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Improving mobility in urban environments using intelligent transportation technologies
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Improving mobility in urban environments using intelligent transportation technologies
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Content
Improving Mobility in Urban
Environments Using Intelligent
Transportation Technologies
by
Tooraj Rajabioun
A dissertation Presented to the
Faculty of the USC Graduate School
University Of Southern California
In Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
May, 2016
Page No: 2
Table of Contents
Abstract ........................................................................................................................................ 10
Chapter 1. Introduction ................................................................................................................. 12
1.1 Motivation ............................................................................................................................... 12
1.2 Literature Review .................................................................................................................... 14
1.3 Contribution ............................................................................................................................ 17
1.4 Outline ..................................................................................................................................... 19
Chapter 2. Lack of Data: Traffic Flow Data Completion ............................................................. 20
2.1 Introduction ............................................................................................................................. 20
2.2 Traffic Flow Data Completion ................................................................................................ 21
2.2.1 Initial Traffic Flow Estimation............................................................................................. 22
2.2.2 Offline Traffic Flow Estimation .......................................................................................... 26
2.2.3 Solution Methodology .......................................................................................................... 29
2.2.4 Online Traffic Flow Estimation ........................................................................................... 31
2.3 Computational Results ............................................................................................................ 32
2.4 Conclusion .............................................................................................................................. 34
Chapter 3. On-Street and Off-Street Parking Availability Prediction ......................................... 36
3.1 Introduction ............................................................................................................................. 36
Page No: 3
3.2 Modeling of Parking Utilization ............................................................................................. 38
3.2.1 Parking Trends ..................................................................................................................... 40
3.2.2 Temporal Correlations ......................................................................................................... 41
3.2.3 Spatial Correlations .............................................................................................................. 42
3.2.4 Parking Availability Model .................................................................................................. 42
3.3 Training of the Model ............................................................................................................. 46
3.3.1 Non-recursive (batch) processing ........................................................................................ 48
3.3.2 Recursive Processing ........................................................................................................... 50
3.4 Prediction of Parking Availability .......................................................................................... 52
3.4.1 Prediction Error .................................................................................................................... 53
3.5 Demonstration ......................................................................................................................... 56
3.5.1 Validation of the Model ....................................................................................................... 56
3.5.2 Accuracy of Predictions ....................................................................................................... 60
3.6 Conclusion .............................................................................................................................. 65
Chapter 4. Adaptive Network-Wide Traffic Signal Control System with Truck Priority ............ 67
4.1 Introduction ............................................................................................................................. 67
4.2 Proposed Model ...................................................................................................................... 71
4.2.1 Multi-intersection Model ..................................................................................................... 74
4.3 Controller ................................................................................................................................ 75
Page No: 4
4.4 Control Algorithm ................................................................................................................... 77
4.5 Simulation Results .................................................................................................................. 78
4.5.1 Scenario 1: ............................................................................................................................ 79
4.5.2 Scenario 2 ............................................................................................................................. 81
4.5.3 Scenario 3 ............................................................................................................................. 81
4.6 Conclusion and Summary ....................................................................................................... 85
Chapter 5. Concluding Remarks and Proposed Future Topics ..................................................... 86
5.1 Final Conclusions .................................................................................................................... 86
5.2 Proposed Future Topics........................................................................................................... 88
5.2.1 Traffic light optimization for different classes of vehicles .................................................. 89
5.2.2 Parking Guiding Systems for Trucks ................................................................................... 90
Bibliography .................................................................................................................................. 91
Page No: 5
List of Figures
Figure 2-1 - Dynamic traffic flow generation ............................................................................... 22
Figure 2-2 - Four stage model ....................................................................................................... 23
Figure 2-3 - Measured and estimated volumes (no initial solution) ............................................. 32
Figure 2-4 - Measured and estimated volumes (initial solution) .................................................. 33
Figure 3-1 - General Scheme of the PGI system with Prediction. Real-time and
historical parking information are used at the centralized computer server to
predict parking availability and provide users with parking recommendations. .......................... 37
Figure 3-2 - Experiments with real-time parking data in order to develop the
model; (a) average on-street parking utilization over all the parking locations on a
weekday and a weekend day. (b) Average on-street parking utilization over all
locations on two Sundays for two different times of the year. (c) Correlation
between occupancy of on-street parking locations for different time lags. Data
collected on 6/27/13 in San Francisco. (d) Correlation between parking location’s
utilization with different distances to each other. The data collected on 6/27/14 at
3:00PM. ......................................................................................................................................... 39
Figure 3-3 - Map of the test region (San Francisco Financial District). The real-
time parking data was obtained from SFPark. The red lines form a grid for the
parking tiles. .................................................................................................................................. 56
Figure 3-4 - Model validation; (a) Increasing the number of samples will reduce
Ω 𝑘 , Δ for ∆= 100. (b) Increasing the number of samples would reduce 𝐴 𝐴𝐴 𝑂 𝑘 2. ................... 58
Page No: 6
Figure 3-5 - comparison of empirical model error histogram with a fitted normal
distribution (1000 samples) shows that we can approximate the process to be
zero-mean Gaussian. ..................................................................................................................... 60
Figure 3-6 - Accuracy of the predictions; (a) Scenario 1 results, prediction errors
for different time of the day, and different prediction horizons. (b) Comparison of
spatio-temporal model and temporal model, using the same data set. (c) Scenario
2 results; analyzing the performance of the prediction algorithm................................................. 62
Figure 3-7 - Scatter plots of the 20-minute-predictions versus the actual parking
availability. Blue dots indicate the spatio-temporal model results, and red dots
indicate the temporal model results. (a) on-street locations. (b) Off-street
locations. Analysis shows that the prediction results are slightly better for the off-
street case. ..................................................................................................................................... 65
Figure 4-1 Block Diagram of the control scheme. The controller uses the data
obtained from sensors to update the traffic model and estimate the future delay.
The control strategy optimizes the traffic signals by solving a set of optimization
problems. ....................................................................................................................................... 70
Figure 4-2 Schematic showing a single intersection. Each link is divided into a
number of segments ...................................................................................................................... 72
Figure 4-3 Structure of the neural network model to predict the overall delay of
the network based on the vehicles information, current state of the traffic signals
and future state of the traffic signals. ............................................................................................ 73
Page No: 7
Figure 4-4 Schematic and neural network structure for predicting delay in the
multi-intersection model. Information from adjacent intersections are fed into the
network in addition to the local traffic data. ................................................................................. 74
Figure 4-5 sample results for traffic light transitions at a junction using the above
control strategy. The first horizontal row shows the planned transitions for the
next 2 minutes which are made at time 0 based on the traffic state observed at the
moment. At t time step 5 (vertical axis), based on the developments in the traffic
situation, the transition plan is updated which is illustrated on the second
horizontal row. The adjustments at each decision time are made by taking a
weighted average of the 3 previous decisions to prevent abrupt changes to the
plan. ............................................................................................................................................... 78
Figure 4-6 Test network confined by four major roads in Long Beach, CA. 15
Traffic signal are shown on the map. ............................................................................................ 79
Figure 4-7 performance of the delay prediction model for different NN size. For
11-node network the MSE is 2.2% while for a 5-node network the MSE increases
to 7.5%. ......................................................................................................................................... 80
Figure 4-8 the input vectors for (a) the special case which agents acts
independently, and (b) the general case which agents consider the effects of
neighboring intersections. ............................................................................................................. 82
Figure 4-9 Scenario 2: comparison of two cases; with communication between
links, and without communications. (a) The MSE of the prediction model in the
case of no-interconnections increases to 11%, where the MSE for the general case
Page No: 8
is near 5%. (b) the average vehicle d delays obtained by actuated traffic signal
controller is 31 seconds, whereas the average delay for the locally optimized
controller (without interconnections) is 26 seconds, and the delay for the general
case (network-wise optimized) is 23 seconds. .............................................................................. 83
Figure 4-10 Scenario 3: the average queue length with actuated controller,
optimized with priority, and optimized without truck priority is 151m, 131m, and
124m respectively. ........................................................................................................................ 84
Page No: 9
List of Tables
Table 2.1 - The RMSPE for each step of the traffic flow data completion ................................... 34
Table 4.1 - the processing is performed on a i7 Intel processor, with an input
vector consists of 116 input variables for each intersection, and a batch of 1000
datasets. ......................................................................................................................................... 80
Table 4.2 - Comparison between different simulated control algorithms. The
second column shows that the proposed truck priority algorithm reduces the total
delay of the network by minimizing average truck stops. ............................................................ 85
Page No: 10
Abstract
Intelligent Transportation Technologies (ITS) have been transformed from a luxury option
to a necessity in most cars during recent years. Today, more mainstream cars are connected
directly to the internet than in the past, via 3G or 4G networks. This phenomenon has
contributed significantly to the mobility, safety, and driver comfort. Connected cars enable
engineers to design more efficient car navigation systems by utilizing the information from
the infrastructure, and at the same time provide researchers with a tremendous amount of
data which were not available before. Our objective is to use the available information
gathered from vehicles and infrastructure to design better navigation systems. Dynamic
intelligent transportation systems not only require accurate traffic and infrastructure data,
but also need to be able to predict the traffic situation in the future. In this report, we use
estimation theory and time series models to overcome the challenges of implementing
intelligent navigation and parking guiding systems.
Obtaining accurate information about current and near-term future traffic flows of all links
in a traffic network has a wide range of applications including traffic forecasting, vehicle
navigation devices, vehicle routing, congestion management, etc. A major problem in
getting traffic flow information in real time is that the vast majority of links are not
equipped with traffic sensors. Another problem is that factors affecting traffic flows such
as accidents, public events, and road closures are often unforeseen, suggesting that traffic
flow forecast is a challenging task.
In this report, we first use a dynamic traffic simulator to generate flows in all links using
available traffic information, estimated demand, and historical traffic data available from
links equipped with sensors. We implement an optimization methodology to adjust the
Page No: 11
origin to destination matrices driving the simulator. We then use the real-time and
estimated traffic data to predict the traffic flows on each link up to 30 minutes ahead. The
prediction algorithm is based on an Auto-Regressive model that adapts itself to
unpredictable events.
As a case study, we predict the flows of a traffic network in San Francisco, CA using a
macroscopic traffic flow simulator. We use Monte Carlo simulations to evaluate our
methodology. Our simulations demonstrate the accuracy of the proposed approach. The
traffic flow prediction errors vary from an average of 2% for 5- minute prediction windows
to 12% for 30-minute windows even in the presence of unpredictable events.
Moreover, Parking Guidance and Information (PGI) systems are becoming important parts
of intelligent transportation systems due to the fact that cars and infrastructure are
becoming more and more connected. One major challenge in developing efficient PGI
systems is the uncertain nature of parking availability in parking facilities (both on-street
and off-street). A reliable PGI system should have the capability of predicting the
availability of parking at the arrival time with reliable accuracy.
In this report, we study the nature of the parking availability data in a big city, and propose
a multivariate autoregressive model which takes into account both temporal and spatial
correlations of parking availability. The model is used to predict parking availability with
high accuracy. The prediction errors are used to recommend the parking location with the
highest probability of having at least one parking spot available at the estimated arrival
time. The results are demonstrated using real time parking data in the areas of San
Francisco and Los Angeles.
Page No: 12
Chapter 1. Introduction
1.1 Motivation
Urban traffic conditions are extremely time-dependent and change drastically during a day.
Urban transportation network is a complex and living organism that is breeding with
inefficiencies. Drivers face inevitable uncertainties during their car trips in the urban
environments. Factors that affect traffic conditions such as accidents, major public events,
and road closures are not only outside the driver’s control, but are also often unforeseen.
Only a few numbers of links in the transportation network are equipped with traffic sensors
which collect real time traffic data. These links include freeways and major streets, and
there is a lack of data for the rest of links in the transportation network. The lack of data
would not only affect drivers in the urban area; it will also have a huge impact on freight
movements especially in the presence of major events which change traffic flows in the
region drastically.
The research starts with addressing traffic flow data completion in which the general
information from the region, historical and real time traffic data are used to estimate traffic
flows for all the links in the transportation network in real time. In the proposed
methodology, traffic flow estimation methodology is classified into three categories
including initial traffic flow estimation, off-line traffic flow estimation, and real-time
traffic flow estimation. A four-stage model based on the general information within the
region such as location of schools and number of students commuting is used to estimate
the initial traffic flows in the transportation network. The column generation method is
Page No: 13
used to solve the offline traffic flow estimation problem, and the real-time traffic flow
estimation is expressed as the least squares problem. A macroscopic traffic flow simulator
is used to generate real-time traffic flows as well as prediction of flows on the road
network, since the macroscopic traffic flow simulator is much faster than the microscopic
traffic flow simulator. The microscopic traffic flow simulator cannot be used in real-time
due to long simulation running time. Traffic events (accident, road closure, disabled car,
etc.) and social events (sport games, concerts, etc.) are also implemented in the
macroscopic traffic flow simulator to evaluate their impacts on the steady flow in the
transportation network.
The level of vehicles connectivity has enabled engineers to develop custom-made
applications/software for each car and has encouraged numerous ITS application scenarios
[1]. As sensors become less expensive, and both cars and infrastructure get more
connected, the amount of available information increases dramatically. The information
from infrastructure and cars along with the computational power which exists in modern
cars will have a great impact on the speedy implementation of different ITS
technologies[2].
One of the important and growing fields in ITS is parking assist and Parking Guidance and
Information (PGI) Systems. Finding parking in large cities is characterized by frustration
and waste of time and money due to the lack of accurate information about where parking
spots are available at the time they are needed. Shoup [3] states that the average searching
time for parking spots in New York city between 11a.m. and 2 p.m. during a weekday is
10.6 minutes. The adventure of looking for parking in a congested environment leads to
circling around blocks causing additional air pollution and fuel consumption and
contributing to additional congestion. In addition, the scarcity of parking locations and the
Page No: 14
changing of parking rules and restrictions make parking more awkward and challenging,
especially to drivers that are not familiar with the area. Most of the issues associated with
finding parking can be solved or reduced by using new technologies.
Most of the issues related to finding parking can be solved or reduced by using new
technologies. Today’s cars are getting more and more connected; from getting real-time
traffic connection for navigational purposes to using social media to make driving
experience more personalized for each driver. In this paper, we propose a system which
takes into advantage existing technologies on vehicles, analyzes the available information
from the infrastructure which range from real-time parking occupancy data to the parking
regulations posted by city managements and provides the driver with guidance regarding
suitable and available parking space. Our approach uses historical and real time data to
make on line predictions regarding the availability of parking spots in chosen areas that are
suitable for the particular driver. Tests carried out in the two major cities of San Francisco
and Los Angeles demonstrates the effectiveness of the proposed methodology.
1.2 Literature Review
The Intelligent Transportation System (ITS) strategies not only depend on the availability
of data, but also on the quality of the estimation models. Most of traffic models use an
Origin-Destination matrix (OD) to describe travel demands. The OD matrix determines the
number of trips from a specific region (zone) to another one within a transportation
network. There exist various OD matrix estimation methods for a transportation network.
Turnquist et al. and Van Zuylen et al. [4] developed the static OD matrix estimation model
based on data from conductive loops embedded on the surface of streets and highways.
Page No: 15
Unfortunately, the loop data is only available for small portion of streets, and there is no
traffic flow data for major part of a transportation network. Therefore, some other source
of information such as surveys and probe vehicles can be taken into account to improve the
OD matrix estimation algorithm. Cascetta [5] demonstrated the OD matrix estimation
algorithm using traffic count data and survey data.
The dynamic OD matrix estimation can be classified into two categories: Assignment-
based OD matrix estimation and non-assignment-based estimation. Traffic flow simulators
are used in the assignment-based OD matrix estimation to generate the link flows based on
the traffic data. Cascetta et al. [6] presented a generalized least square model to minimize
the difference between the traffic data and corresponding simulated flows. Sherali et al. [7]
presented a new methodology that takes into account the cost of the entire transportation
network and generated link flow on each individual link. Gajewski et al. [8] used the least
square error problem to derive the OD matrix according to the real time traffic data. As
noted, bi-level programming has been extensively used to estimate the OD matrix. Zhou et
al. [9] introduced an improved model of the OD matrix estimation using time window
constraints. Balakrishna et al. [10] and Kim [11] developed offline generalized least square
OD matrix estimation methodologies which are the improved versions of Zhou et al.
which was developed in 2003 [9]. Also, the state-space estimation model has been used for
the dynamic OD matrix estimation. Okutani [12], Ashok et al. [13] ,and Antoniou et al.
[14] developed the dynamic OD matrix estimation methodologies based on the state-space
models.
In the non-assignment OD matrix estimation, the traffic flow is generated according to the
transportation flow equations. The non-assignment OD matrix estimation can be used for a
small transportation network such as intersections. Bell [15] presented the OD matrix
Page No: 16
estimation model with regard to the transportation flow equations. Chang et al. [16] also
developed a non-assignment OD matrix estimation model based on traffic flows on a small
section of freeway. Fernandez et al. [17] described several methodologies which are used
to impute of missing traffic flow data from conductor loops on freeways, and Zhong et al.
