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MODELING TRAVEL TIME IN URBAN ARTERIAL
NETWORKS WITH TIME-VARIANT TURNING MOVEMENTS
USING STATE-SPACE NEURAL NETWORKS

presented by

Timothy Joseph Likens

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MODELING TRAVEL TIME IN URBAN ARTERIAL NETWORKS WITH TIME-
VARIANT TURNING MOVEMENTS USING STATE-SPACE NEURAL
NETWORKS
By

Timothy Joseph Likens

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF- SCIENCE
Department of Civil Engineering

2007

ABSTRACT
MODELING TRAVEL TIME IN URBAN ARTERIAL NETWORKS WITH TIME-
VARIANT TURNING MOVEMENTS USING STATE-SPACE NEURAL
NETWORKS
By

Timothy Joseph Likens

Advanced Traffic Management Systems (ATMS) and Advanced Traveler Information
Systems (ATIS) have become integral components in congestion mitigation strategies,
and are dependent on the ability to reliably estimate and predict travel time. Urban
arterial networks are highly complex and dynamic systems for which travel time has been
difficult to accurately model. This is in part due to the impact of turning movements at
signalized intersections on traffic flow, and thus travel time. State-Space Neural Network
(SSNN) models are developed in this thesis to estimate and predict travel time on arterial
links and routes within an arterial network. Separate models are developed respective to
through, right-tum, and left-tum vehicle movements. The data used to model travel time
include variables that are easily collected in the field using existing surveillance
infrastructure such as queue length, flow rate, and average speed. The inclusion of
variable turning movements in the modeling procedure is observed to have a significant
impact on the accuracy of SSNN models developed to estimate and predict travel time,

especially for the right-tum movement.

Keywords: Travel Time Prediction, State-Space Neural Networks, Turning Movements,

Urban Arterial Networks.

Dedicated to Grandma Hojnowski and Grandma Likens, who could not be here to read

the final page.

iii

ACKNOWLEDGEMENTS

I must first thank my fiancee Rebecca Peters for her love and devotion despite the
countless number of days I had to Spend away from her in pursuit of my Master’s Degree.
Her support carried me through some long nights, and motivated me to leap hurdles along
the way. I am also forever gracious to my parents Terry and Jean Likens. The endless
sacrifices they have made for my brother Jeff and I have enabled us to do achieve great
things. May the work we do now pay our debts to them in the form of pride and joy.

I would also like to thank my advisor, Dr. Ghassan Abu-Lebdeh. His continuing
guidance helped me not only to complete this thesis, but also to challenge myself in terms
of my career aspirations. As my goals evolved throughout the course of the Master’s
program, Dr. Abu-Lebdeh was consistently supportive. I wish to also acknowledge Dr.
Francois Dion for the education he provided in the classroom. He taught me a great deal
about modeling and simulation that I will carry with me into my professional career.

Finally, I am greatly appreciative of the support Dr. Rick Lyles gave in bringing me back
to Michigan State University. He has been my mentor at MSU, and his class gave me my
first interest in Transportation Engineering. Dr. Lyles recognized my desire to teach, and
I cannot thank him enough for having me as his Teaching Assistant. I thank him for
placing trust in me to be a part of his classroom, and I will miss being able to drop by his

office for what was always an interesting conversation.

Timothy J. Likens

Michigan State University, 2007

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... vii
LIST OF FIGURES ........................................................................................................ viii
INTRODUCTION .............................................................................................................. 1
CHAPTER 1 — LITERATURE REVIEW .......................................................................... 5
1.1 Travel Time Estimation and Prediction ............................................................... 6
1.2 Travel Time Reliability ........................................................................................ 7
1.3 State-Space Neural Network Application ............................................................ 8
1.4 Conclusions Based on Related Literature ............................................................ 9
CHAPTER 2 — PROBLEM STATEMENT AND RESEARCH OBJECTIVE ................ 10
2.1 Background and Definitions .............................................................................. 11
2.2 Average Travel Time Estimation versus Prediction .......................................... 14
2.3 Formulation for Average Travel Time Estimation and Short-Term Prediction
Models ................................................................................................................ 16
2.4 Factors Impacting Average Travel Time Estimation and Short-Term
Prediction ........................................................................................................... 22
2.5 Modeling Approach for Travel Time Estimation and Short-Term
Prediction ........................................................................................................... 24
CHAPTER 3 — EXPERIMENTAL SET-UP .................................................................... 27
3.1 Overview of VISSIM Microscopic Simulation ................................................. 28
3.2 Static Network Elements .................................................................................... 29
3.3 Stochastic Network Elements ............................................................................ 31
3.4 Time-Dependent Network Elements .................................................................. 32
3.5 Signalized Intersection Control Design Using Synchro .................................... 36
3.6 Data Collection from VISSIM Simulation ........................................................ 37
3.7 VISSIM Simulation Runs and Model Validation .............................................. 39
CHAPTER 4 — MULTIVARIATE LINEAR REGRESSION FOR AVERAGE LINK
SPEED ESTIMATION ..................................................................................................... 40
4.1 Multivariate Linear Regression Methodology ................................................... 40
4.2 Linear Regression Models for Link Speed Estimation ...................................... 44
4.3 Average Speed Estimation for Major Arterials ................................................. 45
4.4 Average Speed Estimation for Minor Arterials ................................................. 52

CHAPTER 5 — STATE-SPACE NEURAL NETWORK MODELING

METHODOLOGY ........................................................................................................... 59
5.1 State-Space Neural Networks .................................................................................. 60
5.2 State-Space Neural Network Topology ............................................................. 61

5.2.1 SSNN Topology for Travel Time Estimation ................................................. 62

5.2.2 SSNN Topology for Travel Time Prediction .................................................. 62
5.3 State-Space Neural Network Training and Testing ................................................. 65
5.4 Application of State-Space Neural Network Models to Travel Time Estimation and

Prediction for Arterial Routes .................................................................................. 68

CHAPTER 6 — RESULTS AND DISCUSSION .............................................................. 70
6.1 Measures of Effectiveness ....................................................................................... 70
6.2 Travel Time Estimation on Arterial Links ............................................................... 73

6.2.1 Results of SSNN Travel Time Estimation Model for Through

Movement ....................................................................................................... 74
6.2.2 Results of SSNN Travel Time Estimation Model for Left-Turn

Movement ....................................................................................................... 76
6.2.3 Results of SSNN Travel Time Estimation Model for Right-Tum

Movement ....................................................................................................... 79

6.3 Travel Time Prediction on Arterial Links ................................................................ 83

6.3.1 Results of SSNN Travel Time Prediction Model for Through

Movement ....................................................................................................... 84
6.3.2 Results of SSNN Travel Time Prediction Model for Left-Tum

Movement ....................................................................................................... 97
6.3.3 Results of SSNN Travel Time Prediction Model for Right-Tum

Movement ..................................................................................................... 109

6.4 Travel Time Prediction on Arterial Routes ............................................................ 121

6.4.1 Results of SSNN Travel Time Prediction Models for Arterial Route

16-25 ............................................................................................................. 122
6.4.2 Results of SSNN Travel Time Prediction Models for Arterial Route

16-3 ............................................................................................................... 125
6.4.3 Results of SSNN Travel Time Prediction Models for Arterial Route

15-2 ............................................................................................................... 128

CHAPTER 7 — CONCLUSIONS AND FUTURE RESEARCH ................................... 132

7.1 Key Findings and Conclusions ........................................................................ 132

7.2 Limitations ....................................................................................................... 135

7.3 Scope of Future Research ................................................................................ 136

APPENDIX ..................................................................................................................... 137

REFERENCES ............................................................................................................... 141

Vi

LIST OF TABLES

Table 2.1 — Variables Considered for Travel Time Estimation and Short-Term
Prediction .......................................................................................................................... 24

Table 4.1 - Candidate Independent Variables for Multivariate Linear Regression ......... 42

Table 5.1 — Input and Output Variables for SSNN Travel Time Estimation Models ........ 66

Table 5.2 — Input and Output Variables for SSNN Travel Time Prediction Models ........ 67
Table 6.1 — Travel Time Estimation Error Statistics for the Through Movement ............ 76
Table 6.2 — Travel Time Estimation Error Statistics for the Left-Turn Movement .......... 78
Table 6.3 — Travel Time Estimation Error Statistics for the Right-Tum Movement ....... 81

Table 6.4 — Mean Absolute Percentage Error Comparison Under Constant and Variable
Turning Movement Percentage Conditions ...................................................................... 82

Table 6.5 — Travel Time Prediction Error Statistics for the Through Movement ............. 86

Table 6.6 — Mean Absolute Percentage Error Comparison for Through Movement Under
Constant and Variable Turning Movement Percentage Conditions ................................. 96

Table 6.7 — Travel Time Prediction Error Statistics for the Left-Tum Movement ........... 99

Table 6.8 — Mean Absolute Percentage Error Comparison for Left-Turn Movement Under
Constant and Variable Turning Movement Percentage Conditions ............................... 108

Table 6.9 — Travel Time Prediction Error Statistics for the Right-Turn Movement ...... 111

Table 6.10 - Mean Absolute Percentage Error Comparison for Right-Tum Movement
Under Constant and Variable Turning Movement Percentage Conditions .................... 120

Vii

LIST OF FIGURES

Figure 2.1 — Time-Space Domain for the Formulation of Average Travel Time
Estimation and Short-term Prediction ............................................................................... 21

Figure 2.2 — Flowchart of Modeling Approach for Travel Time Estimation and Short-
Terrn Prediction ................................................................................................................ 26

Figure 3.1 — Urban Arterial Network Designed in VISSIM for Travel Time Estimation

and Prediction ................................................................................................................... 30
Figure 3.2 — Average Flow Rate Distributions at Entry Points of Arterials ..................... 33
Figure 3.3 — Left-Tum Profiles for Signalized Intersection Approaches ......................... 35
Figure 3.4 —- Right-Tum Profiles for Signalized Intersection Approaches ....................... 35
Figure 3.5 — Typical Signal Timing Plan for Signalized Intersections ............................. 37

Figure 4.1 — Training Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Through Movement on Maj or Arterials ............................ 47

Figure 4.2 — Testing Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Through Movement on Major Arterials ............................ 47

Figure 4.3 -— Training Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Left-Tum Movement on Major Arterials .......................... 48

Figure 4.4 — Testing Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Left-Tum Movement on Major Arterials .......................... 48

Figure 4.5 - Normalized Average Through Speed versus Normalized Average Right-
Tum Speed on Major Arterials ......................................................................................... 50

Figure 4.6 — Training Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Right-Tum Movement on Major Arterials ....................... 51

Figure 4.7 - Testing Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Right-Tum Movement on Major Arterials ....................... 52

Figure 4.8 — Training Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Through Movement on Minor Arterials ........................... 53

viii

Figure 4.9 — Testing Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Through Movement on Minor Arterials ........................... 54

Figure 4.10 —- Training Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Left-Turn Movement on Minor Arterials ......................... 55

Figure 4.11 — Testing Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Left-Turn Movement on Minor Arterials ......................... 55

Figure 4.12 — Training Plot for Normalized Actual Average Speed versus Normalized
Estimated Average Speed for Right-Tum Movement on Minor Arterials ....................... 57

Figure 4.13 —— Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Right-Tum Movement on Minor Arterials ....................... 58
Figure 5.1 — Simple Neural Network Framework ............................................................ 61
Figure 5.2 — State—Space Neural Network Topology for Travel Time Estimation ........... 63
Figure 5.3 — State-Space Neural Network Topology for Travel Time Prediction ............ 64
Figure 5.4 — Sample MATLAB Graphical Output for SSNN Training ............................ 67
Figure 5.5 - Study Routes for the Evaluation of SSNN Model Performance .................. 69

Figure 6.1 — Training Scatter Plot of Actual versus Estimated Travel Time for the
Through Movement .......................................................................................................... 74

Figure 6.2 — Testing Scatter Plot of Actual versus Estimated Travel Time for the Through
Movement ......................................................................................................................... 75

Figure 6.3 — Training Scatter Plot of Actual versus Estimated Travel Time for the Left-
Tum Movement ................................................................................................................ 77

Figure 6.4 — Testing Scatter Plot of Actual versus Estimated Travel Time for the Left-
Tum Movement ................................................................................................................ 78

Figure 6.5 — Training Scatter Plot of Actual versus Estimated Travel Time for the Right-
Tum Movement ................................................................................................................ 79

Figure 6.6 — Testing Scatter Plot of Actual versus Estimated Travel Time for the Right-
Turn Movement ................................................................................................................ 80

Figure 6.7 — Training Scatter Plot of Actual versus Predicted Travel Time for the
Through Movement .......................................................................................................... 84

ix

Figure 6.8 — Testing Scatter Plot of Actual versus Predicted Travel Time for the Through
Movement ......................................................................................................................... 85

Figure 6.9 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Training Set Arterial 6-10 ........................................................................................... 87

Figure 6.10 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Testing Set Arterial 6-10 ............................................................................................. 87

Figure 6.11 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Training Set Arterial 16-20 ......................................................................................... 89

Figure 6.12 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Testing Set Arterial 16-20 ........................................................................................... 89

Figure 6.13 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Training Set Arterial 21-25 ......................................................................................... 91

Figure 6.14 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Testing Set Arterial 21-25 ........................................................................................... 91

Figure 6.15 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Training Set Arterial 3-23 ........................................................................................... 92

Figure 6.16 — Absolute Percentage Error of Through Movement Travel Time Prediction
for Testing Set Arterial 3-23 ............................................................................................. 92

Figure 6.17 — Pattern of Actual versus Predicted Travel Time for Through Movement on
Testing Set Arterial 6-10 ................................................................................................... 94

Figure 6.18 — Pattern of Actual versus Predicted Travel Time for Through Movement on
Testing Set Arterial 16-20 ................................................................................................. 94

Figure 6.19 — Pattern of Actual versus Predicted Travel Time for Through Movement on
Testing Set Arterial 21-25 ................................................................................................. 95

Figure 6.20 — Pattern of Actual versus Predicted Travel Time for Through Movement on
Testing Set Arterial 3-23 ................................................................................................... 95

Figure 6.21 - Training Scatter Plot of Actual versus Predicted Travel Time for the Left-
Tum Movement ................................................................................................................ 97

Figure 6.22 — Testing Scatter Plot of Actual versus Predicted Travel Time for the Left-
Tum Movement ................................................................................................................ 98

Figure 6.23 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction
for Training Set Arterial 6-10 ......................................................................................... 100

Figure 6.24 — Absolute Percentage Error of Left-Turn Movement Travel Time Prediction
for Testing Set Arterial 6-10 ........................................................................................... 101

Figure 6.25 - Absolute Percentage Error of Left-Tum Movement Travel Time Prediction
for Training Set Arterial 16-20 ....................................................................................... 102

Figure 6.26 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction
for Testing Set Arterial 16-20 ......................................................................................... 102

Figure 6.27 — Absolute Percentage Error of Left—Turn Movement Travel Time Prediction
for Training Set Arterial 21-25 ....................................................................................... 103

Figure 6.28 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction
for Testing Set Arterial 21-25 ......................................................................................... 103

Figure 6.29 — Absolute Percentage Error of Lefi-Turn Movement Travel Time Prediction
for Training Set Arterial 3-23 ......................................................................................... 104

Figure 6.30 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction
for Testing Set Arterial 3-23 ........................................................................................... 104

Figure 6.31 — Pattern of Actual versus Predicted Travel Time for Left-Turn Movement on
Testing Set Arterial 6-10 ................................................................................................. 105

Figure 6.32 — Pattern of Actual versus Predicted Travel Time for Left-Tum Movement on
Testing Set Arterial 16-20 ............................................................................................... 106

Figure 6.33 — Pattern of Actual versus Predicted Travel. Time for Left-Tum Movement on
Testing Set Arterial 21-25 ............................................................................................... 106

Figure 6.34 — Pattern of Actual versus Predicted Travel Time for Left-Tum Movement on
Testing Set Arterial 3-23 ................................................................................................. 107

Figure 6.35 — Training Scatter Plot of Actual versus Predicted Travel Time for the Right-
Tum Movement .............................................................................................................. 109

Figure 6.36 — Testing Scatter Plot of Actual versus Predicted Travel Time for the Right-
Tum Movement .............................................................................................................. 110

Figure 6.37 — Absolute Percentage Error of Right-Tum Movement Travel Time
Prediction for Training Set Arterial 6-10 ........................................................................ 112

xi

Figure 6.38 — Absolute Percentage Error of Right-Tum Movement Travel Time
Prediction for Testing Set Arterial 6-10 .......................................................................... 113

Figure 6.39 — Absolute Percentage Error of Right-Tum Movement Travel Time
Prediction for Training Set Arterial 16-20 ...................................................................... 114

Figure 6.40 — Absolute Percentage Error of Right-Tum Movement Travel Time
Prediction for Testing Set Arterial 16-20 ........................................................................ 1 14

Figure 6.41 — Absolute Percentage Error of Right-Tum Movement Travel Time
Prediction for Training Set Arterial 21-25 ...................................................................... 1 15

Figure 6. 42— Absolute Percentage Error of Right- Turn Movement Travel Time
Prediction for Testing Set Arterial 21 —25 ........................................................................ 115

Figure 6.43 — Absolute Percentage Error of Right-Turn Movement Travel Time
Prediction for Training Set Arterial 3-23 ........................................................................ 116

Figure 6.44 - Absolute Percentage Error of Right-Turn Movement Travel Time
Prediction for Testing Set Arterial 3-23 .......................................................................... l 16

Figure 6.45 — Pattern of Actual versus Predicted Travel Time for Right—Tum Movement
on Testing Set Arterial 6-10 ............................................................................................ 117

Figure 6.46 - Pattern of Actual versus Predicted Travel Time for Right-Turn Movement
on Testing Set Arterial 16-20 .......................................................................................... 118

Figure 6.47 — Pattern of Actual versus Predicted Travel Time for Right-Turn Movement
on Testing Set Arterial 21-25 .......................................................................................... 118

Figure 6.48 — Pattern of Actual versus Predicted Travel Time for Right-Tum Movement
on Testing Set Arterial 3-23 ............................................................................................ 119

Figure 6.49 — Pattern of Actual versus Predicted Travel Time for Arterial Route
16-25 ............................................................................................................................... 123

Figure 6.50 — Pattern of Actual versus Predicted Travel Time for Arterial Route
16-25 ............................................................................................................................... 123

Figure 6.51 — Absolute Percentage Error of Travel Time Prediction for Arterial Route
16-25 ............................................................................................................................... 125

Figure 6.52 — Pattern of Actual versus Predicted Travel Time for Arterial Route
16-3 ................................................................................................................................. 126

xii

Figure 6.53 — Pattern of Actual versus Predicted Travel Time for Arterial Route

16-3 ................................................................................................................................. 127
Figure 6.54 — Absolute Percentage Error of Travel Time Prediction for Arterial Route

16-3 ................................................................................................................................. 127
Figure 6.55 — Unique Route for the Validation of SSNN Model Performance .............. 128

Figure 6.56 — Pattern of Actual versus Predicted Travel Time for Arterial Route
15-2 ................................................................................................................................. 129

Figure 6.57 — Pattern of Actual versus Predicted Travel Time for Arterial Route
15-2 ................................................................................................................................. 130

Figure 6.58 — Absolute Percentage Error of Travel Time Prediction for Arterial Route
15-2 ................................................................................................................................. 131

xiii

Introduction

Throughout the United States, traffic congestion is a major problem that continues to
plague urban areas. In 2003, Americans experienced approximately 3.7 billion total
hours of delay and expended 2.3 gallons of additional fuel due to traffic congestion.
Additionally, the impacts Of urban traffic congestion cost Americans over 63 billion
dollars (Schrank and Lomax 2005). In 2006, the United States Department of
Transportation (USDOT) responded to these concerns with the National Strategy to
Reduce Congestion on America ’5 Transportation Network (Mineta 2006). The Federal
Highway Administration (FHWA) has supported this plan through a number of
congestion relief strategies that incorporate Intelligent Transportation Systems (ITS).

ITS has become an integral component of congestion mitigation in recent years,
especially in urban areas. Typically in urban highway networks, physical capacity cannot
be easily increased to reduce congestion due to right-of-way and financial restrictions.
ITS can help to reduce congestion by improving the use of existing physical capacity
through the use of technology and the communication of data within the existing
infrastructure.

Parallel with matters related to ITS and highway congestion, the FHWA has given
attention to issues associated with travel reliability. Due to fluctuating traffic demands,
incidents, weather, and other variable factors, congestion is often dynamic and
unpredictable. This creates an environment where congestion can occur at any time due
to a number of factors. ITS applications must be able to function reliably in this dynamic

environment in order to effectively reduce congestion (Cambridge Systematics 2005).

In order to improve system reliability and reduce congestion, particular ITS applications
such as Advanced Traffic Management Systems (ATMS) and Advanced Traveler
Information Systems (ATIS) have been developed. These systems utilize traffic data to
aid in the management of traffic operations and to communicate important information to
the users of the system. “These tools [ATMS/ATIS] are intended to perform real-time
system-wide traffic estimation and prediction, based on the existing surveillance
system...these intelligent functions are predicated on the availability of reliable and
robust traffic flow models capable of representing the dynamic evolution of traffic over
space and time” (Qin 2006).

