Site and data

All the traffic collision data used in this study are based on State Wide Integrated Traffic Records Systems (SWITRS) that are open-source database published by California Highway Patrol [24]. The SWITRS database records all vehicle collisions occurred on a public road-way. Among the traffic collision in SWITRS database, Traffic Accident Surveillance and Analysis System (TASAS) collected the collision data that occurred on a road that is managed by Caltrans, and this study used the TASAS collision data. The traffic collision data used was collected between 2004 and 2008 from 390 miles of six freeway routes in the San Francisco bay area, including I-80W, I-80E, I-580W, I-580E, I-880N, and I-880S. These routes share similar features, i.e., they are all multi-lane, interstate, urban freeways with high traffic volume. The purpose of the study is to evaluate the reproducibility of fatal collision locations on freeways based on the collision occurrence data and not depending on the influential factors that are changed over time such as vehicle safety systems, roadway conditions, and traffic conditions. Therefore, to evaluate our method by comparing carefully calibrated previous method, we used these datasets applied in the previous studies [14] rather than recent data. Over the five years of the study, 49,159 collisions occurred on the six routes and those data were uploaded as the supporting information in S1 Dataset.

Table 1 shows the total number of collisions for each route by collision severity and year. The length of each route ranged from 50 to 80 miles. The average number of collisions per mile per year for each route varied from 15 to 32. The annual number of collisions for each route decreased over the study period. Table 1 shows the proportions of collision severity levels for each route by year. Since the percentages of fatal, injury, and PDO collisions changed over the years or across routes, HCCL identification based on the total number or number of injury collisions would be different from the identification of reproducible fatal collision locations.

Table 2 shows the results of a chi-square test conducted to determine whether there is a significant association between the proportions of collision severity levels and the year or the route. The left side of Table 2 indicates that there is a significant association between the year and proportions of collision severity levels on all routes. The right side of Table 2 also shows a statistically significant association between the routes and proportions of collision severity levels in all years. In particular, these results indicate that the proportion of fatal collisions would be varied significantly in terms of ratio from year to year in all routes, and from route to route in all years. This tendency indicates that collision severity levels should be considered in identifying the reproducible fatal collisions to evaluate the safety of the site properly.

Reproducible (R) and non-reproducible (N) fatal collision site

To evaluate the performance of the proposed method, the reproducible (R) and non-reproducible (N) fatal collision sites were empirically classified based on the occurrence of additional fatal collisions in subsequent years (i.e., validation year) near where a fatal collision had occurred in the reference year. Fig 2 shows the R and N at I-80W for which the reference year is 2006 and the validation years are 2007 and 2008. Fig 2 also shows a plot of the CRP, which estimates the true collision risk based on spatial patterns of collisions by filtering out statistical fluctuation, with the R and N. The peaks in the CRP plot indicate the locations of local-collision concentrations. Fig 2A through 2F show plots of the CRPs constructed using traffic collision data, CRP_A, and those constructed using FI collision data, CRP_FI, observed along I-80W from 2006 to 2008. The black and white circles in Fig 2A and 2D indicate the locations of fatal collisions, respectively, with four of the ten fatal collisions observed during the reference year indexed from F₁ to F₄. S₁, S₂, and S₃ in the figures indicate site locations, with the site boundaries (dotted vertical lines in Fig 2) determined by selecting l/2 upstream and downstream of the location of the fatal collision in 2006. When another fatal collision had occurred within l, the boundary of the site was expanded by combining the l value with respect to the additional fatal collision. In this study, 1.0 mile was used as the l value, considering the potential for error in the reported collision location introduced when milepost information was entered into the traffic collision report [14] and due to the stochastic nature of the driver reaction to the causative factor of a fatal collision [25]. With l = 1.0 mile in our study sites, the lengths of the fatal collision sites ranged from 1 mile to 1.97 miles. We note how additional fatal collisions occurred in subsequent years within S₃, whereas no additional fatal collisions occurred within S₁ or S₂, as shown in Fig 2. Based on these empirical data, sites that experienced additional fatal collisions in subsequent years were classified as R and the others were classified as N. S’₁ to S’₃ in Fig 2B and 2E indicate the locations of fatal collisions in 2007, which is not a reference year, and additional fatal collisions that occurred in 2008. If the reference year is 2007 and the validation years include 2008, sites S’₁ to S’₃ would be classified as R. In other words, the locations classified as R and N would differ with the reference and validation years. In the later section, we evaluate the proposed method in various combinations of reference and validation years.

