Andrew P. Tarko* Associate Professor of Civil …tarko/research/conf...Waiting for crashes to occur...

ESTIMATING THE FREQUENCY OF CRASHES AS EXTREME TRAFFIC EVENTS

Andrew P. Tarko*Associate Professor of Civil Engineering

Purdue UniversitySchool of Civil Engineering

550 Stadium Mall DrWest Lafayette, IN 47907-2051Email: [email protected]

Tel: (765) 494-5027Fax: (765) 496-7996

*Corresponding Author

Praprut SongchitruksaGraduate Research Assistant

Purdue UniversitySchool of Civil Engineering

550 Stadium Mall DrWest Lafayette, IN 47907-2051

Email: [email protected]: (765) 494-2206Fax: (765) 496-7996

Submitted for presentation at the 84th Annual Meeting of the Transportation Research Board,

January 9-13, 2005, Washington D.C.

Length of manuscript: 7,486 words, 1 table, and 9 figures.

TRB 2005 Annual Meeting CD-ROM Paper revised from original submittal.

Tarko and Songchitruksa 2

ABSTRACT

Recent applications of modern technologies to transportation have increased the need for rapid evaluation of safety; and at the same time, they have presented new possibilities to measuresafety. This paper discusses an extreme value theory approach to safety estimation using observable traffic characteristics. Unlike the past research on surrogate measures of safety, this approach uses traffic characteristics that represent crash-free traffic operations, as well as crash occurrence. The fundamentals of the method are presented, emphasizing how it addresses the issues found in the other approaches. Example estimation for a selected intersection is presented to explain the details of the method. Proposed future research closes the presentation.

It has to be stressed that the purpose of this presentation is to point out an intriguing and attractive new direction in surrogate measures of safety and not to provide unquestionable evidence that the proposed method is valid. We will try to argue that the proposed new direction is worthy of further exploration.

Keywords: extreme value theory, traffic safety, safety evaluation, risk estimation, traffic conflicts



1. INTRODUCTION

The frequency of crashes and their severity are the measures of road safety preferred by transportation agencies and by safety specialists for their unquestionable connection with safety. Unfortunately, due to the random and infrequent occurrence of crashes, collecting crash data at intersections and road segments can require several years of monitoring; and even then, the estimation precision of the crash frequency is sometimes far from desirable. This intricacy hampers safety modeling, evaluation, and enhancement. Identification of hazards and evaluation of safety impacts do not necessarily yield timely and even correct answers, which may result in an inefficient allocation of resources. Waiting for crashes to occur is one of the most undesirable necessities of the current methods.

Modern technologies are increasingly being used in transportation systems. However, technological innovations applied to transportation have a relatively short life compared to the traditional infrastructure, and their effects on safety cannot be easily evaluated using crash-based methods. Recent technological progress has opened unprecedented opportunities for collecting microscopic traffic data that may carry safety information.

Given its serious implications, the fundamental issue of measuring safety has been attracting research for a long time. One direction of research was to try to better understand the properties of crash occurrence to ensure adequate statistical tools for estimating crash frequencies and the wealth of research on this subject is impressive. Over the last three decades, researchers arrived at several important conclusions that laid the groundwork for the modern statistical modeling of crash occurrence. These findings include: Poisson variability of counts over time, negative binomial variability of counts across locations for the same period, the regression-to-mean effect, and Bayesian combination of crash counts and regression estimates (1-5).

The second research direction was aimed at finding a measure of safety more frequent than crashes. A good surrogate measure is one that does not require a long period for collecting data and that is related to crashes to allow its conversion to crash frequency and severity. Numerous surrogate safety measures have been proposed. The most acknowledged ones include traffic conflicts (6-9), critical events (10-11), acceleration noise (12), post-encroachment time (13), and time-integrated time-to-collision (14). None of these was convincingly confirmed for its linkage with crashes. Other proposed measures are: volume, speed, delay, accepted gaps, headways, shock-waves, and deceleration-to-safety-time (15). Although some of the latter measures are safety factors rather than surrogate measures, they are listed to adequately reflect the past work.

Over the past three decades, traffic conflict technique (TCT) attracted the most research attention. The technique has been refined over the years to eliminate the dependence on human judgment present in the original version. The recent practice classifies events as conflicts and determines their severity based on measured time-to-collision (8, 16). The validity of TCT must be judged by how good the TCT is in estimating the expected number of crashes (17) or the adequacy in the correlation between observed conflict counts and crash records (7). Although several researchers reported the successful use of TCT (18-20), some researchers questioned TCT seriously. The evaluation of TCT by Williams (21) failed to establish the relationship between crashes and



conflicts. One of the reasons pointed out by Williams for the inconsistency in the research findings was the lack of standard definitions for either traffic conflicts or crashes.

The next section of this paper presents the authors’ hypothesis as to why the past research had difficulties confirming the existence of a relationship between the proposed surrogate measures and crashes. Then, we introduce a new framework that bridges crashes and other traffic events and allows for estimating crash frequency based on a limited time period of traffic observations. The extreme value theory is proposed for this task, and the method is illustrated using an example of right-angle crashes at a signalized intersection. The closing indicates the future research needed to further advance the proposed method.

