Designing heterogeneous sensor networks for estimating and predicting path travel time dynamics: An...

Transportation Research Part B 00 (2013) 000–000

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Designing Heterogeneous Sensor Networks for Estimating and

Predicting Path Travel Time Dynamics: An Information-Theoretic

Modeling Approach

Tao Xinga, Xuesong Zhoub,* Jeffrey Taylorc

aDepartment of Civil and Environmental Engineering, University of Utah, Salt Lake City, UT 84112-0561,USA [email protected] bSchool of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85287, USA [email protected]

cDepartment of Civil and Environmental Engineering, University of Utah, Salt Lake City, UT 84112-0561,USA [email protected]

Abstract

With a particular emphasis on the end-to-end travel time prediction problem, this paper proposes an information-theoretic sensor

location model that aims to minimize total travel time uncertainties from a set of point, point-to-point and probe sensors in a

traffic network. Based on a Kalman filtering structure, the proposed measurement and uncertainty quantification models

explicitly take into account several important sources of errors in the travel time estimation/prediction process, such as the

uncertainty associated with prior travel time estimates, measurement errors and sampling errors. By considering only critical

paths and limited time intervals, this paper selects a path travel time uncertainty criterion to construct a joint sensor location and

travel time estimation/prediction framework with a unified modeling of both recurring and non-recurring traffic conditions. An

analytical determinant maximization model and heuristic beam-search algorithm are used to find an effective lower bound and

solve the combinatorial sensor selection problem. A number of illustrative examples and one case study are used to demonstrate

the effectiveness of the proposed methodology.

Key words: travel time prediction, sensor network design, automatic vehicle identification sensors, automatic vehicle location sensors

1. Introduction

1.1. Motivation

To provide effective traffic congestion mitigation strategies, transportation planning organizations and traffic

management centers need to (1) reliably estimate and predict network-wide traffic conditions and (2) effectively

inform and divert travelers to avoid recurring and non-recurring congestion. Traffic monitoring systems provide

fundamental data inputs for public agencies to measure time-varying traffic network flow patterns and accordingly

generate coordinated control strategies. Furthermore, reliable end-to-end trip travel time information is critically

needed in a wide range of intelligent transportation system applications, such as personalized route guidance and

pre-trip traveler information provision. In this paper, we focus on a series of critical and challenging modeling issues

in traffic sensor network design, in particular, how to locate different types of detectors to improve path travel time

estimation/prediction accuracy.

Based on the types of measurement data, traffic sensors can be categorized into three groups, namely point

sensors, point-to-point sensors, and probe sensors. Point sensors collect vehicle speed (more precisely, time-mean

T. Xing, X. Zhou, J. Taylor / Transportation Research Part B 00 (2015) 000–000

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speed), volume and road occupancy data at fixed locations. With point detectors having significant failure rates,

existing in-pavement and road-side traffic detectors are typically instrumented on a subset of freeway links. Point-

to-point sensors can track the identities of vehicles through mounted transponder tags, license plate numbers, or

mobile phone Bluetooth signals, as vehicles pass multiple but non-contiguous reader stations. A raw tag read

typically records a vehicle ID number, the related time stamp, and the location. If two readers at different locations

sequentially identify the same probe vehicle, then the corresponding data reads can be fused to calculate the reader-

to-reader travel time and the counts of identified vehicles between instrumented points.

Automatic license plate matching techniques have been used in the traffic surveillance field since the 1970s, and

many statistical and heuristic methods have been proposed to reduce reading errors to provide reliable data

association (Turner et al., 1998). Many feature-based vision and pattern recognition algorithms (e.g., Coifman et al.,

1998) have been developed to track individual vehicle trajectories using camera surveillance data. Radio Frequency

Identification (RFID) technologies first appeared in Automated Vehicle Identification (AVI) applications during the

1980s and has become a mature traffic surveillance technology that produces various traffic measures with high

accuracy and reliability. Currently, many RFID-based AVI systems are widely deployed in road pricing, parking lot

management, and real-time travel time information provision. For instance, prior to 2001, around 51 AVI sites were

installed and approximately 48,000 tags had been distributed to users in San Antonio, United States, which

represents a 5% market penetration rate. Additionally, Houston’s TranStar relies on AVI data to provide current

travel time information (Haas et al., 2001). Many recent studies (e.g., Wasson et al., 2008, Haghani et al., 2010)

started to use mobile phone Media Access Control (MAC) addresses as unique traveler identifiers to track travel

time for vehicles and pedestrians.

Many Automatic Vehicle Location (AVL) technologies, such as Global Positioning System (GPS) and

electronic Distance Measuring Instruments (DMI’s), provide new possibilities for traffic monitoring databases to

semi-continuously obtain detailed passing time and location information along individual vehicle trajectories. As the

personal navigation market grows rapidly, probe data from in-vehicle Personal Navigation Devices (PND) and cell

phones become more readily available for continuous travel time measurement. On the other hand, privacy concerns

and expensive one-time installation costs are two important disadvantages influencing AVL deployment progress.

1.2. Literature review

Essentially, any application of real-time traffic measurements for supporting Advanced Traveler Information

Systems (ATIS) and Advanced Traffic Management Systems (ATMS) functionalities involves the estimation and/or

prediction of traffic states. Depending on underlying traffic process assumptions, the existing traffic state estimation

and prediction models can be classified into three major approaches: (1) approach purely based on statistical

methods, focusing on travel time forecasting, (2) approach based on macroscopic traffic flow models, focusing on

traffic flow estimation on successive segments of a freeway corridor, (3) approach based on dynamic traffic

assignment (DTA) models, focusing on wide-area estimation of origin-destination trip demand and route choice

probabilities so as to predict traffic network flow patterns for links with and without sensors. In this research, we are

interested in how to place different types of sensors to reduce traffic uncertainty for the first statistical method-based

travel time estimation and prediction applications.

In sensor location models for the second approach, significant attention (e.g., Leow et al., 2008, Liu and

Danczyk, 2009, Danczyk and Liu, 2011) has been devoted to placing point detectors along a freeway corridor to

minimize the traffic measurement errors of critical traffic state variables, such as segment density and flow. The

traffic origin-destination (OD) matrix estimation problem is also closely related to the travel time estimation

problem under consideration. To determine the priority of point detector locations, there are a wide range of

selection criteria, including the “traffic flow volume” and “OD coverage” criteria proposed by Lam and Lo (1990),

and a “maximum possible relative error (MPRE)” criterion proposed by Yang et al. (1991) that aims to calculate the

greatest possible deviation from an estimated demand table to the unknown true OD trip demand.

Recently, based on the trace of the a posteriori covariance matrix produced in a Kalman filtering model, Zhou

and List (2010) proposed an information-theoretic framework for locating fixed sensors in the traffic OD demand

estimation problem. Related studies along this line include an early attempt by Eisenman et al. (2006) that uses a

Kalman filtering model to minimize the total demand estimation error in a dynamic traffic simulator, and a recent

study by Fei and Mahmassani (2011) that considers additional criteria, such as OD demand coverage, within a


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multi-objective decision making structure. Furthermore, in the travel time estimation problem, the Kalman filtering

based framework has also been employed by many researchers. For instance, an ensemble Kalman filtering model is

proposed by Work et al. (2008) to estimate freeway travel time with probe measurements.

Several recent studies have been conducted on the sensor location problem from different perspectives. Chen et

al. (2004) studied the AVI reader location problem for both travel time and OD estimation applications. They

presented the following three location section criteria: minimizing the number of AVI readers, maximizing the

coverage of OD pairs, and maximizing the number of AVI readings. To maximize the information captured with

regard to the network traffic conditions under budget constraints, Lu et al. (2006) formulated the roadside server

locating problem as a two-stage problem. The first stage was a sensitivity analysis to identify a subset of links on

which the flows have large variability of travel demand, and more links were gradually selected to maximize the

overall sensor network coverage in the second stage. Sherali et al. (2006) proposed a discrete optimization approach

for locating AVI readers to estimate corridor travel times. They used a quadratic zero-one optimization model to

capture travel time variability along specified trips. In a recent study by Ban et al. (2009), link travel time estimation

errors are selected as the optimization criterion for point sensor location problems, and a dynamic programming-

based solution method is constructed to optimize the location of point sensors on link segments along a corridor. Li

and Ouyang (2011) proposed a reliable sensor location method that considers probabilistic sensor failures, and

developed a Lagrangian relaxation based solution algorithm. By considering uncertainty in the traffic system, Fei et

al. (2013) proposed a two-stage stochastic model to find near-optimal sensor location solutions for OD flow

estimation with random non-recurring traffic events. In another recent study by Mirchandani et al. (2009), two AVI

sensor locating models were proposed: one considering the total vehicle-mileage coverage, while the other focusing

on monitoring the travel time prediction performance. Specifically, for their second model, a Bayesian statistical

theory based method was used to update the travel time estimates, and a two-stage heuristic approach was proposed

for locating AVI sensor. In addition, as a comprehensive review for existing flow-oriented sensor network studies,

Gentili and Mirchandani (2012)’s paper systematically classifies sensor location modeling methods as two major

categories of flow observability and flow estimation, with detailed descriptions and comparisons of several

optimization rules and models.

1.3. Proposed approach

While significant progress has been made in formulating and solving the sensor location problem for travel time

estimation and prediction, a number of challenging theoretical and practical issues remain to be addressed.

First, as one of the most important applications of traffic surveillance systems, real-time traffic

estimation/prediction plays a key role in intelligent transportation systems. However, the optimization criteria used

in the existing sensor location models typically differ from those used in real-time travel time estimation and

prediction applications. Due to the inconsistency between these two models, sensor design plans (e.g., focusing on

network coverage) might not fully recognize the underlying stochastic traffic process and lead to sub-optimal

solutions in the actual traffic information provision stage, particularly in terms of minimizing end-to-end travel time

forecasting errors. For example, an AVI sensor location plan that maximizes sensor coverage does not necessarily

yield the least end-to-end travel time estimation and prediction uncertainty if there are multiple likely paths between

pairs of AVI sensors. As a result, a simplified but unified travel time estimation and prediction model for utilizing

different data sources is critically required as the underlying building block for the sensor network design problem.

Second, since most of the existing studies typically focus on real-time speed estimation errors (when using

measured speed from point sensors to approximate speed on adjacent segments without sensors), they do not

explicitly take into account uncertainty reduction and propagation in a heterogeneous sensor network with both

point and point-to-point travel time measurements, as well as possible error correlation between new and existing

sensors.

