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This is to certify that the
thesis entitled

TRAVEL TIME ESTIMATION AND SHORT-TERM
PREDICTION IN URBAN ARTERIAL NE'IWORKS USING
CONDITIONAL INDEPENDENCE GRAPHS AND STATE-

SPACE NEURAL NETWORKS

presented by

AJAY KUMAR SINGH

has been accepted towards fulfillment
of the requirements for the

Master of degree in CIVIL ENGINEERING
Science

 

 

 

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TRAVEL TIME ESTIMATION AND SHORT-TERM PREDICTION IN URBAN
ARTERIAL NETWORKS USING CONDITIONAL INDEPENDENCE GRAPHS
AND STATE-SPACE NEURAL NETWORKS

By

Ajay Kumar Singh

A THESIS

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of

MASTER OF SCIENCE
Department of Civil and Environmental Engineering

2006

ABSTRACT
TRAVEL TIME ESTIMATION AND SHORT-TERM PREDICTION IN URBAN
ARTERIAL NETWORKS USING CONDITIONAL INDEPENDENCE GRAPHS AND
STATE-SPACE NEURAL NETWORKS
By

Ajay Kumar Singh

An important component of Advanced Traveler Information Systems (ATIS) and
Advanced Traffic Management Systems (ATMS) is the travel time estimation and short-
term prediction on urban arterial networks. This thesis develops robust and efficient
average travel time estimation and short-term prediction model for both congested and
non-congested conditions appearing throughout a day on a network. A State-Space
Neural Network model is proposed. An innovative implementation of Conditional
Independence graph is used to identify the independence and interaction between
observable traffic parameters that are used to estimate and predict the travel time. This
led to the selection of relevant variables from a set of independent variables for travel
time prediction. The predictive and computational performance of the Conditional
Independence graph coupled with State-Space Neural Network outperformed the
traditional State-Space Neural Network model in this study. The travel time estimation
and prediction models are developed for links and routes in an arterial network.

Keywords: Travel Time Estimation, Short-term Prediction, Urban Arterials, Networks,

Conditional Independence Graphs, State-Space Neural Networks.

Copyright by
AJAY KUMAR SINGH
2006

Dedicated to my ignorance and endeavors to overcome it

iv

ACKNOWLEDGEMENTS

I am greatly indebted to my advisor Dr. Ghassan Abu-Lebdeh for his continuous
guidance, support, and patience throughout my graduate studies. I learnt so much from
him in the area of Traffic Operations, Intelligent Traffic Control and modeling and
optimization through his courses and interactions during research. I wish to acknowledge
Dr. Francois Dion and Dr. R.V. Ramamoorthi for their invaluable comments towards
improving this thesis and for the courses they taught to me. I express my sincere regards
to Dr. Gilbert Baladi and Dr. Neeraj Buch who accepted me as a Teaching Assistant for
their courses. I whole-heartedly thank all the faculty members, former and present
graduate students, and the administrative staff of the Department of Civil and
Environmental Engineering for making my graduate studies smooth and pleasant.
I wish to acknowledge the love and support of my parents and sisters. I also thank Richa
for her love and patience throughout this work and further. I owe so much to my friends
Deeps, Devesh, Priya, Tanmay, Shishir, Tithi, and Vinay for their memorable company
and encouragement.

Ajay Kumar Singh
Michigan State University

2006

TABLE OF CONTENTS

LIST OF TABLES .................................................................................... ix

LIST OF FIGURES ................................................................................. xii

CHAPTER 1 — LITERATURE REVIEW ........................................................................... 5
1.1. Travel Time Estimation and Prediction on Freeways ........................................... 5
1.2. Travel Time Estimation and Prediction on Urban Arterials ................................. 7
1.3 Conditional Independence Graphs Applications .................................................. 9
1.4 State-Space Neural Networks Applications .......................................................... 9
l .5 Conclusions ......................................................................................................... 10

CHAPTER 2 - PROBLEM STATEMENT AND THESIS OBJECTIVE ........................ 11
2.1. Background and Definitions ............................................................................... 12
2.2. Difference between Average Travel Time Estimation and Prediction ............... 14
2.3. Formulation for Average Travel Time Estimation and Short-term Prediction...l6
2.4. Factors Affecting Average Travel Time Estimation and Prediction .................. 23
2.5. Modeling Approach for Travel Time Estimation and Short-term Prediction ..... 27

CHAPTER 3 - EXPERIMENTAL SET-UP ..................................................................... 30
3.1. Overview of TSIS-CORSIM and its suitability for this study ............................ 30
3.2. Urban Arterial Network Designed In TSIS-CORSIM ........................................ 33
3.3. Signal Control Optimization using Synchro ....................................................... 37
3.4. Definition of Traffic Variables used in study based on TSIS-CORSIM ............ 38
3.5. Data Collection fi'om TSIS-CORSIM ................................................................. 39
3.6. Miscellaneous Issues in Data Collection using TSIS-CORSIM ......................... 41

CHAPTER 4 — MULTIPLE LINEAR REGRESSION FOR AVERAGE SPEED

ESTIMATION ................................................................................................................... 44
4.1 Utility of Average Speed Estimation for Travel Time Modeling ....................... 44
4.2 Multiple Linear Regression for Average Speed Estimation ............................... 46

4.2.1 Average Speed Estimation for Major Arterials ............................................... 50
4.2.2 Average Speed Estimation for Minor Arterials ............................................... 55

CHAPTER 5 — CONDITIONAL INDEPENDENCE GRAPH BASED ANALYSIS OF

TRAVEL TIME ESTIMATION AND PREDICITION .................................................... 61
5.1 Graphical Models and Conditional Independence Graphs in Applied
Multivariate Statistics ............................................................................................... 62
5.1.1 Overview of Graphical Models and Conditional Independence Graphs ......... 62
5.1.2 Utility and Scope of Graphical Models and Conditional Independence Graphs
in Present Study ............................................................................................... 63

5.1.3 Constructing Conditional Independence Graph and Selection of Parent Input
Variables .......................................................................................................... 65

5.2 Conditional Independence Graph for Average Travel Time, Thru Movement
Estimation and Prediction ................................................................................... 72

5.3 Conditional Independence Graph for Average Travel Time, Left-turn Movement
Estimation and Prediction ........................................................................................ 81

5.4 Conditional Independence Graph for Average Travel Time, Right-turn
Movement Estimation and Prediction ....................................................................... 87

CHAPTER 6 — STATE-SPACE NEURAL NETWORKS MODELING FOR AVERAGE

TRAVEL TIME ESTIMATION AND PREDICITION .................................................... 93
6.1 Modeling Approaches for Travel Time Estimation and Prediction ................... 94
6.2 State-Space Neural Networks ............................................................................ 96

6.2.1 Recurrent Neural Networks ............................................................................. 96
6.2.2 State-Space Neural Networks .......................................................................... 97
6.3 State-Space Neural Networks Topology for Travel Time Estimation and
Prediction ................................................................................................................ 102
6.3.1 SSNN Topology for Travel Time Estimation ................................................ 103
6.3.2 SSNN Topology for Travel Time Prediction ................................................. 107
6.4 State-Space Neural Networks based on Conditional Independence Graphs for
Travel Time Estimation and Prediction .................................................................. 110

6.5 State-Space Neural Networks Training and Testing for Travel Time Estimation
and Prediction ......................................................................................................... 1 13

6.6 Conditional Independence Graph based State-Space Neural Networks for
Arterial Route .......................................................................................................... 117

vii

CHAPTER 7 — RESULTS AND DISCUSSIONS .......................................................... 119

7.1 Measures of Performance ................................................................................. 119
7.2 Results of Travel Time Estimation on Arterial Links ....................................... 122
7.2.1 Results of CI-SSNN-Thru Model on Average Travel Time Estimation for
Thru Movement ............................................................................................................... 123
7.2.2 Results of CI-SSNN-Left Model on Average Travel Time Estimation for
Left-turn Movement ......................................................................................................... 126
7.2.3 Results of CI-SSNN-Right Model on Average Travel Time Estimation for
Right-turn Movement ....................................................................................................... 130
7.3 Comparison of Results for SSNN with and without using Conditional
Independence Graph ........................................................................................................ 133
7.4 Results of Travel Time Prediction on Arterial Links ....................................... 136
7.4.1 Results of CI-SSNN-Thru Model on Training Set ..................................... 136
7.4.2 Results of CI-SSNN-Thru Model on Testing Set ....................................... 141
7.4.3 Results of CI-SSNN-Lefi Model on Training Set ...................................... 149
7.4.4 Results of CI-SSNN-Lefi Model on Testing Set ........................................ 153
7.4.5 Results of CI-SSNN-Right Model on Training Set .................................... 160
7.4.6 Results of CI-SSNN-Right Model on Testing Set ...................................... 164
7.6 Results of Travel Time Prediction on Arterial Route ....................................... 171
CHAPTER 8 — CONCLUSIONS AND SCOPE OF FUTURE RESEARCH ................. 175
8.1 Conclusions and Findings ................................................................................. 175
8.2 Limitations ........................................................................................................ 179
8.3 Scope of Future Research ................................................................................. 180
APPENDIX — MATLAB Code for State-Space Neural Network ................................... 181
REFERENCES ................................................................................................................ 185

viii

LIST OF TABLES

Table 5.1 Sample Correlation Matrix for Average Travel Time Estimation for
Through Movement ................................................................................. 75

Table 5.2 Sample Scaled Inverse Correlation Matrix for Average Travel Time
Estimation for Through Movement ............................................................... 76

Table 5.3 Approximated Samples Scaled Inverse Correlation Matrix for Average
Travel Time Estimation for Thru Movement .................................................... 77

Table 5.4 Sample Correlation Matrix for Average Travel Time Estimation for Left-
Turn Movement ....................................................................................... 83

Table 5.5 Sample Scaled Inverse Correlation Matrix for Average Travel Time
Estimation for Left-Turn Movement ............................................................. 84

Table 5.6 Approximated Sample Scaled Inverse Correlation Matrix for Average
Travel Time Estimation for Left-Tum Movement .............................................. 85

Table 5.7 Sample Correlation Matrix for Average Travel Time Estimation for Right-
Tum Movement ....................................................................................... 89

Table 5.8 Sample Scaled Inverse Correlation Matrix for Average Travel Time
Estimation for Right-Turn Movement ............................................................ 90

Table 5.9 Approximated Sample Scaled Inverse Correlation Matrix for Average
Travel Time Estimation for Right-Tum Movement ............................................ 91

Table 6.1 Number of Hidden and Context Layer Nodes used in SSNN Travel Time
Estimation Models for Through, Left, and Right-Tum Movements ........................ 107

Table 6.2 Number of Hidden and Context Layer Nodes used in SSNN Travel Time
Prediction Models for Through, Left, and Right-Tum Movements ........................ 110

ix

Table 6.3 Input and Output Variables for SSNN Travel Time Estimation Models for
Through, Left, and Right-Turn Movements ................................................... 111

Table 6.4 Input and Output Variables for SSNN Travel Time Prediction Models for
Through, Left, and Right-Turn Movements ................................................... 112

Table 6.5 Training and Testing Sets Composition for SSNN Travel Time Estimation
and Prediction Models ............................................................................ 114

Table 7.1 Performance Measure of CI-SSNN-Thru Model for Travel Time
Estimation for Through Movement on Training Set .......................................... 124

Table 7.2 Performance Measure of CI—SSNN-Thru Model for Travel Time
Estimation for Through Movement on Testing Set ........................................... 125

Table 7.3 Performance Measure of CI-SSNN-Left Model for Travel Time
Estimation for Lefi-Turn Movement on Training Set ......................................... 128

Table 7.4 Performance Measure of CI—SSNN-Left Model for Travel Time
Estimation for Left-Turn Movement on Testing Set .......................................... 130

Table 7.5 Performance Measure of CI-SSNN-Right Model for Travel Time
Estimation for Right-Turn Movement on Training Set ....................................... 131

Table 7.6 Performance Measure of CI-SSNN-Right Model for Travel Time
Estimation for Right-Turn Movement on Testing Set ........................................ 133

Table 7.7 Performance Measure of SSNN-Thru Model (State-Space Neural
Networks without using Conditional Independence Graphs) for Travel Time Estimation
for Through Movement on Training Set ........................................................ 134

Table 7.8 Performance Measure of SSNN-Thru Model (State-Space Neural
Networks without using Conditional Independence Graphs) for Travel Time Estimation
for Through Movement on Testing Set .......................................................... 134

Table 7.9 Performance Measure of CI-SSNN-Thru Model for Travel Time
Prediction for Through Movement on Training Set .......................................... 140

Table 7.10 Performance Measure of CI-SSNN-Thru Model for Travel Time
Prediction for Through Movement on Testing Set ............................................ 149

Table 7.11 Performance Measure of CI-SSNN-Left Model for Travel Time Prediction
for Left-Turn Movement on Training Set ...................................................... 153

Table 7.12 Performance Measure of CI-SSNN-Lefi Model for Travel Time Prediction
for Left-Tum Movement on Testing Set ........................................................ 160

Table 7.13 Performance Measure of CI-SSNN-Right Model for Travel Time
Prediction for Right-Tum Movement on Training Set ....................................... 164

Table 7.14 Performance Measure of CI-SSNN-Right Model for Travel Time
Prediction for Right-Tum Movement on Testing Set ......................................... 171

Table 7.15 Performance Measure of CI-SSNN Model for Travel Time Prediction for
Route 16-25 on Testing Set ....................................................................... 174

xi

LIST OF FIGURES

Figure 2.1 Time — Space Domain of Problem Formulation for Average Travel Time
Estimation and Short-Term Prediction ........................................................... 23

Figure 2.2 Flowchart of Modeling Approach Adopted for Travel Time Estimation
and Short-Term Prediction ......................................................................... 29

Figure 3.1 Urban Arterial Network Designed in TSIS-CORSIM for Travel Time
Estimation and Prediction .......................................................................... 34

Figure 3.2 Average Flow Rate Variations at Entry Nodes along Arterials in the
Network ............................................................................................... 36

Figure 3.3 Exclusive Left-Turn Phase with Leading Green Phasing Scheme for a
Typical Intersection in the Network of Study ................................................... 38

Figure 4.1 Training Plot between Scaled Estimated and Actual Average Speed for
Through Movement on Major Arterials .......................................................... 51

Figure 4.2 Training Plot between Scaled Estimated and Actual Average Speed for
Left-Turn Movement on Major Arterials ......................................................... 52

Figure 4.3 Training Plot between Scaled Estimated and Actual Average Speed for
Right-Tum Movement on Major Arterials ...................................................... 52

Figure 4.4 Testing Plot between Scaled Estimated and Actual Average Speed for
Thru Movement on Major Arterials ............................................................... 53

Figure 4.5 Testing Plot between Scaled Estimated and Actual Average Speed for
Left-Tum Movement on Major Arterials ......................................................... 54

Figure 4.6 Testing Plot between Scaled Estimated and Actual Average Speed for
Right-Tum Movement on Major Arterials ...................................................... 54

xii

Figure 4.7 Training Plot between Scaled Estimated and Actual Average Speed for
Through Movement on Minor Arterials ......................................................... 56

Figure 4.8 Training Plot between Scaled Estimated and Actual Average Speed for
Left-Tum Movement on Minor Arterials ........................................................ 57

Figure 4.9 Training Plot between Scaled Estimated and Actual Average Speed for
Right-Tum Movement on Minor Arterials ...................................................... 57

Figure 4.10 Testing Plot between Scaled Estimated and Actual Average Speed for
Through Movement on Minor Arterials ......................................................... 58

Figure 4.11 Testing Plot between Scaled Estimated and Actual Average Speed for
Left-Turn Movement on Minor Arterials ........................................................ 59

Figure 4.12 Testing Plot between Scaled Estimated and Actual Average Speed for
Right-Tum Movement on Minor Arterials ...................................................... 59

Figure 5.1 A Simple Conditional Independence (CI) Graph for Conditional
Independence Relations of Four Random Variables ........................................... 68

Figure 5.2 Conditional Independence Graph for Average Travel Time Estimation for
Through Movement ................................................................................. 80

Figure 5.3 Conditional Independence Graph for Average Travel Time Estimation for
Left-Tum Movement ............................................................................... 86

Figure 5.4 Conditional Independence Graph for Average Travel Time Estimation for
Right-Turn Movement .............................................................................. 92

Figure 6.1 Conditional Independence Graph Representation of a General State-Space
Model ................................................................................................ 100

Figure 6.2 A Simple Recurrent Neural Network Proposed by Elman with Context
Layer for Short-Term Memory .................................................................. 102

xiii

Figure 6.3 State-Space Neural Network Topology adopted for Travel Time
Estimation on an Arterial Link for Specific Through, Left, and Right-Turn
Movements .......................................................................................... 105

Figure 6.4 State-Space Neural Network Topology adopted for Travel Time Prediction
on an Arterial Link for Specific Through, Left, and Right-Tum Movements ............. 109

Figure 6.5 A Sample Training Graph of State-Space Neural Network Model for this
Study ................................................................................................ 1 16

Figure 6.6 An Example Arterial Route between Intersections l6 and 25 used for
Travel Time Estimation and Prediction ......................................................... 118

Figure 7 .1 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Estimation for Through Movement .............................................. 123

Figure 7.2 Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel
Time Estimation for Through Movement ...................................................... 125

Figure 7.3 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Prediction for Left-Turn Movement ............................................. 126

Figure 7.4 Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel
Time Estimation for Left-Turn Movement ..................................................... 129

Figure 7.5 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Estimation for Right-Tum Movement .......................................... 131

Figure 7.6 Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel
Time Estimation for Right-Tum Movement ................................................... 132

Figure 7.7 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Prediction for Through Movement .............................................. 137

xiv

Figure 7.8 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Through Movement on Training Set 1. . 1 38

Figure 7 .9 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Through Movement on Training Set 2.. ...138

Figure 7.10 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Through Movement on Training Set 3.. ...139

Figure 7.11 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Through Movement on Training Set 4. . ...139

Figure 7.12 Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel
Time Prediction for Thru Movement ............................................................ 142

Figure 7.13 Absolute Percentage Error of Predicted Travel Time vs. Departure Time-
Period for Thru Movement applied on Testing Set 1. ....................................... 143

Figure 7.14 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Thru Movement on Testing Set 2 .......... 144

Figure 7.15 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Through Movement on Testing Set 3 . . ....144

Figure 7.16 Absolute Percentage Error of Predicted Travel Time versus Departure
Time-Period for Travel Time Prediction for Through Movement on Testing Set 4. . ....145

Figure 7.17 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Through Movement on Testing Set 1 ....................................................... 146

Figure 7.18 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Through Movement on Testing Set 2 ....................................................... 146

Figure 7.19 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Through Movement on Testing Set 3 ....................................................... 147

XV

Figure 7.20 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Through Movement on Testing Set 4 ....................................................... 147

Figure 7.21 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Prediction for Left-Turn Movement ............................................. 150

Figure 7.22 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Training Set 1. . .. . . . . ....151

Figure 7.23 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Tum Movement on Training Set 2. . . . . . . . ....151

Figure 7.24 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Training Set 3. . . . . .. . ....152

Figure 7.25 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Training Set 4.. ..... .....152

Figure 7.26 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Prediction for Left-Turn Movement ............................................. 154

Figure 7.27 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Testing Set 1 ............. 155

Figure 7.28 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Testing Set 2 ............. 156

Figure 7.29 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Testing Set 3 ............. 156

Figure 7.30 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Left-Turn Movement on Testing Set 4 ............. 157

Figure 7.31 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Left-Turn Movement on Testing Set 1 ...................................................... 157

xvi

Figure 7.32 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Left-Turn Movement on Testing Set 2 ...................................................... 158

Figure 7.33 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Left-Tum Movement on Testing Set 3 ...................................................... 158

Figure 7.34 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Left-Turn Movement on Testing Set 4 ...................................................... 159

Figure 7.35 Training Set Scatter Plot of Predicted versus Actual Travel Time for
Travel Time Prediction for Right-Tum Movement ........................................... 161

Figure 7.36 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Tum Movement on Training Set 1 .......... 162

Figure 7.37 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Turn Movement on Training Set 2 .......... 162

Figure 7.38 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Tum Movement on Training Set 3 .......... 163

Figure 7.39 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Tum Movement on Training Set 4 .......... 163

Figure 7.40 Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel
Time Prediction for Right-Tum Movement .................................................... 165

Figure 7.41 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Tum Movement on Testing Set 1. . . . . . . . ....166

Figure 7.42 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Turn Movement on Testing Set 2.... . . . . ....167

Figure 7.43 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Tum Movement on Testing Set 3. . .. . . . . ....167

xvii

Figure 7.44 Absolute Percentage Error of Predicted Travel Time versus Departure
Time for Travel Time Prediction for Right-Tum Movement on Testing Set 4.. . . . . . . ....168

Figure 7.45 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Right-Tum Movement on Testing Set 1 ................................................... 169

Figure 7.46 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Right-Tum Movement on Testing Set 2 ................................................... 169

Figure 7.47 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Right-Tum Movement on Testing Set 3 ................................................... 170

Figure 7.48 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Right-Turn Movement on Testing Set 4 ................................................... 170

Figure 7.49 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
for Route 16-25 on Testing Set .................................................................. 172

Figure 7.50 Pattern of Actual and Predicted Travel Time for Travel Time Prediction
on arterial links comprising the Route 16-25 on Testing Set ................................. 173

Figure 7.51 Absolute Percentage Error Distribution for Travel Time Prediction for
Route 16—25 on Testing Set ....................................................................... 174

xviii

Introduction

Traffic congestion is a severe problem on urban arterials and freeways. In 2003,
congestion in urban areas caused 3.7 billion hours of travel delay and 2.3 billion gallons
of wasted fuel. This is an increase of 79 million hours and 69 million gallons from 2002
and costs more than $63 billion (Schrank and Lornax 2005). Advanced Traveler
Information Systems (ATIS) and Advanced Transportation Management Systems
(ATMS) are two important components of Intelligent Transportation Systems (ITS).
ATIS and ATMS are identified as innovative solutions to traffic congestion problem in
urban areas.

Travel time is an easily perceptible measure of traffic conditions on arterials or on
freeways and is equally popular among researchers, traffic engineers, and users of the
transportation system. Travel time is a key input to the traffic operations of the system as
well as transportation planning. It indicates the overall performance of the system, a real-
time measure of traffic congestion, and an input to the traffic management strategies
(Longfoot 1991).

The main requirements of ATIS are: to help drivers make better and informed choices, to
assist drivers in avoiding congestion and unexpected delays, and to reduce the time spent
driving. This requires dissemination of reliable travel time information to drivers about
the current traffic conditions and also about near future when they actually start their
trips. The ATMS must have some sort of predictive capability in order to manage and
control the transportation system in proactive manner: “... for these systems (ATIS and

ATMS) to be effective, the generated strategies should be proactive (i.e. based on

predicted traffic conditions) as opposed to reactive, in order to avoid many undesirable .
effects such as overreaction, which reflects the situation where many travelers react to a
known current traffic condition in a similar fashion resulting in simply transferring the
congestion to another location” (Ben—Akiva, et' a1. 2001).

The traffic management strategies on arterial networks under congested and over-
saturated conditions have been proposed by various researchers. The optimal signal
control is one of the traffic management strategies applicable on arterial networks. Abu-
Lebdeh and Benekohal (2000, 2003) propose traffic management strategies based on
dynamic signal control and queue management on arterials. A recent paper by Chen and
Abu-Lebdeh (2006) presents dynamic-signal dynamic-speed traffic management strategy
for arterials. Wey and Jayakrishnan (2004) present an integer-linear programming
formulation for network traffic control scheme optimization. The studies for dynamic
traffic signal optimization usually minimize the total delay on the traffic network. This
delay is correlated with the travel time on the arterial network.

There is a need of efficient, accurate and reliable travel time estimation and prediction
methodologies for urban arterial networks. A significant amount of research for travel
time estimation and prediction on freeways and highways has been done. But, the area of
urban arterials lacks comparable research. The probable reasons are the vastness of the
arterial system, complexity of traffic behavior on arterials due to presence of signal
control devices, and lack of extensive traffic surveillance systems as compared to
freeways. This implies that travel time prediction on urban arterials should rely only on
easily available data that can be obtained in the field using traffic detection technologies

like loop detectors or other image-processing based technologies. A methodology is

required which can avoid the dependence on extensive traffic surveillance systems for
travel time estimation and prediction.

The complexity of traffic dynamics is another aspect of travel time on arterials which
should be explicitly taken into account in the modeling approach. The traffic dynamics on
arterial networks is more complex than on freeways because of the signal interruptions.
These signal interruptions cause interference to the traffic movement which is not present
in uninterrupted facilities like freeways. The traffic stream approaching a signalized
intersection undergoes a platoon compression, a complete stop during red interval, and
then a platoon expansion after the vehicles cross the signalized intersection. The
interference that is caused by this chain of compression and expansion within small
distances adds to the complexity of process. Moreover, the turning movements and
phasing scheme for each movement causes interference to vehicles throughout the length
of arterial link.

This thesis is based on the state-space notion of traffic that is suitable for urban arterials.
The state-space representation to a complex dynamical system provides insight to model
it better. The State-Space Neural Networks are a generic form of Recurrent Neural
Networks and are used to model the travel time estimation and prediction in this study.
The State-Space Neural Networks or, generally, Artificial Neural Networks (ANN) is an
efficient input-output mapping technique. One of the drawbacks of ANN is that it does
not provide the understanding of the process and the interaction among variables in the
process being modeled. So, Conditional Independence Graphs are used, which are a
powerful statistical technique to analyze the independence and interaction among

variables involved in a process. The combination of Conditional Independence graphs

and State-Space Neural Networks is implemented for the first time in this study. It
provides a robust and efficient modeling framework to estimate and predict travel time
for short-terrn in future. This modeling framework relies on data that can be easily

available in the field and hence has enormous potential in ATIS and ATMS applications.

