Ci Liang
email: ci.liang@railenium.eu.

Mohamed Ghazel

Olivier Cazier

El-Miloudi El-Koursi

Developing accident prediction model for railway level crossings

Keywords: Level crossing safety, Train-car collision, Accident prediction modeling, Statistical analysis;

Railway level crossing (LX) safety continues to be one of the most critical issues for railways, despite an everincreasing focus on improving design and application practices. Accidents at European LXs account for about one-third of the entire railway accidents and result in more than 300 deaths every year in Europe. Due to the non-deterministic causes, the complex operation background and the lack of thorough statistical analysis based on accident/incident data, the risk assessment of LXs remains a challenging task. In the present paper, some LX accident prediction models are developed. Such models allow for highlighting the influence of the main impacting parameters, i.e., the average daily road traffic, the average daily railway traffic, the annual road accidents, the vertical road profile, the horizontal road alignment, the road width, the crossing length, the railway speed limit and the geographic region. The Ordinary Least-Squares (OLS) and Nonlinear Least-Squares (NLS) methods are employed to estimate the respective coefficients of variables in the prediction models, based on the LX accident/incident data. The validation and comparison process is performed through statistical means to examine how well the estimation of the models fits the reality. The outcomes of validation and comparison attest that the improved accident prediction model has statisticbased approbatory quality. Moreover, the improved accident prediction model combined with the NB distribution shows relatively high predictive accuracy of the probability of accident occurrence.

Context and related works

Accidents at railway level crossings (LXs) often give rise to serious material and human damage and hamper railway safety reputation, although the majority of accidents are caused by vehicle driver violations. LX safety is one of the most critical issues for railways which needs to be tackled urgently [START_REF] Ghazel | Using stochastic Petri nets for level-crossing collision risk assessment[END_REF][START_REF] Mekki | Validation of a new functional design of automatic protection systems at level crossings with modelchecking techniques[END_REF][START_REF] Liu | Model-Based Diagnosis of Multi-Track Level Crossing Plants[END_REF]. In 2012, there were more than 118,000 LXs in the 28 countries of the European Union (E.U.) which correspond to an average of 5 LXs per 10 line-km (ERA, 2014). Accidents at European LXs account for about one-third of the entire railway accidents. They result in more than 300 deaths every year in Europe [START_REF] Liu | Model-Based Diagnosis of Multi-Track Level Crossing Plants[END_REF].

In some European countries, accidents at LXs account for up to 50% of railway accidents [START_REF] Ghazel | Two-half-barrier level crossings versus four-half-barrier level crossings: A comparative risk analysis study[END_REF]Evans, 2011b). In the entire E.U. zone, the overall number of deaths per fatal accident in railways from 1990 to 2009 is 4.10, with no apparent long-term change over time (Evans, 2011a). In France, the railway network shows more than 18,000 LXs for 30,000 km of railway lines, which are crossed daily by 16 million vehicles on average, and around 13,000 LXs show heavy road and railway traffic (SNCF Réseau, 2011). Despite numerous measures already taken to improve the LX safety, SNCF Réseau (the French national railway infrastructure manager) counted 100 collisions at LXs which led to 25 deaths in 2014. This number was half the total number of collisions per year at LXs a decade ago, but still too large (SNCF Réseau, 2015). In order to significantly reduce the accidents and their related consequences at LXs, it is crucial to establish a high quality accident prediction model and carry out a thorough analysis to understand the potential reasons for accidents occurring at LXs. Indeed, this paves the way for making appropriate safety diagnoses at LXs. Many existing works dealing