Third, how to quantify the uncertainty increase in an integrated travel time and prediction process, especially

under non-recurring traffic conditions, has not received sufficient attention. Under recurring conditions, the travel

time is more likely to be estimated and predicted accurately for links with sensor measurements. For locations

without sensors, one can resort to historical information (e.g., through limited floating car studies) or adjacent

sensors to approximate traffic conditions. However, under non-recurring traffic conditions due to incidents or

special events, without real-time measurements from impacted locations, traffic management centers or traffic


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information provision companies might still offer biased traffic information based on outdated historical estimates or

incorrect approximations from unimpacted neighboring detectors.

By extending a Kalman filtering-based information theoretic approach proposed by Zhou and List (2010) for

OD demand estimation applications, this research focuses on how to analyze the uncertainty for the travel time

estimation and prediction problem with heterogeneous data sources. Since the classical information theory proposed

by Shannon (1948) on measuring information gain related to signal communications, the sensor location problem

has been an important and active research area in the fields of electrical engineering and information science.

Various measures have been used to quantify the value of sensor information in different sensor network

applications, where the unknown system states (e.g., the position and velocity of targets studied by Hintz and

McVey, 1991) can typically be directly measured by sensors. In comparison, sensing network-wide travel time

patterns is difficult in its own right because point sensors only provide a partial coverage of the entire traffic state.

Using AVI data involves complex spatial and temporal mapping from raw measurements, and AVL data are not

always available on a fixed set of links, especially under an early sensor network deployment stage.

Compared with the sensor design method for OD demand estimation studied by Zhou and List (2010), the

sensor location problem considered in this paper needs to address a number of unique challenges for more complex

network-wide travel time applications. In Zhou and List’s paper, the core state transition process in the offline OD

estimation problem is assumed to be stationary. For the travel time estimation and prediction problem, however, the

time-dependent travel time variables are essentially a mixture of stationary and non-stationary processes with

complicated transition dynamics. In this research, the stochasticity sources of non-recurring traffic conditions need

to be explicitly modeled and further captured through instrumented sensors. Moreover, the time-varying path travel

time matrix considered in this study has a much larger dimension compared to an origin-destination matrix, and

therefore it is very challenging to quantify the information gain from an underlying heterogeneous sensor network.

Compared to the travel time prediction-based sensor location model proposed by Mirchandani et al. (2009), this

study utilizes the Kalman Filtering framework to model both sensor location problem and travel time

estimation/prediction problem with a special focus on inherent model consistency. Moreover, by using the

information uncertainty as the driving force of both analytical and heuristic models, this study proposes a unified

framework that can be applied to heterogeneous data sources. Furthermore, to better capture the effects of dynamic

traffic behaviors, separate travel time prediction models are derived for both recurring and non-recurring traffic

conditions.

A wide range of studies (e.g., Zhang and Rice, 2003; Stathopoulos and Karlaftis, 2003) have been devoted to

travel time prediction using Kalman filtering and Bayesian learning approaches. To extract related statistics from

complex spatial and temporal travel time correlations, a recent study by Fei et al. (2011) extends the structure state

space model proposed by Zhou and Mahmassani (2007) to detect the structural deviations between the current and

historical travel times and apply a polynomial trend filter to construct the transition matrix and predict future travel

time. This paper presents a unified Kalman filtering-based framework under both recurring and non-recurring traffic

conditions. More importantly, a spatial queue-based cumulative flow count diagram is introduced to derive the

important transition matrix for modeling traffic evolution under non-recurring congestion. Different from existing

data-driven or time-series-based methods, this paper aims to derive a series of point-queue-model-based analytical

travel time transition equations which lay out a core modeling building block for quantifying prediction uncertainty

under non-recurring traffic conditions. In addition, with an extended discussion on the measure of uncertainty, this

study proposes both analytical and heuristic approaches that capture the travel time estimation/prediction uncertainty

reduction from different perspectives. We also extend a maximum determinant based analytical approach to show

that the point sensor location problem can be constructed as a convex optimization problem, which can serve as an

effective lower bound for the corresponding integer programming model for the sensor location problem.

The remainder of this paper is organized as follows. The overall framework and notation are described in the

next section. In Sections 3 and 4, a state-dependent transition model and a measurement model are described,

respectively. Section 5 presents a Kalman filtering based travel time estimation and prediction model. A

comprehensive discussion of information measure models is presented in Section 6, followed by Section 7 which

describes the beam-search based sensor design model and solution algorithms. Finally, illustrative example and

numerical experiment results on a test network are shown in Section 8.

2. Notations and modeling framework overview


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We first introduce some key notation used in the travel time prediction and sensor network design problems.

2.1. Notations

Sets and Subscripts:

N = set of nodes.

A = set of links.

m = number of links in set A.

'A = set of links with point sensors (e.g., loop detectors), 'A A .

''N = set of nodes with point-to-point sensors, ''N N .

'''A = set of links with probe sensor data, '''A A .

'n , ''n , '''n = numbers of measurements, respectively, from point sensors, point-to-point sensors and probe

sensors.

n = number of total measurements, ' '' '''n n n n .

t = time index for state variables.

h = travel time prediction horizon.

o = subscript for origin index, o O , O = set of origin nodes.

d = subscript for destination index, d D , D = set of destination nodes.

a,b = subscript for link index, ,a b A .

i, j = subscript for node index, ,i j N .

k, λ = subscript for path.

p(i,j,k) = set of links which belong to path k from origin node i to destination node j.

State and measurement variables

,t ax = travel time of link a at time t.

, , ,t o d kx = travel time on path k from origin o to destination d, at time t.

,'t ay = single travel time measurement from a point sensor on link a, at time t.

, , ,''t i j ky = single travel time measurement from a pair of AVI readers on path k at time t from node i to node j,

where the first and second AVI sensors are located at nodes i and node j, respectively.

,'''t ay = a set of travel time measurements from a probe sensor that contain map-matched travel time records on

links a on path k and time t from node i to node j, where ( , , )a p i j k .

Vector and matrix forms in Kalman filtering framework:

Xt = travel time vector at time t, consisting of m elements xt,a.

tX = a priori estimate of the mean values in the travel time vector at time t, consisting of m elements.

tX = a posteriori estimate of the mean values in the travel time vector at time t, consisting of m elements.

h

tX = historical regular travel time estimates for the corresponding time interval t.

Vt = historical pattern deviation at time t.

Yt = sensor measurement vector at time t, consisting of n elements.

tP = a priori variance covariance matrix of travel time estimate, consisting of (m × m) elements.

tP = a posteriori error covariance matrix, i.e. conditional covariance matrix of estimation errors after including

measurements.

tH = sensor matrix that maps unknown travel times Xt to measurements Yt, consisting of (n × m) elements.

Kt = updating gain matrix, consisting of (n × m) elements, at time t.


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t = autoregressive parameter set at time t.

,t t hL = non-recurring traffic transition matrix from time t to t+h at time t.

ωt = system evolution noise vector for link travel times, ~ (0, )t tN Q .

Qt = system evolution noise variance-covariance matrix, at time t.

εt = combined measurement error term, εt ~ N(0, Rt ) , at time t.

Rt = variance-covariance matrix for measurement errors, at time t.

Parameters and variables used in information measure and sensor design models

, , ,i j k a = path-link incidence coefficient, , , ,i j k a =1 if path k from node i to node j passes through link a, and 0

otherwise.

,'''

t a = stochastic link traversing coefficient for GPS probe vehicles;

,''' 1

t a if GPS probe vehicles pass

through link a at time t, and 0 otherwise.

, , ,t o d kE = path travel time estimation error on path k from origin o to destination d at time t.

, , ,t o d kf = traffic flow volume on path k from origin o to destination d at time t.

α = market penetration rate for vehicles equipped with AVI sensors/tags.

β = market penetration rate for vehicles equipped with AVL sensors.

l = subscript of sensor design solution index.

lS = lth sensor design solution, represented by ', '', ''', ,lS A N A .

*S = optimal sensor design solution.

2.2. Problem statement and assumptions

Consider a traffic network with multiple origins oO and destinations dD, as well as a set of nodes connected

by a set of directed links. We assume the following input data are available:

(1) The prior information on the time-dependent historical travel time pattern for all the links in the network,

including a vector of historical travel time estimates h

tX and the corresponding variance-covariance matrix

tP . For instance, the historical pattern can be obtained through aggregating travel time every 5 minutes for

the same day of the week across different weeks/months.

(2) The current link sets with point sensor and point-to-point AVI sensor data, specified by 'A and "N .

(3) Estimated market penetration rate α for point-to-point AVI sensors.

(4) Estimated market penetration rate β for probe sensors, and set of links with accurate probe data '''A .

The sensor network to be designed and deployed will include additional point sensors and point-to-point

detectors that lead to sensor location sets of 'A and ''N , where ' 'A A , '' ''N N . In the new sensor network,

through GPS map-matching algorithms, GPS probe data can be converted from raw longitude/latitude location

readings to link travel time records on a set of links '''A . In this study, we assume that probe data will be available

through a certain data sharing program (e.g., Herrera and Bayen, 2010), or from vehicles equipped with Internet-

connected GPS navigation systems or GPS-enabled mobile phones. It should also be noticed that, depending on the

underlying map-matching algorithm and data collection mechanism, only a subset of links in a network, denoted by

'''A , can produce reliable GPS map-matching results. For example, it is practically difficult to distinguish driving

vs. walking mode on arterial streets through data from GPS-equipped mobile phones, so one might need to consider

'''A as a set of freeway links (with reliable travel time estimates available).

Another key assumption in our study is that the historical travel time information can be characterized by the

time-dependent a priori historical travel time pattern tX and estimation error variance matrix tP . In practice, if


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point or point-to-point sensor data are available from the existing sensor sets 'A and "N , it is relatively easy to use

archived data to estimate tX and tP on the corresponding links. For links without direct sensor measurements, the

travel time mean estimate can be approximated by using a national or regional travel time index and set the

corresponding variance to a sufficiently large value (to represent high uncertainty). One complicated issue is how to

estimate non-diagonal elements in matrix tP . For simplicity, one can assume zero correlation between links in the

initial travel time covariance matrix, and in the case of a complete lack of historical demand information, we can set 1( ) 0tP .