Chapter 1

Literature Review

The research on travel time estimation and short-term prediction is similar to short-term
traffic forecasting. The short-terrn traffic forecasting includes forecasting of traffic
variables like travel time, average speed, flow rate, occupancy, queue length etc. There
has been a lot of research on travel time prediction for freeways. The application area of
arterial networks is still lagging as compared to that of freeways. Here, travel time
prediction studies have been surveyed for freeways as well as arterial networks. The
techniques of Conditional Independence graphs and State-Space Neural Networks are
used in this study. A brief survey of literature on the applications of these techniques is

presented.

1.1 Travel Time Estimation and Prediction on Freeways

A recent paper by Vlahogianni, et a1. (2004) gives a critical discussion on the short-term
traffic forecasting techniques for freeways and urban arterials for different types of
implementation like Advanced Traveler Information Systems (ATIS) and Advanced
Transportation Management Systems (ATMS). The discussion of methodologies for

traffic forecasting identifies two broad categories: parametric and non-parametric

techniques. This review asserts that the non-parametric modeling techniques like
Artificial Neural Networks (ANN) is promising for traffic forecasting problems and gives
robust and accurate models. Zwet and Rice (2004) propose a travel time prediction
scheme for a freeway section by means of linear regression with time-varying
coefficients. These coefficients are subjected to the time of day and time until start of a
vehicle on a section. The varying coefficients show the dynamic behavior of the traffic
which can also be captured by other advanced parametric and non-parametric techniques.
Ishak and Al-Deek (2002) proposed a non-linear time-series approach to make short-term
predictions of speed using the most recent speed profile at each loop detector station and
then mathematically found travel time from speed. This study is based on freeways data
and also presents statistical analysis to identify the parameters like congestion index,
rolling horizon, prediction horizon and their interaction terms with congestion index to be
significant for model’s performance. Wu, et a1. (2004) use Support Vector Regression to
predict travel time for highways in Taiwan. Chien and Kuchipudi (2003) applied Kalman
Filtering to predict the link and path based travel time based on the time-series data
collected through wireless technologies on freeways.

There is significant volume of studies related to the application of different type of ANN
in short-term travel time prediction on freeways. Mark and Sadek (2004) proposed an
ANN model for freeways to predict the experiential travel time under transient traffic
conditions, including incidents. They found out that speed appears to be the most
influential input variable for travel time prediction on freeways. A special type of
Recurrent Neural Networks (RNN) called State-Space Neural Networks (SSNN) is

employed by van Lint (2004) and van Lint, et al. (2002) for freeways travel time

prediction on the basis of flow and speed. They further show that the analysis of the
internal states and weight configurations of SSNN could develop an internal model that is

closely linked to the underlying traffic process.

1.2 Travel Time Estimation and Prediction on Urban Arterials

Travel time prediction on urban arterials is still lacking efficient models. Sisiopiku and
Rouphail (1994) presented a literature review of methods for travel time estimation on
urban arterials. They listed the traffic parameters used by various researchers that
influence the travel time on arterials and concluded that most of the existing models are
link-specific rather than estimating travel time at a section (path) level. Sisiopiku et al.
(1994) proposed the linear regression equations between travel time and percentage
occupancy for occupancies in a range of approximately 17 to 60 percent. No model for
the congested conditions which results into higher percentage occupancy is reported in
this study. Rajaraman (1998) proposed travel time estimation model that has two
components, overflow queue estimation and travel time calculation. Stathopoulos and
Karlaftis (2003) proposed a multivariate time-series state-space model for traffic flow
prediction on arterials and compare it with other time-series techniques like ARIMA
modeling. They also assert that different model specifications are appropriate for
different time periods of a day. Lin, et al. (2004) decomposed travel time on arterials into
two components, free flow travel time and delay. The total delay on an arterial is further
decomposed into link delay and intersection delay. The travel time prediction model is

based on the flow condition, the proportion of net inflows into the arterial from the cross

streets, and the signal coordination level. The inherent limitation in this model is that it
relies on existing intersection delay formulas which are not suitable for over-saturated
conditions. A recent study reported by Liu, et a1. (2006) presents a State-Space Neural
Networks and the extended Kalman filter (EKF) hybrid model for travel time prediction.
The model is developed for a test bed in the Delft, Netherlands using license plate
cameras and loop detectors for data collection of travel time and flow rates. This model is
based on a particular section of road and is not taking into account the signal control
parameters and the geometries of the arterial. This limits this model’s applicability to a
new arterial network where the Signal control and geometries are different. These studies
identify the need of a generalized travel time prediction model that explicitly takes into
account varying traffic demand, turning movements, signal control, and geometries on
urban arterial networks. The present study overcomes these limitations as the geometries
and signal control are taken as independent variables affecting travel time.

There are some research studies forecasting other traffic variables like average flow rate,
queue length, speed etc. for signalized arterials. Since, travel time is dependent on the
input variables like average flow rate, queue length, speed etc., the forecasting of these
traffic variables can be integrated with travel time estimation and prediction models.
Gang and Su (1995) developed a queue length prediction model at signalized
intersections based on given flow, occupancy, queue length and signal state from past.
Ledoux (1997) also developed models to forecast queue length at signalized intersections
from flow and queue length measurements in past time periods. These models when
integrated with travel time estimation and forecasting models can result in prediction of

more than one traffic parameter for urban arterials.

\

1.3 Conditional Independence Graphs Applications

The Conditional Independence graphs are a component of Bayesian Belief Networks
which have several applications in traffic. engineering. Conditional Independence
relationships and ideas of Bayesian forecasting are proposed by Whittaker, et al. (1997)
for predicting traffic for freeways. A dynamic state-space model is proposed in the study
with optimal state estimation is coming from the Kalman filter. Apart from this reference,
no other paper is found to explicitly use Conditional Independence graph-based models

for short-term traffic forecasting or travel time prediction.

1.4 State-Space Neural Networks Applications

The State-Space Neural Networks have been recently used for short-term travel time
prediction on freeways as well as urban arterials. The studies shown in van Lint (2004)
and van Lint, et al. (2002) use State-Space Neural Networks for freeways travel time
prediction the basis of flow and speed. A hybrid model for urban arterial travel time
prediction has been recently introduced by Liu, et al. (2006). This hybrid model is based
on State-Space Neural Networks and the extended Kalman Filter. This study also finds
the State-Space Neural Networks proposed by these researchers to provide a reliable and

generalized travel time estimation and prediction model.

1.5 Conclusions

The literature survey for this study suggests there is a need to develop a travel time
estimation and prediction model that relies onltraffic parameters that are easily obtained
in field. These traffic parameters can be obtained using basic detection technologies like
loop detectors and image processing-based technologies. The modeling approach should
explicitly take into account the geometries and signal control parameters. The inclusion
of geometries and signal control in the modeling approach will provide a generalized
model applicable to a real arterial network. A State-Space Neural Network (SSNN)
model which has been successfully employed by van Lint (2004) and van Lint, et a1.
(2002) for freeways is found promising here. The SSNN model is expected to capture the
non-linear and complex spatio-temporal relationship between traffic parameters and
travel time on a section of arterial. The Conditional Independence (CI) Graphs is
identified as a statistical tool to understand the interaction and independence among
parameters affecting travel time or any other output in a process. The information
obtained using CI graphs about the process can be used for SSNN modeling. The
combination of CI graphs and SSNN can provide a reliable and efficient travel time

prediction model for arterials.

lO

Chapter 2

Problem Statement and Research Objective

The problem of average travel time estimation and short—term prediction is a basic
component of Advanced Traveler Information Systems (ATIS) and Advanced
Transportation Management Systems (ATMS). With the advancement of traffic
surveillance infi'astructure on urban arterials and freeways, Intelligent Transportation
Systems (ITS) become an integral part of solutions to congestion and transportation
system management.

It is important that reliable and efficient methodologies are developed which can make
the best use of existing traffic surveillance information and improve traffic operations.
This study provides a prediction model that can rely on existing surveillance systems like
loop-detectors and other basic information available about the network like geometries,
and existing signal control scheme for use as input information. Thus, it avoids the
deployment of more sophisticated and costly surveillance systems which are based on
tracking individual vehicles in a traffic network to provide accurate estimates of travel
time and prevalent traffic conditions information.

This chapter introduces the urban arterial networks which is the study area for this work.
The notion of average travel time and its prediction is illustrated. The difference between
travel time estimation and prediction is made clear. The problem formulation of travel

time estimation and short-term prediction provides a conceptual base for modeling the

11

travel time on arterial networks. The dependence of average travel time on the traffic
variables like flow rate, average speed, queue length, geometries and signal control
variables, which shape the traffic conditions on an arterial network is presented. The last
section provides an overview of the different steps in the modeling framework adopted in

this research.

2.1 Background and Definitions

The short-term travel time prediction model developed is based on and is applicable to
urban arterial networks. This section provides the definition of an arterial link, arterial,
and arterial route for which the model is applicable.

An urban arterial network consists of several arterial links aligned in a geo-spatial
manner forming a grid. This grid network can be broadly represented by major and minor
streets crossing each other. The crossings in this network are signalized intersections. The
signalized intersections are main point of operations and control in the network.

First, the definition of an arterial link, arterial, arterial route, and a network is presented

below-

Definition 1: An ‘Arterial Link’ is a two-way length of the road that begins just at the end

of a signalized intersection and extends through (and includes) the next downstream

signalized intersection.

12

Definition 2: An ‘Arterial’ is the set of arterial links which are traversed by a vehicle
remaining in through movement only starting from an origin arterial link to the end of its

destination arterial link.

Definition 3: An ‘Arterial Route’ is the set of arterial links which are traversed by a
vehicle being in through movement as well as taking left and right turns as required

starting from an origin arterial link to the end of its destination arterial link.

Definition 4: An ‘Arterial Network’ is a system which contains all the arterial links in the
area of study. These arterial links are arranged such that the network symbolizes a grid

network having major arterials and minor arterials.

In this thesis, unless otherwise stated, a link means an arterial link, and a route means an
arterial route as stated in above definitions.

An arterial network is traversed by individual vehicles contained in a traffic stream. The
ATIS and ATMS applications are generally dependent on the travel time value of a traffic
stream rather than travel time experienced by an individual vehicle traversing an arterial
link. This notion of travel time of an individual vehicle and all the vehicles following a

traffic stream is stated below.

Definition 5: The ‘Individual Vehicle Travel Time’ is the travel time obtained by tracking

an individual vehicle along an arterial link. Hence, it is the property of an individual

13

vehicle and may not represent the travel time incurred on a particular arterial link or a set

of such links.

Definition 6: The ‘Average Travel Time’ is the average value of the travel time incurred
by individual vehicles if each vehicle can be tracked on an arterial link. The average

travel time is a property of an arterial link indicating the level of service of link.

Once, a prediction model is developed for the average travel time on a link, it can be
easily used for an arterial or an arterial route comprising a set of links. This study aims
for a model to be used for ATIS and ATMS applications to improve the level of service

and operations on an arterial link or an arterial route in the system.

2.2 Difference between Average Travel Time Estimation and Prediction

The travel time studies done by several researchers deal with the estimation of travel time
or its prediction for future unseen traffic conditions. The traffic conditions on an arterial
link at any time period can be represented by a set of traffic variables like average flow
rate, average speed, queue length etc. The term used for the traffic conditions on an
arterial link in this study is the ‘state’ of the link. The notion of state of the system
(arterial network comprising links) is important in this study as it acts as input variable to
estimate and predict the average travel time. The definition of the estimated and predicted

average travel time is important to state here.

14

l

Definition 7: The Estimated Average Travel Time on an arterial link or arterial route is
the ‘experiential’ travel time by the vehicles traversing that link or route. The term
‘experiential’ denotes the travel time which is already experienced by the vehicles that
have departed in the current or past time—periods when the state of the system was known

or estimated.

Definition 8: The Predicted Average Travel Time on an arterial link or arterial route is
the travel time which will be experienced by the vehicles when their departure is in a

future time-period during which the traffic conditions are unseen and unknown.

There is an inherent distinction between the estimation and prediction of average travel
time. The travel time estimation is done when the vehicle has already departed and has
experienced a travel time in its trip. This travel time is a complex function of current or
past state traffic conditions that are signified by independent variables like average flow
rate, average speed, queue length etc. These independent variables can in turn be
observed in the field directly by using traffic detection technology or can be measured
from other known variables, or even can be predicted from the past values of some
variables. The average travel time estimation problem requires a modeling technique
which is able to efficiently model the static input-output relationships (van Lint 2004).

The average travel time prediction is a different problem from the estimation as it needs a
modeling technique to capture the dynamic (time-dependent) relationship among known
traffic variables and future traffic conditions or future travel time. The prediction of travel

time is done using the current state traffic variable values for the travel time that will be

15

experienced in the future, when the future traffic conditions are unknown. Thus, travel
time prediction is more complex than travel time estimation problem as it requires the
consideration of dynamic spatio-temporal relationship that occurs between traffic
variables. This thesis provides the methodology for estimating the average travel time as

well its prediction for the short-term into the future.

2.3 Formulation for Average Travel Time Estimation and Short-Term

Prediction

Let an arterial route on a network consisting of n number of arterial links represented as

X], X 2, X 3,... X". Let tp‘ be the current time-period of departure of a vehicle which starts

from entry-point (upstream intersection) of link X1 and ends its trip after clearing the

i #
intersection that ends link X n. This time-period tp is starting from a time-instant to and

i t t *
ending at time instant t1 . So, tp = [to , t1 ]. The traffic parameters that shape prevailing

traffic conditions (like average flow rate, average Speed, queue length etc.) on the arterial

a:

network in the current departure time-period tp can be obtained in the field through

traffic detection system, or can be estimated through other already obtained traffic
variables.

It is assumed that the traffic conditions are constant within each time-period if the length
of such time-period is taken very small. In many traffic engineering studies and

particularly short-term travel time forecasting, it is assumed that traffic conditions are

16

constant within short time-periods of l to 15 minutes. The time-period of 1 minute
provides high temporal resolution to the data and may be useful if cycle-by-cyele analysis
is required. On the other end, 15 minutes is a low temporal resolution for forecasting as
the unstable traffic conditions during congestion and over-saturation may get overlooked
or averaged due to long time-period. Sisiopiku, et al. (1994) assert in a travel time
estimation study that the effects from cycle failures, short-term events, congestion built
up downstream of the subject link, and so forth cannot be detected using lS-minutes
observation periods. Mark and Sadek (2004) recommend temporal resolution of 5
minutes for travel time forecasting and find that no statistically significant increase in
performance was gained by increasing the temporal resolution of the data-set from 5
minutes time averages to 1 minute time averages. In consideration to studies on travel

time forecasting on freeways and arterials, here, the length of each time period is taken as

*

1:
5 minutes which is called the aggregation interval, (D. Thus, (D = t1 - t0 .
i
Let the known traffic conditions on a link X1 in current departure time-period tp be
i *
V(X1, tp ). The vector V(X1, tp ) which contains the variables depicting traffic conditions

*
on a link X1 in a time-period tp is called ‘State’ of an arterial link. The ‘State’ of an

arterial or arterial route is simply a vector which contains the state of each link
comprising that arterial or route during a given time-period. This generic term for traffic
conditions on a link is very important in the context of travel time estimation and
prediction because it is the set of input variables like average flow rate, average speed,

queue length etc., which are affecting the travel time. It will be seen from now on that the

17

modeling approach is trying to model the function that exists between the state of a link

and travel time on that link.

*
The average travel time to traverse a link X1 during the current departure time-period tp

*
is denoted as T'T (X 1, tp ). The problem of estimation of average travel time can now be

defined using the terminology explained in the earlier paragraphs as follows:

Definition 9: The average travel time estimation on arterial link X1 is the modeling

problem to approximate the underlying function between average travel time

t *
(TT (X 1, tp )) and the state (V(X1, tp )) on link X1 during current departure time-period

a

tp.

The average travel time prediction problem is formulated now after stating the average

travel time estimation problem. Assume that the average travel time is to be predicted for

t
a future time-period which is just starting when the current departure time-period tp ends

*
(i.e. time instant t1 ). A term called ‘prediction horizon’ denoted as A is stated, which is a

short-interval of time in the firture for which the average travel time is to be predicted.

The future departure time-period is denoted as tp+1A = [t0, t1], where, this time-period

begins at to and ends at t1. This implies that the to value of future time-period is same as
* *

t1 , the ending time instant of the current time-period. So, to = t1 . Moreover, the length

of this future departure time-period, tp+1A is simply the addition of prediction horizon

18

*

* II
(A) to the ending time of current time-period tp (t1 ). So, tp+1A = t1 +A. Since, the

future departure time-period of tp+1A is a single multiple of prediction horizon (A), it is

called as one-step future time-period and the predicted value of average travel time for
this one-step future time-period will be called the one-step predicted average travel time.

Conveniently, the prediction horizon (A) can be assumed same as aggregation interval
(O) defined earlier. So, A = O (= 5 minutes, here). This assumption is made so that the
temporal scale at which the state of a link and travel time is analyzed can be same for all
the time-periods whether it is current, past, or future. Moreover, it makes the task of

updating the state of the link and travel time values evolving with time-periods easy.

The predicted average travel time to traverse a link X1 during the one-step future

departure time-period tp+1A is denoted as TT (X 1, tp+1A). The problem of one-step

average travel time prediction can now be defined using the terminology explained in

earlier paragraphs as follows,

Definition 10: The one-step average travel time prediction on an arterial link X1 is the

modeling problem to approximate the underlying function between the one-step predicted

average travel time (TT (X 1, tp+1A)) during one-step future departure time-period, tp+1A,
t t
the current state (V(X1, tp )), and the estimated average travel time TT (X 1, tp )) on link

1:
X1 during current departure time-period tp .

l9

It is important to note in the above definition that the state or traffic conditions and the
estimated average travel time of a link in the current time period are used as inputs for the
one-step predicted average travel time which is expected to occur when unseen and

unknown traffic conditions will happen on that link in the one-step future time-period.
The average travel time prediction for any nth step in the firture can be developed on the
same basis, where n 21. Assume that the time-period in 11th steps ahead in future is
represented as tp-I-nA’ when the nth step predicted average travel time is denoted as 77’
(X 1, tp+nA)' The known information for average travel time prediction for nth step is still
a: It
the state of the link, V(X1, tp ) and estimated travel time, TT (X 1, tp ) at current time-
. "' . . . th
period tp . Moreover, the predicted average travel time till n-l steps are also taken as

Input variables for 11 average travel time prediction. The definition of average travel

. . . th . .
trrne prediction for 11 step In future 13 presented as follows:

Definition 11: The 11th step average travel time prediction on an arterial link X1 is the
modeling problem to approximate the underlying function between the nth step predicted
average travel time (TT (X 1, tp-I-nA» during the nth step future departure time-period,
tp+nA and the current state (V(X1, tp 1.)), the estimated average travel time T'I‘ (X 1, tp") on
link X1 during the current departure time-period tp *, and the predicted average travel time

till the n-lth step.

20

So, the predicted travel time at the nth firture departure time-period = function (current

state, estimated travel time at the cmrent departure time-period, and the predicted travel

time till n-rth firture time-step).

The average travel time prediction for the nth step in the future is called the Multiple-Step

prediction. The Short-terrn prediction of average travel time which is the ultimate aim of

this work can be defined as follows:

Definition 12: The Short-term average travel time prediction is solving the multiple step
average travel time prediction problem. The multiple-step average travel time prediction

is a chain process of prediction from step =1 to step n as required for the system.

This study focuses on n =1 step average travel time prediction problem. The n 21 steps
prediction can be done if required using the same methodology adopted in this study for
n=1 step. However, it is expected that the accuracy may decrease as the prediction is
done farther in n steps in the future. Ishak and Al-Deek (2002) found in a travel time
prediction study on fi'eeways that for longer prediction time-steps, less accuracy in
predictions are observed due to the dynamic nature of traffic conditions and the increased
likelihood of larger deviations fi'om actual conditions. Thus, the short-term average travel
time prediction is done in the present work in terms of one-step prediction only.

Figure 2.1 presents a pictorial form of the above formulation in a time-space domain. The

space domain shows a single link X1 and the time scale shows time-periods ranging from

t
the current departure time-period (tp ) to the n-step time-period in future ( tp+1A_ tp+2A,

21

tp+3A,...) for which the prediction is to be made. This time—space domain shows the

an:
assumed vehicle trajectory in each time period. The current departure time period, tp is

termed as ‘Current Horizon’ and future departure time-periods are termed as ‘Prediction

Horizon’. As shown and consistent with the terminology explained in this section, current

.. :-
horizon is starting from time instant t0 and ending at time-instant t1 forming time
I * * o o o a
period tp (length of tp = aggregation Interval, O). The state of link X1 representing

*
prevalent traffic conditions is shown as V(X1, tp ) in the current horizon. The first-step
*
prediction horizon starts from time-instant to (same as t1 ) and ends at t 1, forming first

step future departure time-period, tp+1A. Similarly, second-step, and third-step prediction

horizons are as Shown. Each future departure time-period is of length A. As said earlier, A

=O.

22

Cirfent First-Step Second-Step Third-Step
II . VPredictionV Prediction V/Predictionw
Horizon Horizon Horizon .
l
I

t

I---

 

A
I

- \‘
Vehicle Trajectories

 

 

 

 

 

 

l I
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I I .
I I
I I
I I
I I
I I
I
I
I I
I I I
to I I
O. I I
a) : I I I
r- I I I I
x I I I I
'2? ,A I I I I
1 a. I I I I
' *- I I I I
:3 I I I I
5 I I I I
> I I I I
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I
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‘17—. ¢—p¢——A——> q—A——><——A—>
to II II 12 t3

 

 

TIme Scale %
Figure 2.1 — Time — Space Domain of Problem Formulation for Average Travel Time

Estimation and Short-Term Prediction.

2.4 Factors Affecting Average Travel Time Estimation and Prediction

The traffic conditions on an arterial link termed as State of the link represent the set of
variables which affect average travel time. Here, the traffic variables are identified whose

values in a particular time-period can affect the average travel time experienced during

23

that time-period. These variables will form the inputs in the modeling approach for both
average travel time estimation as well as prediction.

The average travel time on a link is a complex function of the traffic demand on the link,
supply of the link (usually capacity of the intersection, also, capacity of the link if signal
spacing is large) and other factors like weather conditions, incidents, driver behavior,
vehicle composition, etc. It is obvious that an urban arterial network will have traffic
demand fluctuations depending upon the average flow rate on the network during
different times of the day. Due to this varying traffic demand it is intuitive that average
travel time will be higher for high traffic demand, resulting into low speed and lower for
low traffic demand resulting into higher speed. So, average flow rate and average speed
are required to be added as input variables affecting travel time.

The varying traffic demand at arterial links in a network interacts with the signal control
parameters like cycle length, green splits, offsets and phasing sequence. This interaction
of the traffic demand at a link and signal control scheme at the intersection has different
outcomes which depend on the traffic demand and signal control scheme itself. If the
traffic demand is low (non-congested conditions), the average travel time does not have
wide fluctuations because the demand is much lesser than the capacity. In this scenario,
the adjacent links in the arterial act independently and have very less interaction among
each other. On the other hand, when traffic demand is high (congested and over-saturated
conditions), the interaction causes queue build-up and even queue spill-over at an
intersection which results in affecting the progression on the upstream links. This
suggests including signal control parameters and progression variables like green

interval, red interval, and offsets as input variables affecting travel time. Also, presence

24

of queue length at the beginning of green at signalized intersections should be taken into
account for through and right-turning lanes and left-turning lane separately.

The other complexity in traffic behavior is due to the lane-changing among vehicles
traversing a link. This lane-changing is due to the turning movements on a link and is
associated with the driver behavior. The average travel time on a link is a complex
function of this lane-changing behavior. The number of lane-changes in a link is not
considered as input variable because it is difficult to determine in the field. But, the
turning movements which cause the lane-changing are explicitly accounted in this study.
The average flow rate and average speed should be accounted for through, left and right-
turning movements separately to model average travel time for these three types of
movements.

It is also known that average travel time is a basic function of the geometries of a link,
especially link length and posted speed limit. So, speed limit and estimates of the ideal
travel time for through, left and right-turning movement which are based on link length,
speed limit, and known green intervals for through and left-turning movements should be
included in the study.

The present work aims to find estimation and prediction models for average travel time
on a link for through, left and right-turning movements separately. The variables that act
as independent variables affecting average travel time for through, left and right-turning

movement in this study are specified below:

0 Traffic demand variables —- Average flow rate for through, left and right-turning

movements denoted as VT, VL, and VR respectively.

25

Average speed for through, left and right-turning movements denoted as ST, SL,

and SR respectively

Signal control variables — Green interval for an approach, denoted as G, and offset

between a pair of signalized intersections for an approach, denoted as Off.

Queue Length variables — Queue length at through and right-turning movement

lanes denoted as Qr. Queue length at left-turning lane denoted as QL.

Speed limit posted for a link, denoted as S.L.

Geometries of arterial link variables — Ideal travel time for through and right-
tuming movement, denoted as ITTT. ITTT is the summation of ratio of length of
link to speed limit (S.L.) plus the green interval for the left-turning movement.
Ideal travel time for the left-turning movement is denoted as ITTL. ITTL is the

summation of ratio of link length to speed limit (S.L.) plus the green interval for
the through movement and red interval for an approach. The phasing scheme for

intersections is assumed as exclusive left-tum phase with leading green in this

study. Hence, the definition of ITT T and ITTL is given according to this

assumption of leading green only.

The variables listed above are highly interrelated with each other. For example, average

speed is the result of interaction of average flow rate with signal control parameters and

queue length at signalized intersection. The fluctuations in average travel time resulting

from the interactions discussed above makes it a dynamic and complex spatial-temporal

26

function of the demand and supply of the link under consideration. This thesis aims to
understand the interaction among these factors and model the travel time so that it

accounts for this interaction.

2.5 Modeling Approach for Travel Time Estimation and Short-Term

Prediction

The travel time estimation problem is a first step for prediction of average travel time.
The estimation and short-term prediction of travel time has been done separately for
through, left and right-turning movement on an arterial link. So, three different models
have been developed in this problem, but, the methodology underlying these three models
is based on the same techniques and tools.