[18] compared some of these methodologies. Muralidharan et al. [19] provided the need to
estimate traffic flow for the links without traffic sensor. They presented a model-based
approach to estimate traffic flows on ramps of a freeway. Despite all the mentioned works
above, there exists a need to develop a model-based approach to estimate traffic flow for
all links in a transportation network in real time. Traffic sensors are only available on a
small portion of links in a transportation network. Also, the mentioned works do not take
into account the dynamic nature of link travel time which is non-linear with respect to the
volume assigned to the link.
Several recent studies have investigated the problem of predicting parking availability.
Caicedo et al. [20] proposed a prediction system for parking garages which involves
probabilistic models as well as simulations. Rajabioun et al. [21] implemented a PGI
system which is personalized and takes into account individual drivers preferences and
involves a prediction algorithm based on probability distributions of parking availability.
In another study, Wu et al. [22] proposed a prediction algorithm based on autocorrelations
between different time lags. These studies are focused on the parking lots/garages data,
where the data are available and more predictable. The studies reported in [20], [22] view
the parking data as a one dimensional time-series, however as demonstrated in this report,
parking data have multi-dimensional dependencies and correlations. Applying these
algorithms to the on-street parking case results in considerable errors, mainly due to the
fact that the variance in parking availability for on-street parking is relatively higher than
Page No: 17
the variance for off-street parking. As a result the probability distribution based models are
not accurate enough [9] to capture these changes in on-street parking as they do in off-
street parking.
The prediction of parking availability at the time of arrival is strongly connected with both
the traffic situation and the estimated arrival times based on predicted traffic flows and
travel times. An extensive range of prediction models have been studied in the literature
and used to predict traffic flows in transportation networks. Short-term forecasting models
include non-linear models such as neural network models [23], [24] and linear models
such as Kalman filters [25], [26], state space models [27], [28] Auto-Regressive Integrated
Moving Average Models (ARIMA) [29], [30], and simulation based methods. The nature
of data and the type of application determine the modeling method used for traffic
prediction. Vlahogianni et al. [31] developed a logical flow for selecting the proper
forecasting approach based on the input and output data and the quality of data. Smith et
al. [32] compared autoregressive models, nonparametric regression, neural networks, and
heuristic methods, and concluded that seasonal time series models have better performance
in comparison to other approaches. Smith et al. [32] argues that this result is due to the
fact that the traffic condition data is characteristically stochastic rather than chaotic.
Prediction of parking availability has conceptually strong similarities with traffic flow
prediction and eventually the two have to be integrated in order to come up with a reliable
system.
1.3 Contribution
In the first part, we developed a novel algorithm to estimate traffic flows in all links in a
traffic network where traffic data are unavailable, and use the information to predict short-
term traffic flow for the entire transportation network. A large network in the San
Page No: 18
Francisco area is used to demonstrate the efficiency and accuracy of the methodology.
Monte Carlo simulations are used to account for random effects and uncertainties. The
results demonstrate accurate predictions of traffic flow rates up to 30 minutes ahead of
time under normal operations. In the case of events the prediction algorithm adapts to the
changes and modifies its prediction outputs with good accuracy. This work has been
published:
• A.Abadi, T.Rajabioun, P.Ioannou, “Short-Term Traffic Flow Prediction for Transportation
Networks with Limited Traffic Data”, IEEE Transactions on Intelligent Transportation
Systems, 2014.
Moreover, we developed a novel vector spatio-temporal autoregressive model that can be
used to predict the parking availability for both on-street and off-street parking locations at
the estimated arrival time of the driver. The proposed model considers temporal
correlations of parking availability data, as well as spatial correlations. It is used to
recommend the parking location with the highest probability. We used real-time actual
parking data from San Francisco area to evaluate the results, and verify the model. The
results indicate that the proposed system recommends parking locations to drivers with
high reliability. The accuracy of predictions depends on the time horizon ahead. For a 20
minutes prediction horizon the system was demonstrated to recommend a parking location
to a driver with an accuracy of about 95%. The result of this research is presented as:
• Rajabioun, Tooraj, Brandon Foster, and Petros Ioannou. "Intelligent Parking
Assist" Control & Automation (MED), 2013 21st Mediterranean Conference on. IEEE,
2013.
Page No: 19
• T.Rajabioun, P.Ioannou, “On-Street and Off-Street Parking Availability Prediction Using
Multivariate Spatio-Temporal Models”, Under review in IEEE Transactions on Intelligent
Transportation Systems, 2015.
1.4 Outline
We will begin in Chapter 2 with a traffic flow data completion model. We will describe
the method of real time estimating the link flows and corresponding OD matrices for the
entire links in a transportation network. In Chapter 3, we will discuss a methodology to
find the shortest path for an individual driver. Then, we will discuss on-street and off-
street parking availability prediction using multivariate spatio-temporal in chapter 4. In
chapter 5 we conclude the report with the future steps.
Page No: 20
Chapter 2. Lack of Data: Traffic Flow Data
Completion
2.1 Introduction
The loop conductors embedded on the surface of streets are used to collect traffic flow
data. Unfortunately, the loop conductors are only implemented on the surface of highways
and major streets, and there is a lack of data for majority of roads in the transportation
network. Guiding drivers in the urban transportation network requires using the
methodologies to estimate flows where traffic flow data is unavailable. In traffic flow
simulators, the link traffic flows are evaluated by the dynamic traffic assignment and the
OD matrices. We divide the network into zones; where zones represent certain trips from
the origins to the destinations. The OD matrix contains information regarding the volume
of traffic travelling between any two zones in the network. The aggregate sum of all trips
originating from a zone is referred to as the zone’s production, and the aggregate sum of all
the trips that terminate in a zone is referred to as the zone’s attraction. The OD matrix
indicates travel demand, but it does not specify traveler’s route choices.
One of the most significant challenges in the transportation planning and management is
the task of accurately modeling the true OD matrix for a transportation network. Roadside
traffic studies are not only expensive and laborious, but are also more importantly lengthy.
In a world of changing traffic patterns, results could be meaningless within a few months
of the conclusion of the study; that is, the relevance of the OD matrix estimated may be
short-lived and no longer reflect the reality. The loop conductors can provide real-time and
Page No: 21
round-the-clock information that can be used to estimate and update the transportation
network’s OD matrix dynamically.
To estimate the real-time OD matrix and corresponding link flows, we estimate the initial
traffic flows in the transportation network based on the general information from the
region of interest such as location of businesses and number of commuters. In our
proposed methodology, the initial traffic flow estimation is important since it is used as the
initial solution for the offline traffic flow estimation model. The offline traffic flow
estimation model estimates traffic flows based on data from monitoring the past events and
the historical traffic flow data. Finally, we update the traffic flow estimation results based
on the real-time traffic flow data. This continual adjustment helps to ensure that the
modeled system is as close as possible to the actual traffic conditions. The level of
accuracy of traffic flow estimation and the ability to adjust it in real time have a direct
impact on the quality of our goal output: optimum route in road network and accurate
travel time prediction for an individual driver.
2.2 Traffic Flow Data Completion
The transportation network consists of several elements such as links, nodes, zones, etc.
The nodes are connected by the links, and the links represent streets or freeways. The
zones are places that considerable numbers of people visit such as schools, stadiums,
commercial buildings, and so on. Also, one zone is defined for each residential district.
The OD matrix determines the number of trips within zones in each time interval. Each
OD matrix is assigned for one transportation choice. The links represent the range of paths
(e.g. specific street blocks) between the nodes. The links are characterized by their
maximum allowable speed limit, the number and type of lanes, and their length.
Ultimately, these factors lead to a value of 𝑡 0
which is the travel time along the link given
Page No: 22
free-flow condition. Finally, the nodes represent intersections between the links. The
nodes give drivers choices between certain links; that is, there is often more than one set of
links and nodes enable a driver to travel from one zone to another. When traffic flow on a
certain link exceeds the link’s capacity, congestion ensues.
The dynamic traffic flow estimation is an important issue in the transportation planning.
The physical network model consists of many objects such as links, nodes, zones, etc. The
demand matrix represents the aggregated trips from one zone to another. The traffic flow
on each link can be obtained by applying the traffic assignment on a dynamic traffic flow
simulator with regard to the physical network characteristics and the OD matrix. Clearly,
the OD matrix can be estimated based on the loop conductor traffic flow data. Figure 2.1
depicts the diagram for the dynamic traffic flow generation.
In the Proposed methodology, the traffic flow data completion is modeled in three steps:
An initial traffic flow estimation, an off-line traffic flow estimation, and a real-time traffic
flow estimation as discussed below.
2.2.1 Initial Traffic Flow Estimation
The purpose of the initial OD matrices estimation is to estimate the OD matrices based on
the estimated demand data from the region. For instance, students go to schools at 8 AM
Physical Network
Model
OD Matrix
Dynamic Traffic
Flow Simulator
Link Flows
Figure 2-1 - Dynamic traffic flow generation
Page No: 23
and leave schools at 2 PM. We can roughly estimate the demand of each school in the
morning based on the number of students. We use the four stage model (gravity model) to
estimate the initial OD matrices as explained below and illustrated in Figure 2.2 [33]:
Figure 2-2 - Four stage model
The first stage in the model, referred to as trip generation, is the aggregated travel demand
for each zone. Each zone has a certain production, the number of trips that begin at that
zone, and a certain attraction, the number of trips that culminate at that zone. The
information (i.e. production and attraction) can be obtained through surveys and
information from the region. For instance, in a big sport game, the attraction includes
thousands of cars and possibly hundreds of buses heading to the parking lots adjacent to
the stadium before the starting of the game. The structure is the stadium and one zone is
assigned for each parking lot. The production/attraction ratio of each zone (parking lot) for
this specific structure (stadium) can be estimated based on the capacity of the parking lots.
Also, each residential district is considered as one structure within one zone in the center of
the district. Let us denote the trip production of zone 𝑖 and the trip attraction of zone 𝑗 in
the time interval 𝑛 ∈ {1, … . , 𝑇 }, as 𝑃 𝑖 𝑛 and 𝐴 𝑗 𝑛 respectively. 𝑃 𝑖 𝑛 determines the total
number of trips originating from zone 𝑖 , and 𝐴 𝑗 𝑛 is the total number of trips ending to
zone 𝑗 . The production and attraction of each zone can be expressed as follows:
Trip Generation
Trip
Distribution
Transportation
Choice
Traffic
Assignment
Page No: 24
𝑃 𝑖 𝑛 = � 𝛼 𝑞 𝑛 ( 𝑖 )
𝑞 ∈ 𝑄 𝑝 𝑞 𝑛
∀ 𝑖 ∈ 𝑍 , ∀ 𝑛 ∈ {1, … . , 𝑇 } (2.1)
𝐴 𝑗 𝑛 = � 𝛽 𝑞 𝑛 ( 𝑗 )
𝑞 ∈ 𝑄 𝑎 𝑞 𝑛
∀ 𝑗 ∈ 𝑍 , 𝑛 ∈ {1, … . , 𝑇 } (2.2)
where 𝑎 𝑞 𝑛 and 𝑝 𝑞 𝑛 are the total attraction and production for structure 𝑞 in time interval 𝑛 .
The structure 𝑞 is a subset of 𝑄 which contains all the places in the region of study that
people commute. The attraction/production of each structure is distributed to the adjacent
zones. Each structure has a production/attraction ratio for each zone. 𝛼 𝑞 𝑛 ( 𝑖 ) and 𝛽 𝑞 𝑛 ( 𝑗 )
represent the production/attraction ratios for zones 𝑖 and 𝑗 respectively. The parameter 𝑍
indicates the set of all zones in the network.
The second stage of the model referred to as, trip distribution, determines the number of
trips traverse from one zone to another. The number of trips can be estimated as follows:
𝑉 𝑖 𝑗 𝑛 = 𝜌 𝑖 𝑗 𝑛 . 𝑃 𝑖 𝑛 . 𝐴 𝑗 𝑛 . 𝑒 − 𝑑 𝑖𝑖
∀ 𝑖 , 𝑗 ∈ 𝑍 , ∀ 𝑛 ∈ {1, … . , 𝑇 } (2.3)
where 𝑉 𝑖 𝑗 𝑛 is the number of trips from zone 𝑖 to zone 𝑗 in time interval 𝑛 . The correlation
between two zones can be illustrated in different formats such as exponential, linear, and
so on. In this research, we choose the exponential format since it provides the best fit for
the selected region which is an urban region. 𝑑 𝑖 𝑗 is defined as the shortest distance between
zones 𝑖 and 𝑗 . The parameter 𝜌 𝑖 𝑗 𝑛 is a scaling factor to adjust the total number of trips from
zone 𝑖 to zone 𝑗 in the time interval 𝑛 to satisfy the following constraint equations.
Page No: 25
� 𝑉 𝑖 𝑗 𝑛 = 𝑃 𝑖 𝑛 1 ≤ 𝑗 ≤ 𝑍
∀ 𝑖 ∈ 𝑍 , ∀ 𝑛 ∈ {1, … . , 𝑇 } (2.4)
� 𝑉 𝑖 𝑗 𝑛 = 𝐴 𝑗 𝑛 1 ≤ 𝑖 ≤ 𝑍
∀ 𝑗 ∈ 𝑍 , ∀ 𝑛 ∈ {1, … . , 𝑇 } (2.5)
The next step in Figure 2.2, referred to as, transportation choice stage, determines the
proportion of 𝑉 𝑖 𝑗 𝑛 that uses a specific transportation mode such as bicycles, single or multi-
passenger cars, or public transit vehicle. The proportion of 𝑉 𝑖 𝑗 𝑛 that uses car and truck are
used to generate the OD matrices of trips at different intervals of time. For instance, trucks
represent X% of the total number of vehicles and their proportion is denoted as 𝑉 𝑖 𝑗 , 𝑏 𝑛
where 𝑉 𝑖 𝑗 , 𝑏 𝑛 = 0.01 ∗ 𝑋 ∗ 𝑉 𝑖 𝑗 𝑛 , and the rest of vehicles (100-X)% are cars ( 𝑉 𝑖 𝑗 , 𝑐 𝑛 = 𝑉 𝑖 𝑗 𝑛 −
𝑉 𝑖 𝑗 , 𝑏 𝑛 ) for a pair of zones ( 𝑖 , 𝑗 ). The ratios of trucks and cars are variable depending on the
location of zones.
Finally, we can assign the estimated OD matrices onto the links. The assignment of OD
matrices onto links allows us to calculate the link flows and estimate travel time between
zones. In this stage, we assign 10% of the total demand on the links with the minimum
impedance, and evaluate new link impedance values in each step. The process continues
until the total demand is assigned in the network. It generates the initial routes from origins
to destinations as well as the initial link flows. In the proposed methodology, travel time in
the loaded network is considered as the impedance which can be presented as follows [34]:
𝑡 𝑙 − 𝑙𝑙 𝑙 𝑑 = 𝑡 𝑙 − 𝑓 𝑓𝑓 𝑓 . [1 −
𝑞 𝑙 𝑞 𝑙 − 𝑚 𝑙 𝑚 ]
𝛼
(2.6)
where 𝑡 𝑙 − 𝑓 𝑓𝑓 𝑓 represents travel time of link 𝑙 ∈ 𝐿 in the unloaded network and
𝑞 𝑙 𝑞 𝑙 − 𝑚𝑚𝑚 is the
ratio of the density to the density jam of a link. The density jam for a link occurs when the
Page No: 26
link’s volume and the link’s speed reach zero. The parameter 𝛼 is a negative number less
than 1 which can be estimated using historical traffic data. The capacities of links are
difficult to estimate, since the capacity depends on many parameters as presented below
[35].
𝐶 = ( 𝐶 0
)( 𝑛 ). ( 𝑎 )( 𝑏 )( 𝑐 )( 𝑑 )( 𝑒 ) (2.7)
𝐶 0
: 1900 vehicles per hour per lane
𝑛 : Number of lanes
𝑎 : Lane width factor
𝑏 : Truck factor
𝑐 : Parking factor
𝑑 : Distance to downtown
𝑒 : Bus stop factor
The initial OD matrix and the initial link flows for all the links in a transportation network
are generated via the methodology that was explained in this subsection. The following
subsection provides an optimization algorithm and solution methodology to estimate the
traffic flows based on the historical traffic flow data of loop conductors.
2.2.2 Offline Traffic Flow Estimation
The traffic flow estimation for a large-scale network as it is in our case cannot be done in
the real time due to the computational time. We use an offline traffic flow estimation to
calibrate the parameters in the network, and then apply it for the real-time traffic flow
estimation. In the offline estimation, we aim to evaluate the proportion of demand assigned
on each link in the network. In other words, we derive the Link-to-Link Dividing Ratios
(LLDRs) of the link flows based on the initial traffic flow estimation and the historical link
Page No: 27
flows from the loop conductor traffic flow data. The LLDRs determine the ratio of traffic
flows propagating from a specific link to the adjacent ones. For the majority of the links in
a transportation network, there exist no loop conductor; therefore, we solely rely on the
four stage model and recalibrations of it. We define the offline traffic flow estimation
problem with an optimization formulation by taking into account appropriate constraints.