It should be remarked that, measurements from a point sensor are typically instantaneous speed values observed

at the exact location of the detector. Using a section-level travel time modeling framework (e.g., Lindveld et al.,

2000), a homogenous physical link can be decomposed into multiple cells or sections, with the speed measurement

directly reflecting only the section where the sensor is located. In some previous studies, the link or corridor speed

can be estimated using the section based speed, while cells without sensors use approximated values from adjacent

instrumented sections. As shown in Fig. 1, travel times on section A and D are directly measured using sensors 1

and 2, respectively. Meanwhile, the travel times for sections B, C and E, as well as the entire corridor, are estimated

using upstream and downstream sensors. There are a number of travel time reconstruction approaches, such as

constant speed based methods and trajectory methods (Van Lint and Van der Zijpp, 2003).

Fig. 1. Section-level travel time estimation

In this study, for sections without point sensors, the above mentioned approximation error is modeled as prior

estimation errors, which can be obtained through a historical travel time database by considering other related links

such as adjacent links or links with similar characteristics. Furthermore, our proposed framework can also be easily

generalized to a section-based representation scheme, where a section in Fig. 1 can be viewed as a link in our link-

to-path-oriented modeling structure. In this study, we consider instantaneous path travel time as a sum of link travel

time, that is, , , , ,

( , , )

t o d k t a

a p o d k

x x

, rather than experienced path travel time, which is more realistic but requires

complex recursive computation along the links on a path.

2.3. Modeling procedure

Focusing on predicting end-to-end path travel time applications and considering future availability of GPS

probe data on links '''A , the goal of the sensor location problem is to minimize the overall information uncertainty

TU by locating point and point-to-point sensors in sets 'A and ''N , subject to budget constraints for installation and

maintenance. To systematically present our key modeling components in the proposed sensor design model, we will

sequentially describe the following four modules.

Travel time state transition and measurement modeling module:

In this module a unified model of the travel time state transition is constructed for both recurring and non-

recurring traffic conditions. In addition, measurement models for each type of sensors are discussed with a special

focus on representing measurement errors.

Link travel time estimation and prediction module:

Given prior travel time information tX and tP , with traffic measurement vector tY that includes ,'t ay ,

, , ,''t i j ky and ,"'t ay from sensor location sets 'A , ''N and '''A , the link travel time estimation and prediction module

seeks to update current link travel times tX and their variance-covariance matrix tP using a Kalman filtering

framework that is built on proposed state transition and measurement models.

Information quantification module:


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With prior knowledge on the link travel time estimates tX and tP , the information quantification module aims

to find the single-valued information uncertainty ( )lTU S for the critical path travel times for a sensor design

scenario Sl represented by location sets 'A , ''N , '''A , as well as AVI and AVL market penetration rates α and β .

Sensor network design module:

The sensor network design module uses an approximation approach to find the optimal solution * arg min ( )l lS TU S , subject to budget constraints for installation and maintenance.

For each candidate solution Sl,

this module needs to call the information quantification module to calculate ( )lTU S . The optimal solution *

S

produces optimal location sets 'A and ''N , under predicted AVI and AVL market penetration rates α and β, and

(stochastic) location set '''A travelled through by probe vehicles.

3. State Transition Models

In practice, the time-dependent travel time is usually considered as a non-stationary variable, especially for links

with high travel time variability. To better describe the state transition of travel times, we define a historical pattern

deviation as: h

t t tV X X (1)

where Xt represents the (real-world) travel time vector at time t and Xth denotes the historical travel time vector for

the corresponding time interval t. In our proposed state transition model, the travel time deviation in Eq. (1) is

assumed to be a stationary time series under recurring traffic conditions.

3.1. State-dependent transition models

Under recurring traffic conditions, the travel time deviation in Eq. (1) is modeled by a first-order autoregressive

(AR) formula:

1 1

h h R

t t t t t tX X X X (2)

In Eq. (2), t is an autoregressive coefficient for the structure deviation variable h

t tX X , which has been de-

trended from the historical average. Similar AR or structure deviation models have been proposed within a Kalman

filtering framework by Okutani and Stephanedes (1984), Zhou and Mahmassani (2007) and Fei et al. (2011). ωtR is

the error vector of deviation under recurring conditions and can be assumed as white noise in the autoregressive

model. ωtR needs to be calibrated from historical data. In an extreme case of t = 0, Eq. (2) becomes

1 1

h R

t t tX X , meaning that no significant temporal autocorrelation exists and the real-time travel time 1tX

can be predominantly predicted using the corresponding historical pattern 1

h

tX .

For non-recurring traffic conditions, e.g. due to various irregular demand/capacity changes, the process equation

of the deviation in real-time prediction applications can be written in the following state space form:

1 1 , 1

h h NR

t t t t t t tX X L X X (3)

In Eq. (3), the transition matrix Lt,t+1 denotes the non-recurring process matrix of the deviation, and the process error

ωtNR is considered as a time-dependent normally distributed random noise vector. Lt,t+1 can be derived through a

cumulative flow based model as discussed in appendix.

We notice that Eq. (2) and (3) share a similar structure. By re-grouping variables in Eqs. (2) and (3), a unified

state-dependent transition model can be derived for the non-stationary variable Xt.

1t t t t tX A X B (4)


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, 1

Links under recurring congestion

Links under non-recurring congestion

t

t

t t

AL

(5)

1

h h

t t t tB X A X (6)

~ (0, )t tN Q

In the above unified transition model (4), we use a time-dependent and situation-dependent matrix At to relate

the state at current time t to the next time t+1 under recurring and non-recurring conditions according to Eq. (5). In

particular, at a given time t, links under recurring congestion conditions rely on AR parameters to construct the

corresponding cells of the matrix At. Meanwhile, links under non-recurring delay need to derive transition

parameters , 1t tL from a simple spatial-queue model presented in appendix (for random capacity reduction).

Furthermore, a matrix Bt is introduced as a weighted combination of historical patterns at different time intervals.

The matrix Bt does not depend on variables Xt, and it is assumed to be known at current time interval t in real-time

prediction. Essentially, it can be viewed as control inputs in the modeling framework of Kalman filtering.

A state-dependent process error term ωt is established in Eq. (4) for both recurring and non-recurring conditions.

Under a stationary congestion pattern, this study assumes that ωt follows a normal distribution with zero-mean and a

variance-covariance matrix Qt. Qt corresponds to the variation magnitude of the travel time deviation, and this

variability measure is further determined by multiple external factors such as base bottleneck capacity variations,

traffic incidents, and severe weather conditions. More specifically, the (diagonal) variance elements of the matrix Qt

exhibit the travel time variability/uncertainty of each individual link, while the (non-diagonal) covariance elements

should reveal the spatial correlation relationship (mostly due to queue spillbacks) between adjacent links in a

network. We refer interested readers to a recent study by Min and Wynter (2011) for calibrating spatial correlations

of link travel times.

3.2. Transition matrix under non-recurring conditions

There are many external factors leading to non-recurring delays, such as capacity reductions due to incidents,

severe weather, road construction, as well as demand variations associated with special events. To compute the non-

recurring transition coefficient Lt,t+1 in Eq. (5), Newell’s kinematic wave model (Newell, 1993) is used in this

research to capture forward and backward waves as the result of bottleneck capacity-demand interactions.

Newell’s simplified form of traffic flow models is particularly attractive in establishing theoretically sound and

practically operational traffic transition models on bottlenecks. Interested readers are referred to a number of related

studies on Newell’s kinematic wave model, e.g. the model calibration effort by Hurdle and Son (2000), and

extensions to node merge and diverge cases by Yperman et al. (2005) and Ni et al. (2006). There are many other

potential ways to capture the transition relationship of travel times especially on a network setting, such as

simulation-based Dynamic Traffic Assignment (DTA) methods. We selected Newell’s method in this study because

its simplicity leads to insightful analysis for typical corridor-level bottlenecks.

While the following discussion of this paper is intended to focus on the estimation problem in the Kalman

filtering framework, the detailed derivation procedure for (analytical) transition matrices is described in Appendix

A1, with illustrative case studies on non-recurring conditions under traffic incident and queue spillback. By using

the same approach, other complicated cases or combinations can also be derived for simplified non-recurring

situations.

4. General measurement model

In order to estimate the travel time Xt , a linear measurement model is constructed as:

t t t tY H X , where ~ (0, )t tN R (7)


10

With the measurement model in Eq. (7), the travel times can be estimated using the latest measurements Yt .

Measurement vector Yt with (n × 1) elements is composed of travel time observations from point sensors, point-to-

point sensors and probe sensors. In particular, each row of the vector contains a travel time observation value from

an individual sensor (or sensor pair for AVI) available at time t. An n-by-1 measurement error term εt is assumed to

be zero-mean and normally distributed according to a time-dependent variance matrix Rt.

The mapping matrix Ht , with (n × m) elements, connects unknown link travel time Xt to measurement data Yt .

Particularly, each row in the mapping matrix Ht corresponds to a measurement and each column corresponds to a

physical link in the network. For an element at the uth row and wth column of the matrix Ht , value = 1 indicates that

the uth measurement covers or includes the travel time on the wth link of the network, otherwise the value is equal to

0. Specifically, for links covered with more than one sensor, each sensor has its own measurement and error values

in the corresponding rows of the matrix Yt and εt. For instance, for a link a covered by two sensors 1 and 2, the

measurement model can be written as , ,1 , ,1

,

, ,2 , ,2

1

1

t a t a

t a

t a t a

yx

y

, where yt,a,1 and εt,a,1 are the measurement

and error from sensor 1, respectively.

4.1. Measurement model for different types of sensors

Shown in Eq. (7), a linear measurement model is used to map the measurement vector Yt to the travel time

vector Xt by taking into account measurement error term εt from variant sources.

The following three equations aim to show how Ht is constructed for a specific type of measurement at time t. For a point sensor on link a, each measurement covers one link,

, , ,' ' , 't a t a t ay x a A . (8)

For a pair of point-to-point sensors that capture end-to-end travel time from node i to node j through path k,

, , , , , , , , , ,'' '' , , ''t i j k i j k a t a t i j k

a

y x i j N , (9)

where , , ,i j k a is the path-link incidence coefficient for AVI sensors, , , ,i j k a =1 if path k from node i to node j passes

through link a, and 0 otherwise.

For an AVL sensor/probe on link a,

, , , ,''' ''' ''' , '''t a t a t a t ay x a A ,

(10)

with ,'''t a represents the stochastic link traversing coefficient for GPS probe vehicles. ,''' 1t a if GPS probe

vehicles pass through link a at time t, and 0 otherwise.