The literature survey shows that several modeling techniques have been used for traffic
variables forecasting and in particular travel time estimation and forecasting. The
modeling techniques used in the present work are explained in details in the following
chapters. Here, it is worth presenting a brief framework or flowchart as shown in Figure
2.2 of the entire modeling approach to prepare for the upcoming chapters. This figure
starts with problem formulation which has already been dealt in this chapter. The
conclusion of problem formulation is that travel time should be modeled explicitly for
three types of movements. The further chapters will deal with data-collection using TSIS-
CORSIM, average speed estimation for each type of movement. This estimation of
average speed estimation is needed so as to propose a travel time model that can rely on

readily available data in the field. The understanding of interaction between factors

27

affecting travel time is required and is done using a statistical technique called the
Conditional Independence (CI) graph. This technique helps particularly in refining the
state of a link and in selecting parent input variables. The parent input variables are found
sufficient to estimate and predict the travel time. Using these parent input variables,
State-Space Neural Networks (SSNN) models are developed for travel time estimation on
arterial links. The estimated travel time from the SSNN model, parent input variables
from the CI graph, and the average speed estimates are used to develop SSNN prediction
models for travel time. Once SSNN models are developed for arterial links, the results are

extended for arterial route travel time prediction.

28

 

Problem formulation for average travel time
estimation and short-term prediction

I

Need to develop three models for 01m. left
and right-tum movement on arterial link

I

Arterial network of study and Data-collection
using TSIS-CORSIM

 

 

 

 

 

 

 

 

 

 

 

I
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Average speed estimation for thru. left and
right-tum movement using Multiple Linear
Regression

. dltional independence graph for refining
State of link and sobering parent input
variables

  
     
 
   

State-Space Neural Network models for
travel time estimation for Ihru. left and right-
tum movement on arterial link

   
  
    

   
 

State-Space Neural Network models for
short-term travel time prediction for thru. left
and right-tum movement on arterial link

State—Space Neural Network models for
shortvterm travel time prediction for thin. lo“
and right-tum movement on arterial route

  

  

Figure 2.2 — Flowchart of Modeling Approach Adopted for Travel Time Estimation and

Short-Term Prediction

29

Chapter 3

Experimental Set-up

The experimental set-up in this study consists of an urban arterial network for which
travel time estimation and prediction models are to be developed. This study uses
microscopic traffic simulation software to create a hypothetical arterial network. This
arterial network is designed such that it represents a ‘real arterial network’ as closely as
possible. The experimental set-up considers different traffic variables that affect travel
time. It is tried through the experimental set-up to simulate a real arterial network by
varying geometries of the arterials as well as traffic demand.

The chapter introduces TSIS-CORSIM traffic simulation sofiware and its suitability for
this study. The arterial network as designed and data-collection for travel time study are
illustrated. Also, some issues which are related to the use of CORSLM in this study in

particular are discussed.

3.1 Overview of TSIS-CORSIM and its Suitability for this Study

An urban arterial network is designed in TSIS-CORSIM, Version 5.1. The TSIS-
CORSIM consists of a Graphical-User-Interface (GUI) based traffic network and

simulation input editor known as TRAFED. CORSIM is the microscopic traffic

30

simulation module of the TSIS-CORSIM. The microscopic simulation can be viewed in
the animation and graphics module of TSIS-CORSIM known as TRAFVU.

The travel time estimation and prediction models for urban arterial networks are still in
the developing stage. The travel time estimation and prediction requires analyzing the
interaction between a set of traffic variables which are supposed to affect the travel time.
So, a large number of experiments should be performed with changing the traffic
parameters like geometries of the network, signal control parameters, and traffic demand,
which is difficult to perform in the field and quite expensive. All these constraints require
use of traffic simulation software to perform the required experiments. Due to the above
mentioned requirements and limitations, TSIS-CORSIM is used for this study.

CORSIM is microscopic traffic simulation software that tracks individual vehicle on a
network based on car-following logic and gap-acceptance lane-changing model. Its
microscopic approach helps accurately modeling the traffic movement and steps which
result due to the control scheme at signalized intersections and varying traffic demand.
Almost all possible geometry conditions on the field can be simulated in CORSIM and
effect of incidents and other interruptions to the traffic movement can be duly accounted
for.

The primary features of CORSIM that made it suitable for the present study are listed as

follows:

0 CORSIM has an ability to model the time-varying traffic flow throughout a

sequence of ‘time-periods’ as specified by the user. Inside each ‘time-period’

value, smaller slices of time termed as ‘time-intervals’ are specified. CORSIM

31

calculates the different traffic parameters like traffic flow, average speed etc in
every time-interval. Mostly, the value of time-interval is fixed as one minute. The
traffic variables which are calculated at each of the time-interval is aggregated or
averaged for each time-period. The signal control parameters can also be varied at
each time-period in CORSIM, if some study needs to analyze the effect of
variation of signal control scheme for varying traffic demand.

CORSIM has a unique advantage to model the over-saturated traffic conditions as
compared to other empirical and analytical methods (Owen, et a1. 2000). The car-
following logic inherent in CORSIM simulates the stop and go nature of traffic
during heavy traffic demand as well as the queuing delay caused by signal
interruptions. The over-saturated conditions resulting in queue spill-over when the
queue at a downstream intersection interferes with the traffic movement on the
upstream link is also accounted in CORSIM. The present study requires traffic
demand to vary from uncongested conditions (low-volume) to the congested
conditions (high-volume) and again come back to the uncongested conditions.
This variation in traffic flow can be seen throughout the day on an urban arterial.
CORSIM captures this variation in traffic flow and provides the traffic parameters
values for each time period.

CORSIM produces an output file having detailed Measures of Effectiveness
(MOE’s) for a study. This study requires finding out traffic variables which affect
the travel time. CORSIM provides extensive report of MOE’s which helps to

analyze the travel time process.

32

3.2 Urban Arterial Network Designed in TSIS-CORSIM

An urban arterial network is deigned in TSIS-CORSIM in this study. The network is
shown in Figure 5.1 that consists of 25 signalized intersections. These signalized
intersections are numbered from 1 to 25 and are shown as smooth-lined circles. The other
intersections which are numbered from 26-45 and shown as dashed circles are the entry
and exit nodes for this arterial network. There are five arterials running in east-westbound
direction. These are numbered as 1-5, 6-10, 11-15, 16-20, and 21-25. These arterials are
major arterials as they are designed to serve higher traffic demand than the other cross-
arterials. The cross-arterials running along north-southbound direction are numbered as
1-21, 2-22, 3-23, 4-24, and 5-25. These arterials are minor arterials as are designed to
serve lesser traffic demand as compared to major arterials. Each arterial consists of four
arterial links bounded by signalized intersections on either side. For example, 1-6, 6-11,
11-16, and 16-21 are four links in arterial 1-21. It is to be noted that the notion of major
and minor arterials as taken in this study depends on the network in question. The
directions of major and minor arterials may change as the network varies. But, it is not
critical in this study as the major and minor arterials are considered equally in the
modeling approach. Hence, here, it is just a matter of naming the arterials as major and
minor based on traffic demand served.

The geometry of links comprising each arterial is varied to account for the variations
possible in any real arterial network. Moreover, this will lead to study the variation of
travel time with variables that represent geometry of links. The link length varies fi'om

1500 ft to 6000 it as shown in the Figure 5.1. All links have two lanes in each direction

33

and each intersection approach has a left-turn pocket or bay to accommodate the left-
tuming traffic. The speed limit posted on arterial links is also varied and is kept as 30

miles per hour for small links (link length of 1500 ft) and 45 miles per hour for the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

remaining links.
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C.) @924 49 O 4’5.)

Figure 3.1 - Urban Arterial Network Designed in TSIS-CORSIM for Travel Time

Estimation and Prediction.

34

The provision of time-interval and time-period in CORSIM is used in this study to
simulate varying traffic conditions that can be as close to reality as possible. There are
total 19 time-periods each of 5 minutes duration. Hence, each simulation run was made
for 95 minutes or 5700 seconds (19*5 minutes). As said earlier, CORSIM calculates
values of traffic variables at each time-interval which is specified as 1 minute here. So,
the values obtained every one minute are aggregated or averaged for a time-period.

The variation of traffic demand in the network is achieved by varying the traffic flow in
every time-period at the entry links of the network. The variation of traffic flow in
general begins from an uncongested state to a congested state and then gradually back to
the uncongested conditions. The traffic flow variation for each arterial is different but is
kept the same for both directions in an arterial. For example, the traffic flow variation is
different for arterials 6-10 and 16-20 but the same variation is used for eastbound and
westbound directions of both arterials. The traffic flow variation in the minor arterials is
different from that on the major arterials but is kept the same for all the minor arterials.
Figure 5.2 shows the traffic flow variation at five major arterials (arterials 1-5, 6-10, 11-
15, 16-20, and 21-25) and one representative minor arterial (arterial 3-23). As can be seen
the average flow rate is increasing from as low as 250 vehicles per hour to a maximum of
2300 vehicles per hour. Figure 5.2 clearly shows that enough variability in traffic flow is
accounted for which makes travel time estimation and prediction on this network a
challenging problem similar to that in field.

This study explicitly considers turning movements at signalized intersections. Each

intersection is having average flow rate for through, left and right-turning movement. At

35

each intersection, left-turning traffic is kept as 20% and right-turning as 10% of total
traffic flow which is input at the entry node of each arterial.
The signal control in the network is optimized using Synchro which is a macroscopic

traffic software. The signal optimization and control phasing scheme are covered in the

next section.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2600 .
-o—Artenal 1-5
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2200 . . ‘ ‘. " VOIUIne
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Time-Scale (seconds)
Figure 3.2 — Average Flow Rate Variations at Entry Nodes along Arterials in the

Network.

36

3.3 Signal Control Optimization using Synchro

Synchro is a software application for optimizing traffic signal timing and performing
capacity analysis. The software optimizes splits, offsets, and cycle lengths for individual
intersections, arterial, or a complete network (http://www.trafficware.eom/synchro.htm).

Its wide popularity among researchers and practitioners and its efficiency makes it
suitable for signal control optimization for this study as well. The optimized signal
control scheme is obtained from Synchro for the urban arterial network shown in Figure
5.1. This optimized signal control scheme is then exported to TSIS-CORSIM. The cycle
length, green splits, and offsets are optimized at intersection and network level for the
peak-hour traffic flow in the network. The pre-timed signals are incorporated in this study
which is installed in most of the urban networks. The phasing scheme given by Synchro
for a typical intersection (Intersection 9) in the network is shown in figure 5.3. This
phasing scheme is an exclusive left-tum phase with leading green which has four phases.
Since, the left-turn bays or pockets are provided in the geometries of network, so, this

type of phasing is logical to follow for all the intersections.

37

 

Splits and Phases: 9: Int

 

 

 

 

 

 

 

 

 

 

Figure 3.3 — Exclusive Left-Turn Phase with Leading Green Phasing Scheme for a

Typical Intersection in the Network of Study.

The signal optimization in Synchro allows green splits, offsets to be taken as explanatory
variables affecting travel time in this study. These variables are easily available to a
traffic agency responsible for a traffic network and are very easy to use these in the

model here.

3.4 Definition of Traffic Variables used in the Study based on TSIS-

CORSIM

The TSIS-CORSIM tracks each individual vehicle and aggregates the values of traffic
variables for each time-period. So, the values of traffic variables are space-mean values
or true values that are representative of traffic conditions on an entire link. The definition
of traffic variables involved in the present study as given by CORSIM is stated here

(User’s Guide, Traffic Software Integrated System (TSIS), Version 5.1):

38

0 Vehicle Trips - The number of vehicles that have been discharged from the link
since the beginning of the simulation.

0 Average Volume (vehicles per hour) - Total vehicle trips divided by the
simulation time.

0 Vehicle Miles - The vehicle trips times the length of link.

0 Total Time (vehicle-minute) - Total time on the link for all vehicles.

0 Average Speed (miles per hour) — Total vehicle miles divided by the total travel
time.

0 Maximum Queue Length by lane (vehicle) - The maximum queue length that was
observed on a lane since the beginning of the simulation (i.e., 100 percentile).

0 Total Time per vehicle (seconds per vehicle) - The average travel time on a link
for each vehicle, calculated by taking the total travel time and dividing it by the
number of vehicle trips.

It is to be noted that some variables are given by CORSIM for specific time-periods and
others as cumulative for the entire simulation time. All the variables used are finally

obtained for specific time-periods by using the above definitions.

3.5 Data Collection from TSIS-CORSIM

The data set having average travel time (seconds per vehicle) and the traffic variables
affecting average travel time are obtained from CORSIM. The traffic variables that affect

travel time are already mentioned in Chapter 2 (see Section 2.4).

39

A total of two CORSIM runs were done for data collection with different random seeds.
The random seeds are varied in CORSIM runs for headway seed, vehicle seed, and traffic
seed. The difference in random seeds provides data-sets which represent different traffic
demand patterns for the same network. The data set obtained from the first CORSIM run
is termed as ‘training set’ and the second run data set as ‘testing set’. In all modeling
steps (i.e. multiple linear regression of average speed, Conditional Independence graphs
analysis, and State-Space Neural Networks modeling) presented in Chapters 4-6, models
are developed from the training set only, and are tested using the testing set to evaluate
the efficiency of the models when new traffic patterns are presented (i.e. model’s ability
to generalize).

The modeling steps are limited to four arterials selected from the arterial network shown
in Figure 5.1. The selected arterials are 6-10, 16-20, 21-25, and 3-23. The entire modeling
deals with traffic conditions and travel times on the links of these arterials only. This is
done just to reduce the amount of time and effort needed to account for the entire
network. The consideration of these four arterials is sufficient to evaluate the efficiency
of the models developed as the variability needed in geometries, traffic demand, and
signal control is met with these four arterials.

As stated earlier (Section 3.2), each CORSIM simulation run was made for 95 minutes
(5700 seconds), where data is averaged every 5 minutes. So, 19 data-points are available
for each arterial link in one run. Here, four arterials mentioned in the above paragraph are
selected where each arterial consists of four links. So, a total of 304 data-points are
available for training and the same number for testing sets (19 data-points per link * 4

links per arterial * 4 arterials). All 304 data-points are used as the data-set in the

40

modeling steps except for the State-Space Neural Network model that was developed for
travel time prediction. The reason is that for first 5 minutes of the simulation on each
link, the travel time can not be predicted as there are no values of traffic variables for the
past 5 minutes time period earlier than the current time period (because it is the first time-
period of simulation). So, 18 data-points are used for each link for travel time prediction,
which gives a total 288 data-points for training and a similar number for testing sets (18

data-points per link * 4 links per arterial * 4 arterials).

3.6 Miscellaneous Issues in Data Collection using TSIS-CORSIM

Some issues are identified and discussed here which are encountered during the data
collection for the study. These are as follows:

0 Calibration and validation of CORSIM — An important aspect of all computer
simulation programs including CORSIM are calibration and validation whenever
the model is applied to field or a new site. This fact has been acknowledged in the
discussion of application of simulation programs (May, 1990). Sisiopiku et al.
(1994) present calibration of NETSIM (component of CORSIM for surface
streets) with field studies. They have found that observed values in the field for
variables like occupancy, capacity, flows, and delays are different from those
obtained from simulation. But, the differences between simulated and observed
values are due primarily to insufficient model calibration rather than actual
deficiency in the simulation code to properly model traffic processes. Thus, this

study also believes that traffic patterns simulated by CORSIM are approximate

41

depiction of the field situations. It is recommended to calibrate the values of
traffic variables especially average travel time for field studies wherever the
proposed model is applied.

Randomness of traflic patterns and number of random seeds — This study
develops travel time model using one CORSIM run data and tests it on dataset
obtained from a second run with different random seeds. Usually, the studies
using CORSIM simulation average data from about 2-10 random seed runs. The
model in this study does not need to estimate a value of travel time per se which is
subjected to specific traffic conditions. The idea is to obtain a dynamic model
which can learn the firnctional relationships between travel time and the varying
traffic conditions. A single run in CORSIM is sufficient to fulfill this requirement.
Even in a single run, 304 data-points are obtained where each data-point
represents the travel time as an outcome of other traffic variables. A total of 16
arterial links are used in training and testing sets to model travel time for any
arterial link in question. The testing set considered is a different traffic demand
pattern where the model is tested. Thus, sufficient variability has been taken into
account in this study to develop travel time estimation and prediction model.
Maximum Queue Length in CORSIM — According to the definition of maximum
queue length noted earlier (Section 3.4); CORSIM does not give maximum queue
length for each time-period. It identifies the maximum value of queue length
observed from the beginning of simulation even for an intermediate time-period.
This maximum queue length is not a good estimate of queue length in the

situation when simulation observes high traffic demand in the starting time-

42

periods resulting into a maximum queue length. For later time-periods, even when
traffic demand lowers, the maximum queue length stays the same. This does not
affect the model much as the inclusion of maximum queue length is done with
other traffic variables which act as inputs to the model. Hence, it is required that
during application of this model in the field, queue length should be duly
considered and accurate value of queue length is obtained. A more accurate value
of queue length will increase the efficiency and accuracy of the proposed travel
time model in this study.

Concluding this chapter, it can be said that this experimental set-up presents all the

complexity and variation which are needed to account for in any travel time study. The

following chapters deal with the modeling approach using data collected from this

experimental set-up.

43

Chapter 4

Multiple Linear Regression for Average Speed Estimation

The multiple linear regression is done here to obtain estimated average speed on arterials.
The difficulty of obtaining average speed from loop detectors is duly considered in this
approach. The proposed model provides estimated speed which is obtained by fitting a
multiple linear regression with other traffic variables which are easy to obtain in the field.
The true average speed that was obtained from CORSIM simulation is replaced by the
estimated average speed obtained here. Using this known relationship in further modeling
steps, the travel time can be estimated and predicted by using the estimated average speed

instead of the average speed obtained from the field.

4.1 Utility of Average Speed Estimation for Travel Time Modeling

Average speed is an important traffic variable affecting travel time on an arterial link.
The travel time prediction models developed for freeways consider speed as one of the
input traffic parameters (for example, see vanLint 2004 and Mark and Sadek 2004).
Several travel time estimation models developed for urban arterials as reviewed by

Sisiopiku and Rouphail (1994) use speed as an input traffic parameter.

The average speed obtained from TSIS-CORSIM in this study can be considered as the
true average speed, as it is calculated by tracking individual vehicles in the simulation.
The average speed is a traffic parameter which is difficult to observe or estimate in the
field. Some of the loop detectors calculate speed on a link. The recommended location of
loop detectors for speed estimation in the field for arterial networks is specified in Traffic
Detector Handbook (1990). But, this speed is limited to the location of the detector itself.
For example, if the detector for estimating speed is located close to the stop line of the
signalized intersection, the estimated speed is of lower value. This is because the vehicles
slow' down and even completely stop when they approach a signalized intersection during
red interval. On the other hand, if a detector is installed midway on a link, the estimated
speed will be a higher value because of the low interference to the vehicle from the signal
control and neighboring vehicles in the traffic stream. The average speed will be a true
average speed, if and only if each individual vehicle is tracked and their acceleration and
deceleration profiles are obtained. This task is fairly complicated and expensive in the
field. Moreover, it is recognized that average speed is different for different types of
movement on an arterial link. That is to say that average speed for through, left and right-
turning vehicles is different. Recognizing the complexity in finding average speed in the
field for the entire traffic stream neglecting their turning movement, it is more complex to
find average speed for each traffic movement.

This study aims to provide a modeling approach for travel time estimation and prediction
which relies only on easily available data in the field. So, it is required that average speed

is calculated from other traffic variables that are easy to obtain from existing traffic

detection technologies. The traffic variables other than average speed (like VT, VL, VR,

45

G, Off, Of, QL, S.L., ITTT, and ITTL) affecting travel time can be easily obtained in the

field. So, an intermediate modeling step is required to estimate average speed from other
known and easily obtainable traffic variables. This estimated average is used in the next

modeling steps to model travel time.

4.2 Multiple Linear Regression for Average Speed Estimation

The traffic variables that are easy to obtain in field are VT, VI, VR, G, Oflf Q7; Q1, S.L.,

ITTT, and 1771,. The function between average speed for each type of movement (i.e.

through, left and right-tum) and these traffic variables is to be obtained. The technique

used to model or approximate this function is multiple linear regression. The average

speed for through, left and right-tum movements (ST, 5],. and SR respectively) are termed
as response variables in this case, and the known traffic variables like VT, VL, VR, G, 01?:

Q7, QL, S.L., 1T1" T, and [IT L are termed as predictor variables. Due to the presence of

more than one predictor variable (here 10 predictor variables), multiple linear regression

is a promising choice.

Consider a response variable Y and its value Y ,- in the ith trial. A set of predictor variables
like X1 and X 2 is identified and Xi] and X3 are their respective values in ith trial. The

regression model which gives a relationship between Y and X1 and X 2 is an example of a

multiple linear regression model. Consider the model given below-

46

Yr =flo +131Xr°1 +fl2Xiz+ 5i
It is called a first-order multiple linear regression model. The parameters of this model

are fig, ,8 1, ,82, and the error term is 6;. The parameters ,8 1, fig are sometimes called as

partial regression coefficients because they reflect the partial effect of one predictor
variable when the other predictor variable is included in the model and is held constant

(Kutner, et al. 2004).

Assuming that the expected value of the error, i.e., E { 51-} =0, the regression function is

15:97:50 +31 X1 +132X2
The similar equation can be extended to more than two predictor variables and to 10

variables in the speed estimation here. The multiple linear regression models for this

study which need to be determined are-

ST =130T + fin VT + flzr VL+ flsr VR + .34T 0+ fl5T Off + .36T Q7"r .37T QL+ fier-L- +

fl9r 1771+ firorITTL

SL =fl0T + flIT VT + ,32T VL+ 163T VR + 3410+ 357 Off + .B6T QT+ .377“ QL+ .38TS-L- +

.B9T ITTT+fl10T1TTL

SR =fl0r + [in VT + fizr VL+ flsr‘ VR + .347 0+ fl5r Ofl + .362" Q7"r fin" QL+ fier-L- +

,39T ITTT+ fiIOTITT L

47

The fiij with appropriate notation of i and j are the parameters of these models which are
to be estimated. The value of fig is found by using the method of least squares.

Let the least square criterion be L for the regression model between Y and X1 and X 2, as

stated earlier. The value of L can be written as —
" 2
L = Z(Yi=fi0+fi1 Xi1+fi2XiZ)
i=1
According to the method of least squares, the estimators of fig, fi 1, and fig are those

values that minimize the criterion L for the given sample observations (Kutner, et al.
2004)

The multiple linear regression for average speed estimation was done using SPSS 14.0
for Windows. The stepwise regression method was used which is also called as statistical
regression. Stepwise regression is a way of computing ordinary least squares (OLS) in
stages. In stage one, the predictor (independent) variable best correlated with the response
(dependent) variable is included in the equation. In the second stage, the remaining
independent variable with the highest partial correlation coefficient with the dependent
variable, controlling for the first independent variable, is entered. This process is

repeated, until the addition of a remaining independent variable does not increase square

of correlation coefficient (R2) by a significant amount (or until all independent variables

are entered).
The data-set that is used to obtain a multiple linear regression model for average speed is

the training set as discussed earlier in Chapter 3. Recalling that training set consists of

48

data collected by first run of CORSIM simulation. The training set consists of 304 data
points. The obtained regression model is evaluated by applying it on testing set. The
testing set was obtained from second run of CORSIM simulation.
The training and testing sets are transformed by a linear function to scale in a range of 0.1
and 0.9. This transformation is also done with the dataset when it is used for the
Conditional Independence graph and State-Space Neural Networks as discussed in later
chapters. The scaling of data is required for fast and stable learning in neural networks
(Haykin 1999). The scaling of data is done for average speed estimation here just for the
sake of uniformity in data set so that obtained speed is in the required range and can be
directly fed to next modeling steps.
There are various tests for analysis of appropriateness of regression model and analysis of
variance as explained in Kutner, et al. (2004). These tests are provided by SPSS during
report generation for a model. The following statistical tests and results can be used to
evaluate the efficiency of multiple linear regression models -

o descriptive statistics including mean and standard deviation of set of variables

0 model summary

0 ANOVA

o Coefficients

0 Residual statistics
In the present travel time modeling approach, the travel time estimation and prediction is
done on the basis of the training set and is applied on the testing set. Thus, same approach
is used for evaluating the efficiency of these regression models also. The reliability of

regression model in this study is tested by doing cross-validation i.e. by applying it on a

49

testing set. The performance of regression model on testing set states the efficiency of the
model. The statistical tests mentioned above are usually required for multiple linear
regression model evaluation. However, the use of testing set for evaluating its
performance is assumed enough in the scope of present study.

As noted earlier, three regression models for average speed of through, left and right-tum
movement on arterial links were obtained. The speed estimation models were developed
for the major and minor arterials separately in the arterial network. The major and minor

arterials in the studied arterial network are stated earlier in Chapter 3 (Section 3.2).
4.2.1 Average Speed Estimation for Major Arterials

The average speed estimation for through, left, and right-turn movements on arterial links
is discussed here. First, the average speed estimation for through movement vehicles on
major arterials is presented. It is noted again that the average speed is scaled by a linear
function to lie in a range of 0.1 and 0.9. The training set is used for the regression model
development. The following regression models are obtained by using stepwise regression

method for average speed estimation on major arterials-

sT= 0.444- 0.462 QT+ 0.317S.L. + 0.593 117T- 0.398 ITTL
sL = 0.165 - 0.073 VT— 0.159 VL + 0.261 G — 0.07 Ofl— 0.07 QL+0.I57S.L. +0403 IYTL

SR - 0.458- 0.086 G - 0.396 QT + 0.336 S.L. +0. 649 ITI‘T— 0.602 1777,

50

The efficiency of the estimation models is first evaluated for the training set. This is done
by plotting best-fit line between scaled estimated and actual average speed. Figures 4.1,
4.2, and 4.3 show the plots for through, left and right -turn movement on major arterials.
The R2 values are as shown in these figures. The R2 values obtained for the training set
are reasonably well and these models are selected for the purpose here. It is to be
remembered that average speed estimation is an intermediate step in modeling travel
time. There are other variables also which act as input to travel time model. Hence, a

reasonable estimate of average speed was required and this purpose is fulfilled here.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.3 — a
/
0.2 1 A

0.1 k--

 

0.9
3 0.8
_ 0 Sealed
8. .______ Average
8 0'7 Speed
_ (miles per
E 0.6 —»——4—~ hour)
i _. . _
g ‘ 0 0 .. —B%t-Fit
g 0.4 «r ., , ° Line
< /o : . y = 0.8404x + 0.0938
g . A ‘3' R2=0.7865

 

 

 

0 0.2 0.4 0.6 0.8 1
Sealed Actual Average Speed (mlles per hour)

Figure 4.1 — Training Plot between Scaled Estimated and Actual Average Speed for

Through Movement on Major Arterials.