Naïve Bayesian approach for detecting reproducible fatal collision sites (R)

Naïve Bayes is a simple technique for modeling classifiers that assumes conditional independence of the observed features. In our approach, the observed features are the occurrence of fatal collisions along the freeway route in the reference year. Therefore, our approach assumes that the occurrence of each fatal collision is conditionally independent given D, which is a continuous random variable indicating the locations of R. Using Bayes’ theorem, the conditional probability (i.e., posterior probability) of a site being classified as R at D, P(D┃F_1:m), given the m fatal collision locations along the route in the reference year, can be calculated as shown in Eq (1): (1) where P(D) is the prior probability of a site being classified as R at D, and P(F_1:m┃D) is the likelihood of the occurrence of fatal collisions at locations F₁ to F_m given R at D. Since the purpose of the proposed method is to rank sites based their PMP values, and not to develop an unbiased estimate of a site being classified as R, the numerator, P(D)P(F_1:m┃D), is used to rank sites that are likely to be classified as R. The likelihood,P(F_1:m┃D), can be decomposed as shown in Eq (2) with the assumption of the conditional independence of fatal collision occurrences: (2) where ∝ indicates proportionality. The proposed method estimates the PMP of the site being classified as R at D, P(D)P(F_1:m┃D), based on both the observed locations of fatal collisions, F_1:m, and the CRP of the reference year. The sites are then prioritized based on their PMP values regarding the recommendation for in-depth investigation. The CRP is used as a prior probability of a site being classified as R in the procedure for estimating the PMP.

Estimation of prior probability

The CRP estimates the true collision risk based on spatial patterns by filtering out statistical fluctuation. The CRP can be constructed using various filtering techniques such as the weighted moving average technique [20, 21] and ensemble empirical mode decomposition [22]. The level of filtering is an important parameter for constructing the CRP, which can be empirically determined by investigating the changes in the number of critical points where the CRP slope is zero [21] or the spectral attributes of the traffic collision data [22]. In this study, the CRPs were constructed using the process described by Chung et al. [20], which filters out random noise as much as possible without affecting the location of the CRP peaks. More details and numerical examples of CRPs are presented in Chung et al. [20] and Chung et al. [21]. The shape of a CRP indicates the underlying true risk profile, and its value measures the collision frequency per unit distance of roadway. A continuous CRP profile can be used to consider the spatial correlation of traffic collisions and the area under the CRP indicates the expected collision frequency considering the regression-to-the-mean (RTM) phenomenon [26].

The unit of the CRPs shown in Fig 2 is the number of collisions per 0.01 mile per year along the route. A visual inspection of CRP_A in Fig 2A–2C reveals the reproducible patterns over the years, which are consistent with the findings from previous studies [20, 21, 26]. CRP_FI in Fig 2D–2F provides a stronger indicator of fatal collisions than those constructed using PDO, injury, and fatal collisions, CRP_A (i.e., all traffic collisions), as it can lead to fatality depending on the characteristics of the victim: injury collisions may later be changed to fatal collisions when the injured person dies within a certain period of time after the collision [27]. However, the overall reproducibility of CRP_FI compared to CRP_A (see Fig 2A–2C) is smaller. In this study, both CRP_A and CRP_FI were evaluated with respect to the prior probability of a site being classified as R. These two prior probabilities are both used in estimating the PMP of a site being classified as R, based on the naïve Bayes approach.

Estimation of likelihood

Fatal collisions can occur without any external causative factors. However, there may also exist factors that contribute to causing fatal collisions. If such factors exist, one would expect to see fatal collisions occurring near the location in which there are unknown fatal-collision causative factors. To consider the average effect of unknown causative factors (i.e., unobserved heterogeneity along spatial dimensions) of all fatal-collision locations on each route, we assumed the likelihood of the occurrence of observed fatal collisions given R at D, L(D), which is a function of the distance between fatal-collision locations (F_i) and the R location (D), as shown in Eqs (3a) and (3b). The influence of proximity among the fatal collisions on a site being classified as R was modeled using Eq (3b), where d_i is the distance between F_i and D, which indicates that the closer D is to F_i, the greater the effect of those unknown causative factors of F_i. The value of the likelihood function is one if the distance between F_i and D is zero, and its minimum value is zero when F_i and D are far away from each other. Other linear and non-linear functions sharing these properties have been applied, but they all exhibit comparable performance. Therefore, we selected the simplest function that is easy to implement in practice, as shown in Eq (3b). The hypothesis underlying the estimation of likelihood is that the collision rate near R may not be high enough to be detected by other HCCL identification procedures [15], or the reproducible fatal collision may not have a high enough concentration of collisions, as shown in Fig 2. (3a) (3b) The likelihoods of the observed fatal collisions occurring at locations F₁ to F₁₀ are estimated based on D, which is the location of R. This can be further explained with reference to Fig 3A–3D. F₁ to F₁₀ in Fig 3 show the fatal-collision locations that occurred on I-80W in 2006, and P(F┃D = 10) is the likelihood of a fatal collision occurring at F when the assumed location of R is 10. Therefore, the peak in Fig 3A representing P(F | D = 10) at F = 10 is 1 by definition. Likewise, Fig 3B and 3C show the likelihood of the fatal collision occurring at F for locations R = 40 and 60, respectively.

