ABSTRACT 
 
 
 
 
The Title of Document: ARTERIAL PROBABILISTIC TRAFFIC 
MODELING AND REAL-TIME TRAVEL 
TIME PREDICTION WITH VEHICLE 
PROBE DATA USING MACHINE 
LEARNING  
  
 Bahar Zarin, Ph.D., 2018 
  
Directed By: Prof. Ali Haghani, Department of Civil and 
Environmental Engineering 
 
 
 
This study proposes a probabilistic modeling framework for the estimation and 
prediction of link-based arterial travel time distribution using GPS data. The 
spatiotemporal correlations of the network are modeled using a directional acyclic 
graphical model, and several external variables in the prediction model are included to 
yield a better prediction in a variety of situations. This study also aims to investigate 
the effects of each factor on the travel time and the uncertainty associated with it.  
In the proposed model, factors such as weather conditions, seasons, time of day, and 
day of the week are added as external variables in the graphical model. After 
determining the structure of the model, Streaming Variational Bayes (SVB) is used for 
training and parameter inference; this offers a valuable option when constant streaming 
data is utilized. SVB adaptively changes its parameters gradually with a lower 
computational cost, which makes the process less time-consuming and more efficient. 
  
The analysis shows that incorporating external variables can improve the model 
performance. 
The data used in this study is INRIX vehicle trajectory raw data from four months – 
February, June, July, and October of 2015 – which makes it possible to take into 
account the effects of seasons and weather conditions on travel time and its uncertainty. 
One of the products of this study is a framework for vehicle trajectory data cleaning 
process including trip identification, removing outliers, and cleaning the trips data. 
Once the data are cleaned and ready to use, they should be mapped to the roads. The 
Hidden Markov Model (HMM) map matching algorithm is used to map the GPS 
latitude/longitude data to the Open Street Map (OSM) base map and find the traversed 
links between each pair of GPS points of vehicle trajectories.  
Finally, a novel procedure to compare any travel time prediction model with any 
available commercial routing API is proposed and tested to compare the proposed 
model with Google API.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
ARTERIAL PROBABILISTIC TRAFFIC MODELING AND REAL-TIME 
TRAVEL TIME PREDICTION WITH VEHICLE PROBE DATA USING 
MACHINE LEARNING 
 
 
 
By 
 
 
Bahar Zarin 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2018 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Ali Haghani, Chair 
[Professor Larry S. Davis] 
[Professor Paul M. Schonfeld] 
[Dr. Mark Austin] 
[Dr. Cinzia Cirillo]  
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
© Copyright by 
Bahar Zarin 
2018 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
Preface 
This dissertation is composed as a requirement for the degree of Doctorate in 
Philosophy in Civil and Environmental Engineering. 
 
 
 
 
 
 
 
 
 
 
 
 
  
ii 
 
 
 
Dedication 
To my parents, sister, and brothers for their endless love, support, and 
encouragement. 
  
iii 
 
 
 
Acknowledgements 
First and foremost I would like to express my deepest gratitude to my advisor, Dr. Ali 
Haghani who has been a tremendous mentor and supported me both academically and 
personally during my Ph.D. study. I am extremely grateful for all his contributions of 
time, idea, and endless support throughout my Ph.D. research and dissertation. It has 
been the greatest honor to learn from him and be his student.  
Very special thanks to my committee members, Dr. Paul M. Schonfeld, Dr. Larry S. 
Davis, Dr. Cinzia Cirillo, and Dr. Mark Austin for their time, support, and their brilliant 
comments and suggestions to improve my research. It is an absolute honor to have them 
as my committee members.  
I also would like to thank the Center for Transportation Technology (CATT) for giving 
me the opportunity to work on interesting projects. I learned a lot during my two years 
of research there. Additionally, I like to thank the Maryland State Highway for 
providing the data used in this dissertation.  
I am grateful for friendship and encouragement of my graduate research group for the 
tremendous learning experience I gained by being in the group and learning from their 
research. Special thanks to Mahyar for his support and insight during my dissertation. 
I would also like to thank all my friends: Kiana, Ali, Elham, Mahshid, Ali, Elham, 
Samira, Babak, Soroush, Marjan, Mostafa, Sepehr, Ladan, Niloofar, Pouya, Aida, 
Azadeh, Marjan and list goes on, for making my life beautiful and being like a family 
to me.  
Finally, I would like to dedicate my dissertation to my parents, my brothers and my 
sister. Words cannot express how grateful I am for all of the sacrifices they have made 
iv 
 
 
 
for me to get here and all the support and endless love they gave me in my entire life, 
thank you!  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
v 
 
 
 
Table of Contents 
Preface ...................................................................................................................... ii 
Dedication ................................................................................................................ iii 
Acknowledgements .................................................................................................. iv 
Table of Contents ..................................................................................................... vi 
List of Tables ........................................................................................................... ix 
List of Figures ........................................................................................................... x 
Chapter 1: Introduction ............................................................................................. 1 
1.1 Problem Statement .......................................................................................... 3 
1.2 Research Objectives ........................................................................................ 5 
1.3 Research Contributions .................................................................................... 8 
1.4 Research Outline ............................................................................................. 9 
Chapter 2: Literature Review................................................................................... 11 
2.1 Parametric Models ......................................................................................... 12 
2.1.1 Naïve Models ......................................................................................... 12 
2.1.2 Autoregressive Models ........................................................................... 12 
2.1.3 State Space Model .................................................................................. 13 
2.1.4 Hidden Markov Model (HMMs) ............................................................. 14 
2.1.5 Bayesian Networks (BNs) ....................................................................... 14 
2.1.6 A Feedforward Neural Network .............................................................. 15 
2.2 Non-Parametric Models ................................................................................. 16 
2.2.1 K-nearest neighbors ................................................................................ 16 
2.3 Other Methods ............................................................................................... 16 
2.4 Arterial Travel Time ...................................................................................... 17 
2.5 Intersection Delay.......................................................................................... 21 
2.6 Probe Data ..................................................................................................... 23 
2.7 Summaries and Limitations of Previous Studies............................................. 23 
2.7.1 Limitations of Previous Studies............................................................... 25 
Chapter 3: Model Preparations ................................................................................ 27 
3.1 Data Cleaning ................................................................................................ 28 
3.2 Probe Filtering ............................................................................................... 29 
3.2.1 Trip Splitter ............................................................................................ 32 
3.2.1.1 Time Difference Splitter ...................................................................... 32 
3.2.1.2 Way Point Sequence Splitter ................................................................ 33 
3.2.2 Cleaning/Filtering the Trips .................................................................... 33 
3.2.2.1 Same Time Filter ................................................................................. 33 
3.2.2.2 Same Location Filter ............................................................................ 34 
3.2.2.3 Speed Boundary Filter ......................................................................... 34 
3.2.2.4 Idle Filter ............................................................................................. 34 
3.2.2.5 Location Displacement Filter ............................................................... 35 
3.2.2.6 Trip Length Filter ................................................................................ 35 
3.2.2.7 Large Error Size Filter ......................................................................... 36 
3.2.2.8 Shaking Probe Filter ............................................................................ 36 
3.2.3 An Examples .......................................................................................... 36 
vi 
 
 
 
3.3 Map Matching ............................................................................................... 37 
3.3.1 HMM-based Map Matching .................................................................... 38 
3.3.1.1 Measurement Probabilities ................................................................... 39 
3.3.1.2 Transition Probabilities ........................................................................ 39 
3.3.1.3 Optimum Match ................................................................................... 40 
3.3.2 Map Matching Assumptions ................................................................... 40 
3.4 Graphical Model ............................................................................................ 41 
3.4.1 Acyclic Graphical Models....................................................................... 42 
3.4.2 Model Assumptions ................................................................................ 43 
3.4.3 Implemented Graphical Model ................................................................ 44 
3.4.4 Creation of the Network .......................................................................... 46 
3.5 Data ............................................................................................................... 47 
3.5.1 Study Area .............................................................................................. 47 
3.5.2 Open Street Map (OSM) ......................................................................... 48 
3.5.3 INRIX Data ............................................................................................ 49 
3.5.4 Weather Data .......................................................................................... 50 
3.6 Summary ....................................................................................................... 51 
Chapter 4: Graphical Model .................................................................................... 53 
4.1 The External Variables. ................................................................................. 53 
4.1.1 Weather Conditions ................................................................................ 53 
4.1.2 Seasons ................................................................................................... 54 
4.1.3 Day of the Week ..................................................................................... 55 
4.1.4 Time of Day ........................................................................................... 55 
4.1.5 The External Variables ........................................................................... 56 
4.2 Graphical Model with External Variables ...................................................... 57 
4.3 Travel Time Allocation.................................................................................. 59 
4.4 Parameter Estimation ..................................................................................... 61 
4.4.1 The Inference .......................................................................................... 61 
4.4.2 Streaming Variational Bayes (SVB) ........................................................ 63 
4.5 Summary ....................................................................................................... 66 
Chapter 5: Analysis ................................................................................................. 68 
5.1 Case Study .................................................................................................... 68 
5.1.1 Trip Summaries ...................................................................................... 70 
5.2 Developed Models ......................................................................................... 71 
5.2.1 The Initial Developed Graphical Model .................................................. 71 
5.2.2 Baseline Model ....................................................................................... 72 
5.2.3 Test ........................................................................................................ 73 
5.3 Validation of the Proposed Model.................................................................. 74 
5.3.1 The Model without any Variables ........................................................... 74 
5.3.2 The Model with Day of the Week ........................................................... 75 
5.3.3 Travel Time Distribution ........................................................................ 76 
5.4 Model Improvements ..................................................................................... 78 
5.4.1 The Model with All of the External Variables ......................................... 78 
5.4.1 Regression Model ................................................................................... 82 
5.4.2 Comparison Between the Final Model and Improved Base Model........... 84 
5.4.3 Comparison Between the Two Graphical Models.................................... 90 
vii 
 
 
 
5.4.4 The Model Test on a Rainy Day.............................................................. 91 
5.4.1 The Time of Day..................................................................................... 93 
Other Modifications ............................................................................................ 95 
5.4.2 The Graphical Model with Intersection ................................................... 95 
5.4.3 The Graphical Model with Driver Habits Variable .................................. 99 
5.5 Summary ..................................................................................................... 101 
Chapter 6: Comparison and Sensitivity Analysis ................................................... 103 
6.1 Validation .................................................................................................... 103 
6.2 Shape-Matching .......................................................................................... 104 
6.2.1 Hausdorff Distance ............................................................................... 105 
6.2.2 Fr´echet Distance .................................................................................. 106 
6.3 Route Similarity .......................................................................................... 107 
6.4 Google Maps API ........................................................................................ 109 
6.4.1 General Information .............................................................................. 109 
6.4.2 Departure Time ..................................................................................... 109 
6.4.3 Travel Time with Traffic....................................................................... 110 
6.5 Waypoint  Parameter ................................................................................... 111 
6.6 Comparison ................................................................................................. 111 
6.6.1 Case Studies ......................................................................................... 112 
6.6.2 Results .................................................................................................. 113 
6.6.2.1 Off-peak Travel Time Comparison..................................................... 113 
6.6.2.2 PM peak Travel Time Comparison ..................................................... 119 
6.7 Sensitivity Analysis ..................................................................................... 121 
6.7.1 Number of States .................................................................................. 121 
6.7.2 Number of Observations ....................................................................... 124 
6.8 Summary ..................................................................................................... 127 
Chapter 7: Summary and Conclusion .................................................................... 129 
7.1 Summary ..................................................................................................... 129 
7.2 Conclusions ................................................................................................. 134 
7.3 For Future Research .................................................................................... 135 
Bibliography ......................................................................................................... 138 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
viii 
 
 
 
 
List of Tables 
Table 1 Summary of trips used for each month and each day of the week in the 
training data set ....................................................................................................... 71 
Table 2 Summary of GPS points used for each month and each day of the week in the 
training data set ....................................................................................................... 71 
Table 3 Accuracy of predictions for all the weekdays .............................................. 74 
Table 4 Accuracy of predictions based on days of the week for the Baltimore test data
 ............................................................................................................................... 75 
Table 5 Summary of winter predictions based on days of the week for the Baltimore 
test data ................................................................................................................... 85 
Table 6 Summary of summer predictions based on days of the week for the Baltimore 
test data ................................................................................................................... 87 
Table 7 Summary of fall predictions based on days of the week for the Baltimore test 
data ......................................................................................................................... 88 
Table 8 Aggregated average percentage of error for different season ....................... 89 
Table 9 Aggregated average percentage of error for each days of week ................... 89 
Table 10 The comparison of the model with and without weather variable .............. 92 
Table 11 Summary of winter predictions based on days of the week for daytime ..... 94 
Table 12 Google and proposed travel time prediction summary (off-peak) ............ 117 
Table 13 Sample of Google, proposed model travel time prediction and observations 
travel time comparison (off-peak) ......................................................................... 118 
Table 14 Google, proposed model travel time prediction and observations travel time 
comparison (PM peak) .......................................................................................... 120 
Table 15 Sensitivity analysis on the number of states ............................................ 123 
Table 16 Sensitivity analysis on the amount of data ............................................... 125 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ix 
 
 
 
List of Figures 
Figure 1 The change in delay over time for Baltimore, Maryland and DC (Virginia, 
2017) ........................................................................................................................ 2 
Figure 2 The change in commute time over time for Baltimore, Maryland and DC 
(Virginia, 2017) ........................................................................................................ 2 
Figure 3 The conceptual framework of modeling the arterial travel time ................. 28 
Figure 4 Sample of trips in Telenav probe data ........................................................ 30 
Figure 5 Sample of trips in INRIX probe data ......................................................... 31 
Figure 6 The data cleaning Process.......................................................................... 32 
Figure 7 An example of Telenav trip before and after filtering ................................ 37 
Figure 8 An example of INRIX trip before and after filtering .................................. 37 
Figure 9 A Map matching result instance for Baltimore city .................................... 41 
Figure 10 Spatiotemporal model of arterial traffic evolution represented as a 
Graphical model. ..................................................................................................... 46 
Figure 11 Study area, downtown Baltimore ............................................................. 48 
Figure 12 The graphical model with the proposed external variables ....................... 59 
Figure 13 The available data on the area .................................................................. 69 
Figure 14 The available data for the case study area ................................................ 70 
Figure 15 Comparison of prediction error between proposed model and the baseline 
by days of the week ................................................................................................. 76 
Figure 16 The example link in downtown Baltimore ............................................... 77 
Figure 17 The travel time histogram for link 13 ....................................................... 78 
Figure 18 Real-time level of congestion subnetwork of Baltimore downtown. ......... 81 
Figure 19 Real-time relative travel time of Baltimore downtown network. .............. 82 
Figure 20 Comparison of prediction error between proposed model and base model in 
winter ...................................................................................................................... 86 
Figure 21 Comparison of prediction error between proposed model and base model in 
summer ................................................................................................................... 86 
Figure 22 Comparison of prediction error between proposed model and base model in 
fall .......................................................................................................................... 87 
Figure 23 Aggregated average percentage of error for different season .................... 89 
Figure 24 Aggregated average percentage of error for each days of week ................ 90 
Figure 25 Comparison of the proposed model with seasons and weather conditions 
and without them..................................................................................................... 91 
Figure 26 The comparison of the model with and without weather variable ............. 93 
Figure 27 Comparison of prediction error between proposed model and base model 
for day time in winter .............................................................................................. 95 
Figure 28 The intersection movement demonstration ............................................... 97 
Figure 29 The proposed graphical model including the intersections ....................... 98 
Figure 30 Demonstration of path travel time as the sum of links and intersections 
travel times ............................................................................................................. 99 
Figure 31 The proposed graphical model with proposed relative speed variable .... 100 
Figure 32 Hausdorff distance as a measure of similarity ........................................ 106 
Figure 33 Fr´echet distance as a measure of shape similarity ................................. 107 
x 
 
 
 
Figure 34 Route similarity index example (two routes with similarity index of 
70.05%) ................................................................................................................ 108 
Figure 35 The proposed framework for travel time estimation comparison between 
models .................................................................................................................. 112 
Figure 36 Proposed Model, Google API, and observed travel time comparison (off-
peak) ..................................................................................................................... 115 
Figure 37 Google and proposed model Percentage of error trend (off-peak) .......... 117 
Figure 38 Proposed Model, Google API, and observed travel time comparison (PM 
peak) ..................................................................................................................... 120 
Figure 39 Sensitivity analysis on the number of states ........................................... 123 
Figure 40 Sensitivity analysis on the amount of data ............................................. 126 
 
 
xi 
 
 
 
Chapter 1:  Introduction 
Having an accurate and reliable traffic information system is the first step toward 
achieving active congestion control and alleviation, and can help reach a reliable 
network. The traffic information could be used by both travelers and agencies. One of 
the primary uses of travel time information is pre-trip guidance for travelers. This 
information impacts travelers’ decisions regarding departure time, route, and mode 
choice. Travel time information can also be used by transportation agencies for 
Advanced Traffic Management System (ATMS) strategies, Emergency Transportation 
Operations, and Traffic Incident Management. Such applications are increasingly 
important in a world in which urban road transport systems experience increasing 
congestion year after year.  Figure 1 and Figure 2 each demonstrate the increase in the 
congestion level for several states and cities including Washington, D.C. and 
Baltimore, Maryland.  As shown, in these regions, average commute travel time and 
hours of travel delay have consistently increased over time in most cases. 
1 
 
 
 
 
 
Figure 1 The change in delay over time for Baltimore, Maryland and DC (Virginia, 2017) 
 
Figure 2 The change in commute time over time for Baltimore, Maryland and DC (Virginia, 2017) 
 
Arterials are major roads that carry large volumes of traffic each day. In fact, arterials 
serve as the primary route of travel in major cities and as the second leading route of 
travel between cities. Thus, it is critical that traffic information systems offer accurate 
2 
 
 
 
 
information regarding arterial travel time. One of the most important elements for 
providing accurate travel time information in general and, more importantly, for 
arterials is travel time reliability and uncertainty associated with travel time. No matter 
the degree to which influential factors are considered in prediction models, there is 
always a degree of uncertainty associated with travel time predictions. Nevertheless, 
information about travel time uncertainty can be used for risk-averse routing,  reporting 
travel time reliability to users, and for many other purposes.  
1.1  Problem Statement 
Historically, the use of traffic monitoring systems has been limited primarily to 
freeways. Arterial traffic monitoring is particularly challenging because arterials are 
not covered by dedicated sensing infrastructures. Probe vehicle data remains the main 
available source of data for arterials. Because more than 40 percent of U.S. vehicle-
traveled miles occur on arterial roads – and there are not as many studies in this realm 
– new studies to precisely estimate and predict arterial travel time are highly needed. 
Many travel time forecasting models – either for freeways or arterials –provide only 
expected travel time. Still, by modeling statistical distributions of travel times, rather 
than just mean values, there is an opportunity to provide travelers with travel time 
reliability (or probability of on-time arrival) in addition to travel time. This probabilistic 
forecast of travel time is beneficial for risk-averse routing. It could also be used to 
report travel time reliability to users. In this research, we study the probabilistic 
modeling of arterial travel time. 
3 
 
 
 
 
While there have been previous efforts to establish arterial traffic estimation/prediction 
methodologies, which are discussed in existing literature, modeling and estimating 
travel time remains a challenging task. To start, there is a great deal of variation 
associated with travel time caused by factors such as weather, demand uncertainty, 
incidents, and roadway conditions, as well as varying driver behavior. Such variations 
are more significant in arterials as opposed to other roadways because variations 
associated with traffic lights, stop signs, bus stops, left turns, etc. add a higher degree 
of variation. Since it is impossible to fully consider the effects of all of the influential 
factors in the modeling process, there is always some degree of uncertainty associated 
with travel time estimation/prediction. It is thus essential to model this uncertainty and 
establish an estimation of it and how it changes due to the myriad factors.   
Even though it is impossible to offer an exact estimation/prediction of travel time, 
incorporating as many influential factors into the modeling process as possible could 
produce a more accurate and more reliable estimation/prediction. Unfortunately, such 
factors are rarely considered in the modeling processes outlined in existing studies due 
to a lack of data. Low sampling frequency creates difficulties in inferring parameters. 
In fact, there is a direct relationship between the frequency of data and availability of 
data in different conditions, and the ability to include such data in the modeling process. 
This study thereby represents one of the only attempts to incorporate the effects of some 
of these aforementioned factors into the modeling process.  
4 
 
 
 
