Predicting motor vehicle collisions using Bayesian neural network models: an empirical analysis

Statistical models have frequently been used in highway safety studies. They can be utilized for various purposes, including establishing relationships between variables, screening covariates and predicting values. Generalized linear models (GLM) and hierarchical Bayes models (HBM) have been the most common types of model favored by transportation safety analysts. Over the last few years, researchers have proposed the back-propagation neural network (BPNN) model for modeling the phenomenon under study. Compared to GLMs and HBMs, BPNNs have received much less attention in highway safety modeling. The reasons are attributed to the complexity for estimating this kind of model as well as the problem related to "over-fitting" the data. To circumvent the latter problem, some statisticians have proposed the use of Bayesian neural network (BNN) models. These models have been shown to perform better than BPNN models while at the same time reducing the difficulty associated with over-fitting the data. The objective of this study is to evaluate the application of BNN models for predicting motor vehicle crashes. To accomplish this objective, a series of models was estimated using data collected on rural frontage roads in Texas. Three types of models were compared: BPNN, BNN and the negative binomial (NB) regression models. The results of this study show that in general both types of neural network models perform better than the NB regression model in terms of data prediction. Although the BPNN model can occasionally provide better or approximately equivalent prediction performance compared to the BNN model, in most cases its prediction performance is worse than the BNN model. In addition, the data fitting performance of the BPNN model is consistently worse than the BNN model, which suggests that the BNN model has better generalization abilities than the BPNN model and can effectively alleviate the over-fitting problem without significantly compromising the nonlinear approximation ability. The results also show that BNNs could be used for other useful analyses in highway safety, including the development of accident modification factors and for improving the prediction capabilities for evaluating different highway design alternatives.     

Investigating the effect of modeling single-vehicle and multi-vehicle crashes separately on confidence intervals of Poisson-gamma models

Crash prediction models still constitute one of the primary tools for estimating traffic safety. These statistical models play a vital role in various types of safety studies. With a few exceptions, they have often been employed to estimate the number of crashes per unit of time for an entire highway segment or intersection, without distinguishing the influence different sub-groups have on crash risk. The two most important sub-groups that have been identified in the literature are single- and multi-vehicle crashes. Recently, some researchers have noted that developing two distinct models for these two categories of crashes provides better predicting performance than developing models combining both crash categories together. Thus, there is a need to determine whether a significant difference exists for the computation of confidence intervals when a single model is applied rather than two distinct models for single- and multi-vehicle crashes. Building confidence intervals have many important applications in highway safety.This paper investigates the effect of modeling single- and multi-vehicle (head-on and rear-end only) crashes separately versus modeling them together on the prediction of confidence intervals of Poisson-gamma models. Confidence intervals were calculated for total (all severities) crash models and fatal and severe injury crash models. The data used for the comparison analysis were collected on Texas multilane undivided highways for the years 1997-2001. This study shows that modeling single- and multi-vehicle crashes separately predicts larger confidence intervals than modeling them together as a single model. This difference is much larger for fatal and injury crash models than for models for all severity levels. Furthermore, it is found that the single- and multi-vehicle crashes are not independent. Thus, a joint (bivariate) model which accounts for correlation between single- and multi-vehicle crashes is developed and it predicts wider confidence intervals than a univariate model for all severities. Finally, the simulation results show that separate models predict values that are closer to the true confidence intervals, and thus this research supports previous studies that recommended modeling single- and multi-vehicle crashes separately for analyzing highway segments.     

Application of the Conway-Maxwell-Poisson generalized linear model for analyzing motor vehicle crashes