In order to be effective ATMS and ATIS must provide reliable information in terms of
the current and firture states of traffic. Additionally, system managers and users must be
able to easily understand and apply this information to make informed decisions. Chen et
a1. (2002) established that travel time is both important to travelers and a meaningful
measure of performance. For this reason, travel time has emerged as a measure that
drives many ATMS and ATIS applications.

On many urban freeway segments ATMS and ATIS applications have been successfully
developed and deployed. However, these systems are not readily seen on urban arterial
networks. In comparison to free-flow highway systems, traffic flow on arterial networks
is much more complex. Research to effectively estimate and predict travel time on
arterial networks has lagged behind freeway applications due to this complexity. In order
to make ATMS and ATIS applications more viable for urban arterial networks, models

must be developed to estimate and predict travel time accurately and reliably.

Traffic flow on arterial networks is heavily influenced by Signalized intersections. May
(1990) states that, “Signalized intersections are the most critical and complicated
elements of the arterial network system.” At Signalized intersections traffic flow is
interrupted, resulting in queue formation as vehicles stop during the red interval and
platoon expansion as the green interval begins. Turning movements, signal phasing,
shockwave propagation, and other variable factors further complicate arterial traffic flow.
Additionally, the operation of Signalized intersections is temporally dependent as the
operation Of a particular cycle is impacted by the previous cycle, and will impact the
following cycle. This is especially true in congested conditions.

Effective travel time estimation and prediction models must incorporate the variability in
these factors. In order to encourage ATMS and ATIS deployment in urban arterial
networks, models must be developed that utilize existing data collection infrastructure.
In 2004, 70 percent of Signalized intersections in US. metropolitan areas were equipped
with electronic surveillance (ITS Joint Program Office 2005). These forms of electronic
surveillance, such as loop detectors and cameras, are capable of collecting data
representing queue lengths, flow rates, and turning movement percentages. Travel time
estimation and prediction models should utilize these data and corresponding traffic
signal timing data, which are easily collected with existing technology and limited need
for additional infrastructure.

Previous research has been conducted to develop State-Space Neural Network (SSNN)
models to estimate and predict travel time for the Short-term future. Singh (2007)
successfully developed such models specific to right turn, left turn, and through

movements on arterial links based on flow rate, queue length, geometries, and signal

control parameters. However, these models are based a hypothetical arterial network
where turning movement percentages are constant. Turning movement percentages can
severely impact Signalized intersection operation as turning movements impact both
signal operation efficiency and saturation flow rates (Roess et a1. 2004). In actual urban
arterial networks, turning movement flows will vary between intersections and over time.
Therefore, reliable SSNN models must be able to accurately estimate and predict travel
time in an environment where turning movements vary spatially and temporally.

The purpose of this thesis is to explore the impacts of variable right and left turning
movements on travel time estimation and prediction for urban Signalized networks.
SSNN models are developed using methodology similar to the work done by Singh, but
incorporate variable turning movements in the modeling framework. The SSNN models
rely on data that is easily obtained using existing surveillance technologies such as loop
detectors and cameras. The successful estimation and prediction of travel time in this
environment advances the work performed by Singh and proves very promising in urban

arterial ATMS and ATIS applications.

Chapter 1

Literature Review

Accurate travel time estimation and prediction is dependent on the ability to understand
the relationship between travel time and variables that may impact traffic flow.
Intelligent Transportation System (ITS) applications such as Advanced Traffic
Management Systems (ATMS) and Advanced Traveler Information Systems (ATIS)
function based on the results of research to model this relationship. Much research has
been done in this area for freeway applications where traffic flow is uninterrupted. AS a
result, ATMS and ATIS use has become more prevalent in urban freeway systems.
However, in urban arterial networks where traffic signals impact the flow of traffic,
research to date has not been sufficient to support widespread ATMS and ATIS
deployment. The need to advance research efforts regarding travel time estimation and
short-term prediction for interrupted flow systems has been documented for this reason.

Stemming from ongoing research into ITS and congestion mitigation strategies, the
reliability of transportation systems has received increased attention and has become a
focal point of recent initiatives. ITS applications must be reliable in terms of the
transportation service provided and the information presented to users and administrators
of each system. In order to advance ATMS and ATIS deployment into urban arterial
networks, travel time estimation and prediction models driving these systems must

perform accurately and reliably in a variety of conditions. Failure to do so may cause not

only a breakdown in the functionality Of the system, but also losses in public trust in

future ITS endeavors.

1.1 Travel Time Estimation and Prediction

A large number of papers have been published regarding travel time estimation and
prediction for urban freeway systems. As a part of the California Partners for Advanced
Highways and Transit (PATH), Coifman (2002) successfully researched the relationship
between spot speed and freeway link travel time. This enables the estimation of travel
time from single-point dual loop detectors. Chen et al. (2003) developed a strategy to
effectively determine travel times using linear regression for display on California
highway variable message signs. The application of Artificial Neural Networks (ANN)
has also proven successful in estimating and predicting freeway travel times in Huston,
Texas (Park and Rilett 1999). Continuing research in this area serves to improve the
reliability of ATMS and ATIS freeway applications, based on a variety of fundamental
models. This is apparent in performing a literature search on this topic.

The same is not true for urban arterial network applications. The results of most research
efforts seem to address only incremental aspects of arterial travel time estimation and
prediction, and have not yet produced robust models. For example, in a paper for the
82nd Transportation Research Board Annual Meeting, Lin et al. (2003) present the use of
intersection delay parameters to estimate arterial link travel time. However, the model
developed fails to perform accurately in congested networks due to the breakdown of

delay estimates in over-saturated conditions. The usefulness of ATMS and ATIS

applications exists in dealing with congested conditions, so this methodology does not
prove promising.

In a recent paper published in the Transportation Research Record, State-Space Neural
Network (SSNN) modeling is proposed to address the complex spatiotemporal problem
of modeling urban arterial network behavior with respect to travel time prediction (Liu et
al. 2006). This approach seems to better address the state of a link under unstable
congested conditions. However, the model is based on a specific roadway segment and
negates traffic control parameters. In arterial networks, traffic signals heavily influence
capacity, and thus the state of link operations. Robust SSNN models should incorporate
measures of capacity such as traffic signal timing to better capture the relationship
between travel time and traffic states on urban arterials. The work presented in this thesis

is focused on meeting this condition.

1.2 Travel Time Reliability

The Federal Highway Administration (FHWA) has recently given attention to matters
related to the reliability of highway systems in response to national concerns regarding
urban congestion. Growing levels of congestion have resulted in larger variations in
travel time and reduction in the reliability of the system. Through advanced data
collection, processing, and communication, ITS technologies may improve the
correlation between the actual and expected traffic conditions, and thus improve system

reliability. In order to do so, ITS technologies themselves must be reliable.

A specific example of the reliability related to traffic conditions is in ATIS systems
which provide users route guidance and congestion avoidance information. Kantowitz et
al. (1997) published findings on driver acceptance of travel time information in
preparation for early-stage ATIS deployments. In situations where information
communicated to the driver were inaccurate, drivers often lost trust in the information
system and disregarded future communications. Following these findings, travel time
estimation and prediction models are useless, if not detrimental, to traffic Operations

when unreliable.

1.3 State-Space Neural Network Application

As previously referenced, Liu et al. (2006) presented a SSNN modeling procedure to
predict travel time on an arterial roadway segment. Although lacking robustness, this
research proves SSNN modeling as a promising approach to travel time estimation and
prediction. Studies by van Lint (2004) and van Lint et al. (2002) have also proven the
SSNN modeling useful to effectively capture the dynamic spatiotemporal relationship
between measurable traffic parameters such as speed and flow rate, and travel time.

In 2006 Singh developed a SSNN modeling procedure to estimate and predict travel time
for an urban arterial network using traffic data easily collected using existing surveillance
infrastructure. The procedure presented in this thesis involves an examination of the
impacts of variable turning movements within the arterial network on the reliability of the

SSNN models previously developed.

1.4 Conclusions Based on Related Literature

A review of current literature related to the estimation and prediction of travel time
reveals the need for research to improve existing modeling techniques for urban arterial
networks. The operation of arterial networks is complex and dynamic, and research to
date has not produced reliable estimation and prediction models. However, previous
research has established a foundation on which effective models can be developed. State-
Space Neural Networks (SSNN) have been shown to be useful in capturing the dynamic
nature of arterial traffic conditions. Furthermore, easily collectable traffic data such as
average Speed and flow rate have been implemented effectively to explain variations in
travel time. This research is aimed to progress the results of previous findings in order to
advance the prospect of future ATMS and ATIS applications to combat urban arterial

congestion problems.

Chapter 2

Problem Statement and Research Objective

Congestion in urban traffic networks is a problem that continues to cost travelers time
and impact economic productivity. In the absence of readily available physical space and
ample financial resources, the supply of additional physical highway capacity in urban
areas is often not possible. With the advancement of information and communication
technologies Intelligent Transportation Systems (ITS) such as Advanced Traffic
Management Systems (ATMS) and Advanced Traveler Information Systems (ATIS) are
being implemented successfully in urban areas to improve the use of existing physical
capacity and combat congestion.

Travel time is a key variable that describes the state of a transportation system and is
easily understood by the users of the system. For this reason, travel time estimation and
prediction is an essential component of ATMS and ATIS applications. These ITS
solutions are particularly prevalent in urban freeway systems to aid administrators in
solving traffic problems and inform users of upcoming traffic conditions. However,
ATMS and ATIS deployment is not established in the urban arterial setting. In arterial
networks, traffic signals interrupt the flow of traffic, creating variability in traffic
conditions that is not readily understood. Advancements in research to accurately
estimate and predict travel time in Signalized networks is necessary to enable the

successfirl deployment of ATMS and ATIS application in urban arterial networks.

10

The Objective of the work presented in this thesis is to develop models that enable
accurate estimation and short-term prediction of travel time in urban arterial networks for
use in future ATMS and ATIS applications. Furthermore, these models are developed
using input variables that are readily available in the field through existing surveillance
infrastructure. Loop detectors and cameras currently deployed in urban arterial networks
are capable of sensing variables such as flow rate and queue length that can be used to
estimate and predict travel time.

Previous work has been done in this area, producing some promising results. Singh
(2006) developed travel time estimation and short-term prediction models for right turn,
left turn, and through movements in a Simulated urban arterial network. This work did
include assumptions that may impact the robust nature of the models developed, though.
In particular, turning movement percentages were constant between intersections and
over time in the simulation network used to generate data for the development of travel
time estimation and prediction models. In a realistic urban arterial network, turning
movements will vary spatially and temporally based on fluctuations in demand and
proximity to particular origins and destinations. This thesis incorporates variable turning

movements to determine the impact on reliable estimation and prediction of travel time.

2.1 Background and Definitions

The framework presented in this thesis mirrors the framework developed by Singh
(2006), with the exception of turning movement simulation. This is done so that the

impacts of variable turning movements can be understood in comparison with the results

11

previously produced assuming constant turning percentages. The purpose is not to refute
the work completed, but instead to improve the reliability of the modeling technique
employed, in order to advance the reliability of such models for ATMS and ATIS
applications in urban arterial networks.

The framework developed by Singh (2006) and employed in this thesis is presented in
this chapter. The models developed for travel time estimation and prediction are
applicable to urban arterial networks, and so the elements of such networks are defined in

this section. Additionally, the concept of average travel time is presented.

Definition 1: A ‘Signalized Intersection’ is the point where two arterial links meet. A
Signalized intersection is defined by the use of traffic control Signals to manage the flow

of conflicting traffic movements from one arterial link to another.

Definition 2: An ‘Arterial Link’ is a section of road between two Signalized intersections.
For the purpose of travel time estimation, the length of an arterial link begins at the end
of one Signalized intersection and extends through and includes the next downstream

intersection. Arterial links may carry traffic flow in two directions.

Definition 3: An ‘Arterial’ is a sequential group of arterial links that are traversed by

making only a through movement from the entry point to the terminus of the arterial

within the boundaries of the arterial network.

12

Definition 4: An ‘Arterial Route’ consists of a set of arterial links traversed by a vehicle
making any series of left turn, right turn, and through movements from origin to

destination, within the boundaries of the network.

Definition 5: An ‘Arterial Network’ is comprised Of the arterial links and Signalized
intersections within a particular area of study. An urban arterial network typically takes
on a grid formation having both major and minor arterials. ‘Major’ and ‘Minor’ refer to

the relative traffic flows and importance in terms of network operation.

In an arterial network individual vehicles traverse a series of links from origin to
destination. Individual vehicles that travel together along a common arterial link or route
in a common time interval travel in a traffic stream. ATMS and ATIS applications
typically function based on characteristics of the traffic stream, which are defined by
statistical measures, such as the average, of individual vehicle movements. The concept
of average travel time for a traffic stream will be implemented in the model development

strategy for this thesis.

Definition 6: The ‘Average Travel Time’ is the average value of the travel times

experienced by individual vehicles on a particular arterial link during a specific time

interval.

AS defined previously, arterial links are the primary component of any arterial or arterial

route. Therefore, the average travel time for an arterial or arterial route is defined by the

13

average travel time on each subsidiary link. The Objective in developing accurate
estimation and prediction models for average travel time on an arterial link is to provide
the fundamental component for estimation and prediction of average travel time for
higher-level network components, such as arterials and arterial routes. This framework

will provide a more robust tool available for a variety Of ATMS and ATIS applications.

2.2 Average Travel Time Estimation versus Prediction

Average travel time estimation and prediction are both important elements of ATMS and
ATIS applications. Travel time is easily understood by transportation system users, and
has been determined to be a useful measure of performance for administrators. In such
ITS applications, it is useful to know not only current travel times, but also what travel
times will be in the future, so that proactive measures can be taken to avoid congested
conditions.

Both current travel time and travel time in the future are based on the state of the
network, or in other words, the quality of traffic Operations. However, there is an
important difference between estimating current average travel time, and future
prediction. The estimation of travel time is based on known or estimated traffic

parameters, whereas traffic conditions are unknown in the future.

Definition 7: The ‘Estimated Average Travel Time’ is the travel time experienced by

vehicles during a current or past time interval. Average travel time estimation is based on

known or estimated traffic parameters describing the state of traffic for any arterial link.

14

Definition 8: The ‘Predicted Average Travel Time’ is the travel time that iS expected to
be experienced by vehicles during a future time interval. Average travel time prediction
is based on preceding traffic conditions and temporal fluctuations in these conditions, as

future traffic parameters are unknown.

The objective of this thesis is to develop accurate and reliable travel time estimation and
short-terrn prediction models using input variables that are easily collected using existing
infrastructure. The variables that describe and influence travel time in arterial networks
include flow rate, queue length, arterial geometries, and signal timing parameters. Both
travel time estimation and prediction models are developed using these variables,
although the methods differ from one another.

During current or past time intervals, variables describing the state of traffic operations
can be measured and calculated. A static relationship can then be determined between
these input variables and the output variable, average travel time. In other words, models
can be developed to calculate travel time based upon the values of the input variables.
This relationship is not dependent on changes in the value of each input variable over
time, rather on average values during a specified time interval. The accuracy of such
models depends on the strength of correlations between each input variable and actual
average travel time.

Travel time prediction for a future time period is a much more complex problem, as the
values of potential variables influencing travel time are not yet known. For this reason,
prediction of travel time is based on understanding temporal changes in traffic conditions

during previous time intervals. A relationship is then developed between changes in the

15

input variables over time and the resulting changes in Observed travel times. In this
manner, travel time prediction can be performed using data that is easily obtained using
existing infrastructure. The accuracy of prediction models depends on the ability to
understand how temporal fluctuations in past traffic conditions may influence firture
travel times. Methodologies for both estimation and short-term prediction of average

travel time in urban arterial networks are provided in this thesis.

2.3 Formulation for Average Travel Time Estimation and Short-Term

Prediction Models

Let an arterial route on a network consisting of n number of arterial links be represented
as X), X 2, X 3,...Xn. Let tp' be the current time interval of departure for a vehicle which
starts from an entry point at the beginning of link X I and ends its trip after clearing the
intersection that ends link X". This time interval tp' starts from a discrete time to. and ends
at time t), Where tp' = [t0', if]. The time interval tp' is impacted by prevailing traffic
conditions on the arterial network. The traffic parameters that describe traffic conditions
during time tp' can be obtained in the field through traffic detection devices, or can be
estimated through other already obtained traffic variables.

The assumption is made that traffic conditions are constant within each time interval if
the length of such interval is very small. In many traffic engineering studies, particularly
regarding short-term travel time forecasting, it is assumed that traffic conditions are
constant within Short time intervals of 1 to 15 minutes. A time interval of 1 minute

provides high temporal resolution to the data and may be useful if cycle-by-cycle analysis

16

is required. On the other hand, an interval of 15 minutes offers low temporal resolution
for forecasting, as changes in unstable traffic conditions during congestion and over-
saturated conditions may be overlooked or aggregated. Sisiopiku et al. (1994) in a travel
time estimation study assert that effects from cycle failures, short-term events, congestion
built up downstream of the subject link, and so forth cannot be detected using lS-minute
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 is gained by increasing the temporal resolution from 5 minute to 1 minute
intervals. Based on the results of these studies, the length of each time interval for this
thesis is determined to be 5 minutes in duration. This interval is called the aggregation
interval, denoted as (D, where (I5 = t,'- to'.

Let the known traffic conditions on a link X 1 in the current time interval tp'be V(X1, tp).
The vector V(X1, t,,), which contains the variables describing traffic conditions on a link
X1 during time interval tp', is called the ‘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 interval. This generic term for traffic conditions on a
link is very important in the context of travel time estimation and prediction because the
set of input variables such as average flow rate, average Speed, queue length, etc., impact
the state of the link, and therefore travel time. Following this concept, the approach is to
model the function that exists between the link state and travel time on that link.

The average travel time to traverse a link X 1 during time interval tp’ is denoted as
TT(X,, t,,). The estimation of average travel time is defined using the terminology

explained in the previous paragraphs.

17

Definition 9 - The average travel time on arterial link X1 is estimated by approximating
the underlying function between average travel time TT (X 1, tp') and the state V(X,, tp) of

link X1 during time interval tp '.

The formulation of average travel time prediction is derived from the definition of
average travel time estimation. Assume that the average travel time is to be predicted for
a future time interval that is just starting when the current time interval tp' ends at time
instant I; '. A short interval of time in the future, denoted as A, for which the average
travel time is to be predicted is defined as the ‘prediction horizon.’ The future time
interval is defined as rpm; = [t0, (1], where the time interval begins at to and ends at t).
This implies that the to value of future time-period is the same as t, ', the ending time
instant Of the current time interval. Therefore, to = t1 '. Moreover, the length of this future
time interval tp+1A is equal to the addition of the current time interval. to. and the
prediction horizon A, where tp+1A = t,'+ A. The future time interval tprm is a single
multiple of the prediction horizon A, and is called a one-step future time interval. The
predicted value of average travel time for this one-step future time interval will be called
the one-step predicted average travel time.

The prediction horizon A can be assumed to be equal to the aggregation interval (D as
previously defined. So, A = (15 = 5 minutes. This assumption is made so that the
temporal scale at which the state of a link and travel time is analyzed is equal for all time
intervals whether past, current, or future. The predicted average travel time to traverse a
link X1 during the one-step future time-period tp+m is denoted as TT (X 1, tptm). The

problem of one-step average travel time prediction is defined using this terminology.

18

Definition 10 - The one-step average travel time on arterial link X; is predicted by
approximating the underlying function between the one-step predicted average travel

time 77(X1, tp+1A), state V(X ;, t,,), and the estimated average travel time TT(X1, tp').

It is important to highlight that for the above definition the state of a link and the
estimated average travel time for that 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 transpire.

The average travel time prediction for any n’h step in the future can be developed on the
same basis, where n 21. Assume that the time-period in n steps is represented as tp+nA,

where the nth

step predicted average travel time is denoted as 7'1" (X 1_ tptnA). The
determining factors for average travel time prediction for the n’h step are the state of the
link, V(X ,, tp), and estimated travel time, 77’ (X 1, 1p). Additionally, the predicted average

travel times through the n-l step are also taken as input variables for nth average travel

time prediction.

Definition 11 - The 11th step average travel time is predicted by approximating the
underlying function between the nth step predicted average travel time TT (X 1, tp+nA) during
time interval rpm, state V(X1, tp), the estimated average travel time TT (X 1, t,,), and the

predicted average travel time through the n-l step.

Average travel time prediction for the n’h step in the future is referred to as multiple-step

prediction. Multiple-step average travel time prediction involves incremental prediction

19

from the n = 1 step tO the n’h step as required for the system. Each time-step prediction
will result in some deviation from the actual average travel time. As a result, it is
expected that the accuracy of predictions may decrease as the number of increments
increases.