 
1.2  Research Objectives 
The main objective of this study is to provide probabilistic modeling of arterial travel 
time using GPS data, it addresses both travel time estimation/prediction as well as 
uncertainty associated with it. Beyond the variability inherent in any travel time 
estimate due to factors such as driver behavior, there is additional uncertainty 
associated with arterial travel time as the result of traffic lights, and/or stop signs, 
and/or left turns, etc. The proposed model considers some of these variabilities and 
incorporates them into modeling by adding them into the modeling process in efforts 
to yield a more accurate estimation of travel time in the arterials. Further, the proposed 
model aims to capture variability associated with travel time for arterials.  
Almost none of the existing travel time estimation/prediction modeling is 
comprehensive; in most cases, the proposed models center on normal weather 
conditions and specific time(s) of day and day(s) of the week. However, it is important 
to have an accurate travel time estimation/prediction that can be applied to a variety of 
scenarios, such as adverse weather conditions. Currently, even commercial 
estimation/prediction models lack this capability. As such, one of the primary 
objectives of this study is to propose a model that can provide an accurate travel time 
estimation/prediction in such conditions.  
Another important objective of this study is to evaluate the effects of incorporating this 
information in modeling performance. A better performing model with this information 
is needed to improve mobility and network accessibility and alleviate the cost 
5 
 
 
 
 
associated with unreliability. To do so, factors such as days of the week, and time of 
day impact on the travel time and its variability are evaluated.  
In addition to the primary objective of this study, another aim of this study is to provide 
a comprehensive procedure that can be used to conduct such analysis. Thus, for each 
step of this procedure, a separate framework is proposed and tested. The discussed 
procedure starts with the modeling pre-steps such as data cleaning, mapping vehicle 
trajectory data over a map of the observed region, and creating the city network. Then, 
developing a model with all of the proposed variables and testing it on a case study for 
several test days to compare it with other models. The final step is the creation of a 
framework to compare the proposed model with any commercial routing API.   
This study uses raw GPS trajectory probe data in order to consider real-world 
situations. As previously mentioned, preparation of this data is important to this study 
and applications to modeling. Establishing criteria for data selection and developing a 
process for identifying outliers within the data pose challenges. As such, this study 
proposes a comprehensive framework for cleaning the raw probe data. This framework 
includes identifying trips from device-based probe data and applying several filters to 
clean trips and make them ready to be used in the model. The framework could be used 
for any type of vehicle trajectory data. 
One of the important steps for conducting any modeling is validation in order to allow 
for comparisons with other models. This study uses a general method of validation that 
involves splitting the data into training and test datasets and validate the model on the 
test dataset. In addition, this study proposes a framework to compare the model with 
commercial navigation systems. It can be challenging to compare the proposed model 
6 
 
 
 
 
with existing commercial models, such as Google’s navigation system, since neither 
Google’s cost model nor its real-time data is available. As such, as an important follow-
up step for the proposed modeling procedure, this study outlines a new method for 
comparing the model and the trip travel times with a commercial routing API. 
In summary, this study includes the following tasks: 
(1) Propose a comprehensive framework for working with raw vehicle GPS trajectory 
data. The framework includes data selection, data cleaning (trip splitters, probe 
filtering, and removal of outliers), and finally, preparation of the data to be used in the 
training and test models. Furthermore, the proposed framework is tested on two 
different vehicle GPS trajectory datasets. 
(2) Propose a new model for real-time arterial travel time estimation/prediction 
utilizing Streaming Variational Bayes, which can be used with streaming data (for 
small-to-medium-sized networks).  
(3) Investigate the effects of different factors, such as weather conditions, days of the 
week, time of day, and seasons on arterial travel time. 
 (4) Propose a new arterial travel time estimation/prediction framework that can be 
applied in different weather conditions, days of the week, and time of day, and provide 
more accurate and reliable predictions for any conditions. Then, compare the 
performance to a base model as well as to a base model improved by a regression 
model.  
 
7 
 
 
 
 
(5) Propose a new framework for comparing estimation/prediction resulting from the 
discussed model (or any real-time travel time prediction model, in general) with any 
available routing API, such as Google routing API. 
1.3 Research Contributions 
What follows is a list of the primary contributions of this dissertation to the field.  
• We have proposed a framework for cleaning vehicle trajectory data which can 
be used for all types of data availability and is applicable to any study that 
utilizes real-world data.   
• We have proposed a graphical model that takes into account different external 
variables such as weather conditions, seasons, time of day, and day of the week. 
We proposed the use of the Streaming Variational Bayes to make it possible to 
use the graphical model for larger-scale networks and when a number of 
variables are at play. 
• We have evaluated the effects of incorporating each of these variables in the 
modeling and have addressed how such variables can impact the modeling 
process. We have done so by incrementally adding the variables into the model. 
• Finally, we have proposed a new procedure with which one can compare any 
type of travel time prediction model (for any kind of network: freeway, arterial, 
or both) to any commercial routing API, such as Google API. 
8 
 
 
 
 
1.4  Research Outline 
The rest of this dissertation is organized as follows. In the next chapter, we further 
review existing literature related to travel time estimation studies and the methods used 
in travel time estimation/prediction in general as well as in arterial travel time 
estimation/prediction. Chapter 3 mainly explains the preliminary steps taken before the 
modeling process. In this chapter, a new framework for a data cleaning and filtering 
process is proposed and tested on two different datasets. The chapter then addresses the 
next important step before the modeling – the map matching process used to map the 
cleaned data to the network. To do this, different map matching techniques and 
algorithms were reviewed; the map matching algorithm used in this study is discussed 
in greater detail. The last step before the modeling involves the creation of the roads 
network and capturing the roads’ dependencies. The graphical model and the 
relationship between variables are further explained to demonstrate how the city 
network should be created. 
Chapter 4 demonstrates the enhanced graphical model proposed in this study. All the 
variables considered in the model are explained in greater detail, as is the structure of 
the enhanced graphical model and the relationship between variables. The chapter 
continues by proposing using an approximate inference algorithm named Streaming 
Variational Bayes. This algorithm is used in order to make the model applicable for 
larger-sized networks and instances in which more variables need to be incorporated in 
the model. Chapter 5 outlines the performance of the proposed model with all the 
variables and compares the proposed model with an improved base model. Further, the 
9 
 
 
 
 
chapter analyzes the impact of including each of these variables in the model 
performance.  
Chapter 6 explains the new approach for comparing the proposed model (or any other 
model) with a commercial routing API when the associated cost model is not available. 
Finally, the last chapter includes the conclusion and possible future work for this study. 
In the last chapter, the summary of the research, the contributions, and the findings are 
discussed. The limitations of this study and suggestions for future work are also 
discussed in this chapter.   
 
 
 
 
 
 
 
 
 
 
 
 
10 
 
 
 
 
Chapter 2: Literature Review  
In this chapter, some of the general ideas and methods related to estimating travel time 
(both arterial and freeway) are presented. Following that, more detailed literature 
discussing the arterial travel time estimation/prediction, and the associated methods, 
assumptions, and data used are provided. Throughout this dissertation, other specific 
detailed literature is referenced when appropriate. 
There are different approaches that can be applied to the traffic estimation/prediction 
problem (both freeways and arterials). The approaches can be divided into three 
essential clusters: scope, output, and method (Vlahogianni et al., 2004). Regarding 
conceptual output, there are two main trends with different outcomes in the estimation 
of travel times using probe data. One trend is providing expected travel time. Most 
large-scale navigation systems, such as Apple Maps or Google Maps have this 
outcome, which can be applied to any large-traffic network. The other trend involves 
providing probability distributions of travel times and inferring their parameters using 
different statistical methods that can be used for small-to-medium-sized networks, as 
they are computationally intensive (Hunter, 2013). Studies on travel time 
estimation/prediction can also be divided based on their scope, be it microscopic or 
macroscopic. The microscopic modeling method mostly includes a small number of 
intersections and segments assuming complete data availability, including signal 
timing, vehicle counts, etc. In contrast, macroscopic modeling focuses on large scales, 
such as cities or countries. Another way to distinguish between studies on travel time 
estimation is by comparing the methods they use. There are different approaches to 
11 
 
 
 
 
model travel time, such as parametric, and non-parametric (Perry& Greene, 1982). This 
study focuses on those methods that use probe data. Given this, what follows are the 
methods used for travel time estimation, in general, as well as for arterial travel time 
estimation/prediction using probe data.  
2.1  Parametric Models  
The parametric models have strong assumptions about the data which should be 
satisfied, or they can result in inaccurate estimation. These models have a fixed number 
of parameters (Murphy, 2012). 
2.1.1  Naïve Models 
One of the simplest parametric models is the naïve model. The naïve models 
incorporate very simple methods such as historical averaging or smoothing and are 
considered the simplest forecasting models which are used in travel time estimation 
(Farokhi Sadabadi, Hamedi, & Haghani, 2010). Historical average models are merely 
averaging historical data with the assumption that the traffic repeats the same pattern 
over time and these patterns are used for future prediction (Smith & M. Demetsky, 
1997; Farokhi Sadabadi et al., 2010). 
2.1.2  Autoregressive Models 
A time series is defined as a sequence of measurements of the same variable(s) made 
over time intervals (day, week, etc.). The model in which a value from one-time series 
is regressed on its own past values from that time series is called the autoregressive 
12 
 
 
 
 
model. The order of an autoregression in a time series is defined as “the number of 
immediately preceding values used for present value prediction” 
(onlinecourses.science, 2016). In Autoregressive modeling, an autoregressive 
integrated moving average (ARIMA) approach – also referred to as Box-Jenkins– is 
one of the approaches for analyzing and forecasting, and is a specific form of an 
autoregressive moving average (ARMA). Several early studies in freeway traffic 
modeling used ARIMA for volume and occupancy prediction (Ahmed & Cook 1979, 
Levin & Tsao1980). Later, the ARIMA model was used in all kind of traffic parameters 
forecasting studies or comparing the new models with it (Zhang, 2015; Davis & Nihan, 
1990; Hamed et al., 1995; Kamarianakis & Prastacos, 2005; Williams et al., 1998; 
Cetin & Comert, 2006; Karlaftis & Vlahogiann, 2009; Min & Wynter, 2011).  
SARIMA, on the other hand, is Seasonal Auto-Regressive Integrated Moving Average. 
In fact, SARIMA is an ARIMA model that incorporates the seasonal factors in the 
model as well. Some studies tried to capture the seasonal effect by using the SARIMA 
(Kumar &Vanajaksh 2015; Williams & Hoel 2003).  
2.1.3  State Space Model 
A state space model (SSM) models the probabilistic dependency between the latent 
state variable and the observed measurement (Koller & Friedman 2009). One specific 
type of SSMs is the hidden Markov models (HMMs), in which hidden states are 
discrete (Murphy, 2012). The model could be used to analyze both deterministic and 
stochastic dynamic systems that are observed through a stochastic process. Kalman 
filter and Kalman smoothing algorithms are among the most widely used algorithms in 
13 
 
 
 
 
SSM models. There are several studies that have used the state space model for 
predicting traffic both in freeway and arterial networks (Stathopoulos & Karlaftis 2003; 
Ghosh et al., 2009) 
2.1.4 Hidden Markov Model (HMMs) 
In general, in the (first-order) Markov chain or Markov model, the assumption is that 
the present is only dependent on the immediate past. In other words, Markov model 
centers on the idea that the immediate past captures everything we need to know about 
the entire history. Since this is a very strong assumption, higher-order Markov models 
are created whereby the hidden Markov model (HMM) offers another means through 
which to capture long-range correlations between observations. HMM assumes there 
is an underlying hidden process represented by a hidden variable at each time (Murphy, 
2012). 
2.1.5 Bayesian Networks (BNs)   
A Bayesian network– also known as a causal model as well as a directed graphical 
model – is an approach to representing a system and the dependencies using a directed 
graph. Sun et al. (2006) developed a Bayesian network for the urban highways flow 
forecasting by considering the spatial/temporal correlations of traffic flow between 
adjacent links  
There are several studies using this dynamic Bayesian network for arterial travel time 
modeling (Herring, 2010; Hunter et al., 2009; Hofleitner et al., 2012 a; Hofleitner et 
al., 2012 b). Within a dynamic Bayesian network, however, the network is not dynamic; 
14 
 
 
 
 
instead, it is fixed and problem-specific. There are different structures to be defined. 
First, the structure of the first time-slice must be defined, then, the structure between 
two time-slices, and finally, conditional probability distributions (CPDs) governing all 
the network’s random variables must be defined.  
2.1.6 A Feedforward Neural Network  
A Feedforward Neural Network can be used for classification or regression problems. In 
fact, it is a series of logistic regression models on top of each other, and the information 
moves in a forward direction, with a final layer of logistic regression in cases where 
there is classification problem, or linear regression model if there is a regression 
problem. Neural Network models consist of hidden layers and weight matrices from 
input to hidden nodes and weight matrices from hidden nodes to the output. The hidden 
layers learn non-linear combinations of the original inputs. In instances where there is 
no nonlinearity, the model simply becomes a large linear regression model (Murphy, 
2012). 
There are several studies that have proposed different neural network for freeway 
traffic modeling. Zeng & Zhang proposed neural network models for freeway travel 
time forecasting (Zeng & Zhang 2013). Jiang and Adeli (2005) proposed a dynamic 
wavelet neural network model to predict freeway traffic flow and included time of the 
day and the day of the week as the explanatory variables. Even though most of these 
studies are for freeways traffic prediction, there are few studies using the neural 
network for arterial traffic modeling. Yin et al. (2002) used a fuzzy-neural model 
15 
 
 
 
 
(FNM) to predict the traffic flow in an urban street network. Park and Lee (2004) used 
a neural network as a classifier for arterial travel time estimation.  
2.2  Non-Parametric Models 
Unlike parametric models, the nonparametric model does not have a predetermined 
form for the predictor; rather the predictor form is derived from data. Nonparametric 
models do not have a predetermined number of parameters, and they could grow with 
the amount of data (Murphy, 2012). 
2.2.1 K-nearest neighbors 
One of the simplest examples of a non-parametric classifier is the K-nearest neighbor 
(KNN) classifier. This classifier consists of K points in the training set, each test input 
of x class is defined based on counts of members of each class that are nearest to that 
input (Murphy, 2012). Tiesyte and Jensen (2008) used the Nearest-Neighbor Trajectory 
(NNT) technique on real data from buses to predict future movement. The observed 
trajectory in this study is the path taken between two bus stops.  Myung et al. (2011) 
proposed k nearest neighbor method to predict freeway travel times using vehicle 
detector data and an automatic toll collection data. Robinson and Polak (2005) 
proposed the use of the k nearest neighbors technique to estimate urban link travel time. 
2.3 Other Methods 
There are some other studies using parametric models, non-parametric models, or 
combinations of them. Zhang (2015) proposed a gradient boosting tree-based model to 
16 
 
 
 
 
predict freeway travel time. Hamner (2010) proposed the use of context-dependent 
random forest to predict travel time. Leshem and Ritov (2007) used Adaboost 
Algorithm with random forests as a weak learner to predict freeway traffic flow.  Yu et 
al. (2010) applied the support vector machine (SVM) and Kalman filtering to predict 
bus travel time. Herring proposed Bayesian inference as one of the proposed models to 
estimate arterial travel time (2010). 
Huang et al. (2014) proposed a deep learning approach for freeway traffic flow 
prediction. They used the combination of a deep belief network at the bottom and a 
multi-task regression layer at the top of the architecture to model traffic flow. Yu et al. 
(2017) proposed a deep neural network based on long short-term memory to forecast 
freeway peak-hour traffic and extreme condition traffic (for example, post-accident). 
Cui et al. (2018) proposed a deep bidirectional and unidirectional long short-term 
memory neural network architecture to predict freeway network-wide speed.  
2.4 Arterial Travel Time   
Thus far, this paper has outlined the general ideas of, and the methods used in travel 
time estimation. What follows offers a more detailed discussion of the probabilistic 
methods in arterial travel time estimation and prediction.  
Herring (2010) compares different travel time estimation models including the 
regression model in which a logistic regression is used to classify the discrete states 
and STARMA for the travel time estimation/forecasting. Another considered model in 
this study is historical modeling using Bayesian inference for real-time estimation. 
Real-time estimation is calculated by considering the historical outcome as the prior 
17 
 
 
 
 
distribution and updating the parameters of distributions by weighting between the 
parameters from historical data and real-time observation. The last considered model 
is a probabilistic graphical model, in which the spatiotemporal conditional 
dependencies of the arterial traffic are modeled as a probabilistic graphical model. 
Then, Expectation-Maximization (EM) estimation algorithm, which is an iterative 
method to estimate maximum likelihood, is used to learn parameters. The used data is 
probe data from 500 taxis traveling in a small area (322 links) in San Francisco, from 
3 p.m. to 8 p.m. on weekdays.  
Hunter et al. clearly (2009) proposed a probabilistic model of travel times in the arterial 
network, based on low-frequency taxi GPS probes. In their study, an EM estimation 
algorithm is proposed that iterates between travel time allocation and parameter 
estimation. The study assumes that the travel times on the segments are normally 
distributed and travel times on different segments are independent. The data used in 
this study are obtained from a fleet of 50 taxis in San Francisco, however, the 
performance of the model and scale of the network are not mentioned. Hofleitner et al. 
further develop their primary in their later works (Hofleitner et al., 2012a; Hofleitner 
et al., 2012b). 
Hofleitner et al. (2012 b) proposed a density model to scale the partial travel time on the 
partial links (the start GPS point and end GPS point). Also, they integrated the step of 
travel time allocation on the links is into the EM algorithm which means the 
observations are path travel times. In other words, instead of explicitly distributing the 
path travel time to its corresponding links, in their graphical model, the path travel time 
is treated as the same parent for every link inside the path. They employed a particle-
18 
 
 
 
 
filtering inference in the E-step to optimize the model that can also learn how to 
distribute the total travel time to the links implicitly. The model is tested using Global 
Positioning System (GPS) data from a fleet of 500 probe taxis in San Francisco, and 
the studied network included 800 links. 
Hofleitner et al. (2012 a) proposed a hybrid model. In this study, both the distribution 
parameters as well as static parameters of the roadways (such as free-flow velocity or 
traffic signal parameters) are to be estimated. Their primary approach is improved by 
introducing a state variable that represents the number of vehicles that stop (number of 
queuing vehicles) on a link per light (as opposed to binary state variable) and turn 
fractions at intersections. EM algorithm is used for the parameter learning.  The model 
is tested using data from a fleet of 500 probe vehicles in a sub-network of San Francisco 
shown on 769 links.  
Hunter et al. (2013) took a slightly different approach in using a graphical model. Their 
algorithm called “Stop & go” consists of three steps. First, they use maximum 
likelihood to get the number of stops for each link from the observation. Then, they use 
the Markov model (MM) to calculate the state transition probabilities – the probability 
of the number of stops on a link given the number of stops on the preceding links. Next, 
they use the Gaussian Markov Random Field (GMRF), which provides the joint 
distribution of the travel times of neighboring links, given their state. Their findings 
show that the model can be improved by incorporating the variability of travel times 
due to stops in the structure of the model. However, the independence or correlation of 
travel times does not have a significant impact on the model. It should be noted that the 
19 
 
 
 
 
data used in this study is 9.6 measurements per link, which means that the model needs 
to incorporate a high-frequency GPS measurement. 
There are some other studies that propose probabilistic modeling for emergency 
vehicles, such as ambulances, and they mostly model the distribution of travel time on 
either the entire trip or based on links. However, this modeling might not work as well 
for general purposes since the data source is very low-volume and does not reflect 
travel time experiences for non-emergency vehicles (Budge et al., 2010; Westgate et 
al., 2015; Westgate et al., 2013). For instance, Westgate et al. (2016) proposed a 
regression model for travel time distribution for an entire trip over large networks 
including both freeways and arterials, using ambulance data. In another study, 
Westgate et al. (2013) used Bayesian data augmentation to simultaneously estimate the 
paths travel times as well as parameters of travel time distribution for each road 
segment. 
Jenelius and Koutsopoulos (2013) proposed a spatial moving average (SMA) method 
for arterial travel time estimation on heavily traveled routes. In this study, travel time 
mean and variance is a function of network characteristics (speed limit, functional 
Class) and trip conditions (time of day, season, and weather conditions). This study is 
one of the few studies in which the network characteristics and trip conditions are 
considered in the modeling. However, according to the authors, the focus of this study 
is to model the impact and significance of different explanatory factors on the travel 
times rather than developing a high-performance model for the travel time estimation. 
In this study, Stockholm, Sweden probe vehicle data from a taxi fleet is used.  
20 
 
