Abstract
The effects of
traffic incidents on metropolitan freeways extend beyond causing congestion and
delays. Immediate impacts include
decreased productivity, increased pollution and reduced safety on
highways. State and local governments
spend billions of dollars annually on construction projects and Intelligent
Transportation Systems (ITS) in an effort to curb the adverse consequences of
incidents and incident related delays.
Effective
identification and response on highways is one key to reducing the costs
associated with traffic incidents.
Within the context of a prototype incident identification and response
system developed by the University of Virginia’s Systems Technology Integration
Laboratory (STIL), this research aims to develop a statistical approach to
modeling congestion associated with freeway incidents. The ability to predict congestion will
provide more information and greater situational awareness to emergency
responders and traffic managers, and will allow travelers to make more informed
route selection decisions.
By combining data
from multiple sources, it is possible to match incident severity estimates for
freeway incidents with associated traffic flow counts at the time of the
accident. Four metrics for freeway
congestion were derived from the traffic flow data, and these metrics were then
modeled as functions of the incident severity estimates. The results of multiple linear regression
analysis showed that quantifiable relationships exist between the congestion
metrics and incident severity data such as the number of vehicles involved in
an incident as well as the number of serious injuries reported at the
scene.
Introduction
Problem Statement
Situational
awareness in the critical early stages of a traffic incident is the key to
alleviating a great deal of the congestion that plagues America’s major
metropolitan regions (VDOT). Being able
to detect, respond to and clear an incident faster and more efficiently can
reduce congestion related delays from traffic accidents by up to 45% according
to the Virginia Department of Transportation (VDOT). Within the scope of a comprehensive incident detection and
response system, this research aims to model freeway traffic flow metrics by
using measures for incident severity.
Developed by the
University of Virginia’s Systems Technology Integration Laboratory (STIL), the
Intelligent Highway System (IHS) aims to automate and streamline the process of
incident detection and response. By
leveraging the increased sensing and communications capabilities of motor
vehicles, the IHS is able to automatically detect car accidents and instantly
broadcast pertinent data to appropriate authorities in order to begin the
process of response.
One of the goals
of the IHS is to be able to predict the traffic implications of an
incident. An incident is any event that
temporarily reduces roadway capacity, such as accidents, debris, disabled
vehicles, and hazardous material spills (HCM).
Because this work is part of a larger context of an emergency response
system, this study will focus on accidents only.
It is intuitively
clear that the congestion due to an incident that occupies two of three freeway
lanes will cause more congestion than an incident that occupies only one (Smith
and Qin 362). What is not clear is the
extent to which congestion due to an incident is also a function of the
severity of the incident. Severity can
be defined either as the damage done to the vehicles themselves, or as the
degree of injury sustained by the passengers involved in the accident. This research aims to model the relationships
between incident severity and the congestion associated with that
incident. Measures of incident severity
include vehicle damage, the number of vehicles involved, the number of injuries
and the severity of the injuries, among others.
A widely accepted
metric for measuring the effect of incidents on traffic flow is capacity
reduction (Smith and Qin). Capacity is
defined by the Transportation Research Board as the maximum hourly rate at
which persons or vehicles can reasonably be expected to transverse a point or
uniform section of a lane or roadway during a given time period under
prevailing roadway, traffic, and control conditions (TRB). In order to truly measure the capacity
reduction on a roadway, the roadway of interest must be at its capacity before
an incident occurs. For freeways, this
capacity is estimated to be 2200 vehicles per hour per lane (HCM). When the capacity of a roadway is reduced
below the demand on that roadway, perhaps in the event of an accident, the
volume counts decrease for the duration of incidents. This is a reflection of the decreased speed at which drivers are
able to move along the roadway. The
data set used for this research was limited in size and contained only a small
percentage of incidents where flow on the roadway was at or near capacity
before an incident, therefore alternative metrics for quantifying the effects
of an incident were developed.
The obvious
benefits of increased situational awareness are those of better route guidance
and more efficient and accurate re-routing of traffic via Variable Message
Signs on highways (Hobeika). The
information could also be disseminated over the internet to allow travelers to
make more informed decisions regarding their route selection based on predicted
traffic delays.
As described in
the Literature Review, it is apparent that much study has been done regarding
the prediction of traffic conditions following incidents. Significant research has also been conducted
to predict accident severity based on pertinent data. In the past, however, these two issues have been treated as
separate problems and not addressed together in one body of work. There has been little or no research to
bridge the gap between two areas of study that intuitively seem to be
connected. This research aims to close
that gap and determine whether including incident severity estimates is
valuable in the modeling of freeway congestion.
Intelligent Highway System (IHS)
The broader
context for the proposed prediction method is the Intelligent Highway System
(IHS). Developed as a prototype system,
the IHS aims to automate and streamline the process of incident identification
and response. The system leverages the
increased sensing and communications capabilities of new vehicles. With the addition of acceleration sensors, a
Global Positioning System (GPS) receiver and a modem connected to the cellular
network, a vehicle can become a source of useful data at the point of incident.
The
data that can be sent from the vehicle is limited only by the sensing
capabilities of that vehicle. In the
near future, automobiles will be equipped with a wide range of sensors
including seat belt sensors, video cameras and biometric sensors to determine
the condition of the passengers with ever increasing accuracy.
When
an incident occurs, and the accelerometers experience a g-force above a
pre-determined threshold, the vehicle’s computer recognizes an accident
event. This process is similar to
airbag control processes available in automobiles today (German). The computer then broadcasts relevant data
including GPS position and accelerometer outputs to an Emergency Center using
Hypercast, a Java based communications protocol. The Emergency Center can be thought of as an information broker
that broadcasts appropriate messages to appropriate players in the event of an
accident, also using Hypercast. The
Hypercast protocol’s flexibility in terms of enabling message transmission
between wired and wireless users is well documented (Hunter et. al), and was
the logical protocol choice for the application development.
One
of the players that receive data from the Emergency Center is the Crash
Simulation Lab. The Crash Lab runs a
finite element simulation, with the accelerometer data as input, to estimate
the severity of the injuries sustained by the passengers in the crash. The Crash lab then broadcasts the severity
estimates back to the Emergency Center and to area hospitals as well as the
Traffic Analysis Center. The Traffic
Analysis Center is where congestion metrics would be estimated. Currently, the predictive models that exist
do not use severity measures to estimate metrics for congestion
The
IHS is similar to automatic accident notification systems currently available
in vehicles manufactured by General Motors, among others, as a service
available through the manufacturer.
Incident response centers that are a part of the service are
automatically notified in the event of air bag deployment in a subscribing
vehicle, at which point emergency responders can be notified by service
staff. The IHS, however, is a fully
automated, distributed system, that can be modified and upgraded rapidly and
easily without any disruption to the system.
Individual localities can implement an IHS for a given jurisdiction and
stand alone from similar national or even state wide systems. The diagram below shows the flow of
information in the event of an incident.
Figure 1 Information Flow over the HIS
Literature
Review
Freeway Incident Management
When attempting to
improve the large scale system that includes the freeways of a metropolitan
area and its supporting emergency response agencies, it is important to address
certain issues relating to incident identification, response and management in
the current system. The proposed work
aims to model metrics for congestion in terms of incident severity, but the
interaction between the type of response and the capacity reduction/duration of
an incident must be noted. It is also
necessary to examine the emergency response perspective and traffic management
perspective and note their similarities and differences in the event of an
accident on a major freeway. Effective
freeway management can also lead to fewer incidents. According to the Federal Highway Administration’s Traffic
Management Handbook, 13 percent of all peak period crashes were a direct result
of a previous incident (FHWA).