The likelihood of all fatal collisions along the entire route, L(D), were estimated by combining the individual likelihood of the fatal collision, , where m is the total number of fatal collisions in 2006, i.e., the reference year (see Eq (3a)). D is incremented by 0.01 mile and ranged over the entire route. Fig 3D shows according to D over the entire I-80W route. The α (see Eq (3b)) determines how much distance between fatal collisions and D influences P(F_i | D). A higher value of α results in a stronger likelihood of D being near F. A lower value of α indicates a diminished effect of the distance from a reproducible fatal-collision location in determining P(F_i | D). Fig 4 shows how different values of α affect the shape of P(F_i | D). When α = 0, the shape of P(F_i | D) remains horizontal. Higher values of α bring the shape of P(F_i┃D) to a wide peak near D and the value of P(F_i┃D) drops sharply if the distance from D become more than L_c, whereas lower values of α bring the shape of P(F_i┃D) to a narrow peak and the value of P(F_i┃D) drops gently even if the distance from D is larger than L_c. The black circles in Fig 4 show the value of P(F_i┃D) at the postmile = 26 according to α, which indicates that the high value of α makes the value of P(F_i┃D) more sensitive to the distance from D.

Estimation of a proxy measure of the posterior probability

The naïve Bayesian approach is used to estimate the PMP based on likelihood, L(D), and prior probability, P(D). In other words, the combined influence of the proximity of fatal collisions at a site is used as the likelihood, with the CRP as the indicator of the prior probability. P(D) weighted by the L(D) is used to estimate the PMP, as shown in Eq (4). (4) Fig 5B and 5D show the resulting PMP estimated by L(D) with α = 0.2 and the P(D) based on CRP_A (Fig 5A) and CRP_FI (Fig 5C) together with the locations of the fatal collisions in the reference year. The PMPs of S_1, S_2, and S₃ being classified as R are determined by the mean values of PMP within the site boundary, , which consist of the areas under the curve shaded in gray divided by the site length, as shown in Fig 5B and 5D. As shown in Fig 5B, the of S₃ (0.0288) estimated using CRP_A is much higher than those of S₁ (0.0093) and S₂ (0.0187). The estimated using CRP_FI exhibits the same pattern, with S₃ (0.0368) having the highest value of being classified as R, as compared with S₁ (0.0112) and S₂ (0.0110). To determine the sites that should be classified as R, the estimated of each fatal collision site was ranked based on the information available only in the reference year.

Comparison of PMPs for all routes

To compare the PMPs of multiple routes, the PMP of a site being classified as R for a single route (see Eq (4)) is rescaled to PMP_R, as shown in Eq (5): (5) where s and e are the start- and end-points of the route, respectively. The P(D) value (i.e., CRP_A or CRP_FI) indicates the number of total or FI collisions per year per 0.01 mile. The P(D) values of different routes cannot be directly compared because the collision causative factors, e.g., traffic volume, are different. The HSM [7] suggests use of the PSI, i.e., the difference between the expected and observed collision frequencies, as a safety measure to consider the collision causative factors in prioritizing sites to be investigated. This study applies an excess of CRP that is the difference between the CRP and SPF, which indicates the PSI [14]. The CRP can also consider the RTM phenomenon, which cause bias in estimating the benefits of safety countermeasure for HCCLs, by filtering out random fluctuations in the collision frequency [26]. The value of L(D) is a non-increasing function of the number of fatal collisions along the routes because P(F_i┃D) is always less than one (see Eq (3b)). To adjust for the biases introduced by the number of fatal collisions along each route, the PMP_R is estimated based on the rescaled likelihood, .