This paper documents the application of the Conway–Maxwell–Poisson (COM-Poisson) generalized linear model (GLM) for modeling motor vehicle crashes. The COM-Poisson distribution, originally developed in 1962, has recently been re-introduced by statisticians for analyzing count data subjected to over- and under-dispersion. This innovative distribution is an extension of the Poisson distribution. The objectives of this study were to evaluate the application of the COM-Poisson GLM for analyzing motor vehicle crashes and compare the results with the traditional negative binomial (NB) model. The comparison analysis was carried out using the most common functional forms employed by transportation safety analysts, which link crashes to the entering flows at intersections or on segments. To accomplish the objectives of the study, several NB and COM-Poisson GLMs were developed and compared using two datasets. The first dataset contained crash data collected at signalized four-legged intersections in Toronto, Ont. The second dataset included data collected for rural four-lane divided and undivided highways in Texas. Several methods were used to assess the statistical fit and predictive performance of the models. The results of this study show that COM-Poisson GLMs perform as well as NB models in terms of GOF statistics and predictive performance. Given the fact the COM-Poisson distribution can also handle under-dispersed data (while the NB distribution cannot or has difficulties converging), which have sometimes been observed in crash databases, the COM-Poisson GLM offers a better alternative over the NB model for modeling motor vehicle crashes, especially given the important limitations recently documented in the safety literature about the latter type of model.     

Goodness-of-fit testing for accident models with low means

The modeling of relationships between motor vehicle crashes and underlying factors has been investigated for more than three decades. Recently, many highway safety studies have documented the use of negative binomial (NB) regression models. On rare occasions, the Poisson model may be the only alternative especially when crash sample mean is low. Pearson's X² and the scaled deviance (G²) are two common test statistics that have been proposed as measures of goodness-of-fit (GOF) for Poisson or NB models. Unfortunately, transportation safety analysts often deal with crash data that are characterized by low sample mean values. Under such conditions, the traditional test statistics may not perform very well.     

Modeling crash outcome probabilities at rural intersections: application of hierarchical binomial logistic models

It is important to examine the nature of the relationships between roadway, environmental, and traffic factors and motor vehicle crashes, with the aim to improve the collective understanding of causal mechanisms involved in crashes and to better predict their occurrence. Statistical models of motor vehicle crashes are one path of inquiry often used to gain these initial insights. Recent efforts have focused on the estimation of negative binomial and Poisson regression models (and related deviants) due to their relatively good fit to crash data. Of course analysts constantly seek methods that offer greater consistency with the data generating mechanism (motor vehicle crashes in this case), provide better statistical fit, and provide insight into data structure that was previously unavailable. One such opportunity exists with some types of crash data, in particular crash-level data that are collected across roadway segments, intersections, etc. It is argued in this paper that some crash data possess hierarchical structure that has not routinely been exploited. This paper describes the application of binomial multilevel models of crash types using 548 motor vehicle crashes collected from 91 two-lane rural intersections in the state of Georgia. Crash prediction models are estimated for angle, rear-end, and sideswipe (both same direction and opposite direction) crashes. The contributions of the paper are the realization of hierarchical data structure and the application of a theoretically appealing and suitable analysis approach for multilevel data, yielding insights into intersection-related crashes by crash type.     

Modeling motor vehicle crashes using Poisson-gamma models: examining the effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter

There has been considerable research conducted on the development of statistical models for predicting crashes on highway facilities. Despite numerous advancements made for improving the estimation tools of statistical models, the most common probabilistic structure used for modeling motor vehicle crashes remains the traditional Poisson and Poisson-gamma (or Negative Binomial) distribution; when crash data exhibit over-dispersion, the Poisson-gamma model is usually the model of choice most favored by transportation safety modelers. Crash data collected for safety studies often have the unusual attributes of being characterized by low sample mean values. Studies have shown that the goodness-of-fit of statistical models produced from such datasets can be significantly affected. This issue has been defined as the "low mean problem" (LMP). Despite recent developments on methods to circumvent the LMP and test the goodness-of-fit of models developed using such datasets, no work has so far examined how the LMP affects the fixed dispersion parameter of Poisson-gamma models used for modeling motor vehicle crashes. The dispersion parameter plays an important role in many types of safety studies and should, therefore, be reliably estimated. The primary objective of this research project was to verify whether the LMP affects the estimation of the dispersion parameter and, if it is, to determine the magnitude of the problem. The secondary objective consisted of determining the effects of an unreliably estimated dispersion parameter on common analyses performed in highway safety studies. To accomplish the objectives of the study, a series of Poisson-gamma distributions were simulated using different values describing the mean, the dispersion parameter, and the sample size. Three estimators commonly used by transportation safety modelers for estimating the dispersion parameter of Poisson-gamma models were evaluated: the method of moments, the weighted regression, and the maximum likelihood method. In an attempt to complement the outcome of the simulation study, Poisson-gamma models were fitted to crash data collected in Toronto, Ont. characterized by a low sample mean and small sample size. The study shows that a low sample mean combined with a small sample size can seriously affect the estimation of the dispersion parameter, no matter which estimator is used within the estimation process. The probability the dispersion parameter becomes unreliably estimated increases significantly as the sample mean and sample size decrease. Consequently, the results show that an unreliably estimated dispersion parameter can significantly undermine empirical Bayes (EB) estimates as well as the estimation of confidence intervals for the gamma mean and predicted response. The paper ends with recommendations about minimizing the likelihood of producing Poisson-gamma models with an unreliable dispersion parameter for modeling motor vehicle crashes.     