There
is an interaction between the scale of emergency response to a situation and
the congestion as well as the duration of an incident. It can be argued that with all other factors
being equal, an incident with a larger emergency response will cause greater
congestion and result in longer delays than an incident with a smaller response
(Saunders). Additionally, an incident
with inadequate response will have a longer duration, or time to clear, than an
incident with appropriate response because of the lag in arrival of the right
equipment and personnel (Nathanail)..
Therefore, it is in the best interest of motorists that an appropriate
response to an incident is deployed.
The decisions regarding the type of response to an emergency can be made
based on better information with the implementation of a system such as the IHS
that would provide information from the scene such as injury severity and data
that could allow for immediate traffic analysis.
There
are two perspectives of an incident that have potentially competing
objectives. The Emergency response
perspective has the goal of minimizing the time it takes to respond to an
incident with the appropriate equipment and personnel, and the traffic
management perspective has the goal of restoring the normal traffic flow on the
affected roadway (Zograforos, 536).
These somewhat competing objectives have an effect on the scale of the
response and the time it takes to clear an incident, and therefore play a role
in determining the congestion associated with an incident.
The motivation for
improvements in response to freeway incidents is clear when we realize that approximately
650,000 Americans suffer serious injuries in vehicle crashes every year, yet
these injuries occur in less than one percent of all motor vehicle crashes
(Champion). Thus, it is important to be
able to identify the crashes that have the potential for causing life
threatening injuries. By accurately
identifying the small percentage of accidents that may cause serious injuries,
better resource allocation can be achieved, increasing the likelihood of timely
response. It has been established that
the first 60 minutes following an incident, or the “golden hour”, determines
whether a patient will live or die (Champion).
Therefore, quantifiable benefits can be achieved by reducing response
time.
There are three
key stages that make up the response time of a vehicular accident (Evanco):
·
Accident notification time, or the time between the
crash and emergency medical service (EMS) notification. This includes detection time
·
The time between EMS notification and EMS arrival at
the scene of the crash
·
The time between EMS arrival at the scene and arrival
at the hospital
In a statistical
model based study, Evanco et. al . found that the average notification time was
5.2 minutes and an average of 43.8 deaths per year per state in the year
1990. The study examined the effects of
variables such as Vehicle Miles Traveled (VMT), Mean Vehicle Speed (MVS) and
alcohol consumption. The conclusions of
the work emphasize the significant impact of reducing the notification time after
an accident, noting that a reduction in the average notification time 5.2
minutes to 3 minute would result in an 11% reduction in fatalities, and a
reduction to 2 minutes would result in a 15% reduction in fatalities
(Evanco). An automatic notification
system such as The Intelligent Highway System (IHS) could conceivably reduce
the notification time after an accident to a few seconds, resulting in an
immediate reduction in lives lost.
Event Data Recorders
A central
motivation for determining the impact of injury severity on freeway capacity is
the increasing availability of Event Data Recorders (EDRs) in many late model
cars and light trucks (German). EDRs
are the motor vehicle equivalent to the ‘black box’ found in airplanes. While the importance of black boxes in the
event of an aviation incident is well documented, the fact that their
capabilities extend to automobiles is not as widely known.
As
an adjunct to air bag sensing and control systems, EDRs offer a new source of
useful data that can further the understanding of on-road traffic safety
issues. Combined with the widespread
use of mobile communications capabilities, EDRs can provide immediate data for
analysis in the event of an accident.
Considerable roadblocks are present, however, that are slowing the path
to widespread use of EDRs. These
roadblocks include a lack of standardization of the data recorded, storage
formats and means of retrieval. Privacy
concerns must also be addressed before universal usage of EDRs becomes a
reality as well as legal and liability issues.
Widely
used in systems developed for vehicles in North America and Europe, EDRs are
made up of vehicle mounted accelerometers used to monitor the crash pulse of an
accident. An onboard microprocessor
analyzes the vehicle’s acceleration and determines air-bag deployment based on
pre-programmed decision logic. The EDR
makes possible the capture of relevant data for post-crash analysis. Recorded data includes vehicle speed, engine
RPM, throttle position, brake-switch status and seatbelt use. All of these parameters add valuable
insights into the potential for injuries in the event of an accident. The data can be sent for analysis
immediately and the information returned by the analysis can assist in
emergency response decisions and traffic planning strategies.
Post
accident, the data recorded by EDRs can be used by safety researchers, law
enforcement and insurance companies. If
widespread use is adopted, insurance companies will no doubt want access to the
data to assign fault and support legal actions. Such stakeholder interests raise questions of ownership,
accessibility and the use of data in individual cases for purposes other than
safety analysis.
Currently,
General Motors offers a proprietary GM Crash Data Retrieval (CDR) system. Available in GM vehicles, the device is used
to obtain data from vehicles involved in collisions. Use of EDRs is more widespread in research institutions where
data from staged collisions is collected by EDRs as part of crash testing
programs. The EDRs used currently record
change in velocity (ΔV) slightly – but consistently – higher than the
actual ΔV because of the rebounding velocity recorded by the device. The EDRs work very accurately when recording
the crash pulse, which is not affected by the rebound velocity (German).
The
increased availability of EDRs in commercial vehicles makes it necessary to be
able to analyze the data recorded by the devices. By modeling congestion as a function of incident severity, it
would be possible to predict the traffic implications of an accident that
involves a vehicle equipped with an EDR.
Injury Prediction
The
ability of Event Data Recorders (EDRs) to record deceleration in vehicles in
the event of a collision has been utilized to investigate the correlation
between pre-determined acceleration threshold limits for injury and the
potential for passenger injury in a crash (Gabauer). Models for determining injury severity, or the potential for
injury, however, are still imperfect at best and inadequate in the case of
side-swipe and other non-frontal collisions.
One method for determining the potential for injury is mapping peak
acceleration values in the three dimensions to the Acceleration Severity Index
(ASI) (Gabauer). The ASI combines the
three dimensional acceleration components to provide a single numerical value
that indicates the severity of an incident.
The ASI equation is described below (Gabauer):
where 
, 
and 
are the 50-ms average component accelerations, and 
, 
and 
are threshold accelerations for each component. In previous crash studies it has been
determined that ASI is generally indicative of occupant injury in frontal collisions
(Gabauer). There has also been a high
correlation between analysis of EDR results in calculating delta-V and other,
more traditional models for calculating the accepted metric for injury
severity.
While
the methods for analyzing data from EDRs have not yet been perfected, the
devices are capable of providing immediate data that has otherwise never been
available in the event of an accident.
This fact, combined with the increasing availability of EDRs in consumer
automobiles makes it important to be able to make use of the newly available
data.
Freeway Concepts
Freeways
are segments of road that provide uninterrupted flow (TRB). They have no intersections, and access to
and from the freeway is limited to ramps.
This study focuses on freeways only, as they are the least complicated
roadway facility to model when attempting to determine congestion. There are no interactions with
intersections, traffic signals, and other roadway conditions that exist on other
types of facilities. Freeway capacity is
described as “the maximum sustained 15-min flow rate, expressed in passenger
cars per hour per lane, that can be accommodated by a uniform freeway segment
under prevailing traffic and roadway conditions in one direction of flow” (TRB
13-1).