The formulation defined is presented in a time-space diagram in Figure 2.1. The Space
domain is represented as a single link X1 and the time scale includes time intervals
ranging from the current time interval tp'to the nth future time interval tp+1A , twig/3, tp+3A_...
for which a prediction is to be made. The current time period tp' is labeled as the ‘current
horizon’ and each future time period is termed as the ‘prediction horizon.’ Consistent
with the terminology presented in this section, the current horizon begins at time instant
to. and ends at time-instant t1, forming time period tp'. The duration of time interval tp' is
equal to the aggregation interval 45. The state of link X 1 representing prevailing traffic
conditions is shown as V(X ,, t,,) in the current horizon. The first-step prediction horizon
starts from time-instant to (equal to t1) and ends at t,, forming first step future time
interval t,,+/A. Similarly, second-step and third-step prediction horizons are as Shown.
Each future time—period is of length A, where A = (D.

The work presented in this thesis is focused on average travel time prediction for the
first-step horizon, Where n = 1. While prediction for the n > 1 horizon is possible, there is
an inherent level of error that can be expected to propagate through multiple prediction
increments. Models useful in predicting average travel time for the n = 1 step must be
improved first in order to reduce the probability of error propagation for multiple time

intervals.

20

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Figure 2.1 — Time-Space Domain for the Formulation Of Average Travel Time

Estimation and Short-term Prediction.

The intent of this work is to advance previous research regarding short-term travel time
prediction for urban arterial networks and improve existing Single-step prediction models.
Further research is needed to progress these models to be reliable in multiple-step

prediction applications.

21

2.4 Factors Impacting Average Travel Time Estimation and Short-Term

Prediction

Travel time on arterial links can be impacted by a variety of influences. Traffic demand,
roadway capacity, traffic signal operation, weather, incidents, and driver behavior can all
have an effect on travel time. However, only a portion of these variable factors can be
measured and used to estimate and predict travel time. For the purpose of this thesis,
only variables that are easily measured using existing surveillance infrastructure such as
loop detectors and cameras are considered.

The state of an arterial link is determined by the balance between traffic demand and. the
available supply, or capacity, of the link. Throughout a typical day, traffic demand
fluctuates depending on the needs of the system users. In urban arterial networks the
capacity is ofien insufficient to meet peak demands. AS a result, vehicle spacing and
average vehicle speed decreases, causing increases in average travel time. The input
variables used to estimate and predict travel time should include measures of both traffic
demand and link capacity for this reason.

To capture the variability in traffic demand average flow rate can be measured for a given
time interval. According to the Transportation Research Board Highway Capacity
Manual (2000), right and left-tum movements can influence flow rate as turning vehicles
must reduce speed and may be restricted by conflicting movements. For this reason,
average flow rate is measured for right-tum, left-turn, and through movements

independently. Additionally, right-turn-on-red permission at Signalized intersections can

22

alter arterial operation. The flow rate for the crossing approach should also be considered
to better model right-tum travel time.

Traffic Signal operation has a Significant influence on the available capacity and resultant
state of an arterial link. Conflicting traffic demands at each intersection must be serviced
through the allocation of green time to each approach. Additionally, left-tuming vehicles
must often be accommodated by dedicating a specific signal phase for left-tums. Signal
timing parameters such as duration of green and red intervals are therefore important
variables that control traffic flow and thus influence travel time.

In urban arterial networks, link lengths are relatively Short, and an interaction between
adjacent intersections exists. The Signal phasing and timing schemes must be
coordinated to effectively progress vehicles from one intersection through the next. This
coordination is determined by the link length, vehicle speed, and the time difference in
green intervals, or offset, between adjacent intersections. In congested conditions the
relative operation of adjacent intersections can be even more Vital. Queues that form at
intersection approaches can influence progression and travel time. Furthermore, queue
length can increase in excess of the link length, and block upstream intersections.

Travel time estimation and prediction models are developed based on the above
mentioned influences and the ability to collect data representing such variables using
existing surveillance. In order to produce efficient models that can be implemented into a
variety of ATMS and ATIS applications, models are developed for right-turn, left-tum,
and through movements separately. This will allow not only for travel time estimation
for arterial links, but also for arterials and arterial routes. The variables considered in the

modeling procedure are defined in Table 2.1.

23

Table 2.1 — Variables Considered for Travel Time Estimation and Short-Term Prediction.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VT Average flow rate for through movement
Travel VR Average flow rate for right-tum movement
Demand VL Average flow rate for left-tum movement
VX Average flow rate for approach conflicting right-tum movement
Sr Average speed for through vehicles
Speed SR Average speed for right-fuming vehicles
SL Average speed for left-turning vehicles
GT Length of green interval for through and right-tum phases
Signal GL Length of green interval for left-tum phase
Control Off Offset between adjacent Signalized intersections
R Length of red interval for approach
Queue QT Maximum queue length for through and right-turn lanes
Length QL Maximum queue length for dedicated left-turn lanes
Geometric LL Link length

 

2.5 Modeling Approach for Travel Time Estimation and Short-Term

Prediction

AS acknowledged in supporting literature, there is a need for reliable a reliable modeling

procedure for the estimation and prediction of travel time in urban arterial networks. In

this chapter the research objectives to address this need have been identified, the elements

of an arterial network have been defined, the mathematics supporting travel time

estimation and prediction have been formulated, and candidate variables influencing

travel time have been identified. In the following chapters of this thesis the experimental

procedure and the modeling approach are detailed, along with the conclusions resulting

from this research.

24

 

The overall modeling approach begins with the problem statement and identification of
the need for separate models for right-tum, left-tum, and through movements in order to
be robust in ATMS and ATIS applications. In order to examine the relationship between
variables influencing the state of an arterial link and travel time, data are needed for the
candidate input variables and average travel time. These data are generated from a
sample network developed using VISSIM microscopic simulation.

Average vehicle Speed has been identified as an important variable linked to average
travel time. Intuitively, this is straightforward, as speed is a product of distance and time.
Measuring average vehicle speed continuously along an arterial link is difficult however,
as speeds fluctuate along the length of the link, and existing data surveillance devices are
mainly capable of estimating vehicle speed at discrete points along the link. A
multivariate linear regression model is utilized to more reliably estimate average vehicle
Speed along the length of a link given other more accurately obtainable data.

From the data generated using VISSIM and the average link speeds estimated from linear
regression, State-Space Neural Networks (SSNN) models are introduced. Singh (2006)
developed a series of SSNN models for travel time estimation and prediction in urban
arterial networks. The reliability of these models is examined in this thesis under the
influence of variable turning movements. The overall framework presented in this

section is represented by a flow chart in Figure 2.2.

25

 

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

 

 

 

1
Need to develop separate models for
through, left and right-turn movements on
arterial link

 

 

 

 

 

l

Arterial network development and data

collection using VISSIM, incorporating
variable turning movements

 

 

 

 

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

   
  
  
     

   
  
  

Stale-Space Neural Network models for
short-term travel time prediction for through.
left-turn and right-tum movements on
arterial link

State-Space Neural Network models for
short-term travel time prediction for through,
elt-tum and right-tum movements on arteria
route

 

Evaluation of the influence of variable turning
movements on travel time estimation and
short-term prediction

 

 

 

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

Term Prediction.

26

Chapter 3

Experimental Set-up

There are a wide variety of factors that influence travel time on urban arterial networks,
including traffic demand, signal control operations, speed, geometrics, weather, and
driver behavior. The objective of this thesis is to develop travel time estimation and
prediction models based on variables describing these factors that can be easily measured
in the field using existing surveillance devices such as loop detectors and cameras.
Candidate variables have been identified to meet this objective and are defined in Section
2.4 of this thesis.

In order to explore the relationship between travel time and the defined set of input
variables a dataset is needed containing data for each candidate variable and
corresponding travel times. The intent is to develop robust models from this dataset that
can be applied to future Advanced Traffic Management Systems (ATMS) and Advanced
Traveler Information Systems (ATIS). Therefore, it is important that the dataset contain
a large number of cases describing operations on a number of arterial links.

The use of microscopic simulation enables the collection of large amounts of data
describing operations on a number of arterial links without the need for extensive field
data collection procedures. Additionally, microscopic simulation allows for the variation
of network parameters to study the influence of such variations on the robust nature of

the models developed.

27

Previous work has been done to develop travel time estimation and prediction models
using data from microscopic simulation. The work presented in this thesis is based on the
simulation network developed by Singh (2006) and incorporates variable turning
movement percentages into the experimental procedure. The influence of variable
turning movements on the effectiveness of the models developed is examined. The
usefulness of microscopic simulation to fulfill these objectives and the experimental

procedure implemented is described in this chapter.

3.1 Overview of VISSIM Microscopic Simulation

The data used in the development of travel time estimation and prediction models are
generated using VISSIM microscopic simulation software, Version 4.10. In the VISSIM
environment network parameters are defined by the user, including link lengths, number
of lanes, desired vehicle speeds, signal control operations, traffic compositions, flow
rates, and turning movement percentages. Additionally, a variety of methods can be
implemented to collect the necessary data from each simulation run.

The use of VISSIM allows wide latitude in the creation of each system element to meet
the objectives of the network design. The freedom allowed in network design and data
collection makes the use of VISSIM desirable for this experiment. A network is designed
to replicate a realistic urban arterial environment, containing a series of arterial links and
Signalized intersections with spatially and temporally variable flow rates and turning
movement percentages. This set-up replicates the conditions experienced in many urban

arterial networks, where traffic demands change across the network throughout the day.

28

3.2 Static Network Elements

The urban arterial network created in VISSIM for use in this experiment consists of 10
arterials, each consisting of 4 arterial links. The arterials are aligned in a grid pattern as
shown in Figure 3.1. The 5 arterials running in the east-west direction are defined as the
major arterials, and the 5 arterials running in the north—south direction are defined as the
minor arterials, where ‘major’ and ‘minor’ designate the level of traffic demand and
relative importance with respect to signal progression and network operations. Each
arterial has 2 lanes in each direction.

The point where two arterials meet represents a Signalized intersection and is shown in
Figure 3.1 as a solid circle. Each intersection is numbered 1 though 25, where the
number of the outermost intersections designate the label for each arterial. The major
arterials are referred to as arterials 1-5, 6-10, 11-15, 16-20, and 21-25. The minor
arterials are referred to as arterials 1-21, 2-22, 3-23, 2-24, and 5-25. Every approach at
each Signalized intersection has a dedicated left-turn lane with a length of 150 feet, a
through lane, and a shared through / right-turn lane.

In actual arterial networks, the length of an arterial link can be an important factor
influencing travel time. In uncongested conditions link length influences travel time
simply based on the free-flow speed of traffic on the link. In congested conditions
though, the spacing between Signalized intersections can impact traffic progression
through green intervals and heighten the significance of queue formation where queue

spillback can cause blockage. For this reason, link lengths are varied between 1,500 feet

29

and 6,000 feet in the simulation network. The link lengths for each parallel arterial are

shown in Figure 3.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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O
O
('0
11
= > 11.2) <19 41:) =<1?) . .
«.1 It It I I A
O
O
33
15> 117.) 2(1) 119) 420)
:l l "A '11 'll ‘1i'_"' ’
O
O
O
V
.__.i _ us L- L i -
21
_ _) 123;) L29) £353) L23)
2 It I I I ll—’
0
O
N
I v it

Figure 3.1 - Urban Arterial Network Designed in VISSIM for Travel Time Estimation

and Prediction.

3O

3.3 Stochastic Network Elements

In microscopic simulation, vehicle movements are controlled and tracked individually as
determined by the vehicle following and lane changing logic employed by the simulation
tool. The VISSIM program uses a psycho-physical driver behavior model developed by
Wiedemann (1974) where vehicle acceleration and deceleration is based on individual
perception thresholds to vehicles with different speeds. Stochastic distributions are used
to replicate variability of individual driver characteristics. Each individual vehicle in the
network is assigned specific speed, acceleration, and gap sensitivity based on the
distributions determined for the system. Traffic flow is simulated through the movement
of these vehicles following the Wiedemann car-following model and the defined
distributions (PTV 2005).

For the simulation network developed the default distributions for vehicle following, lane
changing, gap acceptance, acceleration, and deceleration are implemented. As the
purpose of the simulation is to generate data, no actual field data exist whereby to further
calibrate the default distributions. However, the default distributions have been
calibrated over time through multiple field studies performed at the Technical University
of Karlsruhe (PTV 2005), and can be used with confidence for the purpose of this thesis.
Although the majority of distributions implemented in the simulation are based on
program default values, desired speed distributions must be defined for vehicles entering
and traversing the network. Vehicles entering the network and traversing most links are
designated a desired speed following a normal distribution with a minimum of 42.3 miles

per hour (mph) and a maximum of 48.5 mph. Vehicles entering any link of length 1,500

31

feet are slowed to a desired speed following a normal distribution with a minimum of
28.0 mph and a maximum of 32.0 mph. This is done to remain consistent with the speed
distribution defined in the network developed by Singh (2006) with an entry speed limit
of 45 mph and a speed limit of 30 mph on links 1,500 feet in length. Although these
distributions do not exactly match the speed limit designations in Singh’s simulation, the
data generated are representative of similar speed distributions for the purpose of
comparison, and the variability in observed vehicle speed can be used as a continuous

input variable to estimate and predict travel time.

3.4 Time-Dependent Network Elements

In a typical urban arterial network, travel demand changes across the network over time.
The change in demand relative to the available capacity causes a fluctuation between
uncongested and congested conditions on each arterial link. In order to reliably estimate
and predict travel time, the models developed must be able to function in all network
states. Therefore, it is important that the flow rate on each arterial vary over the time
period of the simulation.

In the VISSIM environment, flow rate is designated at the entry point of an arterial in the
simulation network. Average flow rate can be designated for particular time intervals
within the simulation. For this experiment the flow rate is varied over time in 5-minute
intervals between each major arterial, and a separate flow rate distribution is assigned to
the set of minor arterials. The flow rate is the same for both directions at the entry points

of each arterial. In general, these flow rate distributions result in a movement from an

32

uncongested, to congested, and back to uncongested state of the network over the course
of 100 minutes. Congested conditions are observed during microscopic simulation runs
where the traffic demand exceeds the supply offered by the number of lanes and signal
timing operation on each arterial. This results in vehicles that can not progress through
an intersection following multiple signal cycles, and queue spillback between
intersections and from left-tum bays into adjacent through lanes. The time-dependent

flow rate distributions for each arterial are shown in Figure 3.2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2400 —o—— Arterial 1-5
/ . — - Entry
2200 Volume
2000
- - I- - Arterial 6-
E 1800 10 Entry
3 1600 Volume
3 1400 — 4» —Arterial 11-
15 Entry
3 1200 Volume
“- 1000
3: 800 . . —-*—Arterial 11-
2 7 , 15 Entry
3 600 / \\ Volume
400 ,7): \ - -I- -Arteria| 16-
200 20 Entry
Volume
0 T f f r ' 7 ‘l f
0° 0° 0° 0 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° 0° manor I
Time-Scale (seconds) Volume

Figure 3.2 - Average Flow Rate Distributions at Entry Points of Arterials.

In addition to changes in flow rate over time across an arterial network, turning
movement percentages can be expected to vary as well. Depending on the time of day

and the location of particular origins and destinations within an urban arterial network,

33

the percentage of right-tum, left-tum, and through movements at intersection approaches
will differ spatially and temporally. The impact of variable turning movements, which
can be expected in a realistic urban arterial network, on the effectiveness of travel time
estimation and prediction models is presented in this thesis. Therefore, such variations
must be incorporated into the simulation network used to generate modeling data.

The simulation network is a generic representation of a typical urban arterial network. In
this case, particular origins, destinations, and travel patterns within the network are not
defined. In order to incorporate variable turning movements into the network, 4 time-
varying profiles, each, for left-turn and right-tum percentages are defined. These right-
turn and left-turn profiles are independently and randomly allocated to each intersection
approach within the network. In this manner, both spatial and temporal variations in
turning percentages are accounted for, while not causing the data generated to reflect a
specific arrangement of travel patterns. The left-tum and right—tum profiles are shown in

Figure 3.3 and Figure 3.4, respectively.

34

 

 

 

 

 

Percentage (%)

 

 

 

 

 

 

 

8
6 I r T— r I 1 I I I I I I I I r I I I r
0000000000000 000000

Time (seconds)

Figure 3.3 — Left-Tum Profiles for Signalized Intersection Approaches.

 

 

 

 

 

 

 

 

18
16 3.;
L -"‘- /'\/7"\ ‘-
14 f “ v '7' ”is.
R Oral, .q / a' \ ‘r’ 'n
g 12 ,: ..’ \._ II. fi'
.' \ -
é . 1.. , :A
E 10 - . ‘ ‘ '
8 A- a K 1' . ,r—I
I. ' g ..
a 8 -.
fl L..-«A-/ I t I
6 «H L hr f
V W ‘1
4
2 7 I I l f T f j I I I I I I I I l

 

0 0 0 0 0 0 0

0000000000000
0000000000000
‘L “0°ébrb°°&lrb?>b?’¢$‘0°‘6\0°

0‘" ‘1, 0
Time (seconds)

Figure 3.4 —Right-Turn Profiles for Signalized Intersection Approaches.

35

- -o- - Profile
L1

- - I- - Profile
L2

—t—- Profile

L3

—I— Profile

L4

—0-— Profile

R1

- i - Profile
R2

- - i- - Profile
R3

—N— Profile
R4

3.5 Signalized Intersection Control Design Using Synchro

For each intersection in the VISSIM simulation network a traffic signal timing plan must
be developed to facilitate the movement of vehicles from one arterial link to another. The
timing plan for each intersection designates the duration of each cycle and the allocation
of green time to conflicting movements. In order to develop an appropriate signal timing
plan for each intersection Synchro, Version 6, is used. Synchro is macroscopic
simulation software that is used to optimize cycle lengths, green splits, signal phasing,
and offsets for a traffic network.

The main input variables needed to perform signal optimization with Synchro are the
flow rate for right-tum, left-turn, and through movements, and the lane configuration of
each intersection approach. Additionally, the desired level and direction of progression
through a series of intersections can be defined. In this experiment, the east-west arterials
are considered the major arterials and carry heavier traffic volumes than the minor north-
south arterials, so offsets are determined in terms of east-west progression.

The signal timing plans implemented in this experiment are fixed-time in nature. In other
words, changing traffic demands do not alter the signal timing for an intersection. Under
this condition, signal timing is designed based on the highest traffic volume typically
experienced during a peak one-hour time period. This ‘peak-hour’ volume is determined
for the VISSIM network based on the flow rate and turning movement profiles described
in this chapter. The signal phasing, cycle length, and green intervals are optimized with

Synchro and input into the VISSIM network.

36

The optimal signal timing plan for the Signalized intersections modeled in this experiment
involves a 4-phase operation, based on the geometric design and traffic demand for each
intersection approach. Relatively large flow rates throughout the network during the
peak-hour result in insufficient gaps to allow left-tuming vehicles to progress without a
protected left-turn phase in each direction. The typical phasing plan developed for each

intersection is shown in Figure 3.5.

 

\ ll l 112 _ __
sac-r» if a: -1 m 1 _ 3;;

05 115

-..‘. 21h. r ‘u‘ #{j‘
3.. l is . .

Figure 3.5 — Typical Signal Timing Plan for Signalized Intersections.

 

 

 

 

 

 

3.6 Data Collection from VISSIM Simulation

The data generated through the VISSIM microsc0pic simulation must be broken down
into specific time intervals in order to provide both ample cases for model development,
and adequate resolution in terms of changing traffic conditions. As stated in Section 2.3
of this thesis, Mark and Sadek (2004) recommend a temporal resolution of 5 minutes for
travel time forecasting. The VISSIM data collection tools report a wide variety of data
types that can be averaged for specific 5-minute intervals over the duration of the

simulation.

37

During each S-minute interval, data representing queue lengths, flow rates, and travel
times are collected for each direction on each arterial link. The simulation is run over the
course of 100 minutes, resulting in 20 time intervals. It is important to note that the first
time interval from to = 0 minutes to t, = 5 minutes is discarded, as at the start of the
simulation the network contains no vehicles. Discarding this first interval allows vehicles
to reach and fill the interior of the network and thus meaningful data is collected during
the subsequent 19 time intervals.

Although VISSIM allows a wide variety of data collection options, the coding procedure
to define the necessary parameters is time consuming. For this reason, three major
arterials and one minor arterial are identified for this study. Data is collected for arterials
6-10, 16-20, 21-25, and 3-23. These arterials are each comprised of 4 links carrying
traffic in two directions, for a total of 32 data collection points. Over the course of 19
time intervals, a total of 608 cases are generated containing data relative to queue length,
flow rate, and travel time.

In addition to the data generated through VISSIM simulation, data corresponding to
signal timings for each arterial link approach are determined for each of the 608 cases.
Signal control data are taken from the timings input to VISSIM from the Synchro
optimization for the length of green interval, red interval, and intersection offset.
Additionally, link lengths are recorded for each arterial link.