The physical transportation network is defined on a set of links and nodes, 𝐴 = ( 𝑁 , 𝐿 )
where N and L represent nodes and links in the network respectively. Let S denote the OD
pairs and 𝜃 be a set of all routes connecting OD pair 𝑠 ∈ 𝑆 . The cost of using route r for
OD pair s at time t, 𝑐 𝑠 , 𝑓 , 𝑡 is variable and depends on the traffic flow of the route at time t.
We define cost as travel time of loaded route which is formulated in (2.6). We also denote
𝐴 ̅ 𝑙 , 𝑛 the mean of observed flow on link 𝑙 in time slot n, and 𝐴 𝑙 , 𝑛 the estimated link flows
with regard to the historical data and information from the region including the initial link
flow estimation. Note that 𝑛 = {1, … . , 𝑇 } and t ∈ [0, 𝑇 ]. Unfortunately, loop conductors
are only available for a small portion of the links in the network; therefore, we use the
initial traffic flow estimation solutions to generate flows on all the links in the network.
Let 𝐴 𝑠 , 𝑓 , 𝑡 be the generated volume of route r for OD pair s at time t. Therefore, we have:
𝐴 𝑙 , 𝑛 = � � � 𝜌 𝑠 , 𝑓 , 𝑛 𝑙 , 𝑡 𝐴 𝑠 , 𝑓 , 𝑡 𝑡 ∈ 𝑇 𝑓 ∈ 𝜃 𝑠 ∈ 𝑆
∀ 𝑙 ∈ 𝐿 ,
𝑛 ∈ {1, … . , 𝑇 } (2.8)
where 𝜌 𝑠 , 𝑓 . 𝑛 𝑙 , 𝑡 is the decision variable, 1 if link 𝑙 is on route r connecting OD pair s during
time t in time slot n and 0 otherwise. Note that 𝐴 𝑙 , 𝑛 must follow flow conservation, so we
have:
Page No: 28
𝐴 𝑙 , 𝑛 = � 𝑝 𝑙 , 𝑘 , 𝑛 𝐴 𝑘 , 𝑛 𝑘 ∈ 𝐿
∀ 𝑙 ∈ 𝐿 ,
𝑛 ∈ {1, … . , 𝑇 } (2.9)
where 𝐴 𝑘 , 𝑛 represents the flow of links feeding 𝑙 and 𝑝 𝑙 , 𝑘 , 𝑛 is defined as the set of LLDRs
of link 𝑘 to 𝑙 . We define 𝜑 𝑛 ( 𝑅 ) as a set of 𝐴 𝑙 , 𝑛 and corresponding 𝐴 𝑠 , 𝑓 , 𝑡 as well as 𝑝 𝑙 , 𝑘 , 𝑛
satisfying (2.8) and (2.9). The historical link flow 𝐴 �
𝑙 , 𝑛 can be expressed as follows.
𝑊 𝐴 𝑊 𝑇 𝑉 𝑙 ( ℎ, 𝑑 ) = �
(| 𝑀 | − 𝑚 + 1)
2
∑ (| 𝑀 | − 𝑘 + 1)
2 𝑀 𝑘 = 1
∑ 𝑊𝑇 𝑉 𝑙 ( ℎ, 𝑑 , 𝑤 , 𝑚 )
𝑤 ∑ 1
𝑤 ∈ 𝑊 𝑀 𝑚 = 1
∀ ℎ ∈ 𝑊 , 𝑑 ∈ 𝐷
(2.10)
where 𝑊 𝐴 𝑊 𝑇 𝑉 is the weighted average hourly traffic volume and 𝑊𝑇 𝑉 is the hourly traffic
volume. In the rest of this research, the historical link flow 𝐴 �
𝑙 , 𝑛 is calculated by (2.10).
Parameters h, d, w, and m represent hour, day, week, and month respectively. It is assumed
that the link traffic flows are recorded for M consecutive months. The offline traffic flow
estimation problem can be formulated as an optimization problem as presented below, and
we call it the master problem.
minimize
𝑣 𝑠 , 𝑟 , 𝑡
𝐴 ( 𝑉 ) = 𝛼 1
� � �
𝐴 �
𝑙 , 𝑛 − 𝐴 𝑙 , 𝑛 𝐴 �
𝑙 , 𝑛 �
2
𝑙 ∈ 𝐿 𝑛 ∈{ 1,.., 𝑇 }
+ 𝛼 2
� � � 𝑐 𝑠 , 𝑓 , 𝑡 𝐴 𝑠 , 𝑓 , 𝑡 𝑡 ∈ 𝑇 𝑓 ∈ 𝜃 𝑠 ∈ 𝑆
(2.11)
subject to 𝐴 𝑠 , 𝑓 , 𝑡 ≥ 0 (2.12)
Objective function (2.11) minimizes the normalized variation between the historical link
flows and the simulated ones as well as the total cost of the network. Constraint (2.12)
imposes non-negative value for link flows. The coefficients 0 ≤ 𝛼 1
, 𝛼 2
≤ 1 weight the
Page No: 29
relative importance of average historical count data and the total cost of the network. We
choose 𝛼 1
to be much larger than 𝛼 2
, since the historical link flows obtained from the loop
detectors are much more reliable than the total cost of the network to derive the LLDRs.
Also, it can be presented in the following format using (2.8).
minimize
𝑣 𝑠 , 𝑟 , 𝑡
𝐴 ( 𝑉 ) = 𝛼 1
� � �
𝐴 �
𝑙 , 𝑛 − ∑ ∑ ∑ 𝜌 𝑠 , 𝑓 , 𝑛 𝑙 , 𝑡 𝐴 𝑠 , 𝑓 , 𝑡 𝑡 ∈ 𝑇 𝑓 ∈ 𝜃 𝑠 ∈ 𝑆 𝐴 �
𝑙 , 𝑛 �
2
𝑙 ∈ 𝐿 𝑛 ∈{ 1,.., 𝑇 }
+ 𝛼 2
� � � 𝑐 𝑠 , 𝑓 , 𝑡 𝐴 𝑠 , 𝑓 , 𝑡 𝑡 ∈ 𝑇 𝑓 ∈ 𝜃 𝑠 ∈ 𝑆
(2.13)
subject to 𝐴 𝑠 , 𝑓 , 𝑡 ≥ 0 (2.14)
Therefore, the link flows are estimated using the offline link flow estimation model for
each time interval. For example, we would like to know what the estimated flow on link l
is on Saturdays, 8-9 AM. During an observation interval such as 1 year, we can
continually adjust the model parameters to develop an accurate model for the traffic flow
estimation.
2.2.3 Solution Methodology
We use the column generation method [36] to solve the problem; therefore, we need to
start with a subset of solution 𝜃 1
⊂ 𝜃 . The restricted master problem is a way to expedite
the process of finding the lowest cost route. The restricted master problem is defined as a
subset of master problem ( 𝜃 1
). The initial solution would be the solution of the four-stage
model as explained in detail in the previous section. The column generation method adds
new routes to the initial subset of solution in the restricted master problem until the
stopping condition is reached. The column generation method reaches a reliable solution in
a large-scale network consisting of a large amount of variables. Initially, the column
Page No: 30
generation method begins with the restricted master problem with respect to only a small
subset of routes (initial solution). Then, it adds new eligible routes corresponding to the
solutions of the sub-problem. The master problem is solved based on the solutions derived
from the restricted master problem for each step in the column generation process. The
restricted master problem is defined as follows:
minimize
𝑣 𝑠 , 𝑟 , 𝑡
𝐴 ( 𝑉 ) = 𝛼 1
� � �
𝐴 �
𝑙 , 𝑛 − 𝐴 𝑙 , 𝑛 𝐴 �
𝑙 , 𝑛 �
2
𝑙 ∈ 𝐿 𝑛 ∈{ 1,.., 𝑇 }
+ 𝛼 2
� � � 𝑐 𝑠 , 𝑓 , 𝑡 𝐴 𝑠 , 𝑓 , 𝑡 𝑡 ∈ 𝑇 𝑓 ∈ 𝜃 1
𝑠 ∈ 𝑆
(2.15)
subject to 𝐴 𝑠 , 𝑓 , 𝑡 ≥ 0 (2.16)
where 𝜃 1
⊂ 𝜃 and 𝐴 𝑙 , 𝑛 = ∑ ∑ ∑ 𝜌 𝑠 , 𝑓 , 𝑛 𝑙 , 𝑡 𝐴 𝑠 , 𝑓 , 𝑡 𝑡 ∈ 𝑇 𝑓 ∈ 𝜃 1
𝑠 ∈ 𝑆 for the restricted master problem.
The initial solution 𝜃 1
is derived from the four-stage model. The column generation
method can be expressed as follows.
Step 1: Find the initial solution
Step 2: Calculate the total cost
Step 3: Add new routes
Step 4: Total cost improved?
Step 5: Continue until the stopping condition is satisfied
Now, we need to find a model to find new eligible routes to add to the initial solution. In
this problem, we define “cost” as a measure of time required to travel from origin to
Page No: 31
destination in the loaded network which is dynamic. Therefore, the minimum cost trip
would also be the shortest and most desirable trip.
𝐴 𝑚 , 𝑝 , 𝑤 =
𝜕𝜕 ( 𝑉 )
𝜕 𝑣 𝑚 , 𝑝 , 𝑤 = 2 𝛼 1
∑ ∑ �
𝑣 �
𝑙 , 𝑛 − ∑ ∑ ∑ 𝜌 𝑠 , 𝑟 , 𝑛 𝑙 , 𝑡 𝑣 𝑠 , 𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟 ∈ 𝜃 𝑠 ∈ 𝑆 𝑣 �
𝑙 , 𝑛 �
𝑙 ∈ 𝐿 𝑛 ∈[ 0, 𝑇 ]
𝜌 𝑚 , 𝑝 , 𝑛 𝑙 , 𝑤 +
𝛼 2
𝑐 𝑚 , 𝑝 , 𝑤
(2.17)
The routes with the negative value of 𝐴 𝑚 , 𝑝 , 𝑤 are eligible routes and are added to the
previous routes. As a result, the problem of finding a new eligible route to add to the
previous routes has changed to the dynamic shortest route problem with new link costs.
The process of adding eligible routes continues until there is no eligible route (stopping
condition).
2.2.4 Online Traffic Flow Estimation
The estimation model must be constantly recalibrated to have a better traffic flow
estimation. However, the results from the previous steps can fill gaps where the real time
data is not available for certain portions of the network. Therefore, we take into account
both the historical and real time data to minimize travel time.
minimize
𝑣 �
𝑙 , 𝑛
� Β
1
� �
𝐴 𝑙 , 𝑓 − 𝐴 �
𝑙 , 𝑛 𝐴 𝑙 , 𝑓 �
2
+ Β
2
� �
𝐴 𝑙 , 𝑛 , ℎ
− 𝐴 �
𝑙 , 𝑛 𝐴 𝑙 , 𝑛 , ℎ
�
2
𝑙 ∈ 𝐿 𝑙 ∈ 𝐿 1
� (2.18)
subject to 𝐴 �
𝑙 , 𝑛 ∈ 𝜑 𝑛 ( 𝑅 ) (2.19)
where 0 < B
2
< B
1
< 1, 𝐴 𝑙 , 𝑓 is real time flow on link 𝑙 for 𝐿 1
⊂ 𝐿 , and 𝐴 𝑙 , 𝑛 , ℎ
is the
estimated link flows based on the offline estimation for link 𝑙 in time interval 𝑛 . It is worth
Page No: 32
noting that we take B1>B2 because of the relative important role of the real-time data in
computing the link flows. The objective is to find 𝐴 �
𝑙 , 𝑛 and the corresponding OD matrix.
The problem is the linearly least-squares problem and strictly convex; therefore, it can be
solved in real time.
2.3 Computational Results
Downtown Los Angeles is selected as testing area, since it is one of the congested regions
in the United States. It consists of 22,659 links, 8,621 nodes, and 573 zones. Figures 2.3
and 2.4 demonstrate the relationship between the estimated link flows and the measured
ones during one hour of a random day by taking into account no initial solutions and initial
solutions respectively.
Figure 2-3 - Measured and estimated volumes (no initial solution)
Page No: 33
Figure 2-4 - Measured and estimated volumes (initial solution)
The Root Mean Square Percentage Error (RMSPE) is defined
as 𝑅 𝑀𝑆 𝑃 𝑅 =
�
1
𝐿 ∑ �
𝑣 �
𝑙 , 𝑛 − 𝑣 𝑙 , 𝑛 𝑣 𝑙 , 𝑛 �
2
𝐿 𝑙 = 1
. The RMSPE between the estimated and the measured
link flows is 0.17 when no initial traffic flow estimation is used; whereas, the RMSPE is
0.09 by using the initial traffic flow estimation in the proposed methodology. The results
demonstrate the important role that the initial traffic flow estimation play in the traffic flow
data completion model. Also, the following Table provides the RMSPE at each stage of the
traffic flow data completion. The random links from the transportation network are
selected to evaluate the proposed methodology.
Page No: 34
Link ID
ITFE
(RMSPE)
OTFE
(RMSPE)
RTFE
(RMSPE)
37 0.12 0.03 0.009
137 0.17 0.07 0.012
1,295 0.19 0.04 0.003
1,509 0.23 0.07 0.009
2,784 0.18 0.09 0
3,765 0.19 0.02 0
5,110 0.15 0.03 0.005
Table 2.1 - The RMSPE for each step of the traffic flow data completion
where the ITFE indicates the initial traffic flow estimation, the OTFE represents the off-
line traffic flow estimation, and the RTFE is the real-time traffic flow estimation. We
assumed that there is no traffic flow data for these links to evaluate the model, but in fact
there exist loop conductors on these links. The results yield the effectiveness of the
proposed methodology by taking into account the initial solutions derived from the four
stage model.
2.4 Conclusion
The complexity of modern transportation networks is so high. Failures in a fraction of the
system will cascade throughout the system and cause delays that impact the entire
transportation system. Management can take place on the macroscopic level in the form of
congestion management technology such as traffic signal coordination, congestion pricing
Page No: 35
systems, and changeable message signs. Management can also occur on the microscopic
level in the form of directions given to individual drivers.
We developed a methodology for the traffic flow data completion. At its core, the
challenge behind the problem is an effective real time estimation of the link flows and
corresponding OD matrices for the entire transportation network. If we can effectively
estimate the travel time from one zone to another one within a network, we can assign
traffic to the links (roads) in the network, and generate traffic flows given roadway
capacities. To accurately estimate traffic flows, we first make the initial traffic flow
estimation based on the general information within the region of study. Then, we take into
account the historical loop conductor traffic flow data to feed the offline estimation model.
Finally, we add the real-time traffic data and try to match the final results with the existing
traffic conditions. The traffic flow data completion model needs continuous and dynamic
adjustment, so that the generated flow in the macroscopic traffic flow simulator (VISUM)
is as much as close to the real world. The computational results demonstrate the important
role of using the initial solutions in the proposed methodology.
Page No: 36
Chapter 3. On-Street and Off-Street Parking Availability
Prediction Using Multivariate Spatio-Temporal Model
3.1 Introduction
In recent years, there have been several research efforts and advancements in PGI) systems [1-
10]. A PGI system is designed to assist drivers in finding parking space more efficiently, by
acquiring the necessary data from infrastructure, processing them, and communicating the results
to the driver in a non-distracting fashion [39]. Studies show that the availability of parking and
traffic condition information to the users lead to a reduction in searching time for parking
locations and consequently to a reduction in air pollution, fuel consumption and in walking
distances [40], [41].
An important obstacle in implementing an effective PGI system is the realization of vehicle to
infrastructure (V2I) and infrastructure to vehicle (I2V) communications. Several methods have
been proposed, ranging from local radio stations to text messages and cellular networks [42],
[43]. Rajabioun et al. [21] implemented a PGI system which uses 4G cellular networks to
communicate with vehicles. The use of these technologies alone however will not achieve the
best possible result with respect to parking availability when needed. One problem which arises
in the on-street parking guiding systems is that the real-time parking availability data is useful
only when the driver is very close to the parking location. This kind of data does not help in
recommending a parking location at the start of the driver’s trip or even in the scheduling of the
trip. This is because parking availability data is not deterministic, and it may change while the
driver is on her way to the parking spot. Given the lack of parking reservation systems,
Page No: 37
implementing a prediction algorithm will solve this issue by guaranteeing (with a certain level of
accuracy) a parking spot at the destination time.
Figure 3-1 - General Scheme of the PGI system with Prediction. Real-time and historical parking
information are used at the centralized computer server to predict parking availability and provide
users with parking recommendations.
While real time parking availability information becomes more accessible as more parking
meters and parking structures are equipped with sensors, predicting the availability of parking
spots when arriving at the destination will remain an important problem. Real-time parking
availability, rules, and pricing information is now available for a number of locations in big US
cities, and it is anticipated to expand to most major urban areas in the US. SFpark [44] and
Streetline, Inc. [45] are examples of companies that provide real time parking availability in San
Francisco and Los Angeles.