For
any given segment of roadway there is a capacity that can be determined. In the event of an accident or excess
demand, the capacity of a freeway segment is reduced temporarily due to the
blockage of lanes. According to the Highway Capacity Manual, an incident does
not have to block a travel lane to create a bottleneck, or temporarily reduce
capacity.
At
any given time, a freeway segment is experiencing one of three conditions, or
flow types. The segment is either
“unsaturated”, in a state of “queue discharge” or “oversaturated flow”. An unsaturated segment is unaffected by
upstream or downstream conditions, with speed ranges of 55 mph to 75 mph. The capacity of the highway is determined
under this condition because it allows for the maximum speed and flow
rate. The queue discharge regime is
characterized by traffic that has just passed through a bottleneck and is
accelerating back to its free-flow speed.
This acceleration generally takes 0.5 to 1 mile downstream from the
bottleneck.
Finally,
the oversaturated flow regime is one that is influenced by the effects of a
downstream bottleneck – a traffic accident, for example. The traffic flow in the congested regime can
vary based on flows and speeds depending on the characteristics of the
bottleneck itself, and queues can grow to several thousand feet upstream of the
bottleneck. These queues differ from
queues at intersections in that vehicles move slowly through the queue with
periods of stopping. The reduced
capacity of a freeway segment is calculated under the oversaturated flow
conditions. By comparing the capacity,
or vehicles per hour per lane under prevailing conditions, to the capacity
during an incident, it is possible to quantify the effect of a traffic incident
on the freeway system.
Ideally,
we would conduct such a comparison for this study as a primary metric. The limited data set that was available for
use, caused by the difficulty in accurately matching incident severity factors
with location data, prevented the use of capacity reduction as a metric. Instead, five other metrics associated with
traffic flow (v/h/l) and their relationships with several independent variables
were examined.
Traffic Prediction
There is a large body of work
devoted to traffic prediction using various inputs and predicting various
traffic conditions, including volume and speed. Smith and Demetsky show that nonparametric regression can be used
to forecast freeway volumes. Using
historical data for a given site, a k-nearest neighbor (KNN) approach is used
to predict traffic volumes over 15 minute intervals. The method proved to be accurate and portable, with some
calibration required to fit the model to specific sites.
Nonparametric
regression has also proven useful in estimating accident duration. Qi and Smith used a KNN model to forecast
the duration of an incident based on time of day, day of week, incident type,
blocked lanes, weather and the number of vehicles involved. One issue addressed by this work was the use
of categorical variables in nonparametric regression. The data set used for the study included many categorical
variables, including incident type and weather conditions. This issue had to be addressed before a
distance measure, or objective function, was developed. As a result, distance matrices were
developed for each categorical variable to allow for a quantitative measure of
how ‘far apart’ each categorical value was from the others, with the values of
the elements in the distance matrix indicating the impact of the independent variables
on the duration.
Alternative
methods for forecasting traffic conditions include time-series modeling, neural
nets, decision trees and linear regression.
The work of Yang et. al. uses auto-regressive models to predict traffic
speed. Using historical freeway flow
data, future speed is forecasted by assigning coefficients of regression to
previous time intervals in an effort to minimize the prediction error. This method may be difficult to adapt to the
proposed work because time-series modeling does not account for external
factors including incidents and their associated severity.
Smith
and Qin found that the capacity reduction due to accidents was greater than the
physical blockage on roadways. As the
number of lanes blocked increases, so too does the capacity reduction. The study also stated a case for the
argument that capacity reduction can be modeled as a random variable.
Injury Severity
The
work by Qi and Smith is an effort to predict accident duration based on
available data. The proposed work aims
to use a similar modeling approach to forecast congestion due to incidents
based on incident severity. By using
sensor data to first gain insight into the severity of the crash itself,
emergency responders can be better prepared in the event of an accident. Here, we examine some of the analytical
techniques for predicting crash severity based on roadway conditions such as
weather, speed, guardrail configurations and roadway geometry.
Kim
et. al. used logistic regression to predict the severity of injury based on
crash type and location. The study
found that the odds of a fatal incident increase greatly for those involved in
head on and roll over crashes. Being
rear ended had the highest likelihood of injury, but the lowest odds of being
killed. Seat belt use was also analyzed
as well as interaction effects between the examined variables. The limitation of a logistic model is the
fact that the response variable is binary.
In this case, fatal/nonfatal crash was the response. While this is a good starting point for
analysis, it would be ideal to have more levels of injury represented in the
response variable.
Many
statistical approaches exist to model crash severity as a function of roadway,
roadside, operational, environmental and other descriptive variables. Regression methods can be used to test the
contribution of these factors to crash severity. Donnell and Mason used pavement surface conditions, use of drugs
or alcohol, presence of an interchange entrance ramp, crash type and traffic
volume to predict the severity of median-related crashes. The method chosen was a logistic regression
model. Similarly, there exist many
crash-type specific models to estimate crash severity, with logistic regression
being the modeling method of choice for most studies. Studies have been conducted to predict the severity of
truck-related incidents, teen-driver related incidents and other specific
categories of crashes. As described by
the discussion of the Intelligent Highway System, it may be possible in the
near future to determine, based on data sent by vehicles involved in accidents,
the type of crash that has occurred. It
would then be feasible for a specific model to be applied to the situation, and
greater accuracy in severity estimation could be achieved. For the purposes of this work, however,
incident severity data, including injury levels, gathered from police reports
will be used to model congestion, as a data set with accelerometer data readings
from accidents is not readily available.
It
is clear that research has been conducted to estimate traffic conditions based
on explanatory variables such as volume in previous time intervals, time of day
and incident occurrence. Much
investigation has also been conducted to predict the severity of injuries in a
car accident based on state variables such as seatbelt usage, and roadway
conditions. A missing link in current
research exists, however, in predicting the traffic implications of an incident
based on measures of incident severity.
Methodology
Data
The data necessary
to conduct this study was consolidated from multiple sources, each filling a
specific need of the study. Incident
data, including date, time and location along with injury severity and other
measures of incident severity were not available in a single data set for
reasons concerning the privacy of individuals involved in the incidents. As a result, data from the National Highway
Traffic Safety Administration (NHTSA) was merged with data available through
the Virginia Transportation Research Council (VTRC).
State
Data System
The State Data
System, a data collection program sponsored by NHTSA, records every roadway
accident in Virginia, among other states.
The data, collected from Police Accident Reports (PARs), includes separate
files for the accident itself, the vehicles involved, and the people
involved. For this study, the SDS data
set for Virginia in the year 2003 was made available by NHTSA. The crash
data file consists of information such as date, time, county and roadway
type.
The vehicle data file describes the damage
sustained by each vehicle in a given accident as well as characteristics of the
vehicle such as direction of travel, number of passengers and whether the
vehicle is classified as private or commercial. Finally, the person
file records the age, injury severity and sex of each person involved in the
incident. Injury severity is classified
on a scale from zero to six, as shown below.
Figure 2 shows a
simplified relational database view of the SDS data. Incidents are identified across the three files through a common
case number (CASENO).
Figure 2:
SDS Relational View
While
the SDS contains a wealth of valuable information regarding traffic incidents,
exact time and location data is not available.
