INTRODUCTION
A fundamental goal of surface transportation agencies is to
provide safe and reliable transportation services to road users. Being
supported by legislation such as SAFETEA-LU (Safe, Accountable, Flexible, Efficient
Transportation Equity Act: A Legacy Users) (1), U.S. transportation agencies have been making a variety of efforts in improving safety and
mobility. Much effort in utilizing advanced transportation technologies such
as Intelligent Transportation Systems (ITS) has been expended mainly for
mobility improvement (2), whereas utilizing these technologies for
safety improvement has not been thoroughly explored. Further, in recent years,
safety seems to be receiving more attention than mobility as the former costs
significantly more than the latter. One study showed that the total cost of
traffic crashes is almost two and half times the cost of traffic congestion (3).
As an essential component of typical traffic
safety studies, considerable effort was expended to identify traffic patterns
that trigger crashes based on crash records and aggregated traffic flow data.
For example, Oh et al. attempted to relate aggregated loop detector data to
crash data using a Bayesian classification and non-parametric density function
(4). They assumed that disruptive traffic flows trigger crashes and
found that a 5-min standard deviation of speeds right before a crash was the
best crash indicator. Golob et al. conducted an analysis with data collected
on highways in southern California and concluded that median speed, speed
variation, mean flow, and flow variation were the best crash indicators based
on a statistical analysis of 30-sec-interval traffic flow data collected 30 min
before the crashes (5). With the same dataset, Kockelman and Ma employed
binomial regression models and found that 30-sec average speed and 5-min speed
variation were highly related to crash occurrence (6).
Abdel-Aty et al. considered spatial variation of
traffic patterns for crash prediction by exploring data obtained from detectors
located upstream and downstream of a crash location (7). They developed
crash prediction models using case-control logistic regression and suggested 5-min
average occupancy upstream and 5-min coefficient of variation in speed
downstream of the crash location 5 to 10 min before crash occurrence as crash
indicators. Lee et al. conducted a similar study using a log-linear model and concluded
that coefficient of variation in speed, spatial variance of speed, and covariance
of volume difference upstream and downstream of crash locations were important crash
indicators (8).
As stated, the analysis methods mostly employed in
the safety studies were based on site-specific crash records and aggregated
traffic flow data. Given that a crash occurs through individual vehicular
interactions, aggregated data often do not adequately capture crash
characteristics. Individual vehicular data could significantly enhance crash
modeling. For example, a large speed variance among vehicles likely indicates
a high crash potential; it might not be if the time headways among the adjacent
vehicles were large enough to lead to crash avoidance. Thus, the use of
individual vehicular information could enhance crash prediction capability. However,
few research efforts have attempted to use individual vehicular data in the
development of crash prediction models mostly because of limitations in obtaining
and using individual vehicular data.
As advanced communication technologies for
vehicle to vehicle and vehicle to infrastructure through dedicated short range
communication, IEEE 802.11p, etc., become available for deployment, individual
vehicular information is going to be available in the near future. It is
expected that individual vehicular information can provide a unique opportunity
to improve safety as well as mobility. Because of the potential benefits, an
innovative concept of integrating information exchanges between vehicles and
infrastructure, known as Vehicle Infrastructure Integration (VII), has been
proposed by federal and state transportation agencies and vehicle manufacturers
(9). In spite of a surge in recent attention with regard to the VII
applications to traffic safety, few research efforts have been made to enhance the
safety as well as the mobility of the surface transportation system. For
example, using individual vehicular information such as vehicular speeds and
headways can provide accurate information that can be used to understand crash risk
in real time, but little is known how to link such information to traffic crash
occurrence.
Park and Yadlapati studied
simulation-based work-zone safety (10). They examined individual
vehicular interactions within work zones in simulation and proposed a crash risk
predictor. Oh et al. proposed a freeway safety index based on real-time
individual vehicular data (11). However, their index was not empirically
validated. Moreover, they simplified a continuous crash risk into a binary
outcome (crash and non-crash), which caused loss of information regarding crash
potential magnitude.
