.
Suhlike observed parameters improve the state estimation quality, e.g., in ase
21
of varying weather onditions, and serve as inidents indiators.
Metho ds that rely on a priori olleted turning prop ortions an b e exp eted to
work well in normal situations but to b e rather problemati during exeptional
events when turning frations deviate from pre-sp eied values. In priniple,
every turning-probability driven approah an b e supplied with a b ehavioral
mo del for the generation of these parameters. However, this alone do es not
larify how to adjust the b ehavioral mo del itself to given measurements. This
problem is onsidered next.
1.2.2 Estimation With Behavioral Mo deling
1.2.2.1 Stati Tra Assignment
The lassial planning metho d for the mo deling of network tra is stati as-
signment. The problem is stated as to assign a given demand of ows between
origin-destination pairs (OD pairs) on the network. Typial assignment riteria
are a Nash equilibrium (all atually used routes for eah OD pair have equal
ost and no unused route has smaller ost; also alled user equilibrium (UE))
or stohasti user equilibrium (SUE; the assignment of OD ows on routes fol-
lows a given distribution whih is based on link ost). In so-alled ongested
assignment, ost on a link is an inreasing funtion of link ow whih is gener-
ated by ows on routes that use the link. Links that are heavily used b eome
exp ensive, thus diverting some of the ow to other routes, e.g., [35℄.
The only way to approximate within-day dynamis by means of stati assign-
ment is to run indep endent simulations on onseutive time slies. Within limits
and in ombination with dynamial mo del omp onents, this approah an b e
integrated into a pratially aeptable system representation for telematis
purp oses, as the following two examples show.
The naming path ow estimator (PFE) is usually asso iated with the approah
prop osed in [17℄. It desrib es a marosopi one-step network observer that
estimates stati path ows from link volume measurements based on a SUE
mo deling assumption in a ongested network [14℄. The estimation problem
is transformed into one of smo oth optimization, whih is iteratively solved.
The mo del has b een enhaned by multiple user lasses and a simple analytial
queuing mo del to represent tra ow dynamis [16℄ and has b een suessfully
implemented in various researh and development pro jets [15℄. The limitations
asso iated with its original assumption of a logit path hoie model (overlapping
path problem, e.g., [18℄) have b een mitigated by the implementation of a C-logit
path hoie mo del [38, 173℄. The PFE's stati UE ounterpart was prop osed in
[157, 159℄ and has b een further advaned in [133, 134℄.
The tra management enter of Berlin (Verkehrsmanagementzentrale VMZ)
also op erates an online tra monitoring system [170℄. The fully marosopi
metho d omprises a substantial numb er of dierent adjustment steps. It predits
measurement tra jetories by a lustering approah similar to that of OLSIM and
uses either a stati or a simplied queue-dynamial mo del to interp olate tra
ows b etween sensors. Route hoie is assumed to b e in a stati UE that is
simulated in time slies of one hour. The assigned OD matrix is seleted based
22
on a similarity measure between urrently prevailing measurements and those
the matrix had previously b een alibrated with [171℄.
A omputationally ostly but metho dologially straightforward approah to
trak route hoie at an aggregate level is to estimate the assignment matrix
itself onurrently with the OD matrix. The resulting estimation problem is
in general highly under-determined, so a prior assignment matrix is inorp o-
rated in muh the same way a prior OD matrix ensures a unique solution to the
ommon OD matrix estimation problem [109, 110℄.
1.2.2.2 Dynami Tra Assignment
The following presentation onentrates on simulation-based approahes to dy-
nami tra assignment (DTA). This is justied by their mirosopi vehile
representation whih is a fundamental mo deling assumption of this thesis. An
overview of DTA that inludes analytial approahes an b e found in [146℄.
Most urrent network loading mo dels use similar tehniques [8, 19, 57, 115, 136℄:
They have individual, deision-making partiles (driver vehile units (DVUs) )
whih usually are sampled from an OD matrix and are moved forward along
links using funtions that in some way or other ouple sp eed to density. Most
mo dels inlude storage apaities on their links, that is, the density of vehiles
is limited and one a link is full, no more vehiles an enter. This implies that
upstream links form queues of vehiles that annot leave the link b eause the
downstream link is full.
Time-dep endent Nash equilibria are omputed on suh mo dels via iterations
[130℄: Start with some version of time-dep endent demand whih gives, for eah
time slot and OD pair, the numb er of vehiles leaving the origin during that time
slot. Have eah vehile follow a pre-omputed route. After the network loading
has run, re-ompute the time-dep endent path hoie information. For example,
give some fration of travelers a new route that would have been fastest in
the last iteration (b est resp onse), or distribute travelers b etween path options
aording to a distribution funtion, e.g., a path size logit or a C-logit model
[18, 38℄. This pro edure is iterated until an approximate xed point is reahed
[132℄.
As noted b efore, a dynami equilibrium is a reasonable assumption for planning
purp oses, while the mo deling of within-day utuations requires additional ef-
forts. Even more in suh a setting, simulation-based approahes are the metho d
of hoie b eause of their inherent ability to deal with individual and sp onta-
neous driver b ehavior.
There are two pro jets in the United States, namely DynaMIT (Dynami Net-
work Assignment for the Management of Information to Travelers, [19, 60℄) and
DYNASMART (Dynami Network Assignment Simulation Model for Advaned
Road Telematis, [61, 115℄), whih pursue oneptually similar approahes. For
illustration, a minimal online state estimation senario is outlined in the follow-
ing. More elaborate desriptions an b e found in [3, 7℄ for DynaMIT and in
[183℄ for DYNASMART.
23
•
Beyond strutural information, both systems require at least a stati OD
matrix and an initial set of tra ounts to prepare their online (within-
day) estimation shemes. They pro eed by estimating a time-dep endent
OD matrix, using metho ds whih are in priniple similar to the seminal
tehniques prop osed in [36℄.
•
In online op erations, either system uses a linear Kalman lter to estimate
the deviation of OD ows from average historial tra jetories. This allows
to inorp orate the latters' strutural information. Both systems apture
the dynamis of a time-dependent OD matrix in the Kalman lter's state
transition equation: DynaMIT assumes that the OD ow deviations follow
a within-day autoregressive pro ess with a priori estimated parameters.
DYNASMART uses a p olynomial trend representation of the OD tra je-
tories, whih yields a linear state equation for the temp oral evolution of
these p olynomials' derivatives. In either ase, the dynamial mo del allows
for a demand predition and (by simulation) for a network-wide predition
of tra onditions.
•
Loading a urrent demand estimate on the network yields a dynami as-
signment matrix that linearly maps OD ows on link ows and thus relates
state variables and tra ounts. This mapping onstitutes the Kalman
lter's measurement equation.
•
Both systems run in a rolling horizon mo de where two pro edures take
turns: (i) The Kalman lter generates a urrent demand estimate based
on the most reent assignment matrix and the urrent measurements.
(ii) The network loading pro edure assigns the estimated demand on the
network in order to predit tra onditions and to provide an up dated
assignment matrix.
•
Both systems use the estimated demand tra jetories of a given day to up-
date a historial OD matrix as a basis for the next day's online estimation
problem. While for DynaMIT various smo othing metho ds are prop osed,
DYNASMART assumes a day-to-day random walk of the true OD ma-
trix, onsiders the demand estimate of a single day as measurement of
this matrix, and up dates the historial OD matrix by another Kalman
lter.
Muh like in the stati ase, a time-dep endent assignment matrix an be es-
timated together with the demand. This results in a signiant state spae
inrease and requires nonlinear ltering tehniques [7℄. The state vetor an
also b e extended by time-dep endent network parameters. This improves the
adaptive properties of the overall monitoring system but again requires non-
linear estimators, various of whih are ompared in [3℄. The inorp oration of
additional data soures suh as prob e vehile samples [4, 183℄ is sub jet of ongo-
ing researh as well as advaned numerial solution algorithms [5, 23℄. Reently,
the DynaMIT system shifted from the Kalman ltering approah to a sparse
least squares solution pro edure [179℄, whih, however, do es not impair the on-
eptual orretness of the outline given ab ove.
24
1.2.2.3 Multi-Agent Tra Simulation
This approah is haraterized by the fully disaggregate representation of trav-
elers throughout the entire mo deling pro ess, while in DTA time-dep endent OD
matries are typially disaggregated and re-aggregated whenever onvenient.
The multi-agent approah is attrative in the tra domain sine it app ears
natural to represent every traveler by a software ob jet, to put these individual
mo dels into a representation of the physial world of mobility, and to observe
the resulting mobility patterns. Due to its strutural resemblane of real-world
pro esses, the metho d is easily ommuniated and inreasingly applied in trans-
p ortation mo deling (see, e.g., the olletion of artiles in [100℄).
Multi-Agent Simulation (MASim) an go b eyond other simulation metho ds by
inluding travelers' goals and ommitments into the modeling. For example, it
is p ossible with MASim to dierentiate b etween a delayed p erson with a free
evening and a delayed p erson with a time-restrited day-are pik-up. MASim
for transp ortation planning appliations typially onsists of the following mod-
ules [10, 11, 65, 130, 149℄:
•
A syntheti p opulation generation mo dule generates, from demographi
data, a syntheti p opulation that, in all its statistial asp ets, orresp onds
to the real population under investigation, while at the same time preserv-
ing privay.
•
An ativity-based demand generation mo dule generates, for eah member
of the syntheti p opulation, omplete daily plans inluding a sequene of
ativities (suh as home, work, shop, leisure), ativity lo ations, and a
temp oral shedule. Conseutive ativities at dierent lo ations generate
the demand for travel.
•
A router module omputes how that demand is atually exeuted on the
network, p ossibly inluding mo de hoie. At this p oint, all syntheti trav-
elers have plans that desrib e what they intend to do.
•
There is now always some kind of mo dule that puts the syntheti travelers
in a simulated version of the physial network and has them exeute their
plans simultaneously. The physial interation in that system generates
ongestion. Dep ending on the sp ei fo us, this simulation has dierent
names: supply simulation, network loading, tra ow simulation.
It is not possible to ompute the system in the linear way indiated above
sine plans dep end on ongestion but ongestion is a onsequene of the plans.
This is solved by iterations that an b e seen as mo deling human day-to-day
learning. This learning takes plae on various time sales. On the long term,
there are asp ets suh as hoie of residene and employment. These and further
harateristis of an agent onstitute onstraints on deisions that take plae
within dimensions of days, suh as ativity sheduling, lo ation hoie, and route
hoie. Although there are no strit temp oral domains for dierent elements of a
plan, a rough distintion with resp et to transp ortation planning and telematis
an b e made by a separation of elements that are mo died only on a day-to-day
basis and those that an b e reonsidered within a day.
25
The estimation of fully disaggregate travel b ehavior from aggregate sensor data
with a multi-agent tra simulation is a novel venture. In order to larify this
statement, the following related yet dierent problems need to b e mentioned:
•
The alibration of a mobility simulation from aggregate sensor data has
b een widely addressed in the literature, e.g., [47, 48, 59, 97, 103, 141, 142℄.
However, these approahes do not arry over to a alibration of the b e-
havioral simulation omp onent (unless one adopts a dierent terminology
than dened in Setion 1.1.2 and attributes, e.g., ar-following parameters
to the b ehavioral mo del).
•
A DTA-based OD matrix estimator aptures various behavioral asp ets,
yet only on an aggregate level. Sine a time-dep endent OD matrix maps
(origin, destination, departure time) tuples on demand levels, it diretly
represents destination and departure time hoie. A motorist OD matrix
reets mo de hoie at least in terms of deisions for or against the ve-
hiular mode. Route hoie, however, onstitutes no additional degree of
freedom but is a funtion of demand dened by the DTA pro edure. The
only exeption to this are the (b ehaviorally stati) path ow estimators
mentioned ab ove.
1.3 Thesis Contribution and Outline
1.3.1 Coneptual Outline
The omplexity of mo dern tra simulation systems renders the tehnologial
design of a exibly appliable estimator a nontrivial task. Extensive prototyp-
ial programming was onduted in order to validate the prop osed metho d's
appliability. Sine the resulting arhiteture struturally reets the estima-
tor's working, it is outlined b efore metho dologial ontributions are desrib ed.
In order to b e ompatible with the prop osed estimator, a tra simulation
system must b e separable into the omp onents shown in Figure 1.1. Most of
the employed terminology is adopted from [27℄.
•
The
mobility simulation
moves individual vehiles along their hosen
routes through the road network. All physial interations o ur within
this omp onent. A linearizable state spae representation of the mobility
simulation must b e available. This dissertation demonstrates that suh a
requirement is ompatible with a mirosopi driver representation.
•
The trip sequene of every vehile in the mobility simulation is hosen
by an individual
agent
that represents the driver of that vehile. The
travel b ehavior
of an agent is realized by one or two further omp onents.
Whenever a deision is required, the agent provides these omponents with
its
individual parameters
.
The
utility funtion
provides an individually parameterized map
from the
network onditions
on the systemati utility of any b e-
havioral alternative available to the agent. This may inlude utilities
26
Figure 1.1: Simulation
Logial struture of a mirosopi tra simulator that is amenable to the prop osed
estimation metho dology. The utility funtion is an optional omp onent that may b e
omitted.
for partial hoies if suh a deomp osition is required by the deision
proto ol. For example, a route hoie deision proto ol may only
request utilities for single links in the network. The utility funtion
is an optional omp onent that may b e omitted.
The (likewise individually parameterized)
deision proto ol
prob-
abilistially generates a single deision based on this utility informa-
tion. If there is no utility funtion, the hoie is diretly based on
the network onditions. A deision proto ol an b e deomp osed in
the two asp ets of
hoie set generation
and
hoie
. It may b e
delib erative
in that the hoie set of available alternatives is one
enumerated b efore a hoie is made. Alternatively, a
reative
searh
may b e implemented that iterates b etween the generation of some al-
ternatives and their evaluation. In either ase, one hoie is nally
realized by the agent.
This struture is indep endent of a partiular planning or telematis ontext. For
exp erimental purp oses, all simulator omponents were exemplarily implemented
similar to the aording omponents of the MATSim (Multi-Agent Transp ort
Simulation To olkit) simulation system [119℄, in the ontext of whih this work
was onduted.
Estimation is based on reasonable mathematial inferene but follows a sim-
ple tehnial logi. As illustrated in Figure 1.2, the simulation struture is not
hanged at all. An
estimator
omp onent is inserted between the deision pro-
to ol and the remaining simulation system. It is implemented transparently in
that it provides unmo died interfaes to both the deision proto ol and the re-
maining system. The estimator ompares the output of the mobility simulation
to
sensor data
from a surveillane system. Based on this omparison, it alters
27
Figure 1.2: Estimation
Estimation is failitated by the addition of a logial wrapp er around the deision
proto ol. All interfaes within the original simulation system remain unhanged.
the data and ontrol ow around the deision proto ol suh that the resulting
agent b ehavior is most plausible given the measurements.
Two small route hoie examples illustrate how this minor system extension
allows to adjust simulated b ehavior:
•
If the surveillane system observes a tra jam where there is none in the
simulation, the estimator inreases the systemati utility of the aording
links until the agents start to favor these links and reate the ongestion
as observed in reality. Vie versa, if there is ongestion in the simulation
but not in reality, the estimator dereases the involved links' utility until
the agents start to avoid the ritial area.
•
Likewise, the estimator an enourage a ertain b ehavioral pattern by
asking the deision proto ol to draw several alternatives in idential on-
ditions for eah agent. From this set of options, the estimator then passes
only those deisions on to the mobility simulation that are most plausible
given the measurements.
Either approah aesses only a subset of the interfaes touhed by the estimator
in Figure 1.2. This further relaxes the strutural requirements on the simulation
system. The apparent simpliity of this approah is onfronted with (i) the
diulties to relate aggregate measurements and individual b ehavior through
nonlinear tra ow dynamis on large networks of general top ology and (ii) the
intention to b e ompatible with a broad variety of b ehavioral implementations.
The software prototype is single-threaded and written in the Java programming
language [84℄. Its interfae-based design relies on standard software design pat-
28
terns [70℄ in order to simplify the (re-)omp osition of available software omp o-
nents. Likewise experimental implementations for the simulation of sp ontaneous
route swithing b ehavior [79, 80℄ and route guidane by feedbak ontrol [154℄
are integrated in the system.
1.3.2 Metho dologial Contribution
This thesis presents a novel approah to the fully disaggregate estimation of
motorist b ehavior with a multi-agent simulation. The problem is solved by a
ombination of
prior
knowledge ab out the driver b ehavior with available mea-
surements into most likely
posterior
estimates of this b ehavior:
•
The prior knowledge ab out the driver behavior onsists of two parts. First,
an individually mo deled agent exhibits likewise individual features that
inuene its b ehavior, e.g., so io eonomi features, preferenes, and infor-
mation availability. Seond, every suh agent has one or more individually
generated plans it adheres to. These plans sp eify what the agent intends
to do during a day.
•
The measurements of aggregate tra features suh as ows, densities or
velo ities are available at a limited set of network lo ations. Beyond link
related quantities, turning move ounts an b e diretly utilized by the
estimator. The amount of measurements may be arbitrarily small sine
the availability of individual plans guarantees an existing solution to the
estimation problem.
Based on this information, arbitrary b ehavioral asp ets ranging from single
route hoie to plan seletion for a whole day are estimated in a fully disag-
gregate manner, agent by agent. Estimation metho ds of dierent omplexity
are prop osed that allow for a problem-sp ei balane b etween omputational
sp eed and estimation preision. Exp erimental results are given and indiate the
estimator's pratial appliability.
The estimator an b e used in a planning ontext (with an underlying equilib-
rium assumption) and for real time tra monitoring (with a b ehavioral model
that aounts for inomplete driver information and sp ontaneous b ehavior). If
within-day estimates are fed bak to a planning system for inremental adjust-
ments on a day-to-day basis, improved prior information for the following day's
online estimation problem an b e generated.
The following results are also onsidered to b e relevant ontributions. They are
obtained as intermediate steps on the way to a working estimator.
•
A marosopi tra ow simulator is developed that is onsistent with
the phenomenology of the ell-transmission model and the requirements
of rst order tra ow theory. It eiently alulates linearized tra
ow dynamis, while its advaned simulation logi upholds a high ompu-
tational p erformane that allows to simulate large networks of arbitrary
top ology. While linearization is required for estimation, the lass of ap-
pliable mobility simulations is not restrited to this partiular mo del.
29
•
A simulation logi is prop osed that runs a marosopi tra ow mo del
based on the travel b ehavior of a fully mirosopi agent p opulation. This
ontribution to the eld of mesosopi mo deling provides a broadly ap-
pliable link b etween b ehavioral mirosimulation and physial marosim-
ulation.
•
A metho d is develop ed that steers the b ehavior of simulated travelers
suh that a general ob jetive funtion of aggregate network onditions is
improved. Sp eially, this result is employed to express and solve one
instane of the b ehavioral state estimation problem. More generally, the
metho d holds promise for further appliations suh as the generation of
road priing strategies.
1.3.3 Struture of Thesis
The remainder of this doument is organized as follows. Chapter 2 desribes the
marosopi mobility simulation. Chapter 3 treats the disaggregate mo deling
of b ehavior. Its rst part desrib es how individual motorists are simulated in a
marosopi mobility simulation. Its seond part sp eies a formalism of driver
b ehavior that is amenable to a mathematial estimator. Chapter 4 formulates
the b ehavioral estimation problem and disusses dierent solution approahes.
Chapter 5 veries the estimator's omputational feasibility for an appliation
of pratially relevant size. Finally, the work is onluded in Chapter 6, and a
disussion of future researh topis is given.
30
Chapter 2
Marosopi Mobility
Simulation
A mo del of physial reality maps demand for travel on network onditions.
Basially, an inverse mapping is needed if travel b ehavior is to be dedued from
these onditions. Suh an inversion do es generally not exist. Alternatively, a
linearization of the mapping is used, and nonlinearities are aounted for in an
iterative manner.
This hapter desrib es a mobility simulation that an b e linearized. A reader
with only a asual interest in tra ow mo deling may skip this material and
ontinue reading at Setion 2.7 without muh loss of ontinuity.
2.1 Design Choies
The neessity of linearization alls for a marosopi mo del. An aggregation of
travelers into homogeneous groups an b e avoided by the b ehavioral simulation
sheme intro dued later in Chapter 3 so that only single-ommo dity tra is
onsidered here.
Sine the exp erimental validation of new phenomenologial prop osals would ex-
eed the sop e of this thesis, the mo del must build on established ndings. This
and the need to realize a large-sale test ase alls for the simplest available
mo del that still aptures the most relevant tra features with reasonable pre-
ision. Arguably, this is the
kinemati wave model
(KWM) [108, 151℄. Within
its phenomenologial limitations, it is able to represent b oth freeway and intra-
urban tra ow. The hoie of this mo del is well justied in light of the
ongoing debate if more omplex mo dels yield a reasonable gain in expressive
p ower [78, 131℄.
For numerial simulation of the KWM, the
el l-transmission model
(CTM)
is adopted [53, 54, 55℄. Various other marosopi mo dels had b een onsid-
ered b efore this hoie was made [73, 76, 86, 101℄. However, one higher or-
der mo dels are exluded from onsideration, the CTM remains as the by far
31
most established mo del, with various appliations, e.g., in freeway ramp meter-
ing and signal optimization [1, 66, 164℄, and thorough exp erimental validations
[28, 126, 127℄. The CTM is losely related to another implementation of the
KWM, the STRADA mo del [29, 30℄. Both approahes base on the numerial
Go dunov solution metho d [102, 106℄.
The mo del must allow to simulate a large and omplex road network, provide lin-
earized tra ow dynamis, and maintain a high omputational performane.
These requirements motivate three in large parts novel adaptations of the CTM:
•
To allow for linearization, all ow alulation rules of the CTM are unied
in a formal alulation sheme, for whih sensitivity analysis is onduted.
•
Sine the original CTM only speies network topologies where at most
three roads meet at an intersetion, its established phenomenology is
transferred to the mo deling of general intersetions.
•
Spatially disretized marosopi mo dels imply a relatively high omputa-
tional ost b eause of their large number of simulated entities. To ensure
feasibility of large-sale appliations, a simulation logi is adopted that
assigns an individual simulation time step duration to every link in the
network. The additional numerial impreision introdued by this mo di-
ation is investigated and is found to b e ountervailed by its omputational
b enets.
A simplied linearization of the CTM has b een desrib ed b efore [125, 126℄. This
approah swithes b etween linear sub-models aording to the ongestion status
of a onsidered freeway streth. It is a simpliation even of the CTM and is not
appliable to network tra. A likewise onstrained linearization is desrib ed in
[165℄. The originality of an earlier ontribution is also aknowledged where CTM
merges and diverges are reombined to generate more omplex intersetions and
a simulation logi with variable time step lengths is enabled by the nesting of
dierently fast tiking ells [104℄.
Some elements of the KWM theory are given in Setion 2.2. Before the CTM
is onsidered, a general and linearizable ow alulation sheme is intro dued
in Setion 2.3. The CTM and its extensions are then expressed in terms of
this formalism in Setion 2.4. The simulation logi on variable time sales is
desrib ed in Setion 2.5, a suitable spatiotemp oral network disretization logi
is prop osed in Setion 2.6, and, nally, a general state spae representation of
the mobility simulation is given in Setion 2.7.
2.2 The Kinemati Wave Mo del
The KWM requires a minimal set of assumptions to mo del tra ow on a linear
road. Denote by
x∈R
a lo ation on that road and by
t∈R
the ontinuous time.
(x, t)
is the loal density of tra (in vehiles
1
(veh) p er length unit),
q(x, t)
1
In the ontext of a marosopi mo del, the notion of a vehile is to b e understo o d as a
marosopi vehile unit.
32
its ow (in vehiles p er time unit), and
v(x, t)
its velo ity. These quantities are
related by the rst onstituent equation of the KWM:
q(x, t) = v(x, t)(x, t).
(2.1)
The seond modeling assumption is that of vehile onservation. On smo oth
onditions, it is expressed by the ontinuity equation
∂
∂t +∂q
∂x = 0.
(2.2)
Finally, lo al ow is sp eied as a funtion only of lo al density. This relation
is usually denoted as the fundamental diagram:
q(x, t) = Q((x, t), x).
(2.3)
Sine these sp eiations an still result in ambiguities, an additional ondition
must b e instrumented to selet the physially relevant solution. Given a onave
fundamental diagram, the priniple of lo al demand and supply provides a on-
venient tehnique to ensure uniqueness [102℄. Denote by
x−
(
x+
) the lo ation
immediately upstream (downstream) of
x
. For every
x
, the lo al ow
q(x, t)
is
then dened as the minimum of lo al ow
demand
∆((x−, t), x−)
and lo al
ow
supply
Σ((x+, t), x+)
:
q(x, t) = min{∆((x−, t), x−),Σ((x+, t), x+)}.
(2.4)
Figure 2.1 illustrates this funtion.
To b egin with, (2.4) reets the self-evident onstraint that lo al tra ow
is b ounded by the ow that an b e dismissed from the immediate upstream
lo ation and by the ow that an b e absorb ed by the immediately downstream
lo ation. But furthermore, the lo al ow is maximized sub jet to these on-
straints. This prop erty enfores the physially relevant solution of the KWM-
mo del [102℄. Phenomenologially, it is a statement of drivers' ride impulse [2℄,
whih is equivalently expressed by the mirosimulation rule for ellular automata
Drive as fast as you an and stop if you have to! [45℄.
Beyond its ability to uniquely apture tra ow along a link, this priniple
also holds for the modeling of general intersetions, as illustrated in Figure 2.2.
In suh a setting, every upstream link
i
provides a demand
∆i(t)
equal to its
greatest p ossible outow towards the intersetion, and every downstream link
j
provides a supply
Σj(t)
equal to its greatest p ossible inow. Additional phe-
nomenologial mo deling is failitated sine these b oundaries alone are generally
not suient to uniquely dene the ows aross an intersetion. However, every
reasonable sp eiation must adhere to the priniple of lo al ow maximization.
2.3 Intersetion Flow Calulation Sheme
This setion desrib es a formalism for intersetion tra ow mo deling denoted
as the
general pro ess of resoure onsumption
(GPRC). Sine sensitivity
33
Figure 2.1: Lo al supply and demand omprise a fundamental diagram
The pieewise linear demand funtion
∆()
onforms to the original sp eiation of
the CTM, where it is denoted as the
sending
funtion. It onsists of an inreasing
part with its slop e equal to the free ow sp eed, and it is limited by the ow apaity
ˆq
. The supply funtion
Σ()
(also onsistent with the original CTM, where it is alled
reeiving
funtion) is also limited by the ow apaity. The slop e of its delining part
equals the bakward wave sp eed and intersets the absissa at the greatest p ossible
density
ˆ
. The minimum of b oth funtions yields a fundamental diagram.
Figure 2.2: A p oint-like intersetion with
I
ingoing and
J
outgoing links
Every upstream link
i
provides a demand
∆i
equal to its greatest p ossible outow
towards the intersetion, and every downstream link
j
provides a supply
Σj
equal to
its greatest p ossible inow.
34
Algorithm 1
General pro ess of resoure onsumption
ξ(0)
is given
D(0) ={i;ξ(0)
i>0}
m= 0
while (
∃i∈D(m):ϕi(D(m))>0
), do {
for all
i∈D(m)
, do:
θ(m)
i=ξ(m)
i/ϕi(D(m))
θ(m)= min
i∈D(m){θ(m)
i}
B(m)= arg min
i∈D(m){θ(m)
i}
ξ(m+1) =ξ(m)−θ(m)ϕ(D(m))
D(m+1) =D(m)\B(m)
m+ +
}
M=m
analysis for the GPRC is available, every intersetion model that onforms to
its sp eiation an b e linearized.
Consider a dynamial pro ess with time step index
m= 0 . . . M
. Every element
ξ(m)
i∈[0,∞)
of its state vetor
ξ(m)= (ξ(m)
i)
is onsidered as a resoure that
is used up during the pro ess. Its rate of onsumption equals a non-negative
and nite value
ϕ(m)
i
, whih is onstant throughout every time step
m
. Denote
the duration of step
m
by
θ(m)
. The pro ess dynamis are then dened by
ξ(m+1) =ξ(m)−θ(m)ϕ(m)
where
ϕ(m)= (ϕ(m)
i)
. The resoures must not
b eome negative suh that all zero states must have a zero onsumption rate
and
θ(m)≤ξ(m)
i/ϕ(m)
i
must hold for all nonzero states
i
.
The set
D(m)={i;ξ(m)
i>0}
ontains all resoures that are stritly p ositive at
the b eginning of step
m
. The pro ess terminates if all elements in
D(m)
have
a zero onsumption rate. Consumption rates only dep end on the set
D(m)
of
urrently
available
resoures suh that
ϕ(m)=ϕ(D(m))
. Consequently, it is
phrased that step
m
is under
regime
D(m)
. The maximum duration of step
m
in exlusive onsideration of resoure
i
is
θ(m)
i=ξ(m)
i/ϕ(m)
i∈(0,∞)
. Sine
every step
m
is sp eied to last until at least one resoure in
D(m)
reahes
a zero value, its duration is
θ(m)= mini∈D(k){θ(m)
i}>0
. The set
B(m)=
arg mini∈D(m){θ(m)
i}
ontains all resoures that run dry at the end of step
m
.
2
This allows to give
D(m+1) =D(m)\B(m)
as an up date equation.
The temp oral asp et of this pro ess is not to b e interpreted physially. Only
its nal state is of relevane to the physial simulation. Algorithm 1 gives an
overview. An eient implementation of the involved integer sets is desrib ed
in App endix A .
Sensitivity analysis for the GPRC is provided in App endix B, where the fol-
lowing result is derived. It ensures linearizability of the subsequently develop ed
tra ow mo del.
If al l onsumption rates are monotonously inreasing with respet to the number
of available resoures, i.e., if
ϕi(D∪{j})≥ϕi(D)∀i, j
, and if the availability
2
The argmin funtion returns the set of all minimizing indies.
35
Figure 2.3: A straight onnetion
The mapping of upstream demands
∆
and downstream supplies
Σ
on GPRC resoures
ξ
is sp eied in (2.5).
of a resoure with a zero onsumption rate does not inuene the proess dy-
namis, i.e., if
ϕi(D∪{i}) = 0 ⇒ϕ(D\{i}) = ϕ(D∪{i})
, then an approximate
Jaobian
∂ξ(M)/∂ξ(0)
an eiently be omputed onurrently with the GPRC.
If, furthermore, the onsumption rates are parameterized with a onstant pa-
rameter vetor
β
and the sensitivities
∂ϕ(D)/∂β
are provided, an approximate
Jaobian
∂ξ(M)/∂β
an be omputed in a likewise eient way.
2.4 Intersetion Sp eiation
The CTM runs in disrete time and spae. Denote the physial simulation
time step length by
T
, the physial simulation time step ounter by
k
, and
the spatial segments of a link as
ells
. A
onnetor
is plaed b etween every
group of adjaent ells. Eah suh onnetor runs a GPRC implementation that
alulates the ow transmissions between these ells.
3
The demand
∆i(k)
of upstream ells
i= 1 . . . I
and the supply
Σj(k)
of down-
stream ells
j= 1 ...J
(b oth in vehiles p er time step duration) are mapp ed on
individual GPRC resoures by
ξ(0)
i(k) = T∆i(k)
for
i
upstream
ξ(0)
I+j(k) = TΣj(k)
for
j
downstream
.
(2.5)
Transmitted vehile ounts and equivalent average out- and inow rates
q
out
i(k)
,
q
in
j(k)
result after the GPRC's termination from
T q
out
i(k) = ξ(0)
i(k)−ξ(M)
i(k)
for
i
upstream
T q
in
j(k) = ξ(0)
I+j(k)−ξ(M)
I+j(k)
for
j
downstream
.
(2.6)
The original CTM ow alulation rules and their ontinuation into a general
intersetion mo del an now b e expressed by appropriate sp eiations of the
resoure onsumption rates
ϕ(D)
.
2.4.1 Straight Connetions
The CTM's basi ow alulation rule states that the numb er of transmitted
vehiles between two sueeding ells equals the minimum of the available ve-
hiles upstream, the available spae downstream, and an upp er ow onstraint.
This is the disrete-time equivalent of (2.4). The aording straight onnetor
3
Sine a
yweight design pattern
is used for implementation [70℄, the numb er of atually
reated GPRC ob jets winds down to the numb er of dierent intersetion top ologies.
36
Figure 2.4: A merge with
I
ingoing links
The mapping of upstream demands
∆
and downstream supplies
Σ
on GPRC resoures
ξ
is sp eied in (2.5).
has one predeessor and one suessor ell. Sp eaking in terms of the GPRC, its
resoure vetor
ξ= (ξ1ξ2)T
is two-dimensional:
ξ1
represents the number of
available upstream vehiles and
ξ2
equals the available downstream spae, f.
Figure 2.3. The sup ersript
T
denotes the transp ose. The resoure onsumption
vetor
ϕ({1,2}) = (1 1)T
(2.7)
orresp onds to the only regime
{1,2}
with a nonzero onsumption rate. The
resulting one-step GPRC run yields an idential vehile transmission as the
original CTM.
2.4.2 Merges
The original CTM allows for merge onnetions between exatly two upstream
ells and one downstream ell. The aording ow alulation rules state that
b oth predeessors are allowed to send all their available vehiles as long as these
an be aepted by the suessor ell. If this is not the ase, the suessor's
available spae is shared b etween the predeessors in a ratio aording to their
priorities
α1∈[0,1]
and
α2= 1 −α1
. If this auses all available vehiles of one
predeessor to b e transmitted but still leaves available spae in the suessor,
this spae is lled up as muh as p ossible with vehiles from the omplementary
predeessor.
In terms of the GPRC, the merge resoure vetor is
ξ= (ξ1ξ2ξ3)T
where
ξ1
and
ξ2
denote the available vehiles in the predeessor ells and
ξ3
equals the
available spae in the suessor ell. The evolution of the pro ess is fully dened
by three non-zero onsumption rate vetors
ϕ({1,2,3}) = (α1α2α1+α2)T
,
ϕ({1,3}) = (α10α1)T
, and
ϕ({2,3}) = (0 α2α2)T
. Here, the priorities do not
have to sum up to
1
but are required to b e stritly p ositive. An insp etion of
the regime sequenes
{1,2,3} → {1,3}
and
{1,2,3} → {2,3}
shows that this
setup yields an idential b ehavior as the original CTM.
General merge onnetors have an arbitrary numb er of
I≥2
predeessor ells, as
shown in Figure 2.4. The rst
I
elements of the aording resoure vetor are the
available vehiles
ξi
in the predeessor ells
i= 1 . . . I
. The available spae
ξI+1
in the suessor ell makes up one additional resoure:
ξ= (ξ1. . . ξIξI+1)T
.
37
Figure 2.5: A diverge with
J
outgoing links
The mapping of upstream demands
∆
and downstream supplies
Σ
on GPRC resoures
ξ
is sp eied in (2.5).