Specifically, time data is limited to the hour of the incident and
location data is limited to the city in which the incident occurred. For the purposes of this study, however,
exact time and location data is necessary in order to conduct an investigation
into traffic volumes and how they are affected by an incident. To bridge this gap, the SDS data were merged
with another data set, the incident data archive provided by the VTRC.
Virginia
Transportation Research Council (VTRC)
The VTRC collects data
pertaining to incidents on interstates in the Northern Virginia and Hampton
Roads areas. Because of the higher
quality of data available in Hampton Roads, this area was chosen to investigate
for the study. The incident data
archive records the date, location and time (to the minute) of accident events,
along with other descriptive data pertaining to the incident. The location data available includes city,
roadway type and roadway name, direction of travel and a location code, which
maps to a one to three mile section of roadway.
Data
Integration
In
order to make a positive match between incidents across the two data sets, the
following criteria were used: date, hour, city and roadway type. The set was narrowed to include only the
cities of Norfolk, Chesapeake and Virginia Beach because of the limited
availability of traffic flow data. This
also narrowed that number of incidents to a more manageable size for analysis
purposes. Additionally, this study
focused on incidents occurring on weekdays as a starting point for the
analysis.
The
lack of precise time data in the SDS set further narrowed the number of
incidents available for investigation.
To reduce the possibility of incorrectly matching incidents across data
sets, SDS incidents that occurred within one hour of each other were eliminated
from the set. This helped ensure
accurate matching of data points even if incident data was rounded to the
nearest hour or incorrectly entered.
Once
completed, the data integration process resulted in 76 usable data points. Incidents for which a positive match was
made across the two accident data sets and for which traffic flow data was
available were considered usable.
Observing the 76 data points reveals that they are a representative
sample of the larger population, and have similar characteristics in terms of
number and type of injuries.
Traffic
Flow Data
The
final piece of data necessary for the study was the traffic flow data available
through the Virginia Transportation Research Council (VTRC). For each incident, volume counts were
collected from the appropriate inductive loop detectors based on the location
of the incident. Loop detectors were
selected by matching the location of the incident with the location of loop
detectors recorded in the VTRC database.
The volume counts were collected beginning thirty minutes before the
time of the incident and ending two hours following cleanup. These volume counts were collected on one
minute intervals, and for analysis purposes 10-minute averages were calculated
to account for the high variability in individual one minute counts (Smith and
Qin). In addition to the volume counts
before, during and after the incident, volume counts were collected for the same
time period one and two weeks prior to the incident and averaged to determine
the historical volume.
Future work should
consider other data including speeds and occupancy counts to gain better
understanding about traffic conditions post-incident. Once the data were collected and the set reduced to incidents for
which good data was available, six metrics for analysis were calculated for
each incident. The following section
details these metrics.
Metrics
To complete the
data set for analysis of the effects of incidents on traffic flow, five metrics
– one independent variable and four dependent variables - were calculated for
each of the 76 incidents. Figure 3 is a
sample chart for one incident included in the study.
Figure 3
This figure shows
the VTRC query output for a typical incident in chart form. We see that the historical volume remains
relatively steady throughout the 180 minute period of data collection. At about the thirty minute mark, the
accident volume curve dips well below the historical volume curve, illustrating
what is assumed to be the impact of the incident, which occurs at t=30. An interesting characteristic of this and
many of the other Time vs. Volume curves is that once the incident is cleared
at t=70, the volume exceeds the historical volume counts for some time, and
later returns to a level close to that which was observed in the prior
weeks. This is the queue discharge
phase that occurs after the incident is cleared, and the characteristics of
this phase may lend further insight into the incident response in terms of how
many vehicles were re-routed. For the
purposes of this work, however, the following values were recorded for each
incident:
Accident Volume – Represented by length B
in Figure 3, the minimum volume during the time of the accident is an indicator
of the number of vehicles that were able to pass through the loop detector in
one minute during accident conditions.
Historical Volume – Represented by
the sum of lengths A and B, this is an indicator of the
historical demand on the freeway segment of interest at the time of the
incident under normal conditions.
Maximum Difference – Represented by length A, this is an indicator of unsatisfied
demand that is assumed to be a result of the incident.
Vehicle Hours Lost – The light shaded
region in the diagram, the area of which was calculated by subtracting the area
under the accident volume curve from the area under the historical volume
curve. The vehicle hours that we assume
would have been supplied under normal conditions were lost as a result of the
accident. This may be interpreted as a
measure of aggregated delay.
Percent Vehicle Hours Lost – The light shaded region was also
calculated as a percent of the total vehicle hours under normal
conditions. This dependent variable was
investigated to normalize for the fact that some of the incidents occurred on
roadways with low historical volumes.
The historical
volume was used as an independent variable, along with the data collected from
the SDS and the VTRC, to model the remaining five metrics described above. The independent variables and their
correlation coefficients are described below
|
Variable Name
|
Description
|
Variable Type
|
|
hist_vphpl
|
Historical volume (v/h/l)
|
Discrete
|
|
numinj
|
Number of injuries reported
|
Discrete
|
|
numveh
|
Number of vehicles involved
|
Discrete
|
|
numpeople
|
Number of people involved
|
Discrete
|
|
vehdam2
|
Number of vehicles with No
Damage
|
Discrete
|
|
vehdam3
|
Number of overturned
vehicles
|
Discrete
|
|
vehdam4
|
Number of vehicles with
motor damage
|
Discrete
|
|
vehdam5
|
Number of vehicles with
undercarriage damage
|
Discrete
|
|
vehdam6
|
Number of totaled vehicles
|
Discrete
|
|
vehdam7
|
Number of vehicles damaged
by fire
|
Discrete
|
|
vehdam8
|
Number of vehicles with
other damage
|
Discrete
|
|
inj0
|
Number of people reporting
no injuries
|
Discrete
|
|
inj1
|
Number dead before report made
|
Discrete
|
|
inj2
|
Number of people bleeding
or that had to be carried from scene
|
Discrete
|
|
inj3
|
Number of people with other
visible injuries and not carried from the scene
|
Discrete
|
|
inj4
|
No visible injury, but
complained of pain or momentary unconsciousness
|
Discrete
|
|
inj5
|
Number of people that died
later (within 30 days)
|
Discrete
|
The use of these
independent variables in exploring the relationship between incident severity
and the related congestion is described in the next section.
Modeling
The objective of
the data analysis portion of this study was to determine the relationships
between the independent variables and the previously developed metrics for
congestion. By developing multiple
linear regression models, it is possible to gain insight into the relationships
that exist among the available data.
Linear models were developed because pair-wise observation of
independent/dependent variable pairs did not reveal any potential logarithmic
or higher order relationships within the data.
Several models are developed for varying subsets of the data and these
models are then tested for their predictive performance on test data points not
used in model development. Analysis was
conducted for the complete set as well as a subset of 30 points with steady and
high historical volumes on the day/time of the accident.
Assumptions
Before
conducting analysis of the data and associated models, a key assumption must be
made regarding the behavior of traffic on the freeway sections studied for this
work. Because the data comes from
multiple cities and multiple loop detectors, we must assume that the historical
volume collected represents the demand for that detector on the given day of
the week at the given time of day. We
can justify this assumption for the context of this study because the goal is
to examine relationships rather than derive a rule or theorem regarding traffic
behavior after an accident as a function of incident severity.