 
 
 
Woodard et al. (2017) proposed a probabilistic estimation of travel time for large 
networks, including both freeways and arterial networks, using Bing Maps data. In this 
work, the travel time for each link is considered as a Gaussian variable conditioned on 
some latent unobserved random variables which depend only on the same link. 
Assuming some probability distributions over the latent random variables, the team 
used the Maximum A Posteriori (MAP) method to find the set of model parameters that 
maximize the posterior distribution. However, the proposed model has the limiting 
assumption that the state of each link is independent of the others. 
Yang et al. (2017) proposed a modified Gaussian mixture model to estimate arterial 
link travel time distributions. Their model could be used when either fixed-location 
sensors or mobile sensors data is available. They used simulation and a small number 
of real-world data to validate their model. 
2.5 Intersection Delay  
Intersections play a critical role in travel time estimation/prediction, but they are rarely 
considered in the route’s travel time estimation/prediction models. In fact, the travel 
time of a trip consists of two parts: road section travel time and intersection delays. 
Intersection delay is the timestamp difference between the times of vehicles entering 
and leaving an intersection. 
Tang et al. (2016) proposed a microscopic method to estimate intersection travel time. 
Their method assumes that the intersection travel pattern is different based on the 
different flow pattern. They divide the traffic flow patterns into four different 
categories: free traffic flow, minor impediment, moderate impediment, and serious 
21 
 
 
 
 
impediment. In their study, they first recognize the flow pattern, and then using the 
linear fuzzy regression method, they obtain the intersection travel time for each 
mentioned flow pattern. 
Ban et al. (2009) proposed a microscopic model to estimate the individual intersection 
delay pattern and validated it using microsimulation. Their proposed model identifies 
the observed sampled travel times measured between upstream and downstream 
locations of a signalized intersection, and it estimates the intersection delay pattern. 
The group’s proposed model includes two steps, the first of which detects the start and 
end of a cycle by considering the queue-forming and discharging process pattern, and 
the second of which uses a least-squares–based linear-fitting algorithm to estimate the 
delay pattern in each cycle. They considered two different flow conditions which are 
normal and oversaturated conditions. 
Hao et al. (2017) proposed a model to estimate vehicle state trajectory that can 
potentially be used to extract travel time and delays on arterial networks. In their model, 
the team considered four different modal activities: acceleration, deceleration, cruising, 
and idling. Further, the group considered all of the possible sequences of the modal 
activities, and the sequence that maximizes the likelihood of the product of probabilities 
for multiple independent events is chosen as the vehicle speed trajectory. Finally, they 
compared their model with linear interpolation as a baseline using Next-Generation 
SIMulation dataset and showed they reached better results compared to the baseline 
method.  
Even though there have been some studies addressing intersection-delay estimation, 
hardly any macroscopic level study of intersection delay estimation has been published. 
22 
 
 
 
 
In other words, there are few studies that consider both link travel times and intersection 
delays together and as part of the travel time estimation or prediction for a whole trip.  
2.6 Probe Data  
One of the most valuable types of data used in transportation studies is probe data. 
Probe data can be used to develop applications that can improve roadway operations, 
planning, and maintenance, and keep travelers informed of travel conditions. Probe 
technology uses location-aware and internet-enabled devices that are either installed in 
vehicles or carried by travelers. The providers of probe data can vary. For instance, the 
source could be GPS-equipped commercial fleet vehicles, such as trucks, package 
delivery vans, taxi vehicles, construction vehicles, or buses. The disadvantage of this 
data, however, that it is based on the specific spatiotemporal travel patterns of these 
vehicles. Another source of probe data is participatory sensing, such as INRIX, 
TomTom, Google, or Nokia – all of which collect location data from GPS-equipped 
smartphones, personal navigation devices, vehicles with embedded GPS, and other 
mobile consumer devices. The source of data in this study is probe data (either mobile 
data or vehicle embedded GPS data) provided by INRIX. 
2.7 Summaries and Limitations of Previous Studies 
In summary, traffic estimation/prediction models using probe data can be categorized 
based on three main aspects: scope, output, and method. Based on the output, there are 
two different potential outcomes for expected travel time and probability distributions. 
With regards to scope, there are two main options:  microscopic (which is based on a 
23 
 
 
 
 
small number of intersections and segments, assuming comprehensive data 
availability) and macroscopic. Finally, based on the method, there are two main 
approaches: parametric, non-parametric. It should be noted that there might be some 
overlaps between these categories.   
Since this study utilizes a macroscopic scale with an output of probabilistic distribution, 
most of the supporting literature reviews stem from those same categories.  Parametric 
models are usually based, in part, on theoretical reasoning and strong assumptions 
about the data. Some of the frequently used parametric models applied to travel time 
prediction are autoregressive models, Hidden Markov Model, and Feedforward Neural 
Network. Non-parametric models, however, often do not carry any assumptions about 
the data, and their parameters grow as the direct result of the amount of data used. One 
of the most frequently used non-parametric models is the k-nearest neighborhood 
model.  
When it comes to real-time arterial travel time prediction, some models are used more 
than the others.  For example, some studies have used simple Bayesian inference or 
regression models to estimate the historic travel time and considered a factor deduced 
from real-time traffic condition that is multiplied by the historic estimation to calculate 
the future prediction.  One of the more sophisticated models used in real-time arterial 
travel time prediction is the graphical model through which one considers the state of 
traffic as a hidden variable in the model and uses EM to solve it.  Other models have 
implemented complex regression models in which the trip and link variables take into 
account some probabilistic assumptions about the variables and use the Likelihood 
estimator (MLE) tor Maximum A Posteriori (MAP) to solve it.  
24 
 
 
 
 
What follows discusses some of the limitations of the reviewed literature and outlines 
how this study is addressing those limitations. 
2.7.1 Limitations of Previous Studies 
Many previous studies rely on either taxi data or bus data (and in some case, ambulance 
data). Although these data are valuable because of their availability, they might not be 
representative of general travel time and, in most cases, these vehicles experience 
higher travel times (Jenelius & Koutsopoulos, 2013). The reason for choosing the 
mentioned probe data is because the other type of high frequency of probe data is 
usually not available for researchers to use.  
Additionally, previous studies rarely consider weather conditions despite the fact that 
the impact of weather on the travel time is critical, however, it is one of the most needed 
areas to focus on since the existing models including the commercial ones (Google 
map, etc.), have the lowest performance in bad weather conditions. Another factor that 
is rarely addressed is taking into account the driver behavior. Many studies are also 
limited by the fact that they treat the intersection as part of the links rather than treat 
intersections as separate entities. Despite the fact that the link and the entering 
intersection are in many ways related, treating these as separate entities might result in 
better traffic modeling. Finally, most existing probabilistic models are applicable only 
to small-sized networks because many of them are not computationally efficient for 
large-road networks and/or large datasets. 
Recognizing these challenges, this study aims to propose a probabilistic model that uses 
vehicle trajectory data and brings different factors, such as weather conditions and 
25 
 
 
 
 
travel behavior, into the modeling process. This study also considers the intersections 
and links separately in order to provide different travel time distributions for each. Last 
but not least, this study uses Streaming Variational Bayes for the inference which 
makes it applicable for large-sized networks as well.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
26 
 
 
 
 
Chapter 3: Model Preparations  
In a perfect world, there are perfect data - and engineers simply need to find the best 
model that can use that data toward a final objective. In line with this thinking, many 
studies try to find cleaned data to use, but this limits the area of study. In fact, in the 
real world, there is always “dirty data” that need to be cleaned prior to any applications 
for modeling. Thus, it is critical to have a data cleaning framework in place, and this 
chapter provides such framework.  
On the other hand, in this study, we have the GPS trajectory data as the observations 
and city roads network as the variables for which we want to predict the travel time. 
Thus, the next step involves mapping these observations (GPS trajectory data) on the 
roads network. Additionally, it is necessary to understand the structure of the city and 
how the road traffic could be interrelated. In order to do this, the main structure of the 
graphical model is explained; this model provides the primary building blocks for the 
model in the next chapter with all the variables. 
This chapter is dedicated to all of the aforementioned preliminary steps taken prior to 
the modeling. Additionally, this chapter discusses input data, including geographic map 
and weather data. Figure 3 illustrates the modeling framework and demonstrates how 
the needed data and model are interrelated.  
27 
 
 
 
 
Pre-Modeling Steps Graphical Model
Raw Probe Parameter 
Data Data Cleaning Map Matching Estimation
Travel Time 
Allocation
OSM
Making a Graph
Other Information
 
Figure 3 The conceptual framework of modeling the arterial travel time 
3.1 Data Cleaning 
Many studies focus on the cleaning of probe data, with different end objectives. Chung 
et al. (2003) proposed a framework to clean the probe data to determine the origin-
destination (OD) pattern of the probes. Their proposed data cleaning framework 
consists of six considerations to find the correct trip destination. The six considerations 
are the gap between parking brake event, the long gap, the gap with unrealistic speed, 
the long stop, the short stop with hazard light, and the U-turn. 
There are other studies that attempt to implement methods to complete an incomplete 
data. Hao et al. (2017) proposed a model to find the vehicle trajectory between two 
observations rather than just merely interpolate between them. 
In this study, a data cleaning process for the vehicle trajectory data with the purpose of 
travel time estimation/prediction is proposed.   
28 
 
 
 
 
The vehicle trajectory data contains the GPS trace information of any vehicle traversing 
a path from an origin to a destination. For each vehicle along its path, there is a GPS 
report (including information such as location, time, speed, etc.) received for every time 
interval (which varies in different datasets).  
However, these GPS reports are most of the time not clean and ready to use. Various 
external causes may degrade the quality of the reports. GPS device may not work as 
expected, the GPS localization may not be accurate in some regions, GPS signal may 
not be fixed in some time intervals, etc. Inaccurate data results in inaccurate modeling.  
Consequently, this proposes a framework to clean the waypoint GPS traces before 
modeling the travel time. This framework deals with problems such as idling, 
impossible speed, impossible sequence, impossible shift in location or time, etc. 
What follows explains the data cleaning process in the most general form, in order to 
allow it to be used for any kind of data. The framework is explained in the most 
comprehensive manner and can be applied to any available data. The cleaning filters 
take into account all possible information, including latitude, longitude, timestamp, 
speed, headway, and GPS error. Also, the process assumes there is no order to the data. 
Based on the availability of data and prior cleaning processes, a subset of this filtering 
might be used. 
3.2 Probe Filtering 
If the provided probe data is raw data that is not cleaned in any way, it can be difficult 
to use for modeling. Use of incorrect data results in the development of an inaccurate 
model. As such, it is essential to have a useful framework and to develop a thorough 
29 
 
 
 
 
cleaning process. What follows is the explanation of the “probe filter” framework, 
which explains the procedure to find tripes, clean the probe data and eliminate noise 
and outliers. The proposed framework is applied to both TeleNav data and INRIX data, 
based on their availabilities. Figure 4 shows an example of TeleNav raw data in the 
Santa Clara, California area. TeleNav data is pulled from different providers;  each 
provider has different standards of quality and uses different time intervals between the 
GPS reports (ranging from one second between reports to a couple of minutes). The 
Telenav data is only used in the data cleaning process. The data is the probe data with 
a one-second interval and the data is device-based. The data are from 2015 to 2017 and 
are for the cities in northern California, including Santa Clara, San Jose, Mountain 
View, and Palo Alto.  
 
 
Figure 4 Sample of trips in Telenav probe data 
Figure 5 shows the example of the INRIX probe data.  As mentioned, from the INRIX 
data, the Baltimore area is selected, and the data covers four months of data with the 
interval of one minute between GPS reports. The INRIX data is not as raw, but it still 
needs to undergo a cleaning process.  
30 
 
 
 
 
 
Figure 5 Sample of trips in INRIX probe data 
If the data is device-based (TeleNav), then the cleaning process involves two different 
steps. The first step centers on finding the trips and the second step centers on cleaning 
the data from each trip. If the data is trip-based (INRIX) then only the second step is 
needed. 
What follows explains each of these two steps:  
1. Finding the trips from the device-based probe data (trip splitter). 
2. Cleaning the trip (applying the rest of the filters). 
Here, all the filters are applied consecutively to the probe data; however, one might 
choose only a few of the filters according to the need, data availability, and the 
objective of the study. Figure 6 shows the cleaning process.  
 
31 
 
 
 
 
Trip Filters
Raw Probe 
Data Trip Cleaned 
Splitters Data
 
Figure 6 The data cleaning Process 
3.2.1 Trip Splitter 
The trip splitter might be used for one of two reasons: the waypoints’ unique identifier 
might be device-based rather that trip-based or the trip might need to be split into more 
than one trip in order to use it in the modeling. The three different criteria to split the 
trips are time, location, and the waypoint (if available).  
3.2.1.1 Time Difference Splitter 
First, if the data are device-based (rather than trip-based), the data are split based on 
the datestamp or timestamp of the data. For instance, the first trip data might be split if 
it is from different days. Then, on the same day a time interval is considered above 
which the probe data is split into different trips (e.g. max time difference threshold = 
120 sec). This time interval depends on the GPS report timestamp (for example, if it is 
one minute or one second between each of the two consecuitive GPS reports). This 
means that the probe data with a time interval over a certain threshold might be split 
32 
 
 
 
Same Time 
Same Location
Speed Boundary
Idle
Location Displacement
Trip Length 
Large Error Size 
Shaking Probe 
 
into two trips. The challenge with this filter, however, is that if we choose a higher 
time-difference threshold, we might incorrectly select two different trips as one and 
then we might lose the whole trip in the cleaning process (“shifted probe filter”). If we 
choose lower threshold, we might split one trip into more trips, but that won’t cause 
any problem since we won’t loose any information aside from the actual origin-
destination of the trip. If the data is already trip-based, there is no need for this 
operation.  
3.2.1.2 Way Point Sequence Splitter 
The last step in the trip splitter process is splitting if the sequence of the waypoints is 
not ordered or when there is a gap in the sequence. This is due to the fact that we limited 
the study area to a boundary, and those trips that go out of the boundary and come back 
again would have a gap in their sequence. Thus, those trips are split every time they 
pass the boundary.  
3.2.2 Cleaning/Filtering the Trips 
3.2.2.1 Same Time Filter 
This filter removes the probes that have the same timestamp. In a way, the first probe 
with a unique timestamp is kept, but all subsequent probes with the same timestamps 
are removed. This filter is a partial filter, which means that only the probes that have 
this duplicative characteristic are removed from the trip data. 
33 
 
 
 
 
3.2.2.2 Same Location Filter 
This filter is also a partial filter; it removes those probes that are based on the same 
location. This process involves keeping the first probe and removing all the subsequent 
probes that have a distance lower than a minimum threshold from the first point (e.g. 
minimum distance between probes of 1 meter).  
3.2.2.3 Speed Boundary Filter 
This filter is used to filter out those trips that feature an impossible speed. The filter 
removes those probes that have speeds registering above the maximum speed (e.g. 
maximum speed = 200 mph). Similarly, if the average trip speed is above the maximum 
or below the minimum speed (e.g. minimum speed = 2 mph and maximum speed = 200 
mph), then the whole trip is removed.This filter could be used even when speed data is 
not available because the speed could be calculated based on the location and 
timestamp data. This filter could be partial, or it could target the entire trip.  
3.2.2.4 Idle Filter  
This filter removes the probes that are identified as idle probes, such as when the speed 
is below a minimum threshold speed, from a trip. Applying this filter means that, if a 
car is idle in the gas station, those probes will be removed; similarly, if time is important 
to us the travel time from origin to destination would be much higher. To address this 
problem, the time difference threshold (trip splitter) could be checked once again. If 
removing the filter causes the time difference between the consecutive probes to be 
more than the time difference threshold, the trip should be split into two trips. Another 
34 
 
 
 
 
potential problem with this filter is the probes behind the traffic light, which are 
removed because they indicate a speed below the threshold (e.g. minimum speed of 1.0 
mph). This filter could be used regardless of whether or not speed data is available 
since the speed could be calculated based on locations and timestamps.  
3.2.2.5 Location Displacement Filter 
This filter removes the whole trip if there is at least one shifted coordinate by more than 
max Distance threshold (e.g. maximum miles between probes = 3 ). This filter is not a 
partial filter and if the conditions are met the whole trip will be removed. The problem 
with this filter could be losing a lot of data if we have a higher threshold of max distance 
and having a lot of shifted probes if we choose the lower max distance. One solution to 
this problem could be to avoid losing this data by either splitting the trips to smaller 
ones, or removing the shifted probes and interpolating between the remaining probes 
or using trajectory identification techniques (Hao et al. 2017) instead of the 
interpolation to improve the accuracy. 
3.2.2.6 Trip Length Filter 
 
This filter removes the entire trip if the probe count is less than a minimum count. The 
count threshold is based on the probe timestamp intervals (if they are 1-second or 1-
minute interval). Here the threshold of 60 probes is used when we have the timestamp 
interval of 1 second and threshold of 5 probes when we have the timestamp interval of 
1 minute (e.g., Min Probes Count = 60). 
35 
 
 
 
 
3.2.2.7 Large Error Size Filter 
This filter is again a partial filter; it removes those probes with GPS error larger or 
equal to a maximum threshold error size from the trip data. Error size/accuracy is a 
variable in the probe data, and it shows the accuracy in meters of the GPS recording. 
(e.g. maximum error size in meters = 200 m) (TeleNav, 2017). This filter could be used 
when the error information is available in the dataset; if it is not, this filter is omitted. 
3.2.2.8 Shaking Probe Filter  
A probe is identified as “shaking” if the previous and next probes have a heading angle 
difference outside the maximum angle difference threshold. This filter removes the 
shaking probes from the trip data, and if the number of shaking probes is too large, the 
whole trip is removed. (maximum percentage of angle difference for shaking probes = 
40%) (Telenav, 2017). This filter could be used when there is a heading; if there is no 
heading, this filter can be omitted. 
3.2.3 An Examples  
The following examples (Figure 7 and Figure 8) show the before and after of the 
filtering. Figure 7 demonstrates how the filtering process solves the displacement probe 
problem by splilliting the trip to two trips. Figure 8 demonstrates how the time-
difference problem is solved by splitting the trip into two trips.  
36 
 
 
 
 
before after 
 
Figure 7 An example of Telenav trip before and after filtering  
 
before after after 
 
Figure 8 An example of INRIX trip before and after filtering 
3.3 Map Matching 
One problem with GPS data is that the data points are never precise enough, so they 
might be off the road. Also, we might not know the traversed links between two GPS 
reports. For example, in our proposed model, the observations that are used as input 
are link-based, however, what we collect from the data are GPS reports that are 
randomly reported. As such, we might have more than one GPS report in one link – 
which is very rare for low-frequency GPS data – or we might have only one GPS report 
37 
 
 
 
 
for several links. Therefore, it is important to find the traversed links between every 
two consecutive GPS reports. To address this, it is important to map these points on the 
road. This problem of matching measured latitude/longitude points to roads, along with 
the path between two points, is called map matching.  
There are a lot of studies about map matching and what follows provides a very brief 
literature review of this concept. These approaches include a geometric approach in 
which the algorithm tries to match a path that has a similar curve to observed locations 
(Greenfeld, 2002; Brakatsoulas et al., 2005; Kim &kim 2001). Another approach uses 
HMM and Kalman filter (Krumm et al., 2007; Newson & Krumm 2009). Some such 
approaches also include the uncertainty associated with the path; however, studies have 
found little benefit to using this in predicting travel time distributions on routes 
(Westgate et al., 2015; Woodard et al., 2017). 
In this study, an HMM-based map matching algorithm, according to the Newson and 
Krumm paper, is implemented (Newson and Krumm, 2009), and it is adapted in 
accordance with this study’s needs. What follows explains the Newson and Krumm 
HMM-based map matching proposed method, as well as assumptions and 
considerations in this study.  
3.3.1 HMM-based Map Matching 
Hidden Markov Model (HMM) map matching finds the most likely point on the map 
for each set of latitude/longitude coordinates, as well as most likely links sequence 
between two consecutive latitude/longitude pairs.  
38 
 
 
 
 
3.3.1.1 Measurement Probabilities 
Due to GPS noise, GPS observations rarely point exactly to a link; but, it is important 
to have GPS points on the links. Relatedly, measurement probability is the likelihood 
that location observation denoted as ot, resulting from a given state li (a link) and based 
on that observation itself. Measurement probability is the likelihood that the 
observation O𝑡 would be observed if the vehicle were actually on road segment l𝑖. For 
a given ot and li, the closest point on the links is denoted as y𝑡, and | ot - y𝑡,𝑖| is the 
distance on the surface between the observed point and the candidate match on the map. 
Ideally, this calculated figure should be as low as possible (Newson and Krumm, 2009). 
3.3.1.2 Transition Probabilities 
Once all the latitude/longitudes are mapped on the links, the next step involves finding 
the traveled links sequence between each consecutive latitude/longitude points pair. 
Transition probabilities are the probability of a vehicle traversing a set of links from 
the first point (i) to the second one (j). If the observation at that time is ot and the next 
observation is ot+1, and their candidate matches are y𝑡,	and	y𝑡+1, j	respectively,	the path 
distance would be | y𝑡,	-y 𝑡+1,j	|path, which is the network distance between y𝑡,𝑖	and	y𝑡+1,j  
(sum of all the links’ lengths from the first point to the second point), which should be 
as small as possible. Based on the Newson and Krumm study (2009), the histogram of 
the absolute distance differences follows an exponential distribution shown in 
Equation1, where 𝛽 is the parameter to be estimated.  
= @A𝑝(𝑑 ) = 𝑒 B>:           (1) >
39 
 
 
 
 
𝑑: = ||𝑜: − 𝑜:F=| − |𝑦:,H − 𝑦:F=,I||        (2) 
3.3.1.3 Optimum Match 
Newson and Krumm (2009) proposed using Viterbi algorithm to find the best point 
matches and path between each pair of consecutive points. The Viterbi algorithm finds 
the match that maximizes the product of the measurement probabilities and transition 
probabilities through dynamic programming (Newson and Krumm, 2009). 
3.3.2 Map Matching Assumptions 
There are several assumptions inherent to the implemented map matching algorithm in 
this study and the used coding. The following are some of them. 
1. When it comes to mapping each GPS latitude and longitude to the map, the priority 
is with intersections. Meaning the intersections are strong candidates to which we 
can map the GPS reports.  
2. If there is more than one observation per link per vehicle, only one is considered 
in the modeling. 
 