For AVI and AVL sensors, a measurement in models (9) or (10) might be referred to as an average value of

multiple raw samples within a certain time period (e.g., from 8:00 AM to 8:15AM). The temporal-spatial coverage

of different types of sensors is illustrated in Fig. 2. For a corridor network with three links (1, 2, 3) and four nodes (a,

b, c, d), one point sensor covers link 2, and two AVI sensors are located at node a and d. As shown in Fig. 2, from

8:00 to 8:30 AM, two GPS probe location sequences (P1 and P2) and two AVI travel time records (A1 and A2) are

received on this corridor, along with three point sensor observations. In practice, the raw data from AVI and AVL

sensors will first be converted to records at corresponding time intervals, and then used in the travel time estimation.

In the example of Fig. 2, the AVI sensor pair covers all three links, with its measurement aggregated from all

records observed in a particular time period. For instance, the AVI measurement at the time interval from 8:15 AM

to 8:30 AM is aggregated from the average of travel time records A1 and A2, and it provides a measurement

equation for instantaneous travel time from a to d : , , ,1 ,2 ,3 , ,'' ''t a d t t t t a dy x x x , where , ''a d N , t is the

time interval between 8:15AM and 8:30AM.

The GPS location sequences P1 and P2 provide three measurement equations:


11

,1 ,1 ,1''' '''t t ty x , ,2 ,2 ,2''' '''t t ty x , ,3 ,3 ,3''' '''t t ty x ,

where ,1'''ty and ,2'''ty are constructed based on single-probe travel time records from P1, and ,3'''ty is an average

travel time based on records P1 and P2, and t is the time interval between 8:00AM and 8:15AM. ,3'''t has a smaller

sampling error than ,1'''t and ,2'''t , as the former includes samples from two GPS probes.

Fig. 2. Spatial-temporal coverage representation for AVI, GPS and point sensors

4.2. Measurement error analysis

The measurement error term ε in the above equations (8-10) are combined terms that reflect the overall effect of

errors from the data conversion, measurement reading and sampling processes, as discussed below.

Data conversion error: Typically, only time-mean speed data are available from a point sensor, and the travel

time value (i.e. space-mean speed) needs to be inferred and approximated from a point speed reading. This

introduces significant data conversion errors, depending on the placement of a point sensor on a link (e.g., the

relative location with respect to the tail of a queue from the downstream node of a link). In addition, a single-loop

detector has to use the observed occupancy and flow counts to calculate the point speed value, which might further

lead to larger approximation errors. Detailed discussions on the above types of data conversion errors can be found

at Rakha and Zhang (2005) and Van Lint (2004). For probe sensors, GPS location data (in terms of longitude,

latitude, point speed, bearing and timestamps) need to be map-matched to specific links in the study network to

estimate corresponding link travel times. This map-matching process again introduces data conversion errors to the

final link travel time estimates.

Sensor reading error: Point sensors that are not carefully calibrated are more likely to generate large

measurement noises. The detection rates of AVI readers are relatively low when vehicle tags are not powered by

batteries. The data quality of GPS location readings depends on the number of satellites that a GPS receiver can

“see” and nearby environment (e.g., urban canyon effect).

Sampling error: Assume , , ,''t i j kg is the number of samples collected through AVI sensors on path k at time t

from node i to node j. As a probe travel time measurement can come from multiple readings, the variance of

sampling error (e.g., for an AVI measurement that is aggregated from , , ,''t i j kg samples) can be described as

, , ,

, , ,

, , ,

var( )var( '' )

''

t i j k

t i j k

t i j k

x

g

(11)

If we assume no correlation between link travel times along path k from node i to node j, then

, , , , ,

( , , ) ( , , )

var( ) var( )t i j k t a t a

a p i j k a p i j k

x x q

.

(12)

Assuming there are a total of , , ,d i j kf vehicles traveling from node i to node j through path k, then the AVI

market penetration rate α can be derived as , , , , , , , , ,''t i j k t i j k t i j kg f . That is, when the market penetration rate

increases, the size of samples also becomes larger, leading to a smaller sampling error and a more reliable travel

time measurement. In practice, due to the lack of data, a single market penetration rate α can be used for a certain

area.

In summary, the magnitude of the combined error εt is determined by a number of external factors, and there are

also possible error correlations among different sensors depending on traffic conditions. For simplicity, the

following analysis assumes the combined errors belong to a normal probability distribution (without autocorrelation)

with zero-mean and a variance-covariance matrix Rt.


12

As point and AVI detectors are installed at fixed locations, the corresponding sensor mapping matrices, denoted

as 'tH and ''tH , typically remain the same within the study time horizon, that is ' 'tH H and '' ''tH H . Even

with more accurate vehicle based link travel time samples, the probe sensors still have two major limitations: low

market penetration rate and stochastic temporal coverage. Specifically, similar to the point-to-point AVI sensor, a

large sampling error is introduced to probe sensor measurements under a low market penetration rate, as shown in

Eq. (11). Moreover, individual travelers with GPS probes can use different paths and links on different days, which

leads to a time-dependent sensor mapping matrix '''tH that consists of stochastic link traversing coefficient ,"'t a

for GPS probe vehicles.

4.3. AVI extension with multiple paths

The point-to-point sensor measurement model in Eq. (9) assumes that all AVI-equipped travelers use only a

single path between each pair of AVI sensors. The following discussion considers a more complex but realistic

situation with multiple used paths between a pair of AVI sensors. For simplicity, we assume the route choice

probabilities for those paths can be computed from a deterministic or stochastic traffic assignment program.

This study adapts a multivariate normal (MVN) distribution to represent the route choice behavior. In particular,

each route choice decision from an individual traveler can be considered as an independent Bernoulli trial from one

of the K possible outcomes (i.e. paths) with probabilities µ1, …, µk,…, µK.

Fig. 3. Two partially overlapping paths between AVI readers at nodes a and c

Consider an example network shown in Fig. 3, where two AVI sensors are located at nodes a and c. Travelers

can take either of two routes: p1 = x1 + x3 or p2 = x2 + x3, where link 3 is shared by these two paths. We now extend

AVI measurement equation (9) from a single-path sensor pair to the following two-path measurement equation

(subscripts t, i, j are omitted for notation simplicity):

1

1 2 2 1 1 2 2 3

3

'' '' '' 1

x

y H X x x x x

x

(13)

where µ1 and µ2 represent the route choice probability for link 1 and 2, with 1 2 1 .

''y is the average travel time from travelers captured by this AVI sensor pair, the contribution from travelers

using route 1 is 1 1p and the contribution from route 2 is

2 2p .

is the error term introduced by sampling variations due to multiple paths.

is the error term associated with using the sample average value to approximate the population mean value X,

as described in Eq. (11).

The variance of multi-choice sampling variation can be calculated through a path travel time point of view:

2 2

1 1 2 2 3 1 1 2 2 1 1 2 2 1 2 1 2var( ) var( ) var( ) var( ) var( ) 2 cov( , )x x x p p p p p p (14)

Assuming there are g AVI samples observed between this AVI sensor pair, and among the g samples, fk drivers

using path k, we can derive E( ) E( )k k k

f g , 2

var( ) var( ) (1 )k k k k

f g g , and

cov( , )k k g .

Thus, Eq. (14) is reduced to

2 1 2

2 1var( ) ( )p pg

(15)


13

First, it is clear that the overlapping portion between different paths does not affect the variance associated with

the multi-choice sampling error, as it is canceled out in 2 1

( )p p . Under perfect deterministic user equilibrium

conditions, all of the used routes have the same travel time, so the var( ) further reduces to zero. Under a more

realistic stochastic user equilibrium assumption, the travel time difference between different traveled routes leads to

an increase in the magnitude of the combined error var( '') var( )R . Similarly, Fig. 4 demonstrates a case

on a corridor with K=3 parallel routes. In general, the variance of the combined error increases as there are more

used paths with significant travel time differences.

2

1 1

, :

1var( ) var( ... ) ( )k k K K k k

k k

p p p p pg

(16)

Fig. 4. Example network with three parallel paths

5. Travel time estimation and prediction models

5.1. Travel time estimate updating

This section discusses the travel time estimation and prediction models as the building block for the (upper-

level) sensor design model. Obviously, there is a wide range of time series-based methods for traffic state

estimation, including autoregressive moving average models and Bayesian learning models, to name a few. The

application of Kalman filtering in dynamic traffic state estimation and prediction starts with early research by

Okutani and Stephanedes (1984). In this section, a Kalman Filtering based estimation and prediction model is

proposed to update link travel times with different types of measurements. Specifically, we want to highlight how a

classical Kalman filtering model can be used to estimate link travel time and, more importantly, the associated

uncertainty propagation with heterogeneous data sources, especially for emerging sensor technologies such as AVI

and AVL.

The targeted variable Xt mentioned in the state transition equation (4) and the measurement equation (7) are

modeled as the true values of travel times for each link. In a state-space travel time estimation/prediction model, we

distinguish two states of the travel time Xt : (1) a priori state before the start of the current time (e.g., before

observations at the current stage are received), corresponding to the predicted travel time tX and uncertainty tP

,

and (2) a posteriori state after taking into account new measurements tY available at time t, corresponding to

estimated travel time tX and uncertainty tP

. We further define the a priori estimate error tX as the difference

between the true travel time vector Xt and the a priori link travel time estimate tX , where t t tX X X . The a

posteriori estimate error tX is the difference between the true travel time vector

tX and the a posteriori link travel

time estimate tX , with t t tX X X . Consequentially, the variance-covariance matrices of the a priori is

expressed as 𝑃𝑡− = 𝑐𝑜𝑣(𝑋𝑡 − 𝑋𝑡

−), and the a posteriori estimate error as 𝑃𝑡+ = 𝑐𝑜𝑣(𝑋𝑡 − 𝑋𝑡

+).

In Kalman filtering, by deriving an optimal Kalman gain factor T T 1( )

t t t t t tK P H H P H R

, a simplified

expression for the estimation error covariance can be derived as

( )t t t tP I K H P (17)

Other formulas for the estimation error covariance are available, for example,

1

1 1 11 1 or T T

t t t t t t t t t tP P H R H P P H R H

. (18)

Eq. (18) provides a convenient approach to evaluate the uncertainty propagation from an information gain

perspective, e.g., P-1. Interested readers are referred to Zhou and List (2010) for more detailed derivations of Kalman

filtering factor and formulas.


14

5.2. Travel time prediction

After updating the travel time mean estimate and its covariance on time t, we now need to predict the travel time

and its uncertainty range for the next time interval t+1. According to the system process model (4), the mean

estimate can be updated to the next time interval with the corresponding transition matrix At.