A straightforward ontinuation of the CTM merge logi is
ϕ(D) = ϕ1(D). . . ϕI(D)
I
X
i=1
ϕi(D)!T
ϕi(D) = αi{i, I + 1} ⊆ D
0
otherwise
,
(2.8)
where
{i, I + 1} ⊆ D
indiates that b oth the upstream ell
i
and the only
downstream ell provide nonzero resoures. For
I= 2
, this repro dues the
original CTM merge. Sine the total vehile transmission is only bounded by
the available upstream vehiles and the downstream spae, ow maximization
is ensured.
A generalization of the CTM merge logi to more than two predeessors has
previously b een referred to as very ompliated [ 86℄. With the GPRC at
hand, this diulty ollapses into speiation (2.8).
2.4.3 Diverges
Diverges of the original CTM split the ow from one predeessor ell into ex-
atly two suessor ells. The splitting frations are denoted by
β1∈[0,1]
and
β2= 1 −β1
. Here, the resoure vetor
ξ= (ξ1ξ2ξ3)T
is omprised of
the single predeessor's available vehiles
ξ1
and the available spae
ξ2
and
ξ3
in the suessor ells. Allowing for only one non-zero onsumption rate ve-
tor
ϕ({1,2,3}) = (1 β1β2)T
implies the assumption of exatly one upstream
lane: If a vehile at the head of the queue on this lane is unable to enter its
downstream ell, it ompletely blo ks the diverge. This logi is reasonable for
large-sale appliations [54, 119℄. The resulting total outow from the prede-
essor is
min{ξ1, ξ2/β1, ξ3/β2}
, just as for the original CTM.
The simulation of
J≥2
suessors for a general diverge, as shown in Figure
2.5, is straightforward by the introdution of an extended resoure vetor
ξ=
(ξ1ξ2...ξ1+J)T
and an aording onsumption rate vetor
ϕ({1,2,...,1 + J}) = (1 β1...βJ)T
(2.9)
for the only non-zero onsumption regime
{1,2,...,1+J}
. For
J= 2
, this yields
idential ow transmissions as the original CTM. The ow is again maximized
sub jet to the availability onstraints and the additional splitting rule.
38
Figure 2.6: A general onnetion with
I
ingoing and
J
outgoing links
The mapping of upstream demands
∆
and downstream supplies
Σ
on GPRC resoures
ξ
is sp eied in (2.5).
Cho osing zero onsumption rates for all regimes but
{1,...,J,1+J}
is neessary
to ensure ontinuity of the ow transmissions with resp et to the turning fra-
tions, whih is required for the linearization of the mo del: If tra ould pass
the diverge unhindered given an unavailable suessor
j
with
βj= 0
, inreasing
βj
by an arbitrarily small amount would instantaneously blo k the diverge. This
disontinuity is avoided by letting the diverge blo k even if
βj= 0
as so on as
suessor
j
b eomes unavailable. This restrition an b e dropp ed if ontinuity
is not required and vanishes anyway in the ombined miro/maro simulation
sheme of the next hapter where all turning frations are guaranteed to b e
stritly p ositive.
2.4.4 General Connetions
A general onnetor is shown in Figure 2.6. Denote by
P={1,...,I}
the
set of its upstream ells, by
S={I+ 1,...,I +J}
the set of its downstream
ells, and by
βij
the presp eied turning fration from predeessor
i
towards
suessor
j
. Given a predeessor onsumption rate
ϕi(D)
, the sp eiation of
suessor oriented onsumption rates
ϕij(D) = βijϕi(D)
maintains onsisteny
with diverge logi (2.9). A priority rule equivalent to merge logi (2.8) is ensured
by letting
ϕi(D) = αi
for all available predeessors
i
as long as the intersetion
is not blo ked by an unavailable suessor. The omplete resoure vetor
ξ=
(ξ1. . .ξIξI+1 ...ξI+J)T
is then onsumed by
ϕ(D) = (ϕ1(D)...ϕI(D)ϕI+1(D)...ϕI+J(D))T
i∈P:ϕi(D) = αii∈D, S ⊆D
0
otherwise
j∈S:ϕj(D) = X
i∈P
βijϕi(D).
(2.10)
Again, all priorities must b e stritly p ositive. The same statements ab out zero
turning frations hold as for a diverge. This general onnetor omprises all
previously dened onnetor typ es as an b e seen from hoosing
I= 1
and/or
J= 1
. Still, it has no immediate ounterpart in the CTM. Its logi results
as the limiting ase of a merge whih is onneted by an innitely short link
to a diverge whose turning frations
βj
result via
βj=PI
i=1 βijqi/PI
i=1 qi
from the ow omp osition
q1,...,qI
transmitted by the merge. No additional
phenomenologial sp eulations are intro dued in this model.
39
It remains to show that the original CTM's onsisteny with the KWM is
maintained, i.e., that speiation (2.10) is still ow-maximizing. In unon-
gested onditions, the intersetion winds down to a linear sup erp osition of
I
diverges and inherits their prop erties. In ongested onditions, the total ow
through the intersetion is limited by at least one downstream ell
j∗
with
Σj∗=PI
i=1 βij∗qi
, aording to (2.9). Assume that
PI
i=1 q′
i>PI
i=1 qi
was
p ossible for an altered onguration
q′
1,...,q′
I
of merging inows. The down-
stream diverge logi still requires
Σj∗≥PI
i=1 βij∗q′
i
, and the merge logi de-
mands
q′
i≥qi
for all
i= 1 . . .I
if more downstream spae b eomes available.
Thus,
Σj∗≥PI
i=1 βij∗q′
i≥PI
i=1 βij∗qi= Σj∗
, whih implies
q′
i=qi
for all
i
.
In onsequene, the general intersetion inherits the ow-maximizing prop erty
of its merge and diverge omp onent.
Sp eiation (2.10) omplies with the GPRC's requirements for linearization,
as stated in Setion 2.3. The relations between demands/supplies and GPRC
resoures (2.5) and b etween GPRC resoures and ow rates (2.6) are already
linear. Combined, this ensures the availability of ow rate sensitivities with
resp et to demands
∆
, supplies
Σ
, and turning prop ortions
β
.
2.5 Simulation Logi
Disrete time network simulation is straightforward if a uniform time step length
T
is used. Every link with maximum velo ity
ˆv
is disassembled into ells of
minimum ell length
=Tˆv.
(2.11)
A simulation step (tik) then onsists of two parts:
1. Every onnetor alulates the vehile transmissions b etween its adjaent
ells.
2. Every ell up dates its o upany aording to these transmissions.
The
o upany
of a ell (link) is dened as the number of vehile units that
are lo ated in that ell (link).
The simulation of a heterogeneous urban network requires relatively small ells
in order to mo del densely meshed regions. This alls for a small
T
and in
turn implies an unneessarily preise mo deling of longer road segments. The
use of larger ells running on the same temp oral grid somewhat mildens this
problem at the ost of a greater numerial disp ersion [55, 102, 127℄. However, a
signiant share of urban network omputations is inurred by the intersetion
logi. Thus, a simulation logi that minimizes the number of simulation tiks
themselves is needed.
The spatiotemp oral dynamis within an isolated link are uniquely dened if
an initial density prole as well as feasible upstream inows and downstream
outows are provided. Given an individually hosen time step length and an
appropriate spatial disretization, the standard CTM logi failitates a KWM-
onsistent simulation. Sine all spatial dynamis are enlosed within the link,
40
it an b e viewed from the outside as a disrete-time, nonlinear, ordinary dy-
namial system with two inputs (in- and outows) and two outputs (upstream
ow supply and downstream ow demand). The same argument holds for in-
dividual ells. Likewise, the intersetion mo del of Setion 2.4.4 alulates ows
onsistently with the KWM. For any hosen time step length, it onstitutes a
memoryless, disrete-time, nonlinear system with its upstream ow demands
and downstream ow supplies as inputs and the resulting ow transmissions as
outputs.
Adopting a tehnial p oint of view, these systems an immediately be linked.
The outputs of systems with a larger time step are held onstant when needed
as inputs for faster tiking systems, and the outputs of faster tiking systems
are integrated/averaged b efore they are fed into slower tiking systems. Sine
suh holding and averaging aet system dynamis mainly in terms of a delay
that is proportional to the involved time step lengths, a reasonable balane
b etween additionally introdued impreision and omputational sp eedup an
b e ahieved. This is onrmed by the exp erimental results given in Setion
2.5.4.
The remainder of this setion details this simulation logi. A ell
i
(onnetor
c
)
is denoted as
due
at disrete
simulation time step
k
if
k
is an integer multiple
of its
individual time step
length
Ti
(
Tc)
. The duration of a simulation time
step is generally assumed to b e 1 seond. Two pro edures are exeuted at every
simulation time step
k
:
1. Every ell
i
that is due aording to its individual time step length
Ti
alulates its supply and demand b oundary from its urrent oupany
and keeps these results onstant for the next
Ti
seonds.
2. Every onnetor
c
that is due aording to its individual time step length
Tc
alulates its average ow rates that hold for the next
Tc
seonds and
noties its adjaent ells of the resulting vehile transmissions.
Setions 2.5.1, 2.5.2, and 2.5.3 detail these steps.
2.5.1 Cell Boundaries
Every ell
i
has exatly one preeding and one sueeding onnetor. Its o -
upany during simulation time step
k
is denoted by
xi(k)∈[0,ˆxi]
where
ˆxi
is its maximum o upany. While the ell has an individual time step length
Ti
, it is emb edded in a system p otentially running at a 1-seond time sale.
This requires its demand
∆i(k)
and supply
Σi(k)
to b e dened at every seond.
Sine these b oundaries are stati funtions only of
i
's o upany, it is suient
to sp eify
xi
in every simulation time step by
xi(rTi+s) = xi(rTi)r∈N, s ∈ {0,...,Ti−1}.
(2.12)
The original CTM b oundary sp eiations an now b e applied:
∆i(k) = min ˆqi,ˆvixi(k)
Li
Σi(k) = min ˆqi,wi(ˆxi−xi(k))
Li
(2.13)
41
where
ˆqi
denotes the ell's ow apaity (in vehiles p er time unit),
Li
its length,
and
wi
its bakward wave speed. These equations an approximately b e lin-
earized with resp et to
xi(k)
if at p oints of non-smo othness the average of
left- and right-sided sensitivity is used. Alternative sp eiations are p ossible
[55, 102℄.
2.5.2 Connetor Flow Rate Up date
Every onnetor
c
has a set
Pc
of preeding ells and a set
Sc
of sueeding
ells. Its individual time step length
Tc
is hosen suh that (i) the onnetor
realulates its ow rates whenever an adjaent ell b oundary hanges and (ii)
the overall omputational load is minimized. This is ahieved by hoosing
Tc
as
the largest ommon divisor of all adjaent ells' time step durations:
Tc=
ld
i∈Pc∪Sc{Ti}.
(2.14)
Arbitrary ell time step durations might yield low omputational savings b e-
ause of p ossibly small
Tc
values resulting from this equation, so they are on-
strained to b e p owers of two. This turns the onnetor time step length into
the minimum of its adjaent ells' time step durations.
2.5.3 Cell State Up date
Even if a ell
i
's state
xi
hanges only every
Ti
seonds, its adjaent onnetors
might run at a higher frequeny. On the nest temp oral sale, this implies
xi(rTi+Ti) = xi(rTi) + 1
s
Ti−1
X
s=0 q
in
i(rTi+s)−q
out
i(rTi+s).
(2.15)
Denote by
pi
(
si
) the preeding (sueeding) onnetor of ell
i
. Beause of
(2.14),
Ti/Tpi
and
Ti/Tsi
are integer values. This allows for the following sim-
pliation:
xi(rTi+Ti) = xi(rTi)
+Tpi
Ti/Tpi−1
X
s=0
q
in
i(rTi+sTpi)
−Tsi
Ti/Tsi−1
X
s=0
q
out
i(rTi+sTsi).
(2.16)
Therefore, it is suient to notify ell
i
every ld
{Tpi, Tsi}
seonds of p ossible
ow rate hanges. This is done indep endently by its upstream and downstream
onnetor every
Tpi
and
Tsi
seonds by transmitting the appropriate addend in
(2.16) to the ell. Sine the ell's b oundaries are held onstant for a p ossibly
longer duration aording to (2.12) and (2.13), the transmitted vehiles are
intermediately ahed by the ell. Equation (2.16) is dierentiable with resp et
to in- and outow rates.
42
Table 2.1: Link parameters in linear test network
max. density 1 veh / 7.5 m
≈
133 veh/km
ow apaity 2000 veh/h
max. veloity 50 km/h
ell length 50 km/h
·
1 s
≈
13.9 m
link length 32 ells/link
·
13.9 m
≈
444 m
2.5.4 Exp erimental Investigation of Simulation Preision
A linear test network is onsidered. It onsists of a sequene of 5 idential links
the parameters of whih are given in Table 2.1. The simulation b oundaries
resemble the onditions in whih the CTM was rst investigated [53℄: A linear
density gradient from zero to maximum density is plaed on the network, with
zero density at its upstream end and maximum density at its downstream end.
No tra is allowed to enter or leave the network. The simulation is run until
a steady state is reahed.
Figure 2.7 shows the resulting spae-time plots in various disretization settings.
Plot 2.7(a) provides a go o d approximation to the exat solution. Initially, two
sho kwaves o ur: an upstream sho kwave moving at p ositive veloity and a
downstream shokwave moving at negative velo ity. They merge in the enter
of the network and p ersist as a stationary density disontinuity with all tra
b eing queued up in the downstream half of the network. For omparison, the
simulation results with a muh oarser but still homogeneous disretization are
shown in plot 2.7(b).
The results with heterogeneous simulation time steps niely reet the working
of the underlying Go dunov method. In every simulation time step, the Go dunov
sheme solves a Riemann problem at all ell b oundaries. Sine ondition (2.11)
ensures that the resulting sho kwaves or rarefation fans do not ross b eyond
one ell during a single time step, these problems an b e solved independently
in a omputationally eient way [102, 106℄. Plaing fast tiking ells next to
slower ells expliitly displays these sho kwaves, as it an b e seen b est in plot
2.7(). While these artifats are unequivo ally owed to the simulation logi on
variable time sales, they are put into relation by plot 2.7(d). It shows the same
result after it has b een averaged on a temp oral grid aording to the largest
involved time step duration. The artifats are niely smeared out while the
original shokwaves are maintained with a preision that is at least omparable
to plot 2.7(b). Analogial statements holds for plots 2.7(e) and 2.7(f ).
These results indiate that the overall simulation error remains in the order of
the largest involved time step duration, as it has b een previously hyp othesized.
Artifats an o ur at the b oundaries b etween slowly and fast tiking ells but
an also be removed by a temp oral averaging of the simulation output b efore
further pro essing. No ampliation of artifats is observed. These exp eriments
annot replae a thorough theoretial investigation. They are, however, onsid-
ered as suient indiations that the simulation logi on variable time sales
p erforms well enough to b e b e applied in the further ourse of this dissertation.
43
(a) All links have a time step duration
of 1 seond and onsist of 32 ells eah.
(b) All links have a time step duration
of 8 seonds and onsist of 4 ells eah.
() All but the seond and fourth link
have an 8 seond time step duration.
(d) The same data as () but averaged
on a temporal grid of 8 seonds.
(e) Only the seond and fourth link have
an 8 seond time step duration.
(f ) The same data as (e) but averaged on
a temp oral grid of 8 seonds.
Figure 2.7: Spae-time plots with variable spatiotemporal disretizations
Colors eno de densities as follows: green is zero density, yellow is half of maximum
density, and red is maximum density. See also Table 2.1 . The parenthesized numbers
b elow the links indiate their individual time step durations.
44
2.6 Network Disretization
2.6.1 Sp eiation
Sp eiations of large road networks usually onsist of an attributed graph
where no des represent intersetions and links represent roads, e.g., [119, 147,
163℄. The ell struture of suh a network an be generated by the following
steps:
1. Cho ose a maximum simulation time step length
ˆ
T
. This
network time
onstant
ompromises b etween a high simulation resolution (small
ˆ
T
)
and a high omputational p erformane (large
ˆ
T
).
2. For every link
a
in the network, do:
(a) Selet the individual time step duration
Ta
of link
a
as large as p os-
sible sub jet to the following onstraints:
•Ta
is stritly p ositive and not larger than
ˆ
T
.
•Ta
is an integer p ower of two.
•
It is required that link
a
an b e partitioned into at least two ells
of equal length
La/2
. Sine (2.11) must hold for eah of these
ells,
Ta≤La/(2ˆva)
is required.
If link
a
is so short that no feasible
Ta
exists, inrease
La
just until
Ta= 1 s
b eomes a feasible solution.
(b) Partition link
a
into
na
idential ells of length
La/na
. In order to
minimize disp ersion, hoose
na
as large as p ossible without violating
ondition (2.11). That is,
na≤La/(ˆvaTa)
must hold. The previous
hoie of
Ta
ensures that this yields at least two ells in link
a
.
3. Plae a onnetor
c
b etween every set of adjaent ells, and alulate its
individual time step length
Tc
via (2.14).
The network entrane of tra is failitated by entry ells in onsisteny with
the original CTM implementation [40℄. Entry ells an hold an arbitrary o u-
pany, have no upstream onnetor, and a maximum outow equal to the entire
o upany that waits in the ell to enter the system. One entry ell is onneted
to the innermost onnetor of every link. The existene of suh a onnetor is
ensured sine every link onsists of at least two ells. A sp eiation of the net-
work exit of tra is p ostp oned to Setion 3.1 where multi-ommo dity tra is
intro dued. The allo ation of demand entry p oints to links and not to nodes is
hosen in onsisteny with the MATSim demand sp eiation [119℄.
2.6.2 Berlin Test Case
The test ase of this thesis is mo deled after the road network of Greater Berlin,
whih is illustrated in Figure 2.8. This network onsists of
1 083
no des and
2 459
unidiretional links. It is quite heterogeneous in that the inner-urban area is
45
Figure 2.8: Ma jor road network of Greater Berlin
The two lippings indiate a loally high network resolution.
46
Figure 2.9: Eet of network time onstant on ell ount
Numb er of ells over
log2(ˆ
T)
. Sine the network geometry has a limiting eet on the
ell sizes,
ˆ
T
values b eyond
26
s do not result in a notably inreased oarsening.
Figure 2.10: Time step duration histogram
Histogram of logarithmi intersetion onnetor time step durations given a network
time onstant of
ˆ
T= 64
s.
47
mo deled in relatively high resolution, whereas the surrounding freeway ring is
omprised of several links that are many kilometers long.
Figure 2.9 shows the eet of the network time onstant
ˆ
T
on the number of
ells in the network. As
ˆ
T
inreases, the number of ells approahes a minimum
value of
2·2 459
. This mirrors the ab ove requirement of at least two ells p er
link. A histogram of intersetion onnetor time step lengths for
ˆ
T= 64
s is
given in Figure 2.10. The high numb er of intersetions with a relatively low
time step duration is owed to the nely meshed interurban network, whih is
preluded from a slower simulation lo k. The relation b etween network time
onstant and omputational p erformane is investigated in Setion 3.1.4.
2.7 State Spae Notation
For greatest generality, the remainder of this thesis is deoupled from sp ei
tra ow mo deling assumptions by the following state spae representation of
the mobility simulation:
x
ms
(0) = x
ms
0
x
ms
(k+ 1) = f
ms
[x
ms
(k),β(k), k].
(2.17)
Vetor
x
ms
(k)
denotes the mobility simulation's
physial state
in time step
k
.
For a spatially disretized rst order mo del, it ontains one element for every
ell
i
in the network:
x
ms
= (xi)
. Single-ommo dity turning frations
β(k) =
(βij(k))
are provided as exogenous parameters to the mo del. Vetor-valued
transition funtion
f
ms
denes the system's evolution through time. It fully
enapsulates the sp eially hosen mobility simulation. The formal mo deling
of demand soures and sinks is p ostp oned to the next hapter.
For the subsequent analysis, it is required that at least approximate Jaobians
∂f
ms
[...,k]/∂x
ms
(k)
and
∂f
ms
[...,k]/∂β(k)
are available. This ondition is
fullled by the mobility simulation prop osed in this hapter sine
•
ell state up date equation (2.16) is linear with resp et to in- and outow
rates,
•
these ow rates an b e linearized with resp et to ell b oundaries and
turning frations, f. (2.5), (2.6), and Setion 2.4.4, and
•
ell b oundary sp eiation (2.13) is linearizable with resp et to the ell
states.
48
Chapter 3
Mirosopi Behavioral
Simulation
This hapter prepares a formal link between individual driver b ehavior and
aggregate harateristis of tra ow.
First, motorist driving deisions are expressed as ontrol measures that at on
a state spae mo del of marosopi tra dynamis. The resulting formalism
is quite general and allows to link dierent marosopi mobility simulations
and mirosopi b ehavioral mo dels. In partiular, it allows to predit the lin-
earized eet of individual driver behavior on global network onditions without
rep eated simulations.
Seond, the deision making pro ess of a driver is formalized in a way that is
ompatible with the aforementioned miro/maro mobility simulation. This rep-
resentation omprises a broad variety of p ossible b ehavioral simulators. Some
more sp ei mo deling approahes are also presented. Apart from their illus-
trative purp ose, they intro due mo deling asp ets that are referred to in later
hapters.
3.1 Coupling Miro- and Marosimulation
Two dierent onepts an b e enountered in the literature on ombined mi-
ro/maro mobility simulations.
Hybrid
approahes link simulations that work on dierent degrees of aggregation
at well-dened lo ations in the network [32, 64℄. This approah is attrative if
the required simulation delity varies spatially but do es not serve the purp ose
of this work where a network-wide marosopi mo del is needed.
Mesosopi
simulations move individual vehiles based on aggregate laws of
motion in order to inrease the omputational performane while retaining a
mirosopi representation of b ehavior [31, 35℄. Simulation-based DTA usually
employs suh mo dels, f. Setion 1.2.2.2 and the referenes therein. Their
ounterpart in physis are
smoothed partile hydrodynamis
(SPH) [124, 155℄.
49
The approah desrib ed here is a mesosopi mo del with a distint marosopi
asp et. In this way, mathematial feasibility (linearization of the marosopi
mo del) and expressive p ower (mirosimulation of b ehavior) are ombined. High
omputational p erformane is maintained by a simulation sheme on variable
time sales.
3.1.1 Representation of Behavioral Heterogeneity
Pursuing a stritly marosopi approah, heterogeneous driver b ehavior ould
b e aptured by splitting tra volumes into partial ows (
ommodities
) with
individual b ehavioral features. For example, destination-b ound ommo dities
would exhibit dierent turning b ehavior at intersetions in order to reah their
destinations. The appliability of this approah is limited by the omputational
ost of traking partial ows for every ommodity on every link in the network.
A mesosopi simulation easily keeps trak of b ehavioral asp ets by attahing
them to individual DVUs. A ontinuation of the mesosopi method towards
somewhat more marosopi mo deling is pursued here. A fully marosopi
representation of the underlying physial mo del is maintained. The b ehav-
ioral information is represented by massless
partiles
that are disp ersed in the
marosopi ow. They drift along with the ow aording to its spatiotemp oral
velo ity eld. If one maintains the marosopi multi-ommo dity p oint of view,
these partiles an b e interpreted as draws from the ommo dity distribution of
the ow entering the network. Commo dity information for any spatiotemp oral
segment of the network an b e reovered by ounting the aording partiles
within that segment.
If one suh partile is dismissed into the system together with the marosopi
ounterpiee of one vehile, an interpretation as a DVU is obvious. However, the
number of partiles is not onstrained by this and an b e hosen as a ompromise
b etween b ehavioral mo deling resolution and omputational p erformane.
3.1.2 Partile Movement
3.1.2.1 Sp eiation
The marosopi tra ow model is required to sp eify a lo al velo ity
vi(k)
in every ell
i
in every time step
k
. The veloity alulation logi employed in
all exp eriments of this thesis is desrib ed in App endix C.
Consider a set of partiles
n= 1 . . . N
(a p opulation of travelers, agents or
vehiles) that are oating through the system. Partiles have no mass insofar
as they do not ontribute to the marosopi oupany in a ell. At the time
of a partile's entrane into the network, an appropriate amount of marosopi
ow is also dismissed into the system, resulting in a mass balane between
partiles and total marosopi oupany.
In any time step
k
of duration
T
, eah partile advanes aording to the lo al
velo ity in its urrent ell. Partile lo ations within a ell are ontinuous vari-
ables and their movement is regarded as ontinuous in time as well: When a
50
Figure 3.1: Partile movement aross ell boundaries
A partile approahes the upstream end of a ongested road segment. The time step
duration is
T= 8
s. The partile needs
5
s to reah the end of ell
i
at
vi= 40
km
/
h.
During the remaining
3
s
,
it advanes another
16.5
m in ell
j
at
vj= 20
km
/
h.
partile rosses a ell b oundary during a single move of duration
T
, it an freely
ho ose its next ell (if there is more than one downstream ell) and ontinue
with the veloity enountered there until its available move time ends. This
pro edure is illustrated in Figure 3.1. The partile evaluates all traversed ells'
velo ities at the start time of its move. In onsequene, this simulation sheme is
impreise in the order of a time step length, just as the marosopi simulation
logi itself.
When a partile reahes its destination, it is removed from the system and an
appropriate amount of marosopi ow is also ltered out of the tra stream
passing the exit lo ation.
3.1.2.2 Simulation on Variable Time Sales
The previous hapter desrib es how a marosopi simulation an b e run with
variable time step lengths for dierent network elements. This approah an b e
extended to the movement of partiles and requires the following ompletion of
the simulation pro edure given in Setion 2.5, p.41. It is illustrated in Figure
3.2.
1. Every ell
i
that is due aording to its individual time step length
Ti
alulates its supply and demand b oundary from its urrent oupany
and keeps these results onstant for the next
Ti
seonds.
2. Eah partile that urrently resides in a ell
i
that is due is moved forward
aording to the following rules:
(a) The partile moves for a duration equal to the time step length
Ti
of
its start ell
i
. It might ross several ells during this move if ell
i
has a larger
Ti
than its downstream ells.
(b) If the partile has used up its time of movement and has arrived in
a ell
j
with
Tj> Ti
, it ontinues its move until it has moved for
an overall duration of
Tj
. This ontinued move never enters another
ell b eause of ondition (2.11) and aounts for the exp eted waiting
time
Tj−Ti
until the partile is again due for movement.
51
Figure 3.2: Partile movement on variable time sales
A homogeneous velo ity eld is assumed so that a orret partile tra jetory is repre-
sented by a straight line in the spae-time plot. The onsidered partile starts its move
in ell
i
at spae-time p oint
P0
. During its initial move of duration
Ti
, it traverses
two small intermediate ells and nally arrives in ell
j
at p oint
P1
. If the move was
nished there, it would not b e ontinued until
Tj−Ti
seonds later from p oint
P′
2
b e-
ause of ell
j
's greater time step length
Tj
. This would b e inorret as the unstraight
blue tra jetory indiates. The partile has to aount for the waiting time on ell
j
by ontinuing its move for another
Tj−Ti
seonds, whih results in the linear and
therefore orret red tra jetory through p oint
P2
.
3. Every onnetor
c
that is due aording to its individual time step length
Tc
alulates its average ow rates that hold for the next
Tc
seonds and
noties its adjaent ells of the resulting vehile transmissions.
Sine the partile still evaluates all traversed ells' velo ities at the start time of
its move, the resulting impreisions remain in the order of the largest involved
time step duration.
3.1.3 Partile Route Choie
3.1.3.1 Sp eiation
Having stated the inuene of marosopi dynamis on individual partiles, the
onverse problem of synhronizing marosopi ows with individual partile
b ehavior is onsidered next.
The route hoie of partile
n
is expressed by a vetor
un(k) = (uij,n(k))
of
turning move indiators
uij,n(k) = 1
if
n
pro eeds from ell
i
to
j
at time step
k
0
otherwise
.
(3.1)
An additional state vetor
x
nt
(k) = (xij(k))
is intro dued. Eah element
xij(k)
represents the aumulated ount of partiles having turned from ell
i
to
j
until
52
time step
k
. The dynamis of these
turning ounters
are dened by
x
nt
(0) = 0
x
nt
(k+ 1) = x
nt
(k) +
N
X
n=1
un(k).
(3.2)
The marosopi turning frations
β(k) = (βij(k))
an now b e sp eied as a
funtion
β(x
nt
(k)) = (βij(x
nt
(k)))
of the turning ounters where
βij(x
nt
(k)) = xij(k)
Plxil(k).
(3.3)
This is a maximum likeliho o d estimator of the turning probabilities if the turning
moves follows a stationary multinomial distribution [87℄. The resulting estimates
an b e diretly fed into the marosopi mo del by a substitution of
β
in (2.17).
In order to avoid undened
0/0
divisions at the b eginning of a simulation, the
turning ounters an b e initialized with small p ositive values instead of all zeros.
While the up date equation in (3.2) assumes stationary turning probabilities,
a straightforward approah to introdue time dep endeny is to dene an ad-
ditional forgetting parameter
w∈(0,1)
in a mo died turning ounter up date
equation
x
nt
(k+ 1) = wx
nt
(k) + (1 −w)
N
X
n=1
un(k).
(3.4)
In the absene of newly observed turning moves, this sheme auses an exp o-
nential forgetting of previously observed ounts. A useful prop erty of this lter
is its innite memory: Even if no partiles arrive at an intersetion for a while,
the turning ounts remain stritly p ositive and thus ensure well-dened ow
splits in (3.3).
One possible problem with (3.4) is the danger of gridlo k. If a tra jam in
one of an intersetion's downstream ells auses all upstream ells' velo ities to
drop, it might take a long time until new partiles reah the intersetion and
provide fresh turning move indiators that reet these drivers' avoidane of the
unavailable outgoing ell. An appropriate gridlo k resolution logi is desrib ed
in App endix D .
A state spae representation of the ombined system (2.17) and (3.4) an now
b e given. Dening
x(k) = x
ms
(k)
x
nt
(k)
(3.5)
and
f[x(k),u1(k). . . uN(k), k] = f
ms
[x
ms
(k),β(x
nt
(k)), k]
wx
nt
(k) + (1 −w)PN
n=1 un(k),
(3.6)
one obtains
x(k+ 1) = f[x(k),u1(k)...uN(k), k].
(3.7)
Aording to the notational onventions of ontrol theory, the turning move
indiators
un
at as ontrol variables in this mo del. In fat, the individual
53
driver b ehavior
steers
the marosopi tra ow.
x
is subsequently denoted
as the
marosopi state
of the mobility simulation. Note that
x
do es not
aount for the mirosopi states of individual partiles. The ombined state
transition funtion
f
is linearizable with resp et to
x
and all
un
b eause of
the linearizability of its onstituting funtions (2.17), (3.3), and (3.4). This
implies that the eet of an agent's route hoie on the marosopi states an
b e linearly predited as the sum of the eets of its turning moves.
The state spae mo del desrib ed so far aptures mobility only within the network
but do es not aount for vehile entries and exits. These extensions require the
more onise formalization of travel demand given in the seond half of this
hapter. Regarding linearizability, it an already b e stated that the marosopi
eet of a partile's entry or exit an b e linearly approximated sine an entry or
exit move orresp onds marosopially merely to a lo al o upany mo diation.
3.1.3.2 Simulation on Variable Time Sales
If the marosopi mobility simulation runs on variable time steps, the rows of
(3.4) are evaluated at likewise variable frequenies:
xij(rTc+s) = xij(rTc)r∈N, s ∈ {0,...,Tc−1}
xij(rTc+Tc) = wcxij(rTc) + (1 −wc)1
Tc
Tc−1
X
s=0
N
X
n=1
uij,n(rTc+s)
(3.8)
where
Tc
is the time step duration of the onnetor
c
that is rossed by turning
move
ij
. An individual weight
wc
is neessary for every suh onnetor in order
to maintain the same degree of averaging for all turning ounters.
If the numb er
PN
n=1 uij,n(k)
of mirosopially simulated
ij
turning moves dur-
ing a single simulation time step is Poissonian with exp etation and variane
λij
, the variane of
xij
as dened in (3.8) approahes
lim
r→∞
VAR
{xij(rTc)}=1−wc
1 + wc
λij
Tc
.
(3.9)
A derivation of this equation an be found in App endix E. The network time
onstant
ˆ
T
dened in Setion 2.6 is now employed to p ostulate that a turning
ounter's variability must b e indep endent of its onnetor's time step length
and, more sp eially, idential to
VAR
1
ˆ
T
ˆ
T−1
X
s=0
N
X
n=1
uij,n(rˆ
T+s)
=λij
ˆ
T.
(3.10)
This variane would result if the turning ounters were averaged non-reursively
on a temp oral grid as oarse as the network time onstant. Equating (3.9) and
(3.10) yields
wc=ˆ
T−Tc
ˆ
T+Tc
.
(3.11)
An innite turning ounter memory is guaranteed if all
Tc
are hosen stritly
smaller than
ˆ
T
. The working of this speiation is illustrated in Figure 3.3.
54
Figure 3.3: Turning ounter dynamis
Three turning ounters (red) with time step durations of 1, 2, and 4 seonds trak a Poissonian signal (blue) for a duration of 100 seonds. The
signal's exp etation jumps from 0 to 5 after 10 seonds and returns to 0 after another 60 seonds. The network time onstant
ˆ
T
is 8 seonds in all
ases. All ounters exhibit a similar variability and sp eed of adaptation.
55
For a simulation time step length of one seond, the requirement of an in-
nite memory ditates a minimum network time onstant of two seonds. Given
this inertia, a preise marosopi traking of individual vehiles is not p os-
sible. However, suh a preision is rather undesirable for the purp ose of this
work. The simulated driver population is an output of MATSim, the mobility
simulation of whih is a queuing model with relatively limited expressive p ower
but a high omputational p erformane [41℄. It aounts for signalized inter-
setions merely by average ow apaity redutions, whih results in relatively
undisturb ed tra streams. Maintaining this mo deling delity, a marosopi
repro dution of individual vehile movements would only intro due additional
disretization noise into (3.7) an utmost undesirable eet sine this mo del is
to b e linearized.
In a planning ontext, a network time onstant of several minutes is a go o d
hoie. It must not be to o large sine otherwise the marosopi mo del even-
tually lo oses trak of the driver behavior. A reasonable upp er b ound for the
network time onstant is the time interval at whih tra information is aver-
aged before it is fed bak to the simulated travelers who in turn reat to this
information by p ossible turning move hanges.
3.1.4 Computational Mo del Investigation
The miro/maro mo del's preision and the aelerating eet of the simulation
logi on variable time sales are investigated. All exp eriments are onduted on
a 1.7 GHz Pentium 4 mahine with 1 GB RAM, using the Sun Java Runtime
Environment 5.0 [84℄.
A syntheti p opulation of
206 353
motorist travelers with omplete daily plans
is available for the Berlin network intro dued in Setion 2.6.2 [153℄. This is a
10 p erent sample of Berlin's true motorist p opulation. Thus, 10 marosopi
vehile units need to b e inserted together with one partile into the simulation.
However, sine the simulations are run on a thinned out version of the full Berlin
network, the use of 2 instead of 10 marosopi vehile units p er partile already
reates realisti ongestion patterns.
The following exp eriments onsider the morning rush hour from 6 to 12 am.
Figure 3.4 shows the total numb er of moving vehiles as a funtion of time. More
than
16 000
partiles, i.e.,
32 000
marosopi vehile units, are onurrently
simulated during the rush hour p eak at approximately 8:30 am.