51

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g 0 Sealed
8. Average
: SQ!”
.. (miles per
3 hour)
i

—Best-Fit
9 Line
i
E
g

 

 

 

 

 

o I j T I
O 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

Figure 4.2 — Training Plot between Scaled Estimated and Actual Average Speed for Left-

Turn Movement on Maj or Arterials.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.9
*3- 0.8 a.--
f 0 Sealed
8. .. __ Average
3 0.7 Speed
r: (miles per
E 0.6 hour)
9 —Best-Flt
Line
g 0.4 -.. .
< 2%“. . . y = 0.7515: + 0.1213
0 3 .. ~ . R2 = 0.6795
' e e» d‘ .b
3 0.2 - . --__.
eOee a e e e
o I I I I
0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

Figure 4.3 — Training Plot between Scaled Estimated and Actual Average Speed for

Right-Tum Movement on Major Arterials.

52

The performance of the average speed estimation models needs to be tested on the testing
set. Figures 4.4, 4.5 and 4.6 show the plots between scaled estimated and scaled actual
average speed for through, left and right-tum movements on major arterials. It is seen that
R2 values for the testing set are better as compared to that of training set. This
observation is just a coincident and may be because the average speed values in the
testing set is similar to those average speed values in the training set which were
accurately captured by the model. The testing set was not used in the model development
and represents different traffic patterns than the training set. This evidence gives support
to use these models for average speed estimation. The proposed average speed estimation
models are assumed as generalized models on the basis of their comparable and even

better efficiency on testing set.

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

g 0.9
f 0 Sealed
8. 0.8 44 -— Average
Speed
3 0 7 w _, (miles per
.5. ' hour)
'0
0.6
i
g 05 —BBSI-FII
Line
2
E 0'4 ° y = 0.8654x + 0.0751
i 03 “:2: v R2 = 0.8582
"BE ‘/ W
0.2 «t -
3 ./ .
g 6.1 _
o I I I I
0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

Figure 4.4 — Testing Plot between Scaled Estimated and Actual Average Speed for

Through Movement on Maj or Arterials.

53

0.8

 

 

 

 

 

 

 

  
 

 

 

 

 

 

 

Scaled Estimated Average Speed (miles per hour)

 

 

 

. y = 0.8721x + 0.0667
R2 = 0.8597
0 T I T I Y I l
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Scaled Actual Average Speed (miles per hour)

Figure 4.5 — Testing Plot between Scaled Estimated and Actual Average Speed for Left-

Tum Movement on Major Arterials.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8
1.?
§ 6.7 .___Mfi
. . e 8 es ee ee 0 Sealed
8. ” Average
3 0.6 Speed
(miles per

3 hour)
3 0.5

,4 —Best-Frt
g 0 Line
“ 0.3 ~» — —
g y=0.748x+0.1149

.. R2 = 0.7521

3 0.2 I ————— —-—
'2 O O
3 0.1

0 r . . .

0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

Figure 4.6 — Testing Plot between Scaled Estimated and Actual Average Speed for Right-

Tum Movement on Major Arterials.

54

4.2.2 Average Speed Estimation for Minor Arterials

The average speed estimation for minor arterials is done separately fiom major arterials.
The minor arterials serve lesser traffic demand as compared to the major arterials. The
resulting average speed is considered to have different dependence on the input variables.
Due to this reason, the minor arterials were considered separately for average speed
estimation to improve the accuracy of the models.

The average speed estimation for through movement vehicles on minor arterial links is
discussed here. It is noted again that the average speed is scaled by a linear function to lie
in the range of 0.1 and 0.9. The training set is used for the regression model development.
The following regression models were obtained by using the stepwise regression method

for average speed estimation on minor arterials-

ST=0.564 - 0.286 VT - 0.115 VR + 0.126S.L. + 1.476ITTT- 1.14IITTL
SL = 0.466 - 0.179 VL — 0.481 0fl+ 0.513 S.L. — 0.104ITTL

SR . 0.575— 0.305 VT - 0.126 VR + 0.096 S.L. +1.04 ITTT - 0.829 ITTL

The efficiency of these estimation models is first evaluated for the training set. This is
done by plotting the best-fit line between scaled estimated average speed and scaled
actual average speed for each type of movement. Figures 4.7, 4.8, and 4.9 show these

plots for through, left and right —turn movements on minor arterials. The R2 values are as

55

shown in the figures. On the basis of R2 values, the proposed average speed estimation

models are found satisfactory.

 

 

 

 

 

0.9
0.8
0 Sealed
Average
0.7 Speed
(miles per
0.6 hour)

 

 

0.5 e
e .
o -—Best-Frt
0 4 A e , ° ° Line

0'3 “‘ n , ..' y = 0.963x + 0.0139
0 2 , . R2 = 0.9199

0.1 -

 

 

 

 

 

 

Scaled Estimated Average Speed (miles per hour)
0

 

 

 

0 r r r
0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

T

Figure 4.7 — Training Plot between Scaled Estimated and Actual Average Speed for

Through Movement on Minor Arterials.

56

0.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘C
3 0.7 l
f 0 Sealed
8. Average
3 0.6 . . Afi Speed
= 0% 0 o .° .. (miles per
g o. 0" . . ° hour)
2 0.5
U)
o_4 —Best-Fit
. . Line
2
<
0.3 _
8 y = 0.6762x + 0.0538
2 0 2 . , . R2 = 0.6965
. a .
E .46.,
3
g 0.1 4»
o I T I I I I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

 

Scaled Actual Average Speed (miles per hour)

Figure 4.8 - Training Plot between Scaled Estimated and Actual Average Speed for Left-

Tum Movement on Minor Arterials.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8
1.? 0.
g 0.7 4. __, +— e’ 02/.
s O . 9 SCSI“
8 ° e. e .9 Average
3 0.6 + . 4; Speed
, ’ ° 0 (milesper
E . ‘e 0 hour)
0
i 0.5 ——— — . .
. O
0 e e
04 . g ° 3 g —Best-Fit
' ’ e. 0 Line
I
‘ 0.3 ~+--— ' r
. . 4° 0 y = 0.7774x + 0.0996
. 0 R2 = 0.7976
3 0.2 — ——+
i; 0.1 4. 4--
o I I I I
0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

 

Figure 4.9 — Training Plot between Scaled Estimated and Actual Average Speed for

Right-Tum Movement on Minor Arterials.

57

The performance of the average speed estimation models is further evaluated on the
testing set. Figures 4.10, 4.11 and 4.12 show the plots between scaled estimated and

scaled actual average speed for through, left, and right-tum movements on minor

. . . 2 .
arterials. It rs again observed that the R values for testing set are better as compared to

that of the training set. The argument given for this observation for major arterials holds
true here also. The testing set was not used in the model development and represents
different traffic patterns than the training set. This evidence gives support to use these
models for average speed estimation. The proposed average speed estimation models are
assumed as generalized models on the basis of their comparable efficiency on the testing

set.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E 0.9 «4+ —
f y/ 0 Sealed
8. 0.8 .477 . . : Average
8 O . 0 Speed
= ‘ (miles per
5 . ' hour)
; e
—Beet-Fit
g Line
5 __ __
3 y = 0.9277x + 0.0426
E R2 = 0.9253
0 I I I I
0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

Figure 4.10 — Testing Plot between Scaled Estimated and Actual Average Speed for

Through Movement on Minor Arterials.

58

 

0.8

 

 

 

 

 

 

 

 

 

 

 

 

0.1

0.7
g 0 Sealed
8. Average
9‘ 0.6 Speed
.3 (miles per
.5. hour)
i 0.5
m

0.4 —Best-Fit
g. Line
9
‘ 0 3
3 .
,5 / y = 0.9505x + 0.0216
g 0 2 . A R2 = 0.9447

 

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Scaled Actual Average Speed (miles per hour)

Figure 4.11 — Testing Plot between Scaled Estimated and Actual Average Speed for Left-

Tum Movement on Minor Arterials.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8
'5‘ /
2 0.7 fl ' .
b 0 .O o 0 Sealed
8. . ~ 0 . Average
3 0.6 +730 Speed
= . e e (miles per
E, . e ° hour)
0 5 ° ‘

' O

E e . 0 e O O
O o .
g 0.4 ° °° ., —Best-F1t
g ‘ e.. 0 Line
0

< O 3 Q 0

° .‘e e
g . . y = 0.79735: + 0.1023
3 0 2 ,o R2 = 0.796

. I L
2
3 0.1 9 —— --

O I fir T I

0 0.2 0.4 0.6 0.8 1

Sealed Actual Average Speed (miles per hour)

Figure 4.12 — Testing Plot between Scaled Estimated and Actual Average Speed for

Right-Tum Movement on Minor Arterials.

59

The average speed estimation models presented in this chapter provide accurate and
reliable estimates of average speed for major and minor arterials. It is also observed that
the average speed estimation for right-turning movement is less capable as compared to
through and left-turning models in terms of accuracy. The reason for lesser accuracy in
right-turning movement models is attributed to the complexity of average speed variation
with traffic demand in these movements. The right-turning movement shares the same
lane and same signal control state as that of through movement on an arterial link. Hence,
none of the input variables except the average flow rate for right-turning movement was
distinguishing the average speed for right-turning movement. The lesser accuracy of
speed estimation for right-turning movement can impact the travel time estimation and
prediction models for right-turning movement. The results of the travel time estimation
and prediction for right-turning movement on arterial links is presented in Chapter 7

where this impact is revisited.

60

Chapter 5

Conditional Independence Graphs Based Analysis of

Travel Time Estimation and Prediction

The Conditional Independence (CI) graph is a founding stone of a series of Probabilistic
Graphical models like the Markov Networks and the Bayesian Belief Networks. This
section uses CI graph as a tool to analyze the interaction and conditional independence
between traffic variables that represent the state of a link and hence influence the travel
time. The application of the CI graph to this problem serves three-fold purposes:
understanding the travel time estimation and prediction process, selecting parent input
variables which refine the state of link, and improving the efficiency of State-Space
Neural Networks which are used as the ultimate modeling tool for this problem.

This chapter begins with an overview of graphical models and CI graphs and presents the
theory behind it. The next sections apply CI graph to analyze average travel time
estimation and prediction for through, left, and right-tum movements on an arterial link.
Much of the content of this chapter is inspired by the text on ‘Graphical Models in

Applied Multivariate Statistics’ by Whittaker (1990).

61

5.1 Graphical Models and Conditional Independence Graphs in Applied

Multivariate Statistics

5.1.1 Overview of Graphical Models and Conditional Independence Graphs

A Graphical model is a family of probability distributions that incorporates a specific set
of conditional independence constraints listed in an independence graph. Graphical
models are a statistical technique which provides a graph depicting conditional
independence among a set of variables in a process which have multivariate normal
distribution. This independence graph is called the Conditional Independence (CI) graph,
simply because it is based on the notion of conditional probability among two sets of
variables given another variable or set of variables. Thus, CI graph is a powerful way of
summarizing the independence and interaction among a set of variables in a process.

The CI graphs are constructed from knowing the pair-wise conditional independencies
among variables in a process. The tool that is needed to interpret this graph is the
Markovian property. The Markovian property based inferences can be made solely after
the construction of CI graph. As can be realized, CI graph provides that information on
how different variables in an underlying process are interacting with each other on the
basis of the conditional independence. It can be further extended to measure the amount
of information contained in one random variable about the values of the other random
variable. This information measure is called the Kullback—Leibler Information

Divergence.

62

The construction of Conditional Independence graph can be further used to develop a
Graphical Gaussian Model, which assumes a multivariate normal distribution among
variables. The graphical Gaussian model can be fitted to the obtained CI graph with
Kullback-Leibler Information Divergence to model the joint probability distribution of
variables. The techniques of Maximum Likelihood estimation and likelihood ratio tests

are used to determine the best fitting parameters in the graphical Gaussian model.

5.1.2 Utility and Scope of Graphical Models and Conditional Independence Graphs

in Present Study

In the previous section, it was noted that the Cl graphs are usually extended to fit a
graphical Gaussian model to obtain the joint probability distribution of variables. The
graphical Gaussian models are themselves a modeling technique for a process which has
normal distribution for the variables. This thesis uses CI graph and estimation of
Kullback-Leibler Information Divergence as a way to understand the process of travel
time estimation and prediction. This study does not aim to fit a graphical Gaussian model
extending the obtained CI graph. The reason is that State-Space Neural Networks
(SSNN) has been found robust and reliable models to approximate complex non-linear
processes like the average travel time estimation and prediction. So, no effort has been
done to obtain the graphical Gaussian models for average travel time estimation and
prediction. Instead, the CI graph and Kullback-Leibler Information Divergence are

combined with SSNN for three—fold reasons:

63

Understanding the travel time estimation and prediction - The SSNN is capable
of modeling a complex non-linear process because of its topology which provides
it with associative memory for capturing temporal fluctuations as well as acting
as an efficient input-output mapping tool. The associative memory of SSNN
captures the state of a link, which is discussed in detail in the next chapter. But, it
will be shown there that this associative memory is hard to decipher and is not
directly correlated with the effect of different input variables. Thus, it prohibits
the modeler from understanding the relationship and interactions among different
variables affecting travel time. This deficiency of Artificial Neural Networks in
general is called as ‘black-box’ nature. The CI graph is used to improve the
understanding of average travel time estimation and prediction process. The
concept of Kullback-Leibler Information Divergence is helpful in evaluating the
amount of information contained in traffic variables about travel time, hence
increasing the understanding of this process.

. Selecting parent input variables - The SSNN in itself does not distinguish if all
variables used are important and needed for modeling the process, or if some
variables are redundant. This deficiency does not provide the modeler any
. information on which variables are significant for modeling the process and
which variables are insignificant. The use of the Markovian property on C1 graph
selects a set of variables termed as parent input variables. These parent input
variables give sufficient information about travel time that remaining variables
are redundant and can be dropped from the input vector. The selection of parent

input variables is covered in the following sections.

64

3. Improving efficiency of SSNN — The efficiency of SSNN may decrease because of
the complexity in the topology which in turn appears because of the inclusion of
more variables than necessary. So, CI graph has been combined for the first time
in this study to reduce the number of variables needed as inputs to model the
average travel time. It is found in this study that the efficiency of the SSNN

actually increased using the information from C1 graph.

5.1.3 Constructing Conditional Independence Graphs and Selection of Parent Input

Variables

This section illustrates applying the CI graph in a process and finding the interaction and
independence between the variables involved, which have a multivariate normal
distribution. First the basic theory of conditional independence and Markovian property is
presented which is used later to interpret a CI graph. The construction steps for obtaining
a CI graph from a dataset is presented. Later on, Markovian properties are used to

interpret CI graph.

Conditional Independence - The theory of independence and conditional independence

between events or random variables is cemented into the very foundations of probability

and statistics theory.

Consider two random vectors X and Y. The conditional density function of X knowing Y

is written as,

65

fXY(x;y)
fX Y (x.'y) =

I IN»
The random vectors X and Y are independent if and only if the joint probability density

function, ny, satisfies

fXY (Ly) =fX (X)fY (y)
for all values of x and y. The relationship is denoted by X-u-Y.
Using the above relation which says that the joint probability density function of X and Y
is the product of their marginal density functions, we get,

fx| 1» (any) = fX(x) for all x.

Thus, X-u-Y if and only if the conditional and the marginal density functions are identical.
Consider three random vectors X, Y, and Z. In a similar way, the random vectors Y and Z

are conditionally independent on X if and only if
fYZlX 0.2.96) =an OH) fZlX (2:16)

for all values of y and z and for all x for which fX(x)>0. This is written as Y-u-ZIX.

Conditional Independence Graph — After stating the meaning of conditional
independence among a set of variables; consider a graph showing these variables and

conditional independencies among them.

Let X = (X 1, X 2 Xk ) denote a vector of random variables. A graph is constructed

considering each of the random variable as a node or vertex. These nodes or vertices are
either connected by an edge or left disconnected based on a certain rule that corresponds

to the conditional independence among two variables knowing remaining all other

66

variables. Let K = {1, 2,..., k} be the corresponding set of vertices which denote the
random variables. The rule followed for connecting the vertices with an edge or leaving
them disconnected in the graph is stated as follows: if a graph is constructed such that
there is no edge between two vertices whenever a pair of variables is independent given
all the remaining variables, the resulting graph is an independence graph, or more
precisely a Conditional Independence graph. Stating a formal definition of conditional

independence graph from Whittaker (1990)-

Definition 1: The conditional independence graph of X is the undirected graph G = (K,

E) where G is the graphical model, K = {1, 2, k} and (i,j) is not in the edge set E if and

only ifolXJ-lxmm.

For Example, taking k = 4; i.e. for X = (X 1, X 2, X3, X4), assume that the following
independence relations are identified among these variables: X 11X3|{X2, X4},

X 11X4|{X 2, X3} and Xz-u-X4l {X 1, X 3}. The CI graph resulting from these relations is

shown in the Figure 5.1. It is clear from this figure that the two variables which are

conditionally independent like X1 and X3 do not have any connecting edge in the graph.

The variables X) and X3 are also separated by the rest of variables (X 2, X4) given which

they are independent. The resulting graph gives a picture of the pattern of dependence or

association between the variables (Whittaker. 1990).

67

 

 

“’9

 

Figure 5.1 - A Simple Conditional Independence (CI) Graph for Conditional

Independence Relations of Four Random Variables.

In the present study, a CI graph is obtained following the construction steps as mentioned
later. Based on C1 graph, the objective is to make some conditional independencies

statements which will be similar to the statements made in the earlier example.

Markovian property - The Figure 5.1 is termed as Markov chain. The independence
relations as specified for this example are based on three Markovian properties. These
properties are listed as below (Whittaker, 1990):
- “the pairwise Markov property: that non-adjacent pairs of variables are
independent conditional on remaining variables;
' the local Markov property: that conditional only on the adjacent variables, any
variable is independent of all the remaining variables; and
I the global Markov property: that any two subsets of variables separated by a third

is independent conditionally only on variables in the third subset.”

68

The details of these Markovian properties are dealt in Whittaker (1990). Here, these are

listed and applied to understand the independence statements that can be made for a CI

graph.

Constructing conditional independence graph — The construction of the CI graph is

presented here given the values of variables involved in the process being modeled.

Suppose there are N observations of a k-dimensional random variable X = (X 1, X2, Xk)

as noted earlier. The steps for constructing a CI graph for this variable X is adapted from

Whittaker (1990) and is presented below:

Step I — Estimate the sample correlation matrix S = corr N(X).

Step 2 — Compute the inverse of the sample correlation matrix, i.e. S4. The diagonal

elements of this sample inverse correlation matrix are interpreted as being the proportion

of variation in the corresponding variable explained by regressing on the remaining or

rest of the variables. More explicitly, each diagonal element equals/ 2 , where R
(1- R )

is the multiple correlation coefficient between that variable and the rest of the variables.
Step 3- Scale the sample inverse correlation matrix such that the diagonal elements are

equal to 1. In this scaled inverse correlation matrix, the off-diagonal elements are the

negatives of partial correlation coefficients, i.e. corr N(Xi, Xj I rest), where ‘rest’ means

rest or remaining of the variables.

69

Step 4 — Set any sufficiently small element of the scaled inverse correlation matrix to
zero.

Step 5 - A graph G = (K, E) is drawn such that there are k nodes representing k
dimensions of variable X, and E denotes the set of edges between the nodes. The rule that
is followed in drawing this graph is that there is no connecting edge between two
variables if their corresponding partial correlation coefficient is zero. The resulting graph

is called a CI graph.

Parent input variables — The selection of parent input variables is one of the aims of
using CI graph in this work. The parent input variables are a set of variables obtained
from the CI graph which have sufficient information contained in them such that the

remaining variables do not add any extra information to travel time estimation and

prediction. For example, take the independence relation X1-u-X3|{X2, X4} used in earlier
example in this section. The parent input variables for modeling X] are X 2 and X4, since

knowing these two variables X1 is conditionally independent of X 3.

Kullback-Leibler Information Divergence — Kullback-Leibler Information Divergence,
1 (fig) is used to evaluate the amount of information available to discriminate between
density functions f and g. The calculation of Kullback-Leibler Information Divergence is
extended to evaluate the amount of information contained in one random variable about
the values of others. In other words, it measures the strength of the specified conditional

independencies shown in a CI graph.

70

Definition 2: The Kullback—Leibler information divergence between two probability

density functions f and g for the random vector X is

f(X)
(f .9) egg“)

where, E is the expectation taken with respect to the density f.

The measure to evaluate the information in one random vector about another is called

information proper, which is directly related to information divergence.

Definition 3 : The information in one random vector about another, the information

proper, is

Inf(X-"-Y) = 1(fxy:fX fr)

where, I is the Kullback-Leibler information divergence.

It is already mentioned that if and only if XJLY, ny = fX fy. So, 1030/; fX fy)=0, i.e.,

Inf(X-“-Y) = O in this case. This implies that if X-u-Y, no information is contained in X
about the values of Y and vice-versa.
The information divergence and the information proper are related to the partial
correlation coefficients (p) obtained fi'om the scaled inverse correlation matrix. The
following equation gives the relation between information proper and p,

p = + {1-eXP (-21nt)} ”2

Inf= -1/2 {log(1- 02)}

71

So, Inf (X-u-erest) = -l/2 {log(l- p2)}, where p = corr N(X, Y I rest).

5.2 Conditional Independence Graphs for Average Travel Time Estimation and

Prediction for Through Movement

This section deals thoroughly with the construction of CI graph for average travel time
estimation for through movement in an arterial link. Once the CI graph for travel time
estimation is obtained, the inferences for travel time prediction are made. In this study,
the CI graphs are developed for the travel time estimation only and the same selected
parent input variables are chosen for travel time prediction modeling also. The travel time
estimation and prediction modeling have the same output, i.e. the average travel time,
with the only difference being whether it is observed in current or future departure time-
period. So, the dependence of travel time on the traffic variables is the focus here
irrespective of the departure time-period in which it is observed. Hence, the selected
parent input variables by constructing the CI graph for travel time estimation can be used
for prediction modeling also.

The variables that represent the state of a link as discussed in Chapter 2 are average flow

rate for through, left, and right-turning movements (VT, V L, and VR respectively);

average speed for through, left, and right turning movements (ST, SL, and SR

respectively); green interval for the approach (G); offsets between the two signalized

intersections forming the link (017); speed limit of link (S.L.), and the ideal travel time for

through and left turn movement (117" T and ITTL, respectively). Chapter 4 deals with the

72

estimation of average speed for each type of movement fi'om basic available data. Since,
the modeling approach in this study relies on basic available data only, estimated values
of average speed are used for modeling here as well as further in the thesis.

Now, the construction of CI graph for average travel time estimation is presented for the
arterial links. The assumption that the distributions of travel time estimation and the
variables composing the state of the link are multivariate normal is made. So, all the data
can be represented with sample mean vector and sample variance-covariance matrix. The
sample correlation matrix is obtained from this variance-covariance matrix because it is
easier to interpret the correlation matrix than the variance-covariance matrix.

The sample correlation matrix obtained from the data of average travel time for through
movement and the state of the link in the current departure time period is presented in
Table 5.1. This sample correlation matrix represents the degree of linear association
between travel time and the independent variables depicting the state of the link, but it
also shows that the independent variables are correlated among themselves. Hence, the
correlation matrix does not provide useful information in the sense that if dependence of
travel time is to be analyzed taking into account inter-correlation among independent
variables.

The inverse matrix of this sample correlation matrix is found and scaled such that the
diagonal elements of the inverse matrix equal to l. The sample inverse correlation matrix
is shown in Table 5.2. The sample inverse correlation matrix gives useful information
which is used for constructing a CI graph. The off-diagonal elements of the scaled sample
inverse correlation matrix are the negatives of the partial correlation coefficients between

the corresponding pair of variables given the remaining variables. If the off-diagonal

73

elements are zero then the corresponding variables are conditionally independent given
the remaining variables. So, this scaled inverse correlation matrix is approximated such
that the elements which have values close to zero (here, <0.l) are put as zero, and the
values 2 0.1 are non-zero entities, symbolized as *. This approximated scaled inverse
correlation matrix is shown in Table 5.3 gives the information about independence

among the variables and can be conveyed by a CI graph.

74

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Assume that the CI graph G = (K, E) where, K represents the nodes and E represents the
edges between these nodes. Each variable in the travel time estimation process is
considered as a node resulting in K = 14. Now, an edge is put between two variables if
there is a non-zero entity (represented by *5) in Table 5.3. Also, no edge is put between
two variables if there is a zero value between them in this table. The obtained graph
following this rule is the CI graph for the average travel time estimation process. The
obtained CI graph is shown in Figure 5.2. As seen in the figure, there are two types of
edges between variables; directed and undirected. The directed edge is drawn if the

general understanding of this process tells that variable at the head of directed edge is

caused by variable at the tail of directed edge. For example, YT T is caused by all other

variables as they represent state of link. Similarly, if general understanding does not able
to figure out the causation, the edge is undirected edge. For example, whether G and Off
cause each other is not clear, so, undirected edge is put in between.

Figure 5.2 shows the pair of variables that are conditionally independent knowing a

variable or set of variables. The point of interest in this figure is the average travel time.