The final variables defined for each data case represent average speeds for right-tum, left-
turn, and through vehicle movements. In VISSIM, individual vehicle movements are
tracked such that average travel time can be measured specifically for right-tum, left-tum,

and through progressions from the start of an arterial link through the end of the next

38

downstream intersection. However, average speed can not be differentiated by
movement for vehicles traversing a link. So, average observed travel times are used to
compute average vehicle speed based on link length for each possible movement along an

arterial link. Average speed for each case is computed in this manner.

3.7 VISSIM Simulation Runs and Model Validation

For the purpose of developing travel time estimation and prediction models, the
generation of 608 data cases is determined to be sufficient to explore the relationship
between travel time and the variables that impact the state of an arterial link. Each case
represents spatial and temporal changes in traffic operations typical of many urban
arterial networks. In total, 14 candidate variables describing link operations are identified
for use in the modeling procedure.

Despite the extensive and variable dataset that is generated using VISSIM, it is important
to realize that in microscopic simulation, the data generated are dependent on the random
seed chosen. The random seed controls stochastic variations in the simulation such as
vehicle generation patterns, and can impact the value of the simulation outcomes. The
models presented in this thesis are developed based on the data generated in the initial
simulation run. This dataset is referred to as the ‘training set.’ Each model is then
validated using data from the second simulation run with an different independent

random seed. The second dataset is referred to as the ‘testing set.’

39

Chapter 4

Multivariate Linear Regression for Average Link Speed Estimation

The average vehicle speed on an arterial link is an important variable linked to travel time
estimation and short-term prediction. Existing surveillance devices such as loop
detectors and cameras are mainly capable of estimating vehicle speeds at discrete
locations on an arterial link. Furthermore, the location of surveillance devices can
heavily influence the accuracy of speed estimation. For example, a camera located in
proximity to an intersection approach may detect speeds that are lower than would be
detected at the midpoint of the link, as vehicles slow and are often stopped with zero
velocity due to traffic signal control. In order to reliably determine travel time along an
arterial link, average speed must be estimated along the entire length of the link. Using
other more reliable data collected using existing surveillance devices, such as queue
length and flow rate, along with signal control parameters and link geometric
information, multivariate linear regression models are presented in this chapter to more

accurately estimate average speeds for arterial links.

4.1 Multivariate Linear Regression Methodology

Multivariate linear regression is a powerful statistical tool used to model the relationship
between a dependent (response) variable and a set of independent (explanatory) variables.

A linear regression equation is developed to mathematically explain this relationship.

40

For example, given the dependent variable Y and independent variables X], X 2, X 3... X",

the linear regression equation is expressed as:

Y =30 +161X1+fl2X2+fl3X3+mflan +5

Each independent variable is assigned a parameter [3,, that is known as the partial
regression coefficient. The partial coefficient designates the statistical influence of each
corresponding independent variable on the dependent variable while the values of the
other independent variables in the model are controlled. The parameter [1’0 is known as
the regression constant and the error term is labeled as (5 (Kutner et al. 2004). Assuming

the expected value of the error coefficient (5 = 0, the regression equation is simplified as:

Y =fl0 +,31X1 +,8;X2+,B3X3+... ,Ban

The purpose of multivariate linear regression in this thesis is to accurately estimate the
average value of speed on an arterial link for inclusion in State-Space Neural Network
(SSNN) models to estimate and predict travel time. SSNN models are developed to
estimate and predict travel time for left-tum, right-tum, and through vehicles. Therefore,
three linear regression models are developed to estimate average link speed for each
possible movement. The average speed value is the dependent variable and is labeled as
SL, SR, and ST for average left-tum, right-tum, and through speeds, respectively.

The candidate independent variables are variables that can be easily measured in the field

using existing surveillance infrastructure. These variables include measures of travel

41

demand, traffic signal control, queue length, and link geometries, and are shown in

Table 4.1.

Table 4.1 — Candidate Independent Variables for Multivariate Linear Regression.

 

VT Average flow rate for through movement

 

Travel VR Average flow rate for right-tum movement

 

Demand V L Average flow rate for left-tum movement

 

VX Average flow rate for approach conflicting right-turn movement

 

Gr Length of green interval for through and right-tum phases

 

Signal GL Length of green interval for left-tum phase
Control Off Offset between adjacent Signalized intersections

 

 

R Length of red interval for approach

 

Queue QT Maximum queue length for through and right-tum lanes
Length QL Maximum queue length for dedicated lefi-turn lanes
Geometric LL Link length

 

 

 

 

 

 

 

In order to determine the appropriate independent variables to include in each average
speed estimation linear regression model, SPSS for Windows, Version 15.0 is utilized.
SPSS is a software tool used to analyze a wide variety of statistical measures and develop
statistical models for a given dataset. The data generated using VISSIM microscopic
simulation, including 608 individual cases as described in Chapter 3, are entered into a
spreadsheet within the SPSS program. Each case contains data relative to the dependent
and independent variables defined in this section. Statistical tests are performed on this
dataset to develop regression models to estimate average link speed.

In general, the independent variables in a multivariate linear regression model should be

independent of one another (Kutner et al. 2004). In other words, a change in one

42

independent variable does not correlate with changes in another independent variable.
Following this logic, each independent variable in the model will explain a different
portion of change in the dependent variable. The change relationship between variables
is examined through correlation statistics. Using SPSS, significant correlations between
candidate independent variables are identified. In the following modeling steps
independent variables with strong correlations are not included in the same regression
model.

Multivariate regression models can be developed in a variety of ways once the candidate
independent variables are identified. A linear regression model may include all possible
independent variables. However, in order to develop more accurate and efficient models
that require limited data collection, it is desirable to limit each model to include only key
variables that explain a significant change in the dependent variable, so long as such
models are accurate. A step-wise regression procedure is used to identify the most
influential explanatory variables for use in each model, and filter out those variables that
do not explain a significant amount of variance in the response variable.

Step-wise regression modeling is based on the method of least squares. For the sample
linear regression equation defined previously in this section, the partial coefficients fig, ,6 1,
,82. .. fin are determined by minimizing the value of the square of the regression equation,
L, for the given sample of observations, as defined in the following equation:

n
L = 2m =30 MIX” +flzx2+flzx3+urnxn>2

1:1
In the step-wise procedure the partial coefficients are computed in stages based on the

ability of each independent variable to explain additional variance in the response

43

variable. The independent variable with the strongest correlation to the dependent
variable is included first in the model. The square of the correlation coefficient, R2, is
calculated for each step, where the R2 statistic ranges in value from 0.0 to 1.0. The R2
statistic describes how well the set of independent variables explains the variance in the
dependent variable, where R2 = 1.0 represents a perfect linear relationship. In each step a
single independent variable with the next strongest correlation to the dependent variable

is added to the model, until the R2 statistic is not significantly improved.

4.2 Linear Regression Models for Link Speed Estimation

The multivariate linear regression models to estimate average vehicle speed are derived
from the training dataset generated through the initial VISSIM microscopic simulation
run. This dataset consists of 608 cases representing the state of 16 arterial links. Each
model is then validated by applying the model to the testing dataset generated through
another independent simulation run. In Chapter 5 the modeling approach for the use of
SSNN models is discussed, and the need to normalize the dataset is presented. In order to
maintain consistency in the analysis for this thesis, both the training and testing datasets
are normalized prior to linear regression modeling for speed estimation. The
normalization of data does not alter the correlations between variables, and the results
presented in this chapter will be shown in terms of normalized values.

Multiple trials of linear regression modeling are performed to increase the R2 statistic and
to improve the efficiency of the model through careful selection of candidate variables.

Through the modeling procedure it is found that more accurate models are derived when

44

major and minor arterial links are considered separately. A total of six regression models
are developed, including estimation models for left-tum, right-tum, and through
movement speed for both major and minor arterial groups.

Each linear regression model is derived from the training dataset and the performance of
each model is evaluated based on the resulting R2 statistic. A plot of the actual average
speed observed in simulation versus the average speed estimated using the regression
model is generated to provide a visual interpretation of the performance of the model.
Each model is applied to the testing dataset to validate the performance of the model with
an independent source of data. Scatter plots for the training and testing datasets,
including a line of best fit and resulting R2 statistics, are presented for all six linear

regression models developed.

4.3 Average Speed Estimation Models for Major Arterials

Linear regression models are developed to estimate average vehicle speed for left-tum,
right-turn, and through movements for both major and minor arterials. As defined in
Section 2.1, a major arterial is comprised of a series of consecutive arterial links that
serve relatively large traffic flows and are more significant in terms of network operation
than minor arterials. In the arterial network for this thesis, data is collected on the links
comprising major arterials 6-10, 16-20, and 21-25. The data collected from these
arterials in microscopic simulation include 456 cases and are used to derive the models

presented in this section.

45

The linear regression model for through speed estimation on major arterials is:

ST = .352 + .346 LL - .544 QT + .285 GT + .134 Ojf- .070 VL

The plots of actual versus estimated average through speed are shown in Figures 4.1 and
4.2 for the training and testing datasets, respectively. This model performs reasonably
well, with an R2 statistic of .807 for the training set. The performance of the model is
validated in the testing set, with a similar distribution pattern and a R2 statistic of .781.
Although the model is not perfect, speed estimation is an intermediate step in the
modeling procedure for this thesis, and estimated speed will be one of several variables
included in the SSNN models for travel time estimation and prediction.

The linear regression model for left-tum speed estimation on major arterials is:

SL = .497 + .379 LL - .461 QL - .149 VT+ .120 CL - .160 VL

The plots of actual versus estimated average left-tum speed are shown in Figures 4.3 and
4.4 for the training and testing datasets, respectively. This model performs reasonably
well, with an R2 statistic of .793 for the training set. The performance of the model is

validated in the testing set, with a similar distribution pattern and a R2 statistic of .777.

46

0 Normalized
Through
Movement
Major Arterial

— Linear
Regression

Estimated Average Speed (ft/sec)

y = 0.807x + 0.115
R2 = 0.807

 

0.0 0.2 0.4 0.6 0.8 1.0
Actual Average Speed (ft/sec)

Figure 4.1 — Training Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Through Movement on Major Arterials.

0.3
0.2

1.0
A 0.9
§ 08
P. i 0 Normalized
T; 0.7 Through
g. Movement
10 0.6 MajorArterial
o
3’ 0.5
h
1» —Linear
3: 0'4 Regression
1:
3
N
.E
'6'»
u.|

y = 0.880x + 0.117
R2 = 0.781

0.1
0.0

 

0.0 0.2 0.4 0.6 0.8 1.0
Actual Average Speed (ft/sec)

Figure 4.2 — Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Through Movement on Major Arterials.

47

 

’13

o

a 9 Normalized

; Left-Turn

3 Movement

0. Major Arterial

in

o

a:

E

2 -—Linear

< Regression

1:

8

N O

.E

,2 v

~ -- ——.— -»—— M y = 0.793x + 0.099

R2 = 0.793

0.0 0.2 0.4 0.6 0.8 1.0
Actual Average Speed (ft/sec)

Figure 4.3 — Training Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Left-Tum Movement on Major Arterials.

0.3
0.2

0.9
A 0.8
U
8
a 0.7 0 Normalized
; Left-Tum
g 0.6 Movement
13' Major Arterial
,3, 0.5
m
N .
'5 0.4 —Linear
3: Regression
8
‘6
.E
‘u'i
m

3 0‘
0.1 y=0.685x+0.111

R2 = 0.777

O
O

 

0.0
0.0 0.2 0.4 0.6 0.8 1.0

Actual Average Speed (ft/sec)
Figure 4.4 — Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Left-Tum Movement on Major Arterials.

48

The derivation of the speed estimation model for right-turns requires more in-depth
analysis than the modeling procedure described for through and left-turn speed
estimation. Initial modeling based on the 11 independent variables defined in Table 4.1
results in poor R2 statistics and significant derivations between the training and testing
sets. Further exploration of correlation and residual statistics reveals that the candidate
variables do not capture or explain the variability in right-turn speed to an. acceptable
leveL

In order to more accurately model average right-tum speed, the relationship between
right-tum speed and through speed is explored. In the VISSIM simulation model, there
are no exclusive right-tum lanes. Instead, right-tuming vehicles share the rightmost lane
with through moving vehicles. Right—turning vehicles also share the same green interval
with through vehicles at each intersection approach. Therefore, it is likely that right-
tuming vehicles have a similar average speed and flow in a similar manner as through
vehicles on a particular link.

Figure 4.5 is a scatter plot that displays the relationship between average right-tum
speeds and average through movement speeds on major arterial links. There is a linear
pattern observed for the majority of the points. This line is shown in Figure 4.5 to have a
nearly 1:1 slope (this line is not a regression line), signifying that right-turn and through
movement average speeds are nearly the same on major arterial links. There is a
secondary set of points outlined with an oval in Figure 4.5 that diverge from the typical

relationship, but still appear to have a somewhat parallel linear relationship.

49

 

_l
O

 

‘3 0.9
0
g 0.8
'3 0.7
a
0:, 0-6 0Normalized
.. Speed Major
I? 0'5 Arterials
in 0.4
I!
o 0.3
a:
E 0.2
2
< 0.1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Average Through Speed (ft/sec)
Figure 4.5 — Normalized Average Through Speed versus Normalized Average Right-

Tum Speed on Major Arterials.

The linear relationship between right-turn and through speeds on major arterial links
implies that average through speed is a strong predictor of right-turn speed. For this
reason, the estimated average through speed for each case in the training and testing
datasets is calculated based on the previously defined regression equation. The variable
for estimated average through speed, ST, is included in the step-wise regression procedure
for right-tum speed estimation. In order to avoid correlations between independent
variables, the independent variables included in the through speed estimation model are
excluded from consideration in the right-tum speed estimation model.

The resulting linear regression model for right-tum speed estimation on major arterials is:

SR = .281 + .671 ST - .289 VT

50

The plots of actual versus estimated average right-tum speed are shown in Figures 4.6
and 4.7 for the training and testing datasets, respectively. The performance of the model
is less than desirable, with an R2 statistic of .440 for the training set. Additional statistics
are explored, including correlations between various stratifications of the candidate
independent variables and right-tum speed. However, none of these measures reveal any
explanatory relationships or improvements in the model. The performance of the model

is validated in the testing set, with a similar distribution pattern and a R2 statistic of .389.

0.3
0.2

0.9
g 0.8
m 0 Normalized
E 0'7 Right-Tum
u
o 0.6 Movement.
3. Major Arterial
II)
a, 0.5
m
g 0.4 —Linear _
> Regressron
<
u
3
I!
.§
‘15
11.1

0.1 y = 0.439x + 0.299

R2 = 0.440

 

0.0
0.0 0.2 0.4 0.6 0.8 1.0

Actual Average Speed (ft/sec)
Figure 4.6 — Training Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Right-Tum Movement on Major Arterials.

51

 

0.9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

?
0
fl 0 Normalized
:5" Right-Tum
3 Movement
3 Major Arterial
O
8’
E1 —Linear
< Regression
1:
.9
CU
.§
‘15
m
0.1 y = 0.403x + 0.319
R2 = 0.389
0.0 t I T I
0.0 0.2 0.4 0.6 0.8 1.0

Actual Average Speed (ft/sec)

Figure 4.7 — Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Right-Tum Movement on Major Arterials.

4.4 Average Speed Estimation Models for Minor Arterials

Minor arterials carry lesser average traffic flow than major arterials, but are important
nonetheless in terms of travel time estimation and prediction for an arterial network. In
the arterial network for this thesis, data is collected on the links comprising the minor
arterial 3-23. The data collected on this arterial includes 114 cases and are used to derive
the models presented in this section.

The linear regression model for through speed estimation on minor arterials is:

S;- = .615 + .323 LL - .887 QT

52

The plots of actual versus estimated average through speed are shown in Figures 4.8 and
4.9 for the training and testing datasets, respectively. This model performs reasonably
well, with an R2 statistic of .712 for the training set. The performance of the model is

validated in the testing set, with a similar distribution pattern and a R2 statistic of .729.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.9
0.8
9
g 0.7 0 Normalized
~— Through
3 0.6 Movement
é, Minor Arterial
a, 0.5
a:
N .
'0': 0.4 —L1near
i Regression
‘3 0.3
0
‘6 ° °
.5, 0.2
III
I“ O
0.1 y=0.713x+0.165
R2 = 0.712
0.0 1 . . .
0.0 0.2 0.4 0.6 0.8 1.0

Actual Average Speed (ft/sec)

Figure 4.8 — Training Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Through Movement on Minor Arterials.

53

0.9

A 0.8

0

8

a 0.7

E 0.6

8

a, 0.5

8’

a; 0.4

<

'3 0.3

‘6

g 0.2

13

'“ 0.1
0.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0

I

l

0.2 0.4 0.6

Actual Average Speed (ft/sec)

0.8

0 Normalized
Through
Movement
Minor Arterial

— Linear
Regression

y = 0.732x + 0.183
R2 = 0.729

Figure 4.9 — Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Through Movement on Minor Arterials.

The linear regression model for left-tum speed estimation on minor arterials is:

SL = .453 + .387 LL - .843 QL + .268 CL - .218 VL

The plots of actual versus estimated average left-tum speed are shown in Figures 4.10

and 4.11 for the training and testing datasets, respectively.

This model performs

reasonably well, with an R2 statistic of .728 for the training set. The performance of the

model is validated in the testing set, with a similar distribution pattern and a R2 statistic of

.629.

54

0 Normalized
Left-Turn
Movement
Minor
Arterial

— Linear
Regression

Estimated Average Speed (ft/sec)

 

y = 0.692x + 0.150
R2 = 0.728

 

0.0 0.2 0.4 0.6 0.8
Actual Average Speed (ft/sec)

Figure 4.10 —— Training Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Left-Tum Movement on Minor Arterials.

0 Normalized
Left-Turn
Movement
Minor Arterial

— Linear
Regression

 

Estimated Average Speed (ftlsec)

y = 0.558x + 0.193
R2 = 0.629

 

0.0 0.2 0.4 0.6 0.8 1.0
Actual Average Speed (ft/sec)

Figure 4.11 - Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Left-Tum Movement on Minor Arterials.

55

The derivation of the speed estimation model for right-turns on minor arterials follows
the same procedure as described for major arterials. The initial set of candidate variables
results in poor R2 statistics and significant derivations between the training and testing
sets. Further exploration of the data pertaining to minor arterials reveals a strong
correlation between average right-tum speed and average through movement speed. The

resulting linear regression model for right-tum speed estimation on minor arterials is:

SR = .028 + .526 ST + .341 R - .209 VX + .112 Off

This model is similar to the right-tum speed estimation model for major arterials, with the
addition of explanatory variables corresponding to the red interval for the approach and
the conflicting through volume on the crossing major arterial. These variables are likely
correlated to right-tum speed on minor arterials because of the influence on right-turn—on-
red movements. On major arterials heavier flows are experienced, and a heavier volume
of through traffic may block right-tuming vehicles from turning during the red interval
more frequently than on minor arterials where flows are lighter. Thus, on minor arterials
there may be an advanced opportunity to turn right-on-red, and these variables are
included in the right-tum speed estimation model.

The plots of actual versus estimated average right-turn speed are shown in Figures 4.12
and 4.13 for the training and testing datasets, respectively. The performance of the model
is less than desirable, with an R2 statistic of .416 for the training set. Additional statistics
are explored, including correlations between various stratifications of the candidate

independent variables and right-tum speed. However, none of these measures reveal any

56

explanatory relationships or improvements in the model. The performance of the model

is validated in the testing set, with a similar distribution pattern and a R2 statistic of .316.

 

 

 

 

 

 

 

 

 

0.8
g 0.7
"’ 0 Normalized
g 0'6 Right-Turn
'8 Movement
5% 05 Minor Arterial
0
g' 0.4 -
it —Linear
< 0.3 ~ Regression
'0
2 e
G
E 0.2
it"

0‘ ‘“_ "‘ " _"’— ’--'“_ __ " y=0.416x+0.293

R2 = 0.416
0.0 1
0.0 0.2 0.4 0.6 0.8 1.0

Actual Average Speed (ft/sec)

Figure 4.12 — Training Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Right-Tum Movement on Minor Arterials.

Although the predictive capability of right-turn speed estimation models for major and
minor arterials is not as accurate as for the left-turn and through movement, the impact on
travel time estimation is not yet clear. As previously stated, estimated average speed is
only one of the input variables used in travel time estimation and prediction.
Additionally, the overlying goal of this thesis to is to create models that can accurately
predict travel time for a route within an arterial network. Despite the inaccuracy of right-
tum speed estimation models, the SSNN models may be able to overcome this deficiency

in terms of overall travel time estimation and prediction capability.

57

0.8

 

 

 

 

 

 

 

 

 

 

§ 0 Normalized
E Right-Tum
I; Movement
3 Minor
(3- Arterial
8
g —Linear
2' Regression
u
8
N
E 0.2
‘17»
“‘ 0-1 r —' y = 0.319x + 0.355
R?- = 0.316
0.0 . . .
0.0 0.2 0.4 0.6 0.8 1.0

Actual Average Speed (ft/sec)
Figure 4.13 — Testing Plot for Normalized Actual Average Speed versus Normalized

Estimated Average Speed for Right-Tum Movement on Minor Arterials.