In an attempt to develop statistical models
predicting crashes on highway, various count data models have been applied. Zero-inflated
count models have been introduced to deal with an excess of zero counts in
crash prediction modeling. They assume that zero crash counts are generated
from two processes distinguishing sampling zeros from structural zero. Given
that no roadway segments are absolutely safe from crashes, these models with
structural zero have an unrealistic assumption. Unlike zero-inflated models,
zero-hurdle models are capable of addressing excessive zero counts without such
an assumption (12). One popular example that describes the conceptual
difference between zero-inflated and zero-hurdle models is the case with the
following question: “How many fish did you get?” Zero responses would be either
those who never try to get fish or those who did but did not get fish. In the
case where those who never tried were included in the study, zero-inflated
models are conceptually appropriate. However, if all zero responses are from
those who tried but did not get fish, zero-hurdle models are appropriate. In
spite of the conceptual relevance to crash data, hurdle count models have not
been applied in traffic safety analysis.
To this end, this paper presents an approach to
predict crash occurrence using individual vehicular information. Crash
prediction models including zero-inflated hurdle models were developed on the
basis of crash data and individual vehicular information collected from
inductive loop detectors on urban interstate highways in Virginia.
METHODOLOGY
A new crash risk predictor, called safe headway distance
(SHD), is proposed to calculate the crash potential between two consecutive vehicles.
An aggregated SHD is computed by summing only negative SHD values, which
indicate potential crash risks, over at a given time interval and a roadway
segment is used as an input to a crash prediction model. Several count data
models were employed
Safe Headway Distance Equation
A traffic conflict can be defined in several ways depending
on the research purpose and design (10). This paper defines the
conflict as a condition of two consecutively moving vehicles having inadequate
SHD such that the following vehicle will crash into the leading vehicle when it
makes an unexpected stop. It assumes that the leading vehicle abruptly reacts
to a stimulus resulting in an emergency stopping maneuver while the following
vehicle reacts to such a braking maneuver by the leading vehicle. Based on
these assumptions, a safe (i.e., non-collision) condition of the two consecutive
vehicles can be defined as a condition where a minimum stopping distance of the
leading vehicle is greater than that of the following vehicle. Hence, using
the minimum stopping distance formula (13), the safe condition can be
mathematically expressed as:
Minimum
Stopping Distance (leading vehicle) > Minimum Stopping Distance (following
vehicle) ≡
(Eq.
1)
where VLeading = leading vehicle’s speed
in miles per hour (mph)
VFollowing = following
vehicle’s speed in mph
acc = deceleration rate in ft/sec2
g = gravity acceleration (32.2 ft/sec2)
Gr = grade in percentage
h = time headway in seconds
PRT = perception reaction time of the
following vehicle in seconds.
Using Equation 1, an individual SHD for a pair
of two consecutive vehicles can be defined as a difference of the two minimum
stopping distances of vehicles as follows:
and 
(Eq.
2)
where i = index of a pair of two consecutive vehicles
SHDi=
safe headway distance of the ith vehicle pair
Diffi=
difference of two minimum stopping distances of the ith vehicle pair.
As noted, the individual SHD reflects a crash risk between the
two consecutive vehicles. A larger SHD is translated into a greater deficiency
in safe distance suggesting a crash-prone case.
When individual vehicular data become available
on a real-time basis through advanced technologies such as the VII system, the SHD
can be computed in real time and linked to actual crash occurrences. However,
under the current data collection system, vehicular and crash data are not available
in real time. Thus, to relate the SHD to crash occurrences under the current
system, the use of historical data, especially for crash data, is inevitable.
Moreover, aggregation of such data over a certain time period is convenient to
formulate crash prediction models empirically.