The travel time of the through movement (TT T) is shown directly interacting with eight
variables namely QT, QL, ST, SL, SR, G, S.L., and ITI‘L. Following the global Markovian
property, it can be written as {UT-"- (VT, VL, VR, Off ITTT) I (QT, QL, ST, SL. SR, G,

S.L., ITTU}. The set of variables Q7, Q1, 87, S1, SR, G, S.L., and HT L is identified as the

parent input variables set. It is suggested by definition of conditional independence that
the parent input variables are sufficient to estimate the average travel time and that no

extra information is available in the remaining variables given the parent input variables.

78

After identifying the parent input variables, the concept of Kullback-Leibler information

divergence and information proper can be used to evaluate the amount of information

contained in each of these variables about TTT. The information proper is calculated

using the relation Inf = -1/2 {log(1- p2)}, which is stated earlier. The p values are

obtained from Table 5.2. The information proper is calculated and is shown in Figure 5.2

in parentheses along with the directed edges that are connecting parent input variables to

TIT. This tells that QL and SL have almost the same amount of information about TI’T.
The variables like SR, and ITTL have almost double information about IT T as compared

to QL and SL. The green interval, G and S.L. have the least information about 17’]: Thus,

CI graph not only tells which traffic variables are parent input variables but also their
relative importance to estimate travel time.

It becomes obvious choice to use parent input variables alone instead of all the variables
which were assumed previously to affect the estimation of average travel time. As
discussed earlier in this section, the same parent input variables obtained for the travel
time estimation process are used for prediction modeling also. These parent input
variables will be used as input variables for the next modeling step, i.e. use of the State-
Space Neural Networks (SSNN) for average travel time estimation and prediction of

through movement.

79

 

 

 

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Through Movement

80

5.3 Conditional Independence Graph for Average Travel Time Estimation and

Prediction for Left-Tum Movement

The CI graph for travel time of the left-tum movement is drawn following the same
methodology of the through movement presented in Section 5.2. The sample correlation

matrix is shown in Table 5.4, which shows that almost all the variables have some

correlation with TTL, the average travel time for left-tum movement. The scaled inverse

correlation matrix is shown in Table 5.5. The scaled inverse correlation matrix is
approximated such that elements having values <0.1 are assumed to be zero entities and
elements having values 2 0.1 are non-zero entities, symbolized by *. The approximated

scaled inverse correlation matrix is presented in Table 5.6. Using this table, the CI graph

is constructed for estimation of TTL. This CI graph is presented in Figure 5.3, which can
be interpreted using the global Markovian property as {IT L J‘- (VT, VI, VR, S7, S1, Ofif
IYTT) I (QT, Q1, SR, G, S.L., HT 1)}. So, the parent input variables to be used for

estimation of average travel time, left-tum movement are QT, Q1, SR, G, S.L., ITTL.

After identifying the parent input variables, the information proper is calculated using the

relation Inf = -1/2 {log(l- p2)}, which is stated earlier. The p values are obtained from

Table 5.5. The information proper is calculated and is shown in parentheses in Figure 5.3

along with the directed edges that are connecting the parent input variables to TTL. This
shows that QT, G and S.L. have almost same amount of information about TTT. The

variable Q), has almost triple information about 7T L as compared to QT, G and S.L. The

81

variables SR and 117' L have almost five times information compared to Q7, G and S.L.,

and are most informative about TTL. Thus, the CI graph not only tells which traffic

variables are the parent input variables but also their relative importance to estimate
travel time.

As discussed earlier in the Section 5.2, the same parent input variables obtained for the
travel time estimation process are used for prediction modeling also. These parent input
variables will be used as input variables in the next modeling step, i.e. use of the State-
Space Neural Networks (SSNN) for average travel time estimation and prediction of left-

tum movement.

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Lefi-Tum Movement.

86

5.4 Conditional Independence Graph for Average Travel Time Estimation and

Prediction for Right-Tum Movement

The CI graph for travel time of the right-tum movement is drawn following the same
methodology of the through movement presented in Section 5.2. The sample correlation

matrix is shown in Table 5.7, which shows that almost all variables have some correlation

with 77);, the average travel time for right-turn movement. The scaled inverse correlation

matrix is shown in Table 5.8. The scaled inverse correlation matrix is approximated such
that elements having values <0.1 are assumed to be zero entities and elements having
values 2 0.1 are non-zero entities, symbolized by *. The approximated scaled inverse

correlation matrix is presented in Table 5.9. Using this table, the CI graph is constructed

for estimation of TTR. This CI graph is presented in Figure 5.4, which can be interpreted
using the global Markovian property as {TI‘RJL (V T: VL, VR, Ofif 1T1" T) I (QT, QL. ST, S],
SR, G, S.L., ITT 1)}. So, the parent input variables to be used for estimation of average

travel time for right-tum movement are Q7, Q1, 57, SL. SR, G, S.L., and ITTL.

After identifying the parent input variables, the information proper is calculated using the

relation Inf = -1/2 {log(l- p2)}, which is stated earlier. The p values are obtained from

Table 5.8. The information proper is calculated and is shown in parentheses in Figure 5.4

along with the directed edges that are connecting the parent input variables to T1" R- This
shows that SR and HT L have the most information about TI‘R. The least information is

contained in G and S.L. about TTR. Thus, CI graph not only tells which traffic variables

87

are the parent input variables but also their relative importance to estimate travel time.

As discussed earlier in the Section 5.2, the same parent input variables obtained for the
travel time estimation process are used for prediction modeling also. These parent input
variables will be used as input variables for the next modeling step i.e. use of the State-
Space Neural Networks (SSNN) for average travel time estimation and prediction of

right-tum movement.

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92

Chapter 6

State-Space Neural Networks Modeling for Average Travel

Time Estimation and Prediction

The Conditional Independence (CI) graphs that were developed in the previous chapter
select parent input variables required for travel time estimation and prediction. The next
modeling step is to develop the State-Space Neural Network (S SNN) models to use the
information from the CI graphs to model the travel time process.

This chapter provides a review of modeling approaches that have been used by
researchers for short-term traffic forecasting in general. The SSNN is identified as a
promising tool to model the complex nonlinear dynamic process of travel time on urban
arterial networks. The SSNN models are developed for the smallest unit of arterial
networks, i.e. arterial links. This study develops separate SSNN models based on the
same topology for three types of traffic movements possible on each arterial link i.e.
through, left, and right-tum movements. The topology of the SSNN models is discussed
and found suitable for the study. The training and testing aspects of neural network
modeling is illustrated. Finally, the travel time estimation and prediction for an arterial
route is developed using SSNN models already developed for its components, i.e. arterial

links.

93

6.1 Modeling Approaches for Travel Time Estimation and Prediction

The problem of short-term travel time prediction is similar in methodology as forecasting
traffic conditions on a transportation networks. There has been considerable research on
short-term forecasting of average flow rate, average speed, and queue length. Though,
travel time prediction is similar to the forecasting of other traffic condition indicators or
variables, it is much more complex in formulation. An extensive literature survey has
been presented by Vlahogianni, et al. (2004) discussing modeling approaches for short-
term traffic forecasting in general.

Here, three broad categories are identified in which modeling approaches for short-term
travel time prediction can be classified:

1. Univariate Time-Series Forecasting — This approach is quite popular time-series
forecasting approach used in all disciplines including traffic engineering. The
travel time is considered a random variable and is forecasted by focusing on the
second-order properties only. If Xt is a certain random variable representing travel
time at any time instant t, the expected values of E (X t) and expected products,
E(Xt+h, Xt), t = 1, 2,..., and h = 0 ,1, 2,...are specified. The properties of sequence
Xt is dependent on these expected values and expected products only, and that is
why it is refered to as second-order properties (Brockwell and Davis 2002). Thus,
here the dependence of travel time on other variables is ignored and only focused
on the second-order properties. The Autoregressive (AR), Moving Average (MA),
Autoregressive Moving Average (ARMA), and Autoregressive Integrated Moving

Average (ARIMA) family of time series models are used in this case.

94

2. Multivariate Time-Series Forecasting — This approach considers that the time
series (here, travel time series) is represented as components of some multivariate
time series Xt. Here, not only serial dependence within each component series
{Xti} exists, but also interdependence between the different component series
{Xti} and {10;}, #j (Brockwell and Davis 2002). The multivariate time-series are
modeled using multivariate ARMA processes. The State-Space models and the
associated Kalman recursions also belong to the family of multivariate time-series
forecasting and are powerful approaches.

3. Non-Parametric Modeling- Unlike the two time-series models mentioned earlier,
non-parametric modeling approach does not assume any specific functional form
for the dependent and independent variables. These are data-driven approaches
and can be again broadly classified into two modeling techniques, namely Non-
parametric regression, and Artificial Neural Networks (ANN). Various forms of
ANN have been used for the prediction of traffic conditions indicators and travel
time.

This thesis uses a variant of ANN called State-Space Neural Networks (SSNN), a generic
form of the Recurrent Neural Networks (RNN) to model travel time estimation and

prediction.

95

6.2 State-Space Neural Networks

The State-Space Neural Networks are a generic form of Recurrent Neural Networks. The
utility of SSNN lies in its topology which can be represented by state-space equations.
The notion of state is considered important in the traffic processes including travel time
estimation and prediction. The RNN and SSNN are introduced here and a brief survey of

literature is provided in these topics.

6.2.1 Recurrent Neural Networks

Recurrent Neural Networks (RNN) are a powerful class of Artificial Neural Networks
(ANN) to model non-linear dynamical systems. There have been vast applications of
RNN in system identification, signal processing, and forecasting. The Feed-forward
Neural Networks (FNN), which is a basic and general ANN, captures the system
dynamics by including present state and past state inputs as separate input nodes in the
input layer. However, RNN tends to capture the complex non-linear system dynamics by
involving feedback loop from inputs, outputs, or prediction errors. The text by Mandic
and Chambers (2001) compares RNN with FNN in terms of non-linear dynamic systems
modeling.

The RNN has two functional uses because of its topology (Haykin 1999)-

0 Associate memory and

o Input-output mapping network.

96

Simply speaking, RNN have local feedback loops (recurrent loops) from the hidden layer
or outputs and can be trained as global feed-forward networks. The three types of RNN
which are most popular in the literature are globally recurrent neural networks (GRNN),
output-feedback recurrent neural networks (OFRNN), and fully recurrent neural network
(FRNN) as discussed in Medsker and Jain(l999).

Because of the inherent features of RNN like the recurrent loops with time delays and
input-output mapping networks, a variant of RNN called State-Space Neural Networks

(SSNN) is used in this study.

6.2.2. State-Space Neural Networks

The State-Space Neural Networks (SSNN) is a generic form of RNN which has
underlined state-space representation of the process being modeled. The state-space
representation of a SSNN is mathematically stated as state-space equations. Let a non-

linear time-variant dynamic system to be noise free (Haykin 1999), then,

M”) = 4* (Ward) + Wbu(t))

y(t) = Cx(t)

where,
q, number of unit delays used to feed the output of the hidden layer back to the input
layer (order of the model),

m, number of input variables,

97

p, number of output variables,

t, a discrete time-point,

x(t), a q-by-l vector denoting the state of a nonlinear discrete -time system,
u(t), a m-by-l vector u(t) denoting the input applied to the system,

y(t), a p-by-l vector denoting the corresponding output of the system.

W,, a q-by-q matrix,

Wb, a q-by-(m+1) matrix,

C, a p-by-q matrix; and

(p is a nonlinear mapping function.

These state-space equations suppose that at each discrete time-point t, a vector valued
observation y(t) is related to x(t). The x(t) is termed as state of the system at discrete time-
point t. The second equation representing the relationship between y(t) and x(t) is called
observation equation. The observation is shown as a linear function of state of the system
(C), but this relationship can be captured by a non-linear function too. The first equation
is termed transition equation or state equation as it models the state of the system at any
time-point t. The state of the system at any time-point t+1 depends on the state at past
time-points (t), and external inputs (u(t)) to the system. The function that is modeled
between the state of the current time-point and the state in the past time-points and
external input vector is a non-linear mapping function ((p). The state of the system is not
observed directly; only through a linear or non-linear mapping. Thus, these are the hidden
internal states of the system and they stay in the so-called space hence, are called State-

Space models (l-lonkela 2001 ).

98

The conditional independence graph representation of these state—space equations is
presented by Whittaker, et al. (1997) for better understanding of these equations and
relationship between inputs, state, and output. The conditional independence graph
proposed by Whittaker, et al. (1997) is adapted and is shown in Figure 6.1. The state of
the system at any time-point t +1, x(t+1), is connected to input y(t) and the state of the
previous time-point, x(t). This dependence is shown by a connecting arrow in the figure.
The ‘past’ represents values of past state (like x(t-I), x(t-2)...) and input variables (u(t-I),
u(t-2)...). There is no edge to x(t+1) or to y(t) fi'om the past, which indicates that enough
information is held in the current state x(t) to render past information. This conditional
independence graph shows simple state-space models. In other cases, there can be an
edge from output 0(0) to the input vector (u(t)). Because of its simplicity, this graph is
discussed here. The advanced forms of state-space models will be encountered in further

discussion.

99

 

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Figure 6.1 — Conditional Independence Graph Representation of a General State-Space

Model.

Various types of SSNN models have been presented and studied by researchers. The

main difference is based on the nature of feedback loops inside the topology of a SSNN.

 

F’tasrt

)((t)

 

 

100

 

ya)

 

 

 

Elman (1990) presented a state-space neural network shown in Figure 6.2. In the Elman
network, an additional layer called context layer is included which is connected to the
hidden layer. The hidden layer values with a time-delay activate the context layer. The
dotted lines represent trainable connections whereas the solid arch between hidden and
context units represent a non-trainable connection. This non-trainable connection
between context and hidden layer has a fixed weight of 1.0. The context layer is also a
hidden layer in the sense that it interacts with the nodes that are internal to the system and
not with the outside nodes (i.e. the input nodes). The provision of a context layer is useful
as it stores the hidden layer values of past time-point. In the state-space notion, hidden
and context layers act as the state of the system in current and past time-points
respectively. Thus, the context layer provides a short-term memory to the network which

is useful in learning dynamic systems.

101

 

 

 

 

 

 

 

 

 

 

 

Input Units Context Units

 

 

 

 

 

 

Figure 6.2 — A Simple Recurrent Neural Network Proposed by Elman with Context Layer

for Short-Term Memory.

Jordan (1986) also presented a recurrent neural network where a feedback fiom the

output layer is provided to the context layer.

6.3 State-Space Neural Networks Topology for Travel Time Estimation and

Prediction

In this research, State-Space Neural Network (SSNN) models are designed for the travel
time estimation and prediction problems. The SSNN models for travel time estimation

and prediction are developed separately because of the difference in the two problems in

102

terms of time-instants at which inputs, state of system and output are considered in the
network.

The smallest unit of an urban arterial network is an arterial link, which is bounded by two
signalized intersections one at each end. An arterial link is characterized by the presence
of three types of vehicle movements: through, left and right-turning vehicles. In this
research, a SSNN topology is developed for an arterial link for each of the three

movements.

6.3.1 SSNN Topology for Travel Time Estimation

The SSNN model for travel time estimation is shown in Figure 6.3. The figure shows an
input layer whose input vector consists of input variables (U(t)) at the current departure
time-period t that affect travel time. This input layer is connected to the hidden layer by a
trainable connection, i.e. a connection whose weights are trained by the gradient descent
algorithm. The hidden layer in the network represents the state of the system at the time-
period t. This state at the time-period t activates the nodes in the context layer which
represent the state of system in the past departure time-period t-I. The context layer
nodes store the hidden nodes values with a time-delay of l. The context layer has a
trainable connection with the hidden layer which provides information about the state at
the past time-period to the hidden layer. So, the state of the system (here, an arterial link)
at the current and the past time-periods is modeled as internal hidden and context layers

respectively.

103

The hidden layer is then connected to the output layer by a trainable connection. The
output layer represents the output of the travel time process, which is the average travel
time. Since, the inputs and outputs are both at same time-periods t, this process is
modeling travel time estimation.

Three similar SSNN models are developed, one for each of the traffic movement. These
are termed SSNN-Thru, SSNN-Left, and SSNN-Right models. The output of the SSNN-
Thru model for travel time estimation of the through movement on an arterial link is the
average travel time at the current departure time-period for through movement. Similarly,
the outputs of the SSNN-Left and SSNN-Right models are the average travel time at the
current departure time-period for the left and right-turning traffic, respectively. Apart

from the difference in the outputs of three SSNN models, the inherent topology is same.

104

LINK X

 

 

Output Layer

U(t)-InputVectoronUnkx
attimeinstant't’

X|(t)—Stateothkatttne
Instant ‘t’ for Hidden Node '1'

 

LINK X

X. (H) - State of Link at time
instant ‘t-1’ for Context layer
Node ‘i’

Y (t)-OutputVectorattime
.. _______ instant ‘t'
Trainable
Weights

Input Layer Hidden Layer _ _> Non - Trainable
Weights

 

 

——>-

Figure 6.3 - State-Space Neural Network Topology adopted for Travel Time Estimation

on an Arterial Link for Specific Through, Left, and Right-Tum Movements.

105

One important issue about the topology of SSNN models is the number of nodes in the
hidden and context layers. Since, the models are developed for each arterial link; the state
of a link can be represented by one node only. Hence, there should be a single node in
each of the hidden and context layers. This is depicted in the Elman network also which
assumes a single non-linear node in the hidden and context layers representing the state
of the system. It has been argued by Rivals and Personnaz (1996) that the state variables
in a neural network can be computed with as many nonlinear hidden neurons as required.
It is also argued that in the Elman network, the nonlinear mapping function between the
observed output and the hidden layer is constrained by a single hidden neuron, which
leads generally to a non-optimal state-space representation. Following this argument in
the present SSNN models, the number of hidden neurons (which is equal to those in the
context layer) was selected after testing different numbers ranging fi'om 3 to 10 to obtain
a minimal optimal state-space representation. Hence, the SSNN models for each type of
traffic movement may have different number of hidden and context neurons in their
topology. Figure 6.3 is a generic representation for any type of traffic movement and
shows only 3 neurons in the hidden and context layers. The actual number of hidden and
context neurons for the SSNN models of the different traffic movements are presented in

Table 6.1.

106

Table 6.1 — Number of Hidden and Context Layer Nodes used in SSNN Travel Time

Estimation Models for Through, Left, and Right-Tum Movements

 

 

 

 

SSNN Travel Time Estimation Models Number of Hidden and Context
Layer Nodes

SSNN-Thru 4

SSNN-Left 10

SSNN-Right 10

 

 

 

 

6.3.2 SSNN Topology for Travel Time Prediction

The SSNN topology for short-term travel time prediction is similar to that of travel time
estimation as discussed in detail in the earlier section. This section deals mainly with the
difference in SSNN topology for the travel time prediction process. Figure 6.4 shows the
SSNN topology for the average travel time prediction on an arterial link for a specific
traffic movement. The input node consists of input vectors which are known at current
departure time-period t. The output node is the average travel time which is predicted for
one-step ahead future departure time-period t+I. The hidden nodes store the state of
arterial link for future time-period 1+1, and the past state values for time-period t are
stored in the context layer nodes.

Based on this topology, three SSNN models are developed for travel time prediction for
through, left, and right-turning movements on a link. The numbers of nodes in the hidden

and context layer are shown to be 3 just for the sake of representation. The actual number

107

of hidden and context nodes vary for the three different SSNN models that were
developed for through, left, and right-turning movement. The actual number of hidden
and context neurons for the SSNN models for the different traffic movements are

presented in Table 6.2.

108

Context Layer

LINK X

 

 

OutputLayer

U (t)-InputVectoron Linkx
at time instant ‘t‘

x.(r+1)- State or Link attime
instant 1+1' for Hidden Node
Ci.

 

LINK X

Xi(t)- State of Link attime
Instant ‘t’ for Context layer
Node 'i’

Y (t+1) - Output Vector at

------- time ins-term {Ht'
I ralnabie

dehts

Non - Trainable
Weights

 

 

Input Layer Hidden Layer

Figure 6.4 — State-Space Neural Network Topology adopted for Travel Time Prediction

on an Arterial Link for Specific Through, Left, and Right-Tum Movements.

109

Table 6.2 - Number of Hidden and Context Layer Nodes used in SSNN Travel Time

Prediction Models for Through, Left, and Right-Turn Movements

 

 

 

 

SSNN Travel Time Prediction Models Number of Hidden and Context
Layer Nodes

SSNN-Thru ' 4

SSNN-Left 10

SSNN-Right 10

 

 

 

 

6.4 State-Space Neural Networks based on Conditional Independence

Graphs for Travel Time Estimation and Prediction

In the SSNN topology, the input layer consists of one input node in each SSNN model.
This input node is fed with the input vector which consists of the set of traffic variables
that affect travel time. The Conditional Independence graph application to understand the
travel time process has the effect of refining the independent traffic variables and retains
only those variables that affect travel time. These input variables were called the parent
input variables in Chapter 5. The travel time estimation process is modeled by SSNN
models where the inputs to the model are parent input variables only (instead of all the
traffic variables). The parent input variables were identified in Chapter 5 for average
travel estimation. Table 6.3 lists the parent input variables which act as input vector to the

SSNN models along with the corresponding output of the model.

110

Table 6.3 — Input and Output Variables for SSNN Travel Time Estimation Models for

Through, Left, and Right-Turn Movements

 

SSNN Travel Input Variables (current departure Output Variable
Time period) (current departure
Estimation ‘ period)

Models

 

SSNN-Thru QT» QL ST, SL, SR, G, S.L., and 177’ L YT T

 

SSNN-Left QT, QL, SR, G, S.L., ITTL 7T1.

 

SSNN-Right QT» QL ST, SL- SR» G, S.L., and HT L 77' R

 

 

 

 

 

The SSNN models for travel time estimation use input variables as a set of parent input
variables which were selected using the conditional independence (CI) graph analysis.
The selection of parent input variables is an important component of SSNN models for
travel time estimation process. Here, the SSNN models are based on the CI graphs and
hence, these models are termed CI-SSNN models. Thus, three CI-SSNN models termed
Cl-SSNN-Thru, CI-SSNN-Left, and CI-SSNN-Right are developed for travel time
estimation on an arterial link for the three types of traffic movements.

The SSNN models for travel time prediction also include the same parent input variables
obtained fiom the CI graphs. Additionally, the input vector contains the estimated travel
time of the current departure time-period, which is the output of CI-SSNN models for
travel time estimation. Hence, travel time prediction models are also termed CI-SSNN-

Thru, CI-SSNN-Left, and CI-SSNN-Right for travel time prediction on an arterial link for

111

the three types of traffic movements. Table 6.4 lists the input and output variables for CI-

SSNN models for travel time prediction.

Table 6.4 — Input and Output Variables for SSNN Travel Time Prediction Models for

Through, Left, and Right-Tum Movements

 

SSNN Travel Input Variables (current departure Output Variable
Time Prediction period) (future departure
Models period)

 

SSNN'Thm YT T (estimated), QT, QL, ST, SL- SR. G. IT T

S.L., and [TI],

 

SSNN-Left T1" L (estimated), Q7; Q1,- SR: G, S.L.. n L

ITTL

 

SSNN-Right 77R (estimated), QT, Q1, ST- SL- SR. 0: TT R

S.L., and [TI],

 

 

 

 

 

112

6.5 Training and Testing of State-Space Neural Networks for Travel time

Estimation and Prediction

The CI-SSNN models as discussed in earlier section are trained and tested to evaluate
their performance and efficiency to model travel time. A programming code is written in
MATLAB software to create a Recurrent Neural Network based on the topology
presented earlier. After creating the RNN, the network is trained using a training set. The
input patterns are presented sequentially (retaining the exact sequence of time-periods as
it evolves) rather than as concurrent matrix (implying batch-mode of training), which is
usually done in simple Feed-forward Neural Networks (FNN). This sequential
presentation of the training set allows the network to learn temporal fluctuations in traffic
conditions affecting travel time. The code for the SSNN models used in this study is
provided in Appendix A.

The training and testing sets are obtained from the data-collection done using CORSIM
simulation. The process of obtaining training and testing sets is already covered in
Chapter 3. Here, a review is provided of the composition of these sets. The training and
testing sets consist of data or values of traffic variables for four arterials namely 6—10, 16-
20, 21-25, and 3-23 on the arterial network (refer to Figure 5.1 in Chapter 5). Each of
these arterials contains 4 links. The CORSIM simulation was run two times with different
random seeds. The first simulation run is taken as training set and the second as testing

set.

113

The features of the training and testing sets for travel time estimation and prediction are
presented in Table 6.5. It specifies the CORSIM run and data-points that each set is based
on.

It is noted in this study that no validation set is used apart from the training and testing
sets. The reason that the validation set is generally used in Artificial Neural Networks is
to ensure that the neural network model is not over-trained, so that it performs
comparably well on both training and testing sets. The validation set is not used in this
study due to the lack of data and moreover, sufficient number of trials have been done to
obtain a neural network model that performs comparably well on the training as well as
testing sets. This argument holds good as the analysis of the results of neural network

models in next chapter show the comparable accuracy on training and testing sets.

Table 6.5 — Training and Testing Sets Composition for SSNN Travel Time Estimation

and Prediction Models

 

CI-SSNN Training Set CORSIM simulation Run 1 304 data-points

Travel Time

 

 

 

Estimation . . . .
Testing Set CORSIM srmulatlon Run 2 304 data-pomts

Models

CI-SSNN Training Set CORSIM simulation Run 1 288 data-points

Travel Time

Prediction Testing Set CORSIM simulation Run 2 288 data-points

Models

 

 

 

 

 

 

ll4

There are certain guidelines for training and testing in Artificial Neural Networks as
discussed in Haykin (1999). These guidelines were found usefirl and easy to follow and
they provide an efficient learning and testing of a neural network model. The guidelines
are usually for selecting learning rate, momentum constant, and number of hidden nodes
in the network. These values are obtained using the trial and error approach. Different
values within a range are tried and the selected values are those that provide a smooth
training graph and meet the goal error requirements. The optimum values of learning rate
and momentum constants found for the networks in this study were 0.0001 and 0.9
respectively. The optimmn number of hidden nodes obtained for different CI-SSNN
models were previously shown in Tables 6.1 and 6.2.