3.1.4.1 Preision of Miro/Maro Coupling
The mirosopi b ehavior inuenes the marosopi ow splits via the turn-
ing ounter mehanism, whereas the mirosopi movements are guided by the
marosopi veloity eld. The preision of this miro/maro mo del synhro-
nization is investigated here.
Figure 3.5 shows the mirosopi and marosopi tra density tra jetories
for two seleted links of the Berlin network. Marosopi density is the ratio
of marosopi vehile units on a link to the link's spae apaity. The spae
56
Figure 3.4: Simulated Berlin morning p eak
A simulation of the Berlin morning peak b etween 6 and 12 am. The urve shows the
marosopi number of moving vehiles over time.
apaity of a link is dened as its length times its numb er of lanes. Mirosopi
density is alulated here as the quotient between
twie
the mirosopi partile
ount on a link and its spae apaity. The fator of two aounts for the fat
that one partile represents two vehile units in the given exp erimental setting.
Link (a) is only 25 meters long, whereas link (b) has a length of 1611 meters.
This dierene is reeted in the muh greater variane of the mirosopi den-
sity on the shorter link. Both marosopi density tra jetories trak the miro-
sopi trends with high preision and almost no lag. The strong disretization
noise partiularly on the shorter link is signiantly redued.
In order to avoid arbitrariness, these links were automatially hosen aording
to the following riteria: Link (a) exhibits the largest ratio of density to spae
apaity during the rush hour p eak, whereas link (b) arries the largest total
amount of vehile units, i.e., the largest produt of density and spae apaity,
in the same time interval. That is, the rst riterion prefers small links, and
the seond riterion prefers large links. Both riteria favor ongested links sine
unongested onditions prevail anyway before the rush hour sets in.
The marosopi densities b eyond
133
veh/km indiate that the gridlo k res-
olution mehanism desribed in App endix D atively inuenes the tra dy-
namis. This shows that the purely marosopi gridlok resolution logi is
ompatible with the mirosopi mo del omp onents.
The network time onstant is hosen as large as
5
minutes. This is justied
in light of the
15
minute time bins in whih MATSim averages travel times
b efore feeding them bak to the simulated travelers in its iterative simulation
pro edure, f. Setions 1.2.2.3 and 3.2.2.3.
The dierene b etween this mo del and a typial mesosopi approah is empha-
sized. The presented marosopi tra jetories are not alulated by some kind
57
(a) Mirosopi and marosopi density tra jetory for a short link of
25
m length
under heavy ongestion. The disrete value domain of the mirosopi urve reets
the strong vehile disretization noise. The marosopi urve removes most of this
noise. Unrealistially high mirosopi densities are p ossible b eause of the massless
partiles. The marosopi urve, however, is within b ounds.
(b)
Mirosopi and marosopi density tra jetory for a
1.6
km long link under
heavy ongestion. The disretization noise has a weaker eet sine a greater number
of partiles is averaged in the mirosopi density alulations. The mirosopi signal
trend is traked very well by the marosopi urve.
Figure 3.5: Preision of miro/maro mo del synhronization
58
Figure 3.6: Mean normalized bias and error tra jetories
Mean normalized bias MNB and mean normalized error MNE as dened in (3.12)
and (3.13). The intermediate mirosopi exess in MNB of ab out 1 p er mille is
negligible and owed to the partile entrane mehanism whih puts partiles ahead of
their marosopi ow into the system. Likewise, there is a similar undersho ot as the
partiles leave the system ahead of their marosopi ow at the end of the rush hour.
of mirosopi vehile ount averaging. Rather, they impliitly result from on-
tinuously traked turning frations that guide an appropriate amount of truly
marosopi ow aross eah link.
A network-wide p oint of view is adopted by means of the following two hara-
teristis:
MNB
(k) = 100
|A|X
a∈A
miro
a(k)−
maro
a(k)
ˆ
(3.12)
represents the mean normalized bias where
maro
a(k)
(
miro
a(k)
) is the maro-
sopi (mirosopi) vehile density on link
a
in time step
k
,
ˆ
is the marosopi
jam density of
133
veh/km, and
A
is the set of all links in the network. The
seond harateristi
MNE
(k) = 100
|A|X
a∈A
miro
a(k)−
maro
a(k)
ˆ
(3.13)
is the m ean normalized error.
Figure 3.6 shows that MNB utuates unsystematially around
0
p erent. This
indiates that the mass balane b etween mirosopi and marosopi ow is
well maintained. The maximum value of approximately
3
p erent for MNE is
mo derate and plausible in onsideration of Figure 3.5.
These results show that the miro- and the maro-mo del are well synhronized
despite of their sparse interations. The resulting marosopi tra hara-
teristis exhibit a signiantly lower disretization noise than a simple average
over the mirosopi partiles.
59
Figure 3.7: Mirosopi and marosopi omputation times
Mirosopi and marosopi omputation times over
log2
of the greatest allowed time
step duration. The simulated time span is 6 hours.
3.1.4.2 Computational Performane
The impreisions intro dued by the simulation sheme on variable time sales are
now justied by their ountervailing omputational b enets. The same morning
p eak senario as b efore is onsidered.
The omputational eort for the miro- and for the marosimulation is distin-
guished in the following way. The marosimulation omprises all proesses de-
srib ed in Chapter 2 plus the turning ounter traking desrib ed in Setion 3.1.3.
The mirosimulation omprises the additional op erations neessary to up date
the individual partile lo ations as desrib ed in Setion 3.1.2. In onsequene,
the total omputational eort is the sum of miro- and marosimulation.
Figure 3.7 shows the mirosopi and marosopi omputation time over
log2
of the greatest allowed simulation time step duration, whih is roughly the
same as the network time onstant
ˆ
T
.
1
The overall numb er of omputations is
prop ortional to the numb er of network elements and to the frequenies at whih
these elements are up dated. An inreased
ˆ
T
aets b oth, the element ount
and the alulation frequeny. Thus, the omputation times initially derease
quikly with
ˆ
T
but then stabilize b eause of the geometrial onstraints on the
link and no de time step durations. Cho osing large ells and long time steps
do es not only redue the numb er of marosopi alulations but also dereases
the frequenies at whih the mirosopi partiles are up dated.
Figure 3.8 shows the real time ratio, i.e., the ratio of simulated time to the time
required to run the simulation. The aomplished maximum value is 90. This
1
More preisely, the network time onstant
ˆ
T
is slightly larger than the greatest allowed
simulation time step duration in order to ensure an innite turning ounter memory, f.
Setion 3.1.3.2.
60
Figure 3.8: Real time ratio
Real time ratio over
log2
of the largest simulation time step duration in the network.
These values aount for all op erations of the simulation system and inlude a numb er
of supplementary pro edures. In onsequene, the evaluated running time is slightly
larger than the sum of pure miro- and marosimulation.
shows that the model is ready for real-time simulations of large-sale senarios.
In summary, its omputational eieny is owed to the following properties:
•
The mo del do es not require a realisti numb er of partiles. If, for example,
only a
10
p erent sample of the omplete p opulation is loaded on the
network, the marosopi equivalent of
10
vehiles is inserted into the
system together with every partile. The hosen sample size must b e
large enough to prop erly represent the atual p opulation's b ehavior but
otherwise an b e minimized for high omputational p erformane.
•
The marosopi mobility simulation only moves single-ommodity ows.
No are has to b e taken of partial densities as it would be the ase if
b ehavioral asp ets were represented marosopially.
•
Every link is simulated with a ell size and a time step length that are
optimally adjusted to its harateristis.
Altogether, two results obtained in this setion are useful indep endently of a
state estimation problem. First, it is shown how a general marosopi tra
ow mo del an b e employed to simulate mirosopi travel b ehavior. A useful
feature of this approah is its ability to remove vehile disretization noise.
Seond, the marosopi simulation logi on variable time sales, f. Setion
2.5, is extended towards this miro/maro oupling sheme and exhibits a high
omputational p erformane.
Imp ortant for estimation, the linearizability of state spae mo del (3.7) is main-
tained throughout the entire development. This provides the sensitivity infor-
61
mation that is subsequently applied to predit the linearized eet of an indi-
vidual driver's turning move sequene on the global network onditions without
rep eated simulations.
3.2 Simulation of Drivers' Choies
The rst part of this hapter speies physially observable driver b ehavior as
a sequene of turning moves. In the following, the deisions that preede this
b ehavior are disussed and formalized in a way that allows for a seamless linkage
to the previously desrib ed miro/maro mobility simulation. The resulting
b ehavioral representation is logially ompatible with the estimation algorithm
develop ed in the next hapter and tehnially ompatible with a MATSim-like
simulation system. Sine this dissertation do es not ontribute to the eld of
b ehavioral mo deling itself, the following disussion is kept problem-sp ei and
is not exhaustive from a b ehavioral mo deling p oint of view.
3.2.1 Choie Formalism
It is assumed that, whenever a traveler is faed with a situation that alls for
a deision, this traveler ho oses preisely one element from a nonempty set of
disrete alternatives. The deision making pro ess itself is strutured aording
to the framework given in [21℄:
1. denition of the hoie problem,
2. generation of alternatives,
3. evaluation of attributes of alternatives,
4. hoie,
5. implementation.
These steps are made preise in the remainder of this setion. Note that a
reative deision proto ol as dened in Setion 1.3.1 may rep eat steps 2 and 3
several times b efore a hoie is made.
The disussion omits sp ei mo deling assumptions and algorithmi details that
would b e neessary for the realization of an appliable b ehavioral mo del. This
is justied by the intention to provide an estimator that is ompatible with a
broad range of b ehavioral mo dels and by the rather tehnial assumption that
the estimator is likely to b e attahed to an existing tra simulator, f. Setion
1.3.1. Only a few seleted mo deling asp ets that are referred to in the later
developments are disussed at the end of this hapter.
3.2.1.1 Denition of the Choie Problem
Most of the terminology intro dued here is onsistent with the MATSim sys-
tem sp eiation given in [149℄. However, the underlying oneptions are more
universally appliable to the mo deling of travel behavior and are not onned
to this software.
62
Plans
The ativity and traveling intentions of an individual are denoted as
her plan. For simpliity, only plans for a single day are onsidered. Physially,
a plan desrib es a round trip through the transp ortation network. This round
trip omprises a sequene of legs that onnet intermediate stops during whih
ativities are onduted. The rst and last ativity of a plan typially take plae
at the individual's home lo ation.
Ativities are dened in terms of their typ e (e.g., work, leisure), lo ation (a
link in the network), start time, and end time or presp eied duration. Two
subsequent ativities are onneted by a leg. While in general a leg an b e
asso iated with dierent modes (e.g., ar, publi transp ort, walking), this thesis
onsiders only individual motorist travelers suh that a leg always implies a
vehiular movement through the road network. A motorist leg is parameterized
by origin and destination link, route (a sequene of links that onnets origin and
destination), and departure time. Only a desired arrival time an b e presp eied
sine the atual time of arrival dep ends on the prevailing tra onditions.
When a traveler ho oses her ourse of ation for a given day, she equivalently
ho oses a plan for that day. It is possible to disaggregate the hoie of a plan
into a logial or temporal sequene of deisions [27, 99℄. The latter metho d
is naturally appliable to within-day replanning, where a traveler ontinuously
reonsiders and adjusts her urrent plan aording to pre- and en-trip olleted
information. Formally, the hoie of a plan segment where some degrees of
freedom are xed is not dierent from the hoie of a full plan, and no suh
dierentiation is made in the following. For example, an en-trip route swithing
mo del maintains all ativity loations and timings of the present plan. Equiv-
alently, route swithing an be represented as the hoie of a ompletely new
plan where all degrees of freedom apart from route hoie are onstrained to be
idential to those of the original plan.
Generalized Paths
The oneption of a plan is now formalized in a way that
is amenable to the likewise formal derivation of a b ehavioral estimator.
A simple route
U
onnets two subsequent ativity lo ations. It is dened as a
(physially feasible) sequene of turning moves
U=...u(k−1),u(k),u(k+ 1) ...={u(k)}k
(3.14)
with
u(k)
sp eied in (3.1). The representation of a route as a sequene of
turning moves rather than a sequene of links maintains onsisteny with the
mirosopi driver representation sp eied in the rst half of this hapter. It
an b e thought of as an ordinary edge sequene in an inverted network where
vertexes represent links and edges represent turning moves, f. Figure 3.9. A
sequene of turning moves uniquely denes a sequene of original links, and vie
versa.
The round trip that physially orresp onds to an all-day plan is formalized as
a (yli) path by minor mo diations to the inverted network. Every vertex
v
of the inverted network that represents an ativity lo ation is omplemented
with an additional vertex
v′
that represents the atual exeution of an ativity
at this lo ation. The start of an ativity is then equivalent to a turning move
v→v′
, and its end an be identied by a
v′→v
move. A plan's full sequene
63
Figure 3.9: Route hoie
The original road network is drawn in blue. Three of its links serve as ativity lo ations
(oe, mall, home). The inverted network for route representation is drawn on top in
blak. It represents every original link by a vertex and every p ossible turning move by
an edge.
of ativities and legs now omprises a single round trip through the inverted
network, with yles at the ativity lo ations. Figure 3.10 provides an example.
This formalism simplies notation sine it allows to represent all physially
relevant asp ets of a full plan onsistently with (3.14) in terms of a
generalized
path
U
. If only a plan segment is to be represented, its generalized path segment
also ontains only the orresp onding subset of turning moves. Subsequently, the
notions of a path and a generalized path will b e used synonymially whenever
the ontext allows to distinguish them from a
simple route
that only onnets
two links in the network.
Tra ow mo del (3.7) an b e steered by generalized paths instead of simple
routes without formal mo diation. Sine the eet of entering and exiting
vehiles an b e linearly approximated by this mo del, it is also linearizable with
resp et to the newly intro dued turning moves that represent suh entries and
exits. This implies that the eet of an agent's plan hoie on the marosopi
network onditions an b e linearly predited in the same vein as it has b een
demonstrated for route hoie in Setion 3.1.3.1.
Sine a generalized path
U
is a formal representation of an individual's inten-
tions, it represents an asp et of that individual's
mental state
. Its notation
in terms of the typial ontrol symbol
u
is maintained here sine the largest
p ortion of this thesis deals with the steering eet of driver behavior on maro-
sopi tra dynamis. The denition of a full state spae mo del for a ombined
miro/maro tra system that inludes some kind of mental dynamis is not
neessary for the purp ose of this dissertation.
3.2.1.2 Generation of Alternatives
The
hoie set
of b ehavioral alternatives available to deision maker
n
is de-
noted by
Cn
. The elements of this set are plans, formally represented by (gen-
eralized) paths
U
. It is reasonable to assume that
Cn
is signiantly smaller
than the set of all thinkable plans: The elements in
Cn
must b e ompatible
with the goals and ommitments of a traveler, f. Setion 1.2.2.3. The limited
64
Figure 3.10: Generalized path hoie
The same physial network as shown in Figure
3.9
. Cyles are added to all pos-
sible ativity lo ations. An exemplary plan that onsists of the ativity sequene
home
→
work
→
shop
→
home now onsists of one round trip through the inverted net-
work, with yles at the ativity loations. Its equivalent sequene of vertexes is
h′, h, . . . , o, o′, o, . . . , m, m′, m, . . . , h, h′
.
knowledge of the deision maker exludes all unknown options from onsidera-
tion. Physial, legal, and individual (e.g., nanial, onstitutional) onstraints
further redue the hoie set. If a traveler reonsiders only a segment of her ur-
rent plan, an additional onstraint on
Cn
is that everything but this segment
must remain unhanged in all alternative plans.
It is required that a non-empty hoie set
Cn
is available to every agent
n
in
every situation that alls for a deision. This hoie set may b e sp eied in two
dierent ways, dep ending on the deployed deision proto ol, f. Setion 1.3.1:
•
A reative deision protool inrementally onstruts a set of onsidered
alternatives given a partiular hoie situation. Dierent suhlike sets
may b e generated in rep etitions of otherwise idential onditions b eause
of probabilisti omponents in the generation proedure. In this ase,
Cn
omprises all p ossibly generated alternatives.
•
In a delib erative deision proto ol, the hoie set has typially b een gen-
erated prior to the atual hoie situation. That is,
Cn
is expliitly and
deterministially presrib ed, even if it was originally generated by a ran-
domized algorithm.
The goal of this work is to treat the deision proto ol as muh as a blak b ox as
p ossible. The only requirement implied by the ab ove listing is that there exists
a nonempty set
Cn
of alternatives that ontains all p ossible hoies of agent
n
in a given situation. However, an enumeration of this set is not required.
3.2.1.3 Evaluation of Attributes of Alternatives
The
systemati (deterministi) utility
of an alternative, represented by a
real-valued numb er, is a mo del of the b enets a deision maker exp ets from
65
ho osing this alternative. It reets the deision maker's preferenes. Utility
p ereption an vary among deision makers, and learly utility an dier among
alternatives. Formally, a systemati (deterministi) utility
Vn(U)
is asso iated
with every plan
U
in the hoie set
Cn
of traveler
n
.
The utility of a plan is omprised of two omp onents: p ositive utility for the
exeution of ativities and negative utility (disutility, ost) for travel itself. Typ-
ial asp ets of route (dis)utility are travel time, distane traveled, number of
left-turns, numb er of signalized intersetions, and ontat with inseure neigh-
b orho o ds [18, 20℄. The utility of an ativity varies depending on the type of
ativity, its ontext within the entire plan, and the timing of its exeution [43℄.
If a utility-driven mo deling approah is adopted, it is required that the system-
ati utility for every plan of any agent an be alulated by the utility funtion
shown in Figure 1.1 and that the resulting utility ombines all of the afore-
mentioned (dis)utility omp onents in a single numb er. This evaluation only
has to b e available on request and on a p er-plan basis. It is not required that
the hoie set is enumerated for a omplete evaluation b efore a hoie is made.
Furthermore, if the deision proto ol sequentially omp oses a hoie, e.g., by
inrementally building a plan as a sequene of ativities and legs, the utility
funtion may be limited to an evaluation of the aording plan omponents.
3.2.1.4 Choie
The hoie of a ertain plan (segment) is mo deled non-deterministially. The
probability that deision maker
n
ho oses plan
U ∈ Cn
is denoted by
Pn(U)
.
This hoie distribution may b e parameterized in an agent-sp ei way but
otherwise is required to dep end only on the attributes of the elements in
Cn
. If
the hoie mo del is utility driven, the attributes of a plan must b e represented
by its utility.
A probabilisti hoie logi may represent randomness in human b ehavior or
aount for mo deling impreisions [21℄. The sp ei mo deling assumptions that
underly a partiular deision proto ol are not relevant for the subsequently
develop ed estimation approah b eyond the fat that b ehavior is unertain at
all. Otherwise, there would b e no sop e for a b ehavioral adjustment.
Neither an enumeration of the hoie set nor an expliit (e.g., losed-form)
representation of the implemented hoie distribution need to be available. Only
realizations of hoies must generated by the b ehavioral simulation system.
3.2.1.5 Implementation
The implementation of a hoie requires its realization in the mobility simu-
lation. However, an agent with an imp erfet knowledge of the atual tra
onditions may observe an inonsisteny b etween what it wants to do and what
is physially p ossible. In partiular, the generalized path representation of a
plan omprises a sequene of turning move indiators that presp eify the timing
of every turning move and every entry/exit move in the network. It is unlikely
that the (ongested) tra onditions admit preisely this timing.
66
It therefore is assumed that a plan is
robust
in that it annot b e invalidated
by nite hanges in travel times. An example of a robust plan is one where
(i) the ativities have no xed start time but rather a prespeied duration
and (ii) the legs only sp eify a sequene of links but not the timing of their
entry. Consequently, a one hosen plan an always b e exeuted in the mobility
simulation without further replanning. The MATSim plans are robust in this
regard.
A preise formalization of this situation would require to supplement the mo-
bility simulation (3.7) with another mo del omp onent that up dates the plans
Un={un(k)}k
for all agents
n= 1 . . . N
in every simulation time step
k
suh
that their onsisteny with the physial situation is maintained. However, sine
the atually implemented mobility simulation do es not require the generalized
path abstration at all, the titious existene of suh a mo del omp onent merely
maintains
formal
onsisteny whenever it is stated that
U1...UN
are loaded
on the network or
U1...UN
are fed into the mobility simulation.
The generalized paths
U1...UN
uniquely sp eify b oth the intended and the
implemented driver behavior. Therefore, no formal dierentiation b etween these
asp ets is subsequently made.
3.2.2 Sp ei Mo deling Assumptions
The strutural outline given ab ove is made preise in terms of two fairly dierent
mo deling approahes.
Random utility models
(RU mo dels, RUMs) onstitute a broadly appliable lass
of hoie mo dels that are based on reasonable b ehavioral assumptions and sound
mathematial inferene. The simple mathematial struture of ertain RUMs is
exploited in the derivation of a b ehavioral estimator.
MATSim's b ehavioral mo del basially relies on a dynamial systems assumption
ab out human learning. Sine the resulting mo del b ehavior is dened rather
impliitly through this learning proess, and sine the dynamis of this pro ess
are not yet well-understoo d, MATSim onstitutes a partiularly hallenging
mo del for a b ehavioral estimator.
3.2.2.1 Random Utility Mo dels
RUMs onstitute the mainstay of travel b ehavior mo deling, and a sp ei im-
plementation of the deision proto ol is likely to be based on RU theory [21, 22℄.
The RU mo deling assumptions are outlined b elow.
It is assumed that a deision maker
n
always ho oses the alternative of greatest
p ereived utility from her presp eied hoie set
Cn
. The systemati utility
Vn(U)
onstitutes only an imp erfet mo del of her true utility p ereption. In
order to reet this impreision, a random error omponent
εU,n
is added to the
systemati utility of every alternative
U
. The probability
Pn(U)
that
U
is hosen
thus equals the probability that the random utility of
U
is greatest among all
alternatives:
Pn(U) = Pr(Vn(U) + εU,n ≥Vn(V) + εV,n,∀V ∈ Cn).
(3.15)
67
Closed-form expressions for these hoie probabilities an be obtained for ertain
joint distributions of the error omponents. But even if no suh losed form an
b e found, a simulation of hoies that are onsistent with (3.15) is p ossible. The
pro edure requires (i) to draw a disturbane from the joint error distribution
for all alternatives, p ossibly through a simulation pro edure as desrib ed b elow,
and (ii) to deterministially ho ose the alternative of greatest disturb ed utility.
3.2.2.2 Mo dels of Route Choie
The two ma jor mo deling approahes to route hoie have already b een addressed
in Setion 1.2.2.2: Either route (re)planning is realized by the alulation of a
b est path, or a route is hosen probabilistially from a presp eied hoie set.
Behaviorally, the alulation of a b est path is an idealization. It implies global
network knowledge and an optimal hoie mehanism given a ertain ob jetive
funtion suh as trip travel time. The eetive alulation of a b est path requires
route ost to b e additive in link ost whih ignores existing evidene for nonlinear
ost p ereption. Probabilisti route hoie allows for greater realism. A hoie
set of routes an b e generated in a way that is onsistent with a driver's (usually
limited) knowledge of available alternatives. There is no limitation of link-
additive osts. The random hoie omp onent prop erly reets b ehavioral and
mo deling unertainties [148℄.
Computationally, b est path has an edge over probabilisti hoie. Routing prob-
lems have been intensively studied in omputational siene and eient solution
algorithms are available for problems with link-additive ost [83℄. In ontrast,
probabilisti hoie implies some omputational overhead. Choie set genera-
tion itself is a nontrival task [20, 148℄. Every agent's individual hoie set has
to b e stored and pro essed during simulation, and every alternative needs to be
evaluated for the simulation of a single hoie. Contrarily, the eieny of best
path algorithms is owed to their avoidane of path enumeration [130℄.
The realism of probabilisti hoie and the eieny of routing algorithms an
b e ombined. Sine b est path routing is a ost minimization pro edure, it
an b e applied to mo del a deision maker's rational hoie given a simulated
error of utility p ereption. This oinides with the aforementioned simulation
pro edure for RUMs. In this ontext, it is interesting to insp et a variation
of the route hoie mo del implemented in the MATSim planning simulation.
MATSim mo dels the day-to-day evolution of driver b ehavior as a ontinuous
learning proess. Sp eaking only in terms of routes, a ertain fration of drivers
is allowed to realulate new routes at the b eginning of every simulated day.
These routes are generated based on previously simulated link traversal osts
by a time-dep endent b est path algorithm. The simultaneous exeution of all
routes results in exp eriened osts that are likely to dier from those osts
based on whih the new routes were alulated. This impliitly simulates a
p ereptional error that is idential for all replanning agents and equal to the
dierene b etween the atually exp eriened osts and the osts assumed during
replanning. This logi even avoids the expliit generation of p ereptional errors
but is not derived from RU theory.
The
path size logit
(PS-logit) mo del denes losed-form route hoie proba-
bilities. Its derivation from RU theory an b e found in [67℄. This mo del is
68
Figure 3.11: Three routes example
A simple route hoie example with three alternative routes
A
(omprised of link 1),
B
(omprised of link sequene 2
→
3a), and
C
(omprised of links 2
→
3b). The length
of link 1 is
l
, that of links 3a and 3b is
dl
, and that of link 2 is
l−dl
.
presented here sine its partiular struture allows for some formal manipula-
tions that greatly simplify the b ehavioral estimation problem. PS-logit sp eies
the probability that individual
n
ho oses route
U ∈ Cn
by
Pn(U) = eµVn(U)+ln
PS
n(U)
PV∈CneµVn(V)+ln
PS
n(V)
=
PS
n(U)eµVn(U)
PV∈Cn
PS
n(V)eµVn(V).
(3.16)
It is instrutive to start the disussion with all PS parameters set to one. Then,
sp eiation (3.16) ollapses into the
multinomial logit
(MNL) mo del, the ar-
guably simplest and most popular RUM. The p ositive sale parameter
µ
ontrols
to what degree routes of higher systemati utility are preferred. If
µ→0
, all
routes are hosen with equal probability, whereas
µ→ ∞
deterministially se-
lets a route of maximum utility.
In a route hoie ontext, the ma jor drawbak of MNL is its inability to mo del
situations with overlapping routes. This is most easily demonstrated by an
example. Figure 3.11 shows a simple four-link network. Three routes
A
,
B
, and
C
onnet the leftmost to the rightmost node. All routes have equal utility
¯
V
suh that MNL invariably predits a uniform route split
(P(A)P(B)P(C))
=
(1
/31
/31
/3)
. This is not realisti b eause routes
B
and
C
have a large overlap
and therefore are likely to be p ereived as a single alternative. Behaviorally
reasonable route splits thus approah
(1
/21
/41
/4)
as the overlap of
B
and
C
gets
larger.
PS-logit orrets the MNL mo del by speifying
PS
n(U) = X
a∈ΓU
la
LU
1
PV∈CnδaV
(3.17)
where
ΓU
is the set of all links in route
U
,
la
is the length of link
a
,
LU
is the
length of route
U
, and
δaV
is one if link
a
is ontained in route
V
and zero
otherwise. That is,
PV∈CnδaV
ounts how many routes in
Cn
ontain link
a
.
Eah addend in (3.17) represents the ontribution of a single link to the path size
of route
U
, and PS
(U)
measures to what degree route
U
is p ereived as a distint
alternative. It is one if
U
has no overlap with other routes, and it approahes zero
the greater
U
's overlap with other routes b eomes. A p erfet overlap of routes
B
69
and
C
in the ab ove example yields path sizes
(
PS
(A)
PS
(B)
PS
(C)) = (1 1
/21
/2)
that generate the b ehaviorally reasonable route splits
(1
/21
/41
/4)
when inserted
in (3.16).
The purp oseful nature of these examples is emphasized. Alternative utility
orretion terms and path size denitions have been prop osed in the literature
[38, 67℄ as well as alternative RU mo dels that are not limited to the simple
struture of (3.16) [18, 20, 148℄.
3.2.2.3 Mo dels of Plan Choie
Even with realisti restritions on p ossible ativity sequenes, lo ations, and
timings, and with a likewise restrited route hoie set, the ombinatorial num-
b er of available plans quikly b eomes intratable. For a single day, roughly
1017
alternative b ehavioral patterns p er traveler are estimated in [27℄. It is not
realisti to assume that travelers p ossess the omputational resoures to pro ess
suh a hoie set. However, they do make a deision in some way, and there-
fore it app ears justied to simulate plan hoie by simplifying heuristis that
resemble human deision making [71℄.
This approah is also hosen in the MATSim planning simulation. A traveler's
plan is sored by a utility funtion that omprises p ositive addends for ativity
exeution and negative addends representing travel osts [43℄. Every simulated
traveler strives to maximize its sore by explorative day-to-day learning. This
is realized as a simplied lassier system [149℄: A small set of (typially ve)
alternative plans is memorized by an agent. Every simulated day, one of these
plans is exeuted and the exp eriened sore is memorized. Oasionally, a new
plan is generated, exeuted, and the worst plan is disarded. New plans are
generated by variations of old ones. Routes are realulated as b est paths based
on previously observed link traversal osts [130℄, and ativity timings are hosen
by a variety of heuristis suh as random searh [10℄, reinforement learning [44℄,
and evolutionary algorithms [43, 120℄.
Plan seletion itself is implemented as a simple RU mo del. However, the on-
tinuous hoie set evolution by explorative learning prevents a straightforward
RU interpretation and also ompliates a mapping on the strutural system
requirements that are presupp osed for estimation. There are three diulties.
1. The plan hoie set is variable. If it was xed after a limited numb er of
iterations, the simulation until that p oint ould b e regarded only as a fairly
heavyweight hoie set generation pro ess. However, the limited numb er
of memorized plans in suh a setting (rather a tehnologial problem)
ould raise an issue of b ehavioral variability.
2. Plan hoie is not based on deterministi utilities but on ontinuously
up dated sores. While sore exp etations are tehnially easy to estimate
by reursive averaging, their very existene requires that the simulation
onverges towards a stationary distribution of network onditions. This
prop erty is yet to b e established [132℄.
3. A newly generated plan is immediately seleted for exeution. This is
neessary sine a plan's sore an only b e identied through simulation.
70
Still, this leads to a not yet laried oinidene of hoie set generation
and hoie itself. Again, an o asionally stabilized hoie set would resolve
this issue.
This is not to say that these asp ets of MATSim are inompatible with the pro-
p osals of this dissertation. Rather, they require the more sp eialized treatment
given later in Setion 6.4.5.
MATSim's learning-based approah is a spei instane in a broad mo del range
prop osed in the eld of ativity based demand mo deling, e.g., [27, 98, 99, 172℄,
and the strutural outline given in Setion 3.2.1 is likely to apply to a greater
variety of demand mo dels. Still, the MATSim-related development of this work
naturally suggests a presentation in terms of this system.
Conluding, the seond part of this hapter formalizes a behavioral simulation
system but leaves the b ehavioral mo del itself unsp eied for the most part. This
presentation is not given as an end in itself. The next hapter identies what
b ehavioral estimates are p ossible in this setting.
71
Chapter 4
Estimation
The previous two hapters desrib e a simulation system that onsists of two
omp onents: a mobility simulation and a representation of human b ehavior.
The sp ei prop erties of these omp onents are now exploited in the formulation
and solution of a tra state estimation problem.
As outlined in the introdution, the task is to use spatially and temporally in-
omplete sensor information to reonstrut spatially and temp orally omplete
system state information. Examples for sensors are lo op detetors that measure
ow rates at road ross-setions [91℄, ground- or airb orne ameras that identify
tra densities on road segments [62, 77, 150℄, and oating ars that mea-
sure link veloities [156℄. Only aggregate measurements are onsidered. While
the imp ortane of advaned tra monitoring tehnologies suh as vehile re-
identiation systems is likely to inrease in the future, they are not yet in broad
appliation.
Marosopially, the system states to b e reonstruted are represented by state
vetor sequene
X={x(k)}k
(4.1)
of tra ow model (3.7). This mo del unfolds deterministially given an initial
state
x(0) = x0
and a driver p opulation's b ehavior
U1...UN
. Sine
U1...UN
omprise all aspets of the individual drivers'
mental states
that are neessary
to dene all
marosopi states
X
in the mo del, the
state estimation problem
b eomes to identify ontrol sequenes
U1...UN
that steer
X
towards most likely
values given the available measurements and the b ehavioral a priori knowledge.
The mapping from individual driver b ehavior on marosopi system states is
nonlinear. The prop osed estimator deals with this diulty by repeated lin-
earizations of the marosopi mo del. Sine the mo del is dynamial, this re-
quires to alulate system state sensitivities through simulated time. In result,
the linearized eet of a single driver's deision in any time step
k
on the maro-
sopi states in any later time step
k+ ∆k
an b e predited. Given a distane
measure b etween true and simulated tra onditions, these sensitivities then
provide diretional information for b ehavioral adjustments. Coneptually, this
approah has a ounterpart for example in meteorology, where the linearized
version of a dynamial weather model is denoted as its adjoint mo del. The
72
spatiotemp oral sensitivities it provides are used to iteratively improve the full
mo del's onsisteny with real world observations, e.g., for the purp ose of short-
term weather foreasting [63℄.
The remainder of this hapter is organized in four parts.
First, the problem of how to steer the b ehavior of simulated travelers by sys-
temati manipulation of their utility p ereption is investigated in Setion 4.1.
Apart from b eing of pratial interest itself, this setion prepares a numb er of
tehnial results that simplify the subsequent presentation. This inludes the
aforementioned linearization logi.
Seond, a rst heuristi estimator is prop osed in Setion 4.2. It applies the
previously develop ed metho d to steer agents towards a plausible repro dution
of available sensor data. However, this approah is not yet based on a solid
statistial foundation.
Third, a Bayesian formulation of the estimation problem is given in Setion 4.3.
Starting with a oneptually straightforward but omputationally umb ersome
formulation, various simpliations are adopted that allow for a exible balane
b etween mathematial preision and omputational eieny.
Fourth, Setion 4.4 illustrates the theoretial developments with a small exam-
ple. A test ase of realisti size is p ostp oned to Chapter 5.
4.1 Steering Agent Behavior
The problem is investigated of how to inuene the b ehavior of simulated trav-
elers by hanging their p ereption of systemati utility. The ob jetive aording
to whih agent b ehavior is to b e inuened is represented by a one dierentiable
funtion
Φ(X) =
K
X
k=1
ϕ[x(k), k]
(4.2)
that maps the marosopi system states in simulation time steps
1
through
K
on a real numb er. An improved fulllment of the ob jetive is reeted by an
inrease of this funtion.
This problem statement is related to that of a
dynami system optimal tra
assignment.
The latter seeks to identify a tra pattern that minimizes the
average ost exp eriened by all travelers. It is b ehaviorally not realisti sine it
implies that travelers o op erate in their eorts to minimize ost, but it is a go o d
measure to estimate the greatest eetiveness of a tra system or to identify
optimal ontrol strategies [35, 121℄.
Sine the problem onsidered here is not to attain a strit system optimum but
rather a ompromise b etween individual driver ob jetives and global ob jetive
(4.2), and sine only limited measures to aet agent b ehavior are available, the
notion of a system optimal tra assignment is avoided. The results obtained
here only improve a mirosopi assignment with resp et to a global ob jetive.