In addition to temporal aggregation, the
aggregation needs to be done for a roadway segment. For developing a crash
prediction model relating crashes to the proposed SHD, crash data were
aggregated over a short stretch of highways to produce the number of crashes
suitable for statistical crash modeling. To be consistent with the aggregated
crash data, the SHD were also aggregated along the same road segment over the
same time period.
The segment length for aggregation can vary from
100 ft to several miles, depending on the objective of an analysis. For
example, if the analysis aims at a safety evaluation of a region, the
individual SHD can be aggregated over an entire network within the region. If the
analysis aims at a safety evaluation of a corridor, the SHD can be aggregated along
the corridor. As such a 1-mile segment length, which is typical in traffic
safety engineering studies for road segments, was used for this paper.
For crash prediction models to be estimated
effectively using 1-mile-segment data, crash data need to be accumulated over a
sufficient time period, 6 years in this paper. As the number of crashes over
the 6 years can be divided by a time interval (e.g., 1 hour) within a day and
potentially affect the crash predictions, this paper considers three time
intervals: 15, 30, and 60 min. As a result, three datasets for crash
prediction models were prepared. For example, the number of crashes that occurred
on a particular 1-mile segment can be aggregated by each hour of the day, and
the hourly crashes can be accumulated over the entire crash records. This produces
an aggregated dataset with 24 one-hour crash counts.
Obviously, the SHD needs to be aggregated by
each hour of the day so that the crash prediction model can be developed on the
basis of hourly crash counts and SHD values. The SHD was aggregated by the
three same intervals, 15, 30, and 60 minutes over the 1-mile segment, and
produced an aggregated SHD for each time interval. To predict the number of
crashes for a 1-mile road segment during a certain time interval, the
aggregated SHD was used as a crash risk predictor. These three time intervals
were compared on the basis of the results of the crash prediction models.
Count Data Regression Models
To predict the number of crashes for a roadway segment
during a specific time interval, the aggregated SHD was used as a crash
predictor. It is noted that the count data models are commonly applied for
crash prediction for the following reasons. The number of crashes is a non-negative
integer and typically skewed in its distribution. Moreover, it typically shows
an increasing variance in a relationship with predictors, which is
heteroscedastic. These characteristics are apt to result in inefficient and
biased parameter estimates when a classical linear regression is applied (12).
Count data models such as the Poisson (P) and negative binomial (NB) models are
designed specifically to account for such characteristics.
There exist several count models. The P model,
a standard count data model, assumes that the mean and the variance of a
dependent variable (e.g., crash counts) are equal, called equidispersion.
However, crash counts often show overdispersion (i.e., variance is larger than
mean). In such a case, an NB model is a usual alternative, and it explicitly relaxes
the equidispersion assumption by adding a random term into the P model. In some
cases, excessive zeros in crash counts contribute to an overdispersion. Zero-inflated
and zero-hurdle count models are specifically designed to handle excessive zero
crash counts.
The zero-inflated and zero-hurdle models look similar
in that they integrate a binary response model and a count data model into a
single modeling frame. However, a significant conceptual difference exists between
these two in terms of how zeros were interpreted (14). The
zero-inflated model assumes that there are two types of zeros: structural zeros
and sampling zeros. In contrast, the zero-hurdle model assumes that all zeros
are sampling zeros. A selection between these two seemingly similar models can
be made to some extent based on a study design.
Rose et al. stated that a study design and
purpose should be considered in selecting appropriate models handling excessive
zeros (14). For an example of traffic safety, suppose a crash
prediction model is being developed using crash data collected from 1-mile roadway
segments over 1 year. Because of the short segment length and the relatively
short time period, there likely exists a large number of segments with no crashes
during the year, implying a preponderance of zero counts. By applying the
zero-inflated models to these data, it is assumed that there exist two kinds of
road segments, absolutely safe (i.e., crash-free) segments and crash potential
(i.e., at-risk) segments. In contrast, by applying the zero-hurdle models, it
is assumed that crashes can occur at any segment, implying all segments have crash
potential. As such, it is determined that the zero-hurdle models assuming all
road segments have crash potential are deemed to be more appropriate than the
zero-inflated models. Lord et al. concluded that zero crash data are not
characterized by the two regimes noted in the zero-inflated models (15).