1t t t tX A X B

(19)

As the real-world traffic process evolves to the next time interval, its corresponding travel time uncertainty

should also be updated according to the state process model (4). Specifically, by taking into account the time-

dependent random realizations of traffic demand and capacity, characterized by the system error matrix tQ , we can

update the uncertainty estimate for 1tX

1

T

t t t t tP A P A Q

(20)

With the above uncertainty updating equations (17 & 18), and prediction equation (20), the Kalman filtering

based travel time estimation and prediction model is able to recursively correct the link travel time estimates from

streaming traffic measurements and dynamically adjust the error covariance matrix that indicates the uncertainty

range of the estimation and prediction results.

6. Measure of Uncertainty

One of the fundamental questions in sensor location problems is which criteria should be selected to drive the

core optimization processes. Equations (17, 18 & 20) in the above travel time estimation and prediction model offer

an information-theoretic approach for evaluating a certain heterogeneous sensor deployment strategy. That is, as a

by-product of the proposed Kalman filtering framework, the error matrices P+ and P- provide possible measures to

quantify the uncertainty reduction of the sensor network design. Therefore, by using the error matrices, we can

leverage the inherent consistency of the Kalman filtering framework and design a sensor network that may

potentially optimize the outcome of its main application – travel time estimation and prediction.

The travel time estimation/prediction uncertainty matrices can be analyzed with different approaches. As

discussed in Zhou and List (2010), both the trace and determinant of the error matrix can be used to quantitatively

measure the network wide uncertainty level. Moreover, since both the trace and determinant can be calculated using

the eigenvalues of a matrix, their computational complexities are considered equivalent. Interested readers are

referred to Zhou and List (2010) for more discussions about the comparison of different information or uncertainty

measures.

In this study, two measures of uncertainty are proposed for the travel time estimation application: (1) a

determinant-based analytical approach that optimizes the one-step estimation uncertainty with lower bound

evaluation, and (2) a path travel time oriented heuristic approach that aims to maximize the total uncertainty

reduction of critical OD/paths.

6.1. Analytical Approach with Determinant Maximization of Uncertainty

We first discuss a determinant maximization-based analytical approach to solve the one-step estimation-based

sensor network optimization problem. The uncertainty matrix minimization is a convex optimization problem, and

interested readers are referred to a paper by Vandenberghe et al. (1998) for a discussion on the convexity of the

determinant maximization problem and an efficient interior-point solution algorithm. In this study, we propose an

uncertainty optimization problem by adapting the linear matrix inequality-constrained determinant maximization

problem as the following:


15

1

0 1 1

0 1 1

min log det ( )

. . : ( ) ... 0

... 0

T

m m

m m

c x G x

s t G x G x G x G

F x F x F

(21)

By omitting the term Tc x (the analytical centering problem mentioned in Vandenberghe et al., 1998), a log

determinant problem can be proposed to minimize the estimation uncertainty P+:

min logdet P

(22)

with a strict linear matrix inequality constraint adopted from the estimation error covariance equation (18) to

describe the information/entropy updating:

1 1 1. . : ( ) ( ) 0T

l l l

l

s t P P x H R H

(23)

Problem (22) aims to optimize the estimation uncertainty P+ from a determinant perspective, given the prior

travel time uncertainty condition P- and a set of sensor location candidates Hl. In this model, xl represents the

decision variable for each sensor location design Hl. Consequently, additional constraints 1 0, *l l

l

x x l are

introduced for problem (22), with l* being the pre-defined sensor budget value.

This analytical approach has a significant advantage when applied for optimizing point sensor locations.

Particularly, as each point sensor collects data individually, the corresponding additional information term is

expressed as the matrix format HTR-1H, i.e.: let

10 0

0

0 0

0

0 0

l

L

x

H x

x

, and 0 0l l

H x ,

then we have 1 12

0 0 0

0

0 0

0

0 0 0

T T

l l l l

l l

H R H H R Hx r

. Therefore, each point sensor location strategy

Hl may contain as few as one point sensor, and the variable x may represent the selection of each individual location,

with the predefined constraint value l* as the given budget number of point sensors.

On the other hand, since each additional AVI or AVL sensor could potentially change the structure of the entire

sensor network matrix H, candidate AVI/AVL designs need to be enumerated to generate one sensor scenario for

each Hl, and the constraint value l* should be set as 1.

The proposed determinant maximization problem (22) can be solved using several open source and commercial

packages. In this study, the corresponding numerical experiments for the analytical method are prepared using open-

source Matlab libraries YALMIP (Löfberg, 2004) and SDPT3 (Toh et al., 1999).

One of the advantages of using the semidefinite-quadratic-linear programming package SDPT3 is that it

provides an efficient lower bound evaluation for the solution found. Specifically, by relaxing the decision variables

xl as linear variables between 0 and 1, a primal-dual approach can be applied in the lower bound solver to minimize

the log-determinant of the uncertainty matrix. The resulting uncertainty matrix P+ is calculated from a combination

of linear decision variables (with possible fractional values), and serves as a lower bound for any integer solution set

of xl. An illustrative example and numerical experiments are discussed in Section 8.


16

6.2. Heuristic Approach with Flow Weighted Measure of Total Uncertainty

When selecting the optimal gain factor K to estimate the travel time matrix, the classic Kalman filter aims to

minimize the mean-square error, i.e., the trace of tP . Using a similar approach, this research further proposes a

heuristic method to quantify the travel time uncertainty of the entire network, with an emphasis on critical OD/paths.

To consider the priority of different links, we first denote Ft as the network-wide flow matrix at time t, which

contains the total flow volume of each link. Specifically, each diagonal cell of matrix Ft represents the total flow

count of the corresponding link, while each off-diagonal cell contains the total flow volume shared by the two

correlated links. With flow matrix Ft, the uncertainty measure can be expressed using weighted

estimation/prediction error matrices P- and P+:

+sumest t tTU F P

(24)

where the weighted estimation uncertainty TUest is calculated from the Hadamard product – the element-by-element

product of two same-dimension matrices F and P+, and sum() represents the summation of all elements in the

matrix.

Eq. (24) provides a matrix format of the weighted uncertainty quantification. In practice, with the knowledge of

critical path/OD flow volume, Eq. (24) can be easily reformatted into a set of linear equations. We first denote the

path travel time uncertainty of a specified critical path p(o,d,k) at time t as Et,o,d,k. The travel time uncertainty of path

k from origin o to destination d can be calculated with:

, , , ,

( , , )

var( )t o d k t a

a p o d k

E x

, (25)

where var() represents the variance coefficients of the link travel time estimate matrix tP. If the covariance terms

cov() in tP are also considered, another possible measure of information can be expressed as:

, , , , , ,

( , , ) ; , ( , , )

var( ) 2 cov( , )t o d k t a t a t b

a p o d k a b a b p o d k

E x x x

(26)

For travel time estimation applications, path-level uncertainties are calculated using the estimation uncertainty

matrix tP for a selected set of time intervals. Weighted by the path flow volume of different origin-destination

pairs , , ,t o d kf , the total estimation uncertainty TUest of the network-wide traffic conditions can be determined from

the following equation:

, , , , , ,

, ,

est t o d k t o d k

t o d k

TU f E

(27)

which is equivalent to the matrix format in Eq. (24).

By considering

, , ,

, ,

est

t o d k

t o d k

TU

f, we can interpret the above measure as the average end-to-end path travel

time estimation error for each individual traveler.

Furthermore, we can derive a total forecasting error measure formula, similar to Eq. (27), using the a priori

travel time variance-covariance tP calculated from Eq. (20). However, different from the uncertainty estimation

model in Eq. (17), the stochastic state transition matrix At in Eq. (20) needs to be explicitly considered in order to

quantify the prediction uncertainty. To incorporate different sources of non-recurring delay (e.g., incidents, severe

weather, work zones), their location-specific occurrence probabilities and stochastic characteristics, an Ensemble

Kalman filter (EnKF) is proposed for the one-step prediction procedure. Under the proposed approach, a Monte

Carlo simulation is used to “simulate” the state transition ensembles under different realizations of random capacity


17

drops and capacity reduction durations, and the sample matrices At and the sample variance , ( )t NR eP

over different

ensembles e is then used to approximate the average prediction errors. For more details about EnKF, interested

readers are referred to a widely-cited tutorial paper by Evensen (2003).

Using Eq. (25) or (26), an approximation of the total prediction uncertainty measure is proposed by combining

both the recurring path uncertainty , , , ,t o d k RE and non-recurring path uncertainty , , , , ( )t o d k NR eE with ensemble

probability ,t e for different non-recurring sources or samples/ensembles e.

, , , , , , , , , , , , , ( )

, ,

1pred t o d k t e t o d k R t e t o d k NR e

t o d k e e

TU f E E

(28)

The above total path travel time estimation/prediction uncertainty measures (Eqs. 27 and 28) include three

important components: (1) the sum of diagonal elements in the variance covariance matrix for link travel time; (2)

the sum of the travel time variance for each feasible or critical path in the network; and (3) weights of path flow

volume for different paths. As the path travel time estimation/prediction accuracy (as opposed to individual link

travel times) captures the ultimate information quality requirement for commuters traveling along their end-to-end

trips, this measure of information can serve as a systematic measure of effectiveness (MOE) for network-wide

traffic monitoring applications.

Calculating the uncertainty measures in Eqs. (27) and (28) involves computing P- and P+ for each consecutive

time interval during the planning horizon (e.g., 12 months). However, this day-to-day “simulation” approach is

computationally intensive, especially for regional networks with a large number of links, and obtaining a stable

estimate on the information gain under probabilistic non-recurring conditions requires a large number of samples (or

days). To provide a reasonable approximation for network-wide uncertainty quantification, we adopt the following

simplifying assumptions to estimate the measure of information on a traffic network. (1) This study focuses on a

limited number of representative individual time intervals, rather than a 24-hour daily horizon, and each interval

only involves a one-step estimation and prediction process. (2) Under non-recurring congestion, we focus on the

impact of capacity reduction on individual links, while possible between-link travel time correlations due to queue

spillback are ignored for simplicity. It should be remarked that, the use of traffic flows and ensembles may introduce

additional errors into the total utility calculation. As this study is focused on the travel time estimation and

prediction based sensor designs, the modeling of uncertainty associated with flow estimation and non-recurring

traffic stochasticity is not considered in this paper.

7. Sensor network design model

In this section we propose a beam search-based algorithm to solve the heuristic approach with a flow-weighted

measure of total uncertainty.

7.1. Objective function

The proposed sensor network design model is essentially a special case of the discrete network design problem,

so an integer programming model, shown below, can be constructed to find the optimal sensor location solution.