73
4.1.1 Mo died Utility Pereption
The agents' b ehavior is to b e inuened by a mo diation of their systemati
utility evaluation. Beause of the deision proto ol's probabilisti nature, f.
Setion 3.2.1, there is no guarantee that a single hoie based on suh a modi-
ed utility do es indeed improve the global ob jetive. However, it is reasonable
to assume that, one the eet of agent behavior on the global ob jetive is iden-
tied, a utility mo diation that favors advantageous generalized paths also
leads to hoie distributions that improve the global ob jetive on average. Un-
less otherwise noted, the notion of a path now represents an arbitrary behavioral
pattern ranging from a single route to an all-day plan.
The problem of steering agent behavior is therefore p osed as an ordinary as-
signment problem with mo died systemati utility
Wn(U) = Vn(U) + Φ(X(U1...Un−1,U,Un+1 ...UN))/µ
(4.3)
for every agent
n
and path
U ∈ Cn
. That is, agent
n
evaluates
Φ
as a funtion of
its individual path hoie with the b ehavior of all other agents being xed. The
stritly p ositive parameter
µ
determines the weight of individual utility when
ompared to the global ob jetive. Its hoie is left to the analyst.
This problem statement is given yet indep endently of an estimation problem
and requires no suh interpretation. Sine the subsequently develop ed metho d
to steer simulated travelers holds promise for appliations that go b eyond tra
state estimation, its sp ei deployment for estimation purp oses is p ostp oned
to Setion 4.2.
A straightforward implementation of the ab ove would require the following:
1. Unsteered p opulation b ehavior
U1...UN
is given.
2. For eah agent
n= 1 . . . N
, do:
(a) Replae
Vn
by
Wn
aording to (4.3).
(b) Draw
U′
n
from
Cn
based on
Wn(U)
.
3. Steered p opulation b ehavior is
U′
1...U′
N
.
The following subsetions op erationalize this pro edure.
4.1.2 Linearization of Global Ob jetive Funtion
Every evaluation of
Wn(U)
requires an evaluation of
Φ(X(...U...))
and there-
fore a run of the entire mobility simulation. Sine
Φ
is evaluated separately
by all agents that make deisions based on their mo died utility
Wn(U)
, a
straightforward implementation of this approah is omputationally intratable.
This problem an b e irumvented if the mapping from individual path hoie
U
on
Φ
is linearized. Given
U′=U+ ∆U
, this linearization essentially is
W(U′)≈V(U′) + Φ(X(...U...)) + ∆U · dΦ/dU
. It will turn out that it is
feasible to ompute the sensitivities
dΦ/dU
simultaneously for all agents. In
74
onsequene, it is p ossible to linearly predit the eet of b ehavioral variations
∆U
on the global ob jetive funtion
Φ
for all agents with just one run of the
mobility simulation.
The linearization must aount for the oupling b etween
U
and
X
through
dynamial system onstraint (3.7) that represents the mobility simulation. This
diulty an b e dealt with by well-known metho ds from ontrol theory [101,
138, 145℄. A self-ontained exp osition is given in the following.
Denote
Φ(k) =
K
X
κ=k
ϕ[x(κ), κ]
(4.4)
for
k= 1 . . . K
. This is the remaining ontribution to
Φ(X)
from time step
k
on. It an b e reursively written as
Φ(k) = ϕ[x(k), k] + Φ(k+ 1) k= 1 . . . K −1
ϕ[x(K), K]k=K.
(4.5)
As a rst step, sensitivities with resp et to states are omputed by
dΦ(k)
dx(k)=
∂ϕ[x(k), k]
∂x(k)+dΦ(k+ 1)
dx(k)k= 1 ...K−1
∂ϕ[x(K), K]
∂x(K)k=K.
(4.6)
Sine the interplay b etween variables in dierent time steps is fully dened by
state equation (3.7),
dΦ(k+ 1)
dx(k)=∂f[x(k),u1(k)...uN(k), k]
∂x(k)TdΦ(k+ 1)
dx(k+ 1)
(4.7)
holds for
k < K
, where
x(k+ 1) = f[...]
is used.
Now, sensitivities with resp et to ontrol variables
u1(k)...uN(k)
result from
dΦ(X)
dun(k)=∂f[x(k),u1(k)...uN(k), k]
∂u(k)TdΦ(k+ 1)
dx(k+ 1) .
(4.8)
Here,
∂ϕ[x(k), k]/∂un(k)
disapp ears sine
un(k)
inuenes no state earlier than
x(k+ 1)
.
∂f[...]/∂u(k)
denotes the partial derivative of
f[...]
with resp et to
any
un(k)
, whih is indep endent of
n
. This indep endene allows to entirely
omit the
n
subsript in
Φ
's sensitivities and to subsequently write
dΦ(X)/du(k)
instead of
dΦ(X)/dun(k)
, and it allows to ompute all sensitivities for all agents
simultaneously.
In summary,
dΦ(X)/du(k)
is obtained in a two-pass-pro edure:
1. Using (4.7), solve (4.6) reursively for
k=K . . . 1
. Moving bakwards
through time introdues a far sightedness into the alulations that is
neessary to predit the inuene of present state variations on future
system states.
75
2. Determine the inuene of ontrol variables by (4.8) for
k= 0 . . . K −1
.
Sine this expression is idential for all agents, it needs to b e evaluated
only one for the entire population.
One obtains the following linearization of
Φ(X)
with resp et to
U1...UN
:
Φ(X(U1...UN)) ≈Φ(X0) +
K−1
X
k=0 dΦ(X0)
du(k)TN
X
n=1
(un(k)−u0
n(k))
(4.9)
where
u0
n(k)
is the ontrol vetor of traveler
n
in time step
k
around whih the
linearization takes plae and
X0
is the resulting marosopi state sequene.
Dening the sensitivity sequene
Λ = dΦ(X0)
du(k)k
(4.10)
and the inner pro dut
hΛ,Ui =X
kdΦ(X0)
du(k)T
u(k),
(4.11)
(4.9) an b e rewritten as
Φ(X(U1...UN)) ≈
N
X
n=1hΛ,Uni+
onst (4.12)
where the onstant addend ontains all terms indep endent of
U1...UN
. The
elements of
Λ
are sensitivities of the global ob jetive funtion with resp et to
individual turning moves, and as suh they serve as oeients that are multi-
plied with the turning move indiators ontained in the p opulations' path set
U1...UN
.
Marosopi tra dynamis are linear in go o d approximation with resp et to
a single agent's b ehavior sine individual ontrol variables
uij,n(k)∈ {0,1}
are
small ompared to atual turning ounts in a ongested network. Thus, for a
single agent, a linearization yields a reasonable approximation to the nonlinear
problem, and
Wn(U) = Vn(U) + Φ(X(U1...Un−1,U,Un+1 ...UN))/µ
≈Vn(U) + hΛ,Ui/µ +
onst
(4.13)
holds with go o d preision. The onstant addend is idential for all alternatives
available to an agent. Sine it is reasonable to assume that the preferenes of
a deision maker are not inuened by a onstant shift in the utilities of all
alternatives,
1
Wn(U) = Vn(U) + hΛ,Ui/µ
(4.14)
denes as from now the
mo died utility
of agent
n
's option
U ∈ Cn
. Using
the same
Λ
for all agents reets the fat that the sensitivity of
Φ
to a turning
1
This is always true for RUMs, f. (3.15 ).
76
move (sequene) is indep endent of whih agent is atually moving. Here, the
elements of
Λ
onstitute (up to a saling o eient
µ
) utility orretions for
every single turning move in the network, and the mo died utility of a spei
path is identied by adding up these orretions along that path. This an be
seen most learly if
hΛ,Ui
is fully expanded:
hΛ,Ui =X
kX
ij
dΦ(X0)
duij(k)uij(k).
(4.15)
Only suh omp onents of
Λ
are summed up in
hΛ,Ui
that orresp ond to turning
moves that are atually represented by path
U
through non-zero turning move
indiators. In light of this,
Λ
is denoted either as a sequene of sensitivities or
of utility orretions, dep ending on the ontext.
The ab ove linearization pro edure is onsiderably aelerated if the underlying
mobility simulation runs on variable time sales as prop osed in Setion 2.5.
Sine the mobility simulation's sensitivities vary on the same temp oral grid as
its marosopi states, the overall number of sensitivity evaluations is redued
in the same order as the numb er of ow transmissions during a simulation.
The imp ortane of this omputationally still exp ensive linearization b eomes
lear in omparison with a simplisti approximation. Assume that the maro-
sopi system state
X
is omp osed of vehile o upanies on all road segments
in all time steps. Then, the eet of a vehile's path hoie
U
might app ear pre-
ditable by simply inreasing the oupany of every link in
U
for the duration
of this link's traversal time. In a way, this do es predit the eet of
U
on
X
and thus on
Φ
without any linearization. Still, it do es not apture the global
eet of driver behavior in ongested onditions. A vehile that tries to enter a
ongested link is slowed down, and in turn it slows down all vehiles b ehind it.
That is, it also aets upstream links that are not ontained in its path. A full
linearization of tra ow dynamis aounts for these interdependenies and
thus is sup erior in all but trivially unongested tra onditions.
4.1.3 Consistent Linearization for Many Agents
The linearization of
Φ
relies on the relatively small inuene of a single trav-
eler on the global tra situation. This argument do es not hold if an entire
p opulation is onsidered sine any utility orretion
Λ
that is obtained by a
linearization around a ertain state tra jetory
X0
may result in a p opulation
reation
U1...UN
that auses a signiantly dierent network state tra jetory
X
and thus invalidates the underlying linearization.
For a non-sto hasti planning or telematis simulation, a utility orretion
Λ
is onsistent if the p opulation b ehavior given this
Λ
generates network states
X
suh that a rep eated linearization of
Φ
repro dues the original
Λ
values, f.
Figure 4.1. Formally, a xed p oint of the ombined map sim(ulation), followed
by lin(earization) is required:
Λ =
lin
◦
sim
(Λ)
.
Sine there are sto hasti elements in the simulation, its outome
X
given a
sp ei
Λ
is sto hasti as well, and the repro duibility of
Λ
alls for a likewise
sto hasti interpretation. One may assume that only a randomly distorted map
77
Figure 4.1: Fixed p oint of utility orretions
Consistent utility orretions
Λ
are attained if a linearization of
Φ
around simulation
outome
X
results in the same
Λ
orretions that have previously b een applied in the
simulation.
lin
◦
sim
(Λ) + E
an b e evaluated where
E
is a zero mean disturbane of the
same dimension as
Λ
. Sine no algorithm is known that denitely onverges to
a deterministi
Λ
xed p oint in suhlike noisy onditions for the whole range
of p ossibly implemented simulation mappings, and sine not even the existene
of suh a xed p oint is asertained, a pragmati ourse of ation is taken: The
existene of a xed p oint is merely assumed, and an elementary sto hasti ap-
proximation (SA) metho d is employed for its identiation [26℄.
2
This partiular
metho d is hosen here b eause of its simpliity and larity. Possible algorithmi-
al improvements are indiated in Setion 6.4.1.3.
The prop osed SA approah is outlined in Algorithm 2. It assumes an iterative
simulation logi, whih is equally appliable to a SUE-based planning mo del
and to a telematis model of sp ontaneous and imperfetly informed drivers.
The oneptual dierene is that a SUE deision proto ol typially utilizes all
information from the most reent network loading, whereas a telematis deision
proto ol generates every elementary deision within a plan only based on that
subset of this information that ould have atually been gathered up to the
onsidered point in simulated time [26℄. A full implementation of this algorithm
is exp erimentally investigated in the next hapter.
4.1.4 Behavioral Justiation
Sine the mo died utility deviates from the originally mo deled agent p ereption,
any b ehavior that is based on the mo died utility is not reasonable in itself. A
path
U
that is hosen by traveler
n
based on a mo died utility funtion
Wn
only is onsistent with the b ehavioral mo del if
n
's utility p ereption is indeed
represented by
Wn
instead of the original
Vn
. Thus, the metho d's appliability
dep ends on the p ossibility to reinterpret utility p ereption itself. Three elds
where this is p ossible are identied b elow:
•
The metho d is develop ed with b ehavioral tra state estimation in mind
and is appliable for this purp ose. Given a sp eiation of
Φ
that reets
2
A self-ontained onvergene pro of for the SA method an b e found in [69℄. However, its
requirements annot b e established in the setting onsidered here.
78
Algorithm 2
Steering a p opulation of agents
1. Initialization.
(a) Set iteration ounter
m= 0
.
(b) Fill
¯
Λ(m)
(estimate of
Λ
xed p oint) with all zeros.
2. Simulation.
(a) For all
n= 1 . . . N
, do: Use
Wn(U) = Vn(U) + h¯
Λ(m),Ui/µ
instead
of
Vn(U)
in the deision protool when drawing
U(m)
n
.
(b) Load
U(m)
1...U(m)
N
on the network and obtain
X(m)
.
3. Linearize
Φ(X(m))
and obtain
Λ(m)
.
4. Up date
¯
Λ(m+1) =m
m+ 1 ¯
Λ(m)+1
m+ 1Λ(m)
.
5. If another iteration is desired:
(a) Inrease
m
by one.
(b) Goto step 2.
the quality of measurement reprodution, the resulting
Wn
is interpreted
as an estimate of individual
n
's most likely utility p ereption given these
measurements. Here, the original
Vn
onstitutes a mo del-based a priori
assumption that is orreted by the estimation pro edure suh that
Φ
is
improved. The b elief in the b ehavioral prior information is reeted by
weight parameter
µ
. A disussion of p ossible ambiguities in this interpre-
tation is given in Setion 4.4.3.
•Φ
may also represent a general utility of system op erations. Applying the
ab ove pro edure, the resulting
Λ
o eients dene a toll on all turning
moves in the network. An agent
n
whih hooses its path based on the
resulting
Wn
strives to maximize a weighted ombination of individual
and system utility. Clearly, a physially implementable toll must meet a
number of additional onstraints that are b eyond the sop e of this thesis.
•
An iterative planning simulation requires large amounts of omputation
time. If a sp eiation of
Φ
was found that (i) reets the degree of
suh a simulation's onvergene and (ii) has a vanishing inuene up on
onvergene, it may help to redue the numb er of required iterations until
an equilibrium is reahed. Here, utility p ereption is mo died only during
the transient phase of an iterative algorithm but not in its outome. Still,
this appliation is of rather hypothetial nature sine no suh version of
Φ
is prop osed in this dissertation.
In all ases,
Wn
onstitutes a mo died utility p ereption of driver
n
that is in
one way or the other onsistent with the original assumption of utility-driven b e-
79
havior, and this modiation is generated suh that a problem-spei instane
of
Φ
is improved.
4.2 Heuristi Estimation
A similarity measure b etween simulated and observed sensor data is hosen as
the global ob jetive funtion
Φ
, and the agents are steered towards an inrease
of this funtion.
4.2.1 Mo deling of Aggregate Tra Measurements
A likeliho o d funtion suggests itself to quantify a mo del's measurement t. In
this subsetion, the likelihoo d of aggregate tra measurements is formally
related to individual agent behavior.
Marosopi state spae model (3.7) is supplemented with an output equation
y(k) = g[x(k),ǫ(k)]
(4.16)
that maps system state
x(k)
by a one dierentiable funtion
g
on output ve-
tor
y(k)
of marosopi observables. The latter may inlude ows, veloities,
and densities generated by sensors suh as indutive lo ops, oating ars, and
tra surveillane ameras. The inuene of various soures of error on these
observations is aounted for by random disturbane vetor
ǫ(k)
that turns
y(k)
into a random variable itself. Equation (4.16) denes
y(k)
's probability density
funtion (p.d.f.)
p(y(k)|x(k)) = Zδ(y(k)−g[x(k),ǫ])p(ǫ)dǫ
(4.17)
where
δ
is the Dira funtion and
p(ǫ)
is the known p.d.f. of
ǫ
. A lower-ase
p
generally denotes a p.d.f., whereas an upp er-ase
P
represents a disrete prob-
ability. Subsuming the ab ove expression in terms of tra jetories
Y={y(k)}k
and
X={x(k)}k
yields
p(Y|X) = Y
k
p(y(k)|x(k))
(4.18)
where sto hasti indep endene b etween outputs at dierent time steps is as-
sumed. This is, so far, the not unexp eted result that all spatiotemp oral mea-
surements an b e probabilistially desrib ed if all spatiotemp oral system states
X
are known no b ehavioral information is needed diretly.
Nevertheless, the states
X
are indiretly aused by the p opulation b ehavior
U1...UN
. This allows to dene the b ehavioral likeliho o d
l(U1...UN|Y)
given
the measurements
Y
as a funtion of
U1...UN
:
l(U1...UN|Y) = p(Y|X(U1...UN)).
(4.19)
80
This funtion is linearizable with respet to
U1...UN
if the p.d.f. of
Y
given
X
is dierentiable with resp et to
X
. Frequently, the (likewise linearizable)
log-likeliho o d funtion
L(U1...UN|Y) = ln l(U1...UN|Y)
(4.20)
is also referred to.
Others than link-related measurements are p ossible. Sine the state vetor of
mo del (3.7) ontains smo othed turning ounts, observations of these an be
diretly inorp orated in the output equation. The additional value of suh
measurements is pointed out in the literature review of Setion 1.2.1.
4.2.2 Steering Agents Towards the Measurements
Maximum likeliho o d estimation is the arguably most p opular approah to sta-
tistial parameter identiation, e.g., [140℄. It is an established metho d for the
identiation of OD matries from tra ounts [162℄, and its appliation for
agent-based b ehavioral estimation is ompliated in the same way as traditional
OD matrix estimation: The available numb er of link-related measurements is
usually muh smaller than the number of parameters to be identied the
problem is extremely under-determined.
Typially, a prior OD matrix is integrated in the likelihoo d funtion as a supple-
mentary measurement that resolves this under-determinedness. Sine no suh
prior is available here, a dierent and statistially less rigorous approah is pur-
sued. Algorithm 2 is employed, with its general ob jetive funtion dened as
the measurement log-likelihoo d, i.e.,
Φ(X(U1...UN)) = L(U1...UN|Y).
(4.21)
The resulting overall ob jetive funtion (4.3) of any agent
n
is the weighted sum
Vn(U) + Φ(X(...U...))/µ
of its individual utility funtion and the log-likelihoo d.
The weighting parameter
µ
determines the importane of the behavioral prior
information represented by the original utility p ereption. If
µ
is hosen very
large, the likelihoo d term vanishes and the agent ats in a way that is fully
presp eied by its original utility funtion. The smaller
µ
gets the more weight
is put on the likeliho o d and the more the agent adjusts its b ehavior towards
an inrease of the likeliho o d. While
µ
is used here as a mere weighting param-
eter, the Bayesian problem reformulation given in the next setion enables its
interpretation as a b ehavioral mo del parameter.
Sp eially, if mutually indep endent normal measurement distributions are as-
sumed, (4.21) yields a global ob jetive funtion
Φ(X) = −X
aX
k
(ya(k)−ga[x(k)])2
2σ2
a
(4.22)
where
ya(k)
is the sensor information available for link
a
in time step
k
,
ga[x(k)]
is its simulated exp etation, and
σ2
a
is its variane.
3
This is the arguably sim-
3
The log-likelihoo d of mutually indep endent measurements
ya(k)
is
L(U1. . . UN|Y) =
Pak ln p(ya(k)|x(k)).
Assuming
ya(k) = ga[x(k)] + εa(k),
a normally distributed
εa(k)
with
zero exp etation and variane
σ2
a
implies
p(ya(k)|x(k)) ∝exp[−(ya(k)−ga[x(k)])2/2σ2
a]
.
Consequently,
L(U1. . . UN|Y) = −Pak(ya(k)−ga[x(k)])2/2σ2
a+
onst
.
81
plest approah to the b ehavioral estimation problem: Dene a quadrati dis-
tane measure b etween observed and simulated tra harateristis, ho ose
a reasonable weight parameter
µ
, and let the general metho d to steer agent
b ehavior push the simulation towards a redution of this error funtion.
The partiular assumption of indep endent normal measurements yields an ob-
jetive funtion (4.22) of greatest simpliity. Still, dierent distributional as-
sumptions are feasible. In partiular, orrelated measurements with a known
ovariane struture an b e aounted for in terms of a multivariate (normal)
distribution.
Providing a mo died utility that omprises a weighted sum of individual utility
p ereption and measurement log-likeliho o d to the deision proto ol do es not
result in an overall maximum likeliho o d estimator for two reasons: (i) The indi-
vidual utility addend p ermits no interpretation as a log-likeliho o d omp onent,
and (ii) the deision proto ol draws a hoie instead of deterministially maxi-
mizing the mo died utility. For these reasons, a more systemati derivation of
a statistial estimator is given in the following.
4.3 Bayesian Estimation
Setion 4.1 prepares a general to ol to steer simulated travelers. This to ol fa-
ilitates the prop osal of a rst heuristi estimator in Setion 4.2. Here, the
estimation problem is reonsidered in a statistially more rigorous setting. The
presentation starts with a oneptually straightforward but omputationally
umb ersome formulation. Several simpliations are then adopted that signif-
iantly inrease the omputational feasibility and result in the prop osal of two
op erational estimators. Ultimately, the heuristi estimator is redisovered, this
time, however, with a b etter understanding of its prop erties and limitations.
It has b een stated b efore that aggregate measurements
Y
alone do not provide
suient information for a unique estimate of p opulation b ehavior
U1...UN
sine usually there are many b ehavioral ombinations that generate the same
observations. Here, this problem is resolved by the inorp oration of additional
b ehavioral information in a Bayesian setting. In order to build on a solid foun-
dation, the Bayesian estimator is designed from srath. While some previously
develop ed results suh as the linearization of a log-likeliho o d funtion in dy-
namial onditions are reused in this setion, no onstitutional dep endeny on
the heuristi estimator itself is allowed for.
4.3.1 General Formulation of Estimator
An arbitrary implementation of the deision protool is assumed. It draws
hoies
U ∈ Cn
aording to an individual hoie distribution
Pn(U)
for every
agent
n= 1 . . .N
. Only realizations of this distribution an b e observed, f.
Setion 3.2.1.4.
U
may still represent any of the b ehavioral dimensions desrib ed
in Setion 3.2.1.1, ranging from a single route to an all-day plan. Given mutually
indep endent traveler deisions, the
b ehavioral prior
for the whole p opulation
82
is dened as
P(U1...UN) =
N
Y
n=1
Pn(Un).
(4.23)
The assumption of mutually indep endent hoies is to be understo o d in the
ontext of the iterative simulation logi outlined in Setion 4.1.3 in that (4.23)
desrib es the p opulation's plan hoie distribution in a partiular iteration of
the simulator given the network onditions only from the previous iteration(s).
The available measurements
Y
parameterize a likeliho o d
l(U1...UN|Y)
of the
p opulation's path hoie as speied in (4.19). Bayes' theorem allows to ombine
these two soures of information into a
b ehavioral p osterior
P(U1...UN|Y) = l(U1...UN|Y)P(U1...UN)
PV1∈C1···PVN∈CNl(V1...VN|Y)P(V1...VN),
(4.24)
where the denominator results from
p(Y) = X
V1∈C1··· X
VN∈CN
p(Y|V1...VN)P(V1...VN).
(4.25)
The estimation ob jetive is to have the p opulation ho ose its b ehavior aording
to the p osterior (4.24) instead of the prior (4.23). This an b e enfored if draws
are taken from the prior but are rejeted with a ertain probability that dep ends
on the measurements. Denote by
φ(U1...UN)
the probability to aept a draw
U1...UN
from the prior. If this probability is sp eied by
φ(U1...UN) = l(U1...UN|Y)/D
D≥max
V1∈C1...VN∈CN
l(V1...VN|Y),
(4.26)
then the following aept/rejet pro edure draws from the p osterior:
1. Draw andidate hoies
U1...UN
from the prior (4.23).
2. With probability
1−φ(U1...UN)
, disard the andidates and goto 1.
3. The rst aepted
U1...UN
onstitute a draw from the p osterior (4.24).
The orretness of this simple algorithm is shown by straightforward manipula-
tions. Noting that the overall probability of a rejetion is
φ
rejet
= 1 −X
V1∈C1··· X
VN∈CN
φ(V1...VN)P(V1...VN),
(4.27)
the probability that
U1...UN
is the rst aepted draw is
∞
X
d=0
φd
rejet
φ(U1...UN)P(U1...UN)
=φ(U1...UN)P(U1...UN)
1−φ
rejet
=φ(U1...UN)P(U1...UN)
PV1∈C1···PVN∈CNφ(V1...VN)P(V1...VN)
=P(U1...UN|Y).
(4.28)
83
The b ehavioral p osterior an thus b e generated by suppressing ertain draws
from the prior. Somewhat oarsely expressed: (i) The simulation is run many
times with dierent random seeds, (ii) a large p ortion of these runs is thrown
away, based on the ab ove rejetion riterion, and (iii) the remaining runs are
draws from an aurate Bayesian ombination of the b ehavioral prior and the
measurements.
Although app ealing b eause of its simpliity, this approah is in this form om-
putationally intratable in all but trivial ases. There are two ma jor problems:
1. It is omputationally infeasible to evaluate all p ossible
l(U1...UN|Y)
val-
ues b eforehand sine every suh evaluation requires a full network loading
in order to map
U1...UN
on a marosopi state sequene
X
that enters
the likelihoo d via (4.19). However, these evaluations are required in or-
der to guarantee a feasible denominator for the aeptane probabilities
(4.26). Furthermore, the need for a hoie set enumeration implies that
the estimation logi is aware of this set, whih onstitutes an unwanted
dep endeny of the estimator on mo deling details.
2. Even if the aeptane probabilities' denominator is replaed by an es-
timate in order to mitigate problem 1, a single draw from the p osterior
might still require a substantial numb er of mobility simulation runs sine
every draw from the prior needs to b e loaded on the network at least one
and sine it annot b e guaranteed that an aept o urs after a xed
number of draws from the prior.
In light of these diulties, simplifying assumptions that speed up the sim-
ulation of the posterior are highly desirable even at the ost of some loss in
auray. Two suhlike simplied estimators are proposed in the following two
setions.
4.3.2 Op erational Aept/Rejet Estimator
The Bayesian estimator is onsiderably simplied if the full likelihoo d is replaed
by an approximation. In Setion 4.1.2, a general funtion
Φ
of the marosopi
system states is linearized with resp et to the p opulation's path hoie. Pro-
eeding in this resp et similarly to the heuristi estimator of Setion 4.2.2, this
result is now utilized to linearize the measurement log-likeliho o d. Let
Φ(X(U1...UN)) = L(U1...UN|Y).
(4.29)
A linearization of
Φ
yields the approximation
L(U1...UN|Y)≈
N
X
n=1hΛ,Uni+
onst (4.30)
with the
Λ
o eients dened in (4.10) through (4.12). The resulting likelihoo d
approximation is
l(U1...UN|Y)≈
onst
·
N
Y
n=1
ehΛ,Uni.
(4.31)
84
A substitution of this and the behavioral prior (4.23) in the b ehavioral p osterior
(4.24) yields
P(U1...UN|Y)≈QN
n=1 ehΛ,UniPn(Un)
PV1∈C1···PVN∈CNQN
n=1 ehΛ,VniPn(Vn).
(4.32)
The denominator of this expression requires some attention. It is a sum over
al l
p ossible ombinations of b ehavioral patterns
V1...VN
in the p opulation,
whereas the
eh··· i
terms result from a linearization around a
partiular
maro-
sopi state sequene. The feasibility of this approximation results from the
observation that, even if individuals exhibit variable b ehavior, the resulting
marosopi tra patterns are relatively onentrated in state spae. All de-
terministi tra assignment eorts rely on this assumption. Thus, the ma jority
of b ehavioral draws results in tra patterns over whih a linearization an b e
justied. Behavioral patterns
V1...VN
that generate physial states far away
from this domain are assumed to have suh low probabilities
QN
n=1 Pn(Vn)
that
the aording addends in the denominator an b e negleted.
Applying the distributive law to (4.32), one obtains
P(U1...UN|Y)≈QN
n=1 ehΛ,UniPn(Un)
QN
n=1 PVn∈CnehΛ,VniPn(Vn)
=
N
Y
n=1
ehΛ,UniPn(Un)
PVn∈CnehΛ,VniPn(Vn).
(4.33)
The linearization is b eneial in two ways. First, the p opulation's joint p os-
terior (4.33) is deomp osed into a pro dut of individual p osteriors that an b e
evaluated agent by agent. These individual p osteriors are subsequently denoted
by
Pn(U|Y) = ehΛ,UiPn(U)
PV∈CnehΛ,ViPn(V).
(4.34)
Seond, only a single run of the mobility simulation (plus one alulation of the
Λ
o eients) is needed to parameterize these p osteriors for all agents in the
p opulation.
The aept/rejet pro edure an now b e applied to every agent individually.
The aeptane probability for path
U
from agent
n
's hoie set is dened as
φn(U) = ehΛ,Ui/Dn
Dn≥max
V∈Cn
ehΛ,Vi,
(4.35)
but otherwise the metho d remains unhanged. The only simplifying assump-
tion made here is that the log-likelihoo d an b e linearized with suient prei-
sion. Sine this linearization is likely to b e dierent given either the b ehavioral
prior or the p osterior, an iterative approah similar to the xed p oint searh
of Algorithm 2 is appropriate: Starting from the b ehavioral prior, suessively
improved linearizations are generated from iteration to iteration until a stable
state is reahed where the estimator draws from the b ehavioral p osterior based
85
Algorithm 3
Aept/rejet estimator
1. Initialization.
(a) Set iteration ounter
m= 0
.
(b) Fill
¯
Λ(m)
(estimate of
Λ
xed p oint) with all zeros.
2. Simulation.
(a) For all
n= 1 . . . N
, do:
i. Draw andidate hoie
U(m)
n
from
n
's b ehavioral prior.
ii. With probability
1−φn(U(m)
n)
(where
¯
Λ(m)
is substituted for
Λ
in (4.35)), disard the andidate and goto 2(a)i.
iii. Retain the rst aepted hoie
U(m)
n
.
(b) Load
U(m)
1...U(m)
N
on the network and obtain
X(m)
.
3. Linearize
Φ(X(m))
and obtain
Λ(m)
.
4. Up date
¯
Λ(m+1) =m
m+ 1 ¯
Λ(m)+1
m+ 1Λ(m)
.
5. If another iteration is desired:
(a) Inrease
m
by one.
(b) Goto step 2.
on a linearization that in turn is most appropriate given this very p osterior.
This approah is subsequently denoted as the
aept/rejet (AR) estima-
tor
. It is summarized in Algorithm 3. Again, only a basi SA xed p oint searh
pro edure is deployed for greatest larity.
The typ e of behavior to b e estimated and the prior implemented by the deision
proto ol are arbitrary. Sine a hoie set enumeration is only required to provide
a lower bound for the aeptane probabilities' denominator dened in (4.35), it
an b e avoided if this denominator is treated as a tuning parameter: Cho osing a
large value is likely to omply with the (unknown) lower b ound but also to result
in low aeptane probabilities and inreased omputational ost. Vie versa, a
smaller denominator yields faster but also inreasingly impreise estimates. The
loss in preision an b e appraised by observing the frequeny at whih infeasible
probabilities greater one o ur in (4.35) that need to b e trunated. This provides
a pratially attrative balaning mehanism b etween estimation preision and
omputational eieny, whih do es not rely on a hoie set enumeration.
Computational diulties remain if a b ehavioral draw is exp ensive, e.g., b e-
ause it involves some kind of optimization pro edure, suh as a (randomized)
b est path alulation. One alternative would b e not to disard unwanted draws
but to dupliate desired ones and to use these in a numb er of rep eated hoie
situations. However, sine this would intro due p ossibly unwanted serial or-
relations, it is at odds with the intention to develop a transparent estimation
86
layer. A omputationally more eient yet not as broadly appliable estimator
is presented next.
4.3.3 Op erational Utility-Mo diation Estimator
The b ehavioral p osterior (4.34) for a single agent onstitutes the starting p oint
of this development. It is restated here for ease of referene:
Pn(U|Y) = ehΛ,UiPn(U)
PV∈CnehΛ,ViPn(V).
(4.36)
The PS-logit mo del prepared in Setion 3.2.2.2 is now used as a distributional
assumption ab out the prior hoie probabilities, i.e.,
Pn(U) =
PS
n(U)eµVn(U)
PV∈Cn
PS
n(V)eµVn(V).
(4.37)
Reall that the PS o eients aount for path overlap in a route hoie ontext.
If they are omitted, a plain MNL mo del results. A substitution of (4.37) in (4.36)
yields
Pn(U|Y) =
PS
n(U)eµ(Vn(U)+hΛ,Ui/µ)
PV∈Cn
PS
n(V)eµ(Vn(V)+hΛ,Vi/µ).
(4.38)
This p osterior is struturally idential to its prior. Only the addition of
hΛ,Ui/µ
to
Vn(U)
is dierent. This allows to fore a deision proto ol that implements
a PS-logit prior to immediately draw from the p osterior only by adding a or-
retion term
hΛ,Ui/µ
to every alternative
U
's systemati utility. The PS o e-
ients need not b e known to the estimator for the generation of these orretions.
Consequently, this approah is feasible for all priors that exhibit the funtional
form of the PS-logit mo del, even if the PS o eients result from a dierent
sp eiation than given in (3.17). Suh priors are said to b e of PS-logit stru-
ture. Note that this inludes the plain MNL model.
This approah is subsequently denoted as the
utility-mo diation (UM)
estimator
. Its requirements are more restritive than those of the AR estimator
sine a deision protool of PS-logit struture needs to b e available. However, if
suh a b ehavioral prior is given, the UM estimator and the AR estimator yield
equivalent results sine both rely on the same linearization-based approximation
(4.36) of the p osterior. In this ase, the UM estimator is to b e preferred over
the AR estimator sine it is omputationally more eient in that it rejets no
draws from the prior but immediately draws from the p osterior.
Setion 4.2's estimation heuristi oinides struturally with the UM estimator:
In either ase, the mo died utility is dened by (4.14), and the
Λ
o eients
are identially generated by a linearization of the measurement log-likelihoo d
funtion. The heuristi's weight o eient
µ
oinides with the sale param-
eter of the PS-logit prior. For ompleteness, the UM estimator is sp eied in
Algorithm 4.
87
Algorithm 4
Utility-mo diation estimator
1. Apply Algorithm 2 with the global utility funtion
Φ
dened by (4.21) as
the measurement log-likeliho o d funtion.
2. This estimator has the following prop erties.
(a) It is idential to the heuristi estimator of Setion 4.2.
(b) If the b ehavioral prior is of PS-logit struture, this estimator is equiv-
alent to the AR estimator sp eied in Algorithm 3.
4.3.4 Appliability of Heuristi Estimator
Tehnial ly
, the UM estimator an b e applied in onjuntion with an arbitrary
utility-driven behavioral prior for the estimation of anything from routes to all-
day plans. In suh a general setting, it oinides with the heuristi estimator
of Setion 4.2. This analysis identies the
oneptual
limitations of suh an
approah and thus laries the appliability of the heuristi estimator itself.