Six count data models including the standard P,
NB, zero-inflated P (ZIP), zero-inflated NB (ZINB), zero-hurdle P (ZHP), and zero-hurdle
NB (ZHNB) models were initially considered. The probability distribution
functions of these models are presented in Table 1. A binary logit model is
used to incorporate a probability of structured zeros in the ZIP and ZINB
models and in the ZHP and ZHNB models.
Model Selection
Overdispersion of the crash counts and a percentage of zero
counts were examined in selecting candidate models among the six models presented
earlier. Three tests, the likelihood ratio test, the dispersion parameters
based on deviance, and the regression-based test, were used for testing
overdispersion. Since overdispersion was found by these tests, the P model was
excluded from consideration because of a violation of its equidispersion
assumption. Since there were many zeros in crash counts, zero-inflated and
hurdle models were to be considered as candidate models. Once candidate models
were determined, they were estimated using the maximum likelihood estimation
(MLE) technique.
TABLE 1 Probability
Functions of Count Models
|
Model
|
|
|
For
|
For
|
|
P
|
|
|
|
NB
|
|
|
|
ZIP
|
|
|
|
ZINB
|
|
|
|
ZHP
|
|
|
|
ZHNB
|
|
|
|
where  = crash
count at segment i in time j
 =
vector of explanatory variables of the P and NB models
 = vector of explanatory variables of the binary
logit model
 , logit probability of
being a structured zero for the ZIP and ZINB models and being a zero for the
ZHP and ZHNB models
 = dispersion parameter
of the NB model in the ZINB and ZHNB models
 = vector of
coefficient parameters of the P and NB models
 = vector of coefficient parameters of the binary
logit model.
|
To determine the best-fitted model among the
candidate models, two MLE-based information criteria were used: the Akaike information
criterion (AIC) and the Bayesian information criterion (BIC) as shown in Eqs. 3
and 4, respectively (12).
(Eq.
3)
(Eq. 4)
where L = a log-likelihood value of a model
n
= the number of observations
k
= the number of parameters.
In practice, a model with the lowest AIC and BIC
values is preferred. BIC rule-of-thumb criteria commonly used for model
comparison were used for this study. For example, when the difference in BIC
values of two models is greater than 10, a model with a lower BIC value is very
strongly preferred (16).
In addition to the AIC and BIC values, comparisons
between predicted and observed crash counts (or probabilities) were considered
in the model selection. Plotting the difference between the mean predicted probabilities
from estimated models and the observed proportions of each crash count can be
helpful to compare the performance of the model prediction visually (see Figure
1). The mean probabilities were computed using Eq. 5 (17). Being close
to zero in the probability difference indicates that the model fits the data
well.
(Eq. 5)
where m = non-negative integer number (e.g., crash
count; 0, 1, 2, …)
N = number of observations.
Data Collection
Data were collected from two urban interstates (I-64 and
I-264) in Hampton Roads, Virginia. Two 1-mile-long eastbound segments,
hereinafter called W64-05EB and E264-03EB, were selected from each highway; I-64
and I-264 segments have three and four lanes, respectively. Because the
segments are straight basic highway sections (i.e., no horizontal/vertical
curves and no merging/diverging points within or near the segments), impacts of
geometric design factors were minimal. Individual vehicular data were
collected from loop detector stations placed on the segments using a special
traffic counter. Crash data for these segments were extracted from the Virginia
Department of Transportation (VDOT) crash database and the Smart Travel Lab
(STL) at the University of Virginia. These data are described here.