To determine whether a site can be defined as R, the value is used, i.e., the mean value of PMP_R within the site boundary. Table 3 shows the value estimated by the proposed method for 22 fatal collisions associated with 19 sites on routes I-80W and I-80E in 2006, and the validation years are 2007 and 2008. Site lengths longer than 1.0 mile indicates the occurrence of an additional fatal collision within the boundary of the site, whereas site lengths shorter than 1.0 mile indicate that the fatal collision occurred near the start- or end-points of the route. The rows of Table 3 are arranged in descending order with respect to the estimated using CRP_FI, with the value of α being 0.3. Of the 19 fatal collision locations, five were classified as R, and four Rs were included in the top five values. The overall performances of the six routes were evaluated together, and the findings are reported in the next section.

Comparison of PMPs with SPF-based approaches

The SPF-based approach is a well-established procedures for detecting HCCLs [7, 8]. Conventional SPF-based approaches apply the empirical Bayes (EB) method, which estimates the expected number of collisions based on the dispersion of an SPF. As a safety measure, the PSI is used, i.e., the difference between expected crash frequency based on an EB adjustment and the SPFs [14]. In recent years, to differentiate between collision severity levels to better identify HCCLs, researchers have proposed the calibration of SPF for FI collisions [11, 12] or fatal collisions [13]. Advanced statistical models that consider the unobserved spatial heterogeneity have also been used to improve the SPF [28, 29]. However, the infrequent occurrence of fatal collisions causes a large variance in estimations of their frequency, with distributions of collision frequency skewing excessively toward zero, resulting in erroneous estimations [30]. This incorrect estimation is only exacerbated when estimating PSI because the observed frequency is also so low. Other researchers have proposed the property damage only equivalents (PDOEs) that weight collision frequencies based on collision severity with respect to the associated economic loss [15, 17]. They developed the SPF using PDOEs as a dependent variable, to consider the collision severity levels.

Table 4 shows the methodological considerations of the proposed method compared with previous approaches. Excess of CRP, i.e., the difference between the CRP and SPF, enables consideration of both causative factors and RTM bias, which are addressed by HSM’s traditional SPF-based approach. The spatial heterogeneity due to the spatial dependence of traffic collisions can be considered using the CRP [20]. The likelihood of the occurrence of fatal collisions proposed in this study weights the CRP based on Bayes’ theorem to consider fatal collisions by estimating the average effect of the unknown causative factors of fatal collisions and this likelihood reflects their unobserved spatial heterogeneity [5] for detecting reproducible fatal collision locations.

Performance measures

The performance of the proposed method in classifying sites as R based only on the data observed in the reference year was compared with that of a procedure that investigates fatal collision sites at random, i.e., the ad hoc site investigation procedure (e.g., sites may have been investigated based on the order in which the fatal collisions occurred within a fiscal year or on the number of parties involved). Fig 6 shows the performance of the proposed method in detecting reproducible fatal collision sites for the six routes with 2006 as the reference year and 2007 and 2008 as the validation years. The x-axis represents the number of sites investigated by the proposed method (i.e., top x sites in the order of PMP) and random selection. The y-axis represents the cumulative number of true Rs identified with respect to the number of sites recommended for investigation. The dotted black line marked with x symbols in Fig 6 shows the performance in classifying sites as R with no consideration of the likelihood and prior probability (i.e., random selection). The solid red line with circles shows the performance in classifying sites as R based on . The value was estimated based on the CRP_A (see Fig 6A) and CRP_FI (see Fig 6B), with the likelihood function α = 0.3. The curves in Fig 6 provide information similar to the receiver operating characteristics (ROC), which are used to evaluate classification models [31]. If there are n reproducible fatal collision sites among the N fatal collision sites at a study site, the number of sites investigated is x_i, and the corresponding cumulative number of sites classified as R is y_i, the false negative (i.e., not detecting a true R) is n − y_i, false positive (i.e., flagging a site as R when is not) is x_i—y_i, true positive (TP) is y_i, and true negative is N–n–x_i + y_i. The recall and precision are calculated as shown in Eqs (6) and (7). Note that precision and recall vary with x_i as they have a trade-off relationship. Fig 6 shows the trade-off between precision and recall with respect to x_i and the area under this curve (AUC) indicates the overall performance of the proposed model. The fact that the AUC of the proposed method (i.e., the dotted red lines) is larger than those obtained by random selection (i.e., the dotted black lines) shows that the proposed method outperforms random selection.