In this report, we propose a system which analyzes the available information from the
infrastructure ranging from real-time parking occupancy data to parking limitations/regulations
Page No: 38
and provides the driver with guidance regarding the likelihood of finding parking at the
Estimated Time of Arrival (ETA). Our approach uses historical and real time data to make on-
line predictions regarding the availability of parking in the area where the driver is planning to
park. In our approach, we first analyze the parking availability data characteristics and then,
develop a model for parking availability prediction. We use trending and detrending techniques
employed in time series analysis for financial applications in order to separate the deterministic
part of data from the random parts. A trend is defined by Wu et al. [46] as the moving mean of
the data. After calculating the trend component of the parking data, we can detrend them, which
is defined by Chan et al. [47] as the process of removing the trend of the time series so that the
trend component does not overwhelm the main component. We propose a spatio-temporal model
for parking availability, which can be applied to both on-street and off-street parking locations.
The model is used to predict the availability of parking ahead of time. Tests carried out in the
two major cities of San Francisco and Los Angeles demonstrate the effectiveness of the proposed
methodology.
The report is organized as follows: In part II we study the properties of the parking data, and
discuss the modeling of parking availability, using temporal and spatial correlations motivated
by real examples. In section III, we present a prediction method for parking availability. In part
IV we present a demonstration of our prediction model in a real situation.
3.2 Modeling of Parking Utilization
The parking availability data for a large urban area such as San Francisco, CA plotted in Figure
4.2 reveal the following characteristics: Parking trends are seasonal, there are temporal
correlations between parking availabilities for each parking location, and there exists spatial
correlations between neighboring areas. These characteristics are taken into account in coming
up with the appropriate model for parking availability.
Page No: 39
Figure 3-2 - Experiments with real-time parking data in order to develop the model; (a) average on-
street parking utilization over all the parking locations on a weekday and a weekend day. (b)
Average on-street parking utilization over all locations on two Sundays for two different times of
the year. (c) Correlation between occupancy of on-street parking locations for different time lags.
Data collected on 6/27/13 in San Francisco. (d) Correlation between parking location’s utilization
with different distances to each other. The data collected on 6/27/14 at 3:00PM.
Throughout the report we define a parking spot as the space in which a single vehicle can park
and a parking location an area that consists of one or more parking spots. A parking location can
be an off-street parking garage/lot/valet or a street block with several on-street parking spots.
Parking availability data is usually aggregated and available for a parking location rather than
single parking spots. Therefore, in this report we focus on parking locations instead of individual
parking spots. A parking tile is defined as a group of parking locations in an area. In order to
study the effects of the high/low parking demand in a certain area on the neighboring areas, it is
useful to study a group of parking locations rather than single locations. This approach also
reduces the computational power needed for predictions.
3.2.1 Parking Trends
Figure 4.2-(a) compares the moving average of on-street parking utilization on a weekday
(Thursday) and a weekend day (Sunday) over all the parking locations in the test area (San
Francisco). Note that the moving window is 30-minute long in this case. The utilization of
parking location (block/garage) l at time t is defined by:
𝑈 𝑙 ( 𝑡 ) =
𝑂 𝑙 ( 𝑡 )
𝑃 𝑙 ( 𝑡 )
(3.1)
where 𝑂 𝑙 ( 𝑡 ) and 𝑃 𝑙 ( 𝑡 ) denote the number of occupied spots and total parking spots, respectively,
for parking location l at time t.
Figure 4.2-(a) shows that the parking utilization characteristics are different from day to day,
both in terms of magnitude and shape of the utilization. One possible reason for this phenomenon
is the fact that parking on weekdays is more business oriented in nature while parking on
weekends is more leisure oriented. Figure 4.2-(b) compares the average on-street parking
utilization on two Sundays in two different seasons. It is clear that the parking utilization is also
dependent on seasons with both magnitude and maximum shifted between seasons. These
seasonal changes in the trends occur gradually over several weeks.
Page No: 41
One challenging problem in parking availability modeling is the effect of events (including
social events, accidents, temporary road closures…) on parking availability in a particular area.
An event can have a lasting effect on the parking demand of the area. Such effects are handled in
this report by using both short-term and long-term historical parking data from the area of
interest. In order to do this, we first need to understand both the temporal and spatial
characteristics of the parking availability data which are explained in sections B and C
respectively, and then choose an appropriate model which is described in part II.D. In the next
section we study the correlation of parking availability at different times.
3.2.2 Temporal Correlations
Figure 4.2-(c) shows the correlations between the occupancy of on-street parking locations at
2:00 PM and the occupancy of the same location with different time lags. i.e. correlation between
2:00 PM and 1:59 PM, 1:58 PM…, averaged over all the parking locations in the test area. The
following equation is used to calculate the temporal correlations,
𝑟 ( 𝑡 , 𝑘 ) =
∑ � 𝑈 𝑙 ( 𝑡 ) − 𝑈 �
( 𝑡 ) � � 𝑈 𝑙 ( 𝑡 + 𝑘 ) − 𝑈 �
( 𝑡 + 𝑘 ) �
𝑙 ∈ 𝐿 �
∑ � 𝑈 𝑙 ( 𝑡 ) − 𝑈 �
( 𝑡 ) �
2
𝑙 ∈ 𝐿 ∑ � 𝑈 𝑙 ( 𝑡 + 𝑘 ) − 𝑈 �
( 𝑡 + 𝑘 ) �
2
𝑙 ∈ 𝐿
(3.2)
where 𝑟 ( 𝑡 , 𝑘 ) denotes the correlation coefficients of parking utilization between time t and t+k,
and 𝑈 𝑙 ( 𝑡 ) is the utilization defined in (3.1), and 𝑈 �
( 𝑡 ) ≜ ∑ 𝑈 𝑙 ( 𝑡 )
𝑙 ∈ 𝐿 ∑ 1
𝑙 ∈ 𝐿 ⁄ . 𝐿 = { 𝑙 1
, 𝑙 2
, … } is the
set of all operational parking locations.
The temporal correlation function (3.2) depends on multiple factors including time of the day,
location, day of the week, and season. Figure 4.2-(c) suggests a strong temporal correlation for
parking utilization for small time lags. As expected, the correlation coefficients decrease as the
time lag increases. In the next section we discuss how to incorporate these correlations in the
parking availability model.
Page No: 42
3.2.3 Spatial Correlations
Another important observation that arises from analyzing real data is that a change of demand in
one parking facility affects demand on other nearby parking facilities. Figure 4.2-(d) shows the
correlation of parking utilization between parking facilities with different distances to each other.
The following equation is used to calculate the spatial correlation s(t,d) as,
𝑠 ( 𝑡 , 𝑑 ) =
∑ � 𝑈 𝑙 1
( 𝑡 ) − 𝑈 �
( 𝑡 ) � � 𝑈 𝑙 2
( 𝑡 ) − 𝑈 �
( 𝑡 ) �
( 𝑙 1
, 𝑙 2
) ∈ 𝕃 𝑑 �
∑ �𝑈 𝑙 1
( 𝑡 ) − 𝑈 �
( 𝑡 ) �
2
( 𝑙 1
, 𝑙 2
) ∈ 𝕃 𝑑 ∑ �𝑈 𝑙 2
( 𝑡 ) − 𝑈 �
( 𝑡 ) �
2
( 𝑙 1
, 𝑙 2
) ∈ 𝕃 𝑑
(3.3)
where 𝕃 𝑑 ≜ � � 𝑙 𝑖 , 𝑙 𝑗 � � 𝑙 𝑖 , 𝑙 𝑗 ∈ 𝐿 , 𝑑 𝑖 𝑠𝑡 𝑎 𝑛𝑐𝑒 ( 𝑙 𝑖 , 𝑙 𝑗 ) ≤ 𝑑 � is the set of location pairs whose distance is
less than d miles (mls), and d=0.1 mls, 0.2 mls, etc.
Figure 4.2-(d) shows that in addition to the temporal dependency of parking utilization for one
location, there is a spatial dependency at a certain time between adjacent parking facilities. The
strong spatial correlations of figure 4.2-(d) suggest that in order to model the parking availability
we need to take into account the parking situation in neighboring areas in addition to the parking
situation at the previous time slots. In the next section, we develop a model to incorporate these
dependencies.
3.2.4 Parking Availability Model
The above-mentioned properties motivate us to propose a vector autoregressive model for
parking availability to consider both temporal and spatial correlations simultaneously. Before
proceeding with the description of the model, we define the model variables. Let 𝑎 𝑗 , 𝑡 be the
number of available (empty) parking spots of parking location j at time t, and 𝑝 𝑗 , 𝑡 be the total
number of (operational) parking spots for location j at time t. In general, 𝑝 𝑗 , 𝑡 is time variant due
to the temporary parking limitations including maintenance, street sweeping, accidents, etc. We
define the absolute one-step-flow (the change of parking availability over one time step) as:
Page No: 43
𝑦 𝑗 , 𝑡 ≜ 𝑎 𝑗 , 𝑡 − 𝑎 𝑗 , 𝑡 − 1
(3.4)
Then for the parking tile l consisting of parking locations L, we define the normalized one-step-
flow at time t as:
𝐴 𝑙 , 𝑡 ≜
∑ 𝑦 𝑗 , 𝑡 𝑗 ∈ 𝐿 ∑ 𝑝 𝑗 , 𝑡 𝑗 ∈ 𝐿 (3.5)
The unit of 𝐴 𝑙 , 𝑡 is “average number of cars (entering/leaving the parking tile l) per time step per
spot” (e.g. car/spot/minute). Note that the objective of this study is to forecast parking
availability for individual parking locations, In order to do this, we first predict parking flows for
a group of parking locations (which are considered together as a tile) and in the next step these
flows are used to predict availability for each location. Also, Note that a positive flow 𝐴 means
cars are leaving and negative 𝐴 means that cars are entering the parking location. Finally, the
averaging process consists of two stages: first, averaging over a moving window, and then
averaging over the past W weeks. The average flow 𝐴 ̅ 𝑙 , 𝑡 for parking tile 𝑙 at the specific time 𝑡 is
obtained from:
𝐴 ̅ 𝑙 , 𝑡 =
1
𝑇 ′
� 𝜔 𝑖 � 𝐴 𝑙 , 𝑡 − 𝑖 − 𝑗 𝑇 𝑤 𝑇 ′
− 1
𝑖 = 0
𝑊 𝑗 = 0
(3.6)
where 𝑇 ′
is the length of the moving window in minutes, and 𝑇 𝑤 is the length of a week which
depends on the choice of time step. In our case study, the time step is 1 minute, and hence
𝑇 𝑤 = 10080. W in the number of weeks in consideration (In our case study W=6) . 𝐴 ̅ 𝑙 , 𝑡 denotes
the trend of parking flows for tile l, at time t, and 𝜔 𝑖 ( ∑ 𝜔 𝑖 𝑊 𝑖 = 0
= 1) are positive weights to
emphasize the data of most recent weeks. As explained earlier, the parking data has a seasonal
nature. We consider this seasonal property by averaging the flows of a certain parking tile over
the past W weeks. The parking availability data contains a seasonal trend component. In order to
model the data for forecasting purposes, it is useful to decompose the flows into trend and
Page No: 44
irregular (stochastic) components: 𝐴 𝑙 , 𝑡 = 𝐴 ̅ 𝑙 , 𝑡 + 𝐴 �
𝑙 , 𝑡 . i.e. for parking tile 𝑙 at time 𝑡 , the one-step-
flow 𝐴 𝑙 , 𝑡 defined in (3.5) can be decomposed into trend ( 𝐴 ̅ 𝑙 , 𝑡 ) as defined in (3.6) and stochastic
( 𝐴 �
𝑙 , 𝑡 ) components. In order to calculate the trends we make the following assumptions:
Assumption 1: Temporary parking rules (e.g. street cleaning) of a region do not have a major
effect on the parking “demand” for the area.
Assumption 2: The number of people in the area who drive away because of difficulties in
finding parking is negligible.
The assumptions are made for technical and computational reasons. Consider two cases; if all the
operational parking spaces of a location are occupied, then there is no positive flow (entering
cars) into that location. However, this does not mean that there is zero demand for parking in the
area. In the same way, if a parking location is empty, it does not translate to zero outgoing cars
for the area. Moreover, usually the tile consists of several parking locations which some of them
are not operational for a period of time while others in the same area are operating as normal. In
this case, making these assumptions allow us to automatically exclude the un-operational ones
from the averaging process, and continue as normal. With these assumptions, we can detrend the
parking flows with respect to the parking flow trends of the corresponding time of the day and
day of the week and extract the random components (irregularities) in parking flow of tile l at
time t. We define the differential parking flows 𝐴 �
𝑙 , 𝑡 as
𝐴 �
𝑙 , 𝑡 ≜ 𝐴 𝑙 , 𝑡 − 𝐴 ̅ 𝑙 , 𝑡 (3.7)
where 𝒗 𝒍 , 𝒕 is the real-time parking flows defined in (3.5) and 𝒗 �
𝒍 , 𝒕 is calculated from (3.6).
Vector autoregressive model: Let 𝑽 �
𝒕 represent the 𝑳 × 𝟏 vector consisting of differential flows of
all the parking tiles 1,2, …L at time t, and let 𝑬 𝒕 be a zero-mean random vector consisting of i.i.d
white random processes. i.e.
Page No: 45
𝑉 �
𝑡 ≜ �
𝐴 �
1, 𝑡 𝐴 �
2, 𝑡 ⋮
𝐴 �
𝐿 , 𝑡 � , 𝑅 𝑡 ≜ �
𝑒 1, 𝑡 𝑒 2, 𝑡 ⋮
𝑒 𝐿 , 𝑡 � (3.8)
with
Σ
2
≜ 𝔼 [ 𝑅 𝑡 𝑅 𝑡 ′
] = diag( 𝜎 1
2
, 𝜎 2
2
, … , 𝜎 𝐿 2
) (3.9)
where Σ
2
denotes the covariance matrix, and 𝜎 𝑖 2
is the variance for 𝑒 ̃ 𝑖 . The empirical experiments
(see section V) show that we can approximate 𝑅 𝑡 as a zero-mean Gaussian process. Then, the
following vector autoregressive model describes the spatial and temporal correlations in the
parking flows:
𝑉 �
𝑡 = � 𝐴 𝑚 𝑉 �
𝑡 − 𝑚 𝑀 𝑚 = 1
+ 𝑅 𝑡 (3.10)
where the coefficients 𝐴 𝑚 are 𝐿 × 𝐿 matrices:
𝐴 𝑚 =
⎣
⎢
⎢
⎡
𝛼 1, 1
𝑚 𝛼 1, 2
𝑚 ⋯ 𝛼 1, 𝐿 𝑚 𝛼 2, 1
𝑚 ⋮
𝛼 2, 2
𝑚 ⋮
⋯
⋱
𝛼 2, 𝐿 𝑚 ⋮
𝛼 𝐿 , 1
𝑚 𝛼 𝐿 , 2
𝑚 … 𝛼 𝐿 , 𝐿 𝑚 ⎦
⎥
⎥
⎤
(3.11)
whose coefficients are unknown and need to be estimated. The order M of the model can be
found by using the Akaike information criterion [48]. Let 𝑉 �
𝑡 be an unbiased estimator of 𝑉 �
𝑡 :
𝑉 �
𝑡 = 𝔼 � 𝑉 �
𝑡 � 𝑉 �
𝑡 − 𝑖 , 𝑖 = 1, … 𝑀 �
= � 𝐴 ̂
𝑚 𝑉 �
𝑡 − 𝑚 𝑀 𝑚 = 1
(3.12)
Page No: 46
where 𝐴 ̂
denotes the estimated coefficient matrices. After predicting the parking flows, we can
predict the actual availability of parking locations by using the following equation:
𝑎 �
𝑖 , 𝑡 + 1
= 𝔼 � 𝑎 𝑖 , 𝑡 + 1
| 𝕀 𝑡 �= 𝑎 𝑖 , 𝑡 + 𝑎 �
𝑖 , 𝑡 + 1
(3.13)
where 𝑎 �
𝑖 , 𝑡 + 1
denotes the predicted parking availability (i.e. number of available spots at parking
𝑙 𝑖 at time 𝑡 + 1), 𝕀 𝑡 denotes all the available parking information related to time step t and before,
and 𝑎 �
𝑖 , 𝑡 + 1
= � 𝐴 ̅ 𝑖 , 𝑡 + 1
+ 𝐴 �
𝑖 , 𝑡 + 1
� 𝑝 𝑖 , 𝑡 + 1
is the predicted change in the parking availability between
time steps 𝑡 and 𝑡 + 1. This value is equal to the predicted flow of vehicles (which is normalized
over the total number of spots) multiplied by the total number of operational parking spots of the
particular parking location 𝑝 𝑖 , 𝑡 . Equations (3.4)-(3.13) can be used to predict the parking
availability provided that the model parameters 𝐴 𝑚 , 𝑚 = 1, … , 𝑀 are known. In practice these
parameters are unknown and may vary over time. In the following section we explain how to
estimate these parameters using real time data.