The independent variables with the strongest correlation to the dependent variable are the
same between the average speed estimation models for each of the left-tum, right-tum,
and through movements. This supports the validity of the models in that the relationships
between independent variables and through vehicle speed are similar throughout the
network, and that the relationships observed are not arbitrary or descriptive of only
certain links. This is a promising result leading into the development of SSNN models

for the estimation and prediction of travel time.

58

Chapter 5

State-Space Neural Network Modeling Methodology

Traffic conditions in an arterial network are dynamic and constantly changing in time and
space based on a variety of influences. Traffic control devices, traffic demand, weather,
incidents, available capacity, and a number of other variable factors can impact the state
of an arterial link. In 2006, Singh established that Artificial Neural Networks (ANN) are
useful to model the dynamic relationship between traffic conditions and travel time.
Singh developed a series of State-Space Neural Network (SSNN) models, which are a
variation of ANN, to estimate and predict travel time for right-turn, left-tum, and through
movements on arterial links. The results presented by Singh prove that the modeling
framework developed is useful for the purpose of this thesis.

While these models were shown to be successful in the results presented by Singh (2007),
the data used to develop the models did not account for variations in turning movements
at intersection approaches. The operation of a Signalized intersection is crucial in
determining the state of the arterial links that feed the intersection. Turning movement
patterns vary over time and between intersections in a typical arterial network. In order
to develop SSNN models that are truly robust and applicable to the urban arterial
environment, dynamic turning movement patterns must be incorporated into the
modeling procedure. Variable turning movements are incorporated into the microscopic

simulation used in this experiment as described in Chapter 3.

59

" .‘l 'E'" 1.

 

 

The dynamic relationship between travel time and the state of an arterial link is expected
to be influenced due to the incorporation of variable turning movements. The success of
the SSNN modeling framework in estimating and predicting travel time depends on the
ability of the models to adapt to a more realistic situation where turning movement

percentages are not constant.

5.1 State-Space Neural Networks

State-Space Neural Networks (SSNN) are a generic form of ANN that can be used to
model dynamic non-linear systems, such as the state of traffic in an arterial network.
SSNN modeling incorporates local feedback loops that allow the model to “learn” from
variations in the input data over a series of iterations. Elman (1990) developed a general
recurrent neural network framework consisting of an input layer, a hidden layer, a context
layer, and an output layer, as shown in Figure 5.1.

In this framework, the dotted lines represent trainable connections whereby the
appropriate weights between layers are learned. The solid line represents a non-trainable
connection, which enables the context layer to store values from the hidden layer for the
past time period. For dynamic systems, this context layer provides a built-in short-term

memory for the model to learn from previous iterations.

60

 

Output Layer

 

 

 

A
l
I
l
|
l
1
1

 

Hidden Layer

 

 

 

1‘
I
l
I
I
1
J

 

 

 

Input Layer 8 Context Layer

 

 

 

 

 

 

Figure 5.1 — Simple Neural Network Framework.

5.2 State-Space Neural Network Topology

Based on the general framework of the neural network model presented in Figure 5.1,
SSNN models are developed for the purpose of estimating and predicting travel time on
urban arterial links. Singh (2006) developed separate modeling frameworks for both
travel time estimation and short-term prediction. This is necessary due to the time
intervals considered in each case. The SSNN models are trained using data representing
the state of several arterial links. Therefore, this topology will produce travel time
estimations and predictions for links, and not routes. Such travel times are aggregated to

estimate and predict travel time for arterials and arterial routes.

61

5.2.1 SSNN Topology for Travel Time Estimation

The SSNN model for travel time estimation consists of an input layer, a hidden layer, a
context layer, and an output layer, following the structure of the Elman model. The input
layer contains an input vector consisting of variables that influence travel time at the
current departure time period t. This vector is designated as U(t). The hidden layer
represents the state of the system at time t, and is designated as X(t). The hidden layer
activates the nodes in the context layer in a single time step. The context layer serves as
the short-term memory for the model, as the context layer represents the state of the
system at time t-I. The output layer represents the average travel time estimated at time
period t for the set of input variables. The SSNN topology for travel time estimation is

shown in Figure 5.2.

5.2.2 SSNN Topology for Travel Time Prediction

The SSNN model for travel time prediction is similar to that of travel time estimation,
except that the time period analyzed is different. The input layer contains an input vector
consisting of variables that influence travel time at the current departure time period t,
including the travel time estimated at time t. The hidden layer for travel time prediction
represents the state of the system at time 1+1, and the context layer represents the state of
the system at time t. The output layer represents the average travel time at time period
t+I, and is therefore a single step travel time prediction. The SSNN topology for travel

time estimation is shown in Figure 5.3.

62

 

 

Context Layer

 

 

 

 

 

 

x
x
z
3
:1
X2 (0
:1
X
x
Z
1':
Input Layer Hidden Layer

Output Layer

U (t) - Input Vector on Link X
at time instant ‘t'

Xi (t) — State of Link at time
instant ‘t' for Hidden Node ‘i'

x;(1-1) - State of Link at time
instant ‘t-1’ for Context layer
Node 1'

Y (t) - Output Vector at time
instant ‘t'

I Trainable
Weights

_ _, Non-Trainable
Weights

Figure 5.2 -— State-Space Neural Network Topology for Travel Time Estimation.

63

Context Layer

LINK X
m

 

 

 

 

X2 (t+1)

 

Output Layer

U (t) - Input Vector on Link X
at time instant ‘t'

Xi (t+1) - State of Link at time

instant ‘t+1' for Hidden Node
I"

v

LINK X

 

Xi (t) — State of Link at time
instant ‘t’ for Context layer
Node ‘i’

Y (t+1) — Output Vector at
time instant ‘t+1'

 

 

Trainable
Weights

—--5

Input Layer “"5999 Layer _ _> Non - Trainable
Weights

Figure 5.3 — State-Space Neural Network Topology for Travel Time Prediction.

64

The topology shown in Figure 5.2 and Figure 5.3 contains exactly 3 nodes in the hidden
and context layers for both the travel time estimation and prediction models. The
framework developed by Elman and shown in Figure 5.1 contains only one node, or
neuron, in both the hidden and context layers. Rivals and Personnaz (1996) present that
the state of the neural network elements can be computed using any number of hidden
neurons depending on the complexity of the system.

Travel time estimation and prediction models are developed for right-tum, left-tum, and
through movement travel time estimation and prediction. The topology defined in Figure
5.2 and 5.3 is a general representation of the framework that is applied for all three
movements. As the complexity in travel time estimation and prediction between through
and turning movements is expected to vary, the number of hidden neurons is also
expected to differ. In order to determine the proper number of hidden nodes for the travel
time estimation and prediction SSNN models, a range of 3 to 10 neurons is tested to
determine the optimal state-space representation for each movement. For both the
estimation and prediction models utilized in this thesis, the number of hidden neurons

equals 4 for through movement, and 10 for right—turn and left-turn movements.

5.3 State-Space Neural Network Training and Testing

The SSNN models for travel time estimation and prediction for left-tum, right-tum, and
through movements are replicated using MATLAB computing software, Version 7.0.
The program code was written by Singh (2006) and has been modified for use in this

thesis. The code was developed following the guidelines for learning rate, momentum

65

constant, and the number of hidden nodes documented by Haykin (1999). A generic
version of the SSNN programming code is provided in Appendix A. For each model, the
training dataset is fed to the network in a sequential pattern corresponding to the time
intervals of the microscopic simulation. This allows the model to learn the temporal
fluctuations in the variables that influence travel time.

The input layer of each SSNN model consists of a single node, which is defined by an
input vector comprised of a set of explanatory variables. The input variables for this
thesis are variables that influence traffic conditions on an arterial link, and that are easily
collected using existing surveillance infrastructure. These variables are defined in
Section 2.1 and include VT, VR, VL, ST, SR, SL, GT, GL, R, Off, QT, QL, and LL. The output
layer of each SSNN model is travel time for either the current or one-step future time

period. The specific input and output variables for each model are defined in Table 5.1

and 5.2.

Table 5.1 — Input and Output Variables for SSNN Travel Time Estimation Models.

 

 

 

 

SSNN Travel Input Variables (current time period) Output Variable
Time Estimation (current time period)
Models
SSNN-Thru VT, VR, VL, ST, GT, R, Off, Q7", and LL. TTT
SSNN—Left VT, VR, V1,, SL, G1,, R, 017, QL, and LL. TTL
SSNN-Right VT, VR, V1,, SR, 07, R, 0f], QT, and LL. TTR

 

 

 

 

 

66

Table 5.2 -— Input and Output Variables for SSNN Travel Time Prediction Models

 

SSN N Travel
Time Prediction
Models

Input Variables (current time period)

Output Variable
(future time period)

 

Estimated TTT, VT, VR, V1, ST, 61, R,

 

 

 

 

 

Ofl, QT, and LL.

SSNN-Thru Qfla QT, and LL. TTT
Estimated 171, V7, VR, V1, 51, 01, R,

SSNN—Left 01% QL, and LL. TTL

SSNN-Right Estimated 771;, V1, VR, VL, SR, 67', R, TTR

 

 

As the training data are fed to the MATLAB neural network program, a training graph is
plotted that details the performance of the model over a series of epochs, or iterations. A

sample training plot is shown in Figure 5.4.

10'

Performance is 0002199137, Goal is 0001

 

10'

 

I I T I

l l .
d
d

 

 

Training-Blue Goal-Black

 

10

 

 

L l
1500 2000

3201 Epochs

I l
500 1000

L
2500

l
3000

Figure 5.4 — Sample MATLAB Graphical Output for SSNN Training.

67

The performance of the model is evaluated based on the mean square error (MSE). As
the performance converges on the specified objective error, the trainable weights
converge on an optimal value. The testing dataset is then used to test the validity of the

optimal weights determined in the training of the model.

5.4 Application of State-Space Neural Network Models to Travel Time

Estimation and Prediction for Arterial Routes

The output data from the SSNN models represent the average travel time for a single
arterial link given a set of input variables representing traffic conditions on that link
during time period t. Advanced Traffic Management Systems (ATMS) and Advanced
Traveler Information Systems (ATIS) function mainly on the ability to estimate and
predict travel time along an arterial or arterial route within a network. Therefore, the
results of arterial link travel time estimation and prediction must be able to be applied to
arterial routes in order for the modeling procedure to fulfill the objectives of this thesis.

Following the definitions outlined in Section 2.1, an arterial route is comprised of a series
of arterial links, and vehicles traversing such a route complete a series of specific through
and/or turning movements from arterial link to arterial link. Therefore, the results of
travel time estimation and prediction for a specific set of arterial links, corresponding to
the appropriate vehicle movements, can be aggregated to estimate and predict travel time

along a specified arterial route.

68

 

Two arterial routes are examined in this thesis to evaluate the predictive capability of the
models developed for the purpose of this thesis. These routes are shown in Figure 5.5, as

routes 16-25 and 16-3, and the results are discussed in Chapter 6.

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Figure 5.5 — Study Routes for the Evaluation of SSNN Model Performance.

69

Chapter 6

Results and Discussion

The performance of each State-Space Neural Network (SSNN) model is evaluated based
on the ability of the model to accurately estimate or predict travel time for an arterial link.
Travel time estimations and predictions produced with each SSNN model are compared
to actual average travel times observed in the microscopic simulation trials. The
performance of the SSNN models is evaluated corresponding to travel times for 16
arterial links. Additionally, the performance of the models in terms of travel time
estimation and prediction for two sample arterial routes is analyzed. These results are

presented in this chapter along with discussion of significant findings.

6.1 Measures of Effectiveness

The results of travel time estimation and prediction are expected to have some associated
error in comparison to actual observed travel times. This error is measured in a number
of ways, including Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and
Mean Absolute Percentage Error (MAPE). Additionally, estimated/predicted travel time
is plotted versus actual travel time to analyze the relationship between them. The R-
squared (R2) statistic is calculated to evaluate the ability of each model to explain. the

variance in travel time on arterial links in the sample network (Kutner et al. 2004).

70

Mean Absolute Error (MAE)

The MAE statistic represents the average absolute value of the error between
estimated/predicted and actual travel time for an arterial link for the duration of the study
period. The value of error is calculated as the difference between estimated/predicted and
actual travel times for each arterial link during a specific time interval t. The absolute
value of the error is calculated and averaged for time period to through tn. The value of
MAE is found for each arterial link and is specific to the magnitude and range of the
average travel times in a particular study. Therefore, MAE is only used to evaluate the

performance of the models in this study, and not for comparison with previous works.

Root Mean Square Error (RMSE)

The RMSE statistic represents the square root of the average squared error values for an
arterial link for the duration of the study period. The value of error is calculated as the
difference between estimated/predicted and actual travel times for each arterial link
during a specific time interval t. The value of the error is squared and averaged for time
period to through 1,, to calculate the Mean Square Error (MSE). The square root of the
MSE is taken to determine the RMSE for each arterial link. Similar to MAE, RMSE is
specific to the magnitude and range of the average travel times in a particular study, and

is only used to evaluate the performance of the models in this thesis.

71

 

Mean Absolute Percentage Error (MAPE)

The MAPE statistic represents the proportion of the average value of error between
estimated/predicted and actual travel times, and the actual average travel time. The value
of error is calculated as the difference between estimated/predicted and actual travel
times for each arterial link during a specific time interval t. The absolute value of the
error is calculated and averaged for time period to through t,,. The value of MAPE is the
percentage of the average absolute error versus the actual average travel time for an
arterial link. The MAPE statistic is a relative measure of the error associated with a

prediction, and is used to compare the results of this thesis with the results presented by

Singh (2006).

Square of the Correlation Coefficient (R2)

The correlation coefficient, r, is a measure of the linear association between a dependent
variable and a set of independent variables. The correlation coefficient can have a
positive or negative value depending on the relationship between the response and
explanatory variables, and defines the slope of the regression line. The square of the
correlation coefficient, R2, is a measure of the ability of a set of independent variables to
explain the observed variation in the dependent variable. The R2 statistic ranges in value
from 0.0 to 1.0, where a value of 1.0 signifies that all of the variation in the dependent

variable is explained by the independent variables in the model.

72

For each travel time estimation/prediction model the estimated/predicted travel time is
plotted versus the actual travel time and the R2 statistic is calculated. In the ideal case the
scatter plot follows a perfectly straight linear pattern and the R2 statistic equals 1.0. As
the objective of this thesis is to develop models that accurately estimate and predict travel
time in comparison to actual travel times observed, the plotted travel time values should

follow a linear pattern, and the R2 statistic should be close to 1.0.

6.2 Travel Time Estimation on Arterial Links

Three separate models are developed for travel time estimation respective to right-tum,
left-tum, and through movements on arterial links. These models are referred to as
SSNN-Right, SSNN-Left, and SSNN-Through, respectively. The data contained in the
training set corresponding to the input variables defined in Section 5 .3 for each model are
fed to the appropriate SSNN. The models learn the relationship between travel time and
the states of an arterial link based on the training data, and are then applied to traffic
conditions represented in the testing dataset. The resulting output from each model is the
estimated travel time for the current time period t, for both the training and testing
datasets. The estimated travel time will be used as an input variable for travel time

prediction, and so the accuracy of these models must be explored.

73

6.2.1 Results of SSN N Travel Time Estimation Model for Through Movement

The results of travel time estimation for the through vehicle movement are presented in
this section for both the training and testing datasets. The scatter plot of estimated

average speed versus actual average speed for the training set is shown in Figure 6.1.

300.0

250.0 , 1:

 

 

 

 

 

0 Through
2000 Movement
150.0 — Linear
Regression

100.0

 

 

 

 

50.0 1
y = 0.911x + 9.452

R2 = 0.907

 

 

Estimated Average Travel Time (seconds)

0.0

 

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Actual Average Travel Time (seconds)

Figure 6.1 — Training Scatter Plot of Actual versus Estimated Travel Time for the

Through Movement.

The R2 statistic for the training set is .907, and a nearly linear relationship is observed.
This suggests that the model has efficiently learned the patterns influencing travel time
estimation for the through vehicle movement. The model is applied to the independent
traffic conditions captured in the testing dataset to validate the model. The scatter plot of
estimated average speed versus actual average speed for the testing set is shown in Figure

6.2

74

450.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9

g 400.0

8

2, 350.0 0 Through

E 3000 Movement

§ 250.0 _

': —-L1near

a 200.0 Regression

8

1; 150.0 .-

<

B 100.0 —

I!

g 50.0— y=0.983x+12.202

In 2

m R =0.888
0.0

0.0 100.0 200.0 300.0 400.0 500.0
Actual Average Travel Time (seconds)

Figure 6.2 — Testing Scatter Plot of Actual versus Estimated Travel Time for the Through

Movement.

The R2 statistic for the testing set is .888. This shows that the model is able to perform in
traffic conditions other than those used to train the model. Error statistics for the training
and testing sets are also generated and are displayed in Table 6.1. The error statistics are
slightly higher for the testing set travel time estimations. This implies that the SSNN-
Through model for travel time estimation may have learned some patterns that are only
present in the training dataset. Overall though, these statistics reveal that the model
performs reasonably well, with a MAPE of 13.8 percent for the testing set. A closer
analysis of the testing plot shows that as actual travel times increase, the model actually
overestimates travel time slightly for the through movement, except for a small number

of cases where the actual travel time exceeds 250.0 seconds.

75

Table 6.1 — Travel Time Estimation Error Statistics for the Through Movement.

 

 

 

 

 

 

 

 

 

 

 

Arterial (5:010:19 (5:23:55) MAPE (%)
6-10 10.2 14.4 9.9
.2” 16-20 7.6 10.1 8.5
g 2125 6.8 9.0 7.5
; 3-23 10.0 13.3 10.1
Total 8.6 11.7 9.0
6-10 13.3 18.3 12.3
19:11 16-20 13.6 17.7 12.8
g 2125 14.3 19.2 14.0
1- 3-23 17.5 22.9 16.0
Total 14.7 19.5 13.8

 

 

 

 

 

 

 

6.2.2 Results of SSNN Travel Time Estimation Model for Left-Turn Movement

The results of travel time estimation for the left-turn movement are presented in this
section for both the training and testing datasets. The scatter plot of estimated average
speed versus actual average speed for the training set is shown in Figure 6.3. The R2
statistic for the training set is .813, and the majority of points follow a linear pattern.
More specifically, for actual travel times lesser than 200 seconds, most points are
clustered near the regression line. Only for larger travel times, likely resulting from
congested conditions, does the relationship between estimated and actual speed become

slightly less linear.

76

 

 

 

o Left—Turn
Movement

 

 

 

 

--Linear
Regression

 

 

 

 

 

y = 0.816x + 25.752
R2 = 0.813

 

 

 

 

Estimated Average Travel Time (seconds)

 

T

0.0 100.0 200.0 300.0 400.0 500.0 600.0
Actual Average Travel Time (seconds)

Figure 6.3 -— Training Scatter Plot of Actual versus Estimated Travel Time for the Left-

Tum Movement.

The model is applied to the independent traffic conditions captured in the testing dataset
to validate the model. The scatter plot of estimated average speed versus actual average
speed for the testing set is shown in Figure 6.4. The R2 statistic for the testing set is .796,
and the testing plot follows a similar pattern as the training plot. Error statistics for the
training and testing sets are also generated and are displayed in Table 6.2.

The error statistics for the left-tum movement reveal similar information about the
validity of the left-tum model as for the through movement estimation model. Overall,

the model performs reasonably well, with a MAPE of 16.2 percent for the testing set.

77

500.0

 

’8’
§ 450.0
3 400.0
a o Left-Turn
E 350'0 Movement
— 300.0
3
,1: 250.0
to —Linear
1? 200-0 Regression
o
E 150.0
B 100.0
N
E y = 0.613x + 42.094
.3 50.0 2
m R =0.796
0.0
0.0 200.0 400.0 600.0 800.0

Actual Average Travel Time (seconds)
Figure 6.4 — Testing Scatter Plot of Actual versus Estimated Travel Time for the Lefi-

Turn Movement.

Table 6.2 — Travel Time Estimation Error Statistics for the Left-Turn Movement.

 

 

 

 

 

 

 

 

 

 

 

Arterial (8:33:35) (£33359 MAPE (%)
6-10 26.1 38.0 16.3
an 16—20 18.6 27.8 14.8
3% 21-25 19.2 25.6 14.6
F 3—23 19.9 21.7 13.6
Total 20.7 20.8 14.8
6-10 27.3 46.6 16.6
on 16-20 22.2 34.9 15.3
g 21-25 23.4 33.0 15.3
[- 3-23 28.7 44.8 17.6
Total 25.4 39.8 16.2

 

 

 

 

 

 

 

78

6.2.3 Results of SSN N Travel Time Estimation Model for Right-Turn Movement

The results of travel time estimation for the right-turn movement are presented in this
section for both the training and testing datasets. The scatter plot of estimated average

speed versus actual average speed for the training set is shown in Figure 6.5.