(6) (7) Based on the above evaluation, the practical impact of the proposed method is as follows. In Fig 6, 11 Rs were classified out of the total 40 fatal collision sites along the six routes. Note that if the government agency uses the proposed method with CRP_A (see Fig 6A), it will have to investigate 16 of 40 sites (40.0%) to identify 8 of the 11 reproducible fatal collision sites (72.7%). To identify all Rs, 29 sites will have to be investigated by CRP_A. Using CRP_FI (see Fig 6B), the agency will also have to select 18 sites to identify 8 out of 11 Rs, but 30 sites will need to be investigated to identify all Rs.

Comparison of performance with SPF-based approach

To compare the proposed method with the SPF-based approach that uses the EB method [32], we applied recently developed SPFs to the study site [14] to detect reproducible fatal collision sites. SPF-based approaches prioritize fatal collision locations using PSI, which is the difference between the expected and observed collision frequencies. Kwon et al. [14] developed SPFs for each type of roadway group, considering rural or urban, arterial or highway, and the number of lanes, and their SPFs were found to better fit the data for the study sites than the SPFs currently used by Caltrans. As an explanatory variable, Kwon’s SPF uses the average annual daily traffic (AADT) volume, although other additional variables such as shoulder width and horizontal curves [11] can improve the SPF performance. This study uses Kwon’s SPF for two reasons: i) Caltrans and practitioner currently uses this simple type of SPF based on AADT and roadway group [14] because developing its own database to keep track of the values of additional roadway information is cost-prohibitive and ii) only traffic collision and AADT data are required for practical applications like the proposed method. We used the SPF calibrated for FI crashes in each highway group and estimated the PSI for fatal collision sites using the EB method. Comparison results of the proposed method and SPF-based approach are provided in the following subsections with sensitivity analyses.

Sensitivity analysis of likelihood parameter

The sensitivity of the likelihood parameter α, which adjusts the influence of the proximity of fatal collisions, is examined by applying α values from 0 to 0.7 for the six routes in the same reference and validation years as those in Fig 6. The sensitivity was evaluated by the area under the receiver operating characteristics (AUROC) curve, which is a well-known performance measure for classification models [31]. The AUROC shows the competitiveness of the classification model as compared with random guessing, similar to the area under the red and black lines in Fig 6.

An AUROC value higher than 0.5 means the model outperforms random guessing, with the best score possible being 1.0. Fig 7 shows the performance evaluation with α for the CRP_A and CRP_FI priors, both of which outperformed random guessing with AUROC values higher than 0.5. The proposed method also outperforms the SPF-based approach, which obtained a 0.53 AUROC value only slightly better than random guessing. For a lower value of α, which indicates a weak influence of fatal collision locations on the classification of sites as R, the CRP_FI outperforms the CRP_A, although the CRP_A performs better for a higher value of α. The best performances were obtained by the CRP_FI prior, with a 0.75 AUROC value for α = 0.3 and 0.4 and the CRP_A prior with a 0.75 AUROC value for α = 0.6. The performances of CRP_A and CRP_FI with various α values were comparable, which shows that the proposed method is robust with respect to the parameter α and the prior probability.

Sensitivity analysis for reference and validation years

Reproducible fatal collision sites are classified based on the occurrence of additional fatal collisions in the validation years near where the fatal collision occurred in the reference year. Since traffic collisions are rare and random events, there could be multiple reference and validation years. Although a multiyear time period takes advantage of the RTM phenomenon [32], it causes within-period variation in the presence of unobserved heterogeneity [30]. To evaluate the generalization performance of the proposed method with respect to the reference and validation years, we evaluated our method under various conditions, as shown in Table 5. For example, sites are classified as R in Condition 1 when additional fatal collisions occurred in 2007 or 2008 near where the fatal collision occurred in 2006. In the case of Condition 3, among the fatal collision locations in 2005 or 2006, sites are classified as R when additional fatal collisions occurred at these locations in 2007 or 2008. If there are multiple reference years, the CRP, i.e., the prior probability of sites being classified as R, is constructed from the sum of the collisions in multiple years.