Assume that deision maker
n
disp oses of a hoie set
Cn
and that presp ei-
ed utilities
V0
n(U)
for every
U ∈ Cn
are given. Based on these utilities, the
deision proto ol draws from well-dened but to the estimator unknown hoie
probabilities
P0
n(U)
. These hoie probabilities an b e p erfetly repro dued by
a mo del of PS-logit struture if the PS o eients are re-dened as
PS
n(U) = P0
n(U)
eµV 0
n(U).
(4.39)
The resulting hoie probabilities are
Pn(U) = P0
n(U)eµ(Vn(U)−V0
n(U))
PV∈CnP0
n(V)eµ(Vn(V)−V0
n(V))
(4.40)
suh that
Vn(U) = V0
n(U)
results in
Pn(U) = P0
n(U)
for all
U ∈ Cn
. Lo osely
sp eaking, any b ehavioral prior an be approximated up to 0th order in this way.
The adequay of this approximation for others than the presp eied utilities
only dep ends on the approximated prior's elastiities, i.e., the way relative utility
hanges indue relative hanges in the hoie probabilities.
The elastiities of the PS-logit hoie probabilities with resp et to deterministi
utilities are struturally idential to those of the MNL mo del:
∂Pn(U)
∂Vn(V)
Vn(V)
Pn(U)=(µVn(U)(1 −Pn(U)) U=V
−µVn(V)Pn(V)
otherwise.
(4.41)
In partiular, if alternative
V
b eomes more (less) attrative, its inreased (de-
reased) hoie probability redues (inreases) the hoie probabilities of all
other alternatives
U 6=V
by the same relative amount.
Reall that the UM estimator funtions without expliit knowledge of the PS o-
eients. This implies that an appliation of the UM estimator an be justied
88
Figure 4.2: Three routes example, rep eated
A simple route hoie example with three alternative routes
A
(omprised of link 1),
B
(omprised of link sequene 2
→
3a), and
C
(omprised of links 2
→
3b).
by approximation (4.40) even if the
P0
n
and
V0
n
values that (re-)dene the PS
o eients in (4.39) are unknown. However, it is required that the elastiities of
the prior hoie distribution are suiently well aptured by (4.41). Sine the
UM estimator's working oinides with that of Setion 4.2's heuristi estimator,
idential limitations hold for that heuristi.
4.4 Illustrative Example
The prop osed estimators are illustrated with a simple example. For larity, only
a route hoie problem is onsidered, and stationary onditions are assumed
instead of a full dynamial mo del.
4.4.1 Senario Desription
The example network of Setion 3.2.2.2 is reonsidered. It is rep eated in Figure
4.2. A hoie set of three routes
A,B,
and
C
onnets the origin no de at the
very left to the destination no de at the very right. The systemati utility of all
routes is identially and invariably
¯
V
. The assumption of a onstant systemati
utility is adequate either in unongested onditions or in a telematis setting
where drivers are a priori unaware of atually prevailing network onditions.
(An example with an underlying equilibrium assumption is given in the next
hapter.)
Sine routes
B
and
C
have almost p erfet overlap, a b ehaviorally reasonable
route split is
(P(A)P(B)P(C))=(1
/21
/41
/4)
. However, for the purp ose of this
example, a plain MNL mo del that does not aount for route overlap is hosen
as the b ehavioral prior:
P(U)∝eµ¯
V,U=A,B,C,
(4.42)
where
µ, ¯
V= 1
in all numerial exp eriments. This results in prior route splits
(P(A)P(B)P(C)) = (1
/31
/31
/3).
(4.43)
The mo del is mirosopi in that every departing driver
n= 1 . . . N
individually
ho oses a route. Sine stationary onditions are assumed, a traveler's turning
89
move sequene
Un={un}
and the resulting state sequene
X={x}
only
onsist of a single vetor eah:
un= (uA,n uB,n uC,n)T
(4.44)
x= (xAxBxC)T.
(4.45)
The elements of
u
indiate a driver's initial turn into route
A
,
B
or
C
:
u=
(1 0 0)T
represents the hoie of route
A
,
u= (0 1 0)T
stands for route
B
,
and
(0 0 1)T
indiates route
C
. Sine no tra ow dynamis are modeled, the
network states are dened as the total route volumes
x=
N
X
n=1
un.
(4.46)
A single ow sensor is lo ated on route
A
. Its output
y
is mo deled by the
measurement equation
y=xA+ǫ
(4.47)
where
ǫ
is a normal error with zero mean and
σ2
variane. The resulting log-
likeliho o d (4.20) of p opulation route hoie
U1...UN
given measurement se-
quene
Y={y}
is
L(U1...UN|Y) = −(y−xA)2
2σ2
=−y−PN
n=1uA,n2
2σ2.
(4.48)
A linearization of this funtion with resp et to individual route hoie is easier
than in the general ase of Setion 4.1.2 sine no dynamial onstraints are
involved. Maintaining the formalism of that setion,
Φ(X(U1...UN))
is dened
to b e
L(U1...UN|Y)
,
Φ
is linearized, and (4.10) yields a sequene
Λ = ((y−x0
A)/σ20 0)T
(4.49)
of
Φ
's sensitivities evaluated at a state sequene
X0={x0}
. Aording to
(4.11), the approximate eet of a single agent that ho oses route
A
,
B
or
C
on
the log-likelihoo d is
hΛ,Ai = (y−x0
A)/σ2
hΛ,Bi = 0
hΛ,Ci = 0.
(4.50)
These expressions aount for the eet of adding an agent to a route but
ignore the eet of removing it from its previously hosen route. This is feasible
b eause, one the eet of route hoie is linearized, removing an agent from
its original route does not hange the linear eet of its reassignment to a new
route. Sine every hoie implies that any previous hoie is disarded, only the
newly made hoie is relevant for estimation. Formally, the eet of disarding
an outdated hoie is subsumed in the onstant addend of (4.12).
90
4.4.2 Aept/Rejet Estimator
The hoie set
{A,B,C}
is known and sampling from the prior (4.42) is easy,
so the AR estimator an b e applied without diulty. Sine all agents have
idential hoie sets, the aeptane probabilities (4.35) are likewise idential
for all agents:
φ(A) = ehΛ,Ai/D =e(y−x0
A)/σ2/D
φ(B) = ehΛ,Bi/D = 1/D
φ(C) = ehΛ,Ci/D = 1/D
D= max{e(y−x0
A)/σ2,1}.
(4.51)
That is, draws of route
A
are preferred over those of routes
B
and
C
if the
exp onent in
φ(A)
is p ositive, and they are suppressed if it is negative. Sine a
p ositive exp onent indiates that less vehiles than measured are simulated on
route
A
and a negative exp onent indiates that to o many simulated vehiles
ho ose this route, the AR mehanism funtions like a ontroller that works
against the measurement error.
The aeptane probabilities of routes
B
and
C
are equal. This reets the lak of
measurement information that ould justify a preferene for either route. The
equal aeptane probabilities in onjuntion with the onstant deterministi
utilities also imply that the prior ratio of the hoie probabilities for
B
and
C
is not aeted by estimation. (If, however, the deterministi utilities were a
funtion of the route volumes, the deision proto ol may reat to a hange in
estimated tra onditions with a likewise hanged ratio of
B
's and
C
's hoie
probabilities.)
An adopted version of Algorithm 3 that aounts for the simplied mobility
simulation and the homogeneous driver p opulation of this example is given
b elow.
1. Initialization.
(a) Set iteration ounter
m= 0
.
(b) Fill
¯
Λ(m)
(estimate of
Λ
xed p oint) with all zeros.
2. Simulation.
(a) Calulate aeptane probabilities
φ(m)(U)
for
U=A,B,C
(where
¯
Λ(m)
is substituted for
Λ
in (4.51)).
(b) For
n= 1 . . . N
, do:
i. Draw andidate route
U(m)
n
from the prior (4.43).
ii. With probability
1−φ(m)(U(m)
n)
, disard the andidate and goto
step 2(b)i.
iii. Retain the rst aepted hoie
U(m)
n
.
() As a stationary surrogate for a full network loading, use (4.46) to
map
U(m)
1...U(m)
N
on
X(m)
.
91
3. Linearize the log-likeliho o d funtion by ( 4.49) and obtain
Λ(m)
.
4. Up date
¯
Λ(m+1) =m
m+ 1 ¯
Λ(m)+1
m+ 1Λ(m)
.
5. If another iteration is desired:
(a) Inrease
m
by one.
(b) Goto step 2.
For simulative investigations, a total demand of
N= 1000
drivers is gen-
erated, and a single measurement
yA= 500
is assumed on route
A
. This
value is what one would exp et on average if a mo del was used that real-
istially aounts for route overlap by distributing the demand aording to
(P(A)P(B)P(C))=(1
/21
/41
/4)
.
The estimation onvergene of 100 AR iterations for dierent measurement vari-
anes
σ2= 1000
,
100
, and
10
is illustrated in Figure 4.3. The realisti volumes
of 500 vehiles on route
A
and 250 vehiles on routes
B
and
C
are repro dued
b etter with dereasing
σ2
. An improved measurement repro dution omes at
the ost of a lengthened settling time until the estimator draws from an appar-
ently stable p osterior. This is owed to the log-likeliho o d's inreased steepness
that ompliates the identiation of a xed p oint. The ratio of route
B
and
C
's
share is not inuened by the estimation, as it has b een previously hypothesized.
The p erentage of aepted draws is 92%, 74%, and 64% for
σ2= 1000
,
100
, and
10
. The smaller the measurement variane the more pronouned the dierene
b etween prior and p osterior and the more draws from the prior need to b e
rejeted to generate the p osterior. The numb er of draws required by the AR
estimator generally inreases the more the likeliho o d ontradits the prior.
4.4.3 Utility-Mo diation Estimator
The UM estimator speied in Algorithm 4 is employed. The same experimental
setting as for the AR estimator is hosen, and the same adjustments are made
in order to aount for the simplied nature of this example. Sine every draw
based on the modied utilities is aepted, the omputational overhead of the
AR estimator is avoided. Furthermore, sine the MNL prior route hoie dis-
tribution (4.42) is of PS-logit struture, the resulting estimates are draws from
an idential p osterior distribution as for the AR estimator. Their illustration is
therefore omitted.
In this simple example, the utility orretions generated by the UM estima-
tor allow to reonstrut the PS o eients that are disregarded in the plain
MNL prior (4.42): Given
P(U)∝eµ¯
V
, the UM estimator generates a posterior
P(U|Y)∝ehΛ,Uieµ¯
V
, f. (4.36). Comparing this to a hyp othetial PS-logit prior
P(U)∝
PS
(U)eµ¯
V
that prop erly aounts for route overlap, one noties that
ehΛ,Ui
an indeed b e onsidered as an estimate of PS
(U)
.
Figure 4.4 plots
eh¯
Λ(m),Ui
for
U=A,B,C
over the iteration ounter
m
. Appar-
ently, these values onverge towards
(eh¯
Λ(∞),Ai eh¯
Λ(∞),Bi eh¯
Λ(∞),Ci) = (2 1 1)
for
92
Figure 4.3: Measurement t
Estimated route volumes over the iteration ounter for various measurement varianes. An inreasing b elief in the measurement results in a loser
repro dution of the true route splits but also in a lengthened settling time.
93
Figure 4.4: Estimated path sizes
Tra jetories of path size estimates
eh¯
Λ(m),Ui
for
U=A,B,C
over iteration ounter
m
. For dereasing
σ2
, these estimates approah values that are
prop ortional to the real path sizes based on whih the utilized measurement was generated.
94
small measurement varianes. This is a merely saled version of the path size o-
eients
(
PS
(A)
PS
(B)
PS
(C)) = (1 1
/21
/2)
that were derived for this senario
in Setion 3.2.2.2. These path sizes yield the plausible route hoie probabilities
(P(A)P(B)P(C))=(1
/21
/41
/4)
based on whih the utilized measurement was
generated.
It was hyp othesized in Setion 4.1.4 that an estimated utility mo diation ap-
tures those systemati features of an alternative that are not inluded in its
original utility. However, in the present example, systemati utility is p erfetly
mo deled, and the UM estimator only aounts for the overlap of routes
B
and
C
. This shows, given a RUM-based deision protool, that the orretion terms
only represent unmodeled systemati utilities if all orrelations in the utility
errors are prop erly mo deled. Otherwise, unmo deled orrelations may also b e
aounted for by the estimator. In general, a distint interpretation of the re-
sults is imp ossible. Still, this observation do es not impair the orretness of the
estimated p osterior distributions themselves.
Conluding, this hapter provides a numb er of metho ds for the estimation of
individual-level motorist b ehavior. All metho ds have the same Bayesian origin
but dier in their adopted simpliations. A small example laries the prop osed
algorithms. A large test ase is investigated in the next hapter.
95
Chapter 5
Test Case
This hapter investigates the appliability of the prop osed estimation approah
to a syntheti senario of pratially relevant size. It fo uses on omputational
feasibility and logial orretness. Sine various simpliations are neessary
to implement the test ase, its limitations likewise onne the sop e of these
investigations. However, the results learly establish that the estimator exhibits
suient preision, robustness, and omputational p erformane to b e studied
in more realisti settings and in onjuntion with more sophistiated modeling
omp onents.
5.1 Exp erimental Overall Setting
5.1.1 Senario Desription
A senario onsists of two omp onents: (i) invariable settings that desrib e
the strutural features of this test ase and (ii) a partiular hoie of variable
settings.
5.1.1.1 Invariable Settings
All exp eriments utilize the Berlin network desrib ed in Setion 2.6.2. The re-
sp etive driver p opulation is introdued in Setion 3.1.4. Behavioral estimation
for a
206 353
-agent p opulation on a
2 459
-link network is a nontrivial problem.
All exp eriments are onstrained to the time span from 6 to 9 am. This interval
exhibits the most variable tra onditions b eause of the morning rush hour.
Sine only plaeholder omp onents for the b ehavioral simulator are available, the
sole degree of freedom onsidered here is route hoie. That is, all b ehavioral
asp ets apart from route hoie are retained unhanged in the original plans
generated by MATSim. This setting is motivated in two ways. First, MATSim's
basi approah to route hoie is relatively simple to simulate but at the same
time non-trivial from an estimation point of view, f. Setion 3.2.2.2. Seond,
route hoie an b e generalized to plan hoie by minor modiations to the
96
Figure 5.1: Inner-urban part of Berlin
A time-independent toll of 0.24 EUR/km is harged on the olored links.
original network, f. Setion 3.2.1.1. This suggests that an eetive route hoie
estimator is likely to b e appliable in a more general setting as well.
In all experiments, a time-indep endent toll of 0.24 EUR/km is harged in the
ity enter shown in Figure 5.1, and no toll is harged outside of this area. The
unitless utility of a route
U
is
Vn(U) = −
tt
(U)−
toll
(U)
VOT
n/1
s (5.1)
where tt
(U)
is the travel time on route
U
, toll
(U)
is the toll aumulated along
route
U
, and VOT
n
is individual
n
's value of time in EUR/h. For omparison,
the eet of a 0.24 EUR/km toll is equivalent to a travel time inrease by one
the free-ow travel time given a 12 EUR/h VOT and a 50 km/h sp eed limit.
5.1.1.2 Variable Settings
Combining the invariable settings given ab ove with a partiular VOT denes
a
senario
. For simpliity, it is assumed that all drivers within one senario
have an idential value of time, i.e., VOT
n=
VOT
, n = 1 . . . N
. Clearly, this
setting disregards a multi-agent mo del's prominent advantage of apturing a
heterogeneous driver p opulation. However, the purp ose of these exp eriments
is not to re-iterate the well-known features of a multi-agent simulation but to
investigate an estimator's p erformane in ontrolled onditions. A homogeneous
VOT simplies the setup of the exp eriments and their interpretation. Sine
VOT is an agent-sp ei parameter that is entirely transparent to the estimator,
no oneptual diulty exists in estimating the b ehavior of a p opulation that is
heterogeneous in this regard. Finally, no VOT information is ontained in the
syntheti p opulation available for this dissertation anyway b eause the urrent
MATSim implementation provides no suh information.
97
Dep ending on the partiular mo deling assumptions, a
planning senario
and
a
telematis senario
an b e distinguished onsistently with the terminology
of Setion 1.1.3: If drivers are aware of a reently implemented toll but not yet
of the resulting hanges in tra onditions, the hitherto prevailing equilibrium
onditions are invalidated and a transient phase emerges. This senario an
only be represented by a telematis simulation that does not rely on a (S)UE
assumption. If drivers are aware of the toll but also have learned the resulting
hanges in tra patterns, the transient phase stabilizes again. This senario
an b e addressed by a planning simulation the equilibrium assumption of whih
is approximately satised here.
5.1.2 Simulation and Estimation Logi
The following two subsetions elab orate on the applied simulation and estima-
tion logi. The simulator is desrib ed rst. Sine the estimator wraps around
an existing simulation system, f. Figure 1.2, the simulator is entirely indep en-
dent of the subsequently seleted estimation approah.
5.1.2.1 Simulation
Tra ow dynamis are represented by the mobility simulation desrib ed
in Chapters 2 and 3.1. For b ehavioral simulation, the simple logi outlined
in Setion 3.2.2.2 is applied with minor mo diations. Basially, 10 p erent
of all agents realulate a new route in every iteration. Only pre-trip route
(re)planning is onsidered.
1
The implemented deision proto ol exeutes hoie
set generation and hoie in a delib erative manner, f. Setion 1.3.1.
Whenever an agent starts a trip, it has one already generated route
U
at hand.
This is either the route hosen in the previous iteration or, at the initial iteration,
the route provided in the MATSim plans le. The agent also is aware of the most
reently observed travel times. An alternative route is generated by
randomly
ho osing a VOT from the set
{6,12,18,∞}
(all in EUR/h) and running a time-
dep endent b est path algorithm that maximizes the resulting generalized utility
sp eied in (5.1). The innite VOT serves as a notational proxy for a no-toll
ase sine it eetively eliminates the toll addend from the utility. The newly
alulated route is denoted by
V
. This yields a hoie set of two elements: the
original route
U
and the new route
V
.
The agent then selets from
{U,V}
the route of higher utility based on the sim-
ulated senario's
atual
VOT and the most reently observed tra onditions.
Sine the tra onditions vary from iteration to iteration, this hoie may not
b e optimal in hindsight.
This mo del is hosen beause of its similarity to the original MATSim route
replanning logi. Altogether, a single iteration of this simple DTA simulator
onsists of two steps, and rep eated exeutions of these iterations onstitute a
simulation run:
1
The sole onsideration of pre-trip replanning keeps the mo deling simple. The estimator
itself is appliable to en-trip replanning as well, f. Setion 4.3.1.
98
1. For all agents
n= 1 . . . N
, do: With probability 0.9, maintain
n
's route.
Otherwise, generate an alternative route based on a randomly generated
VOT and the most reently observed travel times, and selet the b etter
one of these two alternatives aording to the senario's atual VOT.
2. Load all agents on the network.
This pro edure an b e applied to simulate b oth a planning and a telematis
senario. The planning senario assumes that drivers learn from iteration to
iteration. If one lo oks at
relaxed
iterations only, i.e., suh iterations where
tra onditions have attained a stable distribution, then an alternative inter-
pretation is that the situation of interest is one where drivers are aware of global
tra onditions. This is realized if route-replanning is based on the previous
iteration's travel times. For a telematis senario, however, it is neessary to run
iterations while drivers remain on their initial level of knowledge. This knowl-
edge is generated b eforehand by running many iterations of a relaxed planning
simulation and saving the travel times of every iteration. These travel times are
then used by replanning travelers in the iterated telematis simulation.
Even this simple simulator exhibits fairly omplex dynamis. Sine an elab orate
analysis of these dynamis is b eyond the sop e of this dissertation, the notion
of a relaxed simulation that reahes stable network onditions is to b e un-
dersto o d informally and only in a given exp erimental ontext. Consequently,
all onvergene statements regarding the subsequently desrib ed simulation-
based estimator are of likewise exp erimental nature.
5.1.2.2 Estimation
The estimator adjusts a
prior senario
to measurements that are observations
from a
true senario
. (Measurement generation is desrib ed further b elow.)
The prior and the true senario only dier in their VOT. The true senario rep-
resents a
syntheti reality
that would in a real-world appliation b e replaed
by reality itself.
At this stage of researh, a real-world test ase would rather obsure than larify
the estimator's working sine (i) no guidelines for its appliation are yet avail-
able, (ii) unontrollable error soures would ompliate an interpretation of the
estimation results, and (iii) only a simulated reality is p erfetly observable for
a omparison to its estimated ounterpart. Furthermore, merely an outdated
Berlin network and driver p opulation are available sine the MATSim researh
eorts shifted towards the ity of Zurih around the b eginning of 2007. This
hange ourred to o late to b e traed by this researh.
The UM estimator is applied in all exp eriments. This is required by the impliit
nature of the b ehavioral mo del. As explained in Setion 3.2.2.2, route realu-
lations based on a previous iteration's travel times mo del a p ereptional error
that do es not b eome observable until the next network loading is exeuted.
Sine this error is generated in hindsight, there is no variability within a single
hoie situation. The AR estimator is generally not appliable to this type of
99
b est resp onse simulation.
2
Furthermore, sine no PS-logit route hoie mo del
is used, only a heuristi appliation of the UM estimator is p ossible. This also
puts its robustness with regard to a b ehavioral prior that is not guaranteed to
b e of PS-logit struture to test, f. Setion 4.3.4.
Sine the UM estimator is tehnially equivalent to the heuristi estimator of
Setion 4.2.2, the following presentation is given in terms of the latter. The
heuristi estimator adds a global utility funtion
Φ
to the individual utility of
every agent, where
Φ
is a similarity measure b etween simulated and observed
sensor data. More preisely, the estimator replaes any driver
n
's original utility
p ereption
Vn(U)
as dened in (5.1) by a mo died utility
Wn(U) = Vn(U) +
hΛ,Ui/µ
where the seond addend is a linearized and saled version of
Φ
. In all
subsequent exp eriments,
Φ
is sp eied by
Φ(X) = −X
aX
k
(ya(k)−ga[x(k)])2
2σ2
(5.2)
where
ya(k)
is a measurement on sensor-equipp ed link
a
in time step
k
and
ga[x(k)]
is its simulated ounterpart. An interpretation of this funtion as the
log-likeliho o d of mutually indep endent normal measurements with idential vari-
anes
σ2
is possible but, in light of the overall heuristi setting, not mandatory.
Φ
is eetively saled by
σ−2
. Sine this multipliation an b e applied either
b efore or after the linearization, it is assumed that the
Λ
values result from a
linearization of
Φ(X) = −Pak(ya(k)−ga[x(k)])2/2
and that the
σ2
parameter
is aounted for afterwards:
Wn(U) = Vn(U) + hΛ,Ui
µσ2.
(5.3)
Only the pro dut of
µ
and
σ2
is relevant to the estimation problem. Sine it
reets the b elief in the prior information represented by the original utility
p ereption
Vn(U)
, it is subsequently represented by a prior weight
w
prior
=qµσ2.
(5.4)
For interpretation, given a unit sale parameter
µ
,
w
prior
is equivalent to a
normal measurement's standard deviation. An exp erimental parameter tuning
approah is adopted for its seletion. This also is likely to b e the ourse of ation
in a real-world appliation [171℄.
The estimation logi approahes a xed-p oint of the
Λ
values by means of the
SA algorithm desrib ed in Setion 4.1.3. This pro edure iterates b etween a
linearization of (5.2) and an iteration of the tra simulator. That is, in every
iteration of the estimator, 10 perent of all departing agents replan based on
the most reently obtained utility orretions, a single network loading is run,
and the utility orretions are immediately updated. The omplete estimation
logi is given b elow:
2
Sp eaking in terms of the partiularly hosen model: The route hoie set is generated
based on a randomized VOT one p er iteration, but it is xed throughout that iteration.
That is, rep eated b est resp onse hoies within a single iteration invariably yield the same
result.
100
1. Initialization.
(a) Set iteration ounter
m= 0
.
(b) Fill
¯
Λ(m)
(estimate of
Λ
xed p oint) with all zeros.
2. Simulation.
(a) For all
n= 1 . . . N
, do with probability 0.1:
i. Choie set generation. Generate an alternative route based on a
randomly generated VOT and the most reent travel times.
ii. Choie. Evaluate
Wn(U) = Vn(U) + h¯
Λ(m),Ui/w2
prior
instead of
Vn(U)
when seleting
U(m)
n
.
Vn(U)
is evaluated based on the
prior senario's atual VOT.
(b) Load
U(m)
1...U(m)
N
on the network and obtain
X(m)
.
3. Linearize
Φ(X(m))
and obtain
Λ(m)
.
4. Up date
¯
Λ(m+1) =m
m+ 1 ¯
Λ(m)+1
m+ 1Λ(m)
.
5. If another iteration is desired:
(a) Inrease
m
by one.
(b) Goto step 2.
Note that the hoie set generation is based on the
original
utility
Vn
and a
randomized
VOT, whereas the hoie is based on the
modied
utility
Wn
and
the prior senario's
atual
VOT. This ensures that every one in a while the
hoie set ontains a route that is onsistent with the true senario's VOT.
3
The
question thus b eomes in how far the estimator, given the ab ove set of b ehavioral
alternatives but only a limited number of measurements, an pull the system
away from the wrong VOT of the prior senario towards the orret VOT of
the true senario.
If there are no measurements, the
Λ
o eients are invariably zero and the ab ove
algorithm merely rep eats steps 2a and 2b. That is, it funtions as a simulator
that, up on stabilization in relaxed onditions, pro dues a sequene of draws from
the b ehavioral prior distribution. As measurements b eome available, nonzero
Λ
values result, and the estimator stabilizes in dierent relaxed onditions.
Every iteration then generates a draw from the b ehavioral p osterior given the
partiular prior senario and the available measurements from the true senario.
Tehnially, the estimation problem is to identify a xed point of the
Λ
o ef-
ients. Sine the mapping from
Λ
on itself is eetively from
Λ
on
X
on
Λ
,
f. Figure 4.1, the existene of a
Λ
xed p oint indiates the existene of a
X
xed p oint, and vie versa. This justies the exlusive evaluation of the readily
interpretable system states
X
to monitor the estimator's onvergene, as it is
desrib ed in the next setion.
3
There is no guarantee that running a b est path algorithm diretly on mo died link utilities
ever pro dues a likewise realisti alternative. Setion 5.4 elab orates on this matter.
101
Figure 5.2: Exemplary sensor loations
50 automatially seleted sensor lo ations. One ow sensor is lo ated in the enter of
eah olored link.
5.1.3 Sensor and Validation Data
The estimator utilizes a limited amount of ow measurements as sensor data.
The estimation results are validated based on network-wide o upany informa-
tion.
5.1.3.1 Sensor Data
Flow measurements, i.e., tra ounts at road ross-setions p er time interval,
are used in all exp eriments as synthetially generated
sensor data
. The term
measurement data
is equivalently used. All suh data is averaged in 5 minute
time bins.
For every estimation exp eriment, 50 sensor lo ations are seleted based on a
omparison of the tra onditions in the aording prior and true senario.
The lo ations are automatially hosen by a simple to ol that prefers links on
whih the average ow dierene b etween b oth senarios is largest and at the
same time seeks to maintain indep endent measurement lo ations. Sensor lo-
ations are hosen for all senarios individually in order to provide equally
advantageous preonditions for b etter omparability. An example of suhlike
generated loations is given in Figure 5.2. The true tra onditions utilized
by this pro edure are of ourse unknown in a real-world appliation, where,
however, presp eied sensor lo ations an b e exp eted to b e available.
The mapping from driver behavior on tra ows is nonlinear. In partiular,
the intermediate mapping from tra densities on ow rates is ambiguous in
that every non-maximum ow an b e explained by two dierent densities, f.
Setion 2.2. Sine the estimation is based on rep eated linearizations, suh non-
linearities inrease the danger of lo al onvergene. Therefore, an additional
102
soure of information is employed. Even a simple single-lo op detetor do es not
only measure ow rates but also the fration of time it is overed by a vehi-
le. This information is likely to b e to o noisy to provide immediately useful
tra density information, but it do es allow to distinguish free and ongested
tra onditions [49℄. The estimator uses this information in its linearization
step where it reognizes that in unongested onditions the log-likelihoo d of any
measurement is only sensitive to the upstream tra situation and in ongested
onditions it is only sensitive to the downstream situation.
5.1.3.2 Validation Data
One may argue that an appraisal of the estimation quality should be diretly
based on routes. However, sine every agent may ho ose any yle-free route to-
wards its destination, it is unlikely that an estimated and a true route oinide.
In priniple, the measure of route overlap prop osed in [148℄ is appliable here.
Still, the ontinuous variability of the simulated tra onditions and of the
resulting routes ompliates suh a omparison, and a more viable validation
approah is at hand: In the onsidered mo del, simulated ows result deter-
ministially from marosopi system states, whih in turn are onsequenes
of mirosopi driver behavior, f. Setion 3.1.3. Marosopi link o upan-
ies thus onstitute intermediate states that are easy to proess and interpret.
4
Sine the route hoie model is based on travel times whih are deterministi-
ally dep endent on marosopi link states, an estimator that repro dues link
states well is likely to also generate realisti routes. Partiularly, the b ehavioral
mo del plaeholder is by design suiently restrited to unequivo ally asribe
any systemati hange in aggregate tra onditions to the b ehavioral aspet
of toll-avoidane.
In onsequene, network-wide o upany information, i.e., the average numb er
of vehile units on every link in every 5-minute time bin, is used as the
valida-
tion data
based on whih global tra onditions are ompared. More general
exp eriments are likely to also all for more p owerful behavioral monitoring to ols,
whih onstitutes a researh question in its own right.
5.1.3.3 Quantitative Error Measures
The notion of a
run
is subsequently used as a generi term for b oth a simulation
run and an estimation run. The dierene of a run to a
referene data set
is
evaluated in terms of a ro ot mean square error measure
RMS
(m)
z[
run
] = sPa∈APk(z(m)
a(k)−z
ref
a(k))2
K|A|
(5.5)
where
z(m)
a(k)
is the onsidered tra harateristi (ow, o upany) of the
studied run in iteration
m
on link
a
in time bin
k
, and
z
ref
a(k)
is the aording
referene value.
K
is the total numb er of time bins and
A
is the set of links for
4
The hitherto used notion of o upany as the number of vehile units lo ated in a ell or
link is not to b e onfused with the ommon notion of an indutive lo op's o upany as the
fration of time it is overed by a vehile. The latter is not employed here.
103
whih tra harateristis are evaluated. [run℄ is a shortut for the evaluated
run. Unique referene data sets are used in all planning exp eriments and in
all telematis exp eriments resp etively. Whenever the dep endeny of RMS on
iteration ounter
m
is omitted, the last RMS value in a presp eied sequene
of iterations is referred to.
It is frequently required to ompare a run's (reursively) averaged harateristis
¯z(m)
a(k) =
z(m)
a(k)m < m0
1
m−m0+ 1
m
X
m′=m0
z(m′)
a(k)m≥m0
(5.6)
to the referene data, where
m0
is always hosen large enough to ensure that
the onsidered run reahes a stable distribution of network onditions before the
averaging starts. This allows for the denition of an additional error measure
RMSA
(m)
z[
run
] = sPa∈APk(¯z(m)
a(k)−z
ref
a(k))2
K|A|,
(5.7)
where the only dierene to RMS
z
is that
z(m)
a(k)
is now replaed by the average
value
¯z(m)
a(k)
.
The following partiular error measures are used.
•
The
measurement error
RMSA
q
is an instane of (5.7) that represents
the deviation of an estimation run from its measurement data set. That is,
the referene data used here is idential to the measurement data used for
estimation. Consequently, only the ow rates at the presp eied 50 sensor
lo ations are evaluated. Note that the measurement error is basially a
saled version of
√−Φ
, f. (5.2). Its unit is veh/h, whih is subsequently
omitted for brevity.
•
The
validation error
RMS(A)
x
is an instane of (5.5) or (5.7) that rep-
resents the deviation of a simulation run or an estimation run from its
validation data set. At this, it ompares the o upanies on all links in
the network. Its unit is veh, whih also is subsequently omitted.
5.2 Planning Exp eriments (Equilibrium Situation)
A planning-like setting is onsidered rst. SUE onditions are mo deled by pro-
viding global knowledge about the previous iteration's tra onditions to all
replanning agents in the iterative DTA pro edure desrib ed in Setion 5.1.2.1.
All exp eriments use sensor data from a true senario that is based on one par-
tiular VOT, whereas the prior senario assumed by the estimator is based on
a dierent VOT.
The exp eriments given here examine the logial orretness and overall preision
of the estimator. Sine omputational p erformane is not of primary onern
in an oine planning appliation, its investigation is p ostp oned to Setion 5.3
where a telematis ase study in simulated online onditions is desrib ed.
104
Figure 5.3: RMS
x
and RMSA
x
[6 EUR/h VOT simulation℄
Three simulation runs of 500 iterations eah are onduted in order to investigate
the stability of the 6 EUR/h VOT senario. The utuating RMS
x
values eetively
represent the Eulidean distane b etween the referene data and the simulated o u-
panies of a partiular iteration. The reursive state averaging is turned on after 100
iterations suh that a smo oth RMSA
x
urve branhes o eah RMS
x
urve.
5.2.1 Senario Generation
Given the ab ove overall settings, one planning simulation is run for a senario
without toll, and three further simulations are run for toll-senarios with VOTs
of 6, 12, and 18 EUR/h. Eah simulation is exeuted for 500 iterations. These
initial runs are subsequently denoted as the no-toll and the 6 (12,18) EUR/h
VOT
referene simulations
of their resp etive planning senarios. Link ows
and o upanies are averaged over the last 400 iterations of eah referene sim-
ulation aording to (5.6). These average values onstitute the referene data
sets for all RMS and RMSA error measures given in the subsequent planning
exp eriments, f. Setion 5.1.3.3.
5.2.1.1 Investigation of Senario Stability
To test the robustness of this set-up, another three simulations are run for every
senario. They are ompared to their resp etive referene senario by traking
the validation errors RMS
x
and RMSA
x
over 500 iterations, as shown in Figures
5.3 through 5.6.
All exp eriments start with an idential plans le. This results in dierent tran-
sients during the rst iterations. Sine these transients represent no relaxed
network onditions, the reursive state averaging is turned on not b efore it-
eration 100 where a RMSA
x
urve branhes o eah RMS
x
urve. Sine this
branhing in onjuntion with muh smoother dynamis is harateristi for all
RMSA
x
urves, they are not expliitly lab eled in the plots.
The RMSA
x
urves approah small values when ompared to their RMS
x
oun-
terparts. This indiates that all simulations for a partiular VOT attain similar
105
Figure 5.6: RMS
x
and RMSA
x
[no-toll simulation℄
Three simulations of the no-toll senario. See Figure 5.3 for further explanations.
average system states. The RMS
x
urves stabilize at a onstant degree of vari-
ability. A visual insp etion shows a positive auto-orrelation within eah urve.
This results from the simulation logi that invariably opies 90 p erent of all
routes from one iteration to the next. Altogether, the network states exhibit a
utuating and p ossibly yling b ehavior. Sine no systemati drift is observed,
onvergene towards a stable state distribution annot b e disproved.