Individual Vehicular Data
Individual vehicular data (i.e., speed and time headway) were
collected using PEEK’s ADR-3000 automatic traffic counters connected to
inductive loop detector stations under normal weather conditions on two
weekdays at W64-05EB and three weekdays at E264-03EB between March and April
2005. The speed and time headway data of all vehicles passing over the loop
detectors per each lane were collected for at least one 24-hour time period.
To minimize the impact of suspicious vehicular data attributable to malfunctioning
of traffic counter and/or communication errors, the individual vehicular data were
filtered. Collected individual vehicular speeds and headways were used to
compute an individual SHD, and then each SHD was aggregated into15-, 30-, and
60-min intervals. This resulted in an aggregated SHD by each time interval. An
average value of SHD by each time interval was used as an aggregate SHD for crash
prediction models.
Crash Data
Two sources of crash data were available for the study sites,
the VDOT crash data based on the police crash report and the STL crash data
based on CCTV cameras and the incident management system maintained by the Center
for Transportation Studies at the University of Virginia. The two crash
datasets were similar in nature. The VDOT crash data have reliable information
on collision types and major reasons of crashes, but the crash occurrence time recorded
by the police officer is often inaccurate. The STL crash data have a fairly accurate
crash occurrence time observed by the traffic management center operators via CCTV
cameras and the incident management system, yet they do not contain other crash
information such as collision type. Since the proposed crash prediction models
required collision types and accurate crash times, the two crash datasets were analyzed
together to produce the final dataset. As discussed earlier, crash counts were
aggregated by 15-, 30-, and 60-min intervals for each of the two sites.
The rear-end collision type, known as the most
frequent, appears to be most suitable for an application of the proposed SHD,
although the SHD can be applied to other collision types including sideswipe.
To include a sufficient number of crash counts for modeling, 6 years
(2000-2005) of crash data were used. Rear-end crashes for these years accounted
for 62.7 percent (161 of 256) crashes at E264-03EB and 37.1 percent (33 of 90)
crashes at W64-05EB on weekdays under normal weather condition.
None of crash datasets contained crash lane
information. In addition, at times, individual vehicular data were not
available on certain lanes because of the malfunction of detectors or
communication errors. As such, the number of crashes for each lane was
estimated in proportion to a lane utilization factor (i.e., a percentage of
traffic using a lane at each segment). It was assumed that crash occurrences
were proportional to the traffic volume using each lane for the case of rear-end
crashes. The lane utilization factor was computed from historical traffic flow
data for these segments. For example, historical data revealed that 34.7 and
40.2 percent of traffic volume on the E264-03EB segment used Lane 1 (left-most
lane) and Lane 2 (middle lane), respectively. Thus, 34.7 and 40.2 percent of
all crashes were assumed to have occurred in Lane 1 and Lane 2, respectively.
RESULTS AND DISCUSSION
Three final datasets by 15-, 30-, and 60-min intervals were prepared
for estimating crash prediction models, with 192, 96, and 48 observations,
respectively. Since this paper aims to show empirically the usefulness of the
SHD for predicting crash occurrences, only the aggregated SHD entered into the
models as an explanatory variable. In the end, one final model was selected
for each time interval. An application example is provided for a better
understanding of how to interpret and utilize the final models.
Excessive Zeros and Overdispersion
Table 2 shows summary statistics for each of the three
intervals. Overdispersion was found in the 30- and 60-min interval datasets,
and considerable portions of zero counts were found in all three datasets.
These findings led to the selection of four candidate models, ZIP, ZINB, ZHP,
and ZHNB, which were estimated using each of the three interval datasets,
resulting in a total of 12 estimated models (four models Í three datasets).
TABLE 2
Crash Counts by Time Intervals and Tests for Overdispersion
|
Time Interval
|
No. of Observations
|
No. of Zeros
(% Total)
|
Min.