It must be stressed that the purpose of this presentation is to point out an intriguing and attractive new direction in surrogate measures of safety and not to provide unquestionable evidence that the proposed method is valid. None of the numerous past publications on alternative surrogate measures accomplished this over nearly 30 years. We will try to argue that the proposed new direction is worth exploring.

2. PROPOSED FRAMEWORK

Although the introduction of time-to-collision has helped eliminate some of the issues of early surrogate measures, we claim that the main source of the past difficulties to demonstrate any transferable linkage between the frequencies of surrogate events and crashes is the deficiency of the relationship form proposed in the past. This point is of fundamental importance and we will further discuss this, starting with the well-known concept of pyramid of events.

Any vehicle interaction, or traffic event, has a certain proximity to collision. Some of the events,such as undisturbed passages through an intersection, are much further from a crash event than a near-miss situation where only a rapid evasive maneuver prevents a collision. The wide spectrum of crashes and non-crash events can be viewed as a pyramid shown in Figure 1 (22). “Safer” and more frequent events occupy the pyramid’s lower part while more “dangerous” events are located above. Past research attempted to demonstrate that the frequency of collisions (the top of the pyramid) and the frequency of other events are related in the form of the following expression:

,C k E= ⋅ (1)

where C = crash frequency (crashes/year), k = coefficient (typically estimated with regression) and E = surrogate measure; here frequency of non-crash events (vehicle passages, traffic conflicts, etc.) E is sometimes called exposure, particularly when E is the number of vehicle passages.

Before we further discuss the proposed model, let us clarify the measurement of crash proximity by the use of a specific proximity measure. Among many possibilities, two appealing measures of proximity mentioned in the literature are time-to-collision and post-encroachment time (13). The first measure is the time remaining to a hypothetical collision at the moment when a driver initiates an evasive maneuver, assuming the driver proceeds at the unchanged speed if the evasive



maneuver is not undertaken. This measure takes low values for severe events (higher parts of the pyramid of events) and applies them to any type of collision. Unfortunately, a single and low value may represent a crash and a non-crash event depending on the effectiveness of the evasive maneuver. Some evasive maneuvers initiated 0.5 second prior to the collision may be successful while some others may not. In some circumstances, the time-to-collision may not exist if none of the involved drivers attempt to avoid the collision. Definitive boundaries of event classes in the pyramid shown in Figure 1 do not exist for the time-to-collision measure as defined.

The second measure discussed here is post-encroachment time, which is defined as the time between the moments when the first vehicle leaves the conflict spot and the second vehicle enters it. The post-encroachment time is applicable to collisions between vehicles moving along different paths, for example, crossing lanes of traffic.. It has a definitive boundary of zero seconds between a crash and other events. Let us select post-encroachment time to measure the crash proximity of traffic events and put it in the statistical context. The pyramid of events can be replaced with a distribution of post-encroachment times – a statistical representation of traffic event continuum (Figure 2). The crash area corresponds to the frequency of crashes; the dangerous-event area corresponds to the frequency of dangerous events, and so on.

Let us consider the relationship presented in Equation (1). It postulates that the frequency of crashes is proportional to the frequency of a specific class of events, e.g., dangerous events. The postulated relationship meets the intuitively acceptable boundary condition of zero crash frequency at a location with zero frequency of dangerous events. For the sake of convenience,without loss of generality, let us assume that both frequencies are expressed in the same units, for example, events per year. Dividing the left-hand side of Equation (1) by the frequency of all traffic events yields the crash area under the probability density curve in Figure 2. Similarly, dividing the right-hand side of Equation (1) by the frequency of all events yields the dangerous-event area in Figure 2. Thus, Equation (1) implies that the crash and dangerous-event areas in Figure 2 are proportional across similar locations. There is no basis for this claim. In fact, our preliminary results indicate that different locations, even of the same type, exhibit significantly different distributions of post-encroachment times. Consequently, the relationship in Equation (1) calibrated for some locations cannot be transferred to other locations without adding additional variability to the crash frequency estimates. This fact is already addressed in safety performance functions where the E variable is traffic volume and other covariates are typically present in Equation (1) to reduce the variability of the estimate of C.

Instead of assuming a constant ratio for frequencies in Equation (1) or adding additional covariates to better explain the variability of the C estimates, we propose to estimate this ratio for a location from data collected at that location. Equation (1) has to be slightly modified to better incorporate the known statistical properties of crashes.

C R E= ⋅ (2)

where C = crash frequency (crashes/year), R = likelihood of crash associated with unit exposure, and E = exposure (exposure units/year).

For now, let us assume that unit exposure is a single traffic event. The exposure in Equation (2) includes all the events represented by the post-encroachment times shorter than a pre-determined value. Thus, exposure also includes crashes. This change in the definition of E allows interpreting R as a proportion of traffic events that are crashes. The crash occurrence can now be viewed as a success in the process of independently drawn traffic events represented by E. Since the



likelihood of success (crash) is small and the exposure is large, the number of successes (crashes) is known to follow Poisson variability.