The training graph is plotted by MATLAB between the Mean Square Error (MSE) and
the number of iterations the model has run. When the MSE value meets the specified goal
error (0.001 in this study), the network is assumed to have converged to an optimum
weight configurations. The model is tested using the trained neural network with the
obtained optimum weight configuration. A criterion of an efficient training by a neural
network is to see the training graph by doing different trials with the learning rate,
momentum constant and hidden node values. A training graph which is smooth without
many fluctuations represents an efficient learning or training of the network. Figure 6.5
shows a training graph which is representative of an efficient training for the Cl-SSNN
models in this study. In this figure, the X-axis represents the epochs or iterations and the

Y-axis represents the mean square error of training.

115

_ Performance is 00319204. Goal is 0.001
10 , T l I I I I T

 

10'? I

1 Llllll

 

A ‘A-“‘—-‘L— A _L- _ A- A
" __ A m ‘-_

V'—

 

Training-Blue Goal-Black
I

10 _ :

 

 

 

0 1000 2000 3000 4000 5000 6000 7000
' ‘ ' ‘ " 7801 Epochs

 

 

Figure 6.5 — A Sample Training Graph of State-Space Neural Network Model for this

Study.

In closing, three CI-SSNN models were developed for the average travel time estimation.
Similarly, three CI-SSNN models were developed for average travel time prediction.
Hence, each of the three traffic movements has its own CI-SSNN models that provide the
travel time value of the specific movement. Once the CI-SSNN models are developed for
arterial links, they can be extended to any route that includes the selected arterial links.

The results of the CI-SSNN models for arterial links are discussed in the next chapter.

116

6.6 Conditional Independence Graph based State-Space Neural Networks for

Arterial Route

ATIS and ATMS applications are generally interested in travel time along an arterial
route rather than an arterial link. After developing CI-SSNN models for travel time
estimation and prediction for each type of traffic movement on arterial links, the models
can extended to provide results for an arterial route. An arterial route which consists of a
finite number of links included in the study can be specified. The CI-SSNN models
which are developed for each type of traffic movement on arterial links can be used for
travel time estimation and prediction for an arterial route. The travel time on an arterial
route is simply the summation of travel time for links comprising that route taking into
consideration the movement that is required to traverse that route.

The example arterial route which is used in this study is shown in Figure 6.6 with dark
arrows on the urban arterial network. On this route, a vehicle starts its trip at origin
intersection l6 and wants to get to destination intersection 25. The route that is taken is
marked by turning movements required to traverse the route. The vehicle is considered as
a through movement vehicle on arterial link 16-17, right-turning vehicle on arterial link
17-18, left-turning vehicle on arterial link 18-23, through movement vehicle on arterial
link 23-24, and through movement vehicle on arterial link 24-25. So, the travel time
estimation or prediction problem for this route uses the appropriate CI-SSNN models for
the required traffic movements. The travel time on this route is simply the summation of

the average travel time for the specified turning movements.

117

Hence, this study shows that if travel time for all three types of turning movements is
modeled by CI-SSNN models, it can be easily implemented for any route in question.
The results of implementation of CI-SSNN models for an arterial route are discussed in

the next chapter.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

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I.
0
0
”44000113 I ~
‘35) 11> $1) a i 313?.)
’ II " II II
P‘ y A. v_‘ "—0 L: r =
Ga) 1 > 10 _<8> $9) $10) (35‘
' '1 '4: 'Il ’ ’1: 'n "
-4 "a“: L— : "7“: L: 7‘ ‘
(35; _ 11 <1 ) 113) $14) 115) L35)
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(‘9- Q3) (22) (23) ( ) 50)
/~‘ I“ ([3] "\ -~

/
‘ l
\../ s-’ \.,’ s-’ x...’

Figure 6.6 — An Example Arterial Route between Intersections l6 and 25 used for Travel

Time Estimation and Prediction.

118

Chapter 7

Results and Discussions

This chapter presents the results of the State-Space Neural Network (SSNN) based on the
Conditional Independence (CI) Graph (here, called as CI-SSNN model) for the arterial
links and arterial route travel time estimation and prediction. The measures of
performance used for analyzing the efficiency of the models are specified. The
performance of the CI-SSNN models are presented for through, left, and right turning
vehicles on an arterial link for travel time estimation and prediction. A comparison of
SSNN model with and without using CI graphs shows the benefit of using) the CI graphs
in the modeling fi'arnework. The last section illustrates the performance of the models for

travel time prediction on the example arterial route.

7.1. Measures of Performance

The measures of performance used for evaluating the accuracy and predictive
performance of the CI-SSNN models are stated below (Kutner, et al. 2004 and Zhang and

Rice 2003):

119

Square of Correlation Coefficient (R2)-

A commonly used measure of performance is the coefficient of determination (R2). The
coefficient of determination is interpreted as the proportionate reduction of total variation
associated with the use of predictor variable X (here, actual travel time). A larger R2
value means that more the total variation of the dependent variable Y (here, predicted
travel time) is reduced by introducing the predictor variable X.

A measure of linear association between Y and X when both Y and X are random is the

coefficient of correlation (r). The coefficient of correlation is the square root of

coefficient of determination with positive/negative value attributed to it (i.e. i \I R2 ). The
positive/negative value determines whether the slope of regression line is positive or

negative.

The square of the coefficient of correlation (r2 = (i J—R—Z )2 = R2) is chosen here as the
measure of performance. It is expected that the scatter plot of the predicted travel time
versus actual travel time should be a linear line with slope of one and intercept of zero.
Thus, R2 should be close to one in an ideal condition when the predicted travel time fiom

the model is exactly the same as the actual travel time observed.

Mean Absolute Error (MAE) —

Let a (t, TpRED (t)) = TACTUAL (t) - TPRED (t) be the error of TpRED (t), where TACNALU) and

TpRED (t) are the actual and the predicted average travel time at any time period t

respectively. The Mean Absolute Error (MAE) is the expected value of the absolute value

120

of the error, i.e. E |a(t)| over the sample space of the average travel times at all time-

periods. Thus, MAE measures the average of the absolute prediction error of average
travel time over the entire temporal spectrum on a link. The MAE of the training or
testing set implies the MAE for all the links considered in the set. The value of MAE is
specific to the magnitude and the range of the average travel time in this study and should
be viewed in that context. It is not advisable to compare the MAE values from two test

studies with different magnitudes and range of travel times.

Root Mean Square Error (RMSE) —

Let ,8 (t, TPRED (t)) = (Tamar (t) - Tau-:0 (t))2 be the Square Error (SE) of TpRED (t). The
Mean Square Error (MSE) is the expected value of the Square Error, i.e. E I ,B(t)| over the

sample space of the average travel time at all time-periods. The Root Mean Square Error
(RMSE) is the square root of the E|fl(t)| measuring the root mean square of the
prediction error of the average travel time over the entire temporal spectrum on a link.

The RMSE of the training or testing set implies the RMSE for all the links considered in

the set. The RMSE values are also specific to this study like MAE.

Mean Absolute Percentage Error (MAPE) —

TACI‘UAL(t) - TPRED(t)

Let y (t, Tease (t)) = TAU-ML (t)

 

x100 be the Percentage Error (PE) of TpRED (t).

The Mean Absolute Percentage Error (MAPE) is the expected value of the PE, i.e.

121

E|y(t)| over the sample space of the average travel time at all time-periods. Thus, MAPE

measures the absolute value of the percentage prediction error of average travel time over
the entire temporal spectrum on a link. The MAPE of training or testing set implies the
MAPE for all the links considered in the. set. The MAPE is a unit free value unlike MAE
and RMSE, and can be compared with the MAPE values obtained in other studies

irrespective of the magnitude and the range of the average travel time.

7.2 Results of Travel Time Estimation on Arterial Links

The Conditional Independence (CI) graph analysis of average travel time for through,
left, and right-turn movements resulted in selection of parent input variables. These
parent input variables are fed as inputs to the State-Space Neural Network (SSNN) model
for travel time estimation. Since, the SSNN model is based on parent input variables from
C1 graph, the resulting model are called the CI-SSNN model. The three different Cl-
SSNN models are proposed for average travel time estimation for through, left, and right-
tum movements on arterial links, termed CI-SSNN-Thru, CI-SSNN-Left, and CI-SSNN-
Right, respectively. These models learn the travel time pattern from the traffic conditions
in a training set and are applied on different traffic conditions in a testing set.

The output of the CI-SSNN models is the estimated travel time for the current departure
time-period for the training and testing sets. The utility of these models is to obtain the
estimated travel time values which are then used as an additional input to the travel time

prediction models

122

7.2.1 Results of CI-SSNN-Thru Travel Time Estimation Model for Through

Movement

The results of the CI-SSNN-Thru model for travel time estimation of the training and
testing sets are presented here. First, the correlation coefficient and goodness of fit
between predicted travel time (model-generated estimated travel time) and actual travel
time is presented in Figure 7.1 for the training set. The R2 value is 0.9 which shows

efficient learning of the model using the training set.

 

 

 

 

   
 

 

 

 

 

 

 

 

 

 

A 250.00

E

O

f.

> / .

3 200.00 H .___bi,- ——I—e I Predrctedvs
: Actual

5 Travel Time
0 (Thru

§ 15000 d___,_ 7 Movement)
§ ' "

:. —Best-Fit Line
§ 100.00 e3 - - lire-33$?
... _ .

3 y - 0.92017x + 9.1205 Travel Time.
a R = 0.9062 Thru

.: Movement)
1': 50.00 + —

O

L5

-o

2

a.

0.00 - -
0.00 50.00 100.00 150.00 200.00 250.00

 

fl f

Actual Travel Tlme, seconds (Thru movement)

Figure 7.1 - Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Estimation for Through Movement.

The results in terms of performance measures like MAE, RMSE, and MAPE for the CI-

SSNN-Thru model on arterial links comprising the training set is presented in Table 7.1.

123

The model based on the training set is providing good results with a Mean Absolute

Percentage Error value of 7.6%.

Table 7.1 — Performance Measure of CI-SSNN-Thru Model for Travel Time Estimation

for Through Movement on Training Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 6.6 8.4 7.6

16-20 6.3 8.4 7.2

21-25 8.1 11.4 7.6

3-23 6.9 9.1 8.1

Total Training Set 6.9 9.4 7.6

 

 

 

 

 

The model is then applied to the testing set which includes the same arterials but different
traffic demand pattern. The correlation coefficient and scatter plot between predicted
travel time (model-generated estimated travel time) and actual travel time is presented in
Figure 7.2 for the testing set. The R2 value of 0.9 is obtained which is the same as that of
the training set. This shows that the CI-SSNN-Thru model performed with equal
efficiency on both training and testing sets.

The results of the CI-SSNN-Thru model on testing set are presented in terms of
performance measures in Table 7 .2. The performance measure values for the testing set
are comparable to that of the training set. This shows the robustness and efficiency of the

CI—SSNN-Thru model for the average travel time estimation for through movement on an

arterial link.

124

250.00

 

 

200.00

 

I Predicted vs
Actual
. Travel Time
I (Thru
'_ Movement)

— Best-Fit Line
(Predicted
vs. Actual

y = 0.9244x + 7.4545 Travel Time.

R2 = 0.9085 Thru

Movement)

150.00

 

   

100.00

 

 

 

 

$0.00 <

 

 

Predicted Travel Time, seconds (T hru movement)

 

 

 

0.00 r - - -
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time, seconds (Thru movement)

Figure 7.2 — Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Estimation for Through Movement.

Table 7.2 — Performance Measure of CI-SSNN-Thru Model for Travel Time Estimation

for Through Movement on Testing Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 7.3 10.3 7.7

16-20 5.6 7.0 6.5

21 -25 6.2 9.0 6.5

3-23 7.1 9.0 8.5

Total Testing Set 6.6 8.9 7.3

 

 

 

 

 

125

7.2.2 Results of CI—SSNN-Left Travel Time Estimation Model for Left-Turn

Movement

The results of the CI-SSNN-Lefi model for travel time estimation using the training and
testing sets are presented. The correlation coefficient and goodness of fit between the
predicted travel time and actual travel time is presented in Figure 7.3 for the training set.

The R2 value is 0.87 which shows efficient learning of the model using the training set.

 

 

 

 

 

 

 

    

 

 

 

 

 

 

250.00
E
I Predicted vs
200.00 A Actual
' Travel Time
E (Left-tum
3 Movement)
§ 150.00
E 100.00
— Best-Fit Liner
E . y = 0.8674x + 16.408 (Predicted
§ R2 = 0.8683 vs. Actual
Travel Time.
50'00 Left-tum
2 Movement)
0.00 - 7 - r
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time (Left-tum Movement)

Figure 7.3 — Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Left-Turn Movement.

It is also noted in Figure 7.3 that the travel time values are falling in two clusters. One
cluster has travel time values less than 150 seconds and the second cluster has travel time
values greater than 150 seconds. The formation of these clusters pertains to the geometry

and signal control scheme at the signalized intersections in the network. It was observed

126

in the data that the travel time values for longer arterial links (link length of 6000 ft and
4000 ft) are generally greater than 150 seconds. The travel time values for the links
having smaller length are lesser than 150 seconds. It is also to be noted that this cluster
formation is not observed in the scatter plots of through movement (see Figures 7.1 and
7.2). This is because of the exclusive left-turn phasing with lead green that was adopted
for the signal control in the network. The traffic conditions as viewed in the CORSIM
microscopic simulation showed that majority of the left-tum vehicles could not clear the
intersection during the lead left-turn phase in which they arrived at the intersection
mostly during congested conditions. They had to wait for the next left-turn green in the
next signal cycle. This increases the waiting time (and hence the travel time) for the left-
turn traffic movement at signalized intersections in particular for longer links. This
explains the higher travel time for left-tum traffic movement especially in longer arterial
links.

The proposed CI-SSNN-Left model is expected to capture the different ranges of the
travel time that pertains to geometries and signal control phasing scheme. It will be
observed in the later detailed discussion of the results of this model that the travel time
was estimated with a similar efficiency for varied link lengths and signal control
schemes. This strengthens the claim that the proposed travel time models are generalized
and explicitly account for geometrics and signal control state of the arterial links.

The results in terms of performance measures like MAE, RMSE, and MAPE for the CI-
SSNN-Left model on arterial links comprising the training set is presented in Table 7.3.
The model using the training set provides good results with a Mean Absolute Percentage

Error value of 7.5%.

127

Table 7.3 — Performance Measure of CI-SSNN-Left Model for Travel Time Estimation

for Left-Tum Movement on Training Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 9.2 11.7 8.0

16-20 8.9 11.2 7.4

21-25 8.4 11.1 7.3

3-23 8.4 10.9 7.4

Total Training Set 8.7 11.2 7.5

 

 

 

 

 

The model developed based on the training set is then applied to the testing set. The
correlation coefficient and scatter plot between predicted travel time (model-generated
estimated travel time) and actual travel time is presented in Figure 7.4 for the testing set.
The R2 value is 0.72, which is lower a bit than that of the training set. One reason that is
evident from the graph is that the model produces poor results when the actual travel time
is greater than 250 seconds. Comparing Figures 7.3 and 7.4, it is also clear that none of
the travel time value in the training set is more than 250 seconds. So, it is obvious that the
CI-SSNN-Left model performed efficiently on both sets within the observed range of
data-points in the training set. But, when the model was to extrapolate (because the large
travel time values were not present during training), it did not perform well. Neural
networks in general and also SSNN are not able to perform well in extrapolation. Thus,
during application of this model, it is advisable that the training set should include the
extreme boundary values, i.e. maximum and minimum travel time possible so that, the

model does not have to extrapolate.

128

Another observation in Figure 7.4 is the presence of two distinct clusters of travel time
values. This observation was also noted in Figure 7.3, where it was discussed. The same
explanation holds well here also.

The results of the CI—SSNN-Left model on the testing set are presented in terms of
appropriate performance measures in Table 7.4. The performance measure values for the
testing set is comparable to that of the training set except arterial 6-10 where the travel
time values are found to be more than 250 seconds. This shows the robustness and
efficiency of the CI-SSNN-Left model for average travel time estimation for left-tum

movements on an arterial link.

 

300.00

250.00 / - Predictedvs
Actual
Travel Time
(Left-tum
200.00 y I _ Movement)
I . g .

—Best-Fit Line
Predicted
y = 0.7038x + 35.965 (
R2 - 0 7129 vs' Aqua.
' - Travel Time.
50.00 Left-tum
Movement)

 

 

 

150.00

 

 

  

100.00 4

 

Predicted Travel Time (Left-tum Movement)

 

 

 

 

 

0.00 r r 7— r r r -
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00

 

Actual Travel Tlme (Left-turn Movement)

Figure 7.4 — Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Estimation for Left-Turn Movement.

129

Table 7.4 — Performance Measure of CI-SSNN-Left Model for Travel Time Estimation

for Left-Turn Movement on Testing Set

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 16.8 ' 32.4 10.6

16-20 9.0 11.9 7.8

21-25 10.3 13.2 8.2

3-23 9.7 15.8 7.6

Total Testing Set 11.5 20.1 8.6

 

 

 

 

 

 

7.2.3 Results of CI-SSNN-Right Travel Time Estimation Model for Right-Turn

Movement

The results of the Cl-SSNN-Right model for the travel time estimation using the training
and testing sets are presented. The correlation coefficient and goodness of fit between
predicted travel time (model-generated estimated travel time) and actual travel time is
presented in Figure 7.5 for the training set. The R2 value is 0.91 which shows efficient
learning of the model based on the training set. Moreover, this R2 value is comparable to
that of the CI-SSNN-Thru model which is logical as through and right-tum movements
share the same green interval at the intersection unlike left-turn vehicles which have
separate green interval as well as a separate left-turn pocket which typically results in
different queue lengths.

The results in terms of performance measures like MAE, RMSE, and MAPE for the CI-

SSNN-Right model on arterials using the training set is presented in Table 7.5. The

130

model based on the training set gives good results with a Mean Absolute Percentage

 

 

 

 

 

 

 
    

 

 

 

 

 

 

 

 

 

 

Error value of 8.4%.
250.00
E I Predicted
3 vs Actual
E 20000 d 7 __ __ Travel Time
in ' ' (Right-turn
. Movement)
8
5 15000 Mi:
8 E ’
O O
.- .5-
3 3 —Best-Fit
3 2100.00 ~ , Line
5 I y = 0.9086x + 8.5204 (waffle?
'- ' R2 = 0 9116 vs“ a
'3 - Travel .
‘6 50.00 + Time, Right-
i': tum
2 Movement)
0.
0.00 T I I I
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time, seconds (Right-turn Movement)

Figure 7.5 — Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Estimation for Right-Tum Movement.

Table 7.5 — Performance Measure of CI-SSNN-Right Model for Travel Time Estimation

for Right-Tum Movement on Training Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 5.7 7.8 6.6

16-20 8.1 10.7 9.5

21-25 8.4 10.7 8.9

3-23 7.1 9.8 8.7

Total Training Set 7.3 9.8 8.4

 

 

 

 

 

The model produced using the training set is applied to the testing set. The correlation

coefficient and scatter plot between predicted travel time (model-generated estimated

131

travel time) and the actual travel time is presented in Figure 7.6 for the testing set. The R2
value of 0.83 is obtained which is comparable to that of the testing set. But, this is still a
satisfactory outcome taking into consideration that the testing set is a different traffic

demand pattern which was not presented to the model while training.

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

250.00
. Predicted
E vs Actual
35 / Travel
a 200.00 Time
E " (Right-tum
e; Movement)
3
S
O C
'1 a
‘E‘ ; —Best-Flt
; g Llne(Predic
3 100.00 ted vs.
I
g y = 1.0265x + 3.1661 $32:
2 R2 = 0.8345 Time.
e-o _ nght-tum
43 50.00 Movement)
5
0.00 r . . .
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time, seconds (Right-turn Movement)

Figure 7.6 — Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Estimation for Right-Turn Movement.

The results of the CI-SSNN-Right model using the testing set are presented in terms of
performance measures in Table 7.6. The MAPE values for training and the testing set are
8.4% and 14.0%, respectively. This shows the robustness and efficiency of the CI-SSNN-
Right model for average travel time estimation for right-tum movements on an arterial

link.

132

Table 7.6 — Performance Measure of CI-SSNN-Right Model for Travel Time Estimation

for Right-Tum Movement on Testing Set

 

 

 

 

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 11.9 15.5 13.4

16-20 11.3 13.6 14.7

21-25 13.8 18.0 15.0

3-23 10.4 13.8 13.1

Total Testing Set 11.8 15.3 14.0

 

The above results give satisfactory values of the estimated average travel time for current
time-period for through, left, and right-tum movements on an arterial link. The estimated
travel time for the current departure time-period is added to the parent input variables
used for travel time prediction. The next section deals with the average travel time

prediction for through, left, and right-tum movements.

7.3 Comparison of Results for State-Space Neural Networks with and

without using Conditional Independence Graphs

The travel time estimation done in the earlier section is based on a CI-SSNN model, i.e.
SSNN model combined with Cl graph. It is evident that use of CI graph reduces the
number of input variables required for modeling of travel time. It is still to be proven that
SSNN based on C1 graph produces better or comparable results to that of an ordinary
SSNN model that is typically used by researchers.

Here, travel time estimation is done by using an ordinary SSNN model without using the

CI graph. So, the input vector consists of all traffic variables: VT, VL, Va, 57, SL, SR, G,

133

Off S.L., ITTT and ITTL. These input variables are fed to a SSNN-Thru model for
estimation of average travel time for through movement. The results are compared with
the CI-SSNN-Thru model which was presented in Section 7.2.1. If the results of the CI-
SSNN-Thru model are better or comparable to those of the SSNN-Thru model, the use of
the CI graph for improving the efficiency of SSNN models is justified and recommended.
The results obtained for travel time estimation for through movement using the training

and testing sets are presented in Tables 7.7 and 7.8, respectively.

Table 7.7 - Performance Measure of SSNN-Thru Model (State-Space Neural Networks
without using Conditional Independence Graphs) for Travel Time Estimation for

Through Movement on Training Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 5.7 7.9 6.0

16-20 6.2 8.2 7.1

21-25 7.1 10.2 6.0

3-23 6.6 9.0 7.6

Total Training Set 6.4 8.8 6.8

 

 

 

 

 

Table 7.8 — Performance Measure of SSNN-Thru Model (State-Space Neural Networks
without using Conditional Independence Graphs) for Travel Time Estimation for

Through Movement on Testing Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 9.3 11.9 11.0

16-20 6.6 8.7 8.4

21 -25 4.7 6.3 5.2

3-23 7.0 9.0 8.5

Total Testing Set 6.9 9.2 8.2

 

 

 

 

134

 

A comparison of Tables 7.7 and 7.8 versus Tables 7.1 and 7.2 does not show much
difference between results of the two sets of models. However, the MAPE values for the
training set is lower in the SSNN-Thru model, which may suggest that inclusion of all
input variables leads to a better learning for the SSNN model. But, the MAPE value of
the testing set is higher for the SSNN-Thru model which may suggests that though
learning is better, the SSNN-Thru model suffers when tested on a different traffic demand
pattern.

The CI-SSNN models use lesser variables than the SSNN models and hence the CI-
SSNN models can easily be declared efficient on this practical basis only. The constraint
of limited amount of data obtained in any experiment or in the field makes the CI-SSNN
models more suitable than SSNN models. Hence, it is recommended that a CI graph
analysis be done prior to using artificial neural networks in general in a process that has
multivariate normal distribution. The comparison done here serves the purpose for travel
time estimation and prediction modeling. The further research in this topic is needed to
theoretically and experimentally draw a sound conclusion.

The comparison of the CI-SSNN and SSNN models is shown for through movement only
but the conclusion can be extended to modeling for left and right-tum movements also.
Hence, travel time prediction on arterial links in the following sections is done using CI-
SSNN models only, and it is assumed that the CI—SSNN model will provide better or

comparable results to those of the SSNN modeling.

135

7.4 Results of Travel Time Prediction on Arterial Links

The three different CI-SSNN models are proposed for average travel time prediction for
through, left, and right-turn movements (hereby, termed as CI-SSNN-Thru, CI—SSNN-
Left, CI-SSNN-Right, respectively) on arterial links. These models for travel time
prediction work similar to the travel time estimation models. The only difference is that
the output in this case is the average travel time for a future departure time-period instead
of the current departure time-period. The parent input variables for average travel time
prediction contains the parent input variables used in the travel time estimation models
developed earlier. The estimated travel time from the travel time estimation models is
also included as an additional input variable for the travel time prediction modeling. The
prediction models learn the travel time pattern from the traffic conditions contained in a

training set and are then applied to entirely different traffic conditions in a testing set.

7.4.1. Results of CI—SSNN-Thru Travel Time Prediction Model for Through

Movement on Training Set

The complex non-linear relationship of average travel time for through movement and the
parent input variables is approximated by the CI-SSNN-Thru model. The predicted
average travel time (model generated) is plotted against the actual travel time (as
obtained from the CORSIM simulation) in Figure 7.7. The R2 value of 0.87 shows that

the proposed CI-SSNN-Thru model is able to learn well from the training set.

136

250.00

 

 

 

 

 

 

  

 

 

 

 

 

 

 

‘5‘
O
E
2 /
o . e___,
E 200'00 e Predicted vs
3 Actual
.5 Travel Time
‘; (Thru
'2 150-00 ‘“ Movement)
8
3
6
_§ 100.00 ~+ ____. — . —Best-Fit Line
g. (Predicted
3 y = 0.82661x + 12.635 vs. Actual
E R = 0.8681 Travel Time.
.. _ Thru
'3 50'00 Movement)
.‘2’
u
2
a.
0.00 I T T fi
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time, seconds (Thru movement)

Figure 7.7 — Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Through Movement.