Application of finite mixture models for vehicle crash data analysis

Developing sound or reliable statistical models for analyzing motor vehicle crashes is very important in highway safety studies. However, a significant difficulty associated with the model development is related to the fact that crash data often exhibit over-dispersion. Sources of dispersion can be varied and are usually unknown to the transportation analysts. These sources could potentially affect the development of negative binomial (NB) regression models, which are often the model of choice in highway safety. To help in this endeavor, this paper documents an alternative formulation that could be used for capturing heterogeneity in crash count models through the use of finite mixture regression models. The finite mixtures of Poisson or NB regression models are especially useful where count data were drawn from heterogeneous populations. These models can help determine sub-populations or groups in the data among others. To evaluate these models, Poisson and NB mixture models were estimated using data collected in Toronto, Ontario. These models were compared to standard NB regression model estimated using the same data. The results of this study show that the dataset seemed to be generated from two distinct sub-populations, each having different regression coefficients and degrees of over-dispersion. Although over-dispersion in crash data can be dealt with in a variety of ways, the mixture model can help provide the nature of the over-dispersion in the data. It is therefore recommended that transportation safety analysts use this type of model before the traditional NB model, especially when the data are suspected to belong to different groups.     

Analyzing angle crashes at unsignalized intersections using machine learning techniques

A recently developed machine learning technique, multivariate adaptive regression splines (MARS), is introduced in this study to predict vehicles’ angle crashes. MARS has a promising prediction power, and does not suffer from interpretation complexity. Negative Binomial (NB) and MARS models were fitted and compared using extensive data collected on unsignalized intersections in Florida. Two models were estimated for angle crash frequency at 3- and 4-legged unsignalized intersections. Treating crash frequency as a continuous response variable for fitting a MARS model was also examined by considering the natural logarithm of the crash frequency. Finally, combining MARS with another machine learning technique (random forest) was explored and discussed. The fitted NB angle crash models showed several significant factors that contribute to angle crash occurrence at unsignalized intersections such as, traffic volume on the major road, the upstream distance to the nearest signalized intersection, the distance between successive unsignalized intersections, median type on the major approach, percentage of trucks on the major approach, size of the intersection and the geographic location within the state. Based on the mean square prediction error (MSPE) assessment criterion, MARS outperformed the corresponding NB models. Also, using MARS for predicting continuous response variables yielded more favorable results than predicting discrete response variables. The generated MARS models showed the most promising results after screening the covariates using random forest. Based on the results of this study, MARS is recommended as an efficient technique for predicting crashes at unsignalized intersections (angle crashes in this study).     

Using support vector machine models for crash injury severity analysis

The study presented in this paper investigated the possibility of using support vector machine (SVM) models for crash injury severity analysis. Based on crash data collected at 326 freeway diverge areas, a SVM model was developed for predicting the injury severity associated with individual crashes. An ordered probit (OP) model was also developed using the same dataset. The research team compared the performance of the SVM model and the OP model. It was found that the SVM model produced better prediction performance for crash injury severity than did the OP model. The percent of correct prediction for the SVM model was found to be 48.8%, which was higher than that produced by the OP model (44.0%). Even though the SVM model may suffer from the multi-class classification problem, it still provides better prediction results for small proportion injury severities than the OP model does.     