Min network-wide total uncertainty TU obtained from Eqs. (27) or (28)

Subject to

Traffic pattern uncertainty propagation constraints Eqs. (17) and (20);

Sensor mapping matrix constraint:

, ,' ' , '' '' , ''' , ''' , ,t a i t a t aH function A x N x A

(29)

Budget constraint:


18

' ' '' '' ''' '''a i

a i

x x x (30)

'ax = 1 if a point sensor is installed on link a, 0 otherwise.

''ix =1 if an AVI sensor (point-to-point sensor) is installed on node i, 0 otherwise.

'''x =1 if probe sensor data are collected/purchased, 0 otherwise.

' , '' = installation and maintenance costs for point sensors and point-to-point sensors.

''' = costs for collecting/purchasing data from probe vehicles such as commercial fleet.

= total available budget for building or extending the sensor network.

tH is determined by the sensor location set ' 'aA x and '' ''iN x , randomly generated link traversing

coefficients for GPS probe vehicles ,

'''t a

, and AVI and AVL market penetration rates α and β. This paper considers

only third-party AVL probe data sources, such as vehicle trajectory data collected from commercial fleets or truck

companies. Such external data streams do not require dedicated sensor infrastructure for public-sector traffic

management agencies, so the costs of locating AVL sensor facilities are not explicitly considered in the above

formulation for simplicity, and the cost term ''' for purchasing probe data sources are typically given by private

data vendors.

7.2. Beam search algorithm

The overarching goal of the above sensor location model is to fuse and incorporate information from spatially

distributed sensors to minimize the network-wide path travel time estimation or prediction uncertainty measure. In

this study, a branch-and-bound search procedure can be used to solve the integer programming problem. To further

reduce the computational complexity, a beam search heuristic algorithm is extended from the study by Zhou and

List (2010) in this study. The description below only highlights the key features of the algorithm for travel time

estimation. Given prior information on the link travel time vector and its estimation error covariance from a

historical database, the proposed algorithm tries to find the best sensor location scenario from a set of sensor

location candidates under particular budget constraints. Based on a breadth-first selection mechanism, the beam

search algorithm branches from candidate nodes level-by-level. At each level, it keeps only a limited number of

promising nodes, and prunes the other nodes permanently to limit the total number of nodes to be examined.

Beam search Algorithm

Step 1: Initialization

Generate link candidate set LC and node candidate set NC for point and point-to-point sensors, respectively;

Set the active node list ANL . Create the root node u with '( ) , "( )A u N u , search level

( ) 0sl u , where u is the search node index. Insert the root node into ANL.

Step 2: Termination criterion

Under one of the two following conditions, stop the process and output the best feasible solution;

(1) All the active nodes in ANL have been visited,

(2) The number of active nodes in ANL is exceeded by a predefined maximum capacity.

Step 3: Node generation and evaluation

For each node u at search level sl in ANL, remove it from ANL, and generate child nodes;

Scan through the link candidate sets LC: if a link a is not in A'(u), generate a new child node v’ where

'( ') '( ) , "( ') "( ), ( ') ( ) 1A v A u a N v N u sl v sl u ;

Scan through the node candidate sets NC: if a node i is not in N''(u), generate a new child node v” where

"( ") "( ) , '( ") '( ), ( ") ( ) 1N v N u i A v A u sl v sl u ;

For each newly generated node v, calculate the objective function with Eq. (27) or (28). If the budget constraint

(30) is satisfied for the newly generated node, add this node v into the ANL.

Step 4: Node filtering


19

Select ψ best nodes from the ANL in the search tree, and go back to Step 2.

In the above beam search algorithm, the total computational time is determined by the number of nodes to be

evaluated, which depends on the beam width ψ and the size of the candidate sensor links/nodes. For each node in the

tree search process, the complexity is determined by the evaluation of the objective function, which is mainly

determined by the calculation of the uncertainty matrix P- and P+.

For a large-scale sensor network design application, we can adopt four solution strategies to reduce the size of

the problem and, therefore, the computational time. (1) One can focus on critical OD pairs with significant volumes.

(2) One can aggregate original OD demand zones into a set of super zones within a manageable size, with this

strategy being especially suitable for a subarea analysis where many OD zones outside the study area can be

consolidated together. (3) We can reduce the size of candidate AVI sensor nodes and point sensor links in order to

decrease the number of search nodes to be evaluated. (4) To provide a tight upper bound and eliminate some non-

promising solutions during the beam search procedure, we can first generate some good feasible solutions based on

the current sensor network and bottleneck links.

8. Illustrative example and numerical experiments

8.1. Illustrative example

In Fig. 5, we present an illustrative example with a 6-node hypothetical transportation network to demonstrate

how the proposed measures of information can systematically evaluate the trade-offs between the accuracy and

placement of individual AVI sensors for path travel time estimation reliability. Using Eqs. (26) and (27), this

example focuses on the measure of estimation uncertainty with the heuristic approach described in Section 6.2. As a

comparison, the results of the proposed analytical method, described in Section 6.1, are also provided with lower

bound evaluation. In addition, subscript time interval t is omitted for simplicity.

As shown in the base case in Fig. 5, there are three traffic analysis zones at nodes a, d and b, and three major

origin-to-destination trips: (1) a to b, (2) a to d and (3) d to b, each with a unit of flow volume. P

(e.g., obtainable

from a historical travel time database with point detectors) leads to a trace of 12 and a determinant of 48. Among the

5 links along the corridor, link 5 from node f to b has the highest uncertainty in terms of link travel time estimation

variance. We can view node b as a downtown area, and the incoming flow from the other two zones creates dramatic

traffic congestion and travel time uncertainty, first on link 5 and then on link 4. For the base case, we can calculate

the variance of path travel time estimates for these three OD pairs, respectively, as 1+2+2+3+4=12, 1+2=3 and

2+3+4=9, leading to a total path travel time estimation uncertainty as TU = 12×1+3×1+9×1 = 24.

In both cases (I) and (II), two AVI sensors are first installed at nodes a and b. In case (I), an additional AVI

sensor is located at node f so that we can obtain two pairs of end-to-end travel time measurements: from node a to

node f, and from node f to node b. The second measurement directly monitors travel time dynamics on link 5. In this

particular example along the linear corridor, the end-to-end travel time statistics from a to b can be explicitly

determined from the above two mutually exclusive observations. In order to avoid double-counting the information

gain for the same data sources, the information quantification module in this study only considers two raw

measurements: from a to f, and from f to b, to update the link travel time variance covariance matrix from P

to P

.

To do so, the measurement error matrix is assumed to be 1 0

0 1

R

, and the mapping matrix

1 1 1 1 0

0 0 0 0 1H

, where the first measurement from a to f covers links 1,2,3 and 4, and the second

measurement from f to b covers link 5. As link 5, with the highest travel time uncertainty, is directly measured from

AVI readings in case (I), its link travel time estimate variance is reduced from 4 to 0.8, but the resulting P+ contains

a large amount of correlation in its link travel time estimates for links 1 to 4. All the path travel time uncertainties

for the three OD pairs have been reduced, and TU = 6.74.


20

In case (II), the third AVI sensor is installed at node d to match the underlying OD trip demand pattern, which

produces sensor mapping matrix1 1 0 0 0

0 0 1 1 1 H

. The resulting P+ still contains two clusters of

correlations corresponding to two individual measurements from a to d and from d to b. The path travel time

estimate variances for the OD pairs from a to d and from d to b are dramatically reduced to 0.75 and 0.9,

respectively. Although link 5 still has a relatively large estimate variance of 2.4 (its overall estimation error

measure), the total travel time estimation uncertainty TU is now 3.3, which is much lower than TU = 6.74 in case (I).

In comparison, the analytical model (22) with log-determinant maximization does not consider the flow weight

on each link, and results in a decision value 0.52 for case (I) and 0.48 for case (II). For the targeted estimation

uncertainty P+, case (I) has a log-determinant value of 0.065 and case (II) of 0.18, with the lower bound of the log-

determinant value being -0.53.

This example shows that by locating and spacing AVI sensors to naturally match the spatial trip patterns of

commuters, case (II) is able to systematically balance the trade-off between the needs for monitoring local traffic

variations and end-to-end trip time dynamics. It is also important to notice that, both cases (I) and (II) have the same

sensor network size and generate the same number of measurements every day, but with different amounts of

estimation uncertainty from a commuter/road user perspective. Thus, simple measures of information, such as traffic

network coverage and the number of measurements, might not be able to quantify the system-wide uncertainty

reduction and information gain for traveler information provision applications.

Fig. 5. Example of locating AVI sensors on a linear corridor

8.2. Sensor location design for traffic estimation with recurring conditions

In this study, we examine the performance of the proposed modeling approach through a set of experiments on a

simplified Irvine, California network, which is comprised of 16 zones, 31 nodes and 80 directed links. This study

considers a single path between each OD pair in this simple network.

All the experiments are performed on a computer system equipped with an Intel Core Duo 1.8GHz CPU and 2

GB memory. Shown in Table 1, a set of critical OD pairs with large flow volumes is selected to estimate the

network-wide path travel time based uncertainty. In this experiment, starting from the zero-sensor case, we

initialized the uncertainty matrix P with large variance values. Specifically, the initial variance of each link is equal

to the square of its own free flow travel time (FFTT). Furthermore, we use 20% and 40% of the FFTT as the

experimental values for the state processing error under recurring and non-recurring traffic conditions, respectively.

In addition, the state transition value used in this example is L = 2.

For the algorithm implementation, a beam search width of 10 is used in the beam search algorithm to reduce the

computational complexity. The total number of nodes in the search tree is the number of additional sensors

multiplied by the beam search width. In our experiments, with standard Matlab matrix calculation functions, it takes

about 30 min to compute 160 nodes in the beam search tree for this small-scale network.

In this section, we examine the proposed information measure model and sensor location algorithm for the

estimation of recurring traffic conditions. With given OD flow and prior uncertainty information, three sensor

location plan scenarios are designed for comparison with the existing sensor network.

We first conduct experiments to compare the existing point sensor network (Fig. 6a) and an optimized point

sensor network plan (Fig. 6b), both with the same number (i.e. 16) of point sensors. Table 1 shows the critical path

travel time estimation errors under those two scenarios. The results show that, comparing the prior travel time

estimation variance, the proposed optimization model can reduce the path travel time estimation variance by an

average of 34.3%, while the existing sensor plan only reduces the same measure by about 16.5%. By factoring in the

OD demand volume (shown in the third column in Table 1), we can compute the proposed measure of information:

the total path travel time estimation variance. The base case with zero sensors produces TU_zero_sensor=114855,

the existing locations reduce TU to 88878 (77.3% of TU_zero_sensor), and the optimized sensor location scenario

using the proposed model further decreases the system-wide uncertainty to TU= 63586 (55.3% of TU_zero_sensor).