Figure 9 shows the results of one of the map matchings for the study area. The map on 
the left shows the observations with the reported GPS points, and the map on the right 
shows the result of map matching and the path that the hidden Markov map matching 
algorithm suggests.   
 
40 
 
 
 
 
 
Figure 9 A Map matching result instance for Baltimore city 
 
3.4 Graphical Model   
A graphical model is a probabilistic model in which conditional dependencies between 
random variables are demonstrated by a graph. The vertices of this graph represent the 
random variables, and the edges show the statistical dependence between the variables. 
Graphical models have been widely applied to different problems due to their intuitive 
representation, the natural way to define independency between variables, and the 
efficient approximate methods developed to perform the inference once the graph 
structure is defined. These models are successfully used in probability theory, machine 
learning, statistics, and, in particular, Bayesian statistics. There are several types of 
graphical models; Bayesian networks and Markov random fields are two of the widely 
used branches of graphical representations of distributions (Koller & Friedman, 2009). 
41 
 
 
 
 
In arterial traffic modeling, the spatiotemporal conditional dependencies of the network 
traffic could be modeled as a probabilistic graphical model.  
3.4.1 Acyclic Graphical Models  
Bayesian Networks serve as graphical models with structures as directed acyclic graph. 
This model represents a factor of the joint probability of all random variables. Equation 
3 is this joint probability. 
𝑝[𝑌=, … , 𝑌M] = ∏HR= 𝑃(𝑌H|𝑝𝑎H)        (3) 
Where pai is the set of parents of node Yi, and 𝑌=,… , 𝑌M are the events. In other words, 
the joint distribution factors become a product of conditional distributions. Once the 
problem is formulated by means of a Bayesian network, the inference problem should 
be solved. In general, the inference can be defined as the problem of evaluating the 
distribution over a set of random variables, given another set of random variables. 
There are three different main classes of inference for Bayesian networks: inferring 
unobserved variables, inferring parameters, and structure learning (Russell & Norvig 
2003). The first two are what we need to work on in this study. While it is possible to 
perform inference in a naïve way by computing all required probability terms, in 
practice this tactic is inefficient. The exact inference, however, can be solved more 
efficiently by means of methods like belief propagation and factor graph propagation 
for singly connected graphs. Moreover, for more complex problems, when the exact 
inference becomes intractable, approximate methods such as loopy belief propagation 
can be applied to reduce the computational complexity.  
42 
 
 
 
 
In parameter learning, the inference variables of interest are the model parameters. 
There are, in general, two cases in which parameter estimation is considered: inference 
when all variables are observed, and inference when variables are partially observed.  
Approaches toward inferring the model parameters can be divided into Maximum 
Likelihood and Bayesian approaches. In the Maximum Likelihood approach, the 
parameters of the model are estimated using the Maximum Likelihood estimator, when 
all variables are observed. When some of the variables are hidden (i.e. unobserved) the 
Expectation Maximization paradigm can be applied to solve the problem. On the other 
hand, the Bayesian approach takes the prior distribution over the parameters into 
account. The prior distribution can be one of any number of different distributions, but 
Drichlet distribution is usually deployed due to its flexibility. This leads to updating 
Drichlet distributions when all variables are observed. When there are hidden variables 
in the model, methods such as the Markov Chain Monte Carlo (MCMC), Viterbi 
algorithm, or Variational Bayes(VB) methods can be applied. Variational Bayes 
methods are among the most efficient Bayesian inference techniques and, in terms of 
time complexity, they are similar to the Expectation Maximization approaches. 
3.4.2 Model Assumptions  
Our base graphical model has the following assumptions: 
1. The travel times of links are normally distributed and travel times on different 
links are independent of each other. 
2. Discrete congestion states: a discrete value of s, indicates the level of 
congestion. This variable is not observable; it is a hidden variable. 
43 
 
 
 
 
3. Conditional independence of link travel times: conditioned on the state of a link, 
the travel time distribution of that link is independent of all other traffic 
variables. In words, link travel times are not correlated across links. This 
assumption is based on computational tractability. 
4. Conditional independence of state transitions: conditioned on the states of the 
spatial neighbors of link l, at time t, the state of link l at time t + 1 is independent 
of all other current link states, all past link states, and all past travel time 
observations. This assumption means that each link is correlated with a few 
neighboring subsets of neighboring links, but remains independent of the rest 
of the network. 
3.4.3 Implemented Graphical Model  
Arterial traffic has space and time components. The spatiotemporal conditional 
dependencies of arterial traffic are modeled using a probabilistic graphical model, or 
Dynamic Bayesian Network. Each node is a representation of a random variable, and 
each arrow represents a dependency. The hatched nodes represent the observed 
variables, and non-hatched nodes represent the hidden variables. In our graph, the 
discrete state of each link l and time period t is denoted Xl,t. Since the state of each link 
for all times is not observable, these variables are considered hidden variables and 
shown by non-hatched nodes. The observed travel time on link l and time period t is 
denoted as Yl,t, shown by hatched nodes. The index d is excluded from the variables 
according to assumption 1. The forward arrows show the spatial and temporal 
dependencies of the network. As shown in Figure 10, each Yl,t has the parent Xl, t which 
44 
 
 
 
 
is a representation of assumption 2 and means that the travel time distribution on each 
link is conditioned solely on its hidden state. Also, each state Xl,t, has parents of Xl,t-1, 
and Xpa(l),t-1which is same as assumption 4, meaning a state on a link at time period t, 
is solely conditioned on the neighboring links’ state at the previous time step. Later, 
this model is improved by including other information with adding observation nodes 
to the model. 
This system is assumed to be a Markov process with unobserved states that evolve over 
two time steps.  It is also referred to as the Coupled Hidden Markov Model (CHMM).  
There are three probabilities that need to be estimated to utilize this model: 
• The initial probability of the state of congestions denoted πl,s: It is necessary to 
estimate the initial state probabilities for each link.  
• The discrete transition probability distribution functions, denoted Pl,t, which 
refer to the transition probability between the states. In other words, the element 
of line m and column s in the matrix of Pl,t (m; s), represents the probability of 
link l to be in state s at time t + 1 given that the neighbors of l are in state m at 
time t. 
•  The distribution of travel time on a link given the state of that link, denoted 
gl,s,t. As mentioned before, the travel time on each link is dependent on the state 
of that link. 
The outcome of this model is prediction/estimation of links travel times at each time 
step. In the implemented model, the dependency among the neighboring links is 
considered as relationships between their traffic states. However, for tractability, this 
45 
 
 
 
 
study assumes the travel time distributions of the links are independent of each other 
given their traffic states. This independency of the links makes it possible to calculate 
the path travel time by summing over the estimated links travel time encountered in 
that path.  
 
 
 
Figure 10 Spatiotemporal model of arterial traffic evolution represented as a Graphical model. 
3.4.4 Creation of the Network 
The network or Dynamic Acyclic Graph (DAG) of the roads is created by creating a 
graph of the street map. Once this graph is created, a variable is assigned to each 
direction of a link, in case the link runs in two directions. There are always two sets of 
46 
 
 
 
 
variables, one in time step t, and the same variable in the time step of t+1. Then, if a 
variable A (link and direction A) has a forward traffic direction to variable B (link and 
direction B), A in time step t would be the parent of B in the time step of t+1.  
3.5 Data 
There are different datasets used in this study: vehicle trajectory GPS probe data from 
INRIX, road network data from Open Street Map (OSM), and weather data and severe 
weather data from different sources. What follows explains each of these datasets in 
more detail. 
3.5.1 Study Area 
The study area at this stage is Baltimore, MD with more than 1,700 links and 1,250 
intersections (Figure 11). The network information is extracted from Planet OSM. 
Planet OSM provides the data from the OSM project, which is normally updated every 
day (Planet OSM, 2016).  Once the data is obtained, a graph is made from this area that 
notes network information, such as links, speed limit, and roadway type. In the graph, 
each edge is a link between two nodes (intersections), which means that an edge could 
be run forward, backward or in two directions, per the direction of traffic.  
The following section explains the OSM briefly. 
47 
 
 
 
 
 
Figure 11 Study area, downtown Baltimore 
3.5.2 Open Street Map (OSM) 
Open Street Map is a map that features free geographic data for the world, and is created 
mostly by users (Open Street Map, 2016). Open Street Map represents the physical 
structure of the world with three main elements: nodes, ways, and relations. Elements 
have their specific tags which give the attributes of that element. Tags are to provide 
more information on a specific feature such as functional types of ways or nodes.  
A node is a core element of the OSM data which has latitude, longitude, and node 
identification. Any point feature, ranging from a building or a traffic light to a fountain 
or ATM, could be represented by a node. Tags can be used to provide more information 
on features, such as building names or the type of an intersection. Some of the most 
48 
 
 
 
 
important nodes in arterial travel time modeling are traffic signals, stop signs, 
crossings, bus-stops, motorway junctions, and turning circles. In the proposed model 
with intersections, this data is used.  
A way is another element of OSM data. It features an ordered list of nodes, and it can 
be open or closed.  Closed ways are ways in which the first and last nodes are the same. 
Ways represent linear features such as roads, barriers, and railroads. Tags provide more 
information, such as the type of the road, the name of the road, etc. The most important 
ways in this study are roads that are named highways with type (tags) of motorway-
link, secondary-link, primary-link, tertiary-link, trunk-link, trunk, motorway, primary, 
secondary, tertiary, unclassified, residential, service, road.  
A relation is also a core data element. Relations are used to represent logical or 
geographic relationships between features. Relations consist of an ordered list of nodes, 
ways, and/or relations. Relations can also show all of a road’s segments (Wiki Open 
Street Map). 
3.5.3 INRIX Data 
The traffic data in this study covers the full months of February, June, July and October 
2015. The data includes trip information as well as trip records waypoints for each trip 
and trip provider details. The trip data consists of a variety of information, including 
start/end location and time, trip provider, vehicle weight class, source of the probe data 
(embedded GPS, mobile device, unknown), and an indicator in the trip data that shows 
the mode of transport (walk, vehicle, and rail). The data include both passenger vehicle 
and truck data, the information in the “vehicle weight class” is used to choose only 
49 
 
 
 
 
passenger cars or light trucks data which are the majority of the trips (Markovic et al. 
2017). The waypoints data includes a trip identifier and sequence Id which notes the 
order of the waypoint within the trip starting with "1" and increases by one unit along 
with the latitude and longitude and their corresponding timestamps. The data consists 
of more than one billion waypoints (around 112 GB) for the available area and more 
than 20 million waypoints (around one GB) for the studied area in Baltimore. 
It should be noted that the vehicle trajectory data is different from the Traffic Message 
Channel (TMC)-based Vehicle Probe Project (VPP) data provided by RITIS (2017). 
RITIS has TMC-based probe data for the arterial network which are provided by three 
vendors: INRIX, HERE, and TomTom. In fact, these providers use their own vehicle 
trajectory data to provide the TMC-based travel time for the arterial network. The 
quality of these arterial TMC-based probe data is assessed by Sharifi et al. (2017). 
3.5.4 Weather Data  
One type of weather data used in this study is “Quality Controlled Local Climatological 
Data (QCLCD),” which is high-quality data consisting of hourly, daily, and monthly 
summaries for more than 1,600 U.S. locations. This data is available from January 1, 
2005, to the current date as of this writing (NOAA, 2015). Other weather data used in 
this study is from the Global Historical Climatology Network (GHCN), which collects 
information from stations in its database but does not offer as much quality control as 
the QCLCD. For those stations that are missing data due to times in which they were 
not functioning or are turned off, the GHCN is used (NOAA Climate, 2015). The last 
dataset used for this study is the adverse weather data set obtained from Storm Events 
50 
 
 
 
 
Database (NOAA Storm Events, 2015). The Storm Events Database contains data for 
storms and other significant weather phenomena, as well as significant meteorological 
events from 1950 to 2016 and features the year discussed in this study.  
3.6 Summary 
This section provided a comprehensive framework for the data cleaning process, which 
includes data splitting and trip filtering. The splitting process is more critical when the 
unique identifier is a device rather than a trip. The splitting process for a probe set with 
a unique identifier is based on the time and location and the waypoint sequence (if 
available). The filtering includes eight different filters and should be applied to each 
split trip. The filters remove the outliers or the whole trip. These filters deal with idle 
probes, speeds lower or higher than the possible speed, probes having the same time or 
same location as each other, very small-length trips, probes for trips with sudden shifts 
in the location, large errors, and probes with sudden or frequent changes in the heading 
(if the heading is available).  
Once the data are cleaned, they need to be mapped to the roads. There are different map 
matching algorithms to do that. Map matching algorithms first find each GPS point on 
the map (measurement) and then find the traversed links between two GPS points 
(transition).  HMM-based Map matching was used in this study. In HMM-based map 
matching, the Viterbi algorithm is used to select the mapped point and the traversed 
link alternative that maximizes the product of the measurement probabilities and 
transition probabilities.  
51 
 
 
 
 
Once the inputs are ready, the structure of the city should be created based on the 
graphical model assumptions. In the graphical model, each link (the roads between two 
intersections) is a variable, and the goal is to find the travel time distribution for each 
of the links (variables) at each time step. The traffic state is hidden variable, and the 
assumption is that travel time on each link is dependent on this state. However, in the 
arterial network roads, travel times are dependent on each other, and this dependency 
is reflected in the states; that is, each link state at time t+1 is dependent on the 
neighboring links at time t. The network of the roads and the adjacency of the links (to 
get the neighboring links) are created and the initial graph is based on them. 
The data used for this study are: geographic map, which is accessed via open-source 
Open Street Map (OSM), the GPS trajectory probe data which is obtained from INRIX 
and Telenav (used only for cleaning procedure), and the weather data which was 
obtained from different data sources provided by NOAH.  
 
 
 
 
 
 
 
 
 
 
52 
 
 
 
 
Chapter 4: Graphical Model  
In recent years, thanks in part to greater availability of data, there have been efforts to 
conduct probabilistic modeling of arterial travel time. It is, however, rare for any such 
models to consider the effects of external variables such as seasons or weather 
conditions. These variables are shown to be influential in both demand or driving 
behavior as well as travel time in freeways. Still, they are rarely considered predictive 
variables in arterial travel time modeling.  
This chapter focuses on developing structures for enhanced graphical models, 
including all variables includign linkes and states variables as well as external variables 
into the modeling.  
This chapter outlines each step of the modeling process, including travel time allocation 
and conducting inference using an approximate inference algorithm. Detailed 
algorithms for each step of the modeling are provided in the next chapter.  
4.1 The External Variables.  
4.1.1 Weather Conditions 
Weather conditions affect travel time on both freeways and arterials network. Studies 
also show that weather conditions significantly affect traffic flow, demand, and travel 
behavior, and consequently travel time and its variations (Qiao et al., 2012; Huang & 
Ran, 2003; Tsirigotis et al., 2011). Qiao et al. (2012) proposed a K nearest neighbor-
integrated model and incorporated the effects of weather in the freeways traffic 
modeling using Bluetooth data. The team demonstrated that their model outperforms 
53 
 
 
 
 
the ARIMA and KNN models in rainy weather conditions. Hranac et al. (2006) 
proposed weather condition adjustment factors that are multiplied by the base clear-
condition variables in order to incorporate weather condition information in traffic 
volume estimations.  
Most of these studies have been conducted for freeways, and there are rarely studies 
that incorporate weather conditions in arterial travel time estimation. Jenelius and 
Koutsopoulos (2013) considered weather conditions as an important variable in their 
modeling, however, their model does not provide real-time travel time 
estimation/prediction model.  
Several types of weather information can be used to study weather impacts on traffic. 
The most frequent are precipitation type, precipitation intensity, visibility, average 
wind speed, and adverse weather. Consideration of these variables relies on data 
availability, and incorporation of more conditions requires the availability of data 
pertaining to each condition.  
In this study, weather conditions are incorporated in real-time arterial travel time 
estimation/prediction. An external variable for weather conditions is added into the 
Dynamic Bayesian Network. The considered variables are precipitation intensity, 
visibility, and average wind speed.  
4.1.2 Seasons  
There are seasonal variations in demand that can affect travel times on freeways and 
arterials. In general, there is a seasonal pattern in traffic demand and travel time; as 
such, including the season as a variable factor could capture this pattern in a given 
54 
 
 
 
 
model. A lot of studies tried to use time series analysis or introducing a seasonality 
coefficient in the model such as, for example, Seasonal Auto Regressive Integrated 
Moving Average (SARIMA) to capture the seasonality effects for the travel time 
prediction. Most of these studies have been done more on freeway travel time modeling 
(Williams et al., 1998; Peng et al. ,2014) and less on arterial travel time modeling 
(Khoei et al., 2013).  
In this study, we utilize four months of data, February in winter, June and July in the 
summer and October in the Fall. Thus, we have at least one month representing each 
season (except for spring). The summer season has the highest data and number of 
observations available since it has two months of data.  
4.1.3 Day of the Week 
Another time-varying feature is the day of the week, as it plays a key role in travel time 
modeling. Traffic demand and, consequently travel time varies depending on the day 
of the week. By including this feature, we are able to capture traffic patterns specific to 
each of the days of the week. This feature is critical to traffic estimation/prediction 
modeling. In this study, a variable for weekdays is considered in the modeling, and 
holidays are excluded from the dataset.  
4.1.4 Time of Day  
Another feature that is key to travel time modeling is the time of day. The traffic 
demand – and even travel behavior and signal phasing – could be different depending 
on the time of day. Time of day could be categorized as morning/AM peak, 
55 
 
 
 
 
afternoon/PM peak, nighttime post-afternoon peak and before morning peak, and 
daytime between morning peak and afternoon peak.  
In this study, a variable, showing time of day, is added to the model in order to capture 
all of the aforementioned values. In general, time variables could capture many of the 
variations in travel time patterns. 
4.1.5 The External Variables 
 
As mentioned before, travel behavior and demand –and, relatedly, travel time 
distributions– vary based on the days of week and holidays, as well as different seasons, 
different time of days, and different weather conditions. As such, these variables are 
incorporated into the model.  By including these variables in the proposed model, we 
are able to provide the links’ travel time distributions in various combinations of these 
conditions. What follows are the variables and their notations.  
 