 

 

 

 

 

 

 

 

 

 

 

A 300.0
CD
'0
2
§ 250.0
7; o Right-Tum
g 200 0 Movement
i— .
3
5
,: 150.0 ‘- Linear
o Regression
2 4
h
g 100.0 1
< 1
'8 1
iii 50.0 ‘
g y = 0.405x + 73.061
tn 2
m R = 0.407
0.0 1 1
0.0 100.0 200.0 300.0 400.0 500.0

Actual Average Travel Time (seconds)

Figure 6.5 — Training Scatter Plot of Actual versus Estimated Travel Time for the Right-

Tum Movement.

The R2 statistic for the training set is .407, and the majority of points are spread in both
directions from the regression line. A linear of cluster of points can be seen where the
actual average travel time is between 50.0 and approximately 150.0. When the actual
travel time increases beyond 150.0 seconds though, the SSNN model fails to accurately

estimate travel time for the right-tum movement. A very similar pattern is observed in

79

the scatter plot of estimated average speed versus actual average speed for the testing set.

The testing scatter plot, with an R2 statistic of .418 is shown in Figure 6.6.

 

   

 

 

 

 

 

 

 

A 350.0
%
c
8 300.0
0
3
2 250.0 « o Right-Tum
1: Movement
2 200.0 —~—-~
L‘
1- —L'
3 150-0 Ineal' .
g Regresswn
0
2 100.01 —
1:
2
E 50.0 , , ' — ———“—“— y = 0.287x + 89.931
'5 R2 = 0.418
111

0.0 . .

0.0 200.0 400.0 600.0 800.0 1000.0

Actual Average Travel Time (seconds)
Figure 6.6 — Testing Scatter Plot of Actual versus Estimated Travel Time for the Right—

Tum Movement.

Error statistics for the training and testing sets are generated for the right-tum estimation
model and are displayed in Table 6.3. These statistics reveal significant errors in the
travel time estimations derived from both the training and testing sets, with a MAPE
between 30 and 40 percent. From these results it is obvious that the variability in right-
turns between intersections and over time has a severe impact on the accuracy of right-
turn speed estimation. An impact of this severity is not observed, though for left-tum

speed estimation, where turning movements are also variable.

80

Table 6.3 — Travel Time Estimation Error Statistics for the Right-Tum Movement.

 

 

 

 

 

 

 

 

 

 

 

Arterial (sxfis) (81:23:55) MAPE (%)
6-10 40.3 57.8 30.6
g0 16-20 30.1 39.9 27.9
I; 21-25 35.3 49.0 31.3
a 3-23 44.8 61.7 31.6
Total 37.62 52.1 30.3
6-10 45.0 64.8 39.06
an 16—20 42.7 59.8 36.6
g 2125 45.9 59.5 43.7
E- 323 59.0 88.7 37.4
Total 48.1 68.2 39.2

 

 

 

 

 

 

 

There are two distinct differences between left-tum and right-tum behaviors in the
modeling procedure that likely result in more accurate travel time estimation for left-
tuming vehicles than for right-turning vehicles. First, the left-turn movement is
accommodated in the microscopic simulation model with a dedicated left-tum bay at each
intersection approach. Second, left-turns are served by a protected signal phase. As a
result, the input variables corresponding to signal control and queue length are more
directly connected to the average flow rate, and thus the average travel time for left-
tuming vehicles.

Right-tuming vehicles share the rightmost travel lane with through moving vehicles.
This creates an interaction between through and right-tuming vehicles, as introduced in
Section 4.3. Although the travel time for through movement should correlate closely
with the travel time for the right-tum movement, the presence of right-turn-on-red

movements changes the typical pattern of right-tum travel times. In the case of constant

turning movement percentages, right-turn-on-red movements may be more consistent,

81

and have a lesser impact on average right-tum speed. When right-turn profiles are varied
between intersections and over time, the right-turn travel time distribution becomes more
unpredictable, and the SSNN model. is unable to accurately capture the patterns between
the state of the link and right-tum speed.

The SSNN models for travel time estimation developed by Singh (2006) were trained and
tested based on data assuming constant turning movement percentages. The resulting
MAPE statistics are shown in Table 6.4 in comparison with the MAPE results derived in
this thesis under similar conditions, but incorporating turning movements that vary over

time and space.

Table 6.4 — Mean Absolute Percentage Error Comparison Under Constant and Variable

Turning Movement Percentage Conditions.

 

 

 

 

 

 

 

 

Movement Constant Variable
Turning % Turning %
2” Through 7.6 9.0
E Left-Turn 7.5 14.8
A h-
G
i [— Right-Turn 8,4 303
L13]
9-
‘5 Through 7.3 13.9
E S?
g Left-Tum 8.6 16.2
E-
Right-Turn 14.0 39.2

 

 

 

 

 

 

The MAPE results between the two scenarios do differ slightly for the through
movement. This is expected as turning vehicles, especially right-tuming vehicles, do
interact with through vehicles in a traffic stream, but do not significantly detract from the

accuracy of travel time estimation for the through movement. The influence of variable

82

turning movement percentages is clearly seen though, in the MAPE statistics for left and
right-tums. The left-tum MAPE is nearly double, and the right-tum MAPE is
approximately triple in the presence of variable turning movement percentages. While it
is unclear at this point how these findings impact travel time estimation and prediction for
arterials and arterial routes, it is determined that variable turning movements significantly
impact the overall accuracy of the SSNN models, especially for right-turn speed

estimation.

6.3 Travel Time Prediction on Arterial Links

Three models are developed for travel time prediction respective to right-tum, left-tum,
and through movements on arterial links. These models are separate from the models
designed to estimate travel time, but the function is the same. The data contained in the
training set corresponding to the input variables defined in Section 5.3 for travel time
prediction are fed to the appropriate SSNN, including the estimated travel time for each
movement. The models learn the relationship between travel time and the states of an
arterial link based on the training data, and are then applied to traffic conditions
represented in the testing dataset. The resulting output from each model is the predicted
travel time for the one-step future time period t+1, for both the training and testing

datasets.

83

6.3.1 Results of SSNN Travel Time Prediction Model for Through Movement

The results of travel time prediction for the through vehicle movement are presented in
this section for both the training and testing datasets. The scatter plot of estimated

average speed versus actual average speed for the training set is shown in Figure 6.7.

 

 

 

 

 

 

 

 

 

 

 

300.0

a?

'0

5 250.0

0

(I)

3 0 Through

'5’ 200.0 Movement

i:

0

>

a 150.0 .

._ --L1near

g, Regression

a 100.0

>

<

'0

0

g 50.0

g y = 0.831x + 17.684

a“. R2=0.837
0.0 4 . .

 

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0
Actual Average Travel Time (seconds)

Figure 6.7 — Training Scatter Plot of Actual versus Predicted Travel Time for the

Through Movement.

The R2 statistic for the training set is .837, and a nearly linear relationship is observed.
This suggests that the model has efficiently learned the patterns influencing travel time
estimation for the through vehicle movement. There are two distinct point clusters that
occur along the regression line. For actual average travel times above approximately 130

seconds, more variability is observed between the actual and predicted travel time. This

84

pattern is strongly correlated with the link length, where travel times below 130 seconds
occur mainly on short links, while the majority of the points above 130 seconds occur on
the longest links in the network. Although the model performs more accurately for
shorter links, the overall performance is reasonable.

The model is applied to the independent traffic conditions captured in the testing dataset
to validate the model. The scatter plot of predicted average speed versus actual average

speed for the testing set is shown in Figure 6.8.

 

 

 

 

 

100.0

 

50.0 -

 

-— y = 0.778x + 20.623
R2 = 0.810

350.0
E
c 300.0
o
0
3
I; 250.0 0 Through
E Movement
I-
; 200.0 ..
E
2 —Linear
9 150.0 Regression
8
>
<
1::
GD
§
‘5
2
n.

 

 

0.0 T
0.0 100.0 200.0 300.0 400.0 500.0

Actual Average Travel Time (seconds)

 

Figure 6.8 — Testing Scatter Plot of Actual versus Predicted Travel Time for the Through

Movement.

The R2 statistic for the testing set is .810. This shows that the model is able to perform in
traffic conditions other than those used to train the model. Clustering is observed above

and below approximately 130.0 seconds, but is not as severe as for the training set. Error

85

statistics for the training and testing sets are also generated and are displayed in Table

6.5.

Table 6.5 — Travel Time Prediction Error Statistics for the Through Movement.

 

 

 

 

 

 

 

 

 

 

 

Amt-1211 (823x318) (81:33:58) MAPE (%)
6-10 11.6 17.9 10.9
a” 16-20 8.65 14.4 9.7
:§ 2125 7.5 11.2 8.2
i 323 13.9 18.6 13.6
Average 10.42 15.5 10.6
6-10 12.8 22.6 10.6
go 16-20 10.5 17.5 9.4
‘5' 21-25 10.2 15.9 10.9
[- 3-23 17.2 27.1 13.2
Average 12.7 20.8 11.0

 

 

 

 

 

 

 

The error statistics reveal that the model performs reasonably well, with a MAPE of 10.6
percent for the training set and 11.0 percent for the testing set. In order to validate this
level of confidence in the perfomiance of the SSNN-Through model, a detailed analysis
of the accuracy of travel time predictions on each arterial link is necessary. The Absolute
Percentage Error (APE) is plotted for each arterial link in S-minute increments for both
the training and testing datasets. Figure 6.9 and Figure 6.10 display APE data for major

arterial 6-10.

86

 

Link 6-7 Link 7-8 Link 8-9 Link 9—10

 

 

 

 

 

 

45 -
A 40 ,
,\°
1' 35 ,
o
t
I“ 30
o
o: IMajor
‘2 25 Arterial
‘1’ 6—10
0
E 20 Through
8 15 Movement
2
8 10
a
<

5

0 ‘ .

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.9 — Absolute Percentage Error of Through Movement Travel Time Prediction

for Training Set Arterial 6- 10.

 

 

 

 

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

35 ~— ,
A 30
a\°
E
g 25
§ 20 lMajor
E Arterial
8 6-10
3 15 Through
% : Movement
‘5 i
3 10 i
U5
.1:
< I

5 1
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.10 - Absolute Percentage Error of Through Movement Travel Time Prediction

for Testing Set Arterial 6-10.

87

The APE for each link comprising arterial 6-10 is generally less than 10 percent for both
the training and testing sets. However, there are some specific links and time increments
for which the performance of the model is not as desirable. On arterial link 6-7 the APE
is consistently in excess of 15 percent, and is as high as 43 percent. For most cases, this
corresponds to an absolute error of 10 seconds or less. From time 80 minutes to 90
minutes though, this percentage equates to approximately one minute of error. If this
level of error is experienced for the entire length of an arterial the predicted travel time
may differ from the actual travel time by 5 minutes or more, which is not acceptable for
ITS applications. On the other hand, it is possible that positive and negative errors may
balance along the length of an arterial or arterial route, thus resulting in an acceptable
overall level of error. The predicted versus actual travel time and the associated error for
two particular routes is analyzed at the end of this chapter.

The through movement APE for major arterial 16-20 are shown in Figure 6.11 for the
training set and Figure 6.12 for the testing set. Similar to arterial 6-10, the APE is
generally less than 10 percent on each link, except for link 16-17 where travel time
prediction errors ranging from 15 to 35 percent are consistently observed in the training
dataset. This level of error is only experienced for three time intervals in the testing set,
though. This may be due to the stochastic nature of arrivals in the microscopic
simulation, where an intense concentration of vehicle arrivals at an intersection decreases
signal operation level. of service and therefore travel time predictability. Nonetheless,
travel time prediction models must be able to operate reliably in a wide range of traffic

conditions to be effective in ITS applications.

88

Link 16-17 Link 17-18 Link 18—19 Link 19-20

 

 

 

 

 

 

 

40

35 .
5
lo- 30
E
a: 25 ' .
8’ IMajor
‘5 Arterial
3 20 16-20
i113 Through
a.
o 15 E Movement
05 E
3 10
.o
<

5

0

5 95 5 95 5 95 5 95
Departure Time Period (minutes)

Figure 6.11 — Absolute Percentage Error of Through Movement Travel Time Prediction

for Training Set Arterial 16-20.

 

 

 

 

 

 

Link 16-17 Link 17-18 Link 18-19 Link 19-20
35
A 30 -
3
E
t 25
ul
3 M '
a 20 I 8101’
“E Arterial
3 16-20
5 15 Through
% Movement
‘5
3 10
10 1
D l
< 2
5 3 g
1 i
l 1
1
0 .
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.12 — Absolute Percentage Error of Through Movement Travel Time Prediction

for Testing Set Arterial 16-20.

89

The APE is plotted for major arterial 21-25 in Figure 6.13 and Figure 6.14. Overall,
travel time prediction is within 10 percent of the actual travel time for all links for both
the training and testing datasets. There are isolated time intervals for which the APE is in
excess of 20 percent. These intervals with a higher range of error are typically
consecutive, and the accuracy of travel time prediction recovers in approximately 15
minutes. This trend is observed for other links in the network and corresponds to sudden
peaks in travel time, which could be due to intersection blockage or other sudden loss in
capacity. As the SSNN model learns patterns in traffic over one time step, a sudden
increase in travel time is not effectively predicted by the model developed for the through
movement. Incorporating multiple time steps into the learning process may improve the
accuracy of these models. The relatively rapid recovery of the predictive capability of
the model is promising though for real-world applications where unpredictable incidents
often impact the flow of traffic.

The APE is plotted for arterial 3-23 in Figure 6.15 and Figure 6.16. The error in travel
time prediction is generally higher for this minor arterial, with average percentage errors
regularly observed between 10 and 15 percent. The patterns in the APE are similar to
those described for the major arterials, and where APE exceeds 25 percent the accuracy

of the model is shown to recover quickly.

90

45

Link 21-22

Link 22-23

Link 23-24

Link 24-25

 

40 l
35
30
25

20

 

Absolute Percentage Error (%)

955

 

 

955

 

95 5

Departure Time Period (minutes)

 

lMam
Arterial
21 —25
Through
Movement

Figure 6.13 — Absolute Percentage Error of Through Movement Travel Time Prediction

for Training Set Arterial 21 —25.

50

Link 21-22

Link 22-23

Link 23-24

Link 24—25

 

45
40 ~
35 -
30 —
25-
20 -
15 —

Absolute Percentage Error (%)

10«

 

 

955

 

955

 

i

955

Departure Time Period (minutes)

 

lem
Arterial
21-25
Through
Movement

Figure 6.14 - Absolute Percentage Error of Through Movement Travel Time Prediction

for Testing Set Arterial 21-25.

91

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

 

 

 

 

 

50

45 <
23 40
I:
g 35
m
8. 30 IMinor
a
'E Arterial
g 25 3—23
3 Through
‘: 20 Movement
L5 15 i
O i 5
«3 10 3
< ii

5 .

0 '

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.15 — Absolute Percentage Error of Through Movement Travel Time Prediction

for Training Set Arterial 3-23.

 

 

 

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

40 . .

35 ,
.1?
a 30
15

25
§ IMinor
E Arterial
3 20 ' 3-23
3 Through
% 15 5 Movement
3
3 10 i
.D 1 1
< =:

5 ~ .

0 1 a

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.16 — Absolute Percentage Error of Through Movement Travel Time Prediction

for Testing Set Arterial 3-23.

92

 

As previously mentioned there are time periods when the percentage error for travel time
prediction is observed to spike, and then improve in a short period of time. Sudden
breakdowns in traffic conditions may cause travel time to become more unpredictable
using single time step prediction models and the variable types utilized in this thesis. For
ITS applications, large discrepancies between actual travel times and those
communicated to users and system administrators are not desirable. For this reason, the
ability of the SSNN models to rapidly detect changing traffic conditions must be
improved.

To further explore the ability of the SSNN-Thru model to learn and follow changing
patterns in traffic conditions and accurately predict travel time, the actual travel time and
predicted travel time are plotted together over time. These results for the testing set are
shown for each arterial in Figures 6.17, 6.18, 6.19, and 6.20.

For each arterial link, the pattern of the predicted travel time closely follows the actual
travel time. The SSNN-Through model successfully models the dynamic relationship
between the state of an arterial link and travel time in the short-term future for the
through movement. Although there are errors observed between the actual and predicted
travel times, the model does show the ability to learn from changes in the input variables
selected in this thesis. Significant errors in travel time prediction are typically isolated,
and do not propagate through future time intervals. Further research to incorporate
multiple time step learning of the SSNN model will likely advance the accuracy of travel

time predictions observed here.

93

Average Travel Time (seconds)

 

 

 

 

 

 

 

 

 

Link 6-7 Link 7-8 Link 8-9 Link 9-10
350
300 _
T ------ Actual
250 - ' Through
Travel
Time
200 «
150 5 _
+Predicted
Through
100 ~ Travel
Time
50 ~
0
95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.17 — Pattern of Actual versus Predicted Travel Time for Through Movement on

Testing Set Arterial 6-10.

Average Travel Time (seconds)

250

Link 16-17

Link 17-18

Link 18-19 Link 19-20

 

200

150

100

 

  

:1: :II-‘Hh ‘h;

 

 

----- Actual
Through
Travel
Time

 

 

—-— Predicted

 

 

 

" r ‘ Through
Travel
50 Time
0 .
95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.18 — Pattern of Actual versus Predicted Travel Time for Through Movement on

Testing Set Arterial 16-20.

94

Average Travel Time (seconds)

Link 21—22
180

Link 22-23

Link 23-24

Link 24-25

 

160
140
120
100
80
60
40
20

 

 

W

 

 

 

 

 

95

5

95

5

95 5 95

Departure Time Period (minutes)

------ Actual
Through
Travel
Time

——I— Predicted
Through
Travel
Time

Figure 6.19 — Pattern of Actual versus Predicted Travel Time for Through Movement on

Testing Set Arterial 21-25.

Average Travel Time (seconds)

 

 

 

 

 

 

 

 

Link 3-8 Link 8-13 Link 13-18 Link 18-23
250
100 T 1.. I _
W
50 - ‘ 4— * '
0
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

 

------ Actual
Through
Travel
Time

—-— Predicted
Through
Travel
Time

Figure 6.20 — Pattern of Actual versus Predicted Travel Time for Through Movement on

Testing Set Arterial 3-23.

95

In conclusion of this section, the results observed for the accuracy of travel time
prediction for the through movement are compared to the results obtained by Singh
This

(2006) under the assumption of constant turning movement percentages.

comparison is shown in Table 6.6.

Table 6.6 - Mean Absolute Percentage Error Comparison for Through Movement Under

Constant and Variable Turning Movement Percentage Conditions.

 

 

 

Movement Constant Variable
Turning% Turning%
.5 Training Through 9.0 10.6
n:
G-
; Testing Through 8.8 11.0

 

 

 

 

 

 

 

Based on this comparison, the incorporation of variable turning movements into the
modeling procedure does have a slight impact on the accuracy of travel time prediction
for the through movement. Similar to the comparison of data from the training set, this
increase in error can be expected due to the interaction of though and turning vehicles in
a traffic stream.

Overall, it is concluded that the influence of variable turning movements is not significant
in terms of travel time prediction for the through movement. For the length of an arterial,
which is slightly less than 3 miles, travel time is predicted on average within 13 seconds
of the actual travel time for vehicles making only a through movement. This is a

promising result for the future use of this model in Intelligent Transportation System

(ITS) applications.

96

6.3.2 Results of SSNN Travel Time Prediction Model for Left-Turn Movement

The results of travel time prediction for the left-tum movement are presented in this
section for both the training and testing datasets. The scatter plot of estimated average

speed versus actual average speed for the training set is shown in Figure 6.21.

 

 

 

 

 

450.0
g?
: 400.0 J
o
0
8 350.0—
I o Left-Turn
E 300.0 < Movement
7;; 250.0 -
1': —Linear
3%, 200.0 - Regression
L-
1; 150.0 -
<
B 100.0 «
O
'3 50.0 < y = 0.754x + 31.213
5 R2 = 0.758
0.0

0.0 100.0 200.0 300.0 400.0 500.0 600.0
Actual Average Travel Time (seconds)

Figure 6.21 — Training Scatter Plot of Actual versus Predicted Travel Time for the Left-

Tum Movement.

The R2 statistic for the training set is .758, which is less accurate the results obtained for
the through movement. There is a cluster of points observed for cases corresponding to
an actual average travel time of less than 200.0 seconds. For these cases the travel time

for the left-tum movement is predicted with more accuracy than for cases where the

97

actual travel time exceeds 200.0 seconds. In the latter circumstance, the relationship
between predicted and actual travel time is more dispersed and less linear.

The model is applied to the independent traffic conditions captured in the testing dataset
to validate the performance of the model. The scatter plot of predicted average speed

versus actual average speed for the testing set is shown in Figure 6.22.

 

 

 

 

 

600.0
13‘
1:
S
0 500.0 1
8
I; o Left-Turn
.§ 4000 ‘ Movement
.—
3
«’6
D: 300.0 4 —Linear
o Regression
D)
2
g 200.0 -
<
'0
8
§ 100.0 <
E y = 0.663x + 25.237
2 _
0.0 V R — 0.738
0.0 200.0 400.0 600.0 800.0

Actual Average Travel Time (seconds)
Figure 6.22 — Testing Scatter Plot of Actual versus Predicted Travel Time for the Left-

Tum Movement.