All RMS urves are lo ated ab ove their RMSA ounterparts. However, this
observation do es not prove a systemati dierene between the average system
states and the single-iteration draws. It rather is a onsequene of the hosen
error measures, and the same RMS vs. RMSA onstellation would result even
if the relaxed system states were p erfetly normally distributed: The surfae
of an
(n+ 1)
-dimensional sphere with radius
r
is prop ortional to
rn
. The
probability that a single network state is simulated
r
distane units away from
its exp etation therefore results from an integration of its p.d.f. over a domain
the size of whih is prop ortional to
rn
. Sine the referene data used in RMS(A)
onsists of average network harateristis that approximate this exp etation, a
small RMS value is as unlikely to o ur as a small
r
value, whereas vanishing
RMSA values merely result from the law of large numbers.
5.2.1.2 Measurement and Validation Data Generation
An aurate generation of the syntheti measurements for a single day requires
to take one relaxed iteration of the true senario, to extrat the ow data at
all sensor lo ations, and to randomly disturb this data aording to a distribu-
tional assumption ab out the measurement error. Based on this information, the
(planning) estimator is run with the goal to repro due the true distribution of
tra onditions. In onsequene, an exhaustive validation pro edure must
ompare two full distributions of tra onditions.
Within the sop e of this work, distributions are ompared in terms of their
107
exp etations. The similarity of an estimated and a true distribution of net-
work onditions an thus b e quantied by an RMSA error measure. This error
measure is a random variable itself sine it dep ends on the partiular draw of
measurement data
Y
that is used for estimation, i.e., RMSA
=
RMSA
(Y)
. A
reliable appraisal of the estimation quality would therefore require to gener-
ate a large number of measurement data sets
Y
and to run the estimator for
eah of these sets individually. An exp eted error E
{
RMSA
(Y)}
ould then b e
identied by averaging RMSA
(Y)
over all exp eriments.
Sine strong variability an b e observed in the simulations, many omputa-
tionally demanding exp eriments would b e needed to identify the estimator's
exp eted p erformane.
5
Even if this eort was shouldered, the validity of the
resulting assessment would b e limited by that of the deployed mo del plaehold-
ers. These reservations motivate a less rigorous yet omputationally more viable
approah.
A single, most representative measurement data set is used for eah true se-
nario. The stability analysis of Setion 5.2.1.1 shows that rep eated simulations
of a partiular senario onverge to similar average network states. The initially
generated referene data sets for eah senario are therefore used as sensor and
validation data in all planning exp eriments. Averaging the data instead of av-
eraging the evaluation results is equivalent to the learly idealized assumption
that E
{
RMSA
(Y)} ≈
RMSA
(
E
{Y})
is a feasible approximation.
No additive sensor noise is simulated sine only its zero exp etation app ears in
E
{Y}
. This underlines the idealized exp erimental setting sine the true level of
sensor noise will in reality ertainly impair the estimation p erformane. How-
ever, sine there is no guarantee that the average of many physially p ossible
system states is itself physially feasible, a systemati error may b e intro dued.
These asp ets must b e aounted for when interpreting the estimation results.
This simpliation may even b e realisti in a setting where the sensor data
available for planning purp oses has b een averaged over many days. However,
the eetive motivation for this approah is to limit the degrees of freedom that
need to be exp erimentally investigated. One should reall that the purp ose of
these exp eriments is to demonstrate the estimator's logial orretness. One
this is ahieved, suient oneptual bakground is provided in Chapter 4 for
more extensive investigations in likewise more realisti exp erimental settings.
5.2.1.3 Comparison of Senarios
Figure 5.7 provides an impression of the dierene b etween the syntheti reality
on the one hand and the prior senario assumed later during estimation on the
other hand. It ontains six satterplots that ompare the ow and o upany
data of the 12 EUR/h VOT referene simulation to the 6 EUR/h VOT, the 18
EUR/h VOT, and the no-toll referene simulation.
The rst olumn ompares the referene ow rates and the seond olumn om-
pares the referene o upanies. All satterplots ontain data p oints for
al l
links
5
Reent exp erimental results milden this onern. However, sine these results were ob-
tained to o late to b e aounted for in this dissertation, they are only indiated in this and a
few subsequent fo otnotes.
108
Figure 5.7: Satterplots for omparison of planning referene simulations
The satterplots ompare data from the 12 EUR/h VOT planning ref erene simulation
(on the ordinate) to the other planning referene simulations (on the absissa). The
rst olumn ompares ow rates and the seond olumn ompares o upanies. All
satterplots ontain data p oints for
al l
links in the network. The data p oints apply to
the simulation time interval from 8:30 to 8:35 and represent average values over 400
iterations.
109
in the network. That is, the ow satterplots ontain more information than the
RMSA
q
measurement error, whih only aounts for data at sensor lo ations.
The measurement error indiates to what degree the estimator is able to reon-
strut available sensor data, whereas the satterplots allow for a network-wide
omparison of tra onditions.
All data p oints apply to the simulation time interval from 8:30 to 8:35. At
rst glane, the deviations appear mo derate in onsideration of the broad range
of VOTs. However, reall that all referene data sets are averaged over 400
iterations. An insp etion of the simulation dynamis in Figures 5.3 through 5.6
shows that variability is muh larger without averaging. Using average data
allows to asrib e all p ereptible deviations in the satterplots to systemati
auses.
The ow satterplots in the left olumn give an impression of the amount of
information eetively available to the estimator. The stronger the ow devi-
ations between two senarios the more useful are ow measurements to adjust
one senario to another. Vie versa, if two senarios dier only slightly in their
ows, the estimator has only little information at hand. In all plots, the ows
exhibit no distint bias in that they are sattered unsystematially around the
main diagonal. The reason for this is that route hoie is the only b ehavioral
degree of freedom: Every driver who bypasses the downtown area invariable
drives through the inverse of that area, and vie versa, suh that the ows an
merely b e reallo ated among links.
The seond olumn ontains o upany satterplots. This typ e of data also
denes the RMSA
x
validation error. The degree of variability among dierent
senarios follows the same order as for the ows.
6
However, systemati dier-
enes b etween the senarios an now be observed. Sine the toll is not designed
to maximize tra throughput, it auses inreased ongestion outside the ity
enter. This eet b eomes more pronouned for smaller VOTs, whih mo del a
greater b ehavioral sensitivity to the toll. The nonlinear ongestion eets are
reeted in unsymmetrial plots: The p ositive eet of the toll (less vehiles
downtown) is not as pronouned as its negative ounterpart (more vehiles on
the bypass roads). Suh an eet an b e justied if there are other motives than
ongestion relief for the intro dution of the toll. One should keep in mind that
this is a syntheti senario with no ambition to evaluate road priing strategies
themselves.
5.2.2 Exp erimental Results
12 EUR/h is a reasonable a priori guess for an average VOT. The estimator
therefore adjusts a 12 EUR/h VOT prior senario to the referene measure-
ments of a true no-toll senario, a true 6 EUR/h VOT senario, and a true
18 EUR/h VOT senario. Every estimation run starts with a plans le that is
6
A prominent outlier at o ordinates
(312/175)
in the 6 vs 12 EUR/h VOT o upany
satterplot an b e observed. This is the western segment of Frankfurter Allee, leading
immediately into the toll zone. It has 3 lanes and is almost 3 kilometers long. The lower the
value of time the more drivers try to divert at at the downstream end of this road into the
inreasingly ongested bypasses and ause the observed spillbak.
110
drawn from the 12 EUR/h prior distribution. That is, in the absene of measure-
ments, the estimator immediately draws from the prior, and if measurements
are available, all transients towards the p osterior an be unequivoally asrib ed
to the measurements. Exp eriments with various prior weights
w
prior
as dened
in (5.4) are onduted in order to investigate the estimator's robustness against
sub optimal parameter settings. Three estimation runs are evaluated in every
onguration in order to inrease the statistial reliability of the results.
7
5.2.2.1 Desription of Results
Figure 5.8 shows the resulting error measures over dierent
w
prior
values for
sensor data generated from the 6 EUR/h VOT, the 18 EUR/h VOT, and the
no-toll referene senario. These settings are subsequently denoted as no-toll
estimation and 6(18) EUR/h VOT estimation. Measurement errors RMSA
q
are given in the rst olumn and validation errors RMSA
x
are shown in the
seond olumn. For omparison, error measures for the 12 EUR/h VOT refer-
ene simulation and for the additional three simulation runs onduted in the
stability analysis of Setion 5.2.1.1 are also given in eah diagram. They are
equivalent to running the estimator without sensor input. For ease of ompari-
son, they are re-drawn over every onsidered
w
prior
value in red olor. The three
estimation results p er
w
prior
value are drawn in blue. All exp eriments are run
for 250 iterations. Flow and oupany averaging is started after a settling time
of 50 iterations.
All results are fairly stable in that there is limited variability among rep eated
runs. Often enough, the dots lie on top of eah other and annot be distin-
guished. Repro duible onvergene is a desirable and not at all self-evident
feature for a nonlinear estimator. In these exp eriments, it an b e observed with
go o d preision. However, this result is at least partially owed to the use of
a representative measurement data set in all exp eriments for a partiular true
senario. Another general observation is that the o upany error levels are
relatively small. This is a onsequene of the network-wide p oint of view whih
aounts for many links in the p eriphery that are hardly aeted by the toll.
The rst olumn of Figure 5.8 shows that the measurement error RMSA
q
de-
reases monotonously with
w
prior
. This is plausible: the smaller the b elief in the
b ehavioral mo del the more weight is put on measurement repro dution. The
results dier in the previously hypothesized way in that a large dierene b e-
tween ows in the prior and the true senario provides substantial information
that an b e failitated for estimation, whereas smaller ow dierenes result in
a less foused searh: The 12 EUR/h VOT prior senario is most dierent from
the no-toll senario, less dierent from the 6 EUR/h VOT senario, and least
dierent from the 18 EUR/h VOT senario. Aordingly, the greatest estima-
tion improvements over a plain simulation of the prior are 86%, 63%, and 58%,
resp etively.
7
All results apart from the p erformane b enhmarks of Setion 5.3.3.3 are obtained on a
omputing luster where the no des are equipp ed with AMD 2.6 GHz Opteron pro essors and
have at least 2 GB of RAM. On suh a no de, the omputing time of an estimation run as
desrib ed in this setion is in the order of one day.
111
Figure 5.8: Result overview for planning exp eriments
The left olumn shows measurement errors RMSA
q
and the right olumns shows val-
idation errors RMSA
x
over dierent
w
prior
values for a true 6 EUR/h VOT senario,
a true 18 EUR/h VOT senario, and a true no-toll senario. The three estimation
results p er
w
prior
value are represented by blue dots. For omparison, the error mea-
sures for four plain simulations of the 12 EUR/h VOT prior senario are represented
by red dots. All exp eriments are run for 250 iterations. Flow and o upany averaging
started after a settling time of 50 iterations.
112
The seond olumn of Figure 5.8 shows a non-monotonous relation b etween
w
prior
and the validation error RMSA
x
. As
w
prior
grows, the measurements'
inuene vanishes and the estimation quality graefully deteriorates towards
that of a plain simulation . However, as
w
prior
dereases, a minimum value
of RMSA
x
is invariably enountered, after whih a further derease of
w
prior
results in an inreased validation error. The attained minimum RMSA
x
value
reets the estimator's ability to spatiotemp orally extrap olate the available ow
measurements. The RMSA
x
improvements follow the same order as the RMSA
q
results. When ompared to the 12 EUR/h VOT prior senario, the estimator
ahieves a 48% improvement for the true no-toll senario at
w
prior
= 0.72
or
1.44
, a 36% improvement for the true 6 EUR/h VOT senario at
w
prior
= 2.88
,
and even for the subtle true 18 EUR/h VOT senario a 20% improvement an
b e observed at
w
prior
= 2.88
. The last improvement is partiularly noteworthy
sine fairly little dierene between the 12 and the 18 EUR/h VOT senario
an be identied in Figure 5.7 at all. This indiates that the estimator is quite
preise in that it reognizes even suh subtle dierenes. Reall that all of these
extrap olation results are obtained using only 50 measurement lo ations out of
altogether
2 459
links.
Figures 5.9 and 5.10 provide ow and oupany satterplots that result from
the b est onguration in eah exp erimental setting. Here and subsequently,
the b est onguration orresp onds to the
w
prior
value that yields the smallest
validation error on average. From the aording three estimation runs, the
seond b est is hosen for illustration. The rst olumn of eah gure rep eats
the data obtained during the preparatory simulations, f. Figure 5.7, and the
seond olumn shows the orresp onding estimation results. All data p oints are
averaged over many relaxed iterations suh that all dierenes b etween left and
right olumn an b e asrib ed to a systemati eet of the estimator. Overall,
the visual impression arms the quantitative error measures. Reall that the
previously given RMSA
q
values only aount for the 50 sensor loations, whereas
the ow satterplots ontain data p oints for all links in the network.
5.2.2.2 Disussion of Results
Three explanations an be given for the inreased validation errors at small
w
prior
values. The rst is over-tting. Even if the representative measurements
are not orrupted by sensor noise, their averaging may result in an inonsisteny
with the dynamis of the underlying nonlinear tra ow mo del.
8
The seond
explanation is under-determinedness in ombination with nonlinear dynamis.
There may b e many global tra situations that repro due the measurements
equally well. As the b ehavioral mo del's eet vanishes with dereasing
w
prior
,
insuient b ehavioral information is available as a guidane towards a plausi-
ble solution, and the estimator gets lo ally stuk. This eet is p ossible even
though the ow sensors provide supplementary information ab out free and on-
gested tra onditions sine this data is still insuient to uniquely dene the
tra onditions in the further surroundings of a sensor. Finally, a small
w
prior
eetively ats like a large gain on the log-likelihoo d funtion, and the steepness
of this funtion an have a negative eet on the onvergene of the underlying
8
The reent exp erimental results onrm this hyp othesis.
113
Figure 5.9: Comparison of true and estimated ows (planning)
The rst olumn rep eats the preparatory ow satterplots of Figure 5.7. The se-
ond olumn shows the aording estimation results where the referene ows (on the
absissa) are ompared to their estimated ounterparts (on the ordinate). That is,
every row ontains one satterplot that ompares a partiular true senario to the
prior senario, and it ontains another satterplot that ompares the true senario
to the estimation result. These plots already represent average values suh that all
dierenes b etween left and right olumn an b e asrib ed to a systemati eet of the
estimator.
114
Figure 5.10: Comparison of true and estimated o upanies (planning)
The rst olumn rep eats the preparatory o upany satterplots of Figure 5.7. The
seond olumn shows the aording estimation results where the ref erene o upanies
(on the absissa) are ompared to their estim
ated ounterparts (on the ordinate). See
Figure 5.9 for further explanations.
115
Figure 5.11: RMS
x
and RMSA
x
[6 EUR/h VOT estimation℄
Validation errors over 250 iterations for the three b est exp eriments with a true 6
EUR/h VOT senario. RMS
x
eetively represents the Eulidean distane of the 6
EUR/h VOT referene o upanies to the estimation results of a partiular iteration.
The reursive state averaging is turned on after 50 iterations suh that a smo oth
RMSA
x
urve branhes o eah RMS
x
urve.
SA xed p oint searh algorithm. In either ase, a trustworthy behavioral mo del
that alls for a suiently large
w
prior
avoids the problem.
Rephrasing this observation in more general terms, a go o d state repro dution
dep ends ruially on data and mo deling quality, whih annot b e omp ensated
for by the estimation logi itself. The measurements need to ontain suient
information for a spatiotemp oral extrap olation, and the b ehavioral simulator
must b e struturally orret in that it generates hoies that are ompatible
with the measurements.
Overall, the ahieved measures of estimation quality must b e onsidered in light
of the idealized setting in whih they were obtained. The use of representative
measurement data that is free of sensor errors is an idealization. In a real-world
appliation, the over-tting of ertainly existing measurement errors must b e
avoided. This is likely to require larger
w
prior
values than used here and would
onsequently yield a redued measurement and validation data t. However, it
an b e onluded that the estimator p erforms struturally orret and that the
estimation results in a sp ei appliation will mainly dep end on the available
data and mo deling quality.
5.2.2.3 Estimation Dynamis
Finally, a loser lo ok at the estimation dynamis is provided in Figures 5.11
through 5.13 for the 6 EUR/h estimation, the 18 EUR/h estimation, and the
no-toll estimation. Eah gure shows all three RMS
x
and RMSA
x
tra jetories
for the resp etive b est
w
prior
onguration over 250 iterations. Most RMS
x
tra-
jetories osillate fairly stable in the temp orally auto-orrelated manner known
116
Figure 5.12: RMS
x
and RMSA
x
[18 EUR/h VOT estimation℄
Validation errors over 250 iterations for the three b est exp eriments with a true 18
EUR/h VOT senario. See Figure 5.11 for further explanations.
Figure 5.13: RMS
x
and RMSA
x
[no-toll estimation℄
Validation errors over 250 iterations for the three b est exp eriments with a true no-toll
senario. See Figure 5.11 for further explanations.
117
from the preparatory simulation runs. The eventual outliers, partiularly the
blue urve in Figure 5.11, may b e due to a yet imperfetly relaxed p osterior
distribution. However, similar p erio ds of disarranged dynamis an also b e
found in the preparatory simulations, where no estimation was involved.
All RMSA
x
urves stabilize well in the available 250 iterations. Their sp eed and
reliability of onvergene inreases as the prior and the true senario beome
more similar. The 18 EUR/h VOT estimation onverges fastest, the 6 EUR/h
VOT estimation is somewhat slower yet still very reliable, and the no-toll esti-
mation exhibits the least onsistent onvergene b ehavior. This may result from
the fat that the more distant prior and true senario are the longer the estima-
tor's way through state spae b eomes. In nonlinear onditions, the hane of
branhing o towards dierent lo al solutions is likely to inrease as this way
gets longer.
Altogether, the estimator onsistently generates distint state reonstrution
improvements. It extrats the relevant information out of limited ow mea-
surements even for very subtle dierenes b etween prior and true senario. Its
ability to funtion in the planning-like setting given here shows its appliabil-
ity in onjuntion with a non-deterministi, equilibrium-based dynami tra
simulator.
5.3 Telematis Exp eriments (Non-Equilibrium Sit-
uation)
The seond half of this hapter applies the prop osed estimator in onjuntion
with a telematis model that replaes the hitherto assumed SUE onditions by
an assumption of imperfetly informed drivers. This has a signiant inuene
on the tra onditions when ompared to the planning senario, and the es-
timator has, even under strit running time onstraints, a substantially more
distint eet in this setting.
Exp eriments are onduted in oine and simulated online onditions, f. Setion
1.1.3. In oine onditions, a set of b eforehand olleted measurement data is
pro essed en blo k. In a telematis ontext, this is useful for the ex p ost
analysis of a partiular day. The online estimator runs in a rolling horizon
mo de where the estimation of the tra state for a ertain p oint in time has
only measurements from earlier times available. This setting is harateristi
for a ontinuous tra monitoring problem. The exp eriments in simulated
online onditions allow to investigate the estimator's real time apabilities and
to onlude ab out the senario size its urrent implementation an handle.
5.3.1 Rolling Horizon Estimation
A rolling horizon logi is implemented that runs the estimator in simulated
online onditions. The time p erio d of investigation still is 6 to 9 am. While one
iteration of an oine estimator failitates all measurements from this interval
at one, online onditions imply that the measurements beome available bit by
bit as the simulated real time pro eeds.
118
The online estimation starts at 6:30 simulated real time. Only measurements
until this moment are available. The estimator iteratively adjusts the simulated
driver b ehavior to these measurements aording to the by now established es-
timation logi of Setion 5.1.2.2. During this rst
estimation p erio d
, only a
simulation from 6:00 to 6:30 is iteratively adjusted. After a presp eied numb er
of iterations, the simulated real time is advaned to 6:35, the most reent simu-
lation is ontinued until 7:00 to evaluate the estimator's preditive apabilities,
the measurements from 6:30 to 6:35 beome available, and the next estimation
p erio d from 6:05 to 6:35 b egins. All driver b ehavior until 6:05 is now xed
aording to the last iteration of the previous estimation p erio d.
It is noteworthy that suh a simulation logi is attrative not only for telem-
atis purp oses in online onditions. Being able to iterate ritial time intervals
more frequently than others allows to deploy omputational resoures in a more
fo used way. This also appears useful during the rst iterations of a planning
simulation where the system is far away from an equilibrium. An eventual
sequene of full planning iterations eliminates the arued tendeny of lo al
onvergene. The danger of imp erfet onvergene also needs to b e aounted
for in online estimation and alls for the more elab orate disussion given next.
In rolling horizon estimation, behavior is adjusted only within a limited estima-
tion p erio d that ends at or shortly b efore the urrent p oint in time. As time
pro eeds, this estimation p erio d is also shifted. In the subsequent p erio d, all
driver deisions that have fallen out of the estimation time window are kept xed
at their last values. This is neessitated by the estimation window's onstant
length, whih in turn is enfored by the real time requirement of a onstant
alulation time per estimation p erio d. Sine the estimator ontinues to adjust
b ehavior to measurements, it may hange agent deisions within the given es-
timation p erio d in an attempt to omp ensate for imp erfet estimates at earlier
times.
The problem of sub optimal rolling horizon estimation has already been inves-
tigated for tra monitoring problems with aggregate mo dels [23℄. Sine an
individual-level analysis is pursued here, a b ehaviorally more desriptive point
of view is adopted. The question arises to what degree it is feasible to substitute
the b ehavior of dierent travelers when mathing sensor data without aumu-
lating inorret b ehavioral estimates from one estimation p erio d to the next.
Feasibility is not to b e onfused with individual-level realism no real traveler
aounts for what others do and ompares it to tra ounts. It rather means
that the learly sub optimal b ehavioral preditions for agents that omp ensate
for imp erfet estimates of earlier p erio ds still result in future tra onditions
that are more realisti than an a priori guess without estimation. For example,
distorting the behavior of a few travelers at a ritial time and lo ation in the
network might prevent an unrealisti gridlo k in the simulation. This also pre-
vents the likewise unrealisti reations of many other agents to this gridlo k. In
onsequene, agents that replan in later estimation p erio ds do so in more real-
isti onditions and thus with more realisti results even if no measurements
are aounted for in these later p erio ds.
It is worthwhile to adopt a more formal view on this matter. The b ehavioral
p osterior
P(U1...UN|Y)∝l(U1...UN|Y)P(U1...UN)
(5.8)
119
diers from its prior
P(U1...UN)
only b eause of the information ontained in
the measurement likeliho o d
l(U1...UN|Y)
, f. (4.19) and (4.24). Fixing the
b ehavior of some agents at unreasonable values degrades the estimation quality
by means of this likeliho o d.
This eet an b e substantially mildened by the b ehavioral simulator itself. A
hameleoni b ehavioral prior that admits even highly unrealisti ations with a
low yet non-zero probability is likely to be inappliable in onjuntion with a
sub optimal estimator. If, in sub optimal onditions, the likelihoo d is badly ap-
proximated, the hoie probabilities of implausible ations may b e exessively
inreased. However, if the b ehavioral mo del simply do es not generate implau-
sible ations, i.e., if implausible hoies are seleted with zero probability, no
Bayesian estimator an ever generate a p ositive hoie probability by mere mul-
tipliation in fundamental relation (5.8). The b ehavioral mo del plaeholder used
here is robust in this regard sine it generates alternative routes only based on
reasonable VOT variations. Its simpliity prevents it from ever generating a
strange route that may even b e seleted during estimation b eause of a po or
likeliho o d approximation.
A omputational impliation of these observations relates to the fat that the
estimator linearizes the log-likeliho o d. If the likeliho o d is impreise, there is
little meaning in running a large number of iterations p er estimation p erio d
in order to nally draw from a p osterior that is based on an utmost preise
linearization of the aording log-likeliho o d. The exp eriments of Setion 5.3.3.2
provide more insight into this issue.
5.3.2 Senario Generation
5.3.2.1 Simulation of Imp erfetly Informed Drivers
The rst day after the implementation of the toll is simulated. In this set-
ting, drivers are aware of typial travel times without toll and of the toll itself.
However, they have not yet learned the alterations in tra onditions that
result from other travelers' hanged b ehavior in resp onse to the toll. Suhlike
imp erfetly informed drivers are simulated in the following way.
1. A planning simulation without toll is run. When the simulation attains
relaxed onditions, time-dep endent travel times for all links are written
to le over a long sequene of iterations. The travel time distribution
aptured by these les is used in all subsequent experiments as a repre-
sentation of drivers'
memory
of the no-toll situation.
2. When running the telematis simulation, this sequene of les is pro-
vided to pre-trip replanning travelers instead of the last iteration's travel
times. The travelers base their routing deisions on this memory, plus the
(known) toll. This allows to run the simulation in an iterative manner
and to maintain variability in the tra onditions while avoiding a learn-
ing eet that results if atually simulated travel times are fed bak for
replanning.
120
Figure 5.14: RMS(A)
x
[no-toll planning/telematis simulation℄
The red urves show RMS(A)
x
[no-toll planning simulation℄ and the blue urves show
RSM(A)
x
[no-toll telematis simulation℄ over 500 iterations. The validation data from
the no-toll referene planning senario is used as referene data in all error measures.
Sine the simulations start with an already relaxed plans le, the reursive state av-
eraging is turned on from the very rst.
For estimation, the overall logi of Setion 5.1.2.2 is maintained, only that re-
planning is now based on the previously generated driver memory. The only
strutural dierene between a prior and a true telematis senario is a dierent
VOT. Sine every estimation starts with a plans le that is drawn from a sta-
ble simulation of its resp etive prior senario, all transients during estimation
reet the transition from the prior to the estimated p osterior distribution.
5.3.2.2 Investigation of Senario Stability
Figure 5.14 shows, in red olor, the RMS
x
and RMSA
x
urves for 500 iterations
of a planning simulation in the no-toll ase when ompared to the referene data
for that senario. Sine these iterations start from an already relaxed plans le,
the reursive state averaging is turned on from the very rst. Three further
urve pairs are drawn in blue. They result from an idential set-up as the rst
run, only that the travel times on whih replanning is based are now taken from
the memory les that were written during the rst simulation.
Using the memory les results in an inreased variability of the tra onditions.
This an b e seen from the greater variability of the blue RMSA
x
urves, whih
indiates that the network states are drawn from a wider distribution than in
the initial simulation. The higher overall levels of the blue RMSA
x
urves also
show that a mo derate additional error is intro dued. The higher level of the blue
RMS
x
urves results from the ombination of b oth eets. However, all blue
RMS
x
urves exhibit a similar struture. This shows that, even if the telematis
logi has a side eet on the simulation dynamis, this eet is fairly stable.
The soure of the dierene between the original simulation and the telemat-
is simulations is that the replanning agents are seleted at random in every
121
iteration. That is, even if the available information itself is idential in all sim-
ulations, dierent travelers at dierent lo ations and with dierent destinations
reat to it. The resulting deviations in the tra onditions are not aounted
for by the replanning agents. This an b e seen as an inreased p ereptional
error, whih, in the given setting, also inreases the variability of the resulting
tra onditions.
5.3.2.3 Measurement and Validation Data Generation
The previous setion shows that the dynamis of telematis simulations are even
less well-b ehaved than their planning ounterparts suh that the argumentation
of Setion 5.2.1.2 applies here with even stronger emphasis.
Consequently, representative measurement and validation data sets are again
generated by averaging. That is, a telematis
referene simulation
is run for
the no-toll senario and for the 6,12, and 18 EUR/h VOT senario.
9
Flows and
o upanies are averaged over 400 stable iterations of eah simulation. These
average values onstitute the measurement and validation data sets used as the
referene data in all subsequent evaluations and RMS(A) error measures.
There is a oneptual dierene in the validation of a planning and a telematis
estimator. In a planning appliation, the goal is to estimate a posterior that is
similar to the true
distribution
of tra states (from whih a draw is realized
every day). In a telematis setting, reality onsists of a single day only. Con-
sequently, a telematis p osterior must represent the knowledge ab out a single
realization
of tra onditions only. This dierene is disregarded in the sim-
plied setting onsidered here sine only a single, representative referene data
set is used to validate the planning and the telematis estimator resp etively.
5.3.2.4 Comparison of Senarios
Figure 5.15 ompares ows and oupanies of the 12 EUR/h VOT (telematis)
referene simulation to the 6 EUR/h VOT referene simulation, the 18 EUR/h
VOT referene simulation, and the no-toll referene simulation. Again, all data
p oints are 400-iteration averages, and, again, they apply to the simulated time
interval from 8:30 until 8:35 am.
The 12 EUR/h VOT senario deviates remarkably from the no-toll senario but
do es not dier muh from the other simulations with a non-zero toll. This is a
result of the laking equilibrium assumption: At the rst day of the toll's im-
plementation, the presumably most advantageous route hoie for most drivers
that so far have traversed the toll area is now to avoid it but to bypass it as
sharply as p ossible in order to minimize the inrease in travel time. This, how-
ever, auses an unforeseeable ongestion on the roads that immediately enirle
the toll zone. The no-toll senario is the only senario in whih this ongestion
do es not o ur.
9
The no-toll telematis referene simulation diers somewhat from the no-toll planning
referene simulation b eause of the le-based driver memory in the telematis simulation
logi.
122
Figure 5.15: Satterplots for omparison of telematis referene simulations
The satterplots ompare data from the 12 EUR/h VOT telematis referene simula-
tion (on the ordinate) to the other telematis referene simulations (on the absissa).
The rst olumn ompares ow rates and the seond olumn ompares oupanies.
All satterplots ontain data p oints for
al l
links in the network. The data p oints apply
to the simulation time interval from 8:30 to 8:35. All data p oints represent average
values over 400 iterations.
123
Figure 5.16: Result overview for telematis oine exp eriments
The left diagram shows measurement errors RMSA
q
and the right diagram shows val-
idation errors RMSA
x
over dierent
w
prior
values for a 12 EUR/h VOT prior senario
and a true no-toll senario. The three estimation errors p er
w
prior
value are represented
by blue dots. For omparison, the error measures for three plain simulations of the
prior senario are represented by red dots. All exp eriments are run for 250 iterations.
Flow and o upany averaging is started after a settling time of 50 iterations.
Sine the estimator's ability to trak rather subtle deviations is already demon-
strated in the planning exp eriments, only the no-toll senario is subsequently
used as the syntheti reality. This implies that the real drivers eetively ignore
the toll's eet. Keeping in mind that only the rst day after the installation
of the toll is simulated, suh a b ehavior may either result from unawareness or
from uriosity about the involved tehnial installations. Again, the purpose of
these exp eriments is to sound the apabilities of the estimator, not to disuss
road priing issues themselves.
10
5.3.3 Exp erimental Results
In all telematis exp eriments, the estimator adjusts a 12 EUR/h VOT prior
senario to measurements that are obtained from a true no-toll senario.
5.3.3.1 Oine Estimation
To b egin with, the rolling horizon mo de is not failitated and a sequene of
oine estimations is run over the entire 6 to 9 am time p erio d. Figure 5.16
shows the resulting error measures over dierent
w
prior
values. The measurement
error RMSA
q
is given on the left, and the validation error RMSA
x
is given on
10
The reent exp erimental results indiate that the estimator works equally well if the prior
senario and the syntheti reality are exhanged. Suh a setting, where the real reation to the
toll is muh stronger that a priori exp eted, ould result from an overreation of the drivers
to the toll.
124
the right. The results of three plain simulation runs of the 12 EUR/h VOT
prior senario are represented by red dots, and the three estimation results p er
w
prior
value are drawn in blue. All exp eriments are run for 250 iterations. The
reursive state averaging turned on after a settling p erio d of 50 iterations.
Both, the simulation and the estimation results are very stable; most dots lie on
top of eah other. This even greater stability than in the planning ase despite
of the greater dierene between the prior and the true senario is asrib ed to
the simpler simulation logi that now disp enses with the equilibrium-generating
travel time feedbak b etween subsequent iterations. The estimator generates
remarkable improvements. For
w
prior
= 2.88
, it improves RMSA
q
by 78% and
RMSA
x
by 82% over a plain simulation of the prior senario. The severe onges-
tion of the 12 EUR/h VOT prior senario that do es not o ur in the simulated
reality is suessfully prevented by the estimator. The ows and oupanies
of the best estimation run (seleted aording to the same riterion as in the
planning exp eriments) are opp osed to the referene data for the true senario in
the satterplots of Figure 5.17. Sine these data p oints are averaged over many
iterations, their dierenes leave no doubt ab out the estimator's systemati and
b eneial inuene.
11
5.3.3.2 Online Estimation in Rolling Horizon Mo de
The same estimation problem as b efore is now takled in rolling horizon mo de.
With a real-time appliation in mind, an evaluation of the estimator p erfor-
mane in terms of average system states that are obtained over hundreds of
iterations is now inappropriate. Therefore, only the RMS
x
validation error is
subsequently evaluated. A temp orally disaggregate p oint of view is adopted by
onsidering eah estimation p erio d individually. Preditive apabilities are also
investigated.
A rolling horizon appliation hallenges the estimator more than the previous
oine exp eriments b eause of the dierent use of the travel time memory les.
An idential memory le sequene is used for measurement generation and for
oine estimation. The rolling horizon estimator still uses the same les but
loads a new le in every iteration of every estimation perio d. Sine these les
are now applied in a temp oral ontext that is dierent from the setting in whih
the measurements were generated, any advantage the estimator may have had
during oine estimation is now preluded.
A prior weight of
w
prior
= 2.88
is maintained in all runs sine this setting
ahieved the b est results in the preparatory oine exp eriments. Figure 5.18
provides separate results for every 30-minute estimation p erio d ending at 7
through 9 am. The blue bars represent (from left to right) the RMS
x
validation
errors obtained at the end of 5, 10, 20, 30, 40, and 50 iterations per estimation
p erio d. They are drawn on top of red validation error bars that result from plain
rolling horizon simulations with resp etive iteration numb ers. These simulations
follow an idential logi as the estimator, only that the measurements are not
aounted for.
11
Results of omparable quality were reently obtained in a setting where the sensor data is
not averaged over many iterations but where it is taken from a single iteration of the telematis
simulation that generates the syntheti reality.
125
Figure 5.17: Comparison of true and estimated ows/o upanies (telematis)
The rst row ontains ow satterplots, and the seond row shows oupany satter-
plots. The rst olumn rep eats the no-toll vs. 12 EUR/h VOT satterplots of Figure
5.15 . The seond olumn shows the aording estimation results where the ref erene
data (on the absissa) is ompared to its estim
ated ounterpart (on the ordinate).
That is, every row ontains one satterplot that ompares the true no-toll senario to
the 12 EUR/h VOT prior senario, and it ontains another satterplot that ompares
the true senario to the estimation result. These plots already represent average values
suh that all dierene between left and right olumn an b e asrib ed to a systemati
eet of the estimator.
126
Figure 5.18: RMS
x
[30 min. rolling horizon estimation℄
The blue bars represent (from left to right) validation error measures RMS
x
obtained after 5, 10, 20, 30, 40, and 50 iterations per estimation perio d.
They are drawn on top of red error bars that result from plain rolling horizon simulations with resp etive iteration numb ers.