Study
Limitations
Certain
types of freeway accidents were not included in this study. Namely, there were no fatalities or crashes
involving commercial vehicles and vehicles with hazardous materials. Because these cases make up a relatively small
percentage (less than 0.5%) of all accidents, and likely have vastly different
characteristics from other types of accidents, it would not be realistic to
include them in a study that examines only 76 data points. These special cases likely deserve
individual study rather than being included in a small scale analysis such as
this. Additionally, this study did not
account explicitly for the time of day when an incident occurred. The lack of data prevented such
stratification, but high and low volumes (i.e. rush hour) are accounted for by
including historical volume as an independent variable
Analysis
of the Complete Set
The
first set of models were developed using a training set that was randomly
selected from the complete set of 76 points.
The training set contained 51 points, and the testing – or validation –
set contained the remaining 25 points.
For each of the five proposed metrics, the following steps were
followed:
a. Created
a multiple linear regression model using all available independent variables
b. Conducted
a backwards step-wise refinement process to optimize the adjusted R2
value of the model and create a ‘reduced’ model.
c. Observed
the significant variables
d. Validated
model assumptions
e. Removed
data points that were outliers according to the Cook’s Distance metric for
regression models (if necessary)
f.
Repeated steps (a) and (b) to create a new reduced model.
Maximum
Difference Metric
We
first attempt to model the behavior of the maximum difference metric. Recall that this metric measures the maximum
difference between the volume (v/l/h) observed during the accident and the historical
volume observed by the same loop detector at the same time of day and day of
week as the accident. After conducting
backwards stepwise refinement, Table A.1 shows the best model that could be
created for this metric:
lm(formula = maxdiff_vphpl ~ hist_vphpl + numinj,
data = injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-267.31 -123.59
-21.25 70.09 577.94
Coefficients:
Estimate
Std. Error t value Pr(>|t|)
(Intercept) -143.4662 65.4341 -2.193 0.033222 *
hist_vphpl 0.2934
0.0498 5.891
3.68e-07 ***
numinj 119.8875 31.4845 3.808
0.000398 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 193.1 on 48 degrees of
freedom
Multiple R-Squared: 0.536, Adjusted R-squared: 0.5167
F-statistic: 27.73 on 2 and 48 DF, p-value: 9.907e-09
Table A.1
The
adjusted R2 value for this model is 0.51, showing some correlation
between the independent variables and the congestion metric. We see that the historical volume variable
is very significant, and shows that that the maximum difference between
historical volume and accident volume is about 29.34% of the historical volume. The other variable – also significant – is
the number of injuries in an incident.
The presence of this variable makes intuitive sense because it has a
positive sign and indicates a positive relationship between volume reduction
and the number of injuries in an incident.
Next, we verify that the model assumptions are met. Figure 4 shows model diagnostic charts:
Figure 4
We
see no strong trend in the Fitted vs. Residual plot and a Normal Q-Q plot that
shows normally distributed residuals.
The Cook’s distance plot, a measure of each point’s influence on the overall
model, shows three points that have a high influence (data points 5, 51 and
62). The widely accepted rule of thumb
is that if a point has a Cooks Distance value of 1.0 or higher, we can remove
the point and create the regression model without that point. While none of the points have Cooks Distance
values as high as 1.0, it may be of interest to observe the effect of reporting
the results of removing these points.
Table A.2 shows the best model that was developed after removing the
three points described above:
lm(formula = maxdiff_vphpl ~ hist_vphpl + numinj,
data = injuries.training.reduced)
Residuals:
Min 1Q
Median 3Q Max
-250.00
-93.58 -22.57 72.43
509.63
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) -136.81798 43.95692 -3.113 0.0027
**
hist_vphpl 0.26475 0.03735 7.088 9.22e-10
***
numinj 124.67311 20.31491 6.137
4.68e-08 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 154.4 on 69 degrees of
freedom
Multiple R-Squared: 0.6115, Adjusted R-squared: 0.6002
F-statistic:
54.3 on 2 and 69 DF, p-value:
6.82e-15
Table A.2
We
see that the adjusted R2 increases slightly with little change in the
coefficient estimates for the variables.
The important points to note from the analysis of the maximum difference
metric are the relatively high adjusted R2 along with the positive
relationship with the number of injuries and the positive, fractional relationship
with the historical volume. All these
are intuitive results that support the hypothesis that freeway congestion is a
function of incident severity, assuming that historical volume is an indication
of demand during the accident.
Accident
Volume Metric
The
second metric we examine is the minimum volume along the freeway during the
time needed to clear the accident (incident duration is recorded in the VTRC
database along with the time of incident).
Table A.3 shows the initial results of multiple regression modeling with
an adjusted R2 of 0.803:
lm(formula = acc_vphpl ~ hist_vphpl + numinj, data =
injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-577.94
-70.09 21.25 123.59
267.31
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) 143.4662 65.4341 2.193
0.033222 *
hist_vphpl
0.7067 0.0498 14.190 <
2e-16 ***
numinj -119.8875 31.4845 -3.808
0.000398 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 193.1 on 48 degrees of
freedom
Multiple R-Squared: 0.8109, Adjusted R-squared: 0.803
F-statistic: 102.9 on 2 and 48 DF, p-value: < 2.2e-16
Table A.3
It
is interesting to note that the coefficient estimates are in line with the
estimates for the maximum difference model.
The historical volume has a positive, fractional relationship with a
coefficient of 0.71. The coefficient
for the maximum difference model is 0.29.
The coefficients sum to one, resulting in the intuitively obvious
relationship that the sum of the minimum volume and the maximum difference is
equal to the historical volume. This
relationship exists because the maximum difference is a metric derived from the
historical volume and accident volume measurements, and was included in the
analysis for the purposes of investigating the characteristics of traffic
behavior from as many different perspectives as possible. Figure 5 shows the model diagnostic charts:
Figure 5
We
see no pattern in the Fitted vs. Residuals charts and the residuals are
normally distributed, with some deviation at the tails. The Cook’s Distance chart, however, shows
once again that data points 5, 51 and 62 have high Cook’s Distance values. Again, we report modeling results after
removing these points from the training set in Table A.4. Once again, the adjusted R2 sees
a slight increase along with slight adjustments in the coefficient estimates
Call:
lm(formula = acc_vphpl ~ hist_vphpl + numinj, data =
injuries.training.reduced)
Residuals:
Min 1Q
Median 3Q Max
-509.63
-72.43 22.57 93.58
250.00
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) 136.81798 43.95692 3.113 0.0027
**
hist_vphpl 0.73525 0.03735 19.685 <
2e-16 ***
numinj -124.67311 20.31491 -6.137
4.68e-08 ***
Signif. Codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1
Residual standard error: 154.4 on 69 degrees of
freedom
Multiple R-Squared: 0.8507, Adjusted R-squared: 0.8464
F-statistic: 196.6 on 2 and 69 DF, p-value: < 2.2e-16
Table A.4
Here
we have a very high adjusted R2 value and coefficients that show a
negative relationship between the number of injuries and the volume at the time
of the accident, along with a positive, fractional relationship between the
accident volume and the historical volume.
While we cannot assume causation, the relationships that exist are of
interest for the purposes of this study.