3.3 Training of the Model
In this part, we use the historical parking information to estimate the parameters of the model
(3.10). We define the residual error as the difference between the observed parking flow and the
predicted flow i.e. 𝑉 �
𝑡 − 𝑉 �
𝑡 . The normalized mean square residual error R for the last T time steps
leading to time k is defined as:
𝑅 ( 𝐴 ) = � � 𝑉 �
𝑡 − 𝑉 �
𝑡 �
′
𝑊 � 𝑉 �
𝑡 − 𝑉 �
𝑡 �
𝑘 𝑡 = 𝑘 − 𝑇
= � � 𝑉 �
𝑡 − � 𝐴 ̂
𝑚 𝑉 �
𝑡 − 𝑚 𝑀 𝑚 = 1
�
′
𝑊 � 𝑉 �
𝑡 − � 𝐴 ̂
𝑚 𝑉 �
𝑡 − 𝑚 𝑀 𝑚 = 1
�
𝑘 𝑡 = 𝑘 − 𝑇
(3.14)
Page No: 47
where 𝑊 > 0 is a symmetric normalizing weight matrix. In order to find the parameters of the
model (3.10), we minimize the residuals with respect to 𝐴 = { 𝐴 𝑚 ; 𝑚 = 1, 2, … , 𝑀 }. In the
following sections, we find the parameters by implementing batch processing and recursive
processing schemes. As it will be discussed later, the recursive scheme is more computationally
efficient than the batch (non-recursive) scheme. In this report we first use the non-recursive
scheme to generate an initial estimate of the parameters which are then used as the initial
condition for the parameter estimates generated by the recursive scheme.
We start by rewriting the model (3.10) in the matrix form:
𝑉 �
𝑘 = [ 𝐴 1
⋯ 𝐴 𝑀 ] �
𝑉 �
𝑘 − 1
⋮
𝑉 �
𝑘 − 𝑀 �+ 𝑅 𝑘 (3.15)
Using (3.15) and a window of T+M measurements at times k-T-M+1, k-T-M+2, …., k we can
write the following matrix equation:
Ψ( 𝑘 ) = ΘΦ( 𝑘 − 1) + Ε( 𝑘 ) (3.16)
where
Ψ( 𝑘 ) = [ 𝑉 �
𝑘 − 𝑇 + 1
𝑉 �
𝑘 − 𝑇 + 2
… 𝑉 �
𝑘 ]
Θ = [ 𝐴 1
𝐴 2
… 𝐴 𝑀 ]
Φ( 𝑘 − 1) =
⎣
⎢
⎢
⎡
𝑉 �
𝑘 − 𝑇 𝑉 �
𝑘 − 𝑇 + 1
⋯ 𝑉 �
𝑘 − 1
𝑉 �
𝑘 − 𝑇 − 1
⋮
𝑉 �
𝑘 − 𝑇 ⋮
⋯
⋱
𝑉 �
𝑘 − 2
⋮
𝑉 �
𝑘 − 𝑇 − 𝑀 + 1
𝑉 �
𝑘 − 𝑇 − 𝑀 + 2
… 𝑉 �
𝑘 − 𝑀 ⎦
⎥
⎥
⎤
E( 𝑘 ) = [ 𝑅 𝑘 − 𝑇 + 1
𝑅 𝑘 − 𝑇 + 2
… 𝑅 𝑘 ]
(3.17)
Using the above notation, we can rewrite the residuals (3.14) in the matrix format:
Page No: 48
𝑅 � Θ
�
� = 𝑡 𝑟 � � Ψ( 𝑘 ) − Ψ
�
( 𝑘 ) � ′𝑊 � Ψ( 𝑘 ) − Ψ
�
( 𝑘 ) � �
= 𝑡 𝑟 � �Ψ( 𝑘 ) − Θ
�
Φ( 𝑘 − 1) �
′
𝑊 �Ψ( 𝑘 ) − Θ
�
Φ( 𝑘 − 1) ��
= 𝑡 𝑟 � Ψ( 𝑘 )
′
𝑊 Ψ( 𝑘 ) − 2 Ψ( 𝑘 )
′
𝑊 Θ
�
Φ( 𝑘 − 1) + Φ( 𝑘 − 1) ′ Θ
�
′ 𝑊 Θ
�
Φ( 𝑘 − 1) �
(3.18)
where Ψ
�
( 𝑘 ) and Θ
�
are estimates of Ψ( 𝑘 ) and Θ respectively. The algebraic equation (3.18) can
now be used to obtain an estimate of the unknown parameter matrix Θ by using a non-recursive
or a recursive approach as explained in the following subsections.
3.3.1 Non-recursive (batch) processing
A non-recursive scheme can be formed by letting 𝑇 = 𝑘 in (3.16). A consistent and unbiased
multivariate least square estimator of Θ is derived by taking the derivative of (3.18) and setting
𝑑𝑅 � Θ
�
� 𝑑 Θ
�
⁄ = 0 to obtain [49]:
Θ
�
( 𝑘 ) = Ψ( 𝑘 ) 𝑊 Φ
′
( 𝑘 − 1) � Φ( 𝑘 − 1) 𝑊 Φ
′
( 𝑘 − 1) �
− 1
(3.19)
The unbiasedness of the estimator is established by substituting (3.16) into (3.19) which results
in: (for simplicity we let 𝑊 = 𝐼 )
Θ
�
( 𝑘 ) = [ ΘΦ( 𝑘 − 1) + Ε( 𝑘 )] Φ
′
( 𝑘 − 1) � Φ( 𝑘 − 1) Φ
′
( 𝑘 − 1) �
− 1
= Θ + Ε( 𝑘 ) Φ′ ( 𝑘 − 1)[ Φ( 𝑘 − 1) Φ′ ( 𝑘 − 1)]
− 1
(3.20)
and taking the expectation of (3.20) we obtain:
𝔼 � Θ
�
( 𝑘 ) �= Θ + 𝔼 [ Ε( 𝑘 )] 𝔼 [ Φ′ ( 𝑘 − 1)[ Φ( 𝑘 − 1) Φ′ ( 𝑘 − 1)]
− 1
] = Θ (3.21)
This result is derived using the fact that Ε and Φ are statistically independent and Ε( 𝑘 ) is mean-
zero. In order to establish the consistency (convergence in probability) of the estimator (3.19),
Page No: 49
we use the probability limit (plim) operator defined in [50]: Let { 𝑋 𝑘 } be a sequence of random
variables, then 𝑋 𝑘 converges in probability to 𝑋 ∗
if [50]:
plim 𝑋 𝑘 = 𝑋 ∗
↔ lim
𝑘 → ∞
𝑃 {| 𝑋 𝑘 − 𝑋 ∗
| > 𝜖 } = 0
(3.22)
for every 𝜖 > 0. Applying the plim operator to (3.20) we get:
Plim Θ
�
( 𝑘 ) = Θ + Plim �
Ε( 𝑘 ) Φ ′ ( 𝑘 − 1)
𝑘 �Plim �
Φ( 𝑘 − 1) Φ ′ ( 𝑘 − 1)
𝑘 �
− 1
(3.23)
By assuming that Plim �
Φ( 𝑘 − 1) Φ ′( 𝑘 − 1)
𝑘 � exists and it is invertible, and Plim �
Ε( 𝑘 ) Φ ′( 𝑘 − 1)
𝑘 �= 0
(both assumptions will be verified for the parking data in part V). In order to derive (3.23), we
use the properties of plim [51]: if PlimX
𝑘 = X , PlimY
𝑘 = Y , then
PlimX
𝑘 − 1
= X
− 1
and
PlimX
𝑘 Y
𝑘 = PlimX
𝑘 . PlimY
𝑘 = 𝑋𝑋
(3.24)
We conclude that:
Plim Θ
�
( 𝑘 ) = Θ (3.25)
This proves that Θ
�
( 𝑘 ) converges in probability sense. The condition for the existence of solution
of (3.19) is that the matrix Φ( 𝑘 − 1) Φ ′ ( 𝑘 − 1) is nonsingular. If Φ( 𝑘 − 1) has linearly
independent rows then Φ( 𝑘 − 1) Φ ′ ( 𝑘 − 1) is nonsingular. Note that, since Φ is a 𝑀 𝐿 × 𝑘
matrix, the condition 𝑘 > 𝑀 × 𝐿 must be satisfied in order to have independent rows. Singular
cases are divided into two categories;
Page No: 50
Case 1: a parking tile has a constant flow for a long period of time (more than T). In practice, the
occurrence rate of this case is very low, and it can happen mostly at the hours after midnight,
when there is no significant change in the parking occupancies.
Case 2: two or more parking tiles have exactly the same flows for the whole period T.
In practice, upon receiving a new set of data, the rank of matrix Φ( 𝑘 ) is checked. In the case of
independent rows, based on the dependency category, appropriate action is performed:
In case 1: the specific parking zone is eliminated temporarily from predictions. In case 2: the
specific parking zones are merged into one zone, and the predictions are carried on.
The batch processing scheme explained above has limited applications in on-line systems where
the computation speed is important. This is because a large matrix (ML × ML) needs to be
inverted at each step. In our case the parking data are being fed into the system on a regular
basis, and decisions should be made with minimum delay. Therefore in the following section we
consider a recursive algorithm where the matrix inversion is replaced by a division.
3.3.2 Recursive Processing
The goal of a Parking Guiding System is to update the predictions as the system receives new
parking data. Considering the processing time for the batch processing scheme explained above,
it is preferred to implement a recursive scheme in order to utilize the information which is
received on a constant basis. The updated parameter set at each new time step, is derived using
(3.16) for T=k and the approach of [49].
Θ
�
( 𝑘 + 1) = Θ
�
( 𝑘 ) + � 𝑉 �
𝑘 + 1
− 𝑉 �
𝑘 + 1
� 𝑲 ( 𝑘 + 1) (3.26)
Equation (3.26) calculates the new parameters by combining the just-computed Θ
�
( 𝑘 )with a
correction term which is a linear transformation of the latest estimation error. 𝑉 �
𝑘 + 1
is the Least
Square estimate of 𝑉 �
𝑘 + 1
, given by:
Page No: 51
𝑉 �
𝑘 + 1
= Θ
�
( 𝑘 ) 𝕍 �
𝑘 (3.27)
and the gain matrix 𝑲 ( 𝑘 + 1) is calculated recursively by [49],
𝑲 ( 𝑘 + 1) ≜ � 1 + 𝕍 �
𝑘 ′ 𝑷 𝑘 𝕍 �
𝑘 �
− 1
𝕍 �
𝑘 ′ 𝑷 𝑘 + 1
𝑷 𝑘 + 1
= 𝑷 𝑘 � 𝐼 − 𝕍 �
𝑘 𝑲 ( 𝑘 + 1) �
(3.28)
where
𝕍 �
𝑘 ≜
⎣
⎢
⎢
⎡
𝑉 �
𝑘 𝑉 �
𝑘 − 1
⋮
𝑉 �
𝑘 − 𝑀 + 1
⎦
⎥
⎥
⎤
(3.29)
In the above formulas, the dimensions of matrices are as follows, 𝕍 �
: 𝑀 𝐿 × 1, 𝑃 : 𝑀 𝐿 × 𝑀 𝐿 ,
Θ
�
: 𝐿 × 𝑀 𝐿 , 𝐾 : 1 × 𝑀 𝐿 . In our implemented system the predictions are updated every one minute
as the parking availability data is received. In summary, the estimation process consists of the
following steps:
The initial value of Θ
�
( 𝑘 0
) can be obtained from (3.19), or alternatively, the following initial
values can be used to initiate the process:
Θ
�
( 𝑘 0
) = [ 𝐼 𝐿 ⋯ 𝐼 𝐿 ]
𝑃 𝑘 0
− 1
=
1
𝑎 2
𝐼 𝑀 𝐿
(3.30)
where 𝐼 𝐿 and 𝐼 𝑀 𝐿 are identity matrices of size L and ML respectively, and 𝑎 is a large number
[52].
At each time step, the parameter matrix Θ
�
is updated using the just-received parking data
according to (3.26) and (3.28).
Page No: 52
In part IV, we demonstrate the use of these predictions to predict the actual parking availability
at specific parking locations.
3.4 Prediction of Parking Availability
In this part we use the developed model to predict the parking availability in a fashion which is
usable in PGI systems. In part III, we estimated the parameters of the model
Θ = [ 𝐴 1
𝐴 2
… 𝐴 𝑀 ] using the real-time and historical parking availability information. Now
we will use this to predict the actual availability for different parking locations. In order to make
it easier for computational and scaling purposes, we can write (3.10) in the closed matrix form
𝕍 �
𝑘 = 𝔸𝕍 �
𝑘 − 1
+ 𝔼 𝑘 or:
⎣
⎢
⎢
⎡
𝑉 �
𝑘 𝑉 �
𝑘 − 1
⋮
𝑉 �
𝑘 − 𝑀 + 1
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎢
⎡
𝐴 1
𝐴 2
𝐴 3
… 𝐴 𝑀 𝐼 𝐿 0 0 … 0
0
⋮
𝐼 𝐿 ⋮
0
⋱
…
⋱
0
⋮
0 0 … 𝐼 𝐿 0 ⎦
⎥
⎥
⎥
⎤
⎣
⎢
⎢
⎡
𝑉 �
𝑘 − 1
𝑉 �
𝑘 − 2
⋮
𝑉 �
𝑘 − 𝑀 ⎦
⎥
⎥
⎤
+ �
𝑅 𝑘 𝑅 𝑘 − 1
⋮
𝑅 𝑘 − 𝑀 + 1
� (3.31)
Then, we use the estimator (3.12) to get the matrix form of the one-step estimator as:
𝕍 �
𝑘 + 1| 𝑘 = 𝔼 � 𝕍 �
𝑘 + 1
| 𝕀 𝑘 �= 𝔸 �
( 𝑘 ) 𝕍 �
𝑘 (3.32)
where 𝕍 �
𝑘 + 1| 𝑘 denotes the estimation of 𝕍 �
𝑘 + 1
based on the information up to time k, and 𝕀 𝑘
denotes all the available parking information related to time k and before, and
𝔸 �
( 𝑘 ) =
⎣
⎢
⎢
⎢
⎡
𝐴 ̂
1
𝐴 ̂
2
𝐴 ̂
3
… 𝐴 ̂
𝑀 𝐼 𝐿 0 0 … 0
0
⋮
𝐼 𝐿 ⋮
0
⋱
…
⋱
0
⋮
0 0 … 𝐼 𝐿 0 ⎦
⎥
⎥
⎥
⎤
(3.33)
Note that the first row of 𝔸 �
( 𝑘 ) is Θ
�
( 𝑘 ) which is generated from (3.26)-( 4.28). For notation
simplicity, dependency of 𝐴 ̂
1
, … , 𝐴 ̂
𝑀 on time k is omitted in the rest of the report. Iteratively for
the D-step-ahead predictions we have:
Page No: 53
𝕍 �
𝑘 + 𝐷 | 𝑘 = 𝔼 � 𝕍 �
𝑘 + 𝐷 | 𝕀 𝑘 �= 𝔸 �
( 𝑘 )
𝐷 𝕍 �
𝑘 (3.34)
where D is the number of time steps in the future for which the predictions are performed.
Let 𝑎 𝑖 , 𝑘 and 𝑝 𝑖 , 𝑘 , 𝑖 = 1, … , 𝑟 be the available and the total operational number of parking spots
at time k for parking location i respectively; moreover, let 𝑑 𝑖 , 𝑖 = 1, … , 𝑟 denote the time needed
for the user/vehicle to reach parking location 𝑖 . Then by expanding (3.13), we get the predicted
parking availability 𝑎 �
𝑖 , 𝑘 + 𝑑 𝑖 (i.e. number of available spots) of parking 𝑖 at time 𝑘 + 𝑑 𝑖 as:
𝑎 �
𝑖 , 𝑘 + 𝑑 𝑖 = 𝔼 � 𝑎 𝑖 , 𝑘 + 𝑑 𝑖 | 𝕀 𝑘 �= 𝑎 𝑖 , 𝑘 + 𝑎 �
𝑖 , 𝑘 + 𝑑 𝑖 (3.35)
where 𝑎 �
𝑖 , 𝑘 + 𝑑 𝑖 is the predicted change in the parking availability 𝑎 𝑖 , 𝑘 between time steps 𝑘
(current time) and 𝑘 + 𝑑 𝑖 , given by:
𝑎 �
𝑖 , 𝑘 + 𝑑 𝑖 = � � 𝐴 ̅ 𝑖 , 𝑘 + 𝑗 + 𝐴 �
𝑖 , 𝑘 + 𝑗 � 𝑝 𝑖 , 𝑘 + 𝑗 𝑑 𝑖 𝑗 = 1
(3.36)
where the 𝐴 �
𝑖 , 𝑘 + 𝑗 is the i-th element of the 𝕍 �
𝑘 + 𝑗 | 𝑘 generated by (3.34). The right hand side of
(3.36) is the net change in the number of parking spots. Note that the flows 𝐴 ̅ and 𝐴 � are
normalized over the total number of spots and need to be multiplied by the total number of
operational spots of the particular location 𝑝 to obtain the net change in the available spots.
Equations (3.35)-(3.36) predict the availability of a parking location using the predicted flows
obtained from (3.27) or (3.34).