The R2 statistic for the testing set is .738. This is similar to the accuracy observed for the
training set, and shows that the model is equally as accurate in traffic conditions other
than those used to train the model. Clustering is again observed for actual travel times
below 200 seconds, and the pattern of dispersion along the regression line is similar to

that observed for the training dataset. Travel times in excess of 200 seconds are observed

98

for the left-tum movement on various link lengths, so increases in travel time can be
attributed to congested conditions rather than simply the length of the link. Due to
variations in the percentage of left-turn movements between intersections and over time,
the SSNN model becomes less accurate as congestion builds on network links and actual
travel times increase.

In order to further examine the accuracy of travel time prediction for the left-tum
movement, error statistics for the training and testing sets are generated and are displayed

in Table 6.7.

Table 6.7 — Travel Time Prediction Error Statistics for the Left-Tum Movement.

 

 

 

 

 

 

 

 

 

 

 

Arterial («5236:1319 (81:23:58) MAPE (%)
6—10 31.2 44.6 18.6
a” 16-20 22.9 32.8 17.8
3% 21-25 22.5 31.1 16.7
F 3-23 24.0 33.3 17.5
Average 25.2 35.5 17.6
6-10 36.1 55.2 20.9
go 16—20 25.3 36.5 17.1
g 2125 31.2 45.3 19.6
[- 3-23 32.0 48.1 20.3
Average 31.1 46.3 19.5

 

 

 

 

 

 

 

The error statistics reveal that the influence of variable turning movements is more severe
for left-tum travel time prediction than is observed for the through movement. The
SSNN-Left prediction model performs with a MAPE of 17.6 percent for the training set
and 19.5 percent for the testing set. This level of error is shown later in this chapter to

impact the accuracy of travel time prediction for two routes within the arterial network.

99

The MAPE for each arterial is broken down by link in order to identify any patterns or
specific predictions that may explain the error displayed in Table 6.7. The Absolute
Percentage Error (APE) is plotted for each arterial link in 5-minute increments for both

the training and testing datasets. Figure 6.23 and Figure 6.24 display APE data for major

 

 

 

N
O

.3
O C
0|
1
i_ l
.—
—
—

 

 

 

 

arterial 6-10.
Link 6-7 Link 7-8 Link 8-9 Link 9-10
60 .
g 50 ~
'5
t
“-l 40 1
“é: . I Major
*5 30 -. ff 5 Arterial
g I 6-10
3 W ; Left-Tum
9; Movement
‘5
3
(D
.o
<

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

955 955

 

 

 

95

Departure Time Period (minutes)

Figure 6.23 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Training Set Arterial 6-10.

The APE for each link comprising arterial 6-10 is generally between 10 percent and 20
percent for both the training and testing sets. However, the magnitude of error on each
particular link changes between the training and testing sets. For example, on link 89
the APE exceeds 20 percent for all but two time intervals in the training set, while for
links 6-7 and 7-8 the average error is closer to 10 percent. For the testing set, the APE

for link 8-9 is typically below 15 percent, while the error for links 6-7 and 7-8 is

100

consistently in excess of 15 percent. This discrepancy in the error patterns between the
training and testing sets implies that the patterns in traffic conditions learned to predict
lefi-turn travel time in the training set do not apply in the same manner for the testing set.
The variability in left-turn movements coupled with changing traffic conditions between
datasets causes randomness in the accuracy of travel time predictions that is not desirable

for ITS applications.

 

 

 

Link 6-7 Link 7-8 Link 8-9 Link 9-10
50 .
45 l
E 40 ,
3
IE 35 i
§ 30 lMajor
E : Arterial
3 25 g 610
5 20 - Left-Tum
% . Movement
‘_=‘ 15 « E
3 i
a 3
< 10 T
5 ‘
i i
0 : :
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.24 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Testing Set Arterial 6-10.

The APE is plotted for arterial I6-20 in Figures 6.25 and 6.26, for arterial 21-25 in
Figures 6.27 and 6.28, and for arterial 3-23 in Figures 6.29 and 6.30. The APE is
consistently near 20 percent for each link, and the same discrepancy between the training

and testing sets as described for arterial 6-10 is observed for these arterials.

101

Absolute Percentage Error (%)

Link 16-17

Link 17-18

Link 18—1 9

Link 19-20

 

i
i
s

95

5

 

i

955

 

95 5

Departure Time Period (minutes)

95

 

I Major
Arterial

Movement

Figure 6.25 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Training Set Arterial 16-20.

Absolute Percentage Error (%)

60

Link 16-17

Link 17-18

Link 18-19

Link 19-20

 

50

4O

30

20‘

 

i
i
a
i

95

5

i

955

 

95 5

Departure Time Period (minutes)

95

 

I Major
Arterial
16-20
Left-Turn
Movement

Figure 6.26 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Testing Set Arterial 16-20.

102

Link 21-22 Link 22-23 Link 23-24 Link 2425

 

 

 

 

 

 

1; 60
o\
1'
g 50
Lu
§ 40 IMajor
“E Arterial
3 21-25
5 30 Left-Turn
: Movement
E 20
o
(I)
a
< 10

0

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.27 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Training Set Arterial 21-25.

 

 

 

 

 

Link 21-22 Link 22-23 Link 23-24 Link 24-25
50 :
45 i
E 40 4
5
E 35 -
g: 30 IMajor
E 25 Arterial
3 21-25
3 20 Left-Tum
t Movement
2 15
8 =
.o
< 10 i
5 i
o .
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.28 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Testing Set Arterial 21-25.

103

 

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

 

 

45 -— ——-*~—- ———a’
40 ‘
a?
T: 35 '
g l
LIJ 30 — i
o I
0’ l lMinor
*2 25 ' 1 Arterial
9 3-23
0
3 20 — i Left-Turn
n. I Movement
8 15 ,
3
8
.o 10 =
< E
5 ' E i
o i '
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.29 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Training Set Arterial 3-23.

 

 

 

 

 

 

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

70
a; 60 -
°\
g 50 «
LU
0 .
g 40 IMInor
E Arterial
8 _
a“: 30 ‘ Left-Turn
% Movement
'2' 20 « i
o
tn
.0
< 10

0

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.30 — Absolute Percentage Error of Left-Tum Movement Travel Time Prediction

for Testing Set Arterial 3-23.

104

Inconsistencies in the errors observed between the training and testing sets imply that the
learning of the SSNN-Left model is limited to the traffic conditions observed in the
training set. The influence of variable turning movements and changing traffic
conditions causes a variation in the accuracy of predictions made by the model. To
further explore the ability of the SSNN-Left model to learn and follow changing patterns
in traffic conditions and accurately predict travel time, the actual travel time and
predicted travel time are plotted together over time. These results for the testing set are

shown for each arterial in Figures 6.31, 6.32, 6.33, and 6.34.

 

 

100 i

Link 6-7 Link 7—8 Link 8-9 Link 9-10

500

450 — .
a .l
'2 400 ~ '1' - - -- - Actual
° Left-Tum
§ 350 A Travel
.§
1'. 250 ~
in
5
i: 200 « --I—Predicted
o Left-Turn
5’ 150 A Travel
g Time
<

 

50*

 

 

 

 

 

Departure Time Period (minutes)

Figure 6.31 — Pattern of Actual versus Predicted Travel Time for Left-Tum Movement on

Testing Set Arterial 6-10.

105

Link 16-17 Link 17-18 Link 18-19 Link 19-20

 

 

 

 

 

 

 

 

350
1; 300 .—
2 - - - - Actual
‘3 g Left-Turn
if; 250 Travel
I Time
E 200 w
i:
E 150
': —I—Predicted
a Left—Tum
g 100 , Travel
a; Time
< 50
O
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.32 — Pattern of Actual versus Predicted Travel Time for Left-Tum Movement on

Testing Set Arterial 16-20.

Link 21-22 Link 22-23 Link 23-24 Link 24-25
300

 

- -- - Actual
Left-Turn
Travel
Time

+ Predicted
Left-Turn
Travel
Time

Average Travel Time (seconds)

 

 

 

 

 

 

 

5 95 5 95 5 E 5 if
Departure Time Period (minutes)

Figure 6.33 — Pattern of Actual versus Predicted Travel Time for Lefi-Turn Movement on

Testing Set Arterial 21-25.

106

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

 

 

 

 

400
A 350 «
tn
'2 - - -- - Actual
8 300 ‘ Left-Turn
3 5 Travel
7; 250 . " Time
1-
E . u
'1 200 A A f .— .2":
w I
is
i: 150 - —I—Predicted
o Left-Turn
g 100 - Travel
in Time
2
50 -
O

 

 

 

 

 

 

 

Departure Time Period (minutes)

Figure 6.34 -— Pattern of Actual versus Predicted Travel Time for Left-Tum Movement on

Testing Set Arterial 3-23.

In many cases the predicted travel time is less than the actual travel time, such as on links
6-7, 7-8, 16-17, 21-22, 22-23, and 13-18. Despite the underestimation of travel time, the
predicted travel time is in the same range and follows the same pattern as the actual travel
time for most links. The cause for underestimation in many cases seems to be due to a
delay in the predictive capability of the model. When actual travel time changes, the
change in the prediction made by the model follows one or two time-steps later. This lag
in the reaction of the model to changing traffic conditions is not desirable, and is the
cause of much of the APE observed for the SSNN-Left travel time prediction model.

In conclusion of this section, the results observed for the accuracy of travel time

prediction for the left-tum movement are compared to the results obtained by Singh

107

(2006) under the assumption of constant turning movement percentages. This

comparison is shown in Table 6.8.

Table 6.8 — Mean Absolute Percentage Error Comparison for Left-Tum Movement Under

Constant and Variable Turning Movement Percentage Conditions.

 

 

 

Movement Constant Variable
Turning "/0 Turning "/o
5 Training Left-Turn 7.3 17.6
Li:
a.
g Testing Left-Tum 8.8 19.5

 

 

 

 

 

 

 

Based on this comparison, the incorporation of variable turning movements into the
modeling procedure has a significant impact on the accuracy of travel time prediction for
the left-tum movement. The magnitude of the MAPE is increased 10 to 11 percent as
compared to the results obtained assuming constant turning percentages. The variability
in the left-tum movement influences the accuracy of travel time predictions under
changing traffic conditions, and causes delay in the predictive capability of the model.

The success of Advanced Traffic Management Systems (ATMS) and Advanced Traveler
Information Systems (ATIS) are dependent on the accuracy and reliability of the travel
time prediction models that drive them. In order to be useful for such applications, the
travel time prediction model must be able to adapt quickly and accurately to changing
The addition of multiple time-steps to the learning process of the

traffic conditions.

model may improve the flexibility and reactive capability of the model.

108

6.3.3 Results of SSNN Travel Time Prediction Model for Right-Turn Movement

The results of travel time prediction for the right-turn movement are presented in this
section for both the training and testing datasets. The scatter plot of estimated average

speed versus actual average speed for the training set is shown in Figure 6.35.

250.0

 

 

200.0 -
o Right-Tum
Movement
150.0 ~
—Linear
Regression

 

y = 0.379x + 78.572
R2 = 0.420

Predicted Average Travel Time (seconds)

 

 

 

T T l

0.0 100.0 200.0 300.0 400.0 500.0
Actual Average Travel Time (seconds)

Figure 6.35 — Training Scatter Plot of Actual versus Predicted Travel Time for the Right-

Tum Movement.

The R2 statistic for the training set is .420, which indicates that the average travel time for
the right-tum movement is not accurately predicted. The pattern in the dispersion of
points is similar to that observed for travel time estimation for right-turns, where a linear
pattern can be detected for actual travel times less than 150 seconds. For travel times in
excess of 150 seconds, the SSNN model fails to accurately predict travel time for the

right-tum movement. A very similar pattern is observed in the scatter plot of estimated

109

average speed versus actual average speed for the testing set. The testing scatter plot,

with an R2 statistic of .485 is shown in Figure 6.36.

 

400.0

350.0 -

300.0 — . Right-Tum
Movement

250.0 .

200.0 ~ -—Linear
Regression

150.0 -

 

y = 0.422x + 58.614
R2 = 0.485

Predicted Average Travel Time (seconds)

 

 

 

 

0.0 200.0 400.0 600.0 800.0 1000.0
Actual Average Travel Time (seconds)

Figure 6.36 -— Testing Scatter Plot of Actual versus Predicted Travel Time for the Right-

Tum Movement.

The accuracy observed for the testing set is similar to that of the training set, and shows
that the model performs equally in traffic conditions other than those used to train the
model. Due to variations in the percentage of right-turn movements between
intersections and over time though, the accuracy of travel time predictions is not
sufficient to support ITS applications. In order to further examine the accuracy of travel
time prediction for the right-tum movement, error statistics for the training and testing

sets are generated and are displayed in Table 6.9.

110

 

Table 6.9 — Travel Time Prediction Error Statistics for the Right-Tum Movement.

 

 

 

 

 

 

 

 

 

 

 

Arterial (8:33:38) 81:23:55) MAPE (%)
6-10 42.4 58.1 34.8
3 16-20 31.4 41.4 29.6
.5: 21-25 35.6 48.6 31.5
F 3-23 46.7 59.5 37.0
Average 39.0 51.9 33.2
6-10 40.2 61.4 28.4
go 16-20 40.8 57.2 30.7
g 21-25 35.2 54.5 27.8
i- 323 57.3 88.3 34.6
Average 43.4 65.4 30.4

 

 

 

 

 

 

 

These statistics reveal significant errors in the travel time predictions derived from both
the training and testing sets, with a MAPE of approximately 30 percent. As is stated in
the discussion of travel time estimation for right-turns, it is obvious that the variability in
right-turns between intersections and over time has a severe impact on the accuracy of
right—tum travel time prediction.

Right-turn-on-red maneuvers add to the complexity of travel time prediction for this
movement, especially for this experiment where right-turning vehicles share the
rightmost lane with through vehicles. During the red interval the presence of only one
through vehicle in queue may potentially block vehicles from turning right-on-red, thus
increasing travel time for those vehicles. On the other hand, right-turning vehicles that
are not blocked by a through vehicle in the rightmost lane experience decreased travel
time as they are allowed to progress during the red interval. In congested conditions the
complexity of this situation increases due to the volume of crossing traffic, and travel

time becomes more unpredictable.

111

The MAPE for each arterial is broken down by link in order to identify any patterns or
specific predictions that may explain significant level of error in travel time prediction for
the right-tum movement. The Absolute Percentage Error (APE) is plotted for each
arterial link in S—minute increments for both the training and testing datasets. Figure 6.37

and Figure 6.38 display APE data for major arterial 6-10.

 

 

 

 

 

 

Link 6-7 Link 7-8 Link 8-9 Link 9—10

140
r; 120
o\
E
t 100 —
u:
o .
8 80 manor
'5 Arterial
3 6-10
3 60 ‘ Right-Turn
% Movement
‘2' 40
o
(D
.n
< 20

0 1
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.37 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Training Set Arterial 6-10.

The APE for each link comprising arterial 6-10 is at or in excess of 20 percent during
most time intervals and on all links for both the training and testing sets. No significant
pattern is observed in the APE for this arterial, and the model performs poorly in general
for right-tum travel time prediction. The variability in turning movement percentages
causes absolute percentage errors in excess of 50 percent in some cases, and in one case

the error exceeds the travel time. This level of accuracy is obviously not suitable for

112

ATMS and ATIS applications, where user confidence in the travel time predicted is

 

 

 

 

highly important.
Link 6-7 Link 7—8 Link 8—9 Link 9-10

70 z
A 60
.\°
E
t 50
in
3" 40 I Major
g Arterial
3 6—10
B 30 Right-Turn
% 5 Movement
3 i
3 20 1
ll)
.0
< 10 .

J
0

5 95 5 95 5 95 5 95
Departure Time Period (minutes)

Figure 6.38 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Testing Set Arterial 6-10.

The APE is plotted for arterial 16-20 in Figures 6.39 and 6.40, for arterial 21-25 in
Figures 6.41 and 6.42, and for arterial 3-23 in Figures 6.43 and 6.44. The APE is
consistently in excess of 20 percent for each link, with an average error near 30 percent.
The impact of the errors presented in this section are prevalent in the results for travel

time prediction for arterial routes, and are discussed in more detail in Section 7.4.

113

Link 16-17 Link 17-18 Link 18—19 Link 19-20

 

 

 

 

100

90
=3. 80
b
O
t: 70
in .
§ 60 IMajor
‘E 50 Arterial
8 16-20
5 40_ Right-Turn
% ' Movement
‘5 301
'6
3 20
< a

10 1 ,

1 1
0 i

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.39 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Training Set Arterial 16-20.

 

 

 

 

 

 

Link 16-17 Link 17-18 Link 18—19 Link 19-20

90
A 80 ~
.\°
2’ 70 —
2
m 60 ~
0
or IMajor
«E 50 ' Arterial
0 16-20
0
E 40 Right-Turn
2 30 Movement
3 a
g 20
< i

10 i

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.40 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Testing Set Arterial 16-20.

114

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

 

 

 

 

 

 

120
§ 100
‘E
g
u: 80
0 .
c» IMajor
«I .
E 60 Arterial
8 21-25
E Right-Turn
3 40 Movement
2
o
3 2
< 20 i a

2
0 l

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.41 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Training Set Arterial 21-25.

 

 

 

 

Link 21-22 Link 22-23 Link 23-24 Link 24-25
80 ,
A 70 7
.\°
‘5 60 ,
l:
m 50
3, lMajor
3 Art ‘ l
S 40 ena
0 3 21-25
E Right-Turn
I; 30 ‘ Movement
1 ' E
3 20 i ; E:
2 j; : E
10 I;
o , r
5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.42 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Testing Set Arterial 21-25.

115

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

 

 

 

 

100 7

90« i i
2% 80
'6
t 70
LIJ 2
2'3 50 IMinor
E Arterial
3 50 3-23
5 a Right-Turn
% 40 Movement
*_5 30
o
3 20
<

10

0

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.43 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Training Set Arterial 3-23.

 

 

 

 

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

90 - -

80
$5
T: 70
2
61 60—
0
a: IMinor
*2 50 Arterial
0’ 3-23
0 s
a“: 40 : Right-Turn
1:. Movement
3 30 , i
3 l
o
g 20 g a r
< i

10 l i

o, a.

5 95 5 95 5 95 5 95

Departure Time Period (minutes)

Figure 6.44 — Absolute Percentage Error of Right-Tum Movement Travel Time

Prediction for Testing Set Arterial 3-23.

116

The high level of error associated with travel time prediction for the SSNN-Right model
is due to the influence of variable turning movements between intersections and over
time. The model does not show the capability to accurately learn the patterns in traffic
conditions that are connected to changes in travel time for the right-tum movement. To
further explore the performance of the SSNN-Right model, the actual travel time and
predicted travel time are plotted together over time. These results for the testing set are

shown for each arterial in Figures 6.45, 6.46, 6.47, and 6.48.

 

 

 

 

 

 

 

 

Link 6-7 Link 7-8 Link 8-9 Link 910

350
,.. 300 — _ f
= Wm...
O t v 7 'g ..
3 250 i i. .; Turn
3 -. Travel
2 200 _ Time
p
E
E 150 1 —I—Predicted
l- Right-
0
c» 100 - Turn
‘8 Travel
0
5 Time

50 -
0

Departure Time Period (minutes)

Figure 6.45 — Pattern of Actual versus Predicted Travel Time for Right-Tum Movement

on Testing Set Arterial 6-10.

117

Link 16-17 Link 17-18 Link 18—19 Link 19-20

 

 

 

300
E 250 ' - - - - Actual
5 Right-
3 Turn
3 200 4 Travel
a, .
.§ Time
': 150 i
“>’
g —I—Predicted
'; 100 — Right-
ui Turn
g Travel
5 50 1 Time

0

 

 

 

 

Departure Time Period (minutes)

Figure 6.46 — Pattern of Actual versus Predicted Travel Time for Right-Tum Movement

on Testing Set Arterial 16-20.

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

 

 

300
A > 7" ‘
3 250 '5- - -- - Actual
§ . . .. 1Bight-
.' 1" ,, ;- um
i 200 i - " -. . ', Travel
2 - .. 1. ' Time
: 150 2
E . ' —I—-Predicted
'- 2 ' Right-
3 100 Turn
2 Travel
9 Time
< 50 ,
0

 

 

 

 

 

 

Departure Time Period (minutes)

Figure 6.47 — Pattern of Actual versus Predicted Travel Time for Right-Turn Movement

on Testing Set Arterial 21-25.

118

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

 

600

500 . - - -- - Actual
I 7 Right-
.'. Turn

400 - :2 :2 Travel

.1 , :1 Time

+ Predicted

200 a Right-
Turn
Travel

Average Travel Time (seconds)
0)
O
O

 

 

 

 

 

 

 

Departure Time Period (minutes)

Figure 6.48 — Pattern of Actual versus Predicted Travel Time for Right-Tum Movement

on Testing Set Arterial 3-23.