1. Variables for the summer (June–July), winter (February), and fall (October) 
seasons. 
2. A variable for weather conditions. 
3. A variable for each weekday from Monday to Friday (holiday excluded). 
4. A variable for each time of day: morning peak, evening peak, between 
morning and evening peak, and off-peak nighttime.  
56 
 
 
 
 
As is discussed in the following section, all these variables are the parents of the link 
travel times in the dynamic Bayesian network. This means that, with each set of factors, 
we would have a different distribution for all of the links of the network.  
4.2 Graphical Model with External Variables  
The discussed graphical model depicted in section 3.4.3 can be generalized not only to 
capture the relationships between link states and observations, but also to take 
additional information into account.  
There are additional external variables that can affect the travel time distributions of 
the links. As an example, we can expect to have different conditional distributions for 
a single like under different weather conditions (e.g., sunny or rainy). One of the 
advantages of modeling the travel time distributions using a graphical model is the 
flexibility to represent such conditional dependencies. 
We introduce additional global variables to explicitly represent such information in the 
proposed model, where it is available. These external variables allow the model to 
determine different travel time distributions based on the provided additional 
distributions (e.g., estimating different Gaussian distributions for rainy and sunny 
weather conditions).  
Figure 12 represents the final deployed model. In this figure, 𝑋TUV:WUX,: represents the 
random variable introduced for capturing the weather information at time step 𝑡 . 
Considering the available data sources, in this dissertation, we consider two states for 
the weather; that is 𝑋TUV:WUX,:is a binary random variable, which shows whether in time 
𝑡 the condition was rainy or not. We assume that the random variable comes from a 
57 
 
 
 
 
multinomial distribution with two states. In this study, we assume 𝑋TUV:WUX,:  is 
observed for all time steps.  
The travel time distribution for different links and time steps also varies based on the 
day and season of the year. To model this distribution drift, we introduce two new 
random variables 𝑋YUVYZM and 𝑋@V[ , which denote the season of the year and the day of 
the week. 𝑋YUVYZM is modeled by a multinomial distribution with four discrete states to 
represent each season of the year. 𝑋@V[  is modeled as another multinomial random 
variable and each discrete variable represents a day of the week. These variables are 
observed both during training and inference. In this way, the travel time distribution 
for each link not only depends on the time step, but it also depends on the weather 
condition, the day of the week, and the season of the year 
58 
 
 
 
 
!"# $ ,&'( !"# $ ,& !"# $ ,&)(
!"# $ ,&'( !"# $ ,& !"# $ ,&)(
!$,&'( !$,& !$,&)(
!+,# ,&'( !+,# ,& !+,# ,&)(
*$, &'( *$, & *$, &)(
!-#. !/,#/01
Time	t-1 Time	t Time	t+1
 
Figure 12 The graphical model with the proposed external variables 
 
4.3 Travel Time Allocation 
Once the traversed links between two consecutive GPS points are estimated from map 
matching, the next step is to allocate the total path travel times between the links.  
This can be achieved by maximizing the log-likelihood of the link travel times for each 
observation using equation 4. This optimization could be calculated given the model 
parameters. This means that the probability of link l being in state s, and the parameters 
for travel time distribution of link l for state s should be known; however, we do not 
have this data at this stage. As such, this represents a chicken-or-egg dilemma. The 
optimization has two constraints, shown in equations 5 and 6. First, the sum of travel 
59 
 
 
 
 
times on the links should be equal to the path travel time (travel time between 
consecutive points). Second, the travel time on the links has an upper bound computed 
for each link by using the maximum speed that is realistically possible for the link; this 
maximum speed is 60 mph over the speed limit. 
 
𝐴𝑟𝑔𝑚𝑎𝑥[	a∑e∈gV:W 𝑙𝑛 ∑YYRh 𝑋e,Y𝑔e,Y(𝑦e)i       (4) 
 
𝑆. 𝑡.			 ∑e∈gV:W 𝑦e = 𝑇gV:W         (5) 
𝑆. 𝑡.			𝑦e > 	𝑦e	nVo          (6) 
 
 The travel time is allocated based on the speed limit and length of the links, and then, 
parameters of the model are estimated for the first stream. For the second stream, the 
estimated parameters are used to solve the optimization. The optimization and travel 
time allocation are updated with each stream and the iterations between travel time 
allocation and parameters estimation continue, as long as we have a new stream of data.  
The number of variables in the optimization problem in equation 4 is equal to the 
number of links for the path between two consecutive GPS measurements; this number 
is always a relatively small number, and it makes the optimization problem easy to 
solve using numerical methods. 
60 
 
 
 
 
4.4 Parameter Estimation 
4.4.1 The Inference 
To update the parameters of the model related to each of the defined hidden states, one 
faces a chicken or egg dilemma. To update the parameters related to each of the states, 
the samples – which are more likely from that state – should be determined. However, 
in order to determine these likely samples, the distribution parameters of the model 
should be known. The most frequently used approach to tackle this problem in 
machine-learning is the Expectation-Maximization (EM) approach. 
EM consists of two stages. In the first stage, known as “expectation,” the most likely 
state for each sample is determined based on the current estimated parameters of the 
model. The second stage, dubbed “maximization,”' chooses the best parameters of the 
model given the data and the estimated hidden state from the expectation stage. These 
two stages are repeatedly applied until a convergence criterion is met. Although 
theoretically, the EM algorithm is not guaranteed to converge to the global optima of 
the problem, empirically, the algorithm leads to satisfactory results and is one of the 
most widely used algorithms in the machine-learning community. 
EM provides a practical solution for the aforementioned chicken-or-egg problem. 
However, it is not a truly Bayesian approach, and it follows a frequentist view in the 
expectation stage. In other words, during the expectation stage, it only computes a point 
estimate for the hidden variables (i.e. the most probable value). Another way of solving 
the problem is to stick with the Bayesian approach, even during the expectation stage, 
61 
 
 
 
 
to consider the full posterior distribution for describing the current estimate for hidden 
states. 
In the proposed approach, we consider a generalization of the EM algorithm. For each 
batch, given the seen variables, we use the Variational Bayes approximation inference 
to estimate the posterior distribution of all variables (i.e. finding a set of parameters for 
the posterior distribution which describes the data best). For efficient inference, all 
probability distributions governing the variables are assumed to be a part of the 
exponential family. In other words, all the distributions in our model can be formulated 
as:   
𝑓q(𝑥|𝜃) = ℎ(𝑥)exp	(𝛾(𝜃)𝑆(𝑥) − 𝜆(𝜃))                                                                     (7) 
 
Where h(.) is a function of x, and γ(.) and λ(.) are only functions of θ. S (.) is a function 
of x, known as sufficient statistic, which contains all sufficient information for 
determining the posterior distribution. By these assumptions and by considering 
conjugate priors, one can calculate the posterior distribution in closed form which 
makes the inference very efficient. 
After estimating the posterior of the variables in closed form, we update the parameters 
of the model to match the posterior using the Streaming Variational Bayes and iterate 
these steps through batches (sets of trips) till we visit all batch samples. 
As described earlier, the SVB is used for making the inference, and the training 
model involves iteration between the travel time allocation for a new stream and 
updating the model and the parameters. The proposed algorithm iterates between 
62 
 
 
 
 
finding the most likely state of the network given the model parameters and then uses 
those state estimates to update the most likely model parameters.  
4.4.2 Streaming Variational Bayes (SVB) 
Bayesian inference is time-consuming, especially when it is used for complex models. 
To this end, approximate inference algorithms are usually deployed for real-world 
applications. There are different approaches toward the approximate training of 
Bayesian Networks, including Monte Carlo Markov Chain (MCMC), and Variational 
Inference techniques. MCMC has been used for decades as the primary technique for 
posterior approximation. MCMC forms a Markov Chain on the posterior variable and 
samples from the created chain before finally approximating the posterior with 
empirical estimates computed from the generated samples. However, MCMC becomes 
less usable when the models become more complex or the scale of the data at hand 
becomes large. Variational Inference methods on the other hand use optimization to 
find the approximate posterior. In these approaches, a family of distributions is 
considered to approximate the posterior. Then, an optimization problem is solved to 
find a specific distribution from the family which has the minimum KL-divergence 
with the true posterior. Since the approximate is chosen from a pre-defined family of 
distributions, this family should be defined to be as flexible as necessary. 
The travel time estimation for arterial networks not only leads to complex inference 
models, but it also adds another degree of difficulty to the problem. GPS data from 
users continuously adds live traffic information.  Thus, an effective method should not 
only be able to approximate the posterior distribution effectively, but it should also 
63 
 
 
 
 
change the solution adaptively with minimal computational cost. This incremental 
approach to the problem requires an online learning paradigm for updating the 
parameters of the posterior distribution.  
To address these challenges, we used inference module on a family of approximate 
inference methods called Streaming Variational Bayes. Streaming Variational Bayes 
deploy the Bayes rule to convert a conventional Variational Bayes method into an 
incremental approach. This approach is covered with more detail in the remainder of 
this chapter. 
Assume data is added to the system in i.i.d. batches 𝐵=, 𝐵x, … , 𝐵y where 𝐵H contains 
the samples in batch “𝑖”. Also, let 𝛩 represent the parameters of the model. Assume 
that batches 𝐵=, … , 𝐵{|= have been seen previously and the new batch 𝐵{ is added. To 
have an effective incremental strategy, the parameters estimated from 𝐵=, … , 𝐵{|= 
should be updated efficiently given the newly added batch 𝐵{. This can be achieved by 
directly applying the Bayesian rule. According to the Bayes rule, one can derive the 
following recursive formula: 
𝑝(𝛩|𝐵=, … , 𝐵{) ∝ 𝑝(𝐵{|𝛩)𝑝(𝛩|𝐵=,… , 𝐵{|=)                  (8) 
That is the posterior distribution of the parameters 𝛩  given batches 𝐵= …𝐵{  is 
proportional to the likelihood of the batch 𝐵{ alone and the posterior of the parameters 
given all batches that came prior to batch 𝐵{. 
This formula is the base idea behind the approximate Streaming Variational Bayes 
technique. Let 𝑞(𝛩) = 𝑉𝐵(𝐵, 𝑝(𝛩))  be the naive approximate Variational Bayes 
64 
 
 
 
 
inference method which computes the posterior over the parameters of the model 𝛩, 
given a mini-batch 𝐵 and a prior over the parameters of the model 𝑝(𝛩). Now, let 
𝑞€(𝛩) represent the posterior estimates after seeing mini-batches 𝐵= …𝐵€ i.e. 𝑞€(𝛩) =
	𝑝(𝛩|𝐵= …𝐵€). From equation(8), we can derive a recursive update for computing 
𝑞€(𝛩) based on our estimate from batches 𝐵= …𝐵€|= i.e. 𝑞€|=(𝛩). More precisely, the 
streaming update can be written as follows: 
𝑞€(𝛩) = 	𝑉𝐵(𝐵€, 𝑞€|=(𝛩))          (9) 
This recursive nature makes efficient approximate Streaming Variational Bayes 
method applicable to the travel time estimation problem. Moreover, it is possible to 
compute the actual posterior distribution in a parallel fashion. This is achieved by 
reconsidering the Bayes formula. Assuming mini-batches are i.i.d., we have: 
            
𝑝(𝛩|𝐵=, … , 𝐵y) ∝ 𝑝(𝐵=,… , 𝐵y|𝛩)𝑝(𝛩) =𝑝(𝐵H|𝛩)𝑝(𝛩)
H
𝑝(𝛩|𝐵 )
= [ H𝑝(𝛩) ] 	× 𝑝(𝛩)																																																																									(10) 
H
That is, the problem of computing the posterior given all batches is converted to 
computing the posterior given each of the batches independently and then combining 
the resulting information with Equation (10). Each independent batch update can be 
computed in parallel (Broderick et al., 2013). 
65 
 
 
 
 
Considering Equation (10), a similar formula can be derived when an approximate 
inference method (e.g. Variational Bayes) is used. This approximate update is as 
follows: 
𝑃(𝛩|𝐵= …𝐵€) ≈ 𝑞
…†(†‡,g(ˆ))
€(𝛩) ∝ [ ∏H ] 	× 𝑝(𝛩)                                              (11)g(ˆ)
               
That is the final approximate posterior and can be computed by distributing and 
computing the approximate posterior for each batch independently and then combining 
the results. This leads to an efficient incremental update which is essential for the 
problem at hand. 
4.5 Summary 
This chapter proposed a graphical model in which external variables such as weather 
conditions, season, the day of the week, and time of day are considered in the modeling. 
Each of these variables is shown to be influential in travel time in general.  
In this study, these variables are included in the graphical model as an observed variable 
and as the parent of all the links.  This means we would have different travel time 
distributions for each combination of the mentioned variables. The idea is that, by 
incorporating more of the influential variables, we would yield a better prediction both 
in general as well as in those scenarios. The next chapter demonstrates the model 
performance in most of these scenarios. 
The process of inferring the links ravel-time distribution consists of two steps, travel 
time allocation, and parameter inference. For travel time allocation, an optimization 
model is used to allocate travel time based on the previous model parameters; this refers 
66 
 
 
 
 
to the probability of the link to be in each congestion state and the distribution 
parameters associated with that state. 
For the inference of the parameters based on the allocated travel time, this study 
proposes to use the Streaming Variational Bayes to do approximate Bayesian learning. 
This offers an advantageous possibility of updating the model with streaming data, and 
it is less computationally intensive. The next chapter demonstrates the validation of the 
proposed model and further analysis of the considered variables.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
67 
 
 
 
 
Chapter 5: Analysis  
In Chapter 4 we explained the proposed graphical model with the external variables. 
Any estimation/prediction model should be validated and tested to see how well the 
model is performing. Thus, we apply and test the proposed model on the INRIX probe 
data for a case study, and demonstrate and discuss the results in this chapter. 
This chapter demonstrates the performance of the graphical model with no external 
variables and compares the result with the weighted average model. Then, the 
performance of the proposed model with external variables is illustrated for each day 
of the week, and for each day of the week in each season, and finally in bad weather 
conditions. For each of these scenarios, a representative test day is chosen and the 
trained model is tested and the performance of the model is illustrated in each scenario 
and compared with other models. 
Finally, the chapter ends with two proposed modification of the model which were not 
possible to do in this study, due to lack of enough data. However, the detailed structure 
of the variables and how they should be included in the model is demonstrated and 
discussed. 
5.1 Case Study  
As mentioned earlier, the case study is a subnetwork of Baltimore city. The network 
consists of 1,700 links and 1,250 intersections. The test data are for PM weekday peak 
hours, 4 PM to 6 PM. The considered discrete time in this study is five minutes.  Two 
68 
 
 
 
 
models (proposed model and base model) are developed for PM peak each day of the 
week, and the results are shown as follows. 
Figure 13 demonstrates the available vehicle GPS trajectory data in the Baltimore 
county for a Monday PM peak. As is evident, both the important freeways and 
downtown Baltimore have the highest number of observations.  For the purpose of this 
study and its focus on arterial travel time modeling, downtown Baltimore was chosen 
as the study area. 
Observations
0 - 15
15 - 64
64 - 168
168 - 390
390 - 988
 
Figure 13 The available data on the area 
 
Figure 14 demonstrates the number of observations for the Baltimore downtown, and 
again for a Monday PM peak. Some of the roads have quite a considerable number of 
observations. However, there are some roads for which there is not any observation at 
all.  © OpenStreetMap (and) contributors, CC-BY-SA
 
69 
 
 
 
 
Observations
0 - 1
1 - 32
32 - 67
67 - 116
© OpenStreetMap (and) contributors, CC-BY-SA
 
Figure 14 The available data for the case study area 
5.1.1 Trip Summaries 
The total number of vehicle trips used for the training purpose is 63,419 on the studied 
road network. Table 1 demonstrates the volume of trips used for the training purpose 
by days of the week and by available month. As expected, the number of trips for the 
weekend is much lower than the number of trips for the weekdays since the number of 
trips is less on weekends. The demonstrated data is for PM peak hour (4-6 PM). Table 
2 demonstrates the vehicle GPS points for the training dataset. In total, more than 1 
million GPS points are used to train the proposed model. 
 
 
70 
 
 
 
 
Table 1 Summary of trips used for each month and each day of the week in the training data set 
Time Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total 
June 2,782 3,254 2,507 2,529 2,705 861 802 15,440 
July 2,114 2,328 2,765 2,824 2,544 857 545 13,977 
Feb 2,502 2,491 2,831 2,797 3,059 1,425 970 16,075 
Oct 2,396 2,574 2,673 3,529 3,486 1,980 1,289 17,927 
Total 9,794 10,647 10,776 11,679 11,794 5,123 3,606 63,419 
 
Table 2 Summary of GPS points used for each month and each day of the week in the training data set 
Time Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total 
June 13,510 15,102 13,434 12,537 13,978 4,246 4,047 76,854 
July 9,439 11,157 13,958 13,579 12,186 4,147 2,927 67,393 
Feb 52,435 52,905 80,730 66,830 89,958 70,740 53,534 467,132 
Oct 67,916 77,088 76,907 111,626 113,058 106,314 70,217 623,126 
Total 143,300 156,252 185,029 204,572 229,180 185,447 130,725 1,234,505 
 
5.2 Developed Models 
5.2.1 The Initial Developed Graphical Model 
For the graphical model, we split the data into two data sets. One is used as a real-time 
data set and the other is used as the test data set to determine the performance measures. 
The model uses the average of observed travel time on each link as the current 
observation for the corresponding variable and predicts the travel time for the next time 
71 
 
 
 
 
step using the graphical model shown in the previous chapter. The test data set is used 
to measure the performance of the model and percentage of error for this model.  
To estimate the travel time at each discrete time step, our model performs an 
approximate probabilistic inference on hidden traffic steps shown in equation 12. 
𝑠e,: = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑃(𝑥 = 𝑆∗Š e,: |𝐷h...: ,𝑀)                (12) 
Where, 𝑠e,: is the state of link l at time t to be estimated, 𝑥e,: is hidden state variable and 
S* is the state that makes the probability the highest. 𝐷h...:, 𝑀, is the learned model and 
real-time data.  
Then the Gaussian distribution of that state is used for the estimation is shown below.  
 𝐺(𝜇Y,Ae , 𝜎
Y,A
e )                                                                                                             (13) 
In the next sections, this model is improved by all the external variables, the algorithm 
of training procedure with all the external variables is demonstrated in Algorithm 1.  
5.2.2 Baseline Model 
For the baseline model, we also split the test data into two sets, the real-time data and 
the test data. The baseline model uses the real-time average travel for those links that 
are seen during real-time data. Considering the real-time average or regression as the 
baseline for comparison is a common practice in the literature (Herring,  2009; 
Hofleitner et al., 2012 a; Woodard et at., 2017; Hofleitner et al., 2012 b). Here, the 
baseline model uses the historical prior for those links that are not seen in real-time 
72 
 
 
 
 
data. The test data is used for measuring the performance of the model and the 
percentage of error.  
5.2.3 Test  
As mentioned before, the test data set is used for testing both models. The model can 
also be tested by considering the Bluetooth sensors data as the ground truth travel time 
as proposed by Haghani et al. (2010), for those links with available Bluetooth data. The 
purpose of the test is to estimate the performance of both models and calculate the 
percentage of error for both. To do so, the existing trips GPS point and their timestamps 
in the test data set are used as the ground truth. Both models are used to predict the 
travel time for same trips using the explained procedures. The travel time for each time 
is estimated using equation 14. As mentioned before, this study assumes the 
independency between the links travel time, so, the path travel time in the proposed 
model is the sum of traversed links travel time in the path which can be computed using 
equation 14.  
 
Y
TTgV:WH = 	∑
‡,A‡
H 𝜇e 																			{𝑙 … 𝑙 } ∈ 𝑃𝑎𝑡ℎ                 (14) ‡ = y H
 
The average duration of test trips is above 10 minutes and with different average trip 
lengths for each day of the week. The result of the test is demonstrated in the next 
section.  
73 
 
 
 
 
5.3 Validation of the Proposed Model 
5.3.1 The Model without any Variables  
We implemented the model in the case study and compared the results to the base 
model. At first, we used all the data, without considering the days of the week and 
tested the model for the test data which is also from all days of the week. The 
performance of our model is the same as the base model, and it’s very low. The reason 
could be attributed to the fact that when the days of the week are not considered in 
modeling, the variation is so high that maybe just using the average real-time data could 
be the best approach for the estimation/prediction. The average trip duration is 1068 
seconds (18 minutes). Table 3 demonstrates the baseline error and proposed model 
error in seconds, as well as the percentage of errors for both models.  
 