3.4.1 Prediction Error
In order to use the predictions of parking availabilities (3.35), we need to study the prediction
error, and how this error affects the probability of finding parking. The error of predictions for
parking 𝑙 𝑖 is defined as 𝑒 𝑟𝑟
𝑖 ≜ 𝑎 �
𝑖 , 𝑡 + 𝑑 𝑖 − 𝑎 𝑖 , 𝑡 + 𝑑 𝑖 ∈ ℝ which is a zero-mean Gaussian process;
[ 𝑒 𝑟𝑟
𝑖 ] = 0 , and 𝔼 [ 𝑒 𝑟𝑟
𝑖 2
] = 𝜎 𝑖 2
. Then the probability of error is given by,
Page No: 54
𝑃 𝑖 ( 𝑞 ) ≜ Pr{ 𝑞 − 0.5 ≤ 𝑒 𝑟𝑟
𝑖 < 𝑞 + 0.5}
=
1
� 2 𝜋 𝜎 𝑖 2
� exp( − 𝑥 2
2 𝜎 𝑖 2
⁄ )
𝑞 + 0. 5
𝑞 − 0. 5
d 𝑥
(3.37)
where 𝑃 𝑖 ( 𝑞 ), 𝑞 ∈ ℤ denotes the probability of having an error equal to q in predictions of the
available number of parking spots for parking location 𝑙 𝑖 . Note that having the error of q would
mean that the prediction of available spots at location 𝑙 𝑖 at time spot 𝑡 + 𝑑 𝑖 (wich are made at
time 𝑡 ) is off by the value of 𝑞 spots. In practice, the variances 𝜎 𝑖 2
(error variance for parking
location 𝑙 𝑖 ) can be estimated for different prediction time horizons using the predicted data from
real-time data by applying the following lemma.
Lemma 1: let 𝕀 𝑡 denote all the available information up to time t, and Γ
𝑖 , 𝑖 = 1,2, … be
approximations of the error covariance matrix i-step ahead predictions. Then for large k:
Γ
1
≜ 𝔼 � � 𝑉 �
𝑡 + 1
− 𝑉 �
𝑡 + 1
� � 𝑉 �
𝑡 + 1
− 𝑉 �
𝑡 + 1
� ′ | 𝕀 𝑡 � ≅ Σ
𝑡 2
Γ
2
≜ 𝔼 � � 𝑉 �
𝑡 + 2
− 𝑉 �
𝑡 + 2
� � 𝑉 �
𝑡 + 2
− 𝑉 �
𝑡 + 2
� ′ | 𝕀 𝑡 � ≅ � 𝐼 𝐿 + 𝐴 ̂
1
2
� Σ
𝑡 2
Γ
3
≜ 𝔼 � � 𝑉 �
𝑡 + 3
− 𝑉 �
𝑡 + 3
� � 𝑉 �
𝑡 + 3
− 𝑉 �
𝑡 + 3
� ′ | 𝕀 𝑡 �
≅ � 𝐼 𝐿 + 𝐴 ̂
1
2
+ � 𝐴 ̂
2
+ 𝐴 ̂
1
2
�
2
� Σ
𝑡 2
(3.38)
and so on, where Σ
𝑡 2
is estimated by[53]:
Σ
𝑡 2
=
1
𝑀 � � 𝑉 �
𝑡 − 𝑖 − 𝑉 �
𝑡 − 𝑖 � � 𝑉 �
𝑡 − 𝑖 − 𝑉 �
𝑡 − 𝑖 � ′
𝑀 𝑖 = 0
Proof: using (3.27) and (3.15) we have:
Page No: 55
𝑉 �
𝑡 + 1
− 𝑉 �
𝑡 + 1
= � Θ( 𝑘 ) − Θ
�
( 𝑘 ) � 𝜑 𝑘 + 𝑅 𝑡 (3.39)
If k is large enough, from (3.25), we assume Θ
�
( 𝑘 ) ≈ Θ( 𝑘 ), and by taking the expectation of
(3.39), we have:
𝔼 � � 𝑉 �
𝑡 + 1
− 𝑉 �
𝑡 + 1
� � 𝑉 �
𝑡 + 1
− 𝑉 �
𝑡 + 1
� ′ | 𝕀 𝑡 �= 𝔼 [ 𝑅 𝑡 𝑅 𝑡 ′
] ≅ Σ
2
(3.40)
Γ
2
, Γ
3
, … are calculated similarly by further expanding (3.27) and (3.15).
Since � 𝐴 𝑖 2
�
2
≥ 0, 𝑖 = 1,2, …, we have ‖ Γ
1
‖
2
≤ ‖ Γ
2
‖
2
≤ ‖ Γ
3
‖
2
≤ ⋯. In other words, the
prediction error increases as the prediction horizon increases.
The ultimate goal of a PGI system is to recommend a parking location which is most likely to
have an available spot. Therefore, we need to find the probability of finding at least one empty
spot at the estimated arrival time for each parking location. This probability is given by,
𝜌 𝑖 ≜ Pr � 𝑒 𝑟𝑟
𝑖 < 𝑎 �
𝑖 , 𝑡 + 𝑑 𝑖 − 1 �=
1
� 2 𝜋 𝜎 𝑖 2
� exp( − 𝑥 2
2 𝜎 𝑖 2
⁄ )
𝑙 �
𝑖 , 𝑡 + 𝑑 𝑖 − 1
− ∞
(3.41)
where 𝜎 𝑖 2
is the i-th diagonal element of Γ
𝑑 𝑖 . We explain the use of (3.41) by an example; let
𝐿 = { 𝑙 1
, 𝑙 2
, … , 𝑙 𝑓 } denote the set of parking locations under consideration by a specific user of the
PGI system. By using (3.35) we predict the number of available parking spots of each parking
location � 𝑎 �
𝑙 1
, 𝑡 + 𝑑 1
, … , 𝑎 �
𝑙 𝑟 , 𝑡 + 𝑑 𝑟 � at the estimated arrival times. In order to increase the reliability of
the PGI system, we need to take into account the possible error of the predictions. In the next
step, the probabilities of finding at least one empty spot at the estimated arrival time for different
parking locations � 𝜌 𝑙 1
, … , 𝜌 𝑙 𝑟 � are calculated using (3.41). The parking location with highest 𝜌 𝑖 is
the most likely location to have an empty spot. In the next section we use the actual parking data
to verify the model.
Page No: 56
3.5 Demonstration
We use the data feed provided by SFpark.org to analyze and evaluate the proposed model for
parking availability. Figure 4.3 shows the map of Financial District neighborhood of San
Francisco, CA which is used in this report.
The red lines form a grid of tiles which serve as the parking tiles L1, L2, … ,L16 . The real-time
data is stored in a database, and updated every one minute. The order of the autoregressive model
is chosen as M=50 by observing the temporal correlations for different locations. In this part, we
first validate the proposed autoregressive model, and then evaluate the accuracy of predictions.
Figure 3-3 - Map of the test region (San Francisco Financial District). The real-time parking
data was obtained from SFPark. The red lines form a grid for the parking tiles.
3.5.1 Validation of the Model
Our goal is to use real data to verify the consistency of the estimator, and as a result answer the
question of “how many samples do we need to train the model efficiently?”. In part III.A we
Page No: 57
proved the consistency of the parameter estimator (3.19) by assuming that Plim �
Φ( 𝑘 − 1) Φ ′( 𝑘 − 1)
𝑘 �
exists and it is invertible, and Plim �
Ε( 𝑘 ) Φ ′( 𝑘 − 1)
𝑘 �= 0. In this part, we will verify these
assumptions.
Let us reconsider the autoregressive model (3.12) in the new matrix form of Ψ = ΘΦ + Ε:
Ψ( 𝑘 + 1) = ΘΦ( 𝑘 ) + Ε( 𝑘 ) (3.42)
where k is a variable representing the index of the last available data sample. The model
parameter matrix Θ is unchanged from (3.17), and Ψ( 𝑘 ), Φ( 𝑘 ), and Ε( 𝑘 ) are 𝑙 × 𝑘 , 𝑀 𝑙 × 𝑘 , 𝑙 ×
𝑘 matrices of the following form:
Ψ( 𝑘 ) = [ 𝑉 �
𝑘 𝑉 �
𝑘 − 1
… 𝑉 �
1
]
Φ( 𝑘 ) =
⎣
⎢
⎢
⎡
𝑉 �
𝑘 − 1
𝑉 �
𝑘 − 2
⋯ 𝑉 �
0
𝑉 �
𝑘 − 2
⋮
𝑉 �
𝑘 − 3
⋮
⋯
⋱
0
⋮
𝑉 �
𝑘 − 𝑀 𝑉 �
𝑘 − 𝑀 − 1
… 0 ⎦
⎥
⎥
⎤
E( 𝑘 ) = [ 𝑅 𝑘 𝑅 𝑘 − 1
… 𝑅 1
]
(3.43)
We define the spatio-temporal covariance matrix as,
Σ
Φ
𝑘 ≜
Φ( 𝑘 ) Φ( 𝑘 ) ′
𝑘
(3.44)
In order to show that Σ
Φ
𝑘 converges, we introduce a differential measure:
Ω
𝑘 , Δ
≜ � Σ
Φ
𝑘 − Σ
Φ
𝑘 − ∆
�
2
(3.45)
where k is the number of observations and ∆> 0. Figure 4.4-(a) shows the average of Ω
𝑘 , Δ
for
∆= 100 and 200 < 𝑘 < 1500 over 150 random data sets for a period of one week related to our
test region.
Page No: 58
Figure 3-4 - Model validation; (a) Increasing the number of samples will reduce 𝛀 𝒌, 𝚫 𝐟𝐟 𝐟 ∆=
𝟏𝟏𝟏 . (b) Increasing the number of samples would reduce 𝑨 𝒗 𝑨 � 𝑶 𝒌 �
𝟐 .
We repeated the same experiment with ∆= 1,2, … , 100, and expectedly we observe a similar
result; � Σ
Φ
𝐾 − Σ
Φ
𝐾 − ∆
�
2
→ 0 as 𝑘 → ∞, which implies that Σ
Φ
𝑘 converges to a constant matrix Σ
Φ
.
We also check the independency (orthogonality) of error and observations by defining the
orthogonality measure Ο
𝑘 for k samples as:
Page No: 59
Ο
𝑘 ≜
Φ( 𝑘 )E
�
( 𝑘 ) ′
𝑘
(3.46)
where E
�
( 𝑘 ) is evaluated by,
E
�
( 𝑘 ) = Ψ( 𝑘 ) − Θ
�
( 𝑘 ) Φ( 𝑘 − 1) (3.47)
Figure 4.4-(b) shows the average of ‖ 𝑂 𝑘 ‖
2
over the same set of data as figure 4.4-(a). Note that
the correlation between observations and error converges to zero at a faster rate than Ω. As Σ
Φ
𝑘
converges to a constant matrix and it is invertible (we force Σ
Φ
𝑘 to be nonsingular by removing
the nonsingular cases, as explained in section III.A) and 𝑂 𝑘 converges to zero, we conclude that
the estimator is consistent [49] (converges in probability).
As shown in the above figures it is desirable to have larger number of samples to train the model.
Considering the large scale matrices and computational limitation we chose K=900, i.e. 900
observations are being used to train the model in the batch processing mode.
The error 𝑅 𝑡 in the model (3.10) is assumed to be zero-mean normal process. Figure 4.5
compares the histogram of empirical error obtained from (3.47) with a normal distribution fitted
to the data. The figure confirms that the approximation is intuitively correct.
Page No: 60
Figure 3-5 - comparison of empirical model error histogram with a fitted normal distribution
(1000 samples) shows that we can approximate the process to be zero-mean Gaussian.
3.5.2 Accuracy of Predictions
We use three different scenarios to evaluate the prediction algorithm; Mean Absolute Percentage
Error (MAPE), and recommendation errors.
Scenario1: predictions are made for variable time horizons for random locations based on the
actual historic values. The results are then compared with the actual availability values. The
MAPE is defined as MAPE =
1
𝑛 ∑ �
𝑙 𝑖 − 𝑙 �
𝑖 𝑙 𝑖 �
𝑖 . Where 𝑎 𝑖 and 𝑎 �
𝑖 are actual and predicted availability
respectively, and n is the total number of parking locations Figure 4.6-(a) shows the MAPE
calculated for the predictions made using the data of 6/27/2013 of the San Francisco Financial
District. In this case, the MAPE increases as the prediction horizon grows. The error of
predictions is about 14% for a 20-minute horizon.
In order to compare the performance of our proposed model with other models [20]–[22] which
consider temporal correlations for predictions, we carry out the prediction for the same dataset as
previous example without considering the spatial correlations. In this case all the vector
Page No: 61
calculations in the report reduce to scalars. Figure 4.6-(b) illustrates the difference between
vector spatio-temporal model and scalar temporal model.
Figure 4.6-(b) shows the advantage of considering spatial correlations in modeling parking
availability, where the prediction error is reduced by 5% for 20-minute predictions. The
important point is that these results are related to all parking locations including on-street parking
blocks and off-street parking lots whereas the previous studies have taken into account only
parking lots.
Page No: 62
Figure 3-6 - Accuracy of the predictions; (a) Scenario 1 results, prediction errors for different time
of the day, and different prediction horizons. (b) Comparison of spatio-temporal model and
temporal model, using the same data set. (c) Scenario 2 results; analyzing the performance of the
prediction algorithm.
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Scenario 2: In this scenario we examine the effectiveness of the predictions with respect to
recommendations accuracy. The PGI system recommends the parking location with highest
probability of having at least one available spot to the user. Therefore, it is important to study the
recommendation error which is defined as; I) the system predicts there would be at least one
parking spot at a certain location at the estimated arrival time, however the driver finds no
available spots at the arrival, or II) the system predicts no availability, and the driver finds at
least one spot. In this scenario we consider the sum of these two types of error. These two cases
affect the reliability of the PGI systems adversely. Monte Carlo simulations are used to simulate
drivers with different driving times and different destinations. Note that the recommendation
error is different from the MAPE (in scenario 1). MAPE analysis shows the percentage of error
in the predicted number of available spots, whereas the recommendation error is the percentage
of times that the system makes a mistake in recommending parking to drivers.
The result of this simulation (figure 4.6-c) shows that using the proposed prediction system the
recommendation error improves significantly compared to the case where the prediction system
is not used. The recommendations without prediction (figure 4.6-c) use just the parking
availability data available at the time of decision making, and do not consider the predictions of
parking availability at the time of arrival. This improvement is especially more noticeable for
larger prediction horizons (larger than 5 minutes) were the prediction error is reduced more than
3 times using the prediction model.
Comparing the MAPE (figure 4.6-b) and recommendation error (figure 4.6-c) shows that, since
we recommend the parking location with the highest probability of having at least one parking
spot available (equation 4.41) to the user, the final recommendation error is low even though the
system might have some error in predicting the exact number of parking spots for a location. For
example, for a 20-minute prediction horizon, the 14% MAPE in net predictions (figure 4.6-b)
reduces to 5% error in the final recommendations (figure 4.6-c). Also note that the
recommendation error at 20-minute horizon is almost twice the error for a 5-minute horizon.
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This leads to the idea of updating the recommendations to the drivers as they approach their
destinations.
Scenario 3: In this scenario we compare the prediction accuracy for the on-street and off-street
parking locations. For each type we consider two cases; spatio-temporal model and just the
temporal model. In the latter case, spatial correlations are not considered, and the model (3.10) is
reduced to scalar. In this scenario 9 off-street parking garages, and 350 on-street parking blocks
in the San Francisco Financial District area are considered. Figure 4.7 shows the results.
Analysis shows that the prediction results for off-street parking locations are slightly better than
(by 2.1%) on-street predictions. This is due to the fact that occupancy of parking garages shows
less fluctuation than on-street occupancy.
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Figure 3-7 - Scatter plots of the 20-minute-predictions versus the actual parking availability. Blue
dots indicate the spatio-temporal model results, and red dots indicate the temporal model results.
(a) on-street locations. (b) Off-street locations. Analysis shows that the prediction results are
slightly better for the off-street case.
3.6 Conclusion
Using real-time parking availability data in intelligent transportation systems and parking
guidance systems proves to be challenging due to the fact that parking availability is a stochastic
process. Available parking spots might be taken by the time the driver arrives at the parking
location. Therefore, a model that predicts parking availability at the estimated arrival time will
enhance the performance and improve the acceptance of PGI systems. . In this report, we
developed a vector spatio-temporal autoregressive model that can be used to predict the parking
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availability for both on-street and off-street parking locations at the estimated arrival time of the
driver. The proposed model considers temporal correlations of parking availability data, as well
as spatial correlations. It is used to recommend the parking location with the highest probability
We used real-time actual parking data from San Francisco area to evaluate the results, and verify
the model. The results indicate that the proposed system recommends parking locations to
drivers with high reliability. The accuracy of predictions depends on the time horizon ahead. For
a 20 minutes prediction horizon the system was demonstrated to recommend a parking location
to a driver with an accuracy of about 95%.