21

This clearly demonstrates the advantage of the proposed model in terms of improving end-to-end travel time

estimation accuracy.

Table 1

Critical Path Travel Time Estimation Error under Existing and Optimized Sensor Location Strategies

Fig. 6. Numerical experiment results for regular traffic pattern estimation.

In the next set of numerical experiments, we compare two scenarios with additional sensors installed with the

existing sensors.

1) Add 4 point sensors on unmeasured links with remaining large travel time variances, leading to a total

16+4=20 point sensors, shown in Fig. 6(c);

2) Add 6 AVI readers on major OD-pairs with large volumes, leading to a network with 16 point sensors and 6

AVI readers, shown in Fig. 6(d).

Fig. 7 further compares the measure of uncertainty of different scenarios. It is interesting to observe that,

compared to the optimized scenario (from Fig. 6b) with 16 sensors, the “additional point sensors” scenario (with 20

sensors) does not offer a superior uncertainty reduction performance for different OD pairs. On the other hand,

compared to adding 4 point sensors to cover highly dynamic links, installing 6 additional AVI sensors does not

further improve the path travel time estimation performance dramatically.

Fig. 7. Comparison of different sensor location schemes (for regular traffic estimation)

8.3. Sensor location design for traffic prediction with recurring and non-recurring conditions

Now we implement the proposed algorithm by considering both regular (recurring) and non-recurring traffic

conditions. As discussed in Section 6, the total traffic prediction uncertainty is computed as a probabilistic

combination of recurring and non-recurring uncertainties in Eq. (28). In this illustrative numerical experiment, we

consider incidents as the only non-recurring delay source, with the link-based incident rates shown in Fig. 8(a). For

simplicity, one ensemble is generated for each link with its individual incident probability. The proposed sensor

location algorithm is applied in three scenarios: (1) optimized sensor network with 16 point sensors, (2) current

network with 4 additional point sensors, and (3) current network with 6 additional AVI sensors. Consequentially,

these three sensor network design results are plotted in Figs. 8(b-d). It is interesting to note that, when considering

non-recurring traffic conditions (incidents), the optimized sensor locations (Fig. 8b) are more focused on links with

higher incident rates, compared to the regular pattern estimation based planning result in Fig. 6(b).

Fig. 8. Numerical experiment results under recurring and non-recurring traffic conditions

8.4. Performance Evaluation of Beam Search Heuristic Algorithm

To evaluate the computational efficiency and solution quality of the proposed beam search algorithm for the

sensor location problem, we further performed a comparison between the beam search approach and the lower

bound calculated from the proposed analytical approach, both described in Section 6. Specifically, in this test, the

log-determinant based uncertainty matrix minimization problem (22) was used as the objective function for the

beam search algorithm, so that the two approaches are targeting the same model.

We first applied the analytical model for a point sensor-only design problem, and solved it with the primal-dual

based open-source solver SDPT3. Specifically, the semidefinite-quadratic-linear programming package solves the

overall problem using a continuous range of the decision variables, and its objective function value thus serves as a

lower bound for our original integer problem. Then an approximate solution was generated by rounding non-integer

variables from the optimal solution found by the analytical model. For our testing network with 80 links, the

proposed analytical approach reduced the log-determinant of the estimation uncertainty from -49.226 to -63.439,

with a lower bound of -64.369.


22

We then implemented the proposed beam search algorithm using C# programming language, and found the

same optimal sensor location strategy, with a beam search width ψ of 10. Table 2 shows the log-determinant values

for the two proposed algorithms implemented on networks with different link sizes. The results show that the beam

search heuristic algorithm finds the same best solutions compared to the semidefinite-quadratic-linear programming

package SDPT3, and the solution quality is acceptable compared to the lower bound found by the analytical method.

Additionally, as illustrated in Fig. 9, although the beam search algorithm takes longer computational time than the

proposed analytical approach, it is generally easier to implement in practice, as (1) it does not rely on linear

programming packages and has more flexibility for model development, and (2) it can handle larger networks on the

same computing environment, compared with the analytical approach which can only handle networks with no more

than 160 links in our experiment, which might be also due to the inherent limitation of the SDPT3 solver.

Table 2

Log-determinant values of uncertainty matrix using beam search and analytical approaches

Fig. 9. Computational time for two proposed methods with different network sizes

9. Conclusions

To provide effective congestion mitigation strategies, transportation engineers and planners need to

systematically measure and identify both recurring and non-recurring traffic patterns through a network of sensors.

The collected data are further processed and disseminated for travelers to make smart route and departure decisions.

There are a variety of traditional and emerging traffic monitoring techniques, each with the ability to collect real-

time traffic data in different spatial and temporal resolutions. This study proposes a theoretical framework for the

heterogeneous sensor network design problem. In particular, we focus on how to better construct network-wide

historical travel time databases, which need to characterize both mean and uncertainty of end-to-end path travel time

in a regional network.

A unified Kalman filtering based travel time estimation and prediction framework is first proposed in this

research to integrate heterogeneous data sources and model both recurring and non-recurring traffic evolution.

Specifically, the travel time estimation model starts with the historical travel time database as prior estimates. Point-

to-point sensor data and GPS probe data are mapped to a sequence of link travel times along a set of the most likely

traveled paths. Through an analytical information updating equation derived from Kalman filtering, the variances of

travel times on different links are estimated for possible sensor design solutions with different degrees of sampling

or measurement errors. The variance of travel time estimates for spatially distributed links are further assembled to

calculate the overall path travel time estimation/prediction uncertainty for the entire network as a single-valued

information measure. The proposed uncertainty quantification models and beam search solution algorithm can assist

decision-makers to select and integrate different types of sensors, as well as to determine how, when, and where to

integrate them in an existing traffic sensor infrastructure.

Compared to the early study by Zhou and List (2010), this study has its unique and additional contributions.

First, focusing on the travel time problem, this paper proposed analytical formulas with transition matrix to describe

the travel time prediction under recurring and non-recurring conditions. Second, the information uncertainty of AVI

sensors has been evaluated in a systematic format that is consistent with the travel time estimation/prediction

problems. Third, this paper has an extended discussion on the measure of information. In particular, we proposed

one maximum determinant based analytical approach with lower bound evaluation, and one heuristic approach

which captures the network-wide uncertainty with a focus on the critical OD/paths.

In our on-going research, we plan to expand this methodology in the following ways. First, this study only

focuses on the sensor design problem for better estimating the mean of path travel time, and a natural extension is to

assist sensor design decisions for other network-wide traffic state estimation domains, such as measuring and

forecasting point-to-point travel time reliability, and incident detection probability. Second, under assumptions of

normal distributions for most error terms, the proposed sensor location model is specifically designed for the

minimum path travel time variance criterion, and our future work should consider other crucial factors for real-

world sensor network design, such as allowing log-normally distributed error terms and minimizing maximum

estimation errors. Furthermore, the offline planning-level model developed in this study could be extended to a real-


23

time traffic state estimation and prediction framework with mobile and agile sensors. The numerical experimental

results (for a small-scale network) in this study also demonstrate computational challenges (due to computationally

intensive matrix operations) in applying the proposed information-theoretic sensor location strategy in large-scale

real-world networks, and these challenges call for more future research for developing efficient heuristic and

approximation methods.

Acknowledgement This paper is based on research work partially supported through the TRB SHRP 2 L02 project titled

“Establishing Monitoring Programs for Mobility and Travel Time Reliability”. Special thanks to anonymous

reviewers and our colleagues Jay Przybyla and Hao Lei at the University of Utah for their constructive comments.

The work presented in this paper remains the sole responsibility of the authors.

Appendix

A1. Transition matrix derivation under non-recurring conditions

This study proposes an analytical approach to derive the non-recurring transition coefficient Lt,t+h in Eq. (5). In

this section we demonstrate the derivation of transition matrix/factor with several non-recurring traffic conditions,

including (1) single-link capacity reduction caused by incident and (2) queue spillback from adjacent link. These

two illustrative examples are designed to showcase a set of elementary non-recurring conditions: capacity change,

demand change, and spatial correlation.

A1.1 Transition matrix derivation under non-recurring congestion due to capacity reduction

Corresponding to a recurring congestion with a constant queue discharging rate C, Fig. A1 shows the

cumulative flow diagram with original shifted arrival curve A and departure curve D'. The shaded area represents the

additional delay due to an incident (or any significant capacity reduction), which begins at time s and ends at time e

with a reduced capacity CR . The road capacity is restored back to normal capacity C after time e. In order to derive

the transition coefficient Lt,t+h for updating the travel time of this link, we now examine the additional delay vt and

vt+h for vehicles leaving from the bottleneck at time t and t+h, respectively.

Fig. A1. Cumulative flow count curve for a single bottleneck with reduced capacity due to an incident

A detailed illustration in Fig. A2 showsh

t t tv x x , where tx is the travel delay for a vehicle leaving from the

bottleneck at departure time t under reduced capacity conditions, and tx can be calculated as the horizontal distance

between the shifted cumulative arrival and actual departure time curves A and D. h

tx is the corresponding recurring

delay for the same vehicle (with the actual departure time t under incidents), and it is equivalent to the horizontal

distance between curves A and D'. In Figs. A1 and A2, timestamps t' and t are the original and actual departure times,

respectively, for the vehicle of interest.

Let us denote Δ as the number of vehicles which can be discharged under recurring congestion from s to t'. It is

easy to show that ( ' ) ( )R

C t s C t s through the flow balance, where 'h

tt s x and ( ) tt s x . That

is, ' / ( / )( )R

t s C C C t s . Thus, the travel time deviation term can be determined as

' ( ) ( ' ) ( ) 1R

h

t t t

Cv x x t t t s t s t s

C

.

(A1)

After the incident ending time e, vt becomes a constant value 1RC

e sC

.


24

Fig. A2. A close-up view on capacity reduction from Fig. A1

We can further examine the transition coefficient L in three cases (as shown in Table A1), with a pre-defined

prediction period h (e.g., 15 minutes). According to the non-recurring state transition equation (3), the transition

matrix L (a single value in this example) is derived as the ratio of structure deviation terms between current time t

and future time t+h.