Table 3 Accuracy of predictions for all the weekdays 
All weekdays 
Average test trip 1067.0 
Proposed model error (s) 399.0 
Baseline (s) 397.2 
Proposed model error % 0.37 
Baseline error % 0.37 
 
74 
 
 
 
 
5.3.2 The Model with Day of the Week  
We also provided a model for each of the weekdays. The average test trip duration 
varies on different days, with an average of 1040 seconds (17 minutes). Table 4 
demonstrates the average trip, the average error in seconds for both the baseline model 
and the model proposed in this study, and the percentage of error for both baseline and 
the model proposed in this study by days of the week. The best performance is the 
Monday model. Thursday and Friday have the highest percentage of errors. 
 
Table 4 Accuracy of predictions based on days of the week for the Baltimore test data 
Day of the week Monday Tuesday Wednesday Thursday Friday 
Average test trip 974.4 978.0 1347.0 1060.5 849.7 
Proposed Model error (s) 210.8 248.4 330.3 279.2 221.0 
Baseline error (s) 351.9 490.6 559.1 515.9 446.9 
Proposed Model error % 21.6 25.4 24.5 26.3 26.0 
Baseline % 36.1 50.2 41.5 48.7 52.6 
 
Figure 15 shows the comparison between the base model and the model proposed in 
this study. As it is shown, there is a huge improvement in the percentage of errors when 
using the model proposed in this study. It should be noted that both baseline model and 
our model approximately have the same trend.  This means that for those days on which 
the model proposed in this study has a lower performance and a higher percentage of 
75 
 
 
 
 
error, the baseline model also has a lower performance. This could be due to demand 
and variation in demand for that day of the week. 
 
 
Figure 15 Comparison of prediction error between proposed model and the baseline by days of the week 
 
Even though the model is developed in a way that it can handle all times of the day, 
since peak hour travel time prediction and validation is the most challenging one, here 
just the peak hours results are demonstrated. 
5.3.3 Travel Time Distribution 
To explain the proposed model, one example of link travel time is demonstrated as 
follows. Figure 16 shows the location of the example link for which a histogram is 
created from raw travel time data. The example link is, E Pratt St. in downtown 
76 
 
 
 
 
Baltimore and the length is 223 feet. Figure 17 shows the travel time frequency for the 
example link (link 13), for the Tuesday training data.  
The histogram of the travel time on this link shows two main peaks, one around 19 
seconds and one around 43 seconds. One could identify two distributions with two 
means. The proposed model for the same link and same days also yields two 
distributions, one for uncongested condition (s = 0) with the mean of 21.47 seconds, 
and one for congestion condition (s=1) with the mean of 45.32, which is very close to 
what is observed. 
 
 
Figure 16 The example link in downtown Baltimore  
77 
 
 
 
 
 
Figure 17 The travel time histogram for link 13 
5.4 Model Improvements 
5.4.1 The Model with All of the External Variables   
In this section, we tested the model for each day of the week and each season. The 
developed model has the potential of being tested on any combinations of all of the 
scenarios (time of day, day of week, season, and weather); however, here the results of 
the analysis for one time of day of (PM peak) and three seasons and five weekdays 
(From Monday-Friday) is demonstrated. This means that we selected 15 (five 
weekdays* and three seasons) test days to evaluate the proposed model vs. based model 
improved by the regression model. Algorithm 1 illustrates the step-by-step procedure 
for the training of the proposed model. In this algorithm, 𝕃 represents the set of all links 
in the network, 𝕋 represents the set of time-steps, and 𝕄H denotes the set of reported 
GPS information for the 𝑖 ’th training trip. 𝑋e,:Uo:  represents the observed external 
78 
 
 
 
 
variables for time step 𝑡  and link 𝑙  in the network. 𝑋H,I˜™š		is the 𝑗 ’th reported GPS 
information for the 𝑖’th training trip. We denote the traffic state of link 𝑙 in time step 𝑡 
by random variable 𝑆e,:. 𝒩(𝑙) represents the set of  neighboring links connected to link 
𝑙.  𝜇eY, 𝜎eY  repsents the mean and standard deviation of the normal distribution over link 
𝑙 conditioned on traffic condition 𝑠. 
An example of predicted traffic states and travel time associated with those states is 
also demonstrated as follows. Figure 18 demonstrates the predicted level of congestion 
for downtown Baltimore on October 26th on Monday at 4:50 PM. The green color 
demonstrates the level of congestion “one” and the red color demonstrates the level of 
congestion “two”. It should be noted that the level of congestion “one” is not 
necessarily uncongested and level of congestion “two” is not necessarily congested. 
They could be congested and more congested since both belong to the same time bin 
of PM peak and they are just comparable to each other in that time bin, meaning we 
can say red is more congested than green.  
 
 
 
 
 
 
 
 
79 
 
 
 
 
Algorithm 1. The training procedure. 
Inputs:  
- 𝑋e,:Uo:		∀𝑙 ∈ 𝕃, 𝑡 ∈ 𝕋  observed external variables for each time step 𝑡 and each link 𝑙. 
-	𝑋H,I˜™š		∀𝑖 ∈ ℕ, 𝑗 ∈ 𝕄H	the time of 𝑗Ÿ𝑡ℎ reported GPS data for trip 𝑖 in the training.  
 
Outputs: 
- 𝑃a𝑆e,: = 𝑠 𝑆𝒩(e),:|=i		∀𝑙 ∈ 𝕃, 𝑠 ∈ 𝕋	the probability that link 𝑙 has the discrete state 
𝑠 in time step 𝑡 given the state of its neighbors in time step 𝑡 − 1. 
- 𝜇eY, 𝜎eY 	∀𝑙 ∈ 𝕃, 𝑠 ∈ 𝕊	the mean and standard deviations of the Gaussian distribution for 
link 𝑙 for each discrete state 𝑠. 
 
1: while not converged do 
Ø CREATE A BATCH BY SAMPLING THE TRIPS 
2:   𝔹 ← 𝑠𝑎𝑚𝑝𝑙𝑒(1…ℕ) 
 
Ø ALLOCATE TRAVEL TIMES BY SOLVING THE OPTIMIZATION 
3:   𝑋H,e,:˜™š ← 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒(¥𝑋
H,I e,:
˜™š¦ , ¥𝑃a𝑆 = 𝑠 𝑆𝒩
(e),:|=i¦
e∈𝕃,:∈𝕋) H∈𝔹,I∈𝕄‡
 
Ø EXPECTATION STEP: INFER THE SET OF HIDDEN STATES USING 
VARIATIONAL MESSAGE PASSING 
4:    ℋe,: ← 𝑉𝑀𝑃(𝑋H,e,: , 𝑋e,: , 𝜇e , 𝜎e, 𝑃a𝑆e,: = 𝑠 𝑆𝒩(e),:|=˜™š Uo: Y Y i	)  
 
Ø MAXIMIZATION STEP: PERFORM THE MAXIMIZATION STEP BY 
APPLYING STREAMING VARIATIONAL BAYES 
5:     𝜇e , 𝜎e, 𝑃a𝑆e,: = 𝑠 𝑆𝒩(e),:|=i ← 𝑆𝑉𝐵(ℋ𝑙,𝑡	, 𝑋𝑖,𝑙,𝑡 , 𝑋e,: , 𝜇e , 𝜎e, 𝑃a𝑆e,: = 𝑠 𝑆𝒩(e),:|=Y Y 𝐺𝑃𝑆 Uo: Y Y i) 
 
6: end while 
7: return (𝜇e e e,: 𝒩(e),:|=Y, 𝜎Y , 𝑃a𝑆 = 𝑠 𝑆 i)	∀𝑙 ∈ 𝕃, 𝑠 ∈ 𝕊  
 
  
 
 
 
 
 
 
80 
 
 
 
 
 
Figure 18 Real-time level of congestion subnetwork of Baltimore downtown. 
 
Figure 19 demonstrates the predicted travel time over travel time with the speed limit. 
Green is for values equal or above 0.7 meaning the predicted travel time is close to the 
travel time driving at the speed limit. The black color indicates the relative travel time 
below 0.2 meaning the predicted travel time is one-fifth of the travel time at the speed 
limit. The red and orange are the relative travel time in between two mentioned 
numbers.  
 
81 
 
 
 
 
 
Figure 19 Real-time relative travel time of Baltimore downtown network. 
 
What follows explains how the base model improved by the regression model. Also, 
an Algorithm is dedicated to demonstrate steps of developing this base model.  
5.4.1 Regression Model 
In order to improve the base model a regression model is developed and the proposed 
model is compared to the improved base model. The implemented regression model 
has two different sets of explanatory variables. Trip variables such as weather 
conditions, time of day, day of the week, and season, as well as link explanatory 
variables such as the speed limit, the road type, and the length of the link between the 
two intersections. The goal is to capture the underlying factors behind spatial and 
82 
 
 
 
 
temporal variations in the travel time. The real-time prediction in each time step would 
be a combination of the regression prediction and the real-time observations. Equation 
15 shows the developed regression model and Algorithm 2 demonstrates the detailed 
procedure from travel time allocation to updating the travel time prediction for each 
link using real-time data and the regression model. 
𝑌e,: = 𝑊e e e,: e,: e,:Uo𝑋Uo + 𝑊:XHg𝑋:XHg + 𝑏                                                                               (15) 
In this algorithm, 𝕃	𝑎𝑛𝑑	𝕋 represent the set of all links in the network, and the set of 
time-steps respectively.		𝑋eUo	is the set of explanatory variables for link 𝑙, 𝑋
e,:
:XHg  is the 
set of trip variables for link 𝑙 in time-step 𝑡, 𝑋e,:Z€Y is the set of real-time observations for 
link 𝑙 in time-step 𝑡, and  𝑌e,:		is the travel time for link 𝑙 in time-step 𝑡. We represent 
the size of a set by |. |. 
 
 
 
 
 
 
 
 
 
 
 
83 
 
 
 
 
Algorithm 2. The regression model algorithm 
Inputs:  
- 𝑋eUo		∀𝑙 ∈ 𝕃, 𝑡 ∈ 𝕋	 explanatory variables for each link. 
- 𝑋e,::XHg		∀𝑙 ∈ 𝕃, 𝑡 ∈ 𝕋	 trip variables for each link at each link 𝑙 and time 𝑡. 
-	𝑋e,:Z€Y		∀𝑙 ∈ 𝕃, 𝑡 ∈ 𝕋	  real-time observations for each link at each link 𝑙 and time 𝑡. 
 
Outputs: 
- 𝑌e,:		∀𝑙 ∈ 𝕃, 𝑡 ∈ 𝕋	 travel time for each link 𝑙 and each time step 𝑡. 
 
1: for each 𝑙 ∈ 𝕃 do 
2:     for each 𝑡 ∈ 𝕋	 do 
3: Find 𝑊e , 𝑊e,:Uo :XHg , and 𝑏e,: by solving the regression given training samples 
4:    end for 
5: end for 
 
6: for each 𝑙 ∈ 𝕃 do 
7:     for each 𝑡 ∈ 𝕋	 do 
8: 𝑛Z€Y ← |𝑋
e,:
Z€Y|  
9: if 𝑛Z€Y = 0 then  
10:     𝑌e,: ← 𝑊e 𝑋e +𝑊e,: 𝑋e,: + 𝑏e,:Uo Uo :XHg :XHg  
11: else: 
12:     𝑌e,: ← = ∑Mª«¬ e,:
Mª«¬ MR=
𝑋Z€Y[𝑛] 
13: end if 
14:    end for 
15: end for 
16: return 𝑌e,:			∀	𝑙 ∈ 𝕃	, 𝑡 ∈ 𝕋	     
5.4.2 Comparison Between the Final Model and Improved Base Model  
The average test-trip duration varies on different days and in different seasons. Figure 
20 through Figure 22 and Table 5 through Table 7 demonstrate the average trip, the 
average error in seconds for both the improved base model by regression and the model 
proposed in this study, and the percentage of error for both the improved base model 
and the model proposed in this study by days of the week for each season.  
84 
 
 
 
 
Figure 20 and Table 5 demonstrate the analysis results for winter season and all of the 
weekdays. As it is shown, there is a significant improvement in the proposed model 
with all the variables compared with the previous model across nearly all weekdays. 
Also, there is a huge improvement in the percentage of errors for the base model 
(regression and real-time average). Still, the proposed model significantly outperforms 
the improved base model.  
Table 5 Summary of winter predictions based on days of the week for the Baltimore test data 
Day of the week Monday Tuesday Wednesday Thursday Friday 
Average test trip 544 511 546 518 592 
Proposed Model error (s) 83 120 99 111 117 
Baseline error (s) 138 166 179 152 157 
Proposed Model error % 15.3 24.1 18.5 22.2 19.6 
Baseline % 25.3 31.8 32.3 28.2 26.9 
 
85 
 
 
 
 
35%
30%
25%
20%
15%
10%
5%
0%
Monday Tuesday Wednesday Thursday Friday
Baseline model error % Proposed model error %
 
Figure 20 Comparison of prediction error between proposed model and base model in winter 
Figure 21 and Table 6 demonstrate the same analysis for the summer and Figure 22 and 
Table 7  demonstrate it for the fall. As it is shown there is a general improvement both 
in the base model and the proposed model. 
35%
30%
25%
20%
15%
10%
5%
0%
Monday Tuesday Wednesday Thursday Friday
Baseline model error % Proposed model error %
 
Figure 21 Comparison of prediction error between proposed model and base model in summer 
86 
 
 
 
Prediction Error Prediction Error 
 
Table 6 Summary of summer predictions based on days of the week for the Baltimore test data 
Day of the week Monday Tuesday Wednesday Thursday Friday 
Average test trip 570 529 553 583 602 
Proposed Model error (s) 59 95 95 107 123 
Baseline error (s) 112 152 159 122 191 
Proposed Model error % 10.4 18.5 17.8 18.4 20.1 
Baseline % 19.8 28.0 27.8 20.8 32.2 
 
 
35%
30%
25%
20%
15%
10%
5%
0%
Monday Tuesday Wednesday Thursday Friday
Baseline model error % Proposed model error %
 
Figure 22 Comparison of prediction error between proposed model and base model in fall 
 
 
 
87 
 
 
 
Prediction Error 
 
Table 7 Summary of fall predictions based on days of the week for the Baltimore test data 
Day of the week Monday Tuesday Wednesday Thursday Friday 
Average test trip 502 542 503 550 537 
Proposed Model error (s) 107 89 104 121 108 
Baseline error (s) 132 180 141 163 152 
Proposed Model error % 21.2 17.4 21.1 21.8 20.2 
Baseline % 26.4 31.4 27.6 30.1 28.1 
 
On average, the model performance in the summer is better, with an average percentage 
of error of 17% for all days of the week as shown in Table 8 and Figure 23. This could 
be attributed to the higher number of observation for the summer season. We had two 
months of data for summer, June and July. This also could be attributed to the fact that 
summer has less traffic demand and better weather conditions, so traffic patterns are 
more predictable. The fall and winter average performance for different days of weeks 
are the same, with an average percentage of error of 20%.  
On average, the performance of the proposed model is better on Mondays, with an 
average percentage of error of 15.5% for all the seasons (Table 9 and Figure 24). This 
is followed by Friday and Wednesday. The highest percentage of error is for Thursday 
with the average percentage error of 21 %. This again could be attributed to the lower 
traffic demand on Mondays and better predictions of the traffic pattern for them.  
 
 
 
88 
 
 
 
 
Table 8 Aggregated average percentage of error for different season 
Season Average Percentage of Error % 
Fall 20.3 
Summer 17.0 
Winter 19.9 
 
21.0%
20.0%
19.0%
18.0%
17.0%
16.0%
15.0%
Fal l Summer Winter
Seasons
 
Figure 23 Aggregated average percentage of error for different season 
Table 9 Aggregated average percentage of error for each days of week 
Season Average Percentage of Error % 
Monday 15.6 
Tuesday 20.0 
Wednesday 19.1 
Thursday 20.8 
Friday 20.0 
89 
 
 
 
Percentage of Error
 
 
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
Monday Tuesday Wednesday Thursday Friday
Days of the Week
 
Figure 24 Aggregated average percentage of error for each days of week 
 
5.4.3 Comparison Between the Two Graphical Models   
To demonstrate how incorporating other variables into the graphical model has led to 
better travel time prediction models, the performance of the two graphical models are 
compared. As discussed before, one of these models is a model including the season 
and weather conditions (discussed in section 5.4.1), and the other model is the one with 
only the day of week and time of day variables and not the other two (discussed in 
section 5.2.1).  
The problem is that we have different parentages of error for different days and seasons; 
to overcome this challenge, the average of the percentage of error for all of the 
weekdays are estimated, and a comparison of the two models’ performance is done for 
each day of weekdays. As it is shown in Figure 25, there is an evident improvement for 
90 
 
 
 
Percentage of Error
 
the weekdays. For most of the weekdays, there is more than 5% improvement in the 
percentage of error.  
30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
Monday Tuesday Wednesday Thursday Friday
Model without external variable Model with external variable
 
Figure 25 Comparison of the proposed model with seasons and weather conditions and without them 
 
It should be noted here that some of the test datasets might be in the bad weather 
conditions or good weather conditions. In the next section, in order to specifically 
evaluate the effects of weather conditions variables on the performance of the model, 
another model is trained with all the variables, except for the weather conditions, and 
it is tested on a rainy day. 
5.4.4 The Model Test on a Rainy Day  
As mentioned before, one of the objectives of this study is to develop an integrated 
model that could predict travel times under various weather conditions. Thus, in this 
study, various weather data sources were used, and the variables were included in the 
91 
 
 
 
Percentage of Error
 
graphical model as a trip (global) variable. The test day is Friday, October 9th which 
was rainy during the PM peak hour. To evaluate the proposed model in bad weather 
conditions, a test day in rainy weather conditions was selected and results of the 
proposed model with weather variable inclusion vs without this variable and also base 
model improved by the regression model is demonstrated in Table 10 and Figure 26. 
The average test trip duration is 491 seconds (more than eight minutes). Table 10 
demonstrates the average trip, the average error in seconds for the graphical model 
without weather conditions variable, and including the weather condition variable –  as 
well as the base model with regression. 
Figure 26 shows the comparison between the aforementioned models. As it is shown, 
there is an improvement in the percentage of errors when using the proposed graphical 
model with weather conditions variable compared to the graphical model without it as 
well as compared to the base model improved by a regression model that includes 
weather conditions variable. 
 
Table 10 The comparison of the model with and without weather variable 
Model Percentage of Error % Model error (s) Average test trip 
Model with Rain 22.1 110 491 
Model without Rain 24.0 117 491 
Base model 29.0 141 491 
 
92 
 
 
 
 
30.0%
29.0%
29.0%
28.0%
27.0%
26.0%
25.0%
24.0%
24.0%
23.0%
22.0%
22.0%
21.0%
20.0%
Base model Model with weather Model without weather
 
Figure 26 The comparison of the model with and without weather variable 
5.4.1 The Time of Day   
Even though the analysis is shown for the most challenging time of day, PM peak, the 
proposed graphical model has the capability of providing travel time 
estimation/prediction in any time of day because it includes the time of day variable in 
the model. In this study, the time of day is categorized into bins based on the 
characteristics of traffic. The bins times are chosen in a way that the traffic pattern is 
nearly consistent in each bin, and different from others. 
The time period bins include the following bins:  
• “AM Rush Hour”: weekdays 7 – 9 AM 
• “PM Rush Hour”: weekdays 3 – 6 PM 
• “Nighttime”: Monday-Sunday 7 PM – 6 AM 
• “Weekday Daytime”: remaining times during weekdays 
93 
 
 
 
Percentage of Error
 
• “Weekend”  
Table 11 and Figure 27 demonstrate the performance of the proposed model with all 
the variables for one of the other time bins, weekday daytime bin and winter season, as 
an example. As it is shown in Figure 27, the percentage of error for the proposed model 
is less than the base model. However, there is an improvement in the base model 
performance compare to PM peak base model. This could be attributed to the fact that 
the average or regression model might be able to capture the pattern to some extent 
better for the time bins with less traffic congestion since there is less variability in the 
traffic compared to time bins with more traffic congestion.  
 