127
The estimation and simulation errors rise over time as the tra volumes in-
rease in the morning rush hour. The plain simulation errors do not system-
atially dep end on the number of iterations sine the deployed initial plans le
already results from a stable telematis simulation. A pronouned dierene b e-
tween simulation and estimation an b e observed as the ongestion around the
toll zone b eomes severe in the prior senario. Overall, the estimator redues
RMS
x
by up to 70% in the later p erio ds. Conduting only 5 or 10 iterations
p er estimation p erio d results in lower improvements when ompared to 20 iter-
ations and more. However, running b eyond 20 iterations yields only marginal
improvements.
Figure 5.19 shows the same setup of validation errors as b efore, only that now
the average predition errors over a 0 to 30 minute time interval are given. This
and the previous diagram math temporally in the following way: An estimation
error drawn, e.g., over the 8:30 label is generated at this partiular time and
thus applies to the interval from 8:00 to 8:30. A predition result that is drawn
over the 8:30 lab el is generated at 8:00 for a 30 minute predition window and
onsequently applies to the same interval. A omparison of b oth gures yields
the exp eted diagnosis that the estimation quality is generally higher than the
predition quality. However, an estimation-based predition is learly b etter
than a plain simulation. Again, the predition results for 5 and 10 iterations
p er estimation p erio d are inferior when ompared to those with 20 iterations
and more. The omputational eort of exeuting more than 20 iterations p er
estimation p erio d do es not result in signiantly improved preditions. Overall,
the estimator redues the RMS
x
predition error by 50% to 60% in the later
time p erio ds.
Figures 5.20, 5.21, and 5.22 provide separate RMS
x
plots for the predition in-
tervals from 5 to 10, 15 to 20, and 25 to 30 minutes ahead in time. Here, the
time lab els simply indiate when the predition is made. The quality deterio-
rates graefully as the predition time inreases, starting from a 60% to 65%
improvement for 5 to 10 minutes, attaining 55% to 60% for 15 to 20 minutes, and
yielding around 50% even for the 25 to 30 minute predition. This remarkably
sustained improvement an be traed bak to the rather restrited b ehavioral
degrees of freedom a simulated traveler faes. It also b enets from the fat that
only pre-trip replanning is aounted for suh that a one estimated deision is
maintained for the entire duration of a trip. Finally, the deterministi tra
dynamis ertainly have a p ositive inuene on preditability. However, even
after all these words of reservation, the results show learly that a rolling hori-
zon estimation and predition for this partiular senario is near-optimal if 20
iterations p er 5-minute estimation p erio d are allowed for.
5.3.3.3 Computational Performane
The urrent implementation of the estimator aomplishes 6 iterations p er 5-
minute interval in the given senario. That is, near-optimal results require
another estimation sp eedup of 3 to 4. Given the onsidered problem's size,
this is an enouraging result. After all, one iteration onsists of a 30 minute
tra simulation during the morning rush hour, omprises a b ehavioral model
that relies on time-dep endent b est path alulations, and onduts a omplete
128
Figure 5.19: RMS
x
[0-30 min. rolling horizon predition℄
The blue bars represent (from left to right) 0-30 minute predition error measures RMS
x
obtained after 5, 10, 20, 30, 40, and 50 iterations p er
estimation p erio d. They are drawn on top of red error bars that result from plain rolling horizon simulations with resp etive iteration numbers.
129
Figure 5.20: RMS
x
[5-10 min. rolling horizon predition℄
The blue bars represent (from left to right) 5-10 minute predition error measures RMS
x
obtained after 5, 10, 20, 30, 40, and 50 iterations p er
estimation p erio d. They are drawn on top of red error bars that result from plain rolling horizon simulations with resp etive iteration numbers.
130
Figure 5.21: RMS
x
[15-20 min. rolling horizon predition℄
The blue bars represent (from left to right) 15-20 minute predition error measures RMS
x
obtained after 5, 10, 20, 30, 40, and 50 iterations p er
estimation p erio d. They are drawn on top of red error bars that result from plain rolling horizon simulations with resp etive iteration numbers.
131
Figure 5.22: RMS
x
[25-30 min. rolling horizon predition℄
The blue bars represent (from left to right) 25-30 minute predition error measures RMS
x
obtained after 5, 10, 20, 30, 40, and 50 iterations p er
estimation p erio d. They are drawn on top of red error bars that result from plain rolling horizon simulations with resp etive iteration numbers.
132
spatiotemp oral linearization of the resulting tra dynamis. Even with only
6 iterations p er 5 minutes, the estimator yields substantial improvements when
ompared to the prior senario, whih, however, is likely to b enet from the
simple b ehavioral mo del as explained in Setion 5.3.1.
12
The omputing times are obtained on a 3.2 GHz Pentium 4 stand-alone mahine
with 2 GB of RAM. File i/o onstitutes a ma jor bottlenek in the urrently
single-threaded implementation of the estimator. A large fration of this le
i/o results from the neessity to alulate sensitivities of marosopi system
dynamis bakwards through simulated time, f. Setion 4.1.2. This requires
to store all marosopi states during the simulation and to pro ess them bak-
wards during the linearization. Even if the sparsity of this data b eause of the
simulation sheme on variable time sales is aounted for, f. Setion 2.5, this
adds up to 3.2 MB of binary data p er minute of simulation. Sine the resulting
4 608
MB for a whole day exeed the available RAM of most mahines deployed
in this work, the data is written to hard disk in 5-minute hunks of 16 MB during
the simulation. These les are then reloaded for the linearization. This allows
to estimate the given senario on a mahine with 2 GB of memory. However,
for a limited estimation p erio d of only 30 minutes, the data ould b e kept in
RAM as well. Therefore, the approximate 25% of running time that are spent
waiting for le i/o are omitted when measuring the estimator's omputational
p erformane.
Altogether, the estimator ahieves signiant improvements in a telematis set-
ting. Even if the available senario is somewhat to o large to allow for near-
optimal results in real-time onditions, feasible problems have the same order
of magnitude: Sine the omputational eort rises at least linearly with the
network and p opulation size, a
600+
link senario with
50 000+
agents is imme-
diately approahable by the urrent implementation in real time.
13
A more ex-
tensive prepro essing of the Berlin network illustrated in Figure 2.8 that merges
the many detailed intersetions into single no des might already sue to run
this very senario in real-time.
5.4 Further Disussion
The demonstrated estimator do es not dep end on a hoie set enumeration. This
suggests its appliation for hoie set generation itself. Sine only b est path
alulations are used in the present example, why not run these alulations
diretly based on the mo died utilities instead of rst making a well-informed
guess ab out p ossible routing alternatives and only then hoosing a route based
on these mo diations? To make a long story short: Choie set generation is a
mo deling problem, and tra ounts alone do not provide suient information
to substitute for the strutural information ontained in suh a mo del. However,
12
The reent exp eriments in whih the sensor data is not averaged over many iterations
onverge in roughly half as many iterations but stabilize at somewhat higher error levels.
Apparently, the estimator sp ends signiant amounts of time in the exp eriments given here
trying to extrap olate ontraditory measurements that result from the averaging over many
iterations.
13
The reent results allow for a
1 200+
link senario with
100 000+
agents.
133
this neither implies that tra ounts are useless for hoie set generation nor
that the prop osed estimator is ategorially unsuited for this purp ose.
The onsidered b ehavioral mo del generates its hoie set by running a b est
path algorithm that minimizes travel times whih are generated by the mobility
simulation. These travel times exhibit a partiular orrelation struture that
results from the simulated tra dynamis. This very prop erty enables the
generation of variable routes only based on b est path alulations without ever
resorting to the expliit simulation of a pereptional error by drawing from a
multidimensional travel time distribution with an expliitly known ovariane
matrix.
In ontrast, the estimator only disp oses of lo al measurement information and
pro esses this information in a likewise lo al (linearization-based) manner. If
only few sensors are available, the measurement data is sparsely distributed over
the network. In order to infer a driver's global utility p ereption from this infor-
mation, a mo del is required that aptures the network-wide orrelation of travel
times. In the given simulation system, this orrelation is not aounted for by
the time-dep endent b est path algorithm itself but results from the simulated
travel times based on whih this algorithm is run. If sparse utility orretions
are added to these travel times during the hoie set generation, routes result
that lo ally aount for the orretion terms but globally still adhere to the
orrelation struture of the a priori assumed travel times. If suhlike generated
routes dier suiently from those that atually aused the measurements, the
estimator an only selet among inappropriate prior routes and newly generated
routes that are likewise unrealistially strutured. The result is lo al onver-
gene to a p o or solution.
A visual insp etion of routes that are generated based on estimated utility or-
retions has b een onduted. Their interpretation is diult sine suh routes
invariably aount for b oth travel times and utility orretions. However, a
distint inrease in zig-zagging as one might expet in onsequene of the lo al
utility orretions annot b e observed. Still, even plausibly lo oking routes an
onsist of turning move sequenes that are implausible given a ertain orrela-
tion pattern of the travel times. Within the sop e of this work, it is onluded
that a more rigorous analysis of the simulation-based best-path route hoie
mo del itself is neessary b efore its impliations for the estimation an b e lari-
ed. Reall that this partiular mo del is only implemented as plaeholder and
that the estimator is not onstrained to its deployment.
Again, the ab ove disussion addresses a mo deling problem. The estimator is not
unable to provide useful information for hoie set generation; it just is unable
to solve the generally impossible task of inferring a network-wide utility pat-
tern from arbitrarily few observations. If a mo del was at hand to meaningfully
omplete lo ally estimated utility orretions, hoie set generation ould b e
supp orted by measurements. This typ e of mo del would represent a rather om-
mon asp et of travelers' information pro essing. For example, a radio message
regarding a single onstrution site is likely to motivate a driver to irum-
navigate the surroundings of this site as well sine exp eriene teahes that the
resulting obstrutions are not onentrated at the single lo ation indiated on
the radio. That is, the driver is aware of orrelations in the network onditions.
One might argue that full sensor overage should allow for hoie set generation
134
without further mo deling supp ort. However, this also would require to aount
for measurement orrelations in the likelihoo d funtion. This is avoided here by
ho osing sparse sensor lo ations. Sine travel times are one partiular typ e of
link-related measurements, the problem of orrelation mo deling would not b e
solved but only b e shifted in a dierent ontext. In addition, full sensor overage
annot b e exp eted in real-world onditions.
A meaningful interpretation of the lo al utility orretions in the losing example
of Chapter 4 was p ossible b eause of its simple struture. In the more general
setting onsidered here, suh an interpretation suers from the same problems
as the diret appliation of utility orretions for route generation: Every single
turning move's utility orretion is only meaningful given the b ehavioral mo del
that is used for its identiation. That is, the utility orretions are meaningful
on route level with the route generated based on simulated travel times with
a partiular orrelation struture but not neessarily on turning-move level.
The b ehavioral mo del represents the global ontext that annot b e aptured by
lo al utility orretions. This interplay of modeling and estimation do es not
invalidate the estimator's ability to funtion with an arbitrary implementation
of the b ehavioral simulator. It do es, however, neessitate an interpretation of
the estimation results in terms of the partiular behavioral mo del based on
whih they are obtained.
Summarizing, this hapter demonstrates the prop osed estimator's appliability
in onjuntion with a fully dynamial planning or telematis simulator and
veries its omputational feasibility for a senario of pratially relevant size.
135
Chapter 6
Summary and Outlo ok
This hapter summarizes the present dissertation, highlights its key ndings,
and gives an outlo ok on further researh topis.
6.1 Reapitulation of Work
The goal of this researh is (i) to develop a behavioral tra state estimator for
a multi-agent simulation and (ii) to demonstrate its appliability to a senario
of pratially relevant size. Sine a mo del-based estimation approah is hosen,
exp erimental investigations all for exeutable mo dels of reasonable p erformane
and realism. This applies to b oth the b ehavioral and the physial simulator.
The development of a marosopi tra ow model in Chapter 2 results in
a omputationally eient mobility simulation that is appliable to general
networks and has linearizable dynamis. Its omputational p erformane also
ontributes to an eient solution of the estimation problem itself. The mo del
is enapsulated in a general state spae representation and thus an b e replaed
by a dierent implementation, if required.
This marosopi mobility simulation is ombined with a mirosopi driver
representation in a mathematially tratable way by the mixed miro/maro
simulation logi presented in the rst half of Chapter 3. This logi links any
marosopi mobility simulation that takes ow splits as input parameters to any
mirosopi b ehavioral mo del that generates individual-level turning deisions
at intersetions and network entry/exit p oints. The representation of arbitrary
mobility patterns in terms of suh turning deisions is demonstrated in the
seond half of Chapter 3.
These mo deling eorts establish a linearizable relation b etween individual driver
b ehavior and aggregate tra harateristis. Based on this tehnially pivotal
result, a numb er of b ehavioral estimators is developed in Chapter 4. First,
a heuristi approah is presented. It is based on a more generally appliable
metho d to steer simulated travelers suh that a general ob jetive funtion of
marosopi system states is inreased. For estimation purp oses, this ob jetive
136
funtion is hosen as the log-likeliho o d of the available aggregate sensor data,
and the agents are steered towards a fulllment of the measurements.
Seond, a statistially more rigorous reonsideration of the estimation prob-
lem is given, and two op erational Bayesian estimators are develop ed: (i) The
aept/rejet estimator funtions without further assumptions ab out the b ehav-
ioral prior. Its takes an inreased number of draws from this prior and retains
only a subset of these draws. This subset is representative for the b ehavioral
p osterior. (ii) The utility-modiation estimator adds a orretion term to the
systemati utility of every evaluated alternative. Given a partiular form of
the b ehavioral prior, the simulation system then draws immediately from the
b ehavioral p osterior. The heuristi estimator is found to oinide tehnially
with the UM estimator and an thus b e re-analyzed in the Bayesian setting.
The development of these estimators is aimed at but not tailored to an applia-
tion in onjuntion with the MATSim simulation software. Sine MATSim was
in a transitional p erio d of re-implementation during this work, stable interfaes
ould not b e set up and MATSim's emerging mo deling apabilities ould not b e
failitated. In hindsight, this is not onsidered as a disadvantageous situation.
Sine no predetermined simulator implementation was at hand, no exibility
was given away by restriting the developments towards a partiular system
design. At the time of this writing, an appliation in onjuntion with MATSim
is oneptually and tehnologially feasible. Guidelines for this undertaking are
given in Setion 6.4.5. Still, the estimators' appliability to systems dierent
from MATSim is not hindered by a onnement to this partiular software.
Exp erimental results are presented in Chapter 5. Sine the prop osed estimation
system is of substantial omplexity, it is advisable to obtain a go o d understand-
ing of its working by an initially syntheti test ase that allows for greatest
exp erimental ontrol. It is demonstrated that the metho d is able to adjust
individual-level b ehavior based on a limited amount of tra ounts suh that
a signiantly improved piture of the global tra situation is obtained. The
metho d is found to b e omputationally apable of dealing with senarios of pra-
tially relevant size and to be appliable in b oth a planning and a telematis
setting. The simple b ehavioral mo del plaeholder implemented for exp erimental
purp oses is found to onstitute a ma jor limitation of ontinuative investigations,
and the need for advaned b ehavioral mo deling is aentuated.
Additional real world exp eriments would go b eyond the sop e of this work.
The exp eted eort to prepare and implement suh a test ase is substantial
[129℄. The syntheti exp eriments given here level the ground for this undertak-
ing. Guidane on how to proeed towards real-world exp eriments is provided in
Setion 6.4.1.
6.2 Researh Contributions
The key results of this work are highlighted in this setion. The listing is onned
to novel ontributions to the state of the art.
1. Development of a marosopi mobility simulation with the following fea-
tures:
137
•
phenomenologial onsisteny with the ell-transmission mo del,
•
simulation of no des with an arbitrary numb er of upstream and down-
stream links,
•
approximate linearization of tra ow dynamis with resp et to ell
o upanies (system states) and turning frations (exogenous param-
eters),
•
fast exeution by a simulation logi that runs all network elements
on individual time sales.
2. Development of a ombined miro/maro mobility simulation with the
following features:
•
ompatibility with broad lasses of marosopi tra ow mo dels
and mirosopi driver representations,
•
linearizability in that the eet of any driver's behavior on the global
network onditions an b e linearly predited,
•
omputational eieny in that only a sample of the mirosopi
driver p opulation is required for simulation,
•
omputational eieny by ompatibility with the marosopi sim-
ulation logi on variable time sales,
•
removal of most vehile disretization noise from the marosopi
tra harateristis.
3. Formalization of the physial asp ets of a partial or whole-day plan as a
sequene of turning moves on a slightly expanded network suh that the
linearizability of the global network onditions with resp et to individual
plan hoie is maintained.
4. Development of a general metho d to steer mirosopi agent behavior
suh that a general ob jetive funtion of marosopi tra onditions is
improved.
5. Development of two op erational b ehavioral estimators with the following
ommon features:
•
estimation of fully disaggregate b ehavior from aggregate tra mea-
surements and prior b ehavioral knowledge,
•
ompatibility with a purely simulation-based representation of the
b ehavioral prior information,
•
no requirement of a hoie set enumeration,
•
omputational eieny that allows for an appliation to large se-
narios.
6. In partiular, development of the following distint estimators:
•
an aept/rejet estimator that takes an inreased numb er of draws
from an arbitrary b ehavioral prior and retains only a subset of these
draws that is representative for the b ehavioral p osterior,
138
Figure 6.1: Estimated quantities
Two state estimation problems and two parameter identiation problems are illus-
trated in this gure: (1) estimation of b ehavior (mental states), (2) estimation of
tra onditions (physial states), (3) identiation of physial mo del parameters, (4)
identiation of b ehavioral mo del parameters.
•
a utility-modiation estimator that orrets the systemati utility
of every evaluated alternative suh that, given a ertain struture of
the b ehavioral prior, the simulation system draws immediately from
the b ehavioral p osterior. A heuristi appliation of this estimator for
dierent or unknown priors is p ossible.
7. Exp erimental investigations in a syntheti yet fully dynamial setting with
the following onlusions:
•
Given only a limited amount of tra ounts, the global orretness of
(i) a SUE planning simulation and (ii) a (rolling-horizon) telematis
simulation is onsistently and signiantly improved by the prop osed
estimator;
•
the metho d is apable of handling online estimation problems of pra-
tially relevant size in real time;
•
sine aggregate tra measurements ontain only limited informa-
tion, a struturally orret b ehavioral mo del is essential for go o d
estimator p erformane.
6.3 Classiation of Results
As a transition to some of the further researh topis, Figure 6.1 illustrates
the simulation system in terms of only two omponents, the behavioral mo del
and the mobility simulation. The lower feedbak lo op indiates that not only
b ehavior inuenes tra onditions, but also tra onditions aet b ehavior.
The estimator ompares simulated and real tra onditions and adjusts the
simulation system based on this omparison.
Four dierent typ es of adjustment are identied in this gure. Number 1, esti-
mation of behavior, is treated in this dissertation: The estimation of a plan set
U1...UN
omprises all asp ets of the individual drivers' mental states that are
139
neessary to dene all marosopi states
X
in the mobility simulation. This
estimation approah relies on (i) a deterministi mobility simulation and (ii)
an available parameterization of the underlying b ehavioral and physial mo del
omp onents.
A relaxation of these assumptions leads to the three further estimation tasks
indiated in Figure 6.1. They are: (2) estimation of non-deterministi physi-
al system states, (3) parameter identiation for the mobility simulation, and
(4) parameter identiation for the b ehavioral mo del. Items (2) and (3) are
disussed in Setion 6.4.2, and item (4) is onsidered in Setion 6.4.4.
6.4 Further Researh Topis
Various diretions for future researh are thinkable in ontinuation of this dis-
sertation. This setion strutures these topis and provides guidane on further
developments.
6.4.1 Towards a Real-World Appliation
This work was onduted with a real-world appliation in mind and onse-
quently aounts for typial data requirements, p erformane issues, and mo des
of op eration. The following matters need to b e addressed in the preparation of
a real-world test ase.
6.4.1.1 Mo del Calibration and Validation
Mo del-based state estimation ruially dep ends on strutural mo del orretness.
Only a go o d understanding of reality allows to meaningfully inter- and extrap o-
late the information ontained in limited measurements. This statement equally
applies to the physial and the b ehavioral mo del omp onents.
The prop osed mobility simulation exhibits several novel features: general inter-
setions, variable time sales, and the ombined miro/maro simulation logi.
These developments were neessary to realize an estimator prototype that is
appliable to general senarios of realisti size. While the syntheti nature of
the presented experiments irumvents the need to alibrate and validate the
physial mo del, additional eort in this regard is neessary b efore a real-world
appliation an be attempted. Sine the marosopi mobility simulation is en-
apsulated within a general state spae representation, it may even b e replaed
by an entirely dierent mo del that is more appliable in a partiular setting.
As to b ehavioral mo deling, a struturally orret b ehavioral simulator must b e
externally provided. RUMs are partiularly appliable here beause of their
sophistiated alibration and validation pro edures. However, the estimator
itself is indierent to the applied mo del's degree of mathematization, and a
simple rule-based model is tehnially just as feasible for estimation as a full-
blown RUM.
140
6.4.1.2 Measurement Soures and Sensor Typ es
The exp erimental investigations of this work fo us on ow measurements be-
ause of their predominant role in tra monitoring. However, the general
formalism presented in Setion 4.2.1 allows to utilize a greater variety of sensor
data. As noted there, any aggregate measurement that is a funtion of the state
of a link or a turning ounter an diretly be fed into the estimation pro edure.
If the measurements are not statistially indep endent, their ovariane struture
needs to b e identied b efore the b ehavioral estimator an b e applied.
Some advaned data soures are addressed b elow. While they are not aounted
for in this dissertation, the fully disaggregate b ehavioral mo deling assumption
is at least struturally adequate for their future onsideration.
Any vehile that is equipped with a GPS reeiver an serve as a tra sensor.
If its spatiotemp oral tra jetory is mapp ed on a representation of the underlying
network, a wealth of disaggregate information beomes available that is well
suited for the alibration of a b ehavioral model [67℄. This type of information
may also b e available at a more aggregate level. For example, GPS-equipp ed
taxis typially rep ort their urrent p osition to a dispath enter every few min-
utes. This data an b e transformed into lo al velo ity information, e.g., [156℄,
whih in turn an b e utilized by the prop osed estimator. Unlike tra ounts
from indutive lo ops, suh oating ar data is available at variable lo ations.
It also requires dierent distributional assumptions ab out the derived velo ity
information: A slowly driving vehile might do so for several reasons and thus
is only an imperfet indiator of dense tra. On the other hand, a quikly
advaning vehile is a reliable indiator of unongested tra onditions.
Vehile re-identiation systems provide similar information at a oarser level.
The time span b etween two detetions of a vehile is the sum of all link travel
times along an unobserved route that onnets the two identiation p oints and,
furthermore, inludes the duration of all intermediate stops. In onsequene,
additional mo deling assumptions regarding at least route hoie are neessary
to relate this type of information to the link- or turning move-related states of
a marosopi mobility simulation [4, 183℄.
6.4.1.3 Performane Tuning
The urrently implemented estimator already takles online problems of non-
trivial size. However, further p erformane tuning is p ossible.
Algorithmially, the estimation requires to identify a xed p oint of a nonlinear
and sto hasti mapping that omprises a omplete tra simulator, f. Setion
4.1.3. Only a basi SA pro edure is utilized in this work, and advaned xed
p oint searh algorithms should b e onsidered for this purp ose. The researh on
the onsistent antiipatory route guidane generation problem has pro dued
a numb er of promising results in this regard [26, 51, 52℄.
Op erationally, the estimator is not yet optimized. Its implementation reet its
exp erimental nature that fo uses on exibility and robustness. One a partiu-
lar mo de of op eration is sp eied, this implementation should b e ne-tuned and
141
stripp ed of omputational ballast. For example, the urrently realized rolling-
horizon estimator runs the same SA logi as used in oine op erations indep en-
dently in every estimation p erio d, f. Setion 5.3.1. However, the results of
one estimation p erio d ontain valuable information for the subsequent estima-
tion p erio ds. This information should b e aounted for in a more ne-tuned
implementation.
6.4.2 Combined Behavioral and Physial Estimation
So far, it is assumed that the mobility simulation is modeled without error. A
p ossible relaxation of this assumption is outlined in this setion.
Unertain tra ow dynamis are mo deled by adding a temp orally unorre-
lated zero-mean random disturbane vetor
η(k)
to state equation (2.17):
x
ms
(k+ 1) = f
ms
[x
ms
(k),β(k),η(k), k]
(6.1)
where
x
ms
is the mobility simulation's physial state vetor and
β
represents the
single-ommo dity turning frations. Equation (6.1) replaes the deterministi
tra ow model omp onent of the mixed miro/maro state spae model (3.7).
The relation b etween
x
ms
and the available measurements
y
is represented by
the likewise randomly disturb ed output equation
y(k) = g[x
ms
(k),ǫ(k)],
(6.2)
whih orresp onds to (4.16) without loss of generality. The two above equations
an b e linearized. Given a parameterization
{β(k)}k
, they onstitute a non-
linear, dynamial system that is amenable to the marosopi state estimation
tehniques reviewed in Setion 1.2.
All b ehavioral estimators of this thesis disregard the sto hasti error
η
in (6.1).
Without exeption, they ontain a step in whih
U1...UN
are loaded on the
network and
X
is obtained, f. Algorithms 2 through 4. That is, the b ehav-
ioral estimation problem is solved given a partiular mapping of the b ehavior
U1...UN
on the marosopi states
X
.
The
β
parameters in (6.1) result from the b ehavior of individual partiles in the
mixed miro/maro mobility simulation of Setion 3.1. This partile behavior is
fully determined by a plan set
U1...UN
. The network loading step an therefore
b e replaed by a physial state estimator that formally op erates exlusively on
the mo del sp eiations (6.1) and (6.2) with an externally provided
{β(k)}k
parameterization that is internally generated by an exeution of
U1...UN
. The
physial estimator utilizes the same sensor data
Y={y(k)}k
as the b ehavioral
estimator.
Consequently, the behavioral estimation problem is still solved given a partiu-
lar mapping of the b ehavior
U1...UN
on the marosopi states
X
, only that
this mapping now inorporates a physial state estimation pro edure. This also
enables the traking of time-dep endent physial mo del parameters by an appro-
priate extension of the marosopi state vetor, e.g., [3, 175℄. The straightfor-
wardness of this approah is owed to the minimal interfae between the miro-
sopi and the marosopi mo deling omp onents.
142
6.4.3 Combined Telematis and Planning Estimation
Mutual b enets an b e exp eted if a telematis and a planning estimator are
applied onertedly. Two possibilities to realize suh a oupling are outlined in
this setion. In either ase, it is assumed that an online estimator generates
results on a daily basis that are used to improve the outome of a planning
simulation. This enables the latter to provide improved b ehavioral priors for
the next day's online estimation problem.
The ability to provide an improved prior does not imply that a suhlike ad-
justed planning simulation an also be applied to predit struturally dierent
senarios, where, for example, infrastrutural hanges are onsidered. This abil-
ity would require not only to estimate what hoies are made by the travelers
in a given senario but also to identify the underlying behavioral parameters
that trigger these hoies. This setion only onsiders the problem of how to
adjust a planning simulation for purp oses of inremental online tra moni-
toring. The b ehavioral parameter estimation problem is disussed in subsequent
Setion 6.4.4.
6.4.3.1 Fusion of
Λ
Co eients
The dierene b etween a b ehavioral prior and an estimated p osterior is fully
aptured by the
Λ
o eients. The most straightforward approah to failitate
the o eients
Λd
obtained by the online estimator at a ertain day
d
is to
inorp orate them in baseline o eients
¯
Λ
that are used as starting values in
the next day's online estimation problem. These baseline o eients an also
b e aounted for in a planning mo del if up dated prior information is to b e
simulated. A similar pro edure an b e found in the ontext of OD matrix
estimation where a within-day estimated OD matrix is used to up date a planning
OD matrix, f. Setion 1.2.2.2. Possible up date metho ds are reursive averaging
[7℄ and Kalman ltering. The latter assumes that
¯
Λ
follows a random walk and
that one noisy measurement
Λd
of
¯
Λ
b eomes available p er day [183℄.
6.4.3.2 Choie Set Mo diations
Choie set generation is a omputationally demanding step that is likely to b e
p erformed at least in part oine. In online op erations, omputational onsider-
ations might require a relatively small hoie set p er agent that in onsequene
needs to b e hosen with partiular are. If the online estimator has seleted
a ertain plan rather infrequently, this indiates that this plan is unlikely to
b elong to the onsidered traveler's hoie set and thus should b e replaed by a
more reasonable alternative. This allows for an inremental oine hoie set ad-
justment that should also result in an improved online estimation p erformane.
6.4.4 Behavioral Parameter Estimation
The prop osed estimator also holds promise to provide information ab out param-
eters that underlie the estimated hoies, i.e., to address parameter estimation
143
problem (4) in Figure 6.1. Two suh approahes are disussed in this setion.
1
6.4.4.1 Estimation of Population Parameters
A syntheti p opulation needs to b e reated b efore an agent-based simulation
of tra is p ossible, f. Setion 1.2.2.3. Typially, its generation relies on
a sequene of sampling pro edures where agent parameters are drawn from
b eforehand sp eied distributions that apply to homogeneous subsets of the
p opulation [9℄. For example, the ativity patterns for all male workers of an
urban p opulation may b e drawn from a single distribution, the work lo ations
for all employees that live in a ertain tra zone may b e drawn from yet
another distribution, and so forth.
Sine the distributions that underlie this generation pro edure are themselves
estimates of imp erfet preision, aggregate tra measurements may help to
improve the realism of the syntheti p opulation. Sine this implies that the
sensor data is used to adjust strutural features of the multi-agent model, the
resulting p opulation should b e appliable in a wider variety of senarios that
may onsiderably dier from the onditions in whih the measurements are
obtained. An appliation of the prop osed b ehavioral state estimator for this
purp ose is desrib ed hereafter.
A subset
M⊆ {1. . . N}
of the syntheti p opulation is onsidered. This subset
is homogeneous with resp et to the distribution
PM(θ)
of a ertain p opulation
parameter
θ∈Θ
where
Θ
is a disrete and p ermissibly non-ordinal domain.
Disregarding the sensor data, a single draw of this parameter is assigned to
every individual
n∈M
. All plans of an agent in
M
are thus parameterized
diretly or indiretly by this value. When the simulation is run, the agent
learns individually optimal b ehavioral patterns, and when the iterations have
stabilized, the agent exhibits a reasonable plan hoie distribution given its
partiular
θ
value.
Assume that there is unertainty about the true distribution of
θ
. Sine
M
is
homogeneous with resp et to this distribution, it is feasible to provide every
agent in
M
with two instead of one parameter values, say
θ1
and
θ2
, and to
1
A unied Bayesian formulation of b oth parameter estimation problems onsidered in this
setion was found shortly after the submission of this dissertation. Let the deision protool
b e parameterized with an individual-level parameter vetor
θn
for every agent
n= 1 . . . N
,
denote the individually parameterized hoie distributions by
Pn(Un|θn)
, and assume that a
prior p.d.f.
p(θn)
is available for the parameters. In omplete analogy to the derivation given
in Setion 4.3.2, an individual-level p osterior
pn(Un,θn|Y) = ehΛ,UniPn(Un|θn)p(θn)
PV∈CnehΛ,Vi RPn(V|θ′)p(θ′)dθ′
of agent
n
's joint hoie and parameter distribution given the measurements an be formulated.
The following version of the AR estimator draws from this posterior:
1. Draw
θn
from
p(θn)
.
2. Draw
Un
from
Pn(Un|θn)
.
3. Aept
(Un,θn)
with the original aeptane probability
φn(Un)
dened in (4.35).
Otherwise, goto 1.
Note that this estimator is equally appliable to the identiation of disrete-valued parame-
ters.
144
parameterize one half of its plans with
θ1
and the other half with
θ2
. The re-
sulting parameter o urrenes still follow the original distribution
PM(θ)
in that
the probability that an individual in
M
gets assigned two partiular parameters
θ1
and
θ2
is
PM(θ1)PM(θ2)
.
The estimator now adjusts the p opulation's b ehavior to the sensor data
Y
. The
resulting hoie frequeny of any partiular
θ
value in
M
is
PM(θ|Y)≈1
R|M|
R
X
r=1 X
n∈MI(Ur
n∼θ)
(6.3)
where
r= 1 . . . R
iterations are onsidered,
Ur
n
is the plan seleted by individual
n
in iteration
r
, and
I(U ∼ θ)
is one if plan
U
is parameterized with
θ
and zero
otherwise. This simulated p osterior distribution of
θ
given the measurements
an b e applied to re-sample the parameters of the p opulation subset
M
and to
re-run the estimation. This is repeated until onsisteny of the prior and the
p osterior parameter distribution is attained.
A preaution is neessary to avoid biases in this approah. If there is no sensor
data, the estimator is redued to a plain simulator, and the result of suh a
simulation is that every individual in
M
disards the
θ
value of inferior sub jetive
p erformane. If, for example,
θ
represents a leisure lo ation and all else is equal,
the plans that ontain the more distant leisure lo ation are disarded b eause
they impliate longer travel times. That is, the plan seletion mehanism itself
generates a drift in the parameter distribution.
A remedy to this problem is to split the plan set of every individual aording
to the dierent
θ
values. Every agent in
M
now has two hoie sets
C1
n
(all
elements of whih are parameterized with
θ1
) and
C2
n
(parameterized with
θ2
)
of equal size. When making a deision, the agent rst ho oses a hoie set with
uniform probability and then selets a plan from that set aording to its be-
havioral mo del. In result, the agent exhibits a dual b ehavior. This should not
intro due systemati side eets in the simulation sine the whole subpopula-
tion's parameterization is still onsistent with
PM(θ)
. If now the AR estimator
is applied, all resulting hanges in the
θ
seletion frequenies an be attributed
exlusively to the sensor data. The UM estimator is not appliable here sine
it has no inuene on the uniform distribution used for hoie set seletion.
6.4.4.2 Estimation of RUM Parameters
Typially, the deterministi utility of a RUM is linear in parameters:
Vn(U) = θTxU,n +kU
(6.4)
where
xU,n
is a vetor that represents the features of deision maker
n
and of
alternative
U ∈ Cn
, and
θ
is a vetor of real-valued parameters. The alternative-
sp ei onstant
kU
aptures all hoie-relevant asp ets of
U
that are indep en-
dent of
xU,n
.
The UM estimator of Setion 4.3.3 aets estimated b ehavior via additive utility
orretions:
Wn(U) = Vn(U) + hΛ,Ui/µ
=θTxU,n +kU+hΛ,Ui/µ.
(6.5)
145
That is, the UM estimator eetively adjusts the alternative-sp ei onstants of
an underlying RUM. The preditive p ower of suhlike adjusted RUMs dep ends
on the stability of the alternative-sp ei onstants aross dierent senarios.
If the
θ
parameters themselves admit improvements, an inorp oration of sensor
data into the RUM alibration pro edure is a desirable goal. RUM parameters
are typially identied by maximum likelihoo d estimation [21℄, whih requires
a likeliho o d funtion
l(θ|Y) = p(Y|θ)
to b e available. Noting that the sensor
data
Y
is not diretly dep endent on
θ
, one obtains
l(θ|Y) = X
U1∈C1
... X
UN∈CN
p(Y | U1...UN)P(U1...UN|θ)
=
E
{l(U1...UN|Y)|θ}.