In the proposed framework, estimation of crash frequency includes estimation of the exposure and estimation of the likelihood of a crash associated with the unit exposure. For the sake of brevity, the likelihood of crash will be referred to as “risk.” The risk is defined and estimated inrelation to the exposure unit. The method of estimating the risk may require a specific definition of exposure unit. It can be a traffic event (event-based) or a time interval (time-based). In the latter case, the risk is the likelihood of crash in a unit of time interval. If the time interval is sufficiently short, then the likelihood of two crashes in one interval is negligible. Risk R preserves its meaning as the proportion of the time intervals in one year that experience crashes. Equation (2) is applicable to time-based risk and estimation of time-based exposure is straightforward.

The proposed estimation of crash frequency can be summarized in the following steps: (a) select the crash proximity measure, (b) select the exposure unit, (c) estimate the risk R associated with unit exposure, (d) estimate the exposure E, and (e) estimate the crash frequency C using Equation(2). A method of estimating risk R is presented in subsequent sections.

3. DATA COLLECTION

We will illustrate the proposed method by applying it to a right-angle collision, which is defined as a collision between through vehicles from two adjacent approaches. The potential collision spot of right-angle crashes is relatively static and well detached from the intersection approaches. This characteristic simplifies the manual measurement procedure and is particularly favorable for automated data collection in the future, given the current stage of development in image processing technology.

Now, let us consider the crossing passage event of two conflicting vehicles. Let us define conflict zone as a region where two crossing traffic flows intersect. Conflict spot is an intersection of two vehicle paths in a conflict zone. For instance, a one-way two-lane road crossing with a one-way one-lane road generates one conflict zone and two potential conflict spots for straight lanes. For an intersection where all approaches have only one lane, a conflict zone is equivalent to a conflict spot. Let us define gap time (GT) as the time between the entries into the conflict spot of two vehicles. Any crossing passage event would generate a value of GT. The GT can be broken downinto two components based on the terms coined by Allen et al. (13): encroachment time (ET) and post-encroachment time (PET). ET is the time that the first vehicle entering the conflict spot infringes upon the right-of-way of the second vehicle. PET is the time it takes from the end of the right-of-way infringement of the first vehicle for the second vehicle to reach the conflict spot. The ET and PET can be explained graphically as depicted in Figure 3. GT is a summation of ET and PET.

GT would be a good measure of crash proximity only if the first vehicles occupy the conflict for the same amount of time. This assumption does not hold true in reality because vehicles vary in length and move at different speeds. PET is a better choice. Its value of zero or less represents crash occurrence. Allen et al. (13) also concluded that PET was the most promising indicator among others for its relative ease of measurement and safety implication. PET has a specific



value for any crossing event (consecutive two passages from conflicting directions). Unlike time-to-collision, the PET measurement does not involve human judgment and is relatively easier for automation – an important factor of method practicality.

To demonstrate the approach, we selected the suburban signalized intersection of SR-26 and 18th Street in Lafayette, Indiana, from the list of intersections identified by motorists as unsafe (23). This is a four-leg intersection with one through lane and one left-turn lane on each approach. All left turns are permitted except the eastbound left turn which is protected-permitted. Post-encroachment data were collected by videotaping the traffic at this intersection using the Purdue University van-based mobile traffic laboratory (see Figure 4). The van is equipped with a 42-ft.pneumatic mast with two surveillance cameras. Eight hours of traffic (9:00 AM – 4:00 PM and 4:30 PM – 5:30 PM) were recorded in a digital format on April 8, 2003. All four conflict zones were observable in the camera field of view.

The PET values were measured by watching the video material frame by frame. The manual method was used to reduce the measurement error to a minimum. For the manual measurement method, the recorded video clips at the intersections were digitized at the resolution of 30 fps,which gives the attainable average precision of 1/30 s. Conflict spots were marked on the video monitor with Autoscope virtual detectors (Figure 5). The virtual conflict spots assisted the human reviewer in determining frames where vehicles entered and exited the conflict spots. In a two-hour test of measurement accuracy, two trained observers extracted the PET data from the same video material. Each person configured his/her own conflict spots. The standard error caused by the inter-person variation was 0.2 second. When the two persons used the same pre-configured conflict spots, the variance of the measurement errors was reduced by approximately one-half. The results of the accuracy test were considered satisfactory. A trained observer needed approximately two hours to extract data from one hour of video material. This disadvantage is acceptable in preliminary research when the concept of the method is evaluated but must be overcome in a large-scale study and at the implementation level. More efficient techniques for data collection are beyond the scope of this paper.

In the process of data extraction, we recorded the t1, t2, and t3 as shown in Figure 3 for each crossing event and then computed GT, ET and, PET. PET values larger than eight seconds were not recorded since they are not useful for the estimation purpose. Corresponding PET values were too long to believe that they are in close proximity with crashes. A total of 573 PET data points were extracted from the eight-hour video clips. Figure 6 shows the PET variation over time. In the next section, we will discuss a framework proposed to estimate crash risk R from the collected PET data.

4. EXTREME VALUE THEORY FOR RISK ESTIMATION

Extreme value theory (EVT) has emerged as an important statistical discipline which has found its way to a wide range of applications. Some examples of these include alloy strength prediction, ocean wave modeling, wind engineering, thermodynamics of earthquake, and assessment of meteorological change (24-25). The distinguishing feature of extreme value theory is the capability to model the stochastic behavior of the process that is unusually large or small in nature. This extreme behavior is typically very rare and unobservable within a reasonable amount



of data collection period. The EVT often involves the challenge to estimate the probability of extreme events over an extended period of time given a very short and limited historical data. The extreme value paradigm comprises a principle for model extrapolation based on the implementation of mathematical limits as finite-level approximations. The key implicit assumption of EVT is that the underlying stochastic behavior of the process being modeled is sufficiently smooth to enable extrapolations to unobserved levels. No other credible alternative has been proposed to date.