Table 11 Summary of winter predictions based on days of the week for daytime 
Day of the week Monday Tuesday Wednesday Thursday Friday 
Average test trip 524 549 537 537 545 
Proposed Model error (s) 96 105 99 130 107 
Baseline error (s) 142 136 135 165 156 
Proposed Model error % 18.7 19.8 18.7 20.3 19.8 
Baseline % 26.6 23.9 24.8 26.5 28.5 
 
 
94 
 
 
 
 
30%
25%
20%
15%
10%
5%
0%
Monday Tuesday Wednesday Thursday Friday
Baseline model error % Proposed model error %
 
Figure 27 Comparison of prediction error between proposed model and base model for day time in winter 
 
Other Modifications 
This section focuses on the possible improvements of the graphical model by adding 
as much as information as possible. What follows demonstrates the proposed model 
which includes intersections in the modeling process and the associated graphical 
mode. The section after that proposes an approach by which driver habits and traffic 
conditions could be modeled by introducing a new variable and adding it to the model 
and demonstrating it in the graphical model.  
5.4.2 The Graphical Model with Intersection  
In most of the arterial travel time studies, travel time (expected value or the distribution) 
on the links is provided and the trip travel time is calculated as the sum of the link travel 
times. The alternative, however, is to include travel time on links and delays in the 
95 
 
 
 
Prediction Error 
 
intersections, and calculate trip travel time by adding the link travel times and delays 
at intersections. 
In general, the turn delays can be considered as deterministic or stochastic variables. 
The variation in the delay comes from unobserved changes such as in traffic flows, 
signal phasing, driver behavior, etc. Considering intersection delay as a separate entity 
is very important when we want to distinguish between right turn, left turn, and forward 
movements. If not for these three different movements, the delays at intersections could 
easily be added to the link travel times and there is no need to consider them separately.  
Including the intersection variable in the modeling requires much more data. Because 
there were not enough samples for each of the movements, the proposed model could 
not be tested.  We propose this model for future work when and if there is enough data 
for each movement.  
In this modeling, there are three main challenges to overcome. The first would be to 
determine what variable should be considered in the modeling to represent the reality 
the best. We propose the movement variable. The movement variable is any movement 
from a link to another connected link.  
It could be noted that the movement variable for an intersection is not just one variable. 
Any combination of movement from any incoming link to any outgoing links could be 
considered separately, as shown below. However, one might be interested in just some 
of the movements such as the left turn and through movements. In Figure 28, there are 
two incoming links and two outgoing links, which make the total number of movements 
4 (2*2). The number would be much higher when there are four incoming and four 
outgoing links (16 movements). For this reason, having the appropriate number of 
96 
 
 
 
 
observations is key because all of these movements should be backed by enough 
observations to ensure we have a good model. 
 
 
 
Figure 28 The intersection movement demonstration 
 
The second challenge is to create a new network of the city in which all the intersections 
(movements) are added and the graphical model is made to include the movement 
variables. To ensure this, a dummy link variable is proposed instead of each movement, 
and new parents and children are created based on this new proposed model structure. 
The proceeding nodes Xpa(l,t) (links) are now parents of dummy node Dl,t(intersection) 
and Dl,t is the parent of  Xl,t as shown in Figure 29. 
97 
 
 
 
 
 
Figure 29 The proposed graphical model including the intersections 
 
The third challenge is how to allocate the observation travel time to the movements. 
For the link-based travel time, the travel time is allocated only on the links. When 
considering the intersections, the travel time should be allocated to the combination of 
links and intersection (movements). For that, the same optimization model explained 
in section 4.3 can be used. The optimization needs to be solved to decompose the path 
travel time to links’ travel times and intersections’ (dummy variables) travel times. The 
new bound for lower-bound and upper-bound for the intersections’ travel times is 
proposed based on the intersection type.  
The final proposed model provides the travel time distribution on the links as well as 
travel time (delays) distribution for the intersections, and a trip travel time is the sum 
98 
 
 
 
 
of the traversed link travel times and traversed intersection travel times, as 
demonstrated in Figure 30 
 
 
 
Figure 30 Demonstration of path travel time as the sum of links and intersections travel times 
 
5.4.3 The Graphical Model with Driver Habits Variable   
Another variable that is assumed to have an influence on arterial travel time modeling 
is the driver behavior variable. This study suggests a new variable that can capture 
driver habits and traffic conditions. This is particularly applicable when the vehicles’ 
speeds are also available. If it they are not available, the speed can still be estimated 
based on the time and the link lengths, but might not be as accurate. This variable is an 
observed variable and it provides a way to capture dependency of travel times across 
links. The new suggested variable is shown in equation 16. 
Vup to now= Avg. Vdriver/ Avg. Vspeed limit                           (16) 
99 
 
 
 
 
Figure 31 demonstrates how to include this variable in the model structure. The new 
variable could be used for both the initial graphical model as well as the graphical 
model with intersections. In this study, due to the low number of observations, this 
variable effect could not be tested.  This study suggests these variables for future studies 
for which a higher number of observations is available. This variable also could 
potentially be used to provide the individual travel time prediction based on individual 
driver behavior.  
 
 
Figure 31 The proposed graphical model with proposed relative speed variable 
 
100 
 
 
 
 
5.5 Summary 
This chapter demonstrated the validation and comparison of the model with a base 
model. It also demonstrated that the graphical model performance would improve with 
the addition of the explanatory variables. There is an improvement by adding the days 
of the week variable, seasons, as well as weather conditions, all of which are presented 
by testing the model on a suitable test day. Model percentage of error in most scenarios 
is around or less than 20 %. Testing the model with weather conditions variable on a 
rainy day showed a lower percentage of errors when compared to the graphical model 
without this variable.  
The model is also compared with a base model with regression and a real-time average. 
The regression model includes the link explanatory variables such as speed limit, 
length, etc., and the trip explanatory variables such as seasons, weather conditions, etc. 
Even though there is a significant improvement in the performance of the base model 
when switching from real-time average to the combination of real-time average and 
regression model, the proposed graphical model with all the variables still outperforms 
the base model.  
There were two model modifications that this study aimed to explore but, due to the 
low number of observations and lack of data, the modifications could not be tested to 
see their impact on the performance of the model. Given this, we have left the 
implementation and testing of these modifications for future research. The first 
important modification is the inclusion of the intersection movements, which would 
101 
 
 
 
 
cause the model to have a different delay time for straight movement vs. left turn 
movement.  
 
 
102 
 
 
 
 
Chapter 6:  Comparison and Sensitivity Analysis  
Validation of the model via separate test data is a common way of validating any 
prediction model; however, in reality, one may want to compare a model to existing 
commercial models. To the best of the author’s knowledge, none of the proposed 
models in existing literature have been compared to any commercial prediction model. 
As such, there is not a framework to do so. A comprehensive framework for comparing 
the model proposed in this study (or any model) to commercial travel time providers is 
proposed. The comparison of the proposed model in this study with the Google API, 
and the original observations are demonstrated in this chapter.  
The chapter continues with the sensitivity analysis on a parameter of the proposed 
model as well as available data, and the results of the sensitivity analysis are illustrated 
and discussed.  
6.1 Validation  
Another way to validate the proposed model is to compare the outcome of this study to 
Google estimates. There are two challenges with this approach. First, we do not have 
the current probe data and Google API only provides the travel times for current or 
future time. This poses a big challenge, and there is no way to avoid this obstacle 
without access to current time vehicle trajectory data. However, we can develop a 
general estimate on how the model performs in off-peak hours, when the traffic has a 
lesser effect on travel time. As such, the travel times for the roads network are 
compared to each other. This just gives us an estimate of how our model is performing 
103 
 
 
 
 
in general and without the effect of traffic. It should be noted that we may want to 
compare only those trips’ travel times that have common route with the observed GPS 
trajectory. As such, the second challenge is that the Google-proposed route should be 
the same as the observed trajectory. To address this, we have proposed to use a shape 
similarity algorithm to only select those trips that have the same observed routes as the 
Google routes. Here, the probe vehicle trajectory and Google routes should be 
compared to each other to determine they are the same routes and comparison is 
meaningful.  
6.2 Shape-Matching 
Shape-Matching is an important area of research that has many applications in 
computer vision, and it is the area of arbitrary target object detection. In computer 
graphics, the Hausdorff distance is used to measure the difference between two 
different representations of the same 3D object. It is also used in biological sciences 
for the analysis of protein structures.  Considering we have point sets 𝐴 ∈ 𝑅x  and ∈
𝑅x , the goal is to find the level of similarity between A and B sets. There are several 
algorithms for finding the shape similarity including Hausdorff, Fr´echet distances, and 
Minkowski distance (Veltkamp, 2001). In this section, we try to use these algorithms 
to find the similarities between two routes in three dimensions (Veltkamp, 2001). This 
mathematically means that we want to find a match that has one of the following 
characteristics:  
• Minimizing maximum distance between mapped points  
• Minimizing sum of distances between mapped points 
104 
 
 
 
 
• Minimizing sum of squared distances between mapped points 
6.2.1 Hausdorff Distance 
The Hausdroff distance is a measure of how far two subsets are. It is calculated by 
determining the maximum distance from a set to another set in which each point in a 
set A is matched to the nearest point in set B.  
 
The Hausdorff distance from A to B, h(A, B), is 𝑚𝑎𝑥V∈®𝑚𝑖𝑛€∈†||𝑎 − 𝑏||, 
The Hausdorff distance from B to A, h(B, A), is 𝑚𝑎𝑥€∈†𝑚𝑖𝑛V∈®||𝑏 − 𝑎||, 
The Hausdorff distance between A and B, H(A, B), is max {h(A, B), h(B, A)}. 
(Hausdorff and Frechet distance Lecture, 2017) 
 
Basically, h(A, B), for each point on dataset A, first finds the nearest neighbor point to 
it on dataset B, and finally reports the farthest of these distances. In this way, Hausdorff 
distance measures the mismatch between these two datasets. The idea is shown in 
Figure 32. (Alt & Guibas 1999).  
 
105 
 
 
 
 
A
Min
Min Min Min
Max
Min Min MinB Min
Min
 
Figure 32 Hausdorff distance as a measure of similarity 
 
6.2.2 Fr´echet Distance 
The Fr´echet distance is explained with the following example.  Assume a man is 
walking a dog where the man is walking on one curve and the dog on another curve. 
The minimum length of a leash allowing a dog and its owner to walk along the two 
curves without backtracking is defined as the Fr´echet distance (Figure 33). The 
Fr´echet distance is known to be a better measure of shape-matching for curve or 
surface matching.  
The Fr´echet distance between two curves is calculated as follows: 
 
𝐹(𝐴, 𝐵) = 𝑖𝑛𝑓°,>𝑚𝑎𝑥:∈[h°,=]||𝐴a(𝑡)i − 𝐵(𝛽(𝑡)||                                                              (17)                                   
 
where A, B : [0, 1]	𝑅x are parametrizations of the two curves 
106 
 
 
 
 
 and 𝛼, 𝛽: [0, 1] → [0, 1] range overall continuous and monotone increasing functions. 
(Alt & Godau, 1995; Alt & Godau, 1999) 
 
Figure 33 Fr´echet distance as a measure of shape similarity  
6.3 Route Similarity 
As mentioned before, this study proposes using shape-matching to find the similarities 
between two routes in order to compare the travel times of two similar routes. In this 
study, two algorithms of Fr´echet distance and Hausdorff distance were implemented. 
The Hausdorff distance seems to be a better indication of route similarity and is chosen 
for this aim.  
The implemented algorithm is an approximation to the Hausdorff Distance for two 
geometries. This algorithm measures the degree of similarity between two geometries. 
The computed measure is in the range of [0, 1]. Higher measures indicate a great degree 
of similarity, which means that a measure of 1.0 shows two exact matches and a 
107 
 
 
 
 
measure of 0.0 indicates that the two have essentially no similarity. The algorithm takes 
the geometry of the two routes of interest and returns the normalized Hausdorff metric. 
When comparing travel times, it is important to use the same route; otherwise, the 
comparison is meaningless. Knowing this, we calculate this metric for probe data 
trajectory vs. Google routes. The travel time estimation comparison can be calculated 
between routes that have a similarity score higher than 0.99.  
Figure 34 shows an implementation of the proposed algorithm for a GPS route (in blue) 
vs. Google route (in red) with a 0.7005 (70.05%) similarity index. The origin-
destination is the same for both GPS and Google routes, and it is evident the routes 
have the same path up to a point and they bifurcate after that and meet again at the 
destination.  
 
 
Figure 34 Route similarity index example (two routes with similarity index of 70.05%) 
 
 
108 
 
 
 
 
 
6.4 Google Maps API  
The Google Maps Directions API is a service that estimates directions and travel time 
between locations. The directions between two locations are provided for several 
modes of transportation including transit, driving, walking, and cycling. The challenge 
is that the Google cost model is not available to the public to allow comparisons of 
travel time estimations for the proposed model vs. the Google model. Also, it is not 
possible to acquire all the link combinations with Google API, as it returns only the 
most efficient routes in terms of travel time when calculating directions between two 
locations. For this reason, route similarity is critical to making it possible to compare 
only the routes that are identical.  
6.4.1 General Information 
Google API can provide an estimate for current time or future time. There is a limitation 
of 2,500 free directions requests per day with Google API. There are several parameters 
that can be changed in the URL, some of them are required while others are optional. 
Parameters are separated by the “&” character. Two of the important optional 
parameters for us are “departure_time”, and “traffic_model”. 
6.4.2 Departure Time 
Google uses the parameter “departure_time” to identify a specific travel time; the value 
is the time specified as Unix time, or seconds since midnight, January 1, 1970, UTC.  
109 
 
 
 
 
6.4.3 Travel Time with Traffic 
The other optional parameter to use to collect travel time is “traffic_model,” which can 
pull the following values: best _guess, optimistic, and pessimistic. This parameter’s 
values change the “duration_in_traffic” value.  
traffic_model parameter values: 
1. Best _guess (which is the default factor) is based on historical averages. 
2. Optimistic gives the duration_in_traffic equal to the lowest value in the range 
Google Maps provides (based on observations for a couple cases). 
3. Pessimistic gives the duration_in_traffic equals to the highest value in the range 
Google Maps provides (based on observations for a couple cases). 
The best application of the proposed method would be when there is real-time data, and 
the proposed model could provide a prediction of travel time that could be compared 
with Google API using the Best_guess and Current_time parameters. However, this 
data is not available and this approach could not be taken in this study. Nevertheless, it 
can still be used to find how the model is performing in comparison with Google API. 
In this way, during the off-peak hours, the estimated travel times could be compared, 
removing the real-time effects and the base estimation without traffic. The proposed 
approach could be used in any other studies and most importantly when real-time data 
is available.  
110 
 
 
 
 
6.5 Waypoint  Parameter 
There is an optional parameter in Google API through which one can specify the 
waypoints available for driving, walking and bicycling, but not transit. Using this 
parameter, however, changes the route from the original in most of the cases, either 
because of an observed GPS error or based on Google API logic. As such, it would still 
be necessary to use route similarity. Furthermore, the advantage of using default 
Google routing instead of a giving waypoint is that the route selections could be 
compared as well. This is particularly important when there is also a routing suggestion 
based on an existing cost model. In this way, both the cost model and the routing 
suggestions could be compared. This parameter in Google API collects the specified 
array of waypoints (latitude/longitude coordinate) and alters the route by directing it 
through the specified waypoints. This study tried this feature as well, but it turned out 
using the route comparison and checking for similarity higher than 0.99 is a better 
approach.  
6.6 Comparison 
As mentioned, the comparison of the proposed model and Google travel time 
estimation were conducted for the off-peak hours and equivalent times (same day, 
month, time) and ‘best_guest’ parameters. The proposed model and Google travel time 
estimations are also compared with the actual travel time observed from the probe data. 
The steps that need to be taken for the compression between the proposed model, 
Google model, and the probe data is shown in Figure 35. The indicated steps were 
111 
 
 
 
 
taken, and the results of the comparison for the Baltimore area are demonstrated in the 
following section. 
 
Step 1: Getting the proposed model and Google ETA for the 
same origin/destination and time. 
Step 2: Calculating the route similarity score for routes pairs & 
compare the travel time just for routes with similarity score> 0.99 
Step 3: Compare the proposed model, Google travel time 
estimation and probe data travel time 
 
Figure 35 The proposed framework for travel time estimation comparison between models 
 
6.6.1 Case Studies 
This section demonstrates how the proposed framework is used for the study area to 
compare the proposed model in this study to the Google API and the actual 
observations. As mentioned before, since we do not have access to real-time data, we 
illustrated the framework using historic data.  
To do this, all the historic data for all the training datasets were used as the input for 
the model to be trained. Once the model was trained, the model predictions of travel 
times for the test day again for 12 AM to 6 PM time, and with the same weather 
conditions are obtained. As such, for all the trips that occurred on the test date, we have 
the origin/destination, GPS track, and the proposed model’s predicted travel time. For 
112 
 
 
 
 
all those trips, we acquired the Google estimates as well. In Google API, we entered 
the same day, time of day, day of week and month and same weather condition as the 
test day, and used the “best_guess” parameter as the API input for the future data since 
only current and future times are available through Google, however, If we had the 
current data, we could simply use the current Google API estimation. In this way, we 
obtained the origin-destination, GPS track, and the estimated travel time of Google as 
well.  
The next step was to find those trips with a shape similarity index between the actual 
observations and the Google proposed path. For trips that the observation track and 
Google proposed route are the same, the average of travel times and errors for proposed 
model, the Google, and actual observation is demonstrated in Table 12. The details of 
travel times are shown for a sample of these trips in Table 13. What follows 
demonstrate the results of this comparison between the Google API, proposed model, 
and the observations. 
6.6.2 Results  
6.6.2.1 Off-peak Travel Time Comparison 
Figure 36 demonstrates an example of the comparison between the Google API and 
proposed model estimated travel time and observed travel time. The given trip is on 
Tuesday, in winter, and with fair weather conditions and non-peak hour traffic 
(February 10th at 00:57 AM). 
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The example trip route has a similar route based on a Google API proposed route, which 
means that the shape similarity index between the two routes is higher than 0.99 and 
we can compare the travel times. 
The observed travel time is 500 seconds; whereas, the Google API estimate is 480 
seconds; finally, the proposed model estimation is 508 seconds. In this case, there is a 
4% difference between the observed travel time and Google API estimation, and a 1% 
difference between the observed travel time and proposed model estimation. Also, the 
difference between Google and the proposed model estimation is 0.3 %. As mentioned, 
this is just one example to demonstrate the comparison process. As such, the 
comparison is done for the test dataset and results are demonstrated and discussed as 
follows. 
114 
 
 
 
 
Google
Observation
   
Figure 36 Proposed Model, Google API, and observed travel time comparison (off-peak) 
 
The selected test case is based on trips on a Tuesday in winter under normal weather 
conditions, for midnight to 5 am in the morning, for a travel length of more than five 
minutes. Around 190 cases were randomly selected from which around 40% of them 
had the same route as the Google API route with a shape similarity index of 0.99 and 
above. 
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The sample 90 tested cases have an average trip of 420 seconds, with a maximum of 
608 seconds. From this sample, 10 trips were doubled-checked manually (both probe 
trajectory and timestamps) and all appeared normal.  
Figure 37 Google and proposed model Percentage of error trend demonstrates the travel 
difference from the ground truth trend (they are not necessarily equivalent points) for 
the proposed model and Google. As it is shown, the proposed model has the tendency 
to estimate a higher travel time than the ground truth, and Google’s estimates are lower 
than ground truth 50% of the time.  
As is shown in Table 12, the average estimation error for the proposed model is 20%; 
whereas, the average estimation error for the Google API is 22%. This analysis 
demonstrates the proposed process and the result of the analysis for 90 cases. The 
analysis shows that the proposed model estimation does not yield results that stay far 
from Google API, even though historical data is used. 
 