(Max.)
|
Mean
(Var.)
|
Overdispersion Test
|
|
LR test
(p-value)*
|
Dispersion Parameter Based on Deviance**
|
Regression-based Test
(p-value)***
|
|
15 min
|
192
|
104 (54.2%)
|
0 (4)
|
0.589 (0.641)
|
0.499
|
0.810
|
0.170
|
|
30 min
|
96
|
32
(33.3%)
|
0 (7)
|
1.167 (2.077)
|
0.037
|
1.118
|
0.048
|
|
60 min
|
48
|
5
(10.4%)
|
0 (13)
|
2.313 (7.113)
|
0
|
1.654
|
0.011
|
*Overdispersion concluded if a p-value
of the likelihood ratio (LR) test is smaller than 0.05.
**Overdispersion concluded if a
dispersion parameter is greater than 1.0.
***Overdispersion concluded if a p-value
of the OLS regression coefficient is smaller than 0.05.
Model Selection
As noted, AIC, BIC, and log-likelihood values were computed
and used to compare model performance for each time interval. Table 3 summarizes
the goodness-of-fit measures for the four candidate models.
TABLE 3 AIC,
BIC, and Log-Likelihood of Estimated Model
|
Time Interval
|
Model
|
AIC
|
BIC
|
Log-Likelihood
|
|
15 min
|
ZIP
|
338.7
|
351.7
|
-165.3
|
|
ZINB
|
340.7
|
356.9
|
-165.3
|
|
ZHP
|
333.5
|
350.5
|
-164.8
|
|
ZHNB
|
326.2
|
348.4
|
-161.8
|
|
30 min
|
ZIP
|
249.8
|
260.0
|
-120.9
|
|
ZINB
|
249.5
|
262.3
|
-119.7
|
|
ZHP
|
248.6
|
262.9
|
-122.3
|
|
ZHNB
|
227.4
|
246.2
|
-111.7
|
|
60 min
|
ZIP
|
193.1
|
200.6
|
-92.6
|
|
ZINB
|
182.6
|
192.0
|
-86.3
|
|
ZHP
|
178.6
|
190.1
|
-87.3
|
|
ZHNB
|
149.2
|
164.6
|
-72.6
|
For the 15-min interval, the ZHNB model is the
best based on AIC, BIC, and log-likelihood values because these values are
smaller than those of the other three models. The difference in BIC between the
ZHNB and ZINB models is more than 6, which indicates that the ZHNB model is strongly
preferred over the ZINB model according to the BIC rule-of-thumb criteria (16).
For the 30-min interval, the ZHNB model is also
the best. The AIC and log-likelihood values are much smaller than those of the
other three models. In addition, its BIC value is smaller than those of the
other three models by more than 10. This concludes that the ZHNB model is very
strongly preferred over the others.
For the 60-min interval, the ZHNB model is again
the best. Its BIC value is much smaller than those of the ZIP and ZHP models
by 37 and 25, respectively. Interestingly, the ZHNB model is the best for all
three intervals based on AIC and BIC. Moreover, as the interval increases from
15 to 60 min, the degree of the performance toward the ZHNB model over the ZIP,
ZINB, and ZHP models becomes greater in terms of BIC (i.e., the difference in
BIC values between the ZHNB and the other three becomes larger).
Figure 1 depicts differences between observed
and predicted probabilities of crash counts for the 60-min interval. All four
models look similar at high crash counts, say, 6 or more. However, at low
counts, the ZHNB model outperforms the other three and fits the data particularly
well at very low counts (e.g., zero and 1 crash). This visual comparison
supports the conclusion that the ZHNB model is the best based on AIC and BIC
values. Findings from the visual comparison for the 15- and 30-min intervals,
although the graphs are not presented here, are consistent with the previous
findings.
FIGURE 1 Comparison
of observed and predicted mean probabilities of crash occurrence for 60-min
interval. (ZIP = zero-inflated Poisson model; ZINB = zero-inflated negative
binomial model; ZHP = zero-hurdle Poisson model; ZHNB = zero-hurdle negative
binomial model).