Considering a design wind speed example, a structure designer wants to know the strongest wind expected within 50 and 200 years. The design structure is supposed to withstand the 50-year wind and be at the edge of collapsing in the 200-year wind. Weather data are utilized to estimate the strongest wind. A selected extreme value distribution is fit to daily maximum wind measurements and then used to estimate the strongest wind for the assumed return periods of 50 and 200 years.

There are several alternative distributions to extreme value modeling depending on how we choose to treat the data as extremes. The first method is to block the data into fixed time intervals and to pick only the extremes from each block to estimate the model. The resulting distribution of these extremes is known to follow generalized extreme value distribution (GEV). The second method is the modified version of the first method in that, instead of choosing only the extremes, the r largest values of each block can be used in the modeling. Using the knowledge of the order statistics in the model derivation, this method is called the r largest order statistic model. The third method defines the extremes by selecting the data that exceed the specified threshold. By selecting the appropriate threshold, the threshold excesses are known to follow generalized Pareto distribution (GP). In this paper, we will use PET extremes to estimate risk and frequency of crashes using the GEV model

1/

( ) exp 1 ,z

G z

ξµ

ξσ

−−

= − +

(3)

defined on { : 1 ( - ) / 0}z z µξ σ+ > , where z is negated PET values, µ is a location parameter,σ is a positive scale parameter, and ξ is a shape parameter. When ξ > 0 and ξ < 0, it corresponds to type II and III of extreme value distributions. When ξ = 0, which is interpreted as a limit of GEV as ξ→ 0, ( )G z becomes the Gumbel family (type I),

( ) exp exp , - z .z

G zµ

σ−

= − − ∞ < < ∞

(4)

Estimates of extreme quantiles of the block maximum distribution are obtained by inverting (3) and (4),

{ }

{ }

1 log(1 ) for 0

log log(1 ) for 0

,p

z pp

σ ξµ ξξ

µ σ ξ

−− − − − ≠

=− − − =

(5)



where G(zp) = 1-p. In common terminology, zp is the return level associated with return period 1/p. In other words, the level zp is expected to be exceeded on average once every 1/p periods. More precisely, zp is exceeded by the block maximum in any particular period with probability p.

When ˆ 0ξ < , the upper bound of the return level zp corresponding to zp, p = 0 can be computed as

0ˆˆ ˆˆ / .z µ σ ξ= − (6)

To graph a return level plot, defining yp = -log(1-p), so that

1 for 0

log for 0

.yp

z pσ y p

σ ξµ ξξ

µ ξ

−− − ≠=

− =

(7)

It follows that, if zp is plotted against yp on a logarithmic scale – or equivalently, if zp is plotted against log yp – the plot is linear in case ξ = 0. If ξ < 0, the plot is convex with asymptotic limit as defined in (6). If ξ > 0, the plot is concave and it has no finite bound. The return level plot is useful in highlighting the effect of extrapolation at the tail of the distribution.

The model estimation, statistical inference, model evaluation, model interpretation, and crash estimation methods are presented along with the results using the collected PET data in the next section.

5. ANALYSIS AND RESULTS

To fit the GEV model, we blocked the PET data into 15-minute intervals, thus producing 32 extremes from eight hours of observation. The 15-minute interval selected in this study is not arbitrary but it is a compromise between the uncertainty of model estimates and the validity of the extreme value assumption. Through simulation experiments, we have found that 1-hour interval would be desirable if we have weeks of data for the extreme value modeling. However, due to time and resource constraints, it is not possible to obtain such a large amount of data. The intervals shorter 15 minutes would lead to a bias for model estimates. Therefore, a 15-minute interval was selected in this study to meet all the aforementioned constraints.

In order to have the GEV model described previously applicable to our data, we negated the PET values so that the negative block minima are equivalent to the block maxima. The probability that the PET will be zero or less is equivalent to a risk of experiencing a crash in any time interval. There is a bias-variance trade-off in choosing the length of the time interval. A shorter time interval will produce more data points, resulting in smaller variances in parameter estimates, but it might violate the asymptotic assumption and yield biased estimates.



5.1 MODEL ESTIMATION

Many techniques have been proposed for parameter estimation in extreme value models,including graphical techniques, method of moments, and likelihood-based methods. We selected the likelihood-based method for its all-around utility and adaptability to complex model-building.

Under the assumption that 1,..., mZ Z are independent variables having the GEV distribution, the

log-likelihood function when 0ξ ≠ is

( )1/

1 1

1, , log 1 log 1 1 .

m mi i

i i

z zl m

ξµ µµ σ ξ σ ξ ξ

ξ σ σ

−

= =

− −= − − + + − +∑ ∑

(8)

The case 0ξ = requires a separate treatment using the Gumbel limit of the GEV distribution. The log-likelihood function can then be written as follows

1 1( , ) log exp .

m mi i

i i

z zl m

µ µµ σ σ

σ σ= =

− −= − − − −∑ ∑

(9)

The parameter estimates obtained from for the PET extremes are summarized in Table 1. The standard errors of estimates have been obtained using the standard asymptotic likelihood results.