Investigating driver injury severity patterns in rollover crashes using support vector machine models

Rollover crash is one of the major types of traffic crashes that induce fatal injuries. It is important to investigate the factors that affect rollover crashes and their influence on driver injury severity outcomes. This study employs support vector machine (SVM) models to investigate driver injury severity patterns in rollover crashes based on two-year crash data gathered in New Mexico. The impacts of various explanatory variables are examined in terms of crash and environmental information, vehicle features, and driver demographics and behavior characteristics. A classification and regression tree (CART) model is utilized to identify significant variables and SVM models with polynomial and Gaussian radius basis function (RBF) kernels are used for model performance evaluation. It is shown that the SVM models produce reasonable prediction performance and the polynomial kernel outperforms the Gaussian RBF kernel. Variable impact analysis reveals that factors including comfortable driving environment conditions, driver alcohol or drug involvement, seatbelt use, number of travel lanes, driver demographic features, maximum vehicle damages in crashes, crash time, and crash location are significantly associated with driver incapacitating injuries and fatalities. These findings provide insights for better understanding rollover crash causes and the impacts of various explanatory factors on driver injury severity patterns.     

Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates

Traditionally, transportation safety analysts have used the empirical Bayes (EB) method to improve the estimate of the long-term mean of individual sites; to correct for the regression-to-the-mean (RTM) bias in before-after studies; and to identify hotspots or high risk locations. The EB method combines two different sources of information: (1) the expected number of crashes estimated via crash prediction models, and (2) the observed number of crashes at individual sites. Crash prediction models have traditionally been estimated using a negative binomial (NB) (or Poisson-gamma) modeling framework due to the over-dispersion commonly found in crash data. A weight factor is used to assign the relative influence of each source of information on the EB estimate. This factor is estimated using the mean and variance functions of the NB model. With recent trends that illustrated the dispersion parameter to be dependent upon the covariates of NB models, especially for traffic flow-only models, as well as varying as a function of different time-periods, there is a need to determine how these models may affect EB estimates. The objectives of this study are to examine how commonly used functional forms as well as fixed and time-varying dispersion parameters affect the EB estimates. To accomplish the study objectives, several traffic flow-only crash prediction models were estimated using a sample of rural three-legged intersections located in California. Two types of aggregated and time-specific models were produced: (1) the traditional NB model with a fixed dispersion parameter and (2) the generalized NB model (GNB) with a time-varying dispersion parameter, which is also dependent upon the covariates of the model. Several statistical methods were used to compare the fitting performance of the various functional forms. The results of the study show that the selection of the functional form of NB models has an important effect on EB estimates both in terms of estimated values, weight factors, and dispersion parameters. Time-specific models with a varying dispersion parameter provide better statistical performance in terms of goodness-of-fit (GOF) than aggregated multi-year models. Furthermore, the identification of hazardous sites, using the EB method, can be significantly affected when a GNB model with a time-varying dispersion parameter is used. Thus, erroneously selecting a functional form may lead to select the wrong sites for treatment. The study concludes that transportation safety analysts should not automatically use an existing functional form for modeling motor vehicle crashes without conducting rigorous analyses to estimate the most appropriate functional form linking crashes with traffic flow.     

Modeling young driver motor vehicle crashes: data with extra zeros

Much of the data collected on motor vehicle crashes is count data. The standard Poisson regression approach used to model this type of data does not take into account the fact there are few crash events and hence, many observed zeros. In this paper, we applied the zero-inflated Poisson (ZIP) model (which adjusts for the many observed zeros) and the negative binomial (NB) model to analyze young driver motor vehicle crashes. The results of the ZIP regression model are comparable to those from fitting a NB regression model for general over-dispersion. The findings highlight that driver confidence/adventurousness and the frequency of driving prior to licensing are significant predictors of crash outcome in the first 12 months of driving. We encourage researchers, when analyzing motor vehicle crash data, to consider the empirical frequency distribution first and to apply the ZIP and NB models in the presence of extra zeros due, for example, to under-reporting.     