Vehicle
Hours Lost Metric
Next,
we examine the metric of Vehicle Hours Lost.
While the method of calculating Vehicle Hours Lost employed in this
study is not an accepted method of measuring delays, it was included to fully
investigate the characteristics of the Time vs. Volume chart. Table A.5 shows the best attainable model
for the Vehicle Hours Lost metric.
Call:
lm(formula = arealost ~ hist_vphpl + numveh + inj0 +
inj2, data = injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-882.06 -340.30
-63.96 136.06 2071.48
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) -336.5537 297.8530 -1.130 0.26436
hist_vphpl
0.4236 0.1566 .705 0.00956
**
numveh 293.2180
160.5999 1.826 0.07438
.
inj0 -241.5870 155.0468 -1.558 0.12605
inj2 290.9864 158.6546 1.834 0.07311
.
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 604.4 on 46 degrees of
freedom
Multiple R-Squared: 0.3621, Adjusted R-squared: 0.3066
F-statistic: 6.527 on 4 and 46 DF, p-value: 0.0003018
Table A.5
We
see that the Vehicle Hours Lost can be modeled as a function of the historical
volume, number of vehicles, number of uninjured and number of people with
visible injuries. All of these
coefficients are intuitive in their relationship to the dependent variable, but
the adjusted R2 is low compared to the other metrics. Examination of the model diagnostics in
Figure 6 reveals the existence of some outliers in the Cook’s Distance metric.
Figure 6
When
the potential outliers are removed (points 63 and 62), we see little change in
the adjusted R2, but the metric can now be modeled best using only
the historical volume and the number of severely injured people as shown in
Table A.6. Additionally, the model
assumptions are met, as shown in Figure 7
Call:
lm(formula = arealost ~ hist_vphpl + inj2, data =
injuries.reduced.training)
Residuals:
Min 1Q
Median 3Q Max
-714.49 -239.55
-57.61 90.41 1555.69
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) -194.1899 157.5037 -1.233
0.224007
hist_vphpl 0.4712
0.1296 3.634
0.000713 ***
inj2 287.7378 95.8450 3.002
0.004364 **
---
Signif. codes: 0
'***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 442.9 on 45 degrees of freedom
Multiple R-Squared: 0.3483, Adjusted R-squared: 0.3193
F-statistic: 12.02 on 2 and 45 DF, p-value: 6.555e-05
Table A.6
Figure 7
Percent
Vehicle Hours Lost Metric
In
further exploration of the Vehicle Hours Lost metric, we examine the absolute
number as a percentage of the total vehicle hours during normal conditions on
the given roadway (See previous section for complete explanation of this
metric). Table A.7 shows the modeling
results for this metric
Call:
lm(formula = PercentAreaLost ~ hist_vphpl + inj0 +
inj3 + vehdam3, data =
injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-0.13695 -0.04569 -0.01367 0.03808 0.23207
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) 0.1290534 0.0329257 3.920
0.000293 ***
hist_vphpl 0.0000343 0.0000203 1.690
0.097885 .
inj0 -0.0291254 0.0117225 2.485
0.016672 *
inj3 0.2401116 0.0798890 3.006 0.004283 **
vehdam3 0.6658071
0.0821506 8.105
2.07e-10 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0772 on 46 degrees of
freedom
Multiple R-Squared: 0.665, Adjusted R-squared: 0.6358
F-statistic: 22.82 on 4 and 46 DF, p-value: 1.947e-10
Table A.7
We
see that historical volume, the number of uninjured people, the number of
people with moderate injuries, and the number of overturned vehicles are all
significant indicators of the Percent Vehicle Hours Lost metric with intuitive
coefficients that suggest a positive relationship between incident severity and
the congestion metric. The adjusted R2
of 0.64 is much higher than that attained when modeling Vehicle Hours Lost,
indicating that the demand under normal conditions is an important factor to
consider when modeling the behavior of these roadways.
Summary of Complete Set Analysis
By
modeling the various potential metrics for freeway congestion, we were able to
see the relationships between our independent variables of incident severity
and the dependent indicators of freeway congestion. The table below is a brief summary of the modeling results for
the training data selected from the complete data set:
|
Complete Data Set - Modeling
Coefficients
|
|
|
|
|
Adjusted R squared
|
Historical Volume
|
Number of Injuries
|
Number Uninjured
|
Number of Severe Injuries
|
Number of Minor Injuries
|
Overturned Vehicles
|
|
|
|
|
|
|
|
|
|
|
|
|
Max
Difference
|
0.6
|
0.265
|
124.67
|
-
|
-
|
-
|
-
|
|
|
|
|
|
|
|
|
|
|
|
|
Accident
Volume
|
0.85
|
0.735
|
-124.67
|
-
|
-
|
-
|
-
|
|
|
|
|
|
|
|
|
|
|
|
|
Vehicle
Hours Lost
|
0.32
|
0.47
|
-
|
-
|
287.74
|
-
|
-
|
|
|
|
|
|
|
|
|
|
|
|
|
Percent
Vehicle Hours Lost
|
0.64
|
3.43E-05
|
-
|
-0.29
|
-
|
0.24
|
0.67
|
|
We
see that the historical volume is a significant variable in all of the models,
and various measures of incident severity are used for the different dependent
variables. The number of injuries
proves to be a strong indicator of the impact on volume, while injury
measurements are prevalent in the models relating to Vehicle Hours Lost.
The
complete data set represents a wide range of pre-accident conditions, and
further reduction of the data set was conducted in an attempt to strengthen the
modeling capabilities of the independent variables. The set was reduced to include incidents that were characterized
by relatively high volumes (above 1000 v/h/l) and steady historical volumes
(less than 15% deviation throughout the duration of the incident). This reduced data set was made up of 30
incidents, and the analysis of this set follows
Reduced Data Set
The set of 30 points was partitioned
once again into training and test sets.
This time, the training set contained 25 points with five test set
points. While this set of data is very
small, the goal was to investigate whether stratifying the incidents based on
certain characteristics would make the models more robust. All of the dependent variables were modeled,
and it was found that all of them produced models that had lower adjusted R2
values than when modeling was conducted with the entire set. Additionally, there were several cases of
counter-intuitive variable coefficients that contradicted the findings in the
previous section. Several different
partitions of training/test sets were made, all leading to different
confounding outcomes. These results,
combined with the fact that the models had, on average, 19 degrees of freedom,
made it difficult to make conclusions based on this set. While there may be value in stratifying
based on pre-incident conditions and applying appropriate models for those
conditions, more data and a more thorough investigation of such a method is
necessary, but beyond the scope of this study.
Classification Trees
Another modeling method that was
attempted for the complete data set was the use of classification trees. Specifically, the Classification and
Regression Trees (CART) algorithm was implemented by assigning an index (A thru
F) based on pre-determined partitions to the minimum volume dependent variable. The CART algorithm had a 28%
misclassification rate, and only partitioned the classification tree based on
historical volume. While we cannot say
that classification trees are not a good method for modeling accident volume,
the results of the algorithm on this particular data set were not strong.
In the next section, the performance
of selected models on their training sets is reported and analyzed.