The joint distribution of ( )ˆˆ ˆ, ,µ σ ξ can be approximated by multivariate normal with mean

( ), ,µ σ ξ and variance-covariance matrix equal to the inverse of the observed information matrix

evaluated at the maximum likelihood estimates.

A distribution tail and the consequent safety estimate depend strongly on the shape parameter of the GEV distribution. We use the profile likelihood method (also known as concentrated likelihood) to draw the statistical inference on the shape parameter. The profile log-likelihood of a shape parameter is obtained by fixing a shape parameter and then maximizing the log-likelihood with respect to the remaining parameters. This is repeated for a specified range of a parameter of interest. The corresponding maximized values of the log-likelihood can be used to obtain approximate confidence intervals, which are generally more accurate than those obtained from the delta method. Under suitable regularity conditions and for a large sample,

{ } { }* *

12( ) 2 max ( , , ) max ( , , )

,, ,.pD l lξ µ σ ξ µ σ ξ χ

µ σµ σ ξ= −

: (10)

For shape parameter ξ , { }* *: ( )pC D cα αξ ξ= ≤ is the (1 )α− % confidence interval, where cα

is the (1 )%α− quantile of the 2

1χ distribution. The confidence interval of the shape parameter

using the profile likelihood method is shown in Figure 7.

Due to the sample’s small size, estimated GEV distribution may indicate zero risk which is intuitively incorrect. A risk of crash at a location can be very small but cannot be zero. To be consistent with the non-zero risk assumption the shape parameter ξ has been set at zero. This does



not contradict the results of fitting the GEV distribution because the confidence interval for the shape parameter includes the zero value. The resulted two-parameter Gumbel distribution was fit to the data again. The Gumbel distribution may be used only if it does not contradict the data.

A negative shape parameter would imply the possibility that PET never takes negative or zero values and therefore no crashes occur even in perpetuity. A non-negative shape parameter would imply otherwise. The negative shape parameter is possible, at least in three cases: (a) the location is indeed safe, (b) the location experiences crashes but the sample is small and the estimate is unreliable, and (c) the location experiences crashes but the PET values do not reveal the risk of crashes. The third case will be discussed in the remaining part of the paper for its serious methodological implications.

5.2 MODEL GOODNESS-OF-FIT

The goodness-of-fit of the model can be evaluated using probability, quantile, and return level plots (see Figure 8). These plots are used to assess the model adequacy for the fitted data. With the ordered block maxima (negative PETs) (1) (2) ( )... mz z z≤ ≤ ≤ the empirical distribution

function is defined as

( )( )1

.i

iG z

m′ =

+(11)

The corresponding fitted distribution ( )ˆ ( )iG z is obtained by substituting ( )iz into Equation (4)

evaluated at ˆ ˆ( , )µ σ . A probability plot consists of the points { }( ) ( )ˆ ( ), ( ) ; 1,...,i iG z G z i m′ = .

Similarly, the quantile plot consists of the points 1

( )ˆ , ; 1,...,

1i

iG z i m

m− =

+

. The quantile

plot has an advantage over the probability plot in that the data points are not clustered as ( )iz

increases. Large values of ( )iz are usually a region of most interest in the extreme value

modeling. Any substantial departure from a unit diagonal for both probability and quantile plots indicates a model inadequacy.

The return level plot consists of the locus of points { }ˆ(log , ) : 0 1p py z p< < where ˆpz is the

maximum likelihood estimate of pz . If the Gumbel model is suitable for the data, the model-

based curve and empirical estimates should be in a reasonable agreement. For completeness, the probability density function was plotted against the histogram of the actual data. However, this is generally less informative than the other plots since the shape of the histogram can vary substantially depending on the choice of grouping intervals.

The near-linear pattern of points in the probability and quantile plots in Figure 8 supports the assumption of Gumbel distribution. The return level curve is unbounded as a consequence of the Gumbel limit. Finally, the corresponding density estimate seems consistent with the histogram of the data. All the four diagnostic plots lend support to the fitted Gumbel model.



5.3 EVALUATION OF CRASH FREQUENCY ESTIMATE

The evaluation of the Gumbel model presented in the previous section, although useful in checking the model selection and fitness, does not consider actual crash experience. A more convincing evaluation of the model’s validity is needed. It can be done by comparing the crash frequency estimates produced using the proposed method with the estimates obtained from crash data. We present one possible approach for an example intersection.

First, we have to estimate the frequency of crashes utilizing the estimated Gumbel model. Based on the premise that a PET shorter than zero represents a crash, the risk of experiencing a crash in a 15-minute time interval is equivalent to the probability that a PET is zero or less. The risk estimated by Gumbel assumption is of the form

ˆˆˆ 1 (0) 1 exp expˆ

.R Gµσ

= − = − −

(12)

Note that ˆ1/ R is a return period, or in other words, the expected number of time intervals between consecutive crash occurrences. Using the results listed in Table 1 and Equation (12) the

risk estimate is ( ){ }1 exp exp 4.079 / 0.633 0.001558− − − = .