 
116 
 
 
 
 
1
0.5
0
0 20 40 60 80 100
-0.5
-1
-1.5
Cases
Peoposed Model Google
 
Figure 37 Google and proposed model Percentage of error trend (off-peak) 
 
Table 12 Google and proposed travel time prediction summary (off-peak) 
Variable Value 
Number of Observations  90 
Average Trip Time (s) 420  
Average Proposed Model Average Error (s) 75  
Average Google Average Error (s) 84  
Proposed Model Average Error % 20  
Google Average Error % 22  
 
 
 
 
117 
 
 
 
Percentage of Error
 
Table 13 Sample of Google, proposed model travel time prediction and observations travel time comparison (off-
peak) 
Trip Observation (s) Proposed model (s) Google (s) Google-Obs (s) Proposed-Obs (s) 
1 507 500 480 -20 7 
2 455 322 429 107 132 
3 547 507 444 -63 40 
4 377 362 525 163 15 
5 405 353 380 27 52 
6 420 374 339 -35 46 
7 418 301 415 114 117 
8 429 354 296 -58 75 
9 372 443 315 -128 -71 
10 495 340 468 128 155 
11 606 527 539 12 79 
12 479 580 452 -128 -101 
13 476 450 407 -43 26 
14 436 533 392 -141 -97 
15 377 484 286 -198 -107 
16 465 445 472 27 20 
17 303 360 288 -72 -57 
18 353 414 344 -70 -61 
19 491 540 333 -207 -49 
20 528 435 349 -86 93 
21 328 360 280 -80 -32 
22 491 540 333 -207 -49 
23 390 502 350 -152 -112 
24 390 410 350 -60 -20 
25 318 430 277 -153 -112 
26 461 302 409 107 159 
27 412 344 384 40 68 
28 608 522 465 -57 86 
29 440 372 295 -77 68 
30 600 360 509 149 240 
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6.6.2.2 PM peak Travel Time Comparison  
As mentioned before, to remove the effects of real-time traffic variability, the 
comparison was done during off-peak hours. However, what follows is the 
demonstration of comparison between the proposed model, Google API, and the actual 
travel-time for a sample of observations on a Monday during PM peak hours. The 
sample cases are randomly selected from the test date. Around 30 trips are selected 
from which 11 cases met the comparison criteria. The average percentage of error for 
the proposed model is 10.8 % whereas the Google average percentage of error is 24.4%.  
Figure 38 demonstrates an example of the comparison between the Google API and 
proposed model estimated travel time and observed travel time. The given trip is on 
Monday, in winter, and with fair weather conditions and peak hour traffic. The example 
trip has a similar route as the one proposed by the Google API, meaning that the shape 
similarity index between the two routes is higher than 0.99 and consequently the travel 
times can be compared. 
The observed travel time is 422 seconds; whereas, the Google API estimate is 477 
seconds; and the proposed model estimation is 391 seconds. In this case, there is a 13% 
difference between the observed travel time and Google API estimation, and a 7.4 % 
difference between the observed travel time and proposed model estimation. Table 14 
shows all the 11 cases which are considered in this analysis. 
 
119 
 
 
 
 
Google
Observation
 
Figure 38 Proposed Model, Google API, and observed travel time comparison (PM peak) 
Table 14 Google, proposed model travel time prediction and observations travel time comparison (PM peak) 
Trip Observation (s) Proposed model (s) Google (s) Google-Obs (s) Proposed-Obs (s) 
1 238 220 429 191 -18 
2 414 365 455 41 -49 
3 540 492 598 58 -48 
4 382 421 647 265 39 
5 308 355 474 166 47 
6 422 391 477 55 -31 
7 378 411 620 242 33 
8 630 490 618 -12 -140 
9 233 246 192 -41 14 
10 540 492 598 58 -48 
11 847 779 772 -75 -68 
 
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6.7 Sensitivity Analysis 
Sensitivity analysis is the study of how the output of a mathematical model can be 
sensitive to the inputs and assumptions. In this process, the outcome is recalculated 
under alternative assumptions/inputs to determine the impact of each assumption/input. 
There are several benefits to conducting sensitivity analysis such as understanding the 
potential relationships between input and output variables in the model, evaluating 
the robustness of the results of the model when uncertainty exists, finding any possible 
errors in the model,  and developing a possible enhancement in the model by providing 
better input and so on. This study is an empirical study and the model is a data-driven 
model; as such, there are not many input factors/assumptions in the model. In this 
section, the input factors are explained, and the results of sensitivity analysis are 
demonstrated. 
6.7.1 Number of States  
As mentioned before, here we have a few assumptions on which we can conduct the 
sensitivity analysis, one of which is the fact that we considered only two states of traffic 
congestion. The reason to choose two states of congestion is that in some of the cases, 
two distributions with two different means can be detected in the histogram of the links 
raw travel time data. However, there is no robust empirical study showing that that is 
the case. On the other hand, the more states we have, the more data we need to do the 
analysis, so even if there were more traffic states, we did not think with the available 
amount of data in this study more than two states can be considered. To demonstrate 
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whether or not the higher number of states could result in a different measure of model 
performance, and in order to explain the observation with accuracy, we conducted 
sensitivity analysis on the number of states in the model. 
In this part, the three states and four states are considered for the model. The result for 
the three states is shown in Table 15. As shown in Table 15 and Figure 39, the 
performance of the model utilizing a higher number of states decreased. To ensure that 
the result is not an accident, three different days of week and months are randomly 
selected and the performance of the model with two states and three states are 
compared. Figure 39 also demonstrates this trend of diminishing performance by 
incorporating a higher number of states for several test days.  As mentioned before, this 
could be attributed to the small sample size. For example, utilizing three states means 
that we need to have three distributions for each link, and we need to have enough 
observations in all the cases to have sound data. 
It should be noted that the increasing the number of states results in much higher 
computational complexity for the training part. The initial test on the 4 states case 
showed that training time was very high which made it inapplicable in the real-world 
situation and therefore it is not considered here. Thus, the two-state option seems to be 
a good choice, at least for this amount of data. With a larger amount of data, the number 
of states could be considered to see if it can improve model performance.  
 
 
 
 
122 
 
 
 
 
Table 15 Sensitivity analysis on the number of states 
Test Day Wednesday, Tuesday Friday, 
Season Winter Summer Fall 
Average test trip 546 529 537 
Model with 2 states error (s) 99 95 108 
Model with 3 states error (s) 118 112 132 
Model with 2 states error % 18.5 18.5 20.2 
Model with 3 states error % 21.3 21.0 23.7 
 
 
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
Wednesday, Winter Tuesday, Summer Friday, Fall
2 states 3 states
 
Figure 39 Sensitivity analysis on the number of states 
123 
 
 
 
Percrntagr of Error
 
6.7.2 Number of Observations 
In most of the data-driven models, the model is assumed to be very sensitive to the 
amount of data. In this study, all the available data is used for the model. However, to 
show how sensitive the model is to the amount of data, the sensitivity analysis is 
conducted on the amount of data. For this purpose, since we do not have more data, 
less data is used to see how the model performance could potentially change based on 
the amount of available data. 
In this part, 80% and 50% of the whole data is used for the modeling. 80% (50%) of 
the data is selected from all of the available days to have a better representative data 
(not just using 50% of the days). The test day is randomly selected for this analysis. 
The test day is Friday, October 2nd which is in the fall season and the trained model is 
tested on this day.  
As noted, the model is trained with 80% of the data and 50% of the data and the results 
are demonstrated in the Table 16. As it is shown in Table 16, for the case of the trained 
model with 80% of the data the decrease in the percentage of the error is not significant 
and we can say the performance of the model with 100% of the data and 80% of the 
data, at least on this test day, is the almost the same. However, there is an evident 
improvement in the performance of model comparing the model with 50% of the data 
with models with 80% and 100% of the data. Thus in general, we can say the amount 
of data could have an impact on the model performance. The lower the number of 
observations we have, the lower the performance would be, and the closer the proposed 
124 
 
 
 
 
model performance would be to the base model with real-time average and regression 
model.  
Table 16 Sensitivity analysis on the amount of data 
Variable Value 
Average test trip 537 
Model with 100 % of data error (s) 108 
Model with 80 % of data error (s) 109 
Model with 50 % of data errors (s) 126 
Base model errors (s) 152 
Model with 100 % of data error % 20.2 
Model with 80% of data error % 20.3 
Model with 50 % of data errors % 23.7 
Base model errors % 28.1  
 
This trend is also shown in Figure 40, which illustrates the improvement in the 
performance of the proposed graphical model and its correspondence with the increase 
in the number of observations.  
 
125 
 
 
 
 
24.0%
23.0%
22.0%
21.0%
20.0%
19.0%
18.0%
Model with 100 % of data Model with 80% of data Model with 50 % of data
 
Figure 40 Sensitivity analysis on the amount of data 
 
 
The analysis outcome shows that the model performance could potentially improve 
with more data and the model could be very promising when we have enough samples 
of data. In the future, this model could be applied with more data in order to see how 
performance could be improved. The availability of data is critical not only because the 
analysis shows that the higher amount of data could itself result in higher model 
performance, but also because it makes it possible for other proposed variables such as 
intersections and movements and more weather conditions to be included in the model 
and potentially increase the performance of the model across a variety of conditions. 
 
126 
 
 
 
Percentage Of Error
 
6.8 Summary 
The proposed model to compare this study model and Google API has three main steps. 
The first step involves having a number of trips for the test and changing the Google 
API parameters to match the test trip. The Google API parameters include origin, 
destination, departure time, traffic model, and waypoint, all of which are changed 
according to each trip. Then, the route proposed by Google API and the observation 
are compared to ensure that we are comparing the travel times on the same routes.  To 
achieve this, this study proposes the use of a well-known shape similarity algorithm, 
Hausdorff Distance, and incorporates only those trips for which the Google routes and 
the observation routes are identical. Finally, this study compares the travel times 
provided by the Google API, the proposed model, and the observation. 
The analysis for the sample of trips during off-peak hours shows that, in general, 
Google API and the proposed model have a comparable difference from the ground 
truth (observation) and, in general, Google API overestimates the time in 50% of cases 
and underestimates travel time in 50% of cases. The proposed model, however, has a 
tendency toward overestimation. 
Finally, sensitivity analyses were conducted for two parameters of the model.  One was 
the number of congestion states, which was considered two in this study. To find out 
how the model is sensitive to the states of congestion, three and four congestion states 
were considered and tested. The analysis results showed that, in instances where there 
were a higher number of congestion states, the model had lower performance. This 
could be because of the low number of observations or because two congestion states 
127 
 
 
 
 
are the best in representing the reality. The other parameter that was analyzed was the 
number of available observations.  This analysis showed that, with a lower number of 
observations, we would have lower performance.  This implies that,  with larger amount 
of data, the proposed model’s performance can improve.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Chapter 7: Summary and Conclusion 
In this Chapter, we provide a complete summary of the research. Then we discuss the 
findings of this study. Finally, in conclusion, some of the limitations of this study and 
concrete suggestions for future research are proposed and discussed.  
7.1 Summary 
Travel time prediction is very important to everyday life. Reliable travel time prediction 
helps both road users and system controllers to be informed about the future conditions 
on roadways and enables them to make their best decisions based on that information. 
In a world where we are facing a rapid increase of traffic congestion, accurate Traveler 
Information Systems are increasingly crucial. Accurate and reliable travel time 
information could help to reach control and could alleviate this congestion by providing 
a more reliable network. The first step toward this goal is the creation of accurate real-
time traffic monitoring systems. This is especially important in arterial networks since 
they are major city streets that provide for travel within and between cities. Yet, there 
is not much work that is being done to improve arterial real-time traffic monitoring. 
In general, travel time estimation and prediction are very complex and challenging 
tasks. This is due to unknown variables such as varying driver behaviors, uncertainty 
in demand, weather, incidents, roadway conditions, etc. This is even more challenging 
for arterials since there are other factors playing a role in this uncertainty, such as traffic 
signals and the relation between the links and their effects on each other.  
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Many of these fluctuations could be captured in a travel time predictions model. In fact, 
the goal is to capture these fluctuations and to be able to predict the fluctuations as 
much as possible. Nevertheless, there is still some uncertainty associated with travel 
time that poses challenges for capturing this information. Still, knowing about these 
uncertainties and the range of uncertainties is very important.  
To address the need for travel time predictions, many studies have proposed different 
methodologies. Most of these studies are for freeway travel time prediction, and fewer 
are for arterial travel time prediction.  
An overview of these methods for travel time prediction in general and arterial travel 
time prediction was done in Chapter 2. The approaches to this problem can be 
categorized based on the scope, output, and methodology used. There are two primary 
different outcomes: large-scale navigation systems (mostly deterministic) and 
probability distributions of travel times, mostly for the smaller scale. There are two 
main scopes that are distinguishable in the literature, microscopic (includes a small 
number of intersections) or macroscopic. Finally, in terms of the used methods, the 
approaches can be categorized into three main approaches: parametric, non-parametric, 
and hybrid modeling. The parametric methods, such as Feedforward Neural Network, 
Hidden Markov Model, and Autoregressive Models, have strong model assumptions. 
Comparatively, the non-parametric methods, such as the k-nearest neighbor, require 
fewer assumptions and they grow with the amount of data. Some of the proposed 
probabilistic methods used specifically for arterial travel time prediction, such as 
STARMA, Dynamic Bayesian network using EM algorithm, and Spatial Moving 
Average (SMA), were discussed in more detail.  
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This study proposed a probabilistic model using raw probe data. Using the raw probe 
data for the modeling purpose presents a lot of challenges and requires steps to be taken 
prior to the modeling. However, as of the time of this writing, the author has no 
knowledge of a study that features a framework for each of these steps. Chapter 3 of 
this research provided a framework for each preliminary step needed to be taken before 
the modeling. The steps include the data cleaning framework, map matching of the data 
to the network, and creation of the network. This study proposed a framework to work 
with raw and dirty probe data, including two main steps, trip splitter and trip filtering. 
The three main splitters are Time Difference Splitter, Waypoint Sequence Splitter, and 
Location Difference Splitter. The trip filters are Same Time Filter, Same Location 
Filter, Speed Boundary Filter, Idle Filter, Location Displacement Filter, Trip Length 
Filter, Shaking Probe Filter, Large Error Size Filter, and Shaking Probe Filter. 
Next, the map matching and existing methods for map matching were discussed. This 
study used Hidden Markov Model (HMM) map matching methods to find the match 
that maximizes the multiplication of the likelihoods of the points on the map for each 
latitude/longitude, as well as the maximum likelihood of the sequence between two 
consecutive latitude/longitude pairs.  
Finally, the initial structure of the model was explained. The proposed model in this 
study is an enhanced graphical model.  In the graphical model, it is assumed that there 
is a conditional dependency between random variables, which are demonstrated by a 
graph. 
As such, this chapter explained how a network or the base Dynamic Acyclic Graph 
(DAG) of the roads is created by creating the graph of the street map. In the base DAG, 
131 
 
 
 
 
a variable is assigned to each direction of a link and for time step t and the time step of 
t+1. The state of each link (variable) is conditioned on the neighboring links’ 
(variables) state at the previous time step. Consequently, the most important part of 
creating the network is creating this adjacency matrix of the connection between links.  
The enhanced graphical model was explained in Chapter 4. The enhanced graphical 
model has a global variable for time of the day, the day of the week, season, and weather 
conditions. All of the variations in either demand or travel behavior imposed by each 
of these variables are included in the model, and the model is able to estimate/predict 
travel time on each of these scenarios. The chapter continued with an explanation of 
Streaming Variational Bayes (SVB) algorithm. The SVB is an approximate inference 
algorithm that is used instead of exact Bayesian inference – the latter is very time 
consuming, and it is often inapplicable for complex models. Streaming Variational 
Bayes deploys the Bayes rule to convert a conventional Variational Bayes method into 
an incremental approach. 
The validation of the models and analysis of the variables influence was demonstrated 
in Chapter 5. The results of the enhanced graphical model were demonstrated for 
different scenarios, including different seasons and different days of the week. In most 
of the scenarios, the percentage of error of the model for short-term travel time 
prediction is less than 20%. The graphical models were compared to each other’s with 
incrementally added variables in order to allow the evaluation of the effects of inclusion 
of each variable in the model. The performance of the model was also compared to a 
base model, which is a real-time average, as well as a combination of real-time average 
and a regression model. The proposed model outperformed these two models in all of 
132 
 
 
 
 
the scenarios.  The weather conditions variable was separately evaluated and discussed 
in this chapter, and the accuracy of the model in bad weather conditions was 
demonstrated again for different scenarios.  
The chapter concluded with detailed suggestions on extending the model by bringing 
the intersections and driver behavior into the modeling process. The extended model 
for inclusion of intersections, for example, includes a movement variable in the graph 
for each of the possible movements in the intersection. However, this model could not 
be tested in this research due to a lack of data regarding all the movements. In this 
study, there were enough data to test the original proposed model without the 
extensions. We provided comprehensive details for developing the extended model and 
testing it in future studies pending availability of data.  
Even though the model is evaluated based on the test data, in Chapter 6, this research 
proposed a new method to compare the proposed model (or any real-time prediction 
model) with any publicly available routing API such as Google. The proposed 
framework has three main steps. The first step involves having information from a 
sample trip (including GPS tracks) for the test date, and matching the Google API (or 
any available API) parameters according to each of the trips; it also involves obtaining 
the Google APT prediction/estimation for the same trip. The next step involves 
comparing the observed trip GPS tracks for the proposed model, and the Google API 
routes using Hausdorff Distance to make sure the travel times for the same route are 
being compared. The Hausdorff Distance is a Shape-Matching algorithm and is a 
measure that shows how similar are two subsets which in our case are the GPS tracks. 
Lastly, there is a comparison of the estimations/predictions for those routes that are 
133 
 
 
 
 
identical. The results showed that the proposed model and Google API have a 
comparable difference with the ground truth.  
Sensitivity analysis was done on two factors of the study.  First, we evaluated the 
impact of the higher number of traffic states (three, and four were considered). The 
results of the analysis showed that, with this amount of data, the performance of the 
model decreases when the number of states is increased. Then we analyzed the impact 
of the amount of data available for use in the modeling process. The analysis showed 
that using a smaller amount of data (since we did not have more) could result in lower 
performance of the model, or even make it difficult or impossible to capture the 
influence of a variable at all.  
7.2  Conclusions 
What follows is the list of this study findings: 
• The first step in developing a good model is gaining access to the cleanest data 
available. The cleaned data could result in a better estimation/prediction model. 
This study proposed a framework to clean the data has proven to be effective. 
• Using the Streaming Variational Bayes could make the model usable for large-
scale networks with more variables in the model. 
• There is a time pattern in the travel time that can be captured. This variable could 
be incorporated in the model. 
• There is a seasonal pattern in travel time that can be captured by including this 
variable in the model. 
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• The weather condition does affect either driving behavior or demand, and it 
influences travel time and its uncertainty. The analysis confirms that the travel 
time distribution in bad weather conditions has a higher mean and wider standard 
deviation. 
• There is an average of 2% difference for the Google travel time estimation (with 
the available parameters) and our proposed model travel time estimation. Also, 
there is an average of 20% difference for the Google travel time estimation and 
actual travel time observations. 
• Less amount of data could result in poorer estimation/prediction and lower model 
performance. This means that more data could potentially result in better modeling 
and higher performance. 
• The model has the highest performance when it considers two states for each link’s 
travel time distributions. Considering a higher number of states results in lower 
performance. This could be attributed to the fact that two states can explain the 
observation the best, or it could be that, with the existing sample size, adding more 
variables into the model could decrease model performance. 
7.3 For Future Research 
This research provided contributions to the existing literature in the area of real-time 
arterial travel time prediction. There are some model enhancements that could not be 
accomplished in this study due to a lack of data. The most important enhancement 
would be incorporating intersections in the model. This means that we would 
decompose the trip travel time to link travel times and delays at the intersection. For 
135 
 
 
 
 
this purpose, this research suggested incorporating a movement-based variable in the 
model. By doing so, the most important variation in the intersection based on the type 
of movement (left, right, and straight) could be captured in the modeling. Many 
commercial models still encounter this challenge in producing an accurate 
estimation/prediction waiting time for the traffic signals, specifically when it comes to 
left turns. If and when more data are available as it is becoming increasingly the case, 
this extended model can be developed and tested by using more data.  
This research assumed the links travel time to be independent for the sake of data 
availability, in the future and with the availability of more data, the interdependency 
between the links travel time could be captured by learning the joint travel time 
distributions for neighboring links.   
The other variable that could be considered in the graphical model is the driver behavior 
variable, which could be captured by individual vehicle trip speed. The effects of 
bringing this variable into the graphical model or other models could be evaluated in 
future studies. 
This study also included weather condition as an explanatory variable in the model. 
Two weather conditions were used in this study, however, with greater availability of 
data, more weather condition scenarios (such as heavy rain, heavy snow, storm etc.) 
could be used to expand this variable. This can lead to a more accurate travel time 
estimation/prediction in real-world conditions. Another interesting consideration about 
weather conditions would be the possibility of bringing weather predictions into the 
travel time modeling, as well as the uncertainties associated with it.  
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There are other events that can influence travel time significantly, including incidents, 
work zones, special holidays, etc.. However, enough vehicle probe data are needed for 
each of these events in order for them to be incorporated into the model. With rapidly 
improving technology development, increasing amounts of data are becoming 
available, which makes it possible to include these events in the modeling. Including 
these events could yield more accurate estimation/prediction. 
 
 
137 
 
 
 
 
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