(6.6)
That is, the likelihoo d of
θ
given the sensor data an b e expressed as the exp e-
tation of the available likeliho o d
l(U1...UN|Y) = p(Y | U1...UN)
, f. Setion
4.2.1, given that the p opulation's plan hoie distribution is parameterized by
θ
. A Monte Carlo approximation of this exp etation is p ossible:
E
{l(U1...UN|Y)|θ} ≈ 1
R
R
X
r=1
l(Ur
1...Ur
N|Y)
(6.7)
where
R
is the numb er of draws and
Ur
n
is the plan hosen by individual
n
in simulation
r
given parameter
θ
. Parameter estimation based on a suhlike
simulated likelihoo d is p ossible in priniple [166℄, but it is omputationally ex-
tremely demanding sine every draw requires a full run of the tra simulator.
An interesting question is to what degree a linearization-based approximation
of the network loading proedure an help to aelerate this pro ess.
6.4.5 Integration with MATSim
6.4.5.1 Coneptual Asp ets
Setion 3.2 haraterizes a b ehavioral simulation system that is appliable in
onjuntion with the prop osed estimator. It is observed there that the following
prop erties of the MATSim planning simulation are not immediately ompatible
with this sp eiation:
1. variable plan hoie set,
2. ontinuously up dated (learned) plan utilities (sores),
3. immediate exeution of a newly generated plan.
Problems 1 and 3 are resolved olletively. An invariable hoie set results if an
agent is assumed not only to selet from its urrently memorized plans but also
also from all other plans that an p ossibly b e generated by the MATSim replan-
ning mehanisms desrib ed in Setion 3.2.2.3. The overall probability that a
new plan is generated in a given iteration is denoted by
P
new
. Aordingly, the
seletion probability of any existing plan is
1−P
new
times its hoie probability
146
without plan generation, whereas the seletion probability of any newly gener-
ated plan is
P
new
times its probability of generation. Thus, every agent disposes
of a well-dened (alb eit p ossibly very large) hoie set, and a hoie probability
for eah element in this set exists. Sine neither the expliit availability of these
probabilities nor an enumeration of the hoie set is required, an appliation
of the AR is oneptually feasible at every
single
MATSim iteration. However,
sine the generation of new plans is not utility-driven, the UM estimator is not
appliable here.
Item 2 is related to the strong orrelation b etween
subsequent
MATSim itera-
tions. Travel b ehavior is not simulated based on systemati utilities that are
averaged over a long time horizon but relies more strongly on the most reent
iterations: The sores of exeuted plans are updated by a reursive lter that
has an innite but exp onentially deaying memory. The route realulations
utilize only the most reent iteration's travel times. Thus, even after a large
number of iterations, a situation in whih the tra onditions of subsequent
iterations utuate unorrelatedly around stable average values is unlikely to
o ur. This eet an also b e observed throughout the exp eriments given in
Chapter 5.
The estimation pro edure, however, fundamentally relies on the
Λ
o eients
that represent the sensitivities of the measurement log-likeliho o d to the driver
b ehavior. These sensitivities are averaged over many iterations, f. Setion
4.1.3, and the resulting averages may stabilize even if the overall system ex-
hibits a yli behavior, as it is likely to our in MATSim. Sine this implies a
systemati deviation b etween the atually o urring sensitivities and their av-
erage values, a delined estimator p erformane may result. However, no general
statement ab out MATSim's dynamis an b e made at this p oint.
The AR estimator rep eats a single hoie situation several times. It requires
that rep eated draws are indep endent and identially distributed. This estimator
is not impaired by the orrelation b etween subsequent MATSim iterations as
long as the b ehavioral distribution of an agent is invariable within a single
iteration. MATSim evolves as a Markov pro ess, with its state b eing dened
through the urrent agent memory (in terms of available plans) and the last
iteration's tra onditions (used for the generation of new plans). In every
single iteration, the AR estimator orrets the transition probabilities of this
pro ess in a most plausible way. Thus, it is reasonable to exp et that the
resulting iteration dynamis of MATSim are likewise improved.
The estimator's oneptual ability to funtion even in onjuntion with this
rather untypial model of dynamial tra evolution indiates its exibility
and independene of a spei system design. The following setion exemplies
the tehnial steps that are neessary to assert the ab ove hyp otheses in pratie.
6.4.5.2 Tehnial Asp ets
Several exemplary Java o de snipp ets are provided that represent the arguably
simplest way to attah the estimator to the MATSim system as implemented in
Otob er 2007. For simpliity, only the seletion of full plans is onsidered and
the o de is stripp ed of all oneptually irrelevant elements. Of ourse, various
alternative implementations that ahieve the same eet are thinkable.
147
For the purpose of this presentation, it is suient to speify an agent by a
Person
interfae that provides aess to the set of its available
Plan
instanes.
interfae Person {
Set getPlans();
}
The utility funtion is an implementation of a
SoringFuntion
interfae that
maps a
Plan
on a utility value as p ereived by a partiular
Person
.
interfae SoringFuntion {
double getSore(Plan p, Person n);
}
The deision proto ol is represented by a
PlanSeletor
lass that implements
a
seletPlan(Person, SoringFuntion)
funtion. This funtion returns a
single draw from the
Person
's
Plan
set.
lass PlanSeletor {
Plan seletPlan(Person n, SoringFuntion sF) {
Plan result;
// Choie logi implemented here. Examples:
// * aess hoie set via n.getPlans();
// * evaluate a plan p via sF.getSore(p, n);
return result;
}
}
An appliation of the UM estimator requires to modify the implemented
Soring-
Funtion
. An appropriate tehnique is to implement a wrapp er lass
UMSoring-
Funtion
around the original
SoringFuntion
and to pass this wrapper in-
stead of the original implementation to the
PlanSeletor
.
lass UMSoringFuntion implements SoringFuntion {
SoringFuntion sF;
UMSoringFuntion(SoringFuntion sF) {
this.sF = sF;
}
double getSore(Plan p, Person n) {
return sF.getSore(p, n) +
hΛ,Ui/µ
;
//
U
is turning move sequene of Plan p.
//
hΛ,Ui/µ
addend is defined in (4.14).
}
}
The AR estimator requires a mo diation of the plan seletion logi itself. This
an be realized by funtion overriding. A sub lass
ARPlanSeletor
is derived
from
PlanSeletor
, the
seletPlan(..)
funtion is overridden, and the orig-
inal
PlanSeletor
is replaed by an instane of the
ARPlanSeletor
.
148
lass ARPlanSeletor extends PlanSeletor {
Plan seletPlan(Person n, SoringFuntion sF) {
Plan result;
do {
result = super.seletPlan(n, sF);
} while (Math.random() >=
φn(U)
);
//
U
is turning move sequene of Plan result.
//
φn(U)
is aeptane probability (4.35).
return result;
}
}
Both the
UMSoringFuntion
and the
ARPlanSeletor
need referenes to the
Λ
o eients for the alulation of utility orretions and aeptane proba-
bilities. The linearization logi that generates these o eients is part of the
marosopi mobility simulation. In onjuntion with MATSim, the easiest way
of aessing this data is via les: In every iteration, the b ehavioral simulation
system writes out a le that ontains the seleted plans of all agents. The mobil-
ity simulation then loads these plans, exeutes them, and in turn writes out the
Λ
o eients plus all further data that is required for agent replanning. This
basi implementation suggests itself for rst exp erimental investigations. The
programming eort of a tighter oupling by diret funtion alls would mainly
pay o in terms of an inreased exeution sp eed b eause of the avoided le i/o.
6.4.6 Strutural Mo del Renements
6.4.6.1 Physial Simulation
The miro/maro oupling logi does not dierentiate among vehile typ es.
Within limits, this is p ossible by a sp eiation of dierent marosopi sizes
for passenger ars, truks, buses, and so forth. Continuative modeling may also
dierentiate the dynamis of dierent vehile lasses within the marosopi mo-
bility simulation. This is likely to require a representation of multi-ommo dity
ows within the marosopi mo del omp onent [33℄.
Inner-urban tra ow is dominated by signaling. While the employed mobility
simulation do es not aount for this asp et, the mo deling of signalized interse-
tions has already been demonstrated in onjuntion with a ell-transmission
mo del [1℄. This requires a network mo del at the granularity of individual lanes
in order to avoid unrealisti spill-baks at simulated intersetions that in reality
have turning po kets. In suh a setting, it might prove useful to swith o the
exp onential turning ounter forgetting mehanism (3.4) for the duration of a
red phase.
There is an imp ortant issue regarding adaptive signaling. Adaptive ontrols
may swith strategies based on threshold values and thus may intro due dison-
tinuities in the mobility simulation: A small b ehavioral hange of a single driver
that auses a sensor output to exeed a threshold value might hange the entire
ontrol strategy and thus might have a large eet on the marosopi system
states. However, sine adaptive signaling is sensor driven, the aording sensor
data an b e made available to the estimator as well. This allows to reprodue
149
the true ontrol strategy without error, either by a reonstrution of its logi in
the simulator or by a diret observation of the real signaling. Sine suhlike sim-
ulated signaling is a p erfet image of reality, no adaptivity is neessary within
the mobility simulation suh that its ontinuity with resp et to plan hoie is
preserved.
6.4.6.2 Behavioral Simulation
Flexibility as to dierent behavioral implementations is a main ob jetive of this
work, and few limitations are imp osed on a rened b ehavioral simulator.
Swithing from single-day plans to weekly plans disloses new p otentials for
mid-term foreasting. Sine weekly plans introdue a logial relation b etween
travel behavior at subsequent days, single-day plan estimates provide informa-
tion ab out up oming b ehavior that an b e failitated for predition and, in
partiular, as an improved prior for the next day's estimation problem.
Tra monitoring is not onduted as an end in itself. In online op erations,
a tra predition that is based on the most reent tra state estimate an
b e utilized to provide various information servies to travelers. However, if this
guidane is not arefully hosen, the resulting driver reations might invalidate
the underlying predition. This antiipatory guidane generation problem is
deoupled from the state estimation problem sine all disseminated information
is known up to the present p oint in time at whih the online estimation ends.
In onsequene, the estimator only requires a b ehavioral mo del that prop erly
aounts for the most reently generated guidane, but it is indierent with
resp et to the partiular nature of this guidane [19℄.
150
App endix A
Implementation of GPRC
Integer Sets
The GPRC requires many integer set op erations. Sine all set implementations
provided by the Java Colletions Framework [85℄ rely on ob jet representations
of their elements, they arry a formidable overhead if only primitive typ es are
required. This app endix desrib es a set implementation that is tailored towards
the GPRC.
A GPRC integer set ontains elements from a small value domain
1. . . I +J
where
I
(
J
) is the number of upstream (downstream) links of the onsidered
intersetion. Equivalently, a value domain
0. . . I +J−1
is assumed here in
order to allow for an array-based implementation that starts ounting at zero.
The subsequently provided Java ode fragments onstitute the basis of a lass
NSet
.
publi lass NSet {
// ode fragments here
}
This lass ontains a primitive and two array memb ers of integer type.
private int size;
private final int[℄ values;
private final int[℄ indies;
size
holds the number of entries in a given instane of
NSet
. The rst
size
elds of the
values
-array ontain these entries. If
indies[x℄
equals
-1
, then
x
is not ontained in the set. Otherwise,
indies[x℄
ontains the index of
x
in
values
, that is,
values[indies[x℄℄==x
if
x
is ontained in the set. During
onstrution, b oth arrays are initialized aording to the maximum size
maxSize
allowed for this set.
151
publi NSet(int maxSize) {
size = 0;
values = new int[maxSize℄;
indies = new int[maxSize℄;
for (int i = 0; i < maxSize; i++)
indies[i℄ = -1;
}
This data struture has a onstant memory requirement of
2(I+J)+1
integers.
The following three funtions provide aess to the ontent of this set. Parameter
range heks are omitted for larity.
publi boolean ontains(int value) {
return (indies[value℄ != -1);
}
publi void add(int value) {
if (!ontains(value)) {
indies[value℄ = size;
values[size℄ = value;
size++;
}
}
publi void remove(int value) {
if (ontains(value)) {
size--;
final int removedIndex = indies[value℄;
if (removedIndex != size && size > 0) {
final int movedValue = values[size℄;
values[removedIndex℄ = movedValue;
indies[movedValue℄ = removedIndex;
}
indies[value℄ = -1;
}
}
If only these three funtions were required, a single b o olean array that simply
indiates the existene of an entry would b e roughly twie as eient. However,
an iteration over the elements of suh a set would require to aess every array
entry in order to hek if the aording marker is set. The following imple-
mentation of the
iterator design pattern
[70℄ provides a more eient solution.
It is just as fast as looping only through the rst
size
elements of an array.
This is partiularly advantageous if there are relatively few entries in the data
struture.
publi NSet.Iterator iterator() {
return new NSet.Iterator();
}
152
publi lass Iterator {
private int index;
private Iterator() {
index = 0;
}
publi boolean hasNext() {
return index < size;
}
publi int next() {
return values[index++℄;
}
}
The implementation of
Iterator
as an inner lass of
NSet
is a ommon Java
tehnique that supp orts data enapsulation.
153
App endix B
Sensitivity Analysis for the
GPRC
This app endix provides alulation shemes for
∂ξ(M)/∂ξ(0)
and
∂ξ(M)/∂β
where
ξ(0)
(
ξ(M)
) is the GPRC's initial (nal) state vetor and
β= (βq)
is
a vetor of onstant onsumption rate parameters with an available Jaobian
∂ϕ(...)/∂β
. The notational overlap of
β
with the turning frations
βij
of Se-
tion 2.4 is intended but not required. The omplete notation for the GPRC an
b e found in Setion 2.3.
The subsequent analysis builds on the following preliminaries:
•
If state index
j
is the only element in
B(m)
, then the duration
θ(m)
of step
m
is
θ(m)=ξ(m)
j/ϕj(D(m))
suh that a small variation
δξ(m)
j
of resoure
j
at the b eginning of step
m
implies a likewise small variation
δθ(m)
of
θ(m)
:
B(m)={j} ⇒ δθ(m)=δξ(m)
j/ϕj(D(m)).
(B.1)
•
The onsumption rate of any resoure must b e monotonously inreasing
with the numb er of nonzero resoures:
ϕi(D∪{j})≥ϕi(D)∀i, j.
(B.2)
A resoure is denoted as
blo ked
if it is nonzero but has a zero on-
sumption rate. The monotoniity prop erty implies that (i) available and
previously non-bloked resoures annot blo k from the addition of re-
soures to
D
and (ii) one blo ked resoures stay blo ked sine
D
only
gets redued during a run of the GPRC.
•
The state of a blo ked resoure
i
has no inuene on the resoure on-
sumption rates:
ϕi(D∪{i}) = 0 ⇒ϕ(D\{i}) = ϕ(D∪{i}).
(B.3)
154
Algorithm 5
GPRC sensitivity alulation logi
1. Initialize
∂ξ(0)/∂ξ(0)
and
∂ξ(0)/∂β
. See Setion B.1.
2. At the end of every GPRC step
m= 0,1,...
, do:
(a) Calulate
∂ξ(m+1
/2)/∂ξ(0)
and
∂ξ(m+1
/2)/∂β
. See Setion B.2 and
Algorithm 6.
(b) Calulate
∂ξ(m+1)/∂ξ(0)
and
∂ξ(m+1)/∂β
. See Setion B.3, B.4, and
Algorithm 7.
3. Complete
∂ξ(M)/∂ξ(0)
and
∂ξ(M)/∂β
. See Setion B.5.
Approximations of
∂ξ(M)/∂ξ(0)
and
∂ξ(M)/∂β
are built inrementally while the
GPRC runs through
m= 0 . . . M
. For notational onveniene, these approxi-
mations are denoted by the same symb ols as the exat partial derivatives. Every
step
m
is again split in two segments of equal length
θ(m)/2
, whih neessitates
two sensitivity up dates in every step
m
and the notion of an intermediate step
m+1
/2
. This somewhat inates the presentation but is neessary to handle sit-
uations where several resoures run dry simultaneously. Algorithm 5 provides
an overview. The remainder of this app endix desrib es the details of this logi.
B.1 Initialization of Sensitivities
This is straightforward:
∂ξ(0)/∂ξ(0) =I
(identity matrix) implies that resoures
annot have interated b efore the pro ess has started, and
∂ξ(0)/∂β= 0
(all
zero matrix) states that the onsumption rate parameters
β
annot have had
an inuene b efore the onsumption has taken plae.
B.2 Calulation of
∂ξ(m+1
/2)/∂ξ(0)
and
∂ξ(m+1
/2)/∂β
If
j∈D(m)
, resoure
j
is stritly p ositive at
m+1
/2
. A variation
δξ(m)
j
annot
ause any intermediate regimes but only punhes through to
ξ(m+1
/2)
j=ξ(m)
j−
θ(m)
2ϕj(D(m))
, resulting in
δξ(m+1
/2)
j=δξ(m)
j
, as illustrated in Figure B.1(a). A
variation
δβ[m,m+1
/2]
q
of onsumption rate parameter
βq
that o urs exlusively
during
[m, m+1
/2]
generates
δξ(m+1
/2)
j=−θ(m)
2
∂ϕj(D(m))
∂βq
δβ[m,m+1
/2]
q
, as shown
in Figure B.1(b).
If
j /∈D(m)
, resoure
j
is originally zero during step
m
, whih makes it indierent
to onsumption rate variations and only allows for a positive variation
δξ(m)
j>
0
. If
ϕj(D(m)∪ {j}) = 0
, (B.3) ensures that
j
do es not interat with other
resoures suh that the variation only punhes through to
ξ(m+1
/2)
j
, resulting in
δξ(m+1
/2)
j=δξ(m)
j
, see Figure B.1().
155
(a) (b)
() (d)
Figure B.1: Resoure variations for rst half of GPRC sensitivity alulation
All diagrams show resoure tra jetories over GPRC time. Within eah diagram,
the left arrow represents the ausative variation, and the right arrow represents the
indued variation. Varied resoures are drawn in red, and inuened resoures (if any)
are drawn in blue. Original tra jetories are solid, and their variations are dashed.
156
If
j /∈D(m)
and
ϕj(D(m)∪{j})>0
, resoure
j
runs dry again after its variation
and a new regime
D′=D(m)∪{j}
o urs at the very b eginning of step
m
.
D′
is limited by
B′={j}
suh that (B.1) an b e used to obtain its duration
δθ′=
δξ(m)
j/ϕj(D′)
. During
δθ′
, all resoures
i∈D(m)
are redued by onsumption
rates
ϕi(D′)
instead of
ϕi(D(m))
. Equation (B.2) ensures that these resoures
do not blo k b eause of
j
's addition, whih guarantees ontinuity. This varies
ξ(m+1
/2)
i
by
δξ(m+1
/2)
i= (ϕi(D(m))−ϕi(D′))δθ′
, see Figure B.1(d).
Summarized, the eets of variations
δξ(m)
j
and
δβ[m,m+1
/2]
q
until step
m+1
/2
are:
δξ(m+1
/2)
i
δξ(m)
j
=
I(i=j)j∈D(m)∨ϕj(D′) = 0
ϕi(D(m))−ϕi(D′)
ϕj(D′)
i∈D(m)∧j /∈D(m)
...∧ϕj(D′)>0
0
otherwise
(B.4)
δξ(m+1
/2)
i
δβ[m,m+1
/2]
q
=
−θ(m)
2
∂ϕi(D(m))
∂βq
i∈D(m)
0
otherwise
(B.5)
where
I(A)
is one if
A
is true and zero if
A
is false. The full sensitivities until
step
m+1
/2
an now reursively b e evaluated via
∂ξ(m+1
/2)
i
∂ξ(0) =X
j
δξ(m+1
/2)
i
δξ(m)
j
∂ξ(m)
j
∂ξ(0)
(B.6)
∂ξ(m+1
/2)
i
∂β=δξ(m+1
/2)
i
δβ[m,m+1
/2]+X
j
δξ(m+1
/2)
i
δξ(m)
j
∂ξ(m)
j
∂β.
(B.7)
A alulation sheme for these Jaobians is given in Algorithm 6.
B.3 Calulation of
∂ξ(m+1)/∂ξ(0)
If
j∈D(m+1)
, resoure
j
is stritly p ositive at step
m+ 1
so that any variation
δξ(m+1
/2)
j
only punhes through to
ξ(m+1)
j
. Figure B.1(a) aptures a similar
situation. If
j /∈D(m)
, it originally has run dry b efore regime
D(m)
. A (p ositive)
variation
δξ(m+1
/2)
j
an only o ur if a p ositive variation
δξ(m)
j
has aused the
resoure to blo k. As stated b efore, this implies that
j
will stay blo ked without
inuening other resoures, so the variation
δξ(m+1
/2)
j
only punhes through
to
ξ(m+1)
j
, similarly to Figure B.1(). These ases an b e ombined in that
δξ(m+1)
j=δξ(m+1
/2)
j
holds for
(j∈D(m+1) ∨j /∈D(m))≡j /∈B(m)
.
If
j∈B(m)
, then
ϕj(D(m))
must have b een greater
0
, and therefore
ξ(m+1
/2)
j>0
an b e varied in b oth diretions. A p ositive variation
δξ(m+1
/2)
j
only punhes
through to
ξ(m+1)
j
, see Figure B.2(a). Given a negative variation
δξ(m+1
/2)
j
, a new
regime
D′′ =D(m)\{j}
o urs diretly b efore the end of step
m
, as illustrated
in Figure B.2(b). The new regime
D′′
is limited only by
B′′ ={j}
, so (B.1)
157
Algorithm 6
First half of GPRC sensitivity alulation
for all
j∈D(m)
, do {
∂ξ(m+1
/2)
j
∂ξ(0) =∂ξ(m)
j
∂ξ(0)
∂ξ(m+1
/2)
j
∂β=∂ξ(m)
j
∂β−θ(m)
2
∂ϕj(D(m))
∂β
}
for all
j /∈D(m)
, do {
ϕ′=ϕ(D(m)∪{j})
if (
ϕ′
j= 0
) {
∂ξ(m+1
/2)
j
∂ξ(0) =∂ξ(m)
j
∂ξ(0)
∂ξ(m+1
/2)
j
∂β=∂ξ(m)
j
∂β
} else {
∂ξ(m+1
/2)
j
∂ξ(0) =0
∂ξ(m+1
/2)
j
∂β=0
for all
i∈D(m)
, do {
δξ(m+1
/2)
i
δξ(m)
j
=ϕi(D(m))−ϕ′
i
ϕ′
j
∂ξ(m+1
/2)
i
∂ξ(0) + = δξ(m+1
/2)
i
δξ(m)
j
∂ξ(m)
j
∂ξ(0)
∂ξ(m+1
/2)
i
∂β+ = δξ(m+1
/2)
i
δξ(m)
j
∂ξ(m)
j
∂β
}
}
}
158
(a) (b)
() (d)
Figure B.2: Resoure variations for seond half of GPRC sensitivity alulation
All diagrams show resoure tra jetories over GPRC time. Within eah diagram,
the left arrow represents the ausative variation, and the right arrow represents the
indued variation. Varied resoures are drawn in red, and inuened resoures (if any)
are drawn in blue. Original tra jetories are solid, and their variations are dashed.
159
an be used to obtain its duration
δθ′′ =−δξ(m+1
/2)
j/ϕj(D(m))
. (The negative
sign in this expression is owed to the fat that
δξ(m+1
/2)
j
redues
θ(m)
and that
δθ′′
is the negative of this redution.) During
δθ′′
, all states
i∈D(m)
,
i6=j
,
are redued by onsumption rates
ϕi(D′′)
instead of
ϕi(D(m))
. This varies the
subsequent
ξ(m+1)
i
by
δξ(m+1)
i= (ϕi(D(m))−ϕi(D′′))δθ′′
. If a suhlike aeted
i
b elongs to
B(m)
itself, (B.2) ensures that
ϕi(D(m))≥ϕi(D(m)\{j})
suh that
δξ(m+1)
i≥0
results from a negative variation
δξ(m+1
/2)
j<0
. This eliminates the
p ossibility of additional regime o urrenes at the end of
D′′
.
Averaging the sensitivities for p ositive and negative variations
δξ(m+1
/2)
j
, one
obtains
δξ(m+1)
i
δξ(m+1
/2)
j
=
1i=j /∈B(m)
1/2i=j∈B(m)
ϕi(D′′)−ϕi(D(m))
2ϕj(D(m))i6=j∧i∈D(m)∧j∈B(m)
0
otherwise.
(B.8)
This allows to alulate the full sensitivities via
∂ξ(m+1)
i
∂ξ(0) =X
j
δξ(m+1)
i
δξ(m+1
/2)
j
∂ξ(m+1
/2)
j
∂ξ(0) .
(B.9)
B.4 Calulation of
∂ξ(m+1)/∂β
If
j∈D(m+1)
, resoure
j
is stritly p ositive at step
m+ 1
so that any variation
δβ[m+1
/2,m+1]
q
of parameter
βq
during
[m+1
/2, m + 1]
only aets to
ξ(m+1)
j
.
This yields
δξ(m+1)
j=−θ(m)
2
∂ϕj(D(m))
∂βq
δβ[m+1
/2,m+1]
q
, similarly to the eet
illustrated in Figure B.1(b). If
j /∈D(m)
, it is insensitive to onsumption rate
variations.
If
j∈B(m)
, resoure
j
an b e aeted by a variation
δβ[m+1
/2,m+1]
q
. If this
variation auses a derease
δϕ[m+1
/2,m+1]
j<0
of
j
's onsumption rate,
ξ(m+1)
j
inreases by
δ(m+1)
j=−θ(m)
2
∂ϕj(D(m))
∂βq
δβ[m+1
/2,m+1]
q
, see Figure B.2(). Given
a p ositive
δϕ[m+1
/2,m+1]
j
, resoure
j
is onsumed faster, whih auses a regime
D′′ =D\{j}
to o ur immediately b efore
m+ 1
. The duration of
D′′
is
δθ′′ =−∂
∂βq ξ(m+1
/2)
j
ϕj(D(m))!δβ[m+1
/2,m+1]
q
=ξ(m+1/2)
j
ϕ2
j(D(m))
∂ϕj(D(m))
∂βq
δβ[m+1
/2,m+1]
q
ξ(m+1/2)
j
ϕj(D(m))=θ(m)
2
=θ(m)
2ϕj(D(m))
∂ϕj(D(m))
∂βq
δβ[m+1
/2,m+1]
q,
(B.10)
160
see Figure B.2(d). The eet of
D′′
is idential to that desrib ed in the previous
setion.
Averaging the sensitivities for p ositive and negative variations
δβ[m+1
/2,m+1]
q
,
one obtains
δξ(m+1)
i
δβ[m+1
/2,m+1]
q
=−
θ(m)
2
∂ϕi(D(m))
∂βq
i∈D(m+1)
θ(m)
4
∂ϕi(D(m))
∂βq
i∈B(m)
0
otherwise
...−
θ(m)
2X
j∈B(m)
j6=i
δξ(m+1)
i
δξ(m+1
/2)
j
∂ϕj(D(m))
∂βq
i∈D(m)
0
otherwise
,
(B.11)
where (B.8) ould b e reused b eause of the idential eet of
D′′
in this and the
previous setion.
A alulation of the full sensitivities is now p ossible via
∂ξ(m+1)
i
∂β=δξ(m+1)
i
δβ[m+1
/2,m+1] +X
j
δξ(m+1)
i
δξ(m+1
/2)
j
∂ξ(m+1
/2)
j
∂β.
(B.12)
A logi for the synhronous alulation of the seond half of the state and
parameter sensitivities is given in Algorithm 7.
B.5 Completition of Sensitivities
When the pro ess has terminated at step
M
, the sensitivity alulations are
ompleted by a last run of Algorithm 6 in order to aount for resoure variations
around
m=M
. Beyond
M
, all resoures are either blo ked or zero and require
no further sensitivity up dates.
161
Algorithm 7
Seond half of GPRC sensitivity alulation
for all
i
, do {
if (
i∈B(m)
) {
∂ξ(m+1)
i
∂ξ(0) =1
2
∂ξ(m+1
/2)
i
∂ξ(0)
∂ξ(m+1)
i
∂β=1
2
∂ξ(m+1
/2)
i
∂β−θ(m)
4
∂ϕi(D(m))
∂β
} else {
∂ξ(m+1)
i
∂ξ(0) =∂ξ(m+1
/2)
i
∂ξ(0)
∂ξ(m+1)
i
∂β=∂ξ(m+1
/2)
i
∂β
if (
i∈D(m)
)
∂ξ(m+1)
i
∂β−=θ(m)
2
∂ϕi(D(m))
∂β
}
}
for all
j∈B(m)
, do {
ϕ′′ =ϕ(D(m)\{j})
for all
i∈D(m), i 6=j
, do {
δξ(m+1)
i
δξ(m+1
/2)
j
=ϕ′′
i−ϕi(D(m))
2ϕj(D(m))
∂ξ(m+1)
i
∂ξ(0) + = δξ(m+1)
i
δξ(m+1
/2)
j
∂ξ(m+1
/2)
j
∂ξ(0)
∂ξ(m+1)
i
∂β+ = δξ(m+1)
i
δξ(m+1
/2)
j ∂ξ(m+1
/2)
j
δβ−θ(m)
2
∂ϕj(D(m))
∂β!
}
}
162
App endix C
Calulation of Cell Velo ities
The CTM alulates ow rates diretly from ell o upanies. The elementary
relationship
q=v
is used to determine ell velo ity
v
from ow
q
and density
.
Consider a ell that holds a density
at the b eginning of its next time step
of duration
T
. The ell's length is
L
, and its maximum velo ity is
ˆv
. The
marosopi simulation logi provides in- and outow rates
q
in
and
q
out
(p er
lane) that p ersist for the duration of the next time step. The resulting density
hange is
(q
in
−q
out
)T/L
. A substitution of the average density
+ 0.5(q
in
−
q
out
)T/L
and the average ow
0.5(q
in
+q
out
)
in
v=q/
yields
v=(q
in
+q
out
)
2+ (q
in
−q
out
)T/L.
(C.1)
Two further mo diations are neessary to make this formula operational.
First, this logi fails for an empty network b eause of an undened
0/0
division.
This an b e avoided by the intro dution of small addends
δ > 0
and
δq = ˆvδ
in
v=(q
in
+q
out
) + ˆvδ
2+ (q
in
−q
out
)T/L +δ.
(C.2)
This yields
v= ˆv
for an empty network. For larger o upanies, the mo dia-
tion's inuene vanishes quikly.
Seond, the resulting velo ity is not limited by
ˆv
. Assume that
= 0 ⇒q
out
= 0
and
δ →0
. This yields
v=L/T ≥ˆv
aording to (2.11). Therefore,
v= min ˆv, (q
in
+q
out
) + ˆvδ
2+δ + (q
in
−q
out
)T/L.
(C.3)
The trunation only has an eet during transient dynamis. In stationary
onditions with
q
in
=q
out
=q
, the velo ity b eomes
v=q/
, whih annot
exeed
ˆv
of the fundamental diagram from whih
q
is obtained as a funtion of
.
All exp eriments of this dissertation are based on velo ity denition (C.3). Se-
tion 3.1.4.1 shows that the resulting vehile movements are well-synhronized
with the marosopi ow.
163
App endix D
Gridlo k Resolution
Gridlo k is a known problem in tra simulations that also o urs in reality
[56, 152℄. Sine the mo dels employed in this thesis are relatively simple and
only roughly alibrated, it is hyp othesized that a simulated gridlok is likely
to result from modeling impreisions and thus needs to b e resolved within the
simulation. For this purp ose, a simple mo diation to the tra ow dynamis
of Chapter 2 is subsequently desrib ed.
A minimum velo ity
v
min
that is smaller than the free ow sp eed of any link is
hosen. A reasonable value is the walking sp eed of 4 km/h, whih implies that
taking a ar yields some time savings over walking. Preventing velo ities b elow
v
min
b ounds the network learane time, thus resolves any gridlo k in nite
duration, and redues the risk of gridlo k o urrene by limiting queue spillovers.
The minimum veloity is enfored by two mo diations of the simulation logi.
The following presentation assumes a single-lane ell. For multiple lanes, ow
rates must b e aordingly saled.
First, the upp er ow onstraint of every ell's demand funtion is replaed by
a funtion that inreases linearly with slop e
v
min
, as illustrated in Figure D.1.
This still omplies with the demand/supply logi of the KWM sine onav-
ity is maintained. Phenomenologially, it also has little eet sine all supply
funtions still have a horizontal ow limit.
Seond, it is ensured for every ell
i
with a urrent density
i
that its outow
q
out
i
is not smaller than
v
min
i
. This is equivalent to an enfored demand
∆
min
() = v
min
that is pushed downstream whatever the ongestion level is.
The mo died upp er b ound of the demand funtion ensures that the enfored
demand never exeeds the original demand.
The seond mo diation is not onsistent with the KWM. The lower velo ity
b ound implies that b eyond a ertain density ow is an inreasing funtion of
density even in ongested onditions. Consequently, densities ab ove jam den-
sity are p ossible. Although the resulting fundamental diagram of Figure D.1
has no ounterpiee in reality, the resulting tra dynamis give a satisfatory
impression. The densities in most ells of the network stay in the feasible part
of the fundamental diagram. An inreased ow that is squeezed through rit-
ial setions is observed mainly at bottleneks and roundabouts. These lo al
164
Figure D.1: Mo died fundamental diagram
Eet of gridlo k resolution on the fundamental diagram of a homogeneous road. The
upp er ow onstraint of the demand funtion
∆()
is b ent upwards at the slop e of the
enfored demand
∆
min
()
suh that these two lines do not interset. The minimum
op eration that originally ombines demand and supply is supplemented by a lower
ow b ound that takes eet only at high densities.
ow mo diations avoid the unrealistially heavy spillbaks that may ause a
domino eet of gridlo k throughout the network.
Sine all involved funtions are ontinuous, the gridlo k-resolved tra ow
dynamis an still b e linearized.
165
App endix E
Stationary Limit of Turning
Counter Variane
This app endix derives (3.9) in Setion 3.1.3.2.
First, the variane of the left- and right-hand side of turning ounter state
equation (3.8) is noted:
xij(rTc+Tc) = wcxij(rTc) + (1 −wc)1
Tc
Tc−1
X
s=0
N
X
n=1
uij,n(rTc+s)
⇒
VAR
{xij(rTc+Tc)}=w2
c
VAR
{xij(rTc)}
+(1 −wc)2
T2
c
VAR
(Tc−1
X
s=0
N
X
n=1
uij,n(rTc+s)).
(E.1)
Assuming that
PN
n=1 uij,n(k)
is Poissonian with exp etation and variane
λij
,
the stationary limit of a turning ounter's variane results from the following
manipulations:
VAR
{xij(rTc+Tc)}=w2
c
VAR
{xij(rTc)}+(1 −wc)2
Tc
λij
⇒lim
r→∞
VAR
{xij(rTc+Tc)}=w2
clim
r→∞
VAR
{xij(rTc)}+(1 −wc)2
Tc
λij
⇒lim
r→∞
VAR
{xij(rTc)}=1−wc
1 + wc
λij
Tc
.
(E.2)
166
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