Across each arterial for which right-tum travel time is measured, drastic variations are
observed with sharp increases and decreases in actual travel time. The variability of
turning movements along with permitted right—turn—on-red and interference from through
vehicles causes a highly unpredictable scenario for right—tum travel time. For each
arterial link, the pattern of predicted travel time does follow that of the actual travel time
in a general sense. However, when sudden changes occur in the actual average travel
time, the model is not able to respond adequately. As a result, the actual travel time
oscillates higher and lower while the predicted travel time maintains a more uniform
slope, with minor oscillations. This is especially seen on arterial 16-20.

In order to improve the predictive capability of the SSNN-Right model, multiple time-

steps must be incorporated into the memory of the model. This will allow the model to

119

store more information regarding the change in travel time due to sudden changes in the
state of the arterial link. The model developed in this thesis is unable to react to rapid
changes in traffic conditions that are seemingly inherent with right-tum movements under
the conditions of the network implemented. The influence of variable turning
movements is severe for right-tum travel time prediction, and this influence carries into
the accuracy of travel time prediction for arterial routes.

An additional measure that may improve the performance of the SSNN-Right model may
be to decrease the duration of the time-step from 5 minutes to 1 minute. This would
improve the temporal resolution of the model, and may capture more of the variations
observed in right-tum travel time. It is possible that the aggregation of data in 5-minute
intervals causes an averaging effect in which finite causes of change in right-tum travel
time are lost.

In conclusion of this section, the results observed for the accuracy of travel time
prediction for the right-turn movement are compared to the results obtained by Singh
(2006) under the assumption of constant turning movement percentages. This

comparison is shown in Table 6.8.

Table 6.10 — Mean Absolute Percentage Error Comparison for Right-Tum Movement

Under Constant and Variable Turning Movement Percentage Conditions.

 

 

 

 

 

 

 

 

Movement Constant Variable
Turning % Turning %
é Training Left-Turn 9.6 33.2
E”.
g Testing Left-Turn 15.9 30.4

 

120

 

Based on this comparison, the incorporation of variable turning movements is obviously
detrimental to the performance of the travel time prediction model for the right-tum
movement. The magnitude of the MAPE is 2 to 3 times greater than for the results
obtained assuming constant turning percentages. Actual average travel time for the right-
tum movement changes rapidly and frequently due to the variability in traffic conditions,
and the SSNN model is unable to predict these changes accurately based on the condition
of each arterial link in the current time period. The addition of multiple time-steps to
the learning process of the model may improve the flexibility and reactive capability of
the model. Additional research is needed to identify traffic related variables that better

model the dynamic changes in right-tum travel time.

6.4 Travel Time Prediction on Arterial Routes

The performance of travel time estimation and prediction models is examined in great
detail in reference to arterial links. The state of traffic on each link influences the travel
time on that link, and in congested conditions can impact the travel time on adjoining
links. Travel time on an arterial link is the fundamental unit that contributes to the
overall travel time along an arterial or arterial route, so the accuracy of the models
presented in this thesis are analyzed first at the most fundamental level. In terms of real-
world applications though, the utility of travel time estimation and prediction models is in

determining the travel time along a route within the network of study. In general, system

121

 

users want to know how long it will take to get from origin to destination, not how long it
will take to get form intersection to intersection along the way.

In order to analyze the performance of the SSNN-Through, SSNN-Left, and SSNN-Right
models in reference to an arterial route, two routes are selected within the network as
defined in Figure 5.5 of Section 5.4. The appropriate model is applied to predict the
travel time for each arterial link depending on the movement at the downstream

intersection approach, and the results are aggregated for the length of the route.

6.4.] Results of SSNN Travel Time Prediction Models for Arterial Route 16-25

In Figure 6.49 the actual and predicted average travel time for route 16-25 are plotted
over time for the duration of the study period. The travel times shown represent the time
to traverse the route when departing during the current time interval.

Overall, the predicted travel time follows closely with the pattern of the actual average
travel time, especially from a departure time period of 55 to 90 minutes. There are two
prominent peaks in the actual travel time that are not well predicted by the SSNN models.
In order to gain a better understanding of the cause of the error at departure times of 15
minutes and 30 minutes, a detailed plot of actual versus predicted travel time is generated

respective to each link comprising the route. This plot is displayed in Figure 7.50.

 

850 1
750 . "
— ~ -~ ~ Actual
650 4 Travel
Time
(seconds)

+ Predicted
Travel
Time
(seconds)

.5

U"

Q
I

 

 

Average Travel Time (seconds)
01
0'!
O

00

01

O
l

 

N
0'!
O

l I l l E I I l l I I j

0152025303540455055606570758085909

Departure Time-Period (minutes)

 

5 1

o +—-—

Figure 6.49 — Pattern of Actual versus Predicted Travel Time for Arterial Route 16-25.

Link 16-17 Link 17-18 Link 18-23 Link 23-24 Link 24-25

 

 

 

450
400 i I
a E
“g 350 . 2' ------ Actual
8 :1 Travel
3 300 ~ .1 Time
3' ". (seconds)
,§ 250 -
|._
g 200 ~
; —I—Predicted
g, 150 - Travel
2 , Time
it 100 - h- -- M (seconds)

 

504

 

 

 

 

 

5 95 5 95 5 95:5 95: 5 95

Departure Time Period (minutes)

Figure 6.50 — Pattern of Actual versus Predicted Travel Time for Arterial Route 16-25.

123

In general for each arterial link the predicted travel time follows closely with the actual
travel time. This is especially true for links 16-17, 23-24, and 24-25. These links are
defined by a through movement along the route, so it is logical following the results
observed for the SSNN-Through model that travel time prediction remains accurate for
these links. Link 18-23 is defined by a left-turn movement on this route, and the results
here follow the general observations for the SSNN-Left travel time prediction model in
that the predicted travel time is slightly lower and lags behind changes in the actual travel
time. A right-tum movement defines link 17-18, and the actual travel time oscillates
while the predicted travel time steadily follows the general pattern with significant error.
Moreover, the peaks in the actual travel time observed for link 17-18 correspond to
departure times of 15 minutes and 30 minutes. These spikes in actual travel time for the
right-tum movement on arterial link l7-18 cause the spikes observed for these departure
times in Figure 6.49. Therefore, the error in the travel time prediction for route 16-25 is
caused primarily by the error in right-tum travel time prediction.

In analyzing travel time in terms of arterial links significant percentage errors are
observed for the left-turn and right-tum prediction models tested in this thesis. More
importantly, the error observed in right-tum travel time prediction is seen to propagate
into the accuracy of travel time prediction for an arterial route. Despite average errors in
excess of 30 percent for the right-tum movement, the mean absolute percentage error for
the predicted travel time along route 16-25 is only 8.9 percent. A breakdown in the
percentage error for the route by departure time is shown in Figure 6.51.

Given that the average actual travel time for this route is approximately 9.5 minutes, the

average error in travel time prediction is 50.5 seconds. For an arterial link that is

124

 

approximately 3.5 miles in length, an error in travel time prediction of less than one
minute is promising for the application of SSNN prediction models for Intelligent

Transportation System applications.

]

:3 30
E 25 i
in
a: .
a: . IlAnenal
g 20 Route
3 16-25
3 15 -
n.
3
2 10(
o
in
a
< 51

0 _

 

51015 20 25 30 35 4O 45 5O 55 60 65 70 75 80 85 90
Departure Time-Period (minutes)

Figure 6.51 — Absolute Percentage Error of Travel Time Prediction for Arterial Route

16-25.

6.4.2 Results of SSNN Travel Time Prediction Models for Arterial Route 16-3 I

In Figure 6.52 the actual and predicted average travel times for route 16-3 are plotted
over time for the duration of the study period. Route 16-3 does not contain any right-tum
movements, and there are no significant discrepancies observed between the actual and
predicted travel times for the route. The model accurately follows the changes in travel
time over the duration of the study period; however, underestimates the travel time as the

volume of traffic in the network builds and the actual average travel time increases.

125

 

The cause for the underestimation in travel time is seen in Figure 6.53 where the model

does not respond to a sudden increase in the actual travel time on link 8-3.

 

 

 

 

950 a.
l ,
850 ‘ ,,
8 750 1 —~-—— Actual
3 . Travel
2: Time
E 650 9 (seconds)
'—
3'
5 550 '1
; ,_ . \ —I—Predicted
g, ’ — Travel
2 450 4 Time
2 . (seconds)
350 l
250 I; I . I I ’— I l I I I I I I

O 5101520253035404550556065707580859095
Departure Time-Period (minutes)

Figure 6.52 — Pattern of Actual versus Predicted Travel Time for Arterial Route '16-3.

The overall MAPE of travel time predictions for arterial route 16-3 is 8.1 percent, which
is slightly lower than for arterial route 16-25. This is a positive result that shows the
capability of the model to predict travel time equally well on multiple routes. The

average percentage error is shown for the route by departure time in Figure 6.54.

126

Average Travel Time (seconds)

350 —

300

250

200 -

150 4

100 ,

50

 

Link 16-17 Link 17-18

Link 18-13 Link 13-8 Link 8-3

 

 

 

 

 

O
5

 

 

 

955

Departure Time Period (minutes)

- - -- - Actual

Travel
Time
(seconds)

—-—— Predicted

Travel
Time
(seconds)

Figure 6.53 — Pattern of Actual versus Predicted Travel Time for Arterial Route 16-3.

Absolute Percentage Error (%)

16'
14:

12i

 

A

iiiiiiJi iiiiiil

51015 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Departure Time-Period (minutes)

I Arterial
Route
16-3

Figure 6.54 — Absolute Percentage Error of Travel Time Prediction for Arterial Route

16-3.

6.4.3 Results of SSNN Travel Time Prediction Models for Arterial Route 15-2

The routes discussed in thus far in this chapter are routes that are comprised of arterial
links that are part of the datasets used to train and test the SSNN models. These models
have proven here to be accurate within a 10 percent average error in predicting travel
time over the length of a route. It is important however, to validate the accuracy of these
models on a route that is unique and separate from those used to train the SSNN models.
Figure 6.55 depicts arterial route 15-2, which contains a series of through, left-tum, and

right-tum movements across a set of arterial links that have not been utilized in the model

 

development procedure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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128

In Figure 6.56 the actual and predicted average travel times for route 15-2 are plotted

over time for the duration of the study period.

1050 T
950 7
1
850 i Asia.
Travel
750 Time
(seconds)

   

 

Average Travel Time (seconds)
03
01
O

 

550 .; +Predicted
I Travel
Time
450 ‘ (seconds)
i
l
350 i
|
250 T ‘T T T T T V F T T T T I l

0 5 1015202530354045505560657075808590 95
Departure Time-Period (minutes)

Figure 6.56 — Pattern of Actual versus Predicted Travel Time for Arterial Route 15-2.

For route 15-2 3 very similar pattern is observed as for route 16-3, where the predicted
travel time increases parallel with the increase of actual travel time. For all cases though,
the travel time prediction is below the actual travel time. At a departure time of 70
seconds, the error in travel time estimation for this route is nearly 3 minutes. A more
detailed plot of the travel time on each link comprising this arterial route is shown in
Figure 5.57. Again, for the link defined by the right-tum movement, link 15-14, the
errors in travel time prediction are significant. The difference between predicted and
actual travel time for the right-tum movement on link 15-14 causes the error in travel

time prediction for the route as a whole.

129

 

Link 15-14 Link 14-9 Link 9-4 Link 4-3 Link 3-2

 

300
a 250 — F
2 ' ------ Actual
8 Travel
3 200 ~ Time
3’ (seconds)
.§
t; 150 —
2
E —I—Predicted
a. 100 — Travel
8’ Time
h
2 (seconds)
<

01
O
1

 

 

 

 

 

 

 

 

 

5 95 5 95 5 95 5 95 5 95

Departure Time Perid (minutes)

Figure 6.57 — Pattern of Actual versus Predicted Travel Time for Arterial Route 15-2.

Overall, the predicted travel time has a MAPE of 8.5%. It can be concluded that the
performance of the SSNN models is validated, as the accuracy of travel time predictions
is similar for this case where all arterial links comprising the route are unique to the
training and testing datasets used to train the models. However, improvements in the
modeling topology are needed to further improve the operation of the models for reliable
implementation in ITS technologies, especially in the case of right-tum travel time
prediction. The average percentage error is shown for the route by departure time in

Figure 6.58.

130

18 i

16 ~

14 4 -

12 ‘ lArterial
10 _ Route

Absolute Percentage Error (%)

 

51015 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Departure Time-Period (minutes)

 

Figure 6.58 — Absolute Percentage Error of Travel Time Prediction for Arterial Route

15-2.

131

Chapter 7

Conclusions and Future Research

Travel time estimation and prediction are important components of Intelligent
Transportation Systems (ITS) applications that communicate travel conditions to system
users and administrators. The objective of this thesis is to develop State-Space Neural
Network (SSNN) models that accurately estimate and predict travel time for links and
routes in a Signalized urban arterial network. The research presented is an advancement
of the work completed by Singh (2006) to develop such models.

The modeling procedure incorporated in this thesis involves the introduction of variable
tuming movement percentages to the training and testing of various SSNN models. The
operation of Signalized intersections in an arterial network is a fundamental determinant
of the state of the network. Therefore, it is paramount to consider variations in vehicle
turning movements over time and space within the network. The results discussed
present findings regarding the accuracy of the models and the influence of variable

turning movement percentages on travel time estimation and prediction.

7.] Key Findings and Conclusions

The significant findings from this research on the influence of variable turning

movements on travel time estimation and prediction are as follows:

132

 

Traffic operations in a Signalized urban arterial network are complex and
dynamic. Travel time estimation and prediction must be done using modeling
techniques that are capable of learning the dynamic relationship between travel
time and the state of an arterial link. State-Space Neural Networks have been
proven to be effective in capturing non-linear spatiotemporal changes in traffic
conditions.

Three SSNN models are developed for travel time estimation respective to right-
tum, left-tum, and through vehicle movements on arterial links. Three SSNN
models for travel time prediction are developed following the same framework
including an input layer, a hidden layer, a context layer, and an output layer. The
context layer provides short-term memory for the models to learn the dynamic
relationship between traffic conditions and travel times for each movement.
Travel time estimation and prediction models should be driven by input variables
that are easily collected in the field using existing surveillance infrastructure. The
findings of this thesis prove that variables such as queue length, average speed,
and flow rate are beneficial in travel time estimation and prediction.

Data used to train and test SSNN models for travel time estimation and prediction
must replicate realistic conditions in an urban arterial environment. Turning
movement profiles are not constant within urban arterial networks, and so
simulations used to generate traffic related data should incorporate variable

turning movements.

133

5. Variable turning movement percentages have a significant influence on the
variability of traffic conditions, and therefore the accuracy of travel time
estimation and prediction models.

6. The Mean Absolute Percentage Error (MAPE) for travel times estimated and
predicted using the SSNN model framework developed ranges from 9.0 percent
for the through movement to 39.1 percent for the right-turn movement. The
influence of variable turning movements degrades right-tum travel time
predictability such that the right-turn models are not adequate for ITS
applications.

7. The MAPE for three arterial routes, including one route that is independent of the
datasets used to train and test the SSNN models is between 8.0 and 9.0 percent.
Significant positive and negative errors in travel time prediction for arterial links
cancel out in some cases over the length of an arterial route. Despite the positive
results obtained for arterial route travel time prediction, the errors in link travel
time prediction have a significant impact on the reliability of the SSNN models.

8. The pattern of predicted travel times follows that of the actual travel time over the
duration of the study period for the through movement. The pattern of predicted
travel times for the left-tum movement lags one to two time-steps behind changes
in the actual average travel time. This often results in travel time predictions that
are lower than the actual travel time. Actual travel times for the right-tum
movement are less uniform and more unpredictable. The performance of travel
time prediction models for the right-tum movement often does not accurately

capture changes in actual travel time.

134

 

7.2 Limitations

The limitations of the modeling approach presented in this thesis are as follows:

1.

The proposed methodology does not explicitly account for non-recurring traffic
influences such as weather and incidents. The SSNN models must be able to
adapt to these occurrences in order to be successful in widespread deployment.
The travel time estimation and prediction models proposed are developed based
on data generated from a microscopic simulation that are not calibrated with
actual field data. The implementation of these models in a field test is necessary
to evaluate the readiness of the model for real world ITS applications.

Vehicles completing a left-tum movement must share the rightmost lane with
through moving traffic. The interaction between vehicles coupled with possible
right-tum on red movements causes increased variability in actual travel time for
right-turns. The intersections in the simulation network do not have right-turn
only lanes, which may significantly alter the predictability of right-turn travel
time.

The intersection control scheme utilized in the microscopic simulation network
includes pre-timed signal control only. The variability of actuated signal control

is not accounted for, but may be realistic for modern urban arterial networks.

135

 

 

7.3 Scope of Future Research

Based on the findings and limitations of the work presented in this thesis, the following

points are recommended to further the success of research in this area:

1.

The factors that influence right-turn speed must be better understood and.
modeled. Research is necessary to determine the impact of right-tum on red
permission on right-tum travel time. Additionally, the impact of right-tum only
lanes should be investigated.

Additional research should be performed to identify additional candidate variables
that may improve the accuracy of the SSNN models. There may be additional
variables that can easily be collected in the field that are not included in this
thesis.

State-Space Neural Network models must be advanced to incorporate multiple-
step learning. This may improve the ability of the SSNN models to sense and

react to rapid changes in traffic conditions.

. The time-step resolution of the SSNN models should be further researched.

Models with a shorter time-step and higher resolution may better capture the

variability in travel time caused by variable turning movements.

. The impact of actuated signal control should be investigated for the accuracy of

travel time estimation and prediction models.

136

 

APPENDIX

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(‘Prediction of TT Thru');

load traininoutm ; %loading input file
P = traininout(:,:);

P = rot90(P,l);

ptr3 = P( 2:10,:);ptr3 = con25eq(ptr3); % link input
ttr3 = P( l ,:);ttr3= con25eq(ttr3); % output
for i = 1:576

ptr{l,i} = ptr3{l,i};

end

for i = 1:576

ttr{l,i} = ttr3{1,i};

end

% Loading up State Space Neural Network called here as netl
disp('Loading up network ...');

load netl;

netl = init(netl); % Initializing the network

137

 

% Training Parameters

netl.adaptParam.epochs = 20000;

netl .adaptParam. goal = 0.001 ;

netl .adaptParam.max_fail = 10000 ; % Five Validations to check when error rises

netl .adaptParam.mem_reduc = l; % Full Jacobian Calculated, no memory restrictions
netl.adaptParam.min_grad = le-10; r.
netl .adaptParam.mu = 0.9;

netl .adaptParam.mu_dec = 0.001;

 

netl.adaptParam.mu_inc = 10; '
netl .adaptParam.mu_max = l e 1 0;

net] .adaptParam.show = 100;

netl .adaptParam.time = inf;

netl.adaptparam.lr = 0.1; % Learning Rate

netl .perforchn = 'mse'; % Performance = 'Mean Square Error'
% Training

disp('Training ...');

[net1,tr]= adapt(net1,ptr,ttr);

save netl .MAT netl ;

%Conversion of ttr to concurrent matrix

ttr = seq200n(ttr);

% Results - Network Training

32 = sim(netl ,ptr);

a2 = seq2con(a2);

138

figure(2)

title('Regression Analysis of Training Results: Travel Time Thru'); hold on
[m,b,r] = postreg(a2{ 1,1 },ttr{ 1,1 }); hold off
perftrg1= mse(a2{l,l} - ttr{l,l})

perftrg2 = mae(a2{l,1} - ttr{ 1,1 })

% Testing the network

load testinoutm; %loading testing file
Q = testinout(:,:);

Q = rot90(Q,1);

test3 = Q(2:10,:); test3 = con2seq(test3);

for i = 1:576

test.P{ l,i} = test3 {1,i};

end

testT3 = Q(1,:);testT3 = con25eq(testT3);
for i = 1:576

test.T{ l,i} = testT3{1,i};

end

test.T = seq2con(test.T);

% Results - Network Testing

33 = sim(net1,test.P);

a3 = seq2con(a3);

figure(5)

title('Regression Analysis of Testing Results: Travel Time Thru'); hold on

139

 

[m,b,r] = postreg(a3 { 1,1 },test.T{1,1}); hold off
perfistl = mse(a3{l,l} - test.T{l,l})

perftst2 = mae(a3 { 1,1} - test.T{ 1,1 })

format short;

disp('Network Architecture');

disp('Number of Layers');

disp(netl .numlayers+1);

nneurons = 7;

slayer = size(netl .layers);

 

for i=1:slayer(l,1)

nneurons =[nneurons netl .layers {i} .size];

end

% forming training and testing output files to save

trgoutttthru = a2 { 1 }';

testoutttthru = a3 { l }';

save trgoutttthrudat trgoutttthru -ASCII -DOUBLE -TABS; % saving output
in .dat files.

save testoutttthrudat testoutttthru -ASCII -DOUBLE -TABS;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

140

 

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