The annual crash frequency is a product of the crash risk R and the exposure E. Because the risk is defined as the likelihood of a crash in a 15-minute interval and it was estimated for the daytime conditions, the annual exposure is estimated as the number of daytime hours during one year. Based on the meteorological data, there are approximately 4,413 daytime hours per year for the studied site, or an average of 12.09 hours per day. The annual daytime crash frequency is estimated as

( )ˆ ˆ ˆ 0.001558 4413 hours/year 4 intervals/hour 27.5.C R E= ⋅ = ⋅ ⋅ = (13)

The question now is how well the obtained annual crash frequency compares to the frequency of recorded crashes. The four-year crash records for the 1997-2000 period included 58 crashes, 32 of which were right-angle crashes between vehicles moving straight ahead. To ensure the quality of the data, all the crashes at this location were verified with the police crash reports. Since the data collection is limited to only the daytime period, it was necessary to remove all the nighttime crashes, which reduced the number of studied crashes to 18 for a four-year period or 4.5 right-angle crashes per year. The crash frequency estimated from the PET data is higher than the frequency estimated from crash data.

To complete the comparison, the obtained difference has to be checked for its statistical significance because both of the compared estimates are random variables. The confidence interval for crash frequency estimate from the PET data will be determined as follows. This estimate is a product of the risk estimate and the annual exposure. Because the annual exposure is known with a high degree of certainty, the only source of variability in the crash frequency estimate is the risk estimate. We will consider the risk reciprocal or the return period 1/R. We will use the profile likelihood method to obtain the confidence intervals for the return period, which in turn can be transformed to the confidence intervals for the crash frequency estimate.



Let *

1z

mR

= be a return period associated with a specified return level z*. We can obtain

confidence intervals of z

m ∗ for any specified return level z including z = 0 (crash occurrence

level) using a profile likelihood method. Numerical evaluation of the profile likelihood in this case is not as straightforward as in the case of the shape parameter for the GEV model. This requires a re-parameterization of the GEV model, so that

zm ∗ is one of the model parameters,

after which the profile log-likelihood is obtained by maximization with respect to the remaining parameters in the usual way. For the Gumbel model, the re-parameterization is

1log log 1 .

z

zm

µ σ∗

∗= + − −

(14)

The replacement µ in Equation (9) with Equation (14) leads to a desirable function for z

m ∗ .

Figure 9 shows the profile log-likelihood of the return period evaluated at the zero return level for the fitted Gumbel model. The asymmetric profile surface of the plot confirms the benefit of this method over the delta method, which assumes a symmetric range of confidence intervals about the mean. The horizontal line denotes the log-likelihood value at 95% confidence level, which corresponds to the range of the return period (114, 4647). Dividing the exposure by these return periods yield the range (3.8, 154.8) for the crash frequency estimate obtained from the Gumbel model.

Using the method described in (26), the 95% crash-based Poisson confidence interval of crash frequency estimate is (2.7, 7.1) per year. The two confidence intervals overlap, which indicates that the true crash frequency lies inside the confidence interval of the PET-based crash frequency estimate, which does not provide a basis for rejecting the estimate as incorrect. Although the 95% confidence interval obtained from the model is much larger than the traditional crash-based estimation approach, these preliminary results are quite encouraging. The confidence interval of the proposed method can be narrowed by increasing the period of data collection.

5.4 ADDITIONAL REMARKS ON VALIDATION

This paper intended to propose and demonstrate the first use of extreme value theory, through short-term traffic observations, to estimate crash frequencies.. The obtained results are quiteencouraging, although the evaluation scope falls short of the requirements for model validation. A considerable effort is needed to further investigate various methodological aspects of the proposed approach and its improvement potential. One important prerequisite for the full verification of the method is a practical technique for data collection. Such a method should be cost-effective and accurate. Once available, it will enable data collection in an amount sufficient for strict evaluation of the method.

A central question is how to validate the method. The first thought that comes to mind is to compare the estimated crash frequencies with the true crash frequencies. Although probably acceptable to most safety experts, this answer requires further discussion. First, true crash



frequency (and risk) is changing over time and short-term safety is practically impossible to measure with the existing methods. The crash frequency estimate based on crash statistics is a crude representation of all the safety levels experienced over a long period covered with crash data. On the other hand, the safety frequency (or better risk) estimates produced in the proposed method apply to relatively short periods when the traffic data are collected. How then is the comparison of the estimated and true values possible? A reasonable idea is to estimate the crash risk or frequency for a period that is as representative as possible of the long period with crash data. For example, if traffic data is collected on weekdays during the daytime in non-winter conditions, then crash data should come from a period with similar conditions.

The obtained estimates from crash reports and from traffic characteristics are likely to be subject to strong variability. Thus, comparison of results from a single location is not sufficient. A considerable number of locations are needed in order to to check if the results tend to be biased. Although the variability of the estimates from traffic observables can be controlled to a certainextent by extending the observation period, maximum and practical period lengths are determined by available measurement techniques and data collection costs. A bias detected in validation does not necessarily have to lead to the conclusion that the method is invalid. A modest overestimationcan be explained with the crash underreporting investigated by many authors (27-28).