In summary, the ZHNB model was determined to be final
crash prediction model for the all three intervals, implying that it
outperformed the ZIP, ZINB, and ZNH models. This means that excessive zeros
contribute to an overdispersion, and the assumption regarding two types of
zeros for zero-inflated models is not suitable for the data. Moreover, the
longer time interval was found to offer stronger statistical preference on the
ZHNB model in terms of BIC. However, choosing an appropriate time interval for
aggregating data is also dependent on the study design and purpose.
Final Crash Prediction Model
Table 4 shows the final ZHNB models for the three time
intervals. The aggregated SHD was found to be statistically significant in
both the logit and NB models of the ZHNB model, implying the SHD influences
both whether a crash would occur and how many crashes would occur. A positive
coefficient of the SHD in the logit model implies that an increase in the SHD is
likely to increase the probability that a road segment will have at least one
crash (i.e., crossing the zero hurdle). Meanwhile, a positive coefficient of
the SHD in the NB model means that an increase in the SHD is likely to increase
the expected number of crashes on that segment. This implies that as more
drivers maintain an unsafe distance from their leading vehicle, producing a larger
value of the aggregated SHD, their road segment will have a higher probability
of crash occurrence and more crashes.
TABLE 4 Estimates
of ZHNB Models
|
Time Interval
|
15 min
|
30 min
|
60 min
|
|
Coef.
|
p-value
|
Coef.
|
p-value
|
Coef.
|
p-value
|
|
Logit
|
Agg. SHD*
|
0.000187
|
0.000
|
0.000139
|
0.000
|
0.000568
|
0.149
|
|
Constant
|
-1.545
|
0.000
|
-0.848
|
0.019
|
-0.235
|
0.755
|
|
NB
|
Agg. SHD*
|
0.000195
|
0.007
|
0.000102
|
0.003
|
0.0000527
|
0.006
|
|
Constant
|
-15.49
|
0.591
|
-17.24
|
0.001
|
-2.576
|
0.234
|
|
Ln(alpha)
|
12.60
|
0.662
|
15.55
|
0.003
|
1.624
|
0.466
|
|
No. of Observations
|
192
|
96
|
48
|
|
Log-Likelihood
|
-161.1
|
-111.7
|
-72.6
|
*Agg. SHD is the aggregated SHD, which
was calculated by summing individual SHDs by the specified intervals, 15, 30,
and 60 minutes.
Model Application
Since the SHD in the model was a summation of all negative SHD
values of consecutive-vehicle pairs within the specified time interval, a
straight interpretation of the estimated coefficients of parameters is difficult
to make with respect to operational environments such as average driving
speeds. To illustrate a potential application of the proposed crash prediction
models, the following hypothetical example is given. Suppose a variable
message sign is installed to warn drivers following too close and it affected
drivers’ behavior, resulting in a 10 percent reduction in the aggregated SHDs
for 1 hour while no significant changes in other factors (e.g., volume and
speeds) were observed.
For this example, the study data were used to
provide basic inputs to the final models. The individual vehicular data collected
on weekdays under normal weather condition from two lanes of four on a 1-mile
segment with a 55 mph posted speed limit were used to calculate the aggregated
hourly SHD during three times of day: peak (6 a.m.–9 a.m. and 4 p.m.–7 p.m.),
mid-day (9 a.m.–4.p.m.), and night time (7 p.m.–6 a.m.). The calculated SHD
values for the three times of day were 55,000 ft, 35,000 ft, and 3,000 ft,
respectively. Inputting these SHD values into the ZHNB model developed in this
paper for the 60-min interval resulted in predictions of rear-end crashes for
the three times of day on that segment. Predicted values, probabilities, and expected
numbers of such crashes were calculated with and without the impact of the
variable message sign and are presented in Table 5.