,t h

t t h

t

vL

v

(A2)

As shown in Table A1, with the current time stamp t located in different stages of an incident event, the

transition coefficient L can be analytically derived in different forms. Fig. A3 details an example on how the

transition coefficient L varies according to time after an incident, by assuming the prediction period h is fixed as 15

minutes and a typical incident duration e – s = 30 minutes. It should be noticed that, after the non-recurring

condition ends at time e, the state transition value L will retain a value of 1, and L decreases as the time approaches

the end of the congestion period. Eventually, when the real-time traffic process evolves to congestion time periods

without incidents, then the transition coefficient becomes an autoregressive parameter φt as it enters the regime of

recurring congestion.

Table A1 Derivation of transition matrix L for different time periods

Fig. A3. Illustration of time-dependent transition coefficient L

The above example demonstrates how to derive the non-recurring components of the transition matrix At under

conditions with short-term capacity drops. Similar derivation could be performed for severe weather and work zone

cases. In on-line travel time prediction applications, Lt , as a time-dependent and state-dependent parameter, needs to

be estimated in real-time according to a specific incident duration and degree of capacity reduction. In reality, the

bottleneck discharge capacity is highly dynamic, so estimating the transition matrix using the above analytical

method might not be able to capture the inherent complexity in network traffic, especially under possible queue

spillback conditions. In the planning-level sensor location problem considered in this paper, however, we are more

interested in finding a reasonable estimate on the likely transition matrix so as to establish a Kalman filtering

framework, which will be used to quantify uncertainty associated with different sensor location plans. For instance,

in our experiments presented in Section 8, we consider the following setting for typical incidents, with average

incident duration = (e-s) = 30 min, and the time for making prediction at the middle stage of an incident as (t-s) = 15

min, which leads to a typical value L = 2.

A1.2. Transition matrix for adjacent links with queue spillback

The above example in Section A1.1 considers a simple vertical queue, with an aim of deriving the time-

dependent travel time transition coefficient for an individual link/bottleneck. The following discussions consider a

more realistic, more complicated case with queue spillback from a downstream link (denoted with subscript d) to an

upstream link (subscript u). Our goal is to examine the possible transition matrix Lt,t+h for the upstream link when a

capacity reduction event occurs on the downstream link, as illustrated in Fig. A4. Without loss of generality, we

assume that the recurring congestion on the downstream link has been propagated back to the upstream link during

typical days, which leads to parallel original departure curves Dd’ and Du’, assuming discharge capacity is the same

at both locations. During a day with non-recurring congestion, an incident occurs on the downstream link and

further causes a reduced capacity CR. The queue associated with this non-recurring congestion propagates back to

the upstream link at time s', so the upstream departure curve Du will be also changed.

Fig. A4: Cumulative flow count curve for two adjacent links with queue spillback due to an incident


25

According to Newell’s simplified kinematic wave model (Newell, 1993; and Hurdle and Son, 2000) which

assumes a triangular flow-density relationship, when there is a queue spillback along the backward wave

propagation line from the downstream bottleneck to the upstream link, the departure curve for the upstream link (Du

= Ad) is constrained by the departure curve Dd of the downstream link plus a constant link storage factor (with jam

density kjam times link length ld and number of lanes nd),

, ,u t d t BWTT jam d dD D k l n

(A3)

That is to say, after the queue spills back to the upstream link, the upstream departure curve will be in parallel

with the downstream departure curves due to the constraint in Eq. (A3), with a time shift of backward wave travel

time (BWTT). Consequently, the upstream additional delay at time t+h caused by the incident, denoted as vu,t+h

(point 2 4 in Fig. A4), matches with a downstream additional delay vd,t+h-BWTT (point 1 3). In other words, the

horizontal distance from point 2 to 4 (additional delay on the upstream link) is equivalent to the horizontal distance

from point 1 to 3 (additional delay on the downstream link). In this special case, it is possible to connect the

upstream additional delay at time t+h with the detected downstream additional delay at time t, which leads to a

transition coefficient across two adjacent links by extending Eq. (A2):

, ,

,

, ,

( ) 1

,

( ) 1

R

u t h d t h BWTT

t t h Rd t d t

Ce s

v v C e sL t h BWTT e

v v t sCt s

C

(A4)

where s and e are the starting and ending timestamps of the incident on the downstream link, respectively.

The above discussion can serve as a starting point for analytically deriving possible spatial correlation between

two adjacent links for more complex conditions. However, the detailed changes of cumulative arrival and departure

curves need to be analyzed case-by-case. As a result, in the subsequent sensor design model, we only consider the

uncomplicated point-queue based formulation for L for simplicity.

In summary, the illustrative cases mentioned in Appendix A1 are only used for demonstrating the derivation

procedure of the transition matrix. With the help of Newell’s cumulative flow model, similar derivation could be

performed for a broader set of non-recurring traffic conditions.

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28

List of Figures:















29

A B C D E

Sensor 1 Sensor 2



30

Sp

ace

Time

Point sensor

records

AVI sensor

records

Probe sensor

records

AVI Station

AVI Station

Point Sensor

Station

d

8:00 AM 8:15 AM 8:30 AM

c

b

a

2

3

1

P1

P2

A1 A2



31

a b c3

AVI readernode

1

2



32

a b2

AVI readernode

1

3

4



33



34

1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16

Existing loop detector

Demand zone

(a) Existing point sensor locations

1

2

3

4

5

6

7

8

9

1012

1113

14 15

16

(b) Optimized sensor network locations

(start from a zero-sensor case)

Optimized point sensor location

Demand zone

1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16


Demand zone

Additional loop detector

(c) Sensor network with 4 point sensors in

addition to existing 16 point sensors

1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16


Demand zone

Additional AVI sensor

(d) Sensor network with 6 AVI sensors in




35


1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Index of Critical OD/Path

Path

Tra

vel T

ime E

stim

ation V

ariance

Posterior variance of existing sensors

Posterior variance for optimized sensor network

Posterior variance with addtional loop detectors

Posterior variance with addtional AVI sensors


36

(a) Annual incident rate

1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16

Annual incidents per mile

Demand zone

50

100

150

200

300

(b) Optimized sensor network locations

(start from a zero-sensor case)

1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16

Optimized point sensor location

Demand zone

1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16


Demand zone

Additional loop detector

(c) Sensor network with 4 point sensors in


1

2

3

4

5

6

7

8

9

10

12

11

13

14 15

16


Demand zone

Additional AVI sensor

(d) Sensor network with 6 AVI sensors in




37


80 90 100 110 120 130 140 150 160 1700

1000

2000

3000

4000

5000

6000

7000

Network Link Size

Co

mp

uta

tio

nal

Tim

e (

sec)

Heuristic Beam Search Algorithm

Analytical Approach


38

Cu

mu

lati

ve

Flo

w C

ou

nt

Incident

Duration

Time

C

e

C

s

Shifted

Arrival

Curve ADeparture Curve D

with Incident

Original Departure

Curve D’

Reduced

Capacity CR

t + ht

xt

xt+h

xht+h

vt

vt+h

xht

t'

Shifted

Arrival

Curve A

Original

Departure

Curve D’

Actual Departure

Curve D with

Incident

Additional Delay

due to Incident



39

CΔ

ts

xth

Time

Cu

mu

lati

ve

Flo

w

Co

un

tt'

CR

xt

vt



40


s s+10 s+20 e s+400

1

2

3

4

Time after incident (min)

Tra

nsi

tio

n c

oef

fici

ent

L


41

Cu

mu

lati

ve

Flo

w C

ou

nt

Time

C

C

s' t + h

1

vd,t

BWTT

Upstream Arrival Curve Au

Original Downstream

Departure Curve Dd '

Actual Downstream Departure

Curve Dd with Incident

Additional Delay

due to Incident

CR

vu,t+h

Actual Upstream Departure

Curve Du / Downstream Arrival

Curve Ad , under incident

CR

C

t + h-BWTT

vd,t+h-BWTT

C

t

4

3

2 Original Upstream Departure

Curve Du'



42

List of Tables:

Table 1: Critical Path Travel Time Estimation Error under Existing and Optimized Sensor Location Strategies

Table 2: Log-determinant values of uncertainty matrix using beam search and analytical approaches

Table A1: Derivation of transition matrix L for different time periods


43

Table 1

Critical Path Travel Time Estimation Error under Existing and Optimized Sensor Location Strategies

Origin Destination Hourly

volume

Prior path travel

time estimation

variance

without sensor

Posterior path

travel time

estimation

variance with

existing

sensors

% reduction

in variance

due to

existing

sensors

Posterior path

travel time

estimation

variance with

optimized

sensor

locations

% reduction

in variance

due to

optimized

sensor

locations

1 16 4000 5.87 5.14 12.44% 3.00 48.89%

16 4 6820 5.24 3.94 24.81% 2.80 46.56%

12 4 1152 1.85 1.85 0 1.32 28.65%

4 16 2480 5.23 3.54 32.31% 3.19 39.01%

16 12 832 4.91 3.61 26.48% 3.00 38.90%

15 4 880 2.81 2.60 7.47% 2.07 26.33%

12 16 680 4.90 3.21 34.49% 3.21 34.49%

16 1 4800 5.86 4.28 26.96% 3.00 48.81%

4 15 604 2.81 2.81 0 2.47 12.10%

4 12 444 1.85 1.85 0 1.50 18.92%


44

Table 2

Log-determinant values of uncertainty matrix using beam search and analytical approaches Link size 80 90 100 110 120 130 140 150 160 170

Initial uncertainty -49.22 -52.40 -56.47 -60.11 -61.02 -74.21 -81.97 -90.14 -98.45 -101.62

Beam search optimal -63.44 -66.93 -71.44 -75.49 -76.54 -89.73 -97.62 -105.80 -114.11 -117.28

Approximate solution by

rounding analytical

optimal solution

-63.44 -66.93 -71.44 -75.49 -76.54 -89.73 -97.62 -105.80 -114.11 N/A

Analytical lower bound -64.37 -68.08 -72.83 -76.98 -78.45 -91.71 -99.69 -107.90 -116.24 N/A


45

Table A1 Derivation of transition matrix L for different time periods

Stages

(current time

with respect to

incident event)

Current

Time

t

Prediction

Timestamp

t+h

Detected Additional

Delay

vt

Predicted Additional

Delay

vt+h

Estimated

Transition

Coefficient

,t t hL

Early Stage of

Incident s < t < e-h

s+h < t+h

< e ( ) 1

RCt s

C

( ) 1

RCt h s

C

t h s

t s

Late Stage of

Incident e-h < t < e t+h > e ( ) 1

RCt s

C

1

RCe s

C

e s

t s

Post incident t > e t+h > e+h 1RC

e sC

1

RCe s

C

1

Designing heterogeneous sensor networks for estimating and predicting path travel time dynamics: An...

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