A bias present in the crash frequency estimates may also indicate that the fundamental assumption of extreme values is violated and the collected values are too remote from the crash values. In such a situation, the crash factors may not be fully represented in the collected data. This may happen if the observation period is too short. Two distinct situations may take place. In the first case, some strong but infrequently present safety factors are not present in the observation period. For example, drunken driving brings a high risk of crash and is observed at a specific location occasionally. In such a case, the estimates will understate the risk. In the second case, some factors become effective under close crash proximity and are absent when the traffic event is not dangerous. Corrective human behavior is a good example of such a factor. In this case, the estimates will overstate the risk of crash.

A remedy of the bias is to increase the amount of collected data and select traffic events that are strongly connected to crashes. At its extreme, this approach leads to an observation period so long that actual crashes are observed and the proposed approach becomes the traditional crash-based method. From that point of view, the validation question is not if the method is capable of producing correct answers but rather how long the observation period needs to be to obtain results that are sufficiently accurate. If the required period is too long, then the method becomes impractical. Validation of the method reduces to its practicality check. As mentioned earlier, measurement techniques are critical.

6. CLOSING REMARKS

The proposed method of estimating crash frequencies from traffic measurements bridges crash-based and surrogate measures of safety and promises rapid estimation of crash frequencies from relatively short-term traffic observations. The key component of the method is estimation of the crash risk using the extreme value theory. As opposed to the traditional method for safety evaluation, this method no longer requires a long waiting period to have sufficient amount of



crash data for the analysis. Typical scenarios that the extreme value method can be very useful are when the crash data are (a) not available, e.g. a new or recently modernized intersection, or (b) insufficient for crash-based evaluation.

The method proposed in this study is neither limited to only right-angle crashes nor a specific measure such as post-encroachment times. The proposed method can be applied to other traffic characteristics as well as other types of collision. To apply this method, the key considerations can be summarized into the following steps:

• Traffic characteristics representing crash proximity to the collision type of interest must be defined.

• A valid traffic characteristic must be observable and possess a continuous characteristic that can represent varying risk levels during crash-free operations as well as characterize a collision at extremes.

• A boundary between crash and non-crash events may need to be redefined, depending on the traffic characteristic being considered.

To demonstrate the approach, post-encroachment times (PET) at a selected signalized intersection were analyzed. We first estimated the extreme value model for the PET values in 15-minute intervals in an eight-hour observation period. We demonstrated the adequacy of the developed model through diagnostic plots. Then, we computed the risk of crash in a 15-minute interval. The product of the risk and the corresponding exposure yields the crash frequency estimate.

The profile log-likelihood of the return period at the crash occurrence level was derived to quantify the uncertainty of the crash frequency estimate. The preliminary results are quite encouraging as the four-year average actual crash frequency lies in the 95% confidence interval of the crash frequency estimate. Nevertheless, there are still several challenges yet to be investigated in the modeling procedure.

First, there are several extreme value modeling alternatives other than the generalized extreme value models presented here, which include the threshold models, the r largest order statistic models, and the point process models. The performance evaluation of model alternatives and the criteria for model selection are still to be researched.

In addition, a visual evaluation of the plot in Figure 6 suggests a discernible pattern of variation in extreme values over time. Variations in extreme value behavior at this intersection may be time-varying by themselves, or it is also possible that certain covariates, such as traffic volume, may help explain the process variation after allowances for the time variation. Detailed statistical modeling is needed to disentangle these possible effects. It is possible to treat the PET extreme behavior as a non-stationary process in which the variation pattern could be better explained through the addition of time parameters or certain covariates. The appropriate mathematical structure of the trend and crash estimation procedure has to be explored. To handle the non-stationary in a more elegant way, all the aforementioned classes of extreme value models can be characterized by the theory of point process. The basic idea underlying the point process representation of extremes is to treat the extreme behavior as the two-dimensional non-homogeneous Poisson process.

Next, the optimal size of the data collection has yet to be decided. The given example was based on a limited sample size. The preliminary analyses indicate that this period should be longer,



probably up to several days. The optimal sample size would enable a robust and accurate estimation while minimizing the efforts needed to collect the data.

This paper is thought to present and illustrate a new approach to estimating crash frequencies to encourage more research in this promising approach. The proposed method has to be validated based on extended observation periods and for a sufficient number of various locations. The logical benchmark is the traditional crash-based estimation procedure.

7. ACKNOWLEDGMENT

This work was supported by the Joint Transportation Research Program administered by the Indiana Department of Transportation and Purdue University in research project SPR-2663. The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the data presented herein, and do not necessarily reflect the official views or policies of the Federal Highway Administration and the Indiana Department of Transportation, nor do the contents constitute a standard, specification, or regulation.

8. REFERENCES

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(2) Poch, M. and F. Mannering (1996). “Negative Binomial Analysis of Intersection-Accident Frequencies.” Journal of Transportation Engineering, Vol. 122, No. 2, pp. 105-113.

(3) Abbess, C., D. Jarrett, and C. C. Wright (1981). “Accidents at Blackspots: Estimating the Effectiveness of Remedial Treatment, with Special Reference to the ‘Regression-to-Mean’ Effect.” Traffic Engineering and Control, Vol. 22, No. 10, pp. 535-542.

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(7) Chin, H. C. and S. T. Quek (1997). “Measurement of Traffic Conflicts.” Safety Science, Vol. 26, No. 3, pp. 169-185.