TABLE 5 Safety
Effects of Hypothetical Variable Message Sign (10% Decrease in Aggregated SHD for
60-min Interval)
|
Time of Day
|
Parameter
|
Before
|
After
|
% Change
|
|
Peak
(6 a.m.–9 a.m.
and
4 p.m.–7 p.m.)
|
Agg. SHD (ft)
|
55,000
|
49,500
|
-10.0
|
|
|
3.50E-14
|
7.95E-13
|
2,168.7
|
|
|
0.518
|
0.572
|
10.4
|
|
|
0.482
|
0.428
|
-11.2
|
|
|
4.105
|
3.41
|
-16.9
|
|
Mid-Day
(9 a.m.–4 p.m.)
|
Agg. SHD (ft)
|
35,000
|
31,500
|
-10.0
|
|
|
2.98E-09
|
2.17E-08
|
629.1
|
|
|
0.722
|
0.758
|
5.0
|
|
|
0.278
|
0.242
|
-12.9
|
|
|
2.226
|
2.039
|
-8.4
|
|
Night Time
(7 p.m.–6 a.m.)
|
Agg. SHD (ft)
|
3,000
|
2,700
|
-10.0
|
|
|
0.187
|
0.215
|
14.6
|
|
|
0.776
|
0.75
|
-3.4
|
|
|
0.037
|
0.035
|
-5.4
|
|
|
1.022
|
0.984
|
-3.7
|
The aggregated SHD
values were assumed to be uniformly decreased by 10 percent throughout a day as
a result of the new variable message sign. In response to the installation of
the variable message sign, during a 60-min period in 6 years, the probabilities
of observing no rear-end crash, P (y = 0), were considerably
increased, and the probabilities of observing more than three rear-end crashes,
P (y = 3 or more), were decreased regardless of time of day. Although
percentage changes in the probabilities of zero crashes are enormous for peak
and mid-day times, the probabilities after the sign installed are still practically
zero.
For the peak time period, a 10
percent decrease in the aggregated SHD results in a 16.9 percent reduction in the
expected number of rear-end crashes; for the mid-day and night time time
periods, the reduction is 8.4 percent and 3.7 percent, respectively. This suggests
that the variable message sign achieving a 10 percent reduction in the
aggregated SHD would provide more safety benefits during peak times than during
non-peak times.
CONCLUSIONS AND RECOMMENDATIONS
The major findings of the study were as follows:
·
A
statistically significant positive relationship was found between the
aggregated SHD and the probability of observing rear-end crashes and the number
of such crashes for all three intervals considered: 15, 30, and 60 min. This
indicates that the aggregated SHD was proven to be useful in predicting the
likelihood and number of rear-end crashes in an advanced transportation system.
·
The
ZHNB model outperformed the ZHP, ZIP, and ZINB models for all three intervals,
and its superiority over the three models became stronger as the time interval
increased. This suggests that the excessive zeros contribute to an overdispersion,
and the assumption of two types of zeros in zero-inflated models may not be suitable
for the crash data analysis.
The study showed a promising opportunity in
traffic safety analysis by applying SHD based individual vehicular car
following behavior data in crash prediction. It also showed that the
zero-inflated models with excessive zeros that have gained popularity recently
for analyzing crash data may not be appropriate in that the ZHNB model was
found to be superior to its zero-inflated counterpart: the ZINB model. This
suggests that the presence of excessive zeros does not warrant the use of
zero-inflated models, and characteristics of the zero counts (i.e., existence
of structured zeros in the data) should be considered in selecting appropriate models.
The findings in this study can be used with
future vehicle-infrastructure integration environment. The approach with a vehicle-based
crash predictor would enable traffic engineers to have a reliable safety
evaluation by location, time, and transportation management strategy under
various traffic flow conditions. For example, a traffic engineer could
evaluate the safety impact of typical transportation management strategies at
particular freeway locations prior to their implementation by using well-calibrated
and validated microscopic simulation models equipped with an
individual-vehicle-based crash risk predictor such as the proposed SHD.
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