Results
Accident Volume Model
We begin by
examining the performance of the Accident Volume model on the 25 point testing
data set. The best model (shown below)
was used to predict the Accident Volume metric for the training set points
based on the independent variables, and the predicted volumes were compared to
the observed volumes.
lm(formula = acc_vphpl ~ hist_vphpl + numinj, data =
injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-577.94
-70.09 21.25 123.59
267.31
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) 143.4662 65.4341 2.193 0.033222
*
hist_vphpl
0.7067 0.0498 14.190 <
2e-16 ***
numinj -119.8875 31.4845 -3.808
0.000398 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 193.1 on 48 degrees of freedom
Multiple R-Squared: 0.8109, Adjusted R-squared: 0.803
F-statistic: 102.9 on 2 and 48 DF, p-value: < 2.2e-16
Table R.1
Table R.2 shows the historical
volume and number of injuries for each of the 25 test set points along with the
predicted and actual values for Accident Volume as well as the residuals
(prediction error) for each point.
|
Accident Volume
|
|
|
Injuries
|
Historical
|
Predicted
|
Observed
|
Residual
|
|
0
|
2036.4
|
1627.6
|
1754.4
|
-126.8
|
|
0
|
1784.4
|
1512.4
|
1378.8
|
133.6
|
|
0
|
1798.2
|
1451.5
|
1341.6
|
109.9
|
|
0
|
1493.3
|
1226.1
|
1464.0
|
-237.9
|
|
1
|
1517.4
|
1186.9
|
1474.8
|
-287.9
|
|
1
|
1381.5
|
1101.5
|
1155.0
|
-53.5
|
|
0
|
1249.5
|
1066.3
|
1132.5
|
-66.2
|
|
1
|
1482.0
|
1066.1
|
1014.0
|
52.1
|
|
1
|
1373.0
|
1026.0
|
1242.0
|
-216.0
|
|
1
|
1128.0
|
900.7
|
1081.2
|
-180.5
|
|
0
|
1006.2
|
866.1
|
915.6
|
-49.5
|
|
0
|
1019.4
|
865.6
|
975.6
|
-110.0
|
|
1
|
965.3
|
844.4
|
868.5
|
-24.1
|
|
2
|
1327.5
|
800.0
|
654.0
|
146.0
|
|
1
|
967.5
|
742.9
|
762.0
|
-19.1
|
|
0
|
813.0
|
723.3
|
757.5
|
-34.2
|
|
5
|
1604.4
|
638.4
|
362.4
|
276.0
|
|
0
|
522.0
|
508.2
|
510.0
|
-1.8
|
|
0
|
517.5
|
504.9
|
456.0
|
48.9
|
|
0
|
499.5
|
481.3
|
471.0
|
10.3
|
|
1
|
438.0
|
454.7
|
356.4
|
98.3
|
|
0
|
420.6
|
423.0
|
388.8
|
34.2
|
|
0
|
339.0
|
372.9
|
278.0
|
94.9
|
|
1
|
519.0
|
362.4
|
504.0
|
-141.6
|
|
0
|
199.5
|
259.6
|
171.0
|
88.6
|
Table R.2
Figure 8
shows no trend in the residuals vs. fitted plots.
Figure 8
The error values appear to be
normally distributed, with a mean of -18.3 and a standard deviation of
133.97. To further quantify the
predictive value of this model, the observed Accident Volume metric was given a
classification (A thru F) based on the following values:
|
Volume (v/h/l)
|
Classification
|
Number Classified
|
|
>1700
|
A
|
3
|
|
1400-1700
|
B
|
4
|
|
1100-1400
|
C
|
14
|
|
800-1100
|
D
|
21
|
|
500-800
|
E
|
15
|
|
<500
|
F
|
19
|
|
Total
|
|
76
|
Table R.3
The
model’s predicted values for Accident Volume were then assigned a
classification based on the same ranges as above, and the predicted
classification was compared to the observed classification. The model had a 60% classification rate,
correctly identifying the classification of 15 out of 25 training set
points. Additionally, 100% of the
predicted classifications were within one classification of the observed value.
Next,
the ranges for each classification were adjusted to allow the borders to
overlap. This was done to prevent
misclassifications near the borders and “soften” the ranges for each
classification. The new borders are
described in Table R.3.
|
Volume (v/h/l)
|
Classification
|
|
>1675
|
A
|
|
1375 – 1725
|
B
|
|
1075 – 1425
|
C
|
|
775 – 1125
|
D
|
|
475 – 825
|
E
|
|
<525
|
F
|
Table R.4
When the classification
ranges were modified, the model accurately predicted 17 out of 25
classifications.
Maximum Difference Model
The
performance of the Maximum Difference model (R.5) on the test set was examined
next. Table R.6 shows the number of
injuries, the historical volumes as well as the predicted and observed maximum
differences between historical volume and volume during the accident.
Lm(formula = maxdiff_vphpl ~ hist_vphpl + numinj,
data = injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-267.31 -123.59
-21.25 70.09 577.94
Coefficients:
Estimate
Std. Error t value Pr(>|t|)
(Intercept) -143.4662 65.4341 -2.193 0.033222 *
hist_vphpl 0.2934
0.0498 5.891
3.68e-07 ***
numinj 119.8875 31.4845 3.808
0.000398 ***
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 193.1 on 48 degrees of
freedom
Multiple R-Squared: 0.536, Adjusted R-squared: 0.5167
F-statistic: 27.73 on 2 and 48 DF, p-value: 9.907e-09
Table R.5
|
|
|
Maximum Difference
|
|
|
Injuries
|
Historical
Volume
|
Predicted
|
Observed
|
Residual
|
|
0
|
1784.4
|
259.09
|
405.6
|
-146.5
|
|
0
|
499.5
|
9.43
|
28.5
|
-19.1
|
|
0
|
1798.2
|
390.82
|
456.6
|
-65.8
|
|
0
|
1493.3
|
300.75
|
29.3
|
271.5
|
|
1
|
522.0
|
13.88
|
12.0
|
1.9
|
|
1
|
199.5
|
-79.18
|
28.5
|
-107.7
|
|
0
|
1373.0
|
386.6
|
131.0
|
255.6
|
|
1
|
1381.5
|
284.86
|
226.5
|
58.4
|
|
1
|
1006.2
|
156.89
|
90.6
|
66.3
|
|
1
|
438.0
|
-46.8
|
81.6
|
-128.4
|
|
0
|
1482.0
|
380.3
|
468.0
|
-87.7
|
|
0
|
813.0
|
99.83
|
55.5
|
44.3
|
|
1
|
965.3
|
108.92
|
96.8
|
12.2
|
|
2
|
1604.4
|
930.54
|
1242.0
|
-311.5
|
|
1
|
967.5
|
309.24
|
205.5
|
103.7
|
|
0
|
519.0
|
145.21
|
15.0
|
130.2
|
|
5
|
1128.0
|
184.73
|
46.8
|
137.9
|
|
0
|
2036.4
|
461.16
|
282.0
|
179.2
|
|
0
|
1517.4
|
455.99
|
42.6
|
413.4
|
|
0
|
420.6
|
-13.87
|
31.8
|
-45.7
|
|
1
|
1327.5
|
445.68
|
673.5
|
-227.8
|
|
0
|
517.5
|
12.55
|
61.5
|
-49.0
|
|
0
|
1019.4
|
162.99
|
43.8
|
119.2
|
|
1
|
1249.5
|
234.65
|
117.0
|
117.7
|
|
0
|
339.0
|
-40.17
|
61.0
|
-101.2
|
Table R.6
The
plot of fitted vs. residuals for this model shows a potential upward trend, as
shown in Figure 9. The model performs
better for lower predicted values of the dependent variable, with the range of
error terms increasing as the predicted values increase. The residuals have a mean of 24.85 and a
standard deviation of 162.9. It should
be noted that the model predicts negative values for several points, indicating
an increase in volume after an accident occurs. This makes the model difficult to interpret in some cases, and
indicates that the intercept value for the model may need to be adjusted, or
forced to go through the origin.