An explanatory analysis of driver injury severity in rear-end crashes using a decision table/Naïve Bayes (DTNB) hybrid classifier

Rear-end crashes are a major type of traffic crashes in the U.S. Of practical necessity is a comprehensive examination of its mechanism that results in injuries and fatalities. Decision table (DT) and Naïve Bayes (NB) methods have both been used widely but separately for solving classification problems in multiple areas except for traffic safety research. Based on a two-year rear-end crash dataset, this paper applies a decision table/Naïve Bayes (DTNB) hybrid classifier to select the deterministic attributes and predict driver injury outcomes in rear-end crashes. The test results show that the hybrid classifier performs reasonably well, which was indicated by several performance evaluation measurements, such as accuracy, F-measure, ROC, and AUC. Fifteen significant attributes were found to be significant in predicting driver injury severities, including weather, lighting conditions, road geometry characteristics, driver behavior information, etc. The extracted decision rules demonstrate that heavy vehicle involvement, a comfortable traffic environment, inferior lighting conditions, two-lane rural roadways, vehicle disabled damage, and two-vehicle crashes would increase the likelihood of drivers sustaining fatal injuries. The research limitations on data size, data structure, and result presentation are also summarized. The applied methodology and estimation results provide insights for developing effective countermeasures to alleviate rear-end crash injury severities and improve traffic system safety performance.     

Rainfall effect on single-vehicle crash severities using polychotomous response models

As part of the Wisconsin road weather safety initiative, the objective of this study is to assess the effects of rainfall on the severity of single-vehicle crashes on Wisconsin interstate highways utilizing polychotomous response models.Weather-related factors considered in this study include estimated rainfall intensity for 15 min prior to a crash occurrence, water film depth, temperature, wind speed/direction, stopping sight distance and deficiency of car-following distance at the crash moment. For locations with unknown weather information, data were interpolated using the inverse squared distance method. Non-weather factors such as road geometrics, traffic conditions, collision types, vehicle types, and driver and temporal attributes were also considered. Two types of polychotomous response models were compared: ordinal logistic and sequential logistic regressions. The sequential logistic regression was tested with forward and backward formats. Comparative models were also developed for single vehicle crash severity during clear weather.In conclusion, the backward sequential logistic regression model produced the best results for predicting crash severities in rainy weather where rainfall intensity, wind speed, roadway terrain, driver's gender, and safety belt were found to be statistically significant. Our study also found that the seasonal factor was significant in clear weather. The seasonal factor is a predictor suggesting that inclement weather may affect crash severity. These findings can be used to determine the probabilities of single vehicle crash severity in rainy weather and provide quantitative support on improving road weather safety via weather warning systems, highway facility improvements, and speed limit management.     

Evaluating the road safety effects of a fuel cost increase measure by means of zonal crash prediction modeling

Travel demand management (TDM) consists of a variety of policy measures that affect the transportation system's effectiveness by changing travel behavior. The primary objective to implement such TDM strategies is not to improve traffic safety, although their impact on traffic safety should not be neglected. The main purpose of this study is to evaluate the traffic safety impact of conducting a fuel-cost increase scenario (i.e. increasing the fuel price by 20%) in Flanders, Belgium. Since TDM strategies are usually conducted at an aggregate level, crash prediction models (CPMs) should also be developed at a geographically aggregated level. Therefore zonal crash prediction models (ZCPMs) are considered to present the association between observed crashes in each zone and a set of predictor variables. To this end, an activity-based transportation model framework is applied to produce exposure metrics which will be used in prediction models. This allows us to conduct a more detailed and reliable assessment while TDM strategies are inherently modeled in the activity-based models unlike traditional models in which the impact of TDM strategies are assumed. The crash data used in this study consist of fatal and injury crashes observed between 2004 and 2007. The network and socio-demographic variables are also collected from other sources. In this study, different ZCPMs are developed to predict the number of injury crashes (NOCs) (disaggregated by different severity levels and crash types) for both the null and the fuel-cost increase scenario. The results show a considerable traffic safety benefit of conducting the fuel-cost increase scenario apart from its impact on the reduction of the total vehicle kilometers traveled (VKT). A 20% increase in fuel price is predicted to reduce the annual VKT by 5.02 billion (11.57% of the total annual VKT in Flanders), which causes the total NOCs to decline by 2.83%.     

