ISSN 1055- 1425
January 2011
This work was performed as part of the California PATH Program of the
University of California, in cooperation with the State of California Business,
Transportation, and Housing Agency, Department of Transportation, and the
United States Department of Transportation, Federal Highway Administration.
The contents of this report reflect the views of the authors who are responsible
for the facts and the accuracy of the data presented herein. The contents do not
necessarily reflect the official views or policies of the State of California. This
report does not constitute a standard, specification, or regulation.
Report for Caltrans task order 65A0363
CALIFORNIA PATH PROGRAM
INSTITUTE OF TRANSPORTATION STUDIES
UNIVERSITY OF CALIFORNIA, BERKELEY
Dynamic All- Red Extension
at Signalized Intersection:
Probabilistic Modeling and Algorithm
UCB- ITS- PWP- 2011- 01
California PATH Working Paper
Liping ZHANG, Kun Zhou, Wei- bin Zhang
and James A. Misener
CALIFORNIA PARTNERS FOR ADVANCED TRANSIT AND HIGHWAYS
Dynamic All- Red Extension at Signalized Intersection:
Probabilistic Modeling and Algorithm
Liping ZHANG, Kun Zhou, Wei- bin Zhang and James A. Misener
California PATH, University of California, Berkeley
1357 South 46th st,
Richmond, CA 94804,
f lpzhang, kzhou, wbzhang, misener g @ path. berkeley. edu
1
Acknowledgment
This work was sponsored by the State of California Business, Transportation and Housing
Agency, Department of Transportation ( Caltrans) and U. S. Department of Transportation ( DOT)
as part of the CICAS project ( Caltrans task order 65A0363). The contents of this paper reflect the
views of the authors, who are responsible for the facts and accuracy of the data presented herein.
The contents do not necessarily reflect the official views or policies of the State of California.
2
DISCLAIMER STATEMENT
This document is disseminated in the interest of information exchange. The contents of
this report reflect the views of the authors who are responsible for the facts and
accuracy of the data presented herein. The contents do not necessarily reflect the
official views or policies of the State of California or the Federal Highway Administration.
This publication does not constitute a standard, specification or regulation. This report
does not constitute an endorsement by the Department of any product described herein.
For individuals with sensory disabilities, this document is available in Braille, large print,
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alternate formats, please contact: the Division of Research and Innovation, MS- 83,
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0001.
Abstract
Dynamic All- Red Extension ( DARE) has recently attracted research interest as a non-traditional
intersection collision avoidance method. We propose a probabilistic model to pre-dict
red light running ( RLR) hazard for dynamic all- red extension system. The RLR hazard,
quantified by a predicted encroachment time, has contributory factors including the speed, dis-tance
and car- following status of the violator and the empirical distribution of the entry time of
conflict traffic. An offline data analysis procedure is developed to set the parameters for RLR
hazard prediction. Online- wise, a two- dimensional normal model is developed to predict the
vehicle’s stop- go maneuver based on speeds at advanced detectors. Additionally, unlike most
prediction models which are designed to minimize mean errors, our model identifies two types
of errors, namely the false alarm and missed report. The capability of distinguishing these two
types of errors is crucial to the effectiveness of dynamic systems. To quantify the trade- off be-tween
these two types of errors in the system design, a system operating characteristics ( SOC)
function is then defined. Performance of the proposed model and its prediction algorithm is
evaluated using data collected from a field intersection. At a false alarm rate of 5%, the al-gorithm
reach a correct detection rate of over 65% to over 90% for various legs of the test
intersection. Performance evaluation results showed that the proposed models and algorithms
within the DARE framework can effectively detect the RLR hazards.
Keywords: Traffic Control, Collision Avoidance System, Red- light running, Dynamic all- red ex-tension,
Probabilistic Model of Driver Behaviors
3
Executive Summary
Red light running has been a major cause of intersection injuries and fatalities in the United
States. In 2004 alone, there were 8,619 fatal crashes and 848,000 crashes with people injured, all
caused by RLR. Under the U. S. Department of Transportation ( DOT) initiated Collaborative In-tersection
Collision Avoidance System CICAS program ( Signalized Left Turn Assistance 􀀀 SLTA
and Traffic Signal Adaptation 􀀀 TSA), California PATH program at the University of California,
Berkeley undertake the efforts of designing a system that pro- actively sense the impending hazards
that are caused by RLR and enact dynamic control of the signal by extending the all- red clearance
interval by a few seconds to avoid potential hazardous situation.
In 2007, California PATH setup an observation field intersection at Page Mill road and El
Camino Real at Palo Alto, with 9 Autoscope R  cameras instrumented to collect vehicle movement
data as well as the real- time signal phase. The data collected in 2008 and 2009 have been used to
study the statistical characteristics of the vehicle trajectories approaching the intersection during
the inter- green phase ( including yellow and red). Particularly we have used the collected data
to analyze the following types of vehicle trajectories of ( 1) Red light runners, ( 2) First- to- stop
vehicles, and ( 3) vehicles going through the intersection during yellow, to identify the statistically
significant contributory parameters from the trajectories that can be used for RLR related hazard
prediction purpose.
The data analysis results showed that important contributory factors that need to be included
for accurate RLR hazard prediction should include the single vehicle trajectory parameters such
as speed, time to intersection and acceleration / deceleration, and inter- vehicle status as whether
or not the subject vehicle is following a leading vehicle. A probabilistic model that uses both the
single vehicle parameters and inter- vehicle status has been built for the purpose of RLR hazard
prediction.
Besides the probabilistic prediction model for RLR hazards, we have also developed a frame-work
to optimize the parameter settings for the system using a so- called System Operating Char-acteristic
( SOC) curve, which shows the trade off of two types of errors that could occur to the
system: miss detection and false reports. We used a Neymann- Pearson criteria to gain the opti-mized
parameters that achieve best performance under a constraint of maximum allowable false
alarm rate.
Performance of the proposed system has been evaluated using the collected field data. The
achieved performance ( in terms of the correct detection rate of RLR hazards at different given
levels of false report rates) has been evaluated.
Results of this research laid the basis for a operational dynamic all- red extension system, by
having the following accomplishments:
 Built a set of contributory factors from the vehicle trajectories that can be used to predict
RLR hazards;
 Built probabilistic models of the RLR hazards;
 Developed RLR hazard prediction method with optimized performance; and
4
 Evaluated actual performance of the system using field collected data.
The research results showed that the dynamic all- red extension as designed using the proposed
framework, can predict the RLR hazards with high accuracy while also with a low false report rate
at the same time. This dynamic all- red extension framework is ready for a field operational test
that will address implementation issues related to deployment.