Figure 9
Vehicle Hours Lost Model
The
results of the Vehicle Hours Lost model (Table R.7) were similar to those of
the Maximum Difference Model.
Call:
lm(formula = arealost ~ hist_vphpl + numveh + inj0 +
inj2, data = injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-882.06 -340.30
-63.96 136.06 2071.48
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -336.5537 297.8530 -1.130 0.26436
hist_vphpl 0.4236 0.1566
2.705 0.00956
**
numveh 293.2180
160.5999 1.826 0.07438
.
inj0 -241.5870 155.0468 -1.558 0.12605
inj2 290.9864 158.6546 1.834 0.07311
.
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 604.4 on 46 degrees of
freedom
Multiple R-Squared: 0.3621, Adjusted R-squared: 0.3066
F-statistic: 6.527 on 4 and 46 DF, p-value: 0.0003018
Table R.7
The strength of
the model lies in predicting the lower values, with some negative predicted
values. Again, it may be advantageous
to force the linear model through the origin for interpretive purposes. Table R.8 shows the historical volume,
number of vehicles, number uninjured and number seriously injured along with
the predicted and observed vehicle hours lost for each of the test set
points. Figure 10 shows no trend in the
plot of predicted vs. residual values.
The residuals had a mean of 24.85 and a standard deviation of 162.89.
|
|
|
|
|
Vehicle Hours Lost
|
|
|
Historical
Volume
|
Vehicles
|
Uninjured
|
Seriously
Injured
|
Predicted
|
Observed
|
Residual
|
|
199.47
|
1
|
1
|
0
|
-169.53
|
34.47
|
-204
|
|
339
|
2
|
2
|
0
|
-80.07
|
72
|
-152.07
|
|
438
|
1
|
0
|
1
|
-27.92
|
236.5
|
-264.42
|
|
1604.4
|
2
|
0
|
2
|
1634.57
|
5708
|
-4073.43
|
|
420.6
|
1
|
1
|
0
|
-73.65
|
18.5
|
-92.15
|
|
517.5
|
2
|
2
|
0
|
-2.68
|
16.25
|
-18.93
|
|
499.5
|
1
|
1
|
0
|
-39.44
|
3.75
|
-43.19
|
|
519
|
2
|
2
|
0
|
11.73
|
13.5
|
-1.77
|
|
522
|
2
|
2
|
0
|
-0.72
|
5
|
-5.72
|
|
1327.5
|
3
|
2
|
0
|
413.28
|
1832.5
|
-1419.22
|
|
813
|
2
|
2
|
0
|
125.45
|
15
|
110.45
|
|
967.5
|
2
|
1
|
1
|
1025.25
|
151
|
874.25
|
|
965.25
|
1
|
0
|
1
|
200.69
|
62.75
|
137.94
|
|
1006.2
|
2
|
2
|
0
|
209.22
|
203.75
|
5.47
|
|
1019.4
|
1
|
1
|
0
|
185.98
|
16.75
|
169.23
|
|
1482
|
3
|
2
|
0
|
631.07
|
81.5
|
549.57
|
|
1128
|
3
|
2
|
0
|
90.81
|
19
|
71.81
|
|
1249.5
|
3
|
3
|
0
|
336.04
|
45.6
|
290.44
|
|
1381.5
|
3
|
2
|
0
|
510.26
|
159.5
|
350.76
|
|
1373
|
2
|
1
|
0
|
583.49
|
436.7
|
146.79
|
|
1798.2
|
2
|
2
|
0
|
552.62
|
608.75
|
-56.13
|
|
1784.4
|
4
|
4
|
0
|
185.32
|
348.75
|
-163.43
|
|
1493.25
|
2
|
2
|
0
|
420.4
|
45.5
|
374.9
|
|
1517.4
|
5
|
4
|
0
|
1034.88
|
3140.2
|
-2105.32
|
|
2036.4
|
2
|
2
|
0
|
655.9
|
724.5
|
-68.6
|
Table R.8
Figure 10
Percent Vehicle Hours Lost Model
Finally, we examine the model for
Percent Vehicle Hours Lost. Table R.9
shows the model used for analysis.
Call:
lm(formula = PercentAreaLost ~ hist_vphpl + inj0 +
inj3 + vehdam3, data =
injuries.training)
Residuals:
Min 1Q
Median 3Q Max
-0.13695 -0.04569 -0.01367 0.03808 0.23207
Coefficients:
Estimate Std. Error t
value Pr(>|t|)
(Intercept) 0.1290534 0.0329257 3.920
0.000293 ***
hist_vphpl 0.0000343 0.0000203 1.690
0.097885 .
inj0 -0.0291254 0.0117225 -2.485
0.016672 *
inj3 0.2401116 0.0798890 3.006
0.004283 **
vehdam3 0.6658071
0.0821506 8.105
2.07e-10 ***
---
Signif. codes:
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0772 on 46 degrees of
freedom
Multiple R-Squared: 0.665, Adjusted R-squared: 0.6358
F-statistic: 22.82 on 4 and 46 DF, p-value: 1.947e-10
Table R.9
We see that historical volume, minor injuries and overturned vehicles
contribute to higher loss of vehicle hours.
Additionally, the number of uninjured people reduced the amount of lost
vehicle hours, which may be a counter-intuitive finding because it suggests
that an accident involving a large number of uninjured people would have a very
low loss of vehicle hours. Further
examination of the data, however, shows that the range of this variable is from
zero to five, which makes the coefficient more intuitive for the purposes of
this model. Table R.10 shows the
historical volumes, number uninjured, number with minor injuries, number of
overturned vehicles along with predicted and observed values for percent
vehicle hours lost and associated residuals for each of the 25 test data
points.
|
|
|
|
|
Percent Vehicle Hours Lost
|
|
|
Historical
Volume
|
Uninjured
|
Minor
Injuries
|
Overturned
Vehicles
|
Predicted
|
Observed
|
Residual
|
|
199.47
|
1
|
0
|
0
|
10.06%
|
5.77%
|
4.3%
|
|
339
|
2
|
0
|
0
|
8.09%
|
13.90%
|
-5.8%
|
|
438
|
0
|
0
|
1
|
79.42%
|
12.96%
|
66.5%
|
|
1604.4
|
0
|
3
|
0
|
91.44%
|
57.46%
|
34.0%
|
|
420.6
|
1
|
0
|
0
|
10.79%
|
2.67%
|
8.1%
|
|
517.5
|
2
|
0
|
0
|
8.68%
|
9.63%
|
-0.9%
|
|
499.5
|
1
|
0
|
0
|
11.05%
|
2.26%
|
8.8%
|
|
519
|
2
|
0
|
0
|
8.35%
|
1.75%
|
6.6%
|
|
522
|
2
|
0
|
0
|
8.69%
|
1.32%
|
7.4%
|
|
1327.5
|
2
|
0
|
0
|