Using a reliability process to reduce uncertainty in predicting crashes at unsignalized intersections

The negative binomial (NB) model has been used extensively by traffic safety analysts as a crash prediction model, because it can accommodate the over-dispersion criterion usually exhibited in crash count data. However, the NB model is still a probabilistic model that may benefit from updating the parameters of the covariates to better predict crash frequencies at intersections. The objective of this paper is to examine the effect of updating the parameters of the covariates in the fitted NB model using a Bayesian updating reliability method to more accurately predict crash frequencies at 3-legged and 4-legged unsignalized intersections. For this purpose, data from 433 unsignalized intersections in Orange County, Florida were collected and used in the analysis. Four Bayesian-structure models were examined: (1) a non-informative prior with a log-gamma likelihood function, (2) a non-informative prior with an NB likelihood function, (3) an informative prior with an NB likelihood function, and (4) an informative prior with a log-gamma likelihood function. Standard measures of model effectiveness, such as the Akaike information criterion (AIC), mean absolute deviance (MAD), mean square prediction error (MSPE) and overall prediction accuracy, were used to compare the NB and Bayesian model predictions. Considering only the best estimates of the model parameters (ignoring uncertainty), both the NB and Bayesian models yielded favorable results. However, when considering the standard errors for the fitted parameters as a surrogate measure for measuring uncertainty, the Bayesian methods yielded more promising results. The full Bayesian updating framework using the log-gamma likelihood function for updating parameter estimates of the NB probabilistic models resulted in the least standard error values.     

Identifying repeat DUI crash factors using state crash records

The objective of this study is to identify high risk factors that are closely related to repeat DUI crashes using readily available information from the state crash records. Survival analysis was used and a Cox proportional hazards model was developed using the police-reported crash records in the state of Louisiana. A variety of variables were found to be significant in predicting repeat DUI crashes. The factors included the characteristics of the drivers (gender, race, and age), the types of the vehicle (light truck/pick up truck or other), the characteristics of the crash (hit-and-run, driver violations, and whether the driver is arrested), the type of location (residential area or other), and the characteristics of the roadway (highway type and roadway type). This study provides a comprehensive picture of the repeat DUI crashes. The model can quantitatively predict the relative hazards of repeat DUI crashes. It can be used to identify the characteristics of the crash-involved DUI drivers who are at greatest risk of being involved in a subsequent DUI crash, allowing to apply appropriate remedial measures to reduce the risk.     

Explaining two-lane highway crash rates using land use and hourly exposure

This paper describes the estimation of Poisson regression models for predicting both single and multi-vehicle highway crash rates as a function of traffic density and land use, as well as ambient light conditions and time of day. The study focuses on seventeen rural, two-lane highway segments, each one-half mile in length with varying land use patterns and where actual hourly exposure values are available in the form of observed traffic counts. Land-use effects are represented by the number of driveways of various types on each segment. Hourly exposure is represented for single-vehicle crashes as the total vehicle miles traveled and volume/capacity ratio; for multi-vehicle crashes it is the product of the hourly volumes on the main highway and the roads intersecting it along the study segment. For single-vehicle crashes, the following variables were found to be significant, with a positive or negative effect as noted: daytime (06:00–19:00 h, negative effect), the natural log of the segment volume/capacity ratio (negative), percent of the segment with no passing zones (positive), shoulder width (positive), number of intersections (negative), and driveways (mixed effects by type). Good multi-vehicle crash prediction models had quite different variables: daylight conditions from 10:00–15:00 and 15:00–19:00 h (positive), number of intersections (negative), and driveways (positive for all types). The results show that traffic intensity explains differences in crash rates even when controlling for time of day and light conditions, and that these effects are quite different for single and multi-vehicle crashes. Suggestions for future research are also given.     

Differences in causality factors for single and multi-vehicle crashes on two-lane roads

Past research has found a non-linear relationship between traffic intensity or level of service (LOS) and highway crash rates. This paper investigates this relationship further by including the effects of site characteristics and estimating Poisson regression models for predicting single and multi-vehicle crashes separately. Analysis focuses on rural two-lane highways, with hourly LOS, traffic composition, and highway geometric characteristics as independent variables. The resulting models for single and multi-vehicle crashes have different explanatory variables. Single-vehicle crash rates decrease with increasing traffic intensity (lower LOS), shoulder width and sight distance. Multi-vehicle crash rates increase with the number of signals, the daily single-unit truck percentage, and the shoulder width, and decreased on principal arterials compared to other roadway classes. LOS does not significantly explain variation in the number of multi-vehicle crashes. Ongoing research by the authors is aimed at identifying other site factors, such as driveway density and intersection LOS, that can better explain the differing effects reported here and predict crash rates of both types better.