Table 5 shows the performance of the proposed method for various conditions in classifying sites as R. The proposed method with the CRP_FI prior was evaluated with respect to the likelihood parameter α, and these performances were compared with those of the SPF-based approach using the EB method. For all conditions, the proposed method outperforms the SPF-based approach with the EB method. In condition 4, the performance of the SPF-based approach was worse than random selection (i.e., 0.5 AUROC), which means that the SPF calibrated using FI collisions cannot contribute to the classification of a site as R. In each condition, the best AUROC values ranged from 0.65 to 0.75, with the likelihood parameter α values ranging from 0.1 to 0.3. These results reveal that although the performance of the proposed method could vary with the data used to classify sites as R, it outperforms random selection and the SPF-based approach in detecting Rs.

Although the proposed method successfully detected the reproducible fatal collision locations based on the PMP_R estimated by CRP_A and CRP_FI, the causative factors for those locations requires further exploration to apply effective countermeasures for roadway safety. We investigated the primary collision factors for 145 fatal collisions that occurred on the six routes from 2006 to 2008 and divided the fatal-collision sites into reproducible and non-reproducible locations. The reference year was 2006 and the validation years were 2007 and 2008. Fig 8 shows distributions of the primary collision factors for the reproducible and non-reproducible fatal collision locations, for which a chi-square test was conducted to determine if there is a difference in the frequency of the causative factors of these traffic collisions. Because the chi-square test assumes that no more than 20% of the cell counts are less than five [33], we aggregated three low-incidence collision factors—“Other Violations,” “Improper Driving,” and “Other Than Driver,”—as “Other Violations.” The P-value of the chi-square test was 0.87, which indicates no significant difference in the distributions of primary collision factors between the reproducible and non-reproducible fatal collision locations. This may mean that the primary collision factor cannot be used to classify reproducible fatal collision locations.

Concluding remarks

State agencies are mandated to investigate the locations of fatalities and serious injuries. However, most of existing methods for detecting high-collision-concentration locations do not consider the severity of the collisions, which results in agencies focusing on sites with a high frequency of property damage only collisions while ignoring sites with higher percentages of injury and fatal collisions. Several previous studies have proposed methods for considering the severity of collisions by weighting them with respect to their associated economic losses or the use of a safety performance function (SPF) calibrated to fatal or fatal and injury collisions. However, these approaches can yield a high false-positive rate because fatal collisions that occur randomly and rarely due to the fault of the driver can still dwarf the weight of fatal collisions or cause high variance in the SPF.

To better utilize limited government resources while improving the safety of sites where fatal collisions occur, we developed a method for estimating the proxy measure of the posterior probability (PMP) of a site being classified as R (i.e., a site that is likely to experience an addition fatal collision in the vicinity in the subsequent year). The peaks of the continuous risk profile (CRP), which are reproducible over the years, were used as indicators of the prior probability of reproducible collision locations, and the likelihood of the occurrence of fatal collision was computed by the spatial distribution function for fatal collisions. Then, the likelihood and the prior values were multiplied and normalized to obtain the PMP_R values, which can be compared for different routes. The empirical evaluations indicated that the PMP_R of a site being classified as R can successfully identify sites as R compared with a random selection of sites or the SPF-based approach with the EB method. This finding can assist a state agency to better allocate its limited resources. A sensitivity analysis of the proposed method was also conducted to determine its robustness to some variations of the method, such as the choice of the prior indicator between CRP_A and CRP_FI, the influence parameter, α, and the data used for classifying reproducible fatal collision locations. The results indicate that the proposed method achieves robust performance regardless of these parameters and conditions.

Although advanced vehicle safety technology, advanced highway design, and surrogate safety-measures have developed from the past, the number of total and fatal collisions is still rising, which indicates that the HCCL still remains in the freeway. The proposed method does not find the HCCL based on the complex human and environmental factors but finds the recurrent fatal collision locations using a probabilistic approach based on the locations and number of collisions. In other words, the proposed method can be applied to the recent data since it does not depend on the complex influential factors that change over time. Nevertheless a comparison of the proposed method with other methods using a recent dataset would be further research subject to verify the generalized performance. As the form of the likelihood function was assumed in the present study, evaluating the performance of different likelihood functions will be the subject of future study. This study estimated the likelihood of fatal collision based only on the spatial distribution of fatal collision locations, which is a great advantage for practical application in terms of data availability. However, if significant fatal-collision causative factors are identified, those factors could also be considered in the likelihood function. Therefore, evaluating the impact of other causative factors such as weather, roadway geometry, and traffic characteristics in classifying sites as R [34–37] and other likelihood functions that consider those factors will be the subjects of future study. The authors also plan to further evaluate the performance of the proposed method along rural highways or urban area where the characteristics of fatal collisions can differ from those of this study [38].