A detailed analysis of the learning and performance of the CI-SSNN-Thru model is
needed for the training set. The detailed analysis enables to analyze the model results
with respect to each arterial link contained in the study. The Absolute Percentage Error
(APE) is plotted for all of the four arterials whose data was fed into the model. The plot
of APE against departure time period (5 minutes to 95 minutes on each link) for each
arterial is presented. Figures 7.8, 7.9, 7.10, and 7.11 show these plots for training sets 1,

2, 3, and 4, which represent arterials 6-10, 16-20, 21-25, and 3-23 respectively. The APE

values are mostly lesser than 15% for each individual link in the arterials.

137

 

Link 6-7 Link 7-8 Link 8-9 Link 9-10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4O
35
is 30
5 ..
i 20
lAbsolute
. Percentage
g 15 f Error for
Travel Time
3 10 I I : n I I , (rhm
I I II I |
5 I I I I I I. I I I
. .I.|.|u .111. | ||.|.|||.I.|.|,l|.l. | . II I. l.
5 95 5 95 5 95 5 95

Departure Tlme-Perlod (minutes)
Figure 7.8 — Absolute Percentage Error of Predicted Travel Time versus Departure Time-

Period for Travel Time Prediction for Through Movement on Training Set 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 16-17 Link 17-18 Link 18-19 Link 19-20
40
35 «~—
§ 30
o
E IAbeolute
a 25 5 Percentage
8 Errorfor
§ 5 (Thru
& Movement)
. 15
5
o
3 10 L— - .
“ l
5 , _ _ _ . nl I I _
o I; [III II _ g. g n. l e . LII]... I II | 111.1]: I
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.9 — Absolute Percentage Error of Predicted Travel Time versus Departure Time-

Period for Travel Time Prediction for Through Movement on Training Set 2.

138

Link 21-22 Link 22-23 Link 23-24 Link 24—25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30
A 25
i
lg 20 IAbsolute
Percentage
s Emorfor
TravelTime
§ 15 (Thru
& Movement)
g 10 +7—
3 5 t- _ I n A I I II II , II n I
o Ill]; ”IL III“ |. ,I III lullllllll "I: II, I I
5 955 955 955 95

Departure Tlme-Perlod (mlnutes)
Figure 7.10 — Absolute Percentage Error of Predicted Travel Time versus Departure

Time-Period for Travel Time Prediction for Through Movement on Training Set 3.

Link 3-8 Link 8-13 Link 13—18 Link 18-23

 

 

 

 

40 ..L- __ _____ ____.r__f,g - AbSO'Ute
Percentage
Error for
Travel Time
(Thru
Movement)

 

 

 

 

 

 

 

Absolute Percentage Error (%)
8
l
l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'5" 955 .. 955 955 95

Departure Time-Period (minutes)
Figure 7.11 -— Absolute Percentage Error of Predicted Travel Time versus Departure

Time-Period for Travel Time Prediction for Through Movement on Training Set 4.

139

It is also noted that the CI-SSNN-Thru model is not able to learn well for some links as
depicted from their APE distribution. Links 6-7, 8-9, 16-17, 22-23, 24-25 and 13-18 are
remarkable in this aspect. Though, it can not be inferred that the relatively high APE
distribution on these links is because these links are at a particular location that they act
as a critical link for an arterial. The critical links are those links which have the least
capacity in an arterial or a sequence of links and hence may act as bottlenecks. So, the
relatively high APE distribution of such links may be due to the inherent stochastic nature
of the CORSIM simulation. If there is some unidentified characteristic of a link or traffic
stream which is not explicitly accounted in the model formulation, the APE distribution
of the testing set for the same links should be relatively high as compared to other links.
This argument is revisited in Section 7.4.2., when the testing sets are analyzed.

In conclusion of this section, the measures of performance like MAE, RMSE, and MAPE
of the CI-SSNN-Thru model for the training set are presented in Table 7.9. The model
gives MAPE value of 9% for the training set. It is expected that this model can generalize
and is able to predict the average travel time for through movements when new traffic

demand conditions in the testing set are presented.

Table 7.9 — Performance Measure of CI-SSNN-Thru Model for Travel Time Prediction

for Through Movement on Training Set

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 7.6 10.0 8.4

16—20 7.7 9.7 8.8

21-25 8.9 13.2 8.4

3-23 9.0 11.8 10.7

Total Training Set 8.3 11.2 9.0

 

 

 

 

 

 

140

7.4.2. Results of CI-SSNN-Thru Travel Time Prediction Model for Through

Movement on Testing Set

After analyzing the efficiency of the CI-SSNN-Thru model to learn from the training set,
the model is applied to the testing set. It is expected that the model should perform with
similar efficiency on the traffic pattern of testing set.

The predicted average travel time is plotted against the actual travel time for testing set in
Figure 7.12. The R2 value is obtained as 0.88 which is close to the one obtained for the
training set. Thus, the CI—SSNN-Thru model has generalization ability so that when a
different traffic demand pattern is presented to the model, it predicts the average travel
time with comparable accuracy. This property is a necessary requirement for a reliable

prediction model that can be applied in the field.

141

250.00

20000 T ' ' " I. I Predicted vs
. Actual
Travel Time
. "' (Thru
150.00 -- — ~ ak .

Movement)

 

 

 

 

 

  

 

 

 

 

Predicted Travel Time, seconds (Thru movement)

 

 

 

— Best-Fit Line
100.00 — 7 (Predicted
. l
y = 0.9412x + 7.2594 #33:?“
R2 = 0.877 Thru
50.00 . Movement)
I
0.00 . . . .
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time, seconds (Thru movement)

Figure 7.12 — Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Through Movement.

A detailed analysis of the predictive performance of the CI—SSNN-Thru model is needed
for the testing set to reinforce the claims made in the earlier paragraph. The Absolute
Percentage Error (APE) is plotted for all four arterials that constitute the testing set. The
plots of the APE against future departure time-period (5 minutes to 90 minutes on each
link) for each arterial is presented. Figures 7.13, 7.14, 7.15, and 7.16 show these plots for
training sets 1, 2, 3, and 4, which represent arterials 6-10, 16-20, 21-25, and 3-23,
respectively. These plots are consistent with the training set plots in the sense that the
APE values are mostly lesser than 15% for each individual links in an arterial.

The predictive performance of the proposed Cl-SSNN-Thru model is relatively weaker

for some links where the APE values for some time-periods is as high as 35%. These

142

links are: 6-7, 7-8, 18-19, 23—24, 24-25 and 13-18. It is recalled (as stated in Section
7.4.1) that some links in the training set like 6-7, 8-9, 16-17, 22-23, 24-25 and 13-18 had
relatively larger APE values. Though, all these links do not perform poor in the testing
set as was expected earlier. So, the large values of APE in training and testing sets can be
attributed to the stochastic nature of the CORSIM simulation as stated earlier also in

Section 7.4.1.

 

 

 

 

 

 

 

 

 

45 Link 6-7 Link 7-8 Link 8-9 Link 9—10
40 _ 7 i 7
g 35 - 7 7 7 77 L L
3 3° ' i i a eeeee fl 7 —4 IAbsqute
Percentage
g 25 - 7 7 7 7 7 -7 ,, Errorfor
Travel Time
§ (Thru
K 20 ‘ i r i ’ 7 Movement)
5 15 .7 7 7 7 7 7 7 7 7 77
3 10 — 7 7 7 7 7 _ L
5| I | i | I
o ‘ I I ‘
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.13 — Absolute Percentage Error of Predicted Travel Time vs. Departure Time-

Period for Through Movement applied on Testing Set 1.

143

Link 16—17 Link 17-18 Link 18-19 Link 19-20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30

25 ~— — — 4
1‘.
lg 20 +— — IAbsolute

Percentage

g Errorfor

15 . Travel Time
§ #- (Trim
3 Movement)
g 10
35|I|..I II III In ..

5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.14 -— Absolute Percentage Error of Predicted Travel Time versus Departure

Time-Period for Travel Time Prediction for Through Movement on Testing Set 2.

Link 21-22 Link 22-23 Link 23-24 Link 24-25
30

 

 

?
F
l

 

 

I Absolute
Percentage
Error for
Travel Time
(Thru
Movement)

'8
l
l
l
L
l
l

 

 

 

 

.a
o

 

 

 

 

 

Absolute Percentage Error (%)
a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v v v v v .

vvvvvvvvvvvvvv

95 5 95 5 95

Departure Time-Period (minutes)

 

 

Figure 7.15 — Absolute Percentage Error of Predicted Travel Time versus Departure

Time-Period for Travel Time Prediction for Through Movement on Testing Set 3.

144

Link 3-8 Link 8-13 Link 13-18 Link 18-23

 

(A
O

 

 

 

N
OI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E
g IAbsolute
Percentage
g 2° ‘ 5 Errorfor
g Travel Time
Thru
& 15~ Illiovement)
g
3 10 -- 7
5 _- I.II II I [I |_ n 1
0I- , ...III II.“ ..... I II II . I
955 955 95

 

Departure Time-Period (minutes)

Figure 7.16 — Absolute Percentage Error of Predicted Travel Time versus Departure

Time-Period for Travel Time Prediction for Through Movement on Testing Set 4.

The patterns of the actual and predicted average travel time for the testing set are plotted
to see how closely the predicted values follow the actual travel time fluctuation. Figures
7.17, 7.18, 7.19, and 7.20 show the actual and predicted travel time patterns where future
departure time-period is on the X axis. It is seen that the predicted travel time closely
follows the actual travel time for all the arterial links except for link 24-25 shown in
Figure 7.19. The reason because of this variation between travel time values is not certain
as a similar variation was not observed during the training of the model (see Figure 7.10).
It is already mentioned that the links that give poor results during training do not perform
poor during testing of the model. Hence, one explanation could be the stochastic nature of

the CORSIM simulation.

145

Link 6-7 Link 7-8 Link 8-9 Link 9-10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

250.00
‘a’ 1,
£200.00 «_ .- ----- Actual
g Travel
s Time.
seconds
a (Thru
£150.00 — , Movement)
g ' +Predlcted
g 2 I'-_ Travel
6100.00 -- — 5;", W Time,
5 : 1 : seconds
E .-' = (Thru
g ' L. w" _ -;’.‘- Movement)
3‘ 50.00 ~ ' ..
5
0.00 - - . - - ~ L e - ~ -
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.17 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Through Movement on Testing Set 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

160.00 L” 16-17 Link 17-18 Li_nl_r 18—19 Link 1940
I
A 140.00 ‘*—_ , f.
.. : 5. °°°°° Actual
120.00 ; , . .. , Travel
, . ... . '- ' '. Time.
3 .' - _: - seconds
£100.00 1. ' (Thru
if 3' Movement)
3 80.00 ~ #5. if!” 5.2-“fl..." —I—Predlcted
_ -.. .: Travel
’ _ " 3 Time,
g 60.00 seconds
_ (Thru
E 40.00 Movement)
.—
§ 20.00
9
<
0.00
S 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.18 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Through Movement on Testing Set 2.

146

Link 21-22 Link 22-23 Link 23-24 Link 24-25

 

 

 

 

 

 

 

 

 

 

 

 

160.00

3140.00 «

c

E 0 ----- Actual

2 120.00 7 7 7 Travel

g Time,

2 seconds
€100.00 - 7 (Thru

Movement)

0
u

5 80'00 ‘ ”7 " +Predicted
§ Travel

' Time,
g 6000 5 i7 7 if i 7 i H¥ 77 Hi» W?— 7 seconds
3 ' L. (Thru

5 40 00 - 7 7“, 7 77 7 7 7 77 Movement)
;

0

g 20.00 ~ 7 7 — 7 7 7 7
2

0.00
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.19 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Through Movement on Testing Set 3.

 

 

 

 

 

 

 

 

 

 

 

 

160.00 Link 3-8 Llnk 8-13 Link 13-18 Link 18-23
A 140.00 -
‘E
----- Actual
120.00 ~ Travel
g Time,
3 seconds
E 100.00 (Thru
Movement)
80.00 I V +Predlcted
Travel
- Time,
g 60.00 5 seconds
_ (Thru
E 40 00» 7— 7 7 _ 77 7 7, 1 Movement)
.—
§ 20.00 7 7—— 7— 7 7 7 7 777 7_7 7.7 .. 7
B
<
0.00
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.20 - Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Through Movement on Testing Set 4.

147

The following two inferences can be made about CI-SSNN-Thru model from these

figures:

1.

The predicted average travel time lies in the same range as the actual average
travel time for all arterial links. This confirms that the CI-SSNN-Thru model is a
generalized model which can predict the average travel time within the range that
is typical of the given link. The range of average travel time for any link in
general is a function of variables like link length, and speed limit.

Close inspection of travel time patterns at each arterial link shows that the
predicted average travel time follows the actual average travel time pattern. It
shows that the CI-SSNN-Thru model captures the traffic dynamics and temporal
fluctuations in travel time on each link. This traffic dynamics and temporal nature
of average travel time follows from the varying traffic demand on an arterial link
throughout the day. The model is able to capture this aspect because of its
topology. The hidden and context layers in the SSNN topology provide a short-
terrn associative memory in the network in addition to the input-output mapping

feature inherent in artificial neural networks.

In conclusion of this section, the measures of performance like MAE, RMSE, and MAPE

of the CI-SSNN-Thru model for testing set are presented in Table 7.10. The MAE,

RMSE, and MAPE values in this table are comparable with those obtained for the

training set. This proves the generalization ability of the CI-SSNN-Thru model for short-

term average travel time prediction for through movements on arterial links.

148

Table 7.10 — Performance Measure of CI-SSNN-Thru Model for Travel Time Prediction

for Through Movement on Testing Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 8.9 13.2 9.3

16-20 7.5 9.0 8.8

21-25 7.7 11.1 8.2

3-23 7.5 9.2 8.7

Total Testing Set 7.9 10.7 8.8

 

 

 

 

 

7.4.3 Results of CI-SSNN-Left Travel Time Prediction Model for Left-Tum

Movement on Training Set

The CI-SSNN-Left model is trained using the training set. The efficiency of the model is
analyzed in similar lines with that of the CI-SSNN-Thru model presented in earlier
sections. The R2 value of the predicted versus actual average travel time shown in Figure
7.21 is 0.87. The Figure 7.21 also shows two distinct clusters of travel time values. This
observation was also made in Section 7.2.2 where the travel time estimation modeling for
left-tum movements on arterial links was presented. The explanation given there holds
good here also. The APE distribution for training sets 1, 2, 3, and 4, which represent
arterials 6-10, 16-20, 21-25, and 3-23, is presented in Figures 7.22, 7.23, 7.24, and 7.25,
respectively. The APE distribution is mostly within 15% error range. The arterial links
that have APE values more than 15% to 45% are 16-17, 19-20, 21-22, 24-25, and 13-18.

The detailed analysis of the results is done by calculating the performance measures

shown in Table 7.11. The overall MAPE value of 7.3% shows that the CI-SSNN-Left

149

model is able to learn well the temporal fluctuations in predicted values of average travel

time.

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

250.00
‘5‘ .
g / I Predicted vs
g 200.00 « Actual
Travel Time
g . (Left-tum
g Movement)
Z 150.00 ——
.11
i:
- 100'00 . —Best-Fit Liner
E y = 0.8683x + 16.332 (PreAclicted
2 = vs. ctual
'2 R 0'869 Travel Time.
15 50.00 7— Left-tum
12? Movement)
a
0.00 T . . .
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time (Left-tum Movement)

Figure 7.21 - Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Left-Tum Movement.

150

Link 6-7 Link 7-8 Link 8—9 Link 9—10

 

 

 

 
 
 

 

 

 

 

 

 

 

 

60
50 - 7 77 777 77 77
§
§ 40 -7 77 77 7 77 7
IAbsolute
§ Percentage
‘E 30 ,7 ,7 7, ,, 7,7 7, 7 ,7, , ,7 Errorfor
E Travel Time
& (Left-turn
e Movement)
g 20 77 7 77 7 7 7 7 7 7 7
10 -
0

95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.22 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Tum Movement on Training Set 1.

 

 

 

 

 

 

 

 

 

50 Link 16-17 Link 17-18 Link 18-19 Link 19-20

457 7 7 7 7 7 77 7 77 77 7 7777 777i
A 40 7 7 7 7 7 7 7 7 7 7 77 7 7 7777
ES
3 35 - 77 7 7 7 7 7 7 77 7 7 7 777 77
E

30 7 777777 777 77 7 7 7 7 7 7 7 777 7 777 7 7 7 IAbsolute
s Percentage

25 _ 7 7 7 7 7 7 7 7 7 77 7 7 7 ,7 ,7 Error for
g Travel Time
& (Left-turn
g 20 ‘ T T T T T T T ’ T ’T ’T TT’T TT'T Movement)
3 15 - 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 E

10- 7 77 7 7777777;

0 ... .. _.
95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.23 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Tum Movement on Training Set 2.

151

Link 21-22 Link 22—23 Link 23-24 Link 24-25

 

30

 

 

 

N
01

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E
2 20
“‘ IAbsolute
3 Percentage

15 7 Error for
§ Travel Time
& (Left-tum
. Movement)
3 10

5 I I I - n I I I I L I I II _ 4
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.24 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Tum Movement on Training Set 3.

Link 3-8 Link 8-13 Link 13-18 Link 18-23

 

30

 

 

 

N
or
I
I
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g
E 20 7 7777—— —— 7 777
ru IAbsolute
g Percentage

15 77 7 7 7 Error for
§ Travel Time
1! (Left-tum
g Movement)
.. 10

, .Ir 1 r. r .- ill I

0 III III Irl I ,. Ir IIIII II II III II II. I IIIII I

5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.25 - Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Tum Movement on Training Set 4.

152

Table 7.11 — Performance Measure of CI-SSNN-Lefi Model for Travel Time Prediction

for Left-Turn Movement on Training Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 8.3 10.7 7.0

16-20 8.8 11.6 7.4

21-25 9.5 12.4 8.1

3-23 8.2 10.1 6.9

Total Training Set 8.7 11.2 7.3

 

 

 

 

 

7 .4.4 Results of CI-SSNN-Left Travel Time Prediction Model for Left-Turn

Movement on Testing Set

The CI-SSNN-Left model is now applied on a testing set which has different travel time
patterns than those presented in the training set. The R2 value for a plot between the
predicted versus actual average travel times as shown in Figure 7.26, is 0.73. This value
of R2 for the testing set is lower than that of the training set. One reason of the lower R2
value is attributed to the inability of artificial neural networks to extrapolate. This factor
was acknowledged in Section 7.2.2 also where the CI-SSNN-Lefi model results for travel
time estimation were discussed. It is expected that if training and testing sets have similar

travel time ranges, this model’s accuracy can be significantly improved.

153

 

300.00

250.00 / I Predicted vs

Actual
Travel Time
. (Left-tum
200.00 . - ' — Movement)

 

 

 

 

 

 

  

 

 

1 50.00
—Best—Fit Lin
10°00 ‘ " (Predicted
y = 0.7002x + 36.73 vs. Actual
R2 = 0 729 Travel Time.
50 00 q. ' Left-tum
‘ Movement)

Predicted Travel Time (Left-turn Movement)

 

 

 

 

 

0-00 I l l T f T l
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00

 

Actual Travel Time (Left-tum Movement)

Figure 7.26 — Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Left-Turn Movement.

The APE plots for testing sets 1, 2, 3, and 4, which represent arterials 6-10, 16-20, 21-25,
and 3-23, respectively, are presented in Figures 7.27, 7.28, 7.29, and 7.30. These plots
show that APE distribution is mostly less than 15% error, except for links 7-8, 9-10, 16-
17, 19-20, 24-25, 13-18, and 18-23. The plot of predicted and actual travel time patterns
with departure time-periods for different arterials is presented in Figures 7.31, 7.32, 7.33,
and 7.34. It is seen that the predicted travel time follows the actual travel time
fluctuations in most of the arterials in the testing set. Also, predicted travel time lies
within the same range as that of the actual travel time. It is also observed that there is a

large error for some future departure time-periods as shown in Figure 7.31 and 7.34. The

154

reason is the same as mentioned earlier that the artificial neural networks are incapable of

performing well when the extrapolation is required.

Absolute Percentage Error (%)

Link 6-7

Link 7-8

Link 8-9

 

A
O
i

b)
(I!

(a)
O

N
Ul
A

N
O
1

a
U1
I

_|
o

 

0'1

 

 

9

55 9

55

 

 

 

955

Departure 1 Ime-Penoa (minutes;

 

 

I Absolute
Percentage
Error for
Travel Time
(Left-tum
Movement)

 

 

 

Figure 7.27 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Tum Movement on Testing Set 1.

155

30 Link 1647 Link 17-18 Link 18-19 Link 19-20

 

N
01

 

 

N
O

 

I Absolute
Percentage
Error for
Travel Time
(Left-turn
Movement)

 

 

 

..a
O

 

Absolute Percentage Error (%)
a:

 

 

v v v

5 955 955 955' Y 95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.
—
.
—
.
.
.
—
.
.
.
.
‘—
.
.
_
.
.
‘—
.
.
4
_
.
—
.
——
.
—
.
—_
.
_I
.
—
1
——
.
—
1—
4
J-
—
.
,I
—
.
.
—I
.
,-
—
.
—
.
—
.
_
1
.—
.
I—_
—

 

Departure Time-Period (minutes)

Figure 7.28 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Lefi-Tum Movement on Testing Set 2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

35 Link 21-22 Link 22-23 Link 23-24 Link 24-25

30
S?
T: 25
o
.5
3: IAbsolute

20 Percentage
g Errorfor
§ Travel Time
8 15 (Left-tum
, Movement)
5
g 10

5 |_|| .llllllll ll--l- I I-

0 I. ..,III II I II IIII III, III "III III I II . . I I

5

M
\O

5 955' "’ "T ' 955 ' 95

Departure Time-Period (minutes)

‘ Figure 7.29 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Turn Movement on Testing Set 3.

156

Link 3-8 Link 8-13 Link 13-18 Link 18-23

 

 

 

 

 

 

 

 

.5
01

 

50
45 .P- _
.L 40 -
e".
E 35
u.I 3o IAbsolute
Percentage
25 Error for
Travel Time
& (Len-tum
g 20 Movement)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

Y‘V’V’vv’vvvvvvvvv vvvvvvvvvvvv rvv v Vva vvvv vv- v vvv VYY’TY - vv

5 955 955 955 7 '95

Departure Time-Period (minutes)

 

Figure 7.30 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Left-Turn Movement on Testing Set 4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 6-7 Link 7-8 Link 8-9 Link 9-10
,. 400.00
‘E
g 350 00 m 3. ----- Actual
2 :L Travel
. __ -I L I Time

5 300'00 '. seconds

0 3 ... (Left-mm
3 250.00 ,7 »*————~-—— 5 1. MOM“)
a ‘ 4

§ 200 00 4~ '

. " .... 3-' -O—Predicted
3 150.00 «~- 4 " 3 Tm“
F P. b - i.- Time.
i ’ ... . . . _ _ .. seconds

100.00 " ’_ i - (Lefl‘mm
E .; Movement)
g 50.00
( 0.00 ‘ ‘ A
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.31 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Left-Tum Movement on Testing Set 1.

157

Link 16—17 Link 17-18 Link 18-19 Link 19-20

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

20.00

.. 200.00
‘E
180.00 —— , ,- —-
..-.. ._ .9- , ----- Actual
160.00 ' ' -- T'ave'
E - " 0 Time’
3 _. -_- _ _ _ _ - seconds
..- 140.00 ——— _- : r; .--, (Left-tum
S 120 00 ,. ; :1 -' ’- Movement)
§10000~ A — #A- '7' .
‘ " '-' '- -O-Predicted
:- 80.00 " Trave'
l: Time.
__ 60.00 ——- ‘ seconds
5 (Left—tum
'- 4000 q_ __ Movement)
8
(

 

 

 

 

 

0.00 A A A A A A
5 95 5 95 5 95 5 95

 

Departure Time-Period (minutes)

Figure 7.32 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Left-Turn Movement on Testing Set 2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 21-22 Link 22-23 Link 23-24 Link 24-25
A 250.00
E
g F ----- Actual
: 200.00 -— —- »——- z”: Travel
E : : Time.
3 seconds
3: (Left-tum
3 Movement)
v 150.00
8
g -I-Predicted
g 100.00 ... .._..-. 71 Travel
p . n. " Time.
_ - seconds
3 (Left-tum
£ 50.00 ._ »——- _ Movement)
8.
E
3
0.00 - I
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.33 -- Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Left-Turn Movement on Testing Set 3.

158

Link 3-8 Link 8—13 Link 13—18 Link 18—23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 250.00
E
5 q ----- Actual

200.00 ._ -'1 Travel
6 : '- Time,
3 : ‘.' seconds
‘1. ' 1 (Left-tum
3 150 00 Movement)
g ”- --O-Predicted
g 100.00 Travel
t: Time,
_ seconds
9 (Left-tum
E 50.00 Movement)
2
< 0.00 * ~+

5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.34 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Left-Turn Movement on Testing Set 4.

The detailed analysis of the CI-SSNN-Left model on the testing set is done by noting
suitable performance measures. The performance measures values presented in Table
7.12 show that a MAPE value of 8.8% is obtained for entire testing set. Also, MAE,
RMSE, and MAPE value of arterial 6-10 is higher as compared to other arterials. It is
also because of the same reason that average travel time value was greater than 250
seconds for arterial 6-10 in testing set. Since, the maximum value of travel time
encountered in training set was 250 seconds, so, CI-SSNN-Lefi failed to predict

extrapolated values.