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Chapter 4. Adaptive Network-Wide Traffic Signal Control
System with Truck Priority
Traffic light control is an effective way of controlling traffic at intersections in order to improve
safety and reduce overall delays. The current methods for controlling traffic lights treat all
vehicles the same. In areas such as close to ports and commercial areas where trucks are mixed
with passenger vehicles there is an opportunity to minimize traffic delays for all vehicles by
taking into account the big difference in dynamical characteristics between heavy duty and light
vehicles. In this paper, we present a traffic light control method with the goal of reducing delays
of all the vehicles in the network by giving priority to trucks. Moreover, reducing the number of
stops for trucks can significantly reduce the pavement damage. The presented method
dynamically optimizes the vehicle delays which are approximated by a set of nonlinear
functions. The vehicle delays are modeled using neural networks. The developed signal
controller is adaptive since the delay prediction model is updated once the new data is obtained
from the infrastructure. Simulation results show that in a test area where trucks compose 20% of
all the vehicles in the network, applying the suggested traffic control method will reduce the
average delay of vehicles about 25.2% by reducing the number of truck stops as much as 61%.
4.1 Introduction
Adaptive Traffic Control Systems (ATCS) are traffic management systems which adjust the
timing of traffic signals to adapt to changing traffic patterns and ease traffic congestion. In such
systems, A performance index (PI), e.g. overall delay, number of stops, queue lengths, fuel
consumption or a combination of these parameters is minimized [54]. ATCS optimize traffic
flow on arterial networks with multiple signalized intersections. SCOOT [55] and SCAT [56] are
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prominent and well established ATCS. Systems designed by German researchers are good
examples of ATCS such as MOTION [57] and BALANCE [58]. Improved traffic modeling
techniques developed in the recent years and increase in the computation power, promote
enhancement and further development of sophisticated ATCS. SCOOT minimizes the average
queues by adjusting the signal timings and continuously measuring traffic volumes. The potential
timing plans are evaluated heuristically to adjust the signal timings. On the other hand, SCATS
system does not require modelling. Systems such as SCOOT and SCATS suffer from inefficient
handling of saturated conditions due to inadequate real-time adaptability [59].
Other adaptive systems such as OPAC and RHODES [60]–[62] obtain signal timings by solving
a set of optimization problems in real-time. The performance index to be minimized is the total
intersection delay. These systems suffer exponential complexities that reduce their chances of
being deployed on a large scale [63].
Traffic-responsive urban control (TUC) is another adaptive control strategy [64]. Based on a
store-and-forward modelling of the urban network traffic and using the linear-quadratic
regulatory theory, the design of TUC leads to a multivariate regulator for traffic-responsive,
coordinated network-wide signal control that is particularly suitable also for saturated traffic
conditions. Real-time decisions in TUC cannot be taken more frequently than at the maximum
employed signal cycle. The strategy will need to be redesigned in the case of modifications and
expansions of the controlled network. TUC was compared with a fixed-time signal control
producing reduction in total waiting time and total travel time in the system.
In a large network, if each traffic signal is controlled locally and independent from the rest of the
network, traffic congestion can increase over the whole network. Therefore in order to optimize
the traffic flows effectively, we need to consider the traffic situation and state of the adjacent
intersections for each individual intersection. The dependency of traffic volumes at each
intersection on its neighbors makes it difficult to set the signal timings for a large traffic network
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with multiple intersections. An interesting approach to deal with this traffic signal control
problem is to use distributed control technique involving multiple agents. The goal of a multi-
agent control system is to reduce the traffic congestion for multiple intersections simultaneously.
For effective traffic signal control, such controllers need to adapt themselves continuously. De
Oliveira and Camponogara [65] propose a network of distributed agents to control linear
dynamic systems which are put together by interconnecting linear subsystems with local input
constraints. The framework decomposes the optimization problem obtained from the model
predictive control (MPC) approach into a network of coupled and small sub-problems to be
solved by the agent network. Each agent senses and controls the variables of its intersection,
while communicating with agents in the neighborhood to obtain variables and coordinate their
actions. The proposed approach achieved performance comparable to the TUC system. A real-
time traffic controller is proposed in [66] using a distributed network of agents. The online
learning and update process for each agent is improved by designing a stochastic cooperative
parameter update algorithm. In another study [62], a collaborative reinforcement learning
(RL)algorithm is employed using a local adaptive round robin phase switching model at each
intersection. Each intersection collaborates with adjacent agents in order to learn appropriate
phase timings. In [67], a multi-agent RL was designed to optimize traffic signals at multiple
intersections. The RL systems are trained by waiting times for vehicles and different settings of
the traffic signals Presented results confirm that the proposed algorithm outperforms non-
adaptable traffic light control systems[68].
Another important issue in the traffic control field is implementing priority systems for different
classes of vehicles such as emergency vehicles, bus/transit, and trucks. There are successful
systems implemented for transit systems aiming to reduce the delay times for certain type of
vehicles. For example, RHODES [69] tries to minimize the delay of buses by giving priority to
the buses approaching an intersection, taking into account the number of passengers on each bus
and how late the bus is compared to its schedule.
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Figure 4-1 Block Diagram of the control scheme. The controller uses the data obtained from
sensors to update the traffic model and estimate the future delay. The control strategy optimizes the
traffic signals by solving a set of optimization problems.
Our goal in this paper is to optimize the traffic signal timings in order to minimize the overall
delay of all the vehicles in the network. In order to do so, our system gives priority to trucks in
areas where the number of trucks is relatively high, for example near ports. Studies have shown
that due to slower dynamics of trucks, such as lower acceleration, deceleration and speed, their
contribution to traffic congestion is three to five times higher than other vehicles [70]. Therefore
minimizing the number of truck stops by giving them priority at intersections leads to optimized
overall delay for all the vehicles in the network. Moreover, reducing the number of truck stops
leads to a significant reduction in pavement damages [71]. Here, we propose a multi-agent
distributed traffic signal controller which incorporates the truck data into the optimization
problem. We first develop the delay predictor model. This neural network based model predicts
very short term delays of all the vehicles in the network based on the information of the cars and
trucks and also information obtained from neighboring signals. In the next step, we develop an
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algorithm to optimize the traffic delay (predicted by the neural network). This algorithm
optimizes the next transition time of traffic signals to minimize the delay for each intersection
considering the state of the adjacent intersections and therefore minimizes the overall delay of
the traffic network. Figure 1 shows the block diagram of the control scheme. Our results confirm
that prioritizing trucks at the intersections minimizes the overall delay of the network.
4.2 Proposed Model
We first start with the single intersection, and expand the model to multiple intersections. The
delay is defined as the average of differences between the actual travel times and the free flow
travel times with no stops or slowing (at maximum allowed speed) for all vehicles in the
network. Delay is then predicted using a neural network model. For a single intersection we do
not consider the state of the adjacent intersections, and the problem can be formulated as a
convex optimization problem. Figure 2 illustrates a single intersection. We assume that
information (average speed and number) about trucks and cars is available as two separate sets of
inputs.
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Figure 4-2 Schematic showing a single intersection. Each link is divided into a number of
segments
The problem of controlling a traffic signal in an intersection is handled by an agent. For each
intersection we use a single layer neural network (NN) to predict the average delay of all
vehicles. Each road link is divided into a number of segments (fig. 2). The number and average
speed of trucks and cars for each road segment, the length of queues at each link, current status
of traffic lights as well as the future state of the traffic lights are fed into the neural network
model. The inputs to the NN are as follows:
NTj: Number of trucks in segment j
VTj: average speed of trucks in segment j (km/h)
NCj: Number of cars in segment j
VCj: average speed of cars in segment j (km/h)
LQi: length of queue at link I (m)
Si: current state of traffic signal (green/red)
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S’: future state of the traffic signals (transition time of signals in seconds)
Figure 3 shows the structure of the neural network used to model the time delay:
(4
Figure 4-3 Structure of the neural network model to predict the overall delay of the network based
on the vehicles information, current state of the traffic signals and future state of the traffic signals.
By training the neural network, the delay is modeled as:
𝐷 𝑑 𝑘 = 𝑓 𝑘 � 𝑆 𝑘 , 𝑅 𝑑 𝑘 � = ∑ 𝑎 𝑗 𝑘 𝑦 �𝒘 𝑗 𝑘 𝑇 𝑆 �
𝑁 𝑗 = 1
(4.1)
where 𝐷 𝑑 𝑘 denotes the delay of all the vehicles for the next time period d at time step k, 𝑆 𝑘
denotes the input vector at time step k defined as 𝑆 𝑘 = [ 𝑉 𝑇 , 𝑁 𝑇 , 𝑉 𝐶 , 𝑁 𝐶 , 𝑄 , 𝑆 ], and 𝑅 𝑑 𝑘 is the future
transitions of signal at time step k for the same time period. N denotes the number of hidden
layers in the Neural Network, 𝒘 𝑘 = � 𝑎 1
𝑘 , … , 𝑎 𝑁 𝑘 , 𝒘 1
𝑘 , … , 𝒘 𝑁 𝑘 � is a 1 × 𝑁 𝑤 vector consists of the
weights of the NN, 𝑆 = � 𝑆 𝑘 , 𝑅 𝑑 𝑘 � is the input vector, and 𝑦 ( 𝐴 ) = (1 + 𝑒 − 𝑣 )
− 1
is the logistic
function. The weights 𝒘 𝑘 are updated online by backpropagation using the gradient descent
method [72]:
𝑤 𝑖 𝑘 + 1
= − ∇ 𝐽 𝑘 + 1
+ 𝛼 𝑘 𝑤 𝑖 𝑘 , 𝑖 = 1, … , 𝑁 𝑤 (5.2)
where 𝐽 𝑘 + 1
is the new performance surface and 𝛼 𝑘 is the dynamic learning rate. Traffic light
control relies on finding the arguments 𝑅 𝑑 𝑘 ∗
= argmin 𝐷 𝑑 𝑘 which minimizes the delay with respect
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to constraints on minimum and maximum green light cycles. We describe the solution
methodology in the next section after introducing the multi-intersection scheme.
4.2.1 Multi-intersection Model
Expanding the single-intersection model involves introducing new inputs to the delay model. As
explained in the introduction, to control each signal efficiently, we need information from
adjacent signals in addition to the local traffic data. Figure 4 illustrates the NN with the new
parameters.
Figure 4-4 Schematic and neural network structure for predicting delay in the multi-intersection
model. Information from adjacent intersections are fed into the network in addition to the local
traffic data.
The training of the NN is similar to the single intersection. After the model is trained, a
prediction of the model can be made using:
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𝐷 𝑖 , 𝑑 𝑘 = 𝑓 𝑖 𝑘 � 𝑆 𝑖 𝑘 , 𝑆 𝑖 , 𝑙 𝑘 , 𝑅 𝑖 , 𝑑 𝑘 , 𝑅 𝑖 , 𝑙 𝑘 � = ∑ 𝑎 𝑖 𝑗 𝑘 𝑦 �𝒘 𝑖 𝑗 𝑘 𝑇 𝑆 �
𝑁 𝑗 = 1
(4.3)
where the index 𝑖 denotes the intersection and letter a means the variable belongs to the adjacent
intersections.
4.3 Controller
At each time step agents decide about the next few transitions of the controlled signal. The
decision involves finding the argument to minimize the sum of delays of all intersections
simultaneously:
𝐽 = min
𝑆 ∑ 𝑓 𝑖 𝑘 � 𝑆 𝑖 𝑘 , 𝑆 𝑖 , 𝑙 𝑘 , 𝑅 𝑖 , 𝑑 𝑘 , 𝑅 𝑖 , 𝑙 𝑘 �
𝐼 𝑖 = 1
𝑠 . 𝑡 . 𝐴 𝑖 ≤ 𝐴 𝑖 ≤ 𝑢 𝑖 (4.4)
where 𝐴 𝑖 and 𝑢 𝑖 denote the minimum and maximum green/red times, 𝐴 𝑖 denotes the green time
for each route, and I is the total number of intersections. Since 𝑅 𝑖 , 𝑑 is the information vector
obtained from infrastructure and known to the controller at the decision making time we can
rewrite equation (4) as,
𝐽 = min
𝑆 ∑ 𝑓 𝑖 ( 𝑆 𝑖 , 𝑆 𝑖 𝑙 )
𝐼 𝑖 = 1
𝑠 . 𝑡 . 𝐴 𝑖 ≤ 𝐴 𝑖 ≤ 𝑢 𝑖 (4.5)
Since the neural network model is a sum of sigmoidal functions, we have:
𝑓 𝑖 ( 𝑆 𝑖 , 𝑆 𝑖 𝑙 ) = ∑ 𝑎 𝑖 𝑗 𝑦 ( 𝑤 𝑇 𝑆 )
𝑁 𝑗 = 1
(4.6)
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where 𝑆 ≜ �
𝑆 1
⋮
𝑆 𝐼 � is the state vector of the whole network, and 𝑦 ( 𝐴 ) = (1 + 𝑒 − 𝑣 )
− 1
is the logistic
function. Note that 𝑆 𝑖 ≜ [
𝑠 𝑖 1
… 𝑠 𝑖 𝑁 𝑠 ]
𝑇 . And for further simplicity, by defining 𝐴 𝑖 𝑗 (. ) ≜
𝑎 𝑖 𝑗 𝑦 (. ) the optimization problem becomes:
𝐽 = min
𝑆 ∑ ∑ 𝐴 𝑖 𝑗 � 𝑤 𝑖 𝑗 𝑇 𝑆 �
𝑁 𝑗 = 1
𝐼 𝑖 = 1
𝑠 . 𝑡 . 𝑙 𝑖 𝑗 ≤ 𝑠 𝑖 𝑗 ≤ 𝑢 𝑖 𝑗 (4.7)
where I denotes the total number of network agents, and N is the number of hidden neural
network layers for each agent. At each time step the optimum set of signal transition times is
obtained by solving the above optimization problem. Equation (7) is a linear constrained general
nonlinear optimization problem which consists of a sum of sigmoidal functions. The functions
are not separable in the current form, and thus solving it is not a trivial task. By using the
following linear transformation we obtain a set of separable functions.
𝑍 = 𝑊𝑆 (4.8)
where 𝑍 ≜ [
𝑧 1 1
… 𝑧 1 𝑁 …
𝑧 𝐼 1
… 𝑧 𝐼 𝑁 ]
𝑇 denotes the new state vector, and 𝑊 ≜
[
𝑤 1 1
… 𝑤 1 𝑁 …
𝑤 𝐼 1
… 𝑤 𝐼 𝑁 ]
𝑇 is a 𝐼 𝑁 × 𝐼 𝑁 𝑠 matrix, where 𝑁 𝑠 is the number of states
for each agent. Then the problem becomes:
𝐽 = min
𝑆 ∑ ∑ 𝐴 𝑖 𝑗 � 𝑧 𝑖 𝑗 �
𝑁 𝑗 = 1
𝐼 𝑖 = 1
𝑠 . 𝑡 . 𝑙 𝑖 𝑗 ′
≤ 𝑧 𝑖 𝑗 ≤ 𝑢 𝑖 𝑗 ′
(4.9)
which is a separable nonlinear optimization. The final set of optimization problems reduces to:
𝐽 𝑖 𝑗 = min
𝑆 𝐴 𝑖 𝑗 � 𝑧 𝑖 𝑗 �
𝑠 . 𝑡 . 𝑙 𝑖 𝑗 ′
≤ 𝑧 𝑖 𝑗 ≤ 𝑢 𝑖 𝑗 ′
, 𝑖 = 1, … , 𝐼 , 𝑗 = 1, … , 𝑁 (4.10)
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where 𝑙 𝑖 𝑗 ′
, 𝑢 𝑖 𝑗 ′
are the new sets of constraints obtained by the linear transformation. Note that
𝐴 𝑖 𝑗 � 𝑧 𝑖 𝑗 � = 𝑎 𝑖 𝑗 𝑦 � 𝑧 𝑖 𝑗 �, and depending on the sign of 𝑎 𝑖 𝑗 , 𝐴 𝑖 𝑗 � 𝑧 𝑖 𝑗 � = 𝑎 𝑖 𝑗 𝑦 � 𝑧 𝑖 𝑗 � is a strictly
increasing or strictly decreasing function, given 𝑎 𝑖 𝑗 > 0 or 𝑎 𝑖 𝑗 < 0 respectively, since 𝑦 ( 𝐴 ) =
(1 + 𝑒 − 𝑣 )
− 1
is a strictly increasing function. Therefore,
𝑧 𝑖 𝑗 ∗
= �
𝑙 𝑖 𝑗 ′
𝑎 𝑖 𝑗 > 0
𝑢 𝑖 𝑗 ′
𝑎 𝑖 𝑗 < 0
,
𝑖 = 1, … , 𝐼
𝑗 = 1, … , 𝑁 (4.11)
where 𝑧 𝑖 𝑗 ∗
is the optimum transformed state vector. The actual state vector is obtained by,
𝑆 ∗
= 𝑊 − 1
𝑍 ∗
(4.12)
In order to have unique and invertible transformation the number of states (future transitions of
signals) must be equal to the number of hidden units for each agent. In this case, W is a square
matrix with non-zero elements on the diagonal.