(8) Glauz, W. D. and D. J. Migletz (1980). “Application of Traffic Conflict Analysis at Intersections.” NCHRP Report 219, Transportation Research Board, National Research Council, Washington D. C.

(9) Parker, M. R. and C. V. Zegeer (1989). “Traffic Conflict Technique for Safety and Operation Engineers Guide.” Report FHWA-IP-88-026, FHWA, U.S. Department of Transportation.

(10) Kloeden, C. N., A. J. McLean, V. M. Moore, and G. Ponte (1997). “Traveling Speed and the Risk of Crash Involvement.” NHMRC Road Accident Research Unit, The University of Adelaide.

(11) Porter, B. E., T. D. Berry, and J. Harlow (1999). “A Nationwide Survey of Red Light Running: Measuring Driver Behaviors for the ‘Stop Red Light Running’ Program.” Report, DaimlerChrysler Corporation.

(12) Shoarian-Sattari, K. and D. Powell (1987). “Measured Vehicle Flow Parameters as Predictors in Road Traffic Accident Studies.” Traffic Engineering and Control, Vol. 28, No. 6, pp. 328-335.

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(14) Minderhoud, M. M. and P. H. L. Bovy (2001). “Extended Time-to-Collision Measures for Road Traffic Safety Assessment.” Accident Analysis and Prevention, Vol. 33, No. 1, pp. 89-97.

(15) FHWA (1981). “Highway Safety Evaluation – Procedural Guide.” FHWA Report, FHWA-TS-81-219, Federal Highway Administration, November.

(16) Hayward, J. C. (1972). “Near Miss Determination through the Use of a Scale of Danger.” Report TTSC 7115, The Pennsylvania State University.

(17) Hauer, E. and P. Garder (1986). “Research into the Validity of the Traffic Conflict Technique.” Accident Analysis and Prevention, Vol. 18, No. 6, pp. 471-481.

(18) Zegeer, C. V. and R. C. Deen (1978). “Traffic Conflicts as a Diagnostic Tool in Highway Safety.” Transportation Research Record 667, Transportation Research Board, National Research Council, Washington D. C., pp. 48-55.

(19) Glauz, W. D., K. M. Bauer, and D. J. Migletz (1985). “Expected Traffic Conflict Rates and Their Use in Predicting Accidents.” Transportation Research Record 1026, Transportation Research Board, National Research Council, Washington D. C., pp. 1-12.

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(21) Williams, M. J. (1981). “Validity of the Traffic Conflicts Technique.” Accident Analysis andPrevention, Vol. 13, pp.133-145.

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LIST OF TABLES AND FIGURES

Table 1: Parameter Estimates of the GEV Model for the Negative PET Values

Figure 1: Continuum of Traffic Events

Figure 2: Example Variability of the Post-Encroachment Time

Figure 3: Components of Gap Time: Encroachment and Post-encroachment Time

Figure 4: Purdue University Mobile Traffic Laboratory

Figure 5: Configuration of Conflict Spots for the Data Extraction

Figure 6: 8-Hour Plot of PET Values over Time

Figure 7: The Profile Log-Likelihood of the GEV Shape Parameter

Figure 8: Diagnostic Plots for the PET Extremes Fitted to the Gumbel Model

Figure 9: Profile Log-Likelihood of the Return Period at Zero PET



Table 1: Parameter Estimates of the GEV Model for the Negative PET Values

Unconstrained shape Gumbel type (shape = 0)-4.0568 -4.0930(0.1305) (0.1163)0.6526 0.6333

(0.0945) (0.0850)-0.1069 N/A(0.1520) N/A

Loglikelihood at convergence

-36.248 -36.490

* Standard errors are in parentheses

Location

Scale

Shape

GEV model parameter estimatesItems



NormalEvents

Incipient DangerDangerous Events

Accidents

Figure 1: Continuum of Traffic Events



Figure 2: Example Variability of the Post-Encroachment Time



At t1 At t2 At t3

Conflict spot

Encroachment Time, ET = t 2 - t1; Post-encroachment Time, PET = t3 - t2; Gap time, GT = t3 - t1 = ET + PET

Figure 3: Components of Gap Time: Encroachment and Post-encroachment Time



(a) During the data collection (b) Inside the van

Figure 4: Purdue University Mobile Traffic Laboratory



Figure 5: Configuration of Conflict Spots for the Data Extraction



Intersection: SR-26 @ 18th St, Lafayette, IN (April 8, 2003)

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

7:30 8:30 9:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30

Time Index

Pos

t-E

ncro

achm

ent T

ime

(sec

)

Figure 6: 8-Hour Plot of PET Values over Time



Shape Parameter

Pro

file

Log-

likel

ihoo

d

-0.6 -0.4 -0.2 0.0 0.2 0.4

-41

-40

-39

-38

-37

90 % confidence interval

Figure 7: The Profile Log-Likelihood of the GEV Shape Parameter



Figure 8: Diagnostic Plots for the PET Extremes Fitted to the Gumbel Model



Return Period

Pro

file

Log-

Like

lihoo

d

0 1000 2000 3000 4000 5000 6000

-38.

5-3

8.0

-37.

5-3

7.0

-36.

5

Figure 9: Profile Log-Likelihood of the Return Period at Zero PET


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