159

 

Table 7.12 — Performance Measure of CI-SSNN-Lefi Model for Travel Time Prediction

for Left-Tum Movement on Testing Set

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 15.8 30.0 9.8

16-20 10.1 12.8 8.6

21-25 10.7 13.7 8.5

3-23 10.8 17.6 8.4

Total Testing Set 11.8 19.8 8.8

 

 

 

 

 

7.4.5 Results of CI-SSNN-Right Model for Travel Time Prediction for Rignt-Turn

Movement on Training Set

This section presents the efficiency of the CI-SSNN-Right model for training and its
predictive performance on the testing set for the average travel time prediction of right-
turning vehicles on an arterial link. The plot between the predicted travel time against
actual travel time for right-tum movement is shown in Figure 7.35. The R2 value is 0.88
which reflects an efficient leaming 0f the model. The APE distribution for training sets 1,
2, 3, and 4, which represent arterials 6-10, 16-20, 21-25, and 3-23, respectively, is shown
in Figures 7.36, 7.37, 7.38 and 7.39. The APE values for the arterials during most of the
future departure time-periods are less than 15%. The arterials that have an APE value
more than 15% for some future departure time-periods are 6-7, 16-17, 18-19, 23-24, 24-
25, 8-13, and 13—18. The probable reason for the poor performance of model on these

arterials is the stochastic nature of the CORSIM simulation.

160

The performance measures of the Cl-SSNN-Right model on the training set is shown in
Table 7.13. The MAPE value for the entire training set is 9.6%. This shows an efficient

learning ability of the CI-SSNN-Right model. It is expected that this model should

perform with comparable accuracy when applied on a testing set.

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

250.00

I Predicted
E vs Actual
or Travel Time
1; 200.00 _ (Right-tum
E Movement)
i
g —Best-Fit
i: g Line
E 100.00 ([3dede
g y = 0.8871): + 11.55 VS- Acm'
.— 2 _ 1"an
“g R '- 0-3822 Time. Right-
§ 50.00 — turn
2 . Movement)

0.00 I 1 . .
0.00 50.00 100.00 150.00 200.00 250.00

Actual Travel Time. seconds (Right-tum Movement)

Figure 7.35 — Training Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Right-Turn Movement.

161

Link 6—7 Link 7-8 Link 8—9 Link 9-10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30

25
E
E 20
u:
g IAbsolute
g 15 Percentage
0 Travel Time

(Right-tum
g 10 Movement)
5 I _ I . I I I I II I
5 95 5 95 5 95 5 95

Departure lime-Pence (minutes)

Figure 7.36 - Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Training Set 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 16-17 Link 17-18 Link 18-19 Link 19-20

60

50
E
E 40 ~—
g 30 7 7 IAbsolute
§ Percentage
x Error for

Travel Time

3 20 (Right-tum
3 Movement)

0 ll Viv-1|; II I . I WIIIIII, ”Illllll I I . III I WIIIIEIIIWIJII W||.| III

5 95 5 95 5 95 5 95

Departure T ime-Period (minutes)

Figure 7.37 — Absolute Percentage Error 0f Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Training Set 2.

162

h
0'!

8

(a)
0|

0)
O

N
0|

N
O

15

Absolute Percentage Error (%)

10

Link 21-22

Link 22-23

Link 23—24

Link 24-25

 

 

 

 

 

 

 

IAbsolute
Percentage

 

Error for
Travel Time

 

(Right-tum
Movement)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—.
-
‘ —I
—I
.
—
-
I
-

 

 

 

 

 

-.l
v v

 

 

 

‘ —_
‘ —_I

 

 

95 5“ 95

Departure Time-Period (minutes)

Figure 7.38 - Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Training Set 3.

70

8

0|
0

(A
O

N
0

Absolute Percentage Error (%)
8

.3
O

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 3-8 Link 843 Link 13-18 Link 18-23
IAbsolute
Percentage
Error for
Travel Time
(Right-turn
Movement)
i I. . il I I . . I
IIWIIIIIIIIIIIIIII IIIIIIII I II III I ..IIII II V
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.39 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Training Set 4.

163

 

Table 7.13 — Performance Measure of CI-SSNN—Right Model for Travel Time Prediction

for Right-Tum Movement on Training Set

 

 

 

 

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 6.5 8.5 7 .3

16-20 9.3 11.8 11.0

21-25 9.0 12.1 9.3

3-23 8.8 12.8 10.7

Total Training Set 8.4 11.4 9.6

 

7.4.6 Results of CI-SSNN-Right Travel Time Prediction Model for Travel Time

Prediction for Right-Tum Movement on Testing Set

After training of the CI-SSNN-Right model, it is applied on a testing set to evaluate the
generalization ability of the model. The plot of the predicted versus actual travel time is

shown in Figure 7.40. This plot shows a R2 value of 0.81, which is comparable to that

obtained in the training set.

164

250.00

 

I Predicted
vs Actual
/ Travel
200.00 r Time
(Right-tum
Movement)

 

 

 

150.00

— Best-Fit
Line(Predic
ted vs.

x = 1.0619x + 2.2267 Actual

R2 = 0.8088 Trave'

Time,

Right-turn

Movement)

  
 

Movement)

100.00

 

 

50.00 —

 

 

 

 

Predicted Travel Time. seconds (Right-turn

 

 

0.00 T I . I
0.00 50.00 100.00 150.00 200.00 250.00

 

Actual Travel Time, seconds (nght-turn Movement)

Figure 7.40 -— Testing Set Scatter Plot of Predicted versus Actual Travel Time for Travel

Time Prediction for Right-Tum Movement.

The APE distribution for testing sets 1, 2, 3, and 4, which represent arterials 6—10, l6-20,
21-25, and 3-23, respectively, is shown in Figures 7.41, 7.42, 7.43, and 7.44. The APE
distribution for the predicted travel time shows that most of the arterials in the testing set
have less than 20% percentage error value for predicted travel time. The arterial links
which have higher percentage error for some future departure time—periods are identified
as 6-7, 7-8, 8-9, 16-17, 18-19, 19-20, 21-22, 22-23, 24-25, 8-13, and 3-18. These arterials
show on average higher percentage error than 20% value. It indicates that the travel time
prediction models developed here perform relatively poor for right-turning movements as
compared to through and left-turning movements. The reason for the relatively poor
performance of prediction models for right-turning movements is because the right—

turning traffic shares the same lanes and signal control state as that of through

165

movements. The interference caused by the through movement traffic to the right-tum

movements makes their travel time prediction difficult.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 6-7 Link 7-8 Link 8—9 Link 9—10

60

50
25
E
I: 40
m
g I Absolute
g 30 L Percentage
. Error for
0- | Travel Time
g 20 _ I. (Right-turn
3 Movement)

10 ~

0
5 95 5 95 5 95 5 95

Departure llmO-POI'IOO (mlnutes)

Figure 7.41 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Testing Set 1.

166

Link 21-22 Link 2223 Link 23-24 Link 24-25

 

 

8

 

 

(0
0|

 

 

 

 

IAbsolute
Percentage
Error for
Travel Time
(Right-tum
Movement)

 

N
o
I
1

 

 

 

 

.A
(It

 

Absolute Percentage Error (%)
N on
at O

 

 

 

 

v v v vv yy

5 H ' 955 955' ‘ 95l‘5' b5

Departure Tlme-Pertod (minutes)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

..s
0' O
—_I
—
=

 

 

 

Figure 7.42 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Testing Set 2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 16-17 Link 17-18 Link 18-19 Link 19-20

80

70
is: so
i 50
g IAbsolute
§ 40 Percentage
8 Error for

30 I Travel Time
g (Right-tum
3 Movement)

20 l I

_ I I J _ I I
1oIII Illllllllilll
o],|,.5|,.1|,|7“ I I,|Iu.i_|IIi.II.11 I V.I._Ii||l II I|I1.,.,l.1.I

 

 

5 V 95595 5 v 95 5 5 95
Departure Time-Period (minutes)

Figure 7.43 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Testing Set 3.

167

Link 3—8 Link 8-13 Link 13-18 Link 18—23

 

 

 

 

 

 

 

 

 

 

70

60 ~
§
26' 50 «—
t
in
§ 40
E IAbsolute
2 Percentage
g 30 9» Error for
a Travel Time
§ (Right-turn
E 20 Movement)
1:

 

 

 

v r1 . . v

vvvvvvvvvvvvv

III

5 95 5 95 5 '7 95 5 95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A
O O
—
—
—
—
——
—-

Departure Time-Period (minutes)

Figure 7.44 — Absolute Percentage Error of Predicted Travel Time versus Departure Time

for Travel Time Prediction for Right-Tum Movement on Testing Set 4.

The plots showing pattern of the predicted and actual travel time are presented in Figures
7.45, 7.46, 7.47, and 7.48. It is shown that the predicted travel time is following the actual
travel time pattern except for the arterials which have higher APE distribution as shown
earlier.

The performance measure values of the CI-SSNN-Right model on the testing set are
presented in Table 7.14. The MAPE value for the testing set is 15.9%. It was also
observed in the APE distribution plots presented earlier that the percentage error values
are above 20% for a significant number of arterials. This indicates the relatively poor
predictive performance of travel time modeling when applied to the right-turn

movements.

168

Link 6-7 Link 7-8 Link 8-9 Link 9-10
250.00

 

 

 

 

----- Actual
Travel
Time.
seconds

—— - (Right-turn

Movement)

 

 

 

 

- -l- Predicted
Travel
Time.
seconds
(Right-turn
Movement)

.I'. rho" .

 

 

 

 

Average Travel Time, seconds (nght-tum Movement)

 

 

 

 

 

 

5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.45 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Right-Tum Movement on Testing Set 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 21-22 Link 22-23 Link 23-24 Link 24-25
A 180.00
E
5 160.00 4" - 1
14000 a» _.. __ 4.. ----- Actual
g ‘ r, 3 Travel
‘r - 3'4”. _.;_ '- . .r, i Time.
£120.00 ‘r—'_ “W"?- ... - ' seconds
E 3:. : ‘- (Right-turn
" 100.00 ~— — w —--r~— a; Movement)
g 80.00 «—— —-——- ~—J — .‘ " . ,.3 —a
g“ : 3;, "‘ -O-Predicted
60.00 fi— 3 —~ Travel
E __ ..‘_ .-.~"- Time.
__’_n' ____.k Q seconds
5 40.00 ~ .
(Right-turn
'- Movement)
g 20.00 —~
2
‘ 0.00 A *
5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.46 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Right-Tum Movement on Testing Set 2.

169

Link 16-17 Link 17-18 Link 18-19 Link 19-20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 180.00
‘E
§ 160.00 A“ .—
E 140.00 ~ -- . f 2 ,_ ----- Actual
3 RF 3 , _ ' Travel
‘3‘ _ i 2; 4. 3‘ 3 ‘ Time'
5.12000 «c— -- -.- r . , .— a :, , _.. . seconds
g g‘f : ; "‘ ‘g 9.: 1 (Right-turn
lg 100.00 9' ' -~— — ;. :9; :1. Movement)
g 80.00 —
g. +Predicted
60.00 + Travel
E ' ’ Time,
' seconds
g 40°00 (Right-tum
E Movement)
20.00
2
< 0.00

 

 

 

 

 

5 95 5 95 5 95 5 95

Departure Time-Period (minutes)

Figure 7.47 - Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Right-Tum Movement on Testing Set 3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Link 3-8 Link 8-13 Link 13-18 Link 18-23

A 180.00

E

2

g ----- Actual

§ Travel

‘l‘ Time,

E seconds

a (Right-turn

g Movement)

2‘ +Predicted
Travel

E Time,

g seconds
(Right-turn

'- Movement)

§ 20.00 .- — ~ — ——~ ——

E

< 0.00 - - - A

5 95 5 95 5 95 5 95

Departure Tlme-Period (minutes)

Figure 7.48 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Right-Tum Movement on Testing Set 4.

170

Table 7.14 — Performance Measure of CI-SSNN-Right Model for Travel Time Prediction

for Right-Tum Movement on Testing Set

 

 

 

 

 

 

 

 

 

 

 

 

Arterial MAE RMSE MAPE (%)
(seconds) (seconds)

6-10 13.5 19.5 15.3

16-20 13.6 16.5 17.9

21-25 15.4 20.7 16.9

3-23 10.9 14.9 13.3

Total Testing Set 13.4 18.0 15.9

 

7.5 Results of Travel Time Prediction on Arterial Route

The development of the travel time prediction models for specific traffic movement on
arterial links leads to an easy extension to any selected route in the network. The example
route which is selected in this study has intersection 16 as its origin and intersection 25 as
its destination. This example route was shown earlier in Figure 6.6. The path adopted for
this route covers links 16-17, 17-18, 18-23, 23-24, and 24-25. The way of extending the
CI-SSNN models developed earlier for this route is discussed in Chapter 6. Recalling that
the travel time prediction models developed earlier for specific traffic movement is
applied on each link comprising the route. In this way, the travel time prediction can be
done for arterial route once the models were developed for links.

The results obtained on the testing set are enough to show the predictive performance of
CI-SSNN models applied to this route. Figure 7.49 shows the pattern of the actual and

predicted average travel time on the route with departure time-periods in future. The first

171

future departure time-period is starting from 5 minutes and ending at 10 minutes.
Similarly, the last future departure time-period starts at 90 minutes and ends at 95
minutes. It is seen that the predicted average travel time follows the same pattern of the

actual travel time.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

550.00
525.00
8
C
8
8 50°00 ----- Actual
0' Travel
5; Time.
3 475.00 - seconds
5
|: Fred
0 -+- icted
3 450.00 Travel
2 Time.
< seconds
425.00 + - —~
400.00 T r r r r . l r r l r r r

 

51015202530354045505560657075808590

Departure Tlme-Perlod (minutes)

Figure 7.49 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction for

Route 16-25 on Testing Set.

A detailed pattern of the actual and predicted travel time is presented in Figure 7.50
analyzing arterial links that comprise the example route 16-25. This figure shows that the
predicted travel time closely follows the actual travel time values for arterial links

considering appropriate traffic movement.

172

Link 16—1 7 Link 17-18 Link 18-23 Link 23-24 Link 24-25

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

180.00
160.00 ~> r
3
u
3 140.00 “- --—~
'>’
a ...I --I~-Actua|
E 12000 .. '9: W .' Travel
0 -' , '.' Tune
3 100 00 «» ' -' "
g . . \’
E 80.00 ‘- *‘———~ ~~—-#—r ~_—r .' .1 . F——— -
e ' ". ; 2:
5 ~ . _PrwIth
l- 60.00 ~—- , . .. _ ’i “H” -- Travel
a. I ' 5 “me
E .A, AA. .A-A AAA A
g 40.00
<
20.00 --- rr- ----— .
0.00
5 95 5 95 5 95 5 95 5 95

Deaparture Time-Period (minutes)

Figure 7.50 — Pattern of Actual and Predicted Travel Time for Travel Time Prediction on

arterial links comprising the Route 16-25 on Testing Set.

A further investigation is required to see the Absolute Percentage Error (APE)
distribution for the route. Figure 7.51 shows that APE values for all the departure time-
periods are within 10% value. Table 7.15 lists values of the measures of performance for
this route which shows that the MAPE value is 3.8% for this route.

It is noted that the MAPE value for the travel time prediction on example route is much
lower than those for arterial links presented earlier. The reason is that the average travel
time for the route is obtained in this study by adding the travel time for each link
considering the appropriate traffic movement. At some of the links comprising the route,
the travel time may have been predicted as higher than actual travel time, and for the

other links, the predicted travel time may have been lower than actual travel time. The

173

addition of the travel time on the links comprising the route balances out the negative and
positive errors and hence provides more accurate value of travel time for the route. It is
worth mentioning that the way of finding out the travel time on the route as adopted in
this study is valid as the appropriate traffic movement required to traverse the route on

each link is considered.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10

9
I
E 7
W - I IAbsolute
§6 Percentage
‘é' Enorfor
5 I Time.
% 4 I seconds
‘5 -
33 I“ I

21

“ HIIIIrIHHHnI

0‘ I

51015202530354045505560657075808590

Departure Time-Period (minutes)

Figure 7.51 — Absolute Percentage Error Distribution for Travel Time Prediction for

Route 16-25 on Testing Set.

Table 7.15 — Performance Measure of CI-SSNN Model for Travel Time Prediction for

Route 16-25 on Testing Set

 

Route MAE RMSE MAPE (%)
(seconds) (seconds) .

 

 

 

 

 

 

 

16-25 (testing set) 18.4 23.2 3.8

 

174

Chapter 8

Conclusions and Future Research

The conclusions and findings in this study are summarized. The limitations of the study
and scope of future research for field implementation of the proposed methodology is

discussed.

8.1 Conclusions and Findings

The conclusions and significant findings from this research on travel time estimation and
prediction are as follows:

1. Travel time estimation and prediction on urban arterial networks is a complex
dynamic process with non-linear relationship between traffic parameters like
average flow, average speed, queue length, geometries, etc. and the average travel
time of current and future departure time-periods.

2. Travel time prediction should be combined with the estimation of travel time as
the estimated travel time for current departure time-period acts as an important
input variable along with known traffic parameters for travel time prediction
modeling.

3. The proposed models of travel time estimation and short-term prediction are using

traffic parameters that can be easily obtained or collected in the field by traffic

175

detection technologies like loop detectors and image processing based techniques.
The developed models estimate and predict the travel time using inputs as average
flow rate, queue length, geometries, and signal control parameters only. The
reliance of these models on easily available data makes the models economical
and feasible for field deployment for real-time applications.

. The developed models explicitly take into account turning movements,
geometries as well as signal control scheme on arterial networks. This is a
significant contribution of this study because of the complexity involved in
inclusion of all these parameters.

. The state-space dynamics of traffic behavior in an urban arterial is captured
successfully by using a generic class of Recurrent Neural Networks (RNN)
termed the State-Space Neural Networks (SSNN). The use of SSNN provides
flexibility and efficiency while at the same time modeling the non-linear
dynamical traffic system.

. A pre-processing of observable traffic parameters is required to minimize the
input parameters for SSNN models. A statistical technique called as the
Conditional Independence (CI) graph is introduced for the first time with artificial
neural networks in this research. The analysis of conditional independence and
interaction among observable traffic parameters and the average travel time
increases the understanding of the process as well as improves the predictive
performance of SSNN.

. The SSNN model based on the CI graphs is compared with the typical SSNN

model used by researchers in this study. It shows that the accuracy of the CI

176

10.

graphs based SSNN model is comparable to that of typical SSNN models which
do not use the inferences made by the CI graphs. The CI graphs based SSNN
model is dependent upon lesser input traffic variables as compared to typical
SSNN model. The constraint of limited amount of data obtained in any
experiment or in the field makes the CI graphs based SSNN models a better
choice than the SSNN models. This application of the CI graphs is recommended
not only for the travel time estimation and prediction modeling as done in this
research but also to other traffic modeling problems.

Three SSNN models based on the CI graphs are developed for travel time
estimation for the specific traffic movement of through, left, and right-tum
movement on arterial links. Similarly, three SSNN models based on the CI graphs
are developed for short-term travel time prediction for arterial links. Each single
model is able to capture travel time fluctuations for uncongested, congested, and
the transition stage between these two traffic conditions on arterial links. Thus,
each single SSNN model can perform successfully on arterial networks
throughout the day.

A single SSNN model based on the CI graph is proposed for travel time
estimation and prediction on urban arterials for specific movement which is
efficient for both congested and non-congested traffic conditions occurring
throughout a day.

The CI graph based SSNN models developed for arterial links are extended to

obtain results on any route in arterial network. Thus, the developed travel time

177

ll.

12.

13.

models can be applied on a single arterial link, an entire arterial or a route in a
network.

The Mean Absolute Percentage Error (MAPE) for proposed average travel time
estimation models for arterial links ranges from 6.5% to 15.0%. The MAPE value
for average travel time prediction models ranges from 6.9% to 17.9%. The MAPE
value for the route considered in this study is 3.8%.

The Mean Absolute Percentage Error for average travel time estimation and
prediction for right-tum movement is found more challenging that that of through
and left-tum movements. The reasons for complexity in right-turn movement
travel time modeling is the high interference among through and right-turn
movement vehicles on an arterial.

The travel time estimation and prediction modeling approach in this study is more
generalized as compared to the studies done to date. The proposed models
account for traffic variables like geometries, turning movements, signal control
scheme, which were either assumed constant or not explicitly taken into account
in earlier studies. The proposed models provide prediction for through, left, and
right-turning movements on arterial links and routes, which is a significant

contribution of this study.

178

8.2 Limitations

The limitations of proposed modeling approach are listed as follows —

1. The proposed methodology does not include some factors like weather effects,
incidents etc. These factors are required for a real-time implementation of the
model in field. The flexibility of the modeling approach allows the inclusion of
these factors provided that the data is available.

2. The modeling of travel time estimation and short-term prediction is based on a
hypothetical arterial network created in a microscopic traffic simulation. The
implementation of the models in a real world setting requires calibration and
validation with site data. However, enough variability has been included in
experimental set-up that accounts for a wide range of geometric conditions and

traffic demands.

3. The accuracy of average travel time estimation and prediction models for right-
turning movements is not as good as through and left-turning movements. A high
interference between right-turning traffic and through traffic on arterial links is
observed at signalized intersections in this study. This makes the travel time
modeling for right-turning movements more challenging than other traffic
movements. Thus, the travel time modeling for right-turning movement requires

consideration for improvement of the accuracy.

179

8.3 Scope of Future Research

There are some recommendations and scope of future research which are listed below:

1. The coupling of Conditional Independence graph with Artificial Neural Networks
(ANN) is recommended not only for travel time problem but also to other
engineering problems where ANN is used as a potential modeling or prediction
technique.

2. The short-term travel time prediction in this study focuses on 5 minutes time-
period ahead in future. The travel time prediction for further time-periods in the
future can be done using the same methodology.

3. The real-time implementation of travel time estimation and short-term prediction
model should be done using the same methodology. The author realizes that
issues related to data-collection in the field, data-fusion, and dissemination of

travel time information will pose another challenge to handle.

180

APPENDIX

MATLAB Code for State-Space Neural Network

The Appendix presents the MATLAB code for State-Space Neural Network Model. This
code is written for travel time prediction of thru movement on arterials. Similar code can

be used for other SSNN models changing the parameters.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

display('TRAVEL TIME PREDICTION FOR THRU MOVEMENT');

load traininoutm ; %loading input data file
P = traininout(:,:);
P = rot90(P,1);

ptrl = P( 2:9,:);ptr1 = eon23eq(ptrl); %link 1 input
ttrl = P(1,:);ttr1= con23eq(ttrl); %link 1 output-travel time
for i = 1:288

ptr{1.i} =ptr1{1.i};

end

for i = 1:288

ttr{1,i} = ttr1{1,i};

end

%Loading up State Space Neural Network called here as net]
disp('Loading up network netl ...');

load not];
met] = init(net1); %Initializing the network

%Training Parameters
netl .adaptParam.epochs = 10000;

181

net1.adaptParam.goal = 0.001;
net1.adaptParam.max_fail = 10000
%Five Validations to check when error rises

netl .adaptParam.mem_redue = 1;

%Full J acobian Calculated, no memory restrictions
net1.adaptParam.min_grad = 1e-10;
net1.adaptParam.mu = 0.9; %momentum constant
net1.adaptParam.mu_dec = 0.001 ;
net1.adaptPararn.mu_inc = 10;
net1.adaptParam.mu_max = lelO;
net1.adaptParam.show = 100;
netl .adaptParam.time = inf;

net1.adaptparam.lr = 0.001; % Learning Rate
netl .perforchn = 'mse'; %Perfonnance = 'Mean Square Error'
%Training

disp('Training ...');
[netl ,tr]= adapt(net1 ,ptr,ttr);
save net1.MAT netl ;

%Conversion of ttr to concurrent matrix
ttr = seq200n(ttr);

%Results - Network Training
a2 = sim(net1,ptr);
a2 = seq2con(a2);

figure(1)

title('Regression Analysis of Training Results: Travel Time Thru'); hold on
[m,b,r] = postreg(a2 {1,1 },ttr{1,1}); hold off

perfirgl = mse(a2{1,1} - ttr{1,1})

perfirg2 = mae(a2{1,1} - ttr{1,1})

load testinoutm; %loading testing file
Q = testinout(:,:);

Q = rot90(Q,1);

test] = Q(2:9,:); testl = con23eq(test1);

fori = 1:288
test.P{1,i} = test3{1,i};
end

182

testTl = Q(l,:);testT1 = con25eq(testT1);

for i = 1:288
test.T{1,i} =testT1{1,i};
end

test.T = seq2con(test.T);
%Results - Network Testing

a3 = sim(net1,test.P);

figure(5)

title('Regression Analysis of Testing Results: Travel Time Thru'); hold on
[m,b,r] = postreg(a3 {1,1 },test.T { 1 ,1 }); hold off

perftstl = mse(a3{1,1} - test.T{1,1})

perftst2 = mae(a3{1,1} - test.T{1,1})

format short;
disp('Network Arehitecture');
disp('Number of Layers');
disp(netl .numlayers+1);
nneurons = 5;
slayer = size(netl .layers);
for i=1:slayer(l,1)
nneurons =[nneurons netl .layers {i} .size];
end

%saving input weights files
iwl = netl.IW{1}';
iw2 = netl.IW{2}’;
iw3 = netl.IW{3}';
iw4 = netl.IW{4}';

save iwl .dat iwl -ascii -tabs;
save iw2.dat iw2 -ascii —tabs;
save iw3.dat iw3 -aseii -tabs;
save iw4.dat iw4 -ascii -tabs;

%saving layerweights files
1w1= net1.LW{1,1}';
lw2 = net1.LW {2,2}';
lw3 = netl.LW{3,3}';

183

lw4 = netl .LW {4,4}';

lw4l = net1.LW{4,1}';
lw42 = net1.LW{4,2}’;
lw43 = netl .LW {4,3}';

save lw1.dat lwl -ascii -tabs;
save lw2.dat lw2 ~ascii -tabs;
save lw3.dat lw3 -ascii -tabs;
save 1w4.dat lw4 -ascii ~tabs;
save lw41.dat lw41 -ascii -tabs;
save lw42.dat lw42 -aseii -tabs;
save lw43.dat lw43 -ascii -tabs;

%forming training and testing output files to save
trgoutttthru = a2 {1 }';
testoutttthru = a3 {1 }';

%saving output in .dat files
save trgoutttthru.dat trgoutttthru -ASCII -DOUBLE -TABS;
save testoutttthrudat testoutttthru -ASCII -DOUBLE -TABS;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

184

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