4.4 Control Algorithm
The control procedure of the whole network consists of the following steps:
Step 1: At time step k generate the control vector 𝑆 𝑘 by solving the optimization problem (7).
Step 2: modify the control vector by taking a weighted average of the current and previous
decisions:
𝑆 𝑘 ∗
= ∑ 𝛽 𝑖 𝑆 𝑘 − 𝑖 𝑝 𝑖 = 1
, 0 < 𝛽 𝑝 < ⋯ < 𝛽 1
< 1 (4.13)
Step 3: apply the modified control vector 𝑆 𝑘 ∗
to the signals in the network.
Step 4: update the weights of the model (3).
Step 5: return to step 1 at time step k+1.
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By applying the above control algorithm, at each time step, the future transitions of the signals
are updated by progression of time. Figure 5 shows an example of signal timing progression
generated by simulation:
Figure 4-5 sample results for traffic light transitions at a junction using the above control strategy.
The first horizontal row shows the planned transitions for the next 2 minutes which are made at
time 0 based on the traffic state observed at the moment. At t time step 5 (vertical axis), based on
the developments in the traffic situation, the transition plan is updated which is illustrated on the
second horizontal row. The adjustments at each decision time are made by taking a weighted
average of the 3 previous decisions to prevent abrupt changes to the plan.
4.5 Simulation Results
We evaluate the proposed control algorithm with the microscopic traffic simulator VISSIM. We
chose a road network adjacent to the port of Long Beach which is associated with a high volume
of truck traffic. Our test network, shown in figure 6, consists of 15 signalized intersections which
are controlled by 15 signal controllers, and 99 intersections controlled by stop signs. Road links
are divided into segments each with a length of 60 m and the delay prediction model receives the
vehicle information from each segment. Trucks compose 10% of the total vehicles in the
network, and the flows are generated by the VISSIM dynamic assignment module. The control
algorithm is coded in C++ and integrated with VISSIM using the COM interface. We carried out
multiple tests to evaluate different aspects of the control algorithm. These tests are described in
the following scenarios.
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Figure 4-6 Test network confined by four major roads in Long Beach, CA. 15 Traffic signal
are shown on the map.
4.5.1 Scenario 1:
First we evaluate the delay prediction model obtained by training the neural network. The
training is done by using MATLAB Neural Network Toolbox. Figure 6 shows the prediction
results for the multi-intersection test network. The Mean Squared Error (MSE) of delay
predictions initially improves by increasing the number of hidden layers and nodes. On the other
hand increasing the number of nodes beyond a certain point will cause over-training and
increases MSE. Another important practical issue is the processing time needed to train the NN
which is proportional to the size of the neural network. Table 1 shows the trade-off between
MSE versus process time for our test network. Based on the results shown in this table we
choose a single layer network with 7 nodes, as the MSE of the predictions are acceptable for this
network and adding further layers or nodes to the model increases the process time without
significantly improving the error of delay prediction.
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Figure 4-7 performance of the delay prediction model for different NN size. For 11-node
network the MSE is 2.2% while for a 5-node network the MSE increases to 7.5%.
NN Structure
MSE (%)
Process Time
(s)
Single-layer w/ 3 nodes 11.3 0.70
Single-layer w/ 5 nodes 7.5 0.98
Single-layer w/ 7 nodes 5.0 1.37
Single-layer w/ 9 nodes 3.4 1.92
Single-layer w/ 11 nodes 2.2 2.69
Single-layer w/ 13 nodes 2.2 3.76
2-layer w/ 3 nodes 10.4 1.50
2-layer w/ 5 nodes 6.9 2.10
2-layer w/ 7 nodes 4.6 2.94
2-layer w/ 9 nodes 3.1 4.12
2-layer w/ 11 nodes 2.1 5.76
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2-layer w/ 13 nodes 1.8 8.07
Table 4.1 - the processing is performed on a i7 Intel processor, with an input vector consists of
116 input variables for each intersection, and a batch of 1000 datasets.
4.5.2 Scenario 2
In this scenario we evaluate the performance of the proposed model with regard to the existence
of interconnections between agents. In other words, we compare two modes; one with the
presence of communication between intersections, and one without it. Figure 7 shows the
chromosomes used to feed the neural network in each case.
We also compared these two systems with the performance of the widely deployed actuated
traffic signal controller, where inductive sensors are used to detect vehicles approaching the
intersection. Our simulation results indicate the importance of communication between the
intersections in the performance of the controller. Figure 8-(b) shows that the performance of the
traffic signal controller is enhanced by 12% by means of communications between adjacent
intersections and shows about 26% improvement compared to the actuated traffic signal
controller. Also note that using a network-wise controller with intercommunications between the
agents, the delay of the vehicles tends to be smoother compared to the delay of an actuated signal
controller. Figure 8(a) confirms that making the agents communicate with each other helps
reducing the prediction error.
4.5.3 Scenario 3
In this final part we evaluate the effect of giving priority to trucks on the performance of the
suggested traffic signal controller. To do this, we compare our original delay prediction model
introduced in the previous part (in which the information from trucks and cars are fed into the
NN as separate inputs) with a modified version of the delay prediction model which receives
the averaged data of cars and trucks as a single input set in the neural network, so the network
cannot distinguish between cars and trucks. Figure 9 compares the sum of queue length at all
Page No: 82
intersections in the network for three controllers; actuated controller, optimized with Priority,
and optimized without priority. As it is evident in table 2, both delay and number of stops are
smaller for all vehicle types in our traffic light controller system. Compared to an optimized
system without priority, the use of our controller system leads to 4% lower overall delay and 8%
lower stops per vehicle. The improvement in the average delay time for trucks only, is 6%.
These simulation results indicate that our system outperforms other adaptive multi-agent traffic
signal control systems such as [67] as well as the commonly used actuated control both in terms
of vehicle delays and number of stops in areas where trucks comprise a significant percentage of
the total vehicles.
Figure 4-8 the input vectors for (a) the special case which agents acts independently, and (b)
the general case which agents consider the effects of neighboring intersections.
Figure 4-9 Scenario 2: comparison of two cases; with communication between links, and without
communications. (a) The MSE of the prediction model in the case of no-interconnections increases
to 11%, where the MSE for the general case is near 5%. (b) the average vehicle d delays obtained
by actuated traffic signal controller is 31 seconds, whereas the average delay for the locally
optimized controller (without interconnections) is 26 seconds, and the delay for the general case
(network-wise optimized) is 23 seconds.
Page No: 84
Figure 4-10 Scenario 3: the average queue length with actuated controller, optimized with
priority, and optimized without truck priority is 151m, 131m, and 124m respectively.
Optimized w/
Priority
Optimized w/out
Priority
Actuated Control Fixed-Time
Ave. No. of Truck
Stops
4.34 5.07 6.13 6.89
Ave. No. of car
Stops
5.89 6.34 8.24 8.62
Ave. No. of Stops
for all
5.73 6.22 8.03 8.46
Ave. delay Trucks
(S)
64.13 68.04 74.26 91.80
Ave. delay cars (S) 67.65 70.24 75.95 88.75
Ave. delay all (S) 67.30 70.02 75.78 89.03
Fuel Trucks (g/km) 330.05 350.12 381.85 470.90
Fuel Of Cars (g/km) 110.76 114.76 123.79 143.60
Fuel Of all (g/km) 132.69 138.30 149.60 170.50
Page No: 85
CO2 Emis. all
(g/km)
346.94 361.60 391.14 445.80
NOx Emis. all
(g/km)
0.82 0.86 0.93 1.06
Table 4.2 - Comparison between different simulated control algorithms. The second column shows
that the proposed truck priority algorithm reduces the total delay of the network by minimizing
average truck stops.
4.6 Conclusion and Summary
In this paper, we have presented an adaptive network-wide traffic signal control system with
truck priority suitable for areas where trucks compose a significant portion of the traffic network
vehicles. Our system takes advantage of a single layer neural network model to predict the
average delay of vehicles. Each intersection has its dedicated delay prediction agent which
communicates with agents from other intersections of the network to receive the future and
current status of the traffic lights in the neighboring intersections. The controller system uses the
predicted delay, which is a function of future traffic light transition times and adaptively finds
optimized transition times to minimize the overall delay. The proposed NN prediction model
receives the data from trucks and other vehicles as two separate sets of inputs, and therefore
takes into account the effect of trucks on the overall delay separately from other vehicles. Since
the slow dynamics of trucks is responsible for additional delays at the intersections, the
controller system gives priority to trucks in order to minimize the average delay for all the
vehicles. Our simulation results for a 9 intersection network where trucks comprise 20% of the
traffic load, indicates that applying the suggested traffic control method will reduce the average
delay of all vehicles about 25.2% by reducing the number of truck stops as much as 61%
compared to the actuated control. For areas with higher truck traffic, such as a port area this
improvement will be even more significant.
Chapter 5. Concluding Remarks and Proposed Future
Topics
5.1 Final Conclusions
We proposed a Smart Routing system for personal automobile navigation. At its core, the
challenge behind Smart Routing is an effective estimation of traffic flows in the transportation
network. If we can effectively estimate the travel from zone to zone with a network, we can
assign traffic to the links (roads) in the network and generate traffic flows given roadway
capacities. We then minimize the travel time for a driver travelling from a node in the network
(his origin) to his destination node. To accurately estimate traffic flows, we first make the initial
traffic flow estimation model to use as a control condition against which we will compare
observed traffic flows. We then take into account historical loop detector and traffic count data
as well as data from our monitoring of events to create an offline estimate for each hour of every
day during the week (7 x 24 = 168 hours in total). Finally, we add real-time traffic data and try
and match existing traffic conditions. The traffic flow estimation model needs continuous,
dynamic adjustment so that the navigation system will always be directing the driver along the
path that requires the least time from the driver’s ever-changing origin (as he progresses forward)
to his static destination.
In order to accurately estimate traffic flows, we need to analyze the impact of on-road and off-
road events on links across the network. Detour routes adjacent to certain events may not
necessarily be the time-minimizing routes because drivers avoiding the event may clog the
adjacent links as well. We classify events that create the network as on-road and off-road events.
We further divide these events into planned and unplanned; ideally we can input planned events
into the traffic flow estimation model before they occur. It is fairly straightforward to monitor
the effects of planned on-road events such as road and lane closures since we can compare the
Page No: 87
network with the closure to the baseline of no closure. Accidents are unplanned on-road events
that are, by nature, nearly impossible to plan for; it is almost as difficult to accurately estimate
the accident’s effects. This is because accidents vary in size, intensity, and visibility.
Furthermore, it takes a significant amount of time for accidents to be recorded as such so there is
a chance the network will become congested before we adjust traffic flows because of news of an
accident. Ideally we will be able to preemptively identify characteristic changes in traffic flow
that signify that an accident has occurred. Finally off-road cultural events such as concerts and
sporting events can create huge automobile attraction rates. These events will also release
thousands of automobiles on to the network at their conclusions. When we create the initial and
historical link flows for these off-road events by monitoring major events like USC football
games; however, instead estimating flows every hour, we will shorten our time period to ten-
minute intervals.
The methodology for Smart Routing can be applied to routing cars through the network in the
event of natural disasters and terrorist attacks. Here we will need to set the capacity of several
links or nodes to zero and then run the usual traffic flow estimation model given that many
drivers will be seeking to evacuate the disaster zone. It is this estimation of the effects of major
disruptions on links and nodes that is the subject of the remainder of the chapter.
Using real-time parking availability data in intelligent transportation systems and parking
guidance systems proves to be challenging due to the fact that parking availability is a stochastic
process. Available parking spots might be taken by the time the driver arrives at the parking
location. Therefore, a model that predicts parking availability at the estimated arrival time will
enhance the performance and improve the acceptance of PGI systems. In this report, we
developed a vector spatio-temporal autoregressive model that can be used to predict the parking
availability for both on-street and off-street parking locations at the estimated arrival time of the
driver. The proposed model considers temporal correlations of parking availability data, as well
as spatial correlations. It is used to recommend the parking location with the highest probability.
Page No: 88
We used real-time actual parking data from San Francisco area to evaluate the results, and verify
the model. The results indicate that the proposed system recommends parking locations to
drivers with high reliability. The accuracy of predictions depends on the time horizon ahead. For
a 20 minutes prediction horizon the system was demonstrated to recommend a parking location
to a driver with an accuracy of about 95%.
In Chapter 5, we have presented an adaptive network-wide traffic signal control system with
truck priority suitable for areas where trucks compose a significant portion of the traffic network
vehicles. Our system takes advantage of a single layer neural network model to predict the
average delay of vehicles. Each intersection has its dedicated delay prediction agent which
communicates with agents from other intersections of the network to receive the future and
current status of the traffic lights in the neighboring intersections. The controller system uses the
predicted delay, which is a function of future traffic light transition times and adaptively finds
optimized transition times to minimize the overall delay. The proposed NN prediction model
receives the data from trucks and other vehicles as two separate sets of inputs, and therefore
takes into account the effect of trucks on the overall delay separately from other vehicles. Since
the slow dynamics of trucks is responsible for additional delays at the intersections, the
controller system gives priority to trucks in order to minimize the average delay for all the
vehicles. Our simulation results for a 9 intersection network where trucks comprise 20% of the
traffic load, indicates that applying the suggested traffic control method will reduce the average
delay of all vehicles about 25.2% by reducing the number of truck stops as much as 61%
compared to the actuated control. For areas with higher truck traffic, such as a port area this
improvement will be even more significant.
5.2 Proposed Future Topics
This research discussed in this report will be continued in two areas;
Page No: 89
5.2.1 Traffic light optimization for different classes of vehicles
Traffic light control is an effective way of controlling traffic at intersections in order to improve
safety and reduce overall delays. The current methods for controlling traffic lights treat all
vehicles the same. In areas such as close to ports and commercial areas where trucks are mixed
with passenger vehicles there is an opportunity to minimize traffic delays for all vehicles by
taking into account the big difference in dynamical characteristics between heavy duty and light
vehicles. We will develop a traffic light control method with the goal of reducing delays of all
the vehicles in the network by giving implicit priority to trucks. Moreover, reducing the number
of stops for trucks can significantly reduce the pavement damage. This method will dynamically
optimize the vehicle delays which are approximated by a set of nonlinear functions. The vehicle
delays will be modeled using neural networks. The developed signal controller is adaptive since
the delay prediction model is updated once the new data is obtained from the infrastructure.
This approach can be extended to more vehicle classes such as heavy duty trucks, public transit,
emergency vehicles, and pedestrians. We will be able to optimize the travel times of each class
of vehicles, and therefore optimize the travel times for all the vehicles in the network.
Specifically, Trucks are in general detrimental to traffic due to their slow dynamics and large
size. The time that a truck takes to respond to a traffic light going green, accelerate to cross the
intersection and the time it takes to clear the intersection is close to 3 to 5 times higher than that
of ordinary passenger vehicles. Today’s traffic lights do not take into account the presence of
trucks, which are treated as any other vehicle for traffic light control purposes. As information
and sensor technologies are finding their way to vehicles and traffic management systems,
vehicle classification can no longer be a problem. In other words with today’s communication
and GPS technologies a traffic light could be informed of the different vehicle classes of vehicles
approaching the intersection (position, type and speed) and could take into account their
dynamics in an effort to maximize the flow that goes through in both directions. It may be
Page No: 90
beneficial to all traffic for example to eliminate the delays in trucks decelerating to a stop and
then accelerate to a certain speed at a traffic light by just letting them go through where possible.
5.2.2 Parking Guiding Systems for Trucks
In urban environments, trucks impose great challenges to urban management and planning
authorities, truck drivers, and other drivers. Parking is a main challenge for trucks especially in
areas where trucks are present such as ports and industrial regions. The 2014 annual Top
Industry Issues Survey conveyed by American Transportation Research Institute (ATRI) shows
that truck parking ranked 6th overall but among truck drivers in the survey it ranked 2nd,
eclipsed only by the challenges associated with the Hours-of-Service regulations.
We will tackle the truck parking problem by redesigning the models and algorithms developed
for car parking systems to adjust to the different characteristics of truck parking. Parking for
regular vehicles has a more stochastic nature in compare to the truck parking. This is due to the
fact that truck operations are scheduled and can be tracked.
Page No: 91
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Asset Metadata
Creator
Rajabioun, Tooraj
(author)
Core Title
Improving mobility in urban environments using intelligent transportation technologies
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
04/25/2016
Defense Date
03/03/2016
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
historical time traffic flows,least square method,OAI-PMH Harvest,optimization,parking guidance systems,parking prediction,spatio-temporal models,traffic flow prediction
Format
application/pdf
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Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Ioannou, Petros (
committee chair
), Bogdan, Paul (
committee member
), Dessouky, Maged (
committee member
)
Creator Email
rajabiou@usc.edu,toorajabioun@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-242739
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etd-RajabiounT-4370.pdf (filename),usctheses-c40-242739 (legacy record id)
Legacy Identifier
etd-RajabiounT-4370.pdf
Dmrecord
242739
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Rajabioun, Tooraj
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
historical time traffic flows
least square method
optimization
parking guidance systems
parking prediction
spatio-temporal models
traffic flow prediction