This report is for Caltrans Task order 65A0363 ( Collaborative Intersection Collision Avoid-ance
System).
5
Contents
1 Introduction 9
1.1 Red- Light Running . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Dynamic All- Red Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Dynamic All- Red Extension Outline 11
2.1 Predictive Indicator of RLR Hazard . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 System Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Modeling and Neyman- Pearson Detection of RLR Hazard 13
3.1 Modeling of RLR Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Predictor of RLR Hazard and Two Types of Errors . . . . . . . . . . . . . . . . . 17
3.3 Neyman- Pearson Detection of RLR Hazard . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Approximation of PM- H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.2 Approximation of PFA- H . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 The Implementation of the RLR Hazard Prediction Algorithm 20
4.1 Offline procedure of DARE algorithm . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.1 Estimating the parameters for prediction . . . . . . . . . . . . . . . . . . . 20
4.1.2 Forming the Neyman- Pearson Detection Boundary . . . . . . . . . . . . . 20
4.2 Online Prediction of RLR Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 Arrival time ( time- into- red) estimator . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Formulation of DH ( X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Performance of the RLR Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Data Collection and Processing 24
5.1 Field Setup and Configuration of Point Detectors . . . . . . . . . . . . . . . . . . 24
5.1.1 PrETRLR and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1.2 Minimum sensing distance requirement for DARE system . . . . . . . . . 25
5.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.1 Calibration of the Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.2 Building Trajectories From Point Detectors . . . . . . . . . . . . . . . . . 26
6 Simulation 26
6.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 Simulated System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2.3 Simulated system performance with different hazard settings . . . . . . . . 29
7 Conclusion 29
6
List of Figures
1 Dynamic All Red Extension Algorithm . . . . . . . . . . . . . . . . . . . . . . . 12
2 Example of entry time distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Time Space Diagram of a Conflict Zone and the Calculation of its PrET . . . . . . 15
4 Discrete point sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Bi- variate Normal model of the two hypotheses . . . . . . . . . . . . . . . . . . . 22
6 SOC Curves for RLR Prediction, Using Field Data . . . . . . . . . . . . . . . . . 32
7 SOC Curves For Different Thresholds ( t ) . . . . . . . . . . . . . . . . . . . . . . 33
7
List of Tables
1 Numerical examples of missing rate and false alarm rate . . . . . . . . . . . . . . 18
2 Configuration of the field intersection . . . . . . . . . . . . . . . . . . . . . . . . 24
3 t for each approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Configurations of the Autoscope virtual loops . . . . . . . . . . . . . . . . . . . . 26
5 Prediction algorithm parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8
1 Introduction
1.1 Red- Light Running
Red- light- running ( RLR) is an increasingly major national safety issue at signalized intersec-tions.
In 2004, 2.5 million or 40% of all police- reported crashes in the United States occurred
at or near intersections [ 14]. Of these intersection- related crashes 8,619 or 22.5% were fatal and
848,000 or 46% resulted in injury. The RLR aspect of this problem is known and serious [ 1,5,14],
as approximately 20% of all intersection crashes occur due to signal violation crashes [ 5], resulting
in an estimated monetization of $ 13 billion a year when considering fatalities, lost wages, medical
costs, property damages, and insurance [ 7].
To address the RLR problem, significant effort has been undertaken to conceive engineering
countermeasures. These findings are summarized by Retting et al., and they run the gamut from
optimization of signal timing through removing unwarranted signals [ 17].
One important approach is to extend the yellow interval. It is reported that 1 second more
yellow interval reduces as much as 50% of RLR, especially for those intersections where yellow
intervals were set lower than the Institute of Transportation Engineers ( ITE) standard [ 4]. For high
speed intersections, Green Extension System ( GES) has been developed and research shows that it
helps to effectively reduce RLR caused by dilemma zone [ 2].
Despite of these significant countermeasures, RLR remains to be an outstanding cause for in-tersection
collisions, partly due to the fact that significant portion of the RLR cases feature willful,
intentional running or inattentive platoon following. These RLR scenarios lead to over 50% of the
RLR crashes [ 11], among which most are against Left Turn Across Path from Opposite Direction
( LTAP- OD) vehicles ( usually during the first few seconds after red onset) and Straight Cross Path
( SCP) vehicles ( last until late into red).
1.2 Dynamic All- Red Extension
There have been significant research and development in recent years in the application of
technology to address the intersection collision problem [ 1,6,7,13,14,22]. The central concept of
a dynamic system therein is to actively inform or alert drivers of emerging, impending hazardous
situations.
Extending the all- red interval is an intuitive approach to counteract the RLR related crashes,
especially those caused by willful runners or inattentive runners who enter the intersection during
the first few seconds of red onset. A study carried out by researchers at Iowa State University shows
that extending all- red interval does reduce the crash rate, however the drivers could potentially
adapt to the extension and undo the safety improvement [ 19].
Through applying the aforementioned “ dynamic” concept to RLR violations, a provocative
solution, which potentially addresses the dilemma zone, willful violation and inattentive viola-tion
motivations is to “ dynamically” extend an ” all red” phase. This therefore relies in large part
on detection of potential hazardous situation caused by a signal violator to enact a Dynamic All-
Red Extension ( DARE) - and it is precisely this probabilistic detection and implementation into
an algorithm on which we focus this report. Research reported in this report is for the U. S. De-
9
partment of Transportation ( DOT) initiated Collaborative Intersection Collision Avoidance System
CICAS program ( Signalized Left Turn Assistance 􀀀 SLTA and Traffic Signal Adaptation 􀀀 TSA)
conducted under the auspices of the University of California, Berkeley effort.
To elaborate: the detection ( prediction) of RLR hazardous situation is a crucial part for the
collision avoidance systems, which must be able to sense the dynamic characteristics of a vehicle
approaching the intersection from a reasonable range. A dynamic ” state- map” of the intersection
defined by Zennaro and Misener can then be built upon the sensed components [ 21]. Instead of
collision, which is so rare in both space and time such that its prediction is very difficult, we
propose to predict hazardous situation based on the state map to enable the engineering measures
to protect vehicles affected by the violator [ 2].
We employ a predictive encroachment time to quantify the RLR hazardous situation when
DARE will be triggered if this time lapse is below a threshold, which for a given collision course
can be transformed into a time- into- red threshold for RLR. Thus we address the problem of RLR
hazard prediction with a combination of finding the threshold of time into red for a given colli-sion
course, predicting the vehicle stop- and- go motion plus estimating of the arrival time ( time
into red). The first part, finding the threshold of time into red for RLR hazard is based on data
analysis of the empirical distribution of RLR and the entry time of cross street traffic. The second
part, predicting the stop- go maneuver of vehicles, is a critical part of the system which involves
predicting individual diver behavior at intersections. Sheffi and Mahmassani have introduced a sta-tistical
model wherein the driver decision to stop is normally distributed and a function of ” critical
time” [ 18]. More recent research in driver behavior during the intersection approach has used logit
models [ 3, 8] or empirical models [ 15]. The results in toto show that the stopping probability is a
function of multiple factors, to include the distance- to- intersection, speed and headway [ 3,10,18].
We extend Sheffi and Mahmassani’s probabilistic model to multi- dimensional normal distribution
where measured vehicle- to- intersection factors can be incorporated.
In short, we view the intersection hazard as an evolving event for various distances of the
violator to intersection; therefore, the empirical distribution of entry time of the conflict traffic, the
distance to intersection of violator, speeds of the approaching vehicles and time into yellow ( red)
are important contributory factors to our model.
When applying a prediction algorithm to the collision avoidance system, there are usually two
types of errors: the missing report error when a RLR hazard is missed ( false negative), and also
the false alarm error when proceeding through before a preset critical time or stopping is accepted
as RLR hazard ( false positive). Clearly, in any RLR countermeasure there could be different
consequences for the two types of errors, with the missing report giving a false sense of security
and the false alarm being a nuisance; therefore, a means to minimize and control the balance
is essential for any implementation. As an important note in implementing such a balance, we
consider the a priori probability of hypotheses can be different. For example, at most intersections,
the occurrence of RLR hazards could be significantly less than that of non- violation. Thus the
requirement of false alarm rate should be at or lower than a given threshold.
This is in contrast with the traditional probabilistic analysis for RLR frequency estimation
or dilemma zone prediction, as these do not distinguish the two types of errors. We address our
methodology - or method to control the balance - with the Neyman- Pearson ( NP) principle, which
10
governs the prediction of RLR hazard occurrence based on the empirical observations by maxi-mizing
the probability of detection ( minimizing the missed detection rate) while keeping the false
alarm rate equal to or lower than a given level [ 12]. The capability of adjusting the preset level of
false alarm is important to the effective operation of a dynamic system.
We have restricted the system to use only the real- time point detection sensors. The reason
for restricting the type of sensors is for deployability and cost considerations. Certainly, using
or extending legacy systems makes the conceived system more viable. Therefore, our approach
is to modify existing point detectors to serve as sensing means for RLR hazard prediction or by
using emerging but currently- marketed products. As a case in point in this study, the emulated
speed loops of Autoscope R  cameras are used. They are virtual point detectors which give good
real- time speed measurements. At the same time, the locations of the point detectors are software-reconfigurable,
which makes it possible to optimize the location of the detectors based on empirical
observations.
2 Dynamic All- Red Extension Outline
2.1 Predictive Indicator of RLR Hazard
The system is designed to be ” dynamic”, which should adaptively predict the occurrence of
RLR and its related hazardous situation based on advance detection, and thereafter enacts all-red
extension before all- red interval terminates, when most likely the subject violator has not yet
entered the intersections. We need to develop a predictive indicator to address the severity of
the potential hazardous situation caused by the signal violator. When this indicator shows that
likelihood of collision to happen is high, the DARE will be triggered. The essential part of the
research is to build a probabilistic model and upon which generate the threshold( s) for triggering
the DARE.
Being aware of the rareness of the collision occurrences both in time and space, we adopt
an alternative proximal indicator based on Encroachment Time ( ET). ET is an objective indicator
which is used extensively to quantify the severity of near- miss situations, and which, when below
a certain level, indicates a high likelihood of crashing, the frequency of which is much higher than
that of real crashes. Using conflicts instead of collisions is considered to be a reasonable alternative
evaluation tool [ 20] [ 9].
Traditional ET and other variations of indicators such as Post- Encroachment Time ( PET)
defines scenarios when subject vehicle ( SV) and principal other vehicle ( POV) are both moving ( at
known speeds). While for DARE system, it needs to decide whether or not to extend the all- red
interval before green onset for the conflict traffic when POV is mostly likely still waiting to enter
the intersection with zero speed.
To address this difference, we define a derivative of ET / PET as:
the predicted time lapse between the SV clears the conflict zone and the a priori entry time of
the POV to the conflict zone.
We call this time lapse the ” Predicted Encroachment Time for RLR”, or PrETRLR. PrETRLR
itself is a random variable which is a function of the speed of the SV, distance to conflict zone and
the a priori probability distribution of entry time of the POV.
11
The system extends all- red to avoid collision when
Pr ( PrETRLR < threshold ) < Minimum allowed probability of hazard ; ( 1)
where ” threshold” is a pre- set constant below which the conflict will be considered as hazardous
situation likely to turn into collision.
Assume that violator clears the intersection in constant speed, then the condition defined in
( 1) is equivalent to a condition of
Violator’s time into red < t ; ( 2)
where t is a function of the speed of SV, distance to conflict zone and the a priori distribution of
entry time of POV.
2.2 System Outline
The algorithm of the DARES system, is illustrated in Figure 1. It includes two parts, one is
an offline parameter setting procedure which collects empirical data to get the parameters of the
probabilistic models in ( 1) as well as in RLR prediction which will be stated in detail in next sec-tion.
The second part is the online algorithm which predicts hazardous situations using advanced
point detector outputs and the parameters extracted from the offline procedure.
Discrete sensors
outputs
{ t k , v k , d k }
Tracking ( building
trajectories) and
State Map
Trajectories and car-following
status of Subject
vehicles + trajectories of
other vehicles
Signal phase
and timing
Offline Parameter setting algorithm
RLR Hazard
prediction,
PrET < threshold?
Modeling
parameters
Signal
controller
All- red interface
extension
Optimization
of placement
of sensors
Empirical
distribution
of entry time
Stop- go
prediction
parameters
Figure 1: Dynamic All Red Extension Algorithm
12
3 Modeling and Neyman- Pearson Detection of RLR Hazard
3.1 Modeling of RLR Hazard
To build the probabilistic model of RLR hazard, we hereby define several quantities that are
closely related to RLR and its hazards.
Definition 1 RLR hazard.
A RLR hazard is an event when, according to empirical statistics, the probability of PrETRLR < d 0
( d 0 is a given threshold) is greater than some allowed minimum value Pmin. A commonly used
threshold for ET / PET in safety assessment is d 0 = 1 second.
Definition 2 Clear time of the violator tc and entry time of the conflict vehicle te.
To avoid the needs to define details of vehicle trajectories for different intersection geometry, we
instead define clear time tc and entry time te which can be directly measured given the collision
course ( and thus the conflict zone). tc is the time for the violator to leave the conflict zone and te is
the time when the POV to enter the conflict zone. We define the time into red of the violator to be
t , then we have that ( tc and te are referenced to red onset)
t = tc 􀀀 trc ; ( 3)
where trc is the time duration for the violator to travel from the stop bar to clear the conflict zone.
te has an empirical probability distribution (“ Pr” stands for the probability of an event)
Fte ( x ) = Pr f te < x g ; ( 4)
which can be estimated using histogram of historical data.
The condition in ( 1), given tc known from the speed and distance to conflict zone of the
violator, can be expressed that
Pr f PrETRLR < d 0 j tc g
= Pr f te 􀀀 tc < d 0 j tc g
= Pr f te < tc + d 0 j tc g
= Fte ( tc + d 0 ) : ( 5)
The above probability being less than Pmin is equivalent to
Fte ( tc + d 0 )  Pmin

 t + trc + d 0  F 􀀀 1
te ( Pmin )

 t  F 􀀀 1
te ( Pmin ) 􀀀 trc 􀀀 d 0 : ( 6)
To simplify the evaluation of time- into- red threshold in ( 6), we use the average speed of RLR for
a given approach to get trc,
trc =
dc
¯ vRLR
; ( 7)
13
0 1 2 3 4 5 6 7 8
0
0.05
0.1
0.15
0.2
time ( s)
Normalized Frequency
entry time ( All red duration + offset to green onset)
0 1 2 3 4 5 6 7 8
0
20
40
60
80
100
time ( s)
Percentile (%) F t
e
( t)
Figure 2: Example of entry time distribution
where dc is the distance for the violator to clear the conflict zone from the stop bar and ¯ vRLR is the
average speed of RLR.
In Figure 3, the time space diagram of a conflict zone for which the calculation of the predic-tive
encroachment time using the probabilistic distribution of the entry time and a trajectory of a
RLR is illustrated.
Based on ( 6), we quantify the RLR hazard using the time into red of the violator. Therefore
the essential part of the algorithm will be the prediction of RLR, which is in turn a combination of
the prediction of the vehicle’s stop- go maneuver and the estimation of its arrival time at the conflict
zone.
For RLR hazard prediction, we are only interested in two types of maneuvers : the ” first- to-stop”
and ” go through” ( actually not all the going through vehicles, but those pass the point sensors
during a certain period, say from yellow onset to the first few seconds after red onset). The first-to-
stop vehicles are the ones at each lane that stopped first after the yellow onset. This maneuver
are not driver- based taxonomy. They are coupled with observation of only the vehicles, therefore
the need to define more microscopically driver- related information is obviated.
For each possible collision course, we place point detectors along the track of going through
vehicles to collect data for RLR hazard prediction and along the track of the conflict vehicles ( both
SCP and LTAP- OD) to collect data for empirical distribution of entry time. In the point detectors
along the going through track, we include one at the stop bar and at least one to monitor the
clearance of conflict zone. While for modeling and prediction purposes, we picked up M out of
these ( for the model developed in this paper, M = 2), at distances
f dk1 and dk2 ; dk1 > dk2 g
14
Distance to intersection
Time ( sec)
Stop bar
RLR
Yellow Red onset
Conflict zone
t inred t c
t e
t rc
PrET RLR
Figure 3: Time Space Diagram of a Conflict Zone and the Calculation of its PrET
Figure 4: Discrete point sensors
from the intersection. The detected speeds at these locations are
f v ( k1 ) ; v ( k2 ) g ; ( 8)
and the time instant ( relative to red onset) when these measurements are obtained
f t ( k1 ) ; t ( k2 ) g : ( 9)
15
The time stamps in ( 9) are not independent, assuming that vehicles move at constant acceler-ation
between the detectors. In this case, the data are highly dependent. Thus only the latest data
point t ( k2 ) will be used.
The arrival time at the stop bar estimated from the last detector is then
ˆ tI ( k2 ) = t ( k2 ) + ˆ t2I ( 10)
where ˆ t2I is the estimated travel time from current location to intersection.
First, we present a probabilistic model of the detection problem for the two types of scenarios.
The observations we have are
f v ( k1 ) ; v ( k2 ) ; ˆ tI ( k2 ) g : ( 11)
We have two groups of hypotheses for the go- through ( g) and first- to- stop ( s). Among the go-through
vehicles, the RLR and go- through- in- yellow vehicles only differ in their arrival time at the
intersection, or by definition:
tI > 0 ; RLR ;
tI  0 ; through in Yellow :
( 12)
We define RLR hazard as arriving at an intersection at a time t seconds after red onset, where
t is obtained using ( 3), thus
tI > t ; RLR hazard ;
tI  t ; non- hazard :
( 13)
To further compact the variables used in the model, we define
¯ v =
1
2
( v ( k1 ) + v ( k2 ) ) ;
a ( k2 ) =
v ( k2 ) 􀀀 v ( k1 )
t ( k2 ) 􀀀 t ( k1 )
;
( 14)
the two- dimensional observation
X , f x1 = a ( k2 ) ; x2 = ¯ v g T ; ( 15)
and the clear time tc.
Because many random factors contribute, X can be reasonably hypothesized to be a bi- variate
normal random vector. Thus to formulate the hypotheses, we have
Hs : X  N ( Us ; S s )
Hg : X  N ( Ug ; S g )
( 16)
where Hs and Hg are hypotheses for ” stopping” and ” going”, Us, S s, Ug, S g are the mean and
covariance matrices of X under the two hypotheses, respectively.
16
3.2 Predictor of RLR Hazard and Two Types of Errors
To derive the prediction method for RLR, we generate several definitions regarding RLR
errors.
Definition 3 Predictor of vehicle maneuvers.
A detector which predicts the vehicle maneuvers gives a binary decision between the hypotheses
” stop” or ” go”, or
D ( X ) =  Hs or Hg 	 : ( 17)
which distinguishes two hypotheses: Hs and Hg using X which includes the average speed and
speed difference ( acceleration) of the violator. The detection would present two types of errors,
the probabilities of which are respectively
false alarm: PFA ( g j s ) , P  D ( X ) = Hg j Hs 	 ; ( 18)
and missing: PM ( s j g ) , P  D ( X ) = Hs j Hg 	 : ( 19)
Definition 4 Time to stop bar estimator.
We need to estimate the arrival time at the stop bar when the vehicle passes through the last discrete
sensor at d ( k2 ) . To minimize the chance of missing any RLR hazard due to an improperly estimated
arrival time, we need to construct an estimator ˆ tI ( k2 ) which satisfies the following condition that
Pr  ˆ tI ( k2 )  t j tI > t ; Hg 	 ' 0 : ( 20)
Definition 5 Detector ( predictor) of RLR Hazard.
We denote the detector of RLR hazard as DH ( X ) . It can be defined as
DH ( X ) = ( 1 ; if D ( X ) = Hg and ˆ tI ( k2 ) > t ;
0 ; otherwise :
( 21)
Based on the results of stop- go detector D ( X ) , there are two possible situations when DH misses
a hazard, when a hazard is missed by D ( X ) or when, although D ( X ) reports correctly, the arrival
time estimator misses the hazard. Thus, we can formulate the missing probability as
PM- H , Pr  D ( X ) = Hs j Hg ; tI > t 	 + Pr  D ( X ) = Hg ; ˆ tI ( kM )  t j Hg ; tI > t 	 :
( 22)
The false alarm of RLR hazard happens either when the arrival time predictor falsely indicates
a through vehicle before t seconds into red as a hazard or when the D ( X ) falsely treats a stopping
vehicle as not to stop.
PFA- H , Pr  D ( X ) = Hg ; ˆ tI ( kM ) > t ; tI  t ; Hg 	 + Pr  D ( X ) = Hg ; ˆ tI ( kM ) > t ; Hs 	 :
( 23)
17
Table 1: Numerical examples of missing rate and false alarm rate
Samples Hazard Missed PM- H false re-ports
PFA- H
Hg 600 10 1
10%
5
1% Hs 400 / / 5
It is a noteworthy fact that definitions of probabilities of missing and false alarm in ( 22) and
( 23) are different, the first one being defined using conditional probability while the other one
using un- conditional probability. A numerical example should explains why. For the example in
Table 1, we missed one hazard out of 10 total hazards and 1000 total samples of interest, in which
case the missing rate should be 10% which is the ratio conditioned on the number of hazards only.
While for false alarm rate, since all the samples ( except for hazards) could be dedicated for it, it is
calculated over the whole sample size then.
3.3 Neyman- Pearson Detection of RLR Hazard
It is our wish to quantitatively evaluate the DARE system in terms of the error performance.
Neyman- Pearson criterion is well fitted into this need. It says that we should construct the decision
rule to have maximum probability of correct detection while not allowing the probability of false
alarm to exceed a certain value. In other words, we want to have a system performs in the following
way
max ( 1 􀀀 PM- H ) ;
such that PFA- H  q ;
( 24)
where q is a preset threshold of allowable false alarm rate.
Then the problem turns into how can we evaluate the error probabilities in ( 22) and ( 23)?
The direct evaluation of them, however, are difficult due to their multi- variate nature. To simplify
the problem, we use approximate expressions to the error probabilities and link them to PM ( g j s )
and PFA ( s j g ) , which under the two- dimensional normal model are easier to evaluate.
3.3.1 Approximation of PM- H
There are two terms in its definition ( 22). The second part, while with the assumption ( 20)
that our arrival time estimator should avoid such missing to happen, can be neglected. Also since
our stop- go model depends only on the speed ( ¯ v) and acceleration ( a ( k2 ) ), it can be reasonably
assumed that D ( X ) = Hs is independently distributed to the arrival time for our interested samples
( those arrive late in yellow or early in red). Therefore, we have that
PM- H ' Pr  D ( X ) = Hs j Hg ; tI > t 	 ' Pr  D ( X ) = Hs j Hg 	 = PM ( s j g ) : ( 25)
18
3.3.2 Approximation of PFA- H
PFA- H =
Pr  D ( X ) = Hg | ; t ˆ I ( { k z 2 ) > t ; tI  t ; Hg 	 } ( 1 )
+
Pr |  D ( X ) = Hg { ; z t ˆ I ( k2 ) > t ; Hs 	 } ( 2 )
;
where
( 1 ) ' Pr  D ( X ) = Hg ; Hg 	  Pr f ˆ tI ( k2 ) > t ; tI  t g
= Pr  D ( X ) = Hg j Hg 	  Pr  Hg 	  Pr f ˆ tI ( k2 ) > t ; tI  t g
= ( 1 􀀀 PM ( s j g ) )  Pr  Hg 	  Pr f ˆ tI ( k2 ) > t ; tI  t g :
approximately we have that
( 1 ) ' ( 1 􀀀 PM ( s j g ) )  Pr  Hg 	  Pc ;
where
Pc , Pr f ˆ tI ( k2 ) > t ; tI  t g : ( 26)
We also have that
( 2 ) ' PFA ( s j g )  P ( Hs ) :
Now we have the simplified form
PFA- H ' PFA ( g j s )  P ( Hs ) + ( 1 􀀀 PM ( s j g ) )  Pr  Hg 	  Pc : ( 27)
Now that the performance of the DARE system is linked to the performance of its core al-gorithm:
the stop- go maneuver predictor via the a priori probabilities of Pc, Pr f Hg g and Pr f Hs g which can all be estimated from empirical data. The following diagram shows how the DARE is
optimized using the Neyman- Pearson criterion.
Hazard Detection: PFA- H = q
q = b  P ( Hs ) + ( 1 􀀀 PM ( s j g ) )  Pr f Hg g  Pc
  􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 minPM- H
x ? ?
PM- H ' PM ( s j g )
Stop- go Detection: PFA ( s j g ) = b minimizingPM ( s j g )
􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 ! minPM ( s j g )
19
4 The Implementation of the RLR Hazard Prediction Algorithm
As has been illustrated in Figure 1, the DARE algorithm consists of two parts: the online
prediction and the offline parameter setting.
 Offline parameter setting: We need to collect ( huge amount of) data to get the parameters
in the model. These parameters include the threshold of time- into- red of the violator, t for
each collision course, the a priori probabilities of Stop- Go maneuvers during the observation
window ( P ( Hg, P ( Hs ) ) and also Pc. Then offline optimization should be carried out the
obtain the boundary of stop- go maneuver prediction ( v0 ; a0 ) .
 Online RLR hazard prediction: Online procedure is simple and straightforward. All- red
extension is triggered when a vehicle is predicted to run red light with time into red greater
than given threshold for the most likely collision course.
4.1 Offline procedure of DARE algorithm
4.1.1 Estimating the parameters for prediction
When estimating the algorithm parameters, not only the sensor data at d ( k1 ) ; d ( k2 ) are used
( or X), but also the whole trajectories from the vehicle approaching the entering, entering and
exiting the intersection are used. Note the whole set of sensor outputs of an interesting trajectory as
Zn, where Xn is the X vector ( speed and acceleration) extracted from Zn. We also need trajectories
of conflict traffic f Ym g .
Assume the size of set f Xn j Hg g is Ng and f Xn j Hs g is Ns, respectively. The mean and variance
matrices in Algorithm 1 can be estimated, using
U ˆ g =
1
Ng å n
( Xn j Hg ) ; ( 28)
and
ˆ g =
1
Ng 􀀀 1 å n ( Xn j Hg ) ( XTn
j Hg ) : ( 29)
For Xn j Hs, the estimator is similar.
4.1.2 Forming the Neyman- Pearson Detection Boundary
We define P f X j Hg g and P f X j Hg g as the probability density functions ( PDF) of X under
hypotheses Hs and Hg, respectively ( see ( 16)), b as the false alarm rate for D ( X ) . To form the
Neyman- Pearson detector which minimizes the missing probability while keeping the false alarm
rate not greater than b , the following constrained optimization problem need to be solved:
max  Z X 2 Rf
P f X j Hg g dX  ;
given any Rf, such that Z X 2 Rf
P f X j Hs g dX < b :
( 30)
20
Algorithm 1: Estimating parameters for prediction
input: A set of interesting trajectories f Zn ; 1  n  N g of size N ;
A set of trajectories f Ym ; 1  m  M g from conflict path ;
foreach Zn do
if Zn is RLR then
Xn 2 RLR ;
end
end
Calculate the average speed of RLR ¯ vRLR;
Get the histogram of entry time te using all first- to- enter in green from Yn ;
Get Fte ( x ) from the histogram ;
For SCP and LTAP- OD, use ( 6) to calculate t for each conflict course and choose the
minimum one corresponding to each scenario ;
foreach Zn do
if Zn is going through then
Xn 2 Hg ;
calculate ˆ tI using Xn ;
if ˆ tI > t and tI < t then
Xn 2 f ˆ tI > t ; tI < t g ;
end
else
Xn is first- to- stop ;
end
end
We have the sets f Xn j Hg g ; f Xn j Hs g ;
Estimate P ( Hg ) and P ( Hs ) using their frequency respectively ;
Estimate Pc using the frequency of Xn 2 f ˆ tI > t ; tI < t g ;
Estimate the mean and covariance matrices for hypothesis Hg and Hs ;
Result: t , ˆ P ( Hg ) , ˆ P
( Hs ) , ˆ P
c and N ( U ˆ g ; S ˆ g ) , N ( U ˆ s ; S ˆ s )
21
Given a false alarm rate b , we need to search the observation space to find the solution to
( 30). For the multi- variate detection problem here, there exist various kinds of reasonable deci-sion
boundaries, which usually requires a computationally prohibitive search to find the optimal
solution. For simplicity and clear physical interpretation, we adopt the rectangular boundary
¯ v > v0 ;
a ( k2 ) > a0 ;
( 31)
when we find a sub- optimal solution to the multi- variate optimization problem.
( v( k
2
) − v( k
1
)) /( t( k
2
)− t( k
1
)) ( m/ s2)
( v( k
2
) + v( k
1
))/ 2
¬ H
g
¬ H
s
v
0
a
0
− 6 − 4 − 2 0 2 4 6
5
10
15
20
25
Figure 5: Bi- variate Normal model of the two hypotheses
The Neyman- Pearson detection ( 30) can be formed by finding the boundary variable f v0 ; a0 g according to the given false alarm rate q . That is, to solve the following problem:
f v0 ; a0 g = arg max
¯ vmin < ¯ v < ¯ vmax
amin < a ( k2 ) < amax Z ¯ v > x1
a ( k2 ) > x2
Pr  X j Hg 	 dX ; ( 32)
where
for all x1 2 ( ¯ vmin ; ¯ vmax ) , x2 2 ( amin ; amax ) such that Z ¯ v > x1
a ( k2 ) > x2
Pr f X j Hs g dX  b ; ( 33)
where b is the false alarm rate set according to q such that
q = b  P ( Hs ) + ( 1 􀀀 PM ( s j g ) )  Pr  Hg 	  Pc : ( 34)
To sum up, we have the following algorithm.
22
Algorithm 2: Forming the Neyman- Pearson detection boundary
input: Given maximum false alarm rate q 0 and the parameters obtained in Algorithm 1.
for 0 < b < 1 do
foreach a in ( amin ; amax ) and ¯ v in ( ¯ vmin ; ¯ vmax ) do
Calculate PFA ( g j s ) and PM ( s j g ) using estimated mean and variance ;
if PFA ( g j s )  b then
find the ( a0 ; ¯ v0 ) pair that minimizes PM ( s j g ) ;
end
end
Convert false alarm rate of hazard detection b to q of stop- go predictor using ( 34);
Find the b that makes q ( b ) = q 0 ;
end
Result: ( a0 ; ¯ v0 ) that minimizes PM- H and also keeps PFA- H  q 0.
4.2 Online Prediction of RLR Hazard
4.2.1 Arrival time ( time- into- red) estimator
The arrival time estimator ˆ tI ( k2 ) need to meet the assumption in ( 20). To incorporate the
measured acceleration a ( k2 ) , we adopt an estimator in the following form
ˆ tI ( k2 ) , t ( k2 ) +
d ( k2 )
v ( k2 )
+ r [ a ( k2 ) 􀀀 ¯ a ( k2 ) ] ; ( 35)
where ¯ a ( k2 ) is the average of measured acceleration of the RLR samples collected, r < 0 is a
constant coefficient which rejects some vehicles make to go- through legally by accelerating after
yellow onset, while j r j should be kept small and fine tuned to meet requirement ( 20). Empirical
data show that r = 􀀀 0 : 05s3 = m is a good option ( a is in m = s2 and t in seconds).
4.2.2 Formulation of DH ( X )
Having obtained the optimized decision boundary and arrival time estimator, DH ( X ) can be
formulated as
DH ( X ) = 1 ; if a ( k2 ) > a0 and ¯ v ( k2 ) > v0 and ˆ tI ( k2 ) > t
= 0 ; otherwise :
( 36)
4.3 Performance of the RLR Predictor
The probability of detection ( defined as PD , 1 􀀀 PM) as a function of probability of false
alarm, is frequently used to quantify the performance of detection systems. With radar systems,
where Neyman- Pearson criteria has been extensively used, this function is called the receiver oper-ating
characteristics ( ROC). For an intersection collision avoidance system, we will define a similar
23
system operating characteristics ( SOC) function as
SOC ( q ) , maxPD ( DH ( X ) ) ;
where PFA- H ( DH ( X ) )  q :
( 37)
5 Data Collection and Processing
5.1 Field Setup and Configuration of Point Detectors
Field data are collected using the Autoscope cameras. Nine cameras are installed at three
approaches of an arterial intersection ( Page Mill Road and El Camino Real, Caltrans State Route
82) in the San Francisco Bay Are. We note that the signal phase and timing setting of this intersec-tion
fully conforms to the ITE standard. Three cameras are installed for each approach, covering
respectively the exit area, stop area and advance area. We summarize the configurations into Table
2.
Table 2: Configuration of the field intersection
Page Mill rd. at El Camino Real
East bound West bound South bound
Lanes ( recorded) 2 1 2
Speed Limit ( mph) 30 35 35
Average Speed
( mph)
30 35 35
90% speed less than
( mph)
45 45 40
Yellow Duration ( s) 4 4 4
All- red Interval ( s) 0.5
# Speed Loops
( Exit)/ Lane
1 1 1
# Speed Loops ( ad-vance)
/ Lane
3 3 3
# Speed Loops ( stop
/ conflict entry) /
Lane
3 3 2
95% RLR time- into-red
less than ( s)
2.5s 2.5s 2.5s
Description of the
approach
0 : 15 Mile from
upstream inter-section
No upstream
intersection for
More than 0 : 7
mile upstream
0 : 25 mile from
upstream inter-section,
main
street
As can be seen in Table 2, multiple virtual speed loops are placed at advance area of difference
approaches to monitor the violation and also at stop area to get the conflict entry time behaviors.
24
5.1.1 PrETRLR and t
When there is a signal violator who is going through the intersection, there are possibly mul-tiple
conflict courses. For example, if the RLR is going South bound, first vehicle from each lane
to East bound are all in SCP collision course. Since we are dealing with RLR happening in the first
few seconds in red, the vehicle entering from outmost lane ( furthest from runner) is more likely to
be involved in a hazard.
For each approach, we obtain the empirical distribution of entry time Fte ( x ) ( see example of
South Bound in Figure 2). t is then calculated for each approach using the minimum PrETRLR
among different possible courses. Results are shown in Table 3.
Table 3: t for each approach
Page Mill rd. at El Camino Real 􀀀 t
East bound
( LTAP- OD)
West bound
( SCP)
South bound
( SCP)
Conflict Approach WB SB EB
dc ( feet) 96 110 120
entry distance to
conflict ( feet)
15 15 30
All red interval 0.5s
PrETRLR threshold 1s
F 􀀀 1
te ( Pmin = 0 : 3 ) 5.2s 4.0s 4.2s
t ( Pmin = 0 : 3 ) 1.5s 0.4s 0.5s
5.1.2 Minimum sensing distance requirement for DARE system
We can expect that the closer the distances of the virtual loops for prediction ( namely d ( k1 )
and d ( k2 ) ) are, the higher the prediction accuracy will be. However, there is also a critical require-ment
that when a violator passes the last virtual loop, the system should be able to react ( hold
all- red), most of the time. This requires us to put the sensors far enough to cover most of the RLR,
from which the minimum distance requirement for DARE system is formed as:
the last sensor should be far enough for over 95% of the violator ( assumed to move at speed
limit) to pass it before the termination of all- red interval.
There are always some RLR cases when the violator runs much late into red when there is no
way to hold all- red for that long time. We try to cover most RLR during the first few seconds after
red onset, to include the willful violator and inattentive platoon followers.
Results of the configurations of the virtual loops at the field intersection are summarized in
Table 4. We note that “ distance” indicates the distance of sensor from the stop bar to upstream
direction ( in feet). When less than zero, it indicates a sensor inside the intersection.
25
Table 4: Configurations of the Autoscope virtual loops
Page Mill rd. at El Camino Real : Sensor Configurations
East bound West bound South bound
Average Speed of
RLR
30 mph 35 mph 38 mph
Min distance re-quirement
90 feet 110 ft 110 ft
d ( k1 ) and d ( k2 ) 97,120 139,190 120,195
Locations of All
Speed Loops
- 110, - 15, 0, 15,
40, 63, 97, 120,
190
- 110, - 15, 0, 20,
40, 95, 139, 190
- 120, - 30, 0, 30,
90, 120, 195
5.2 Data Processing
5.2.1 Calibration of the Sensors
The speed loops of Autoscope records the speed when a vehicle arrives at the specific location.
To verify the accuracy of the engineering data generated from the Autoscope processors, we drove
a fully- instrumented vehicle with GPS and data recorder at the test site for two days. The calibrated
results are compared to the wheel speed, showing that the speed errors are usually less than 10%.
To validate the measurements from Autoscope, part of the RLR data are also compared to the
recorded video clips. The comparison shows that the engineering data of Autoscope are consistent
with the raw video data.
5.2.2 Building Trajectories From Point Detectors
To associate the data of discrete point sensors to a vehicle, multiple- hypotheses tracking
( MHT) algorithm developed by Reed is employed [ 16]. MHT is a widely used tracking method
which basically assumes that each discrete measurement could possibly be assigned to a previous
found target ( vehicle) with an a priori probability. Whenever a new measurement is reported by
the sensor, it can be associated with ( assigned to) any previous targets ( vehicles), thus forming a
set of new hypotheses. The MHT algorithm then calculates the evolving probability of each new
hypothesis. Due to the exponential nature of this propagation, usually only the several most likely
hypotheses are kept for further calculation, and finally ( for example, when measurement from exit
area has been associated to some track), hypothesis with highest probability is accepted.
6 Simulation
Computer simulation is carried out to verify the RLR hazard prediction algorithm. Data
collected from the field are used to generate the parameters for each different approach of the site.
Additional data are collected to verify the algorithm.
26
6.1 Simulation Parameters
Data are collected during August to September 2008. The parameters of the algorithm are
obtained using Algorithm 1 and Algorithm 2.
Table 5: Prediction algorithm parameters
Page Mill rd. at El Camino Real
RLR Hazard Occur-rence
12 21 112
Hs Occurrences 912 462 2323
Hg Occurrences 4318 2725 11359
Pc 0.015 0.015 0.018
ˆ P
( Hs ) 0.17 0.14 0.17
ˆ S g  4 : 7 1 : 6
1 : 6 10 : 2   2 : 1 0 : 1
0 : 1 5 : 6   4 : 8 1 : 4
1 : 4 13 : 0 
U ˆ g  0 : 1
15   0 : 5
13 : 2   1 : 4
15 : 2 
ˆ S s  4 : 2 􀀀 1 : 5
􀀀 1 : 5 5 : 2   4 : 1 􀀀 1 : 6
􀀀 1 : 6 5 : 3   4 : 9 􀀀 0 : 1
􀀀 0 : 1 6 : 5 
U ˆ s  􀀀 1 : 9
13   􀀀 1 : 4
12 : 4   􀀀 1 : 1
13 : 3 
6.2 Simulated System Performance
Data collected during October 2008 are used to verify the algorithm. Simulation results using
collected field data are compared to the expected results from the two- dimensional normal model.
6.2.1 Simulation procedure
RLR hazard prediction is simulated using parameters which are generated from Algorithm
1 and Algorithm 2. The data used to generate these model parameters are different than the data
used for simulation, which is the intention of the simulation to also verify the consistency of the
model with the time. The parameters setting procedure generated results are tabulated in Table
5. The RLR hazard parameters, including the t for each approach is summarized in Table 3. The
simulation procedure is described in Algorithm 3.
6.2.2 Simulation results
The actual SOC curves for the field intersection show that the simulated online performance of
prediction algorithm ( Figure 6), which is in good match with the theoretical SOC curves calculated
using the model parameters. It is also noteworthy that the differences in the statistics for different
travel directions lead to different system performance. This difference is most likely caused by
variations in traffic with different approaches.
27
Algorithm 3: Simulation procedure
input: Model parameters listed in Table 5 and Table 3. Training data set ( used to get the
parameters) ;
New data set ( other than the data used for parameter setting) ;
for q = 0 : : : 1 do
Use Algorithm 2 to Find the Neyman- Pearson Detection boundary f ¯ v0 ; a ( k2 ) 0 g with
training data set ;
Form DH ( X ) using f ¯ v0 ; a ( k2 ) 0 g and t from Table 3 ;
foreach Data sample Xn from new data set do
if Xn is hazard and DH ( X ) = 0 then
This is a missing report;
end
if Xn is not hazard and DH ( X ) = 1 then
This is a false alarm ;
end
end
Record the simulated false alarm rate and missing rate ;
Use the two dimensional normal model to calculate ( using numerical integration) the
theoretical missing rate given q ;
end
Result: Simulated SOC curve and the theoretical SOC curve.
28
6.2.3 Simulated system performance with different hazard settings
All the previous simulation results are obtained using one set of threshold for PrETRLR ( one
second) and Pmin = 0 : 3. If these settings are changed, we need to go through the offline parameter
setting procedure to get new parameters for the system. Either increasing the threshold Pmin or
decreasing d 0 will increase the time- into- red threshold t for RLR hazard, which will in turn reduce
the rate when DARE triggering AND the false alarm rate as well ( since the arrival time has less
chance to make a mistake). Through simulation, we also verified this trend. In Figure 7, we plot the
simulation results ( simulated SOC curves) for South Bound with various time- into- red thresholds
( t ), from 0 : 3s to 1 : 5s. When t becomes bigger, the number of RLR hazard occurence ( thus the rate
that DARE system is triggered) is reduced drastically. And the improvement that the false alarm
rate reduces with the increasing of t is also observed.
7 Conclusion
Probabilistic model and prediction algorithm have been developed for a dynamic all- red
extension system, to counteract the intersection safety hazard caused by RLR. We defined the
PrETRLR which is the time lapse between the time for the violator to clear the conflict zone and
the a priori entry time of the first- to- enter vehicle from conflict approach. The hazard, which is
determined by the intersection layout, empirical entry time of conflict traffic and speed of violator,
is linked to an important characteristic of RLR: the time into red of the violator. We then developed
a model using a minimum set of two point sensors to identify the RLR hazard. The prediction of
the hazard is governed by the Neyman- Pearson criterion which gives the algorithm the capability
of distinguishing two types of errors, namely the missing report and false alarm, and balance of
their trade- off.
Field data collection is carried out to study the driver behavior related to RLR hazard and to set
the parameters of the developed model / algorithm. The collected field data also serve the purpose
of verifying the algorithm using simulation. System performance, measured by the trade- off of the
two types of prediction errors, is calculated using field data and compared to the expected results
of the model. A good match is found from this simulation which thereby verified the validity of
the probabilistic model.
Performance of the proposed model and its prediction algorithm is evaluated using data col-lected
from a field intersection. At a false alarm rate of 5%, the proposed algorithm reached correct
detection rates of over 65% for West bound, 80% for South bound and over 90% for East bound,
respectively, at the test intersection. The false alarm rate of 5% at the test intersection is equivalent
to on average one false triggering per more than 3 hours. Performance evaluation results showed
that the proposed DARE framework can effectively detect the RLR hazards.
There are also restrictions of employing DARE to address the RLR collision avoidance prob-lem,
mainly due to the concern that drivers may adapt to the system and the benefits of the system
may be undone. We believe that via improving the hazard detection algorithm we could keep the
triggering rate of DARE to a very low level when only really “ dangerous” hazard will trigger the
system thus minimize the influence on driver behaviors. Also the combination of DARE with
enforcement such as Red- Light running Camera ( RLC) could also be a good intersection safety
29
solution.
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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SOC Curve for RLR prediction Page Mill Rd @ El Camino Real, East Bound
P
fa
Probability of False alarm for RLR Hazard
P
d
Probability of correct detection for RLR Hazard
Simulated SOC curve with field data
Theoretical SOC curve based on empirical statistics
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SOC Curve for RLR prediction Page Mill Rd @ El Camino Real, West Bound
P
fa
Probability of False alarm for RLR Hazard
P
d
Probability of correct detection for RLR Hazard
Simulated SOC curve with field data
Theoretical SOC curve based on empirical statistics
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SOC Curve for RLR prediction Page Mill Rd @ El Camino Real, South Bound
P
fa
Probability of False alarm for RLR Hazard
P
d
Probability of correct detection for RLR Hazard
Simulated SOC curve with field data
Theoretical SOC curve based on empirical statistics
Figure 6: SOC Curves for RLR Prediction, Using Field Data
32
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P
fa
Probability of False alarm for RLR Hazard
P
d
Probability of correct detection for RLR Hazard
SOC ( t = 0.3s)
SOC ( t = 0.5s)
SOC ( t = 0.8s)
SOC ( t = 1.5s)
Figure 7: SOC Curves For Different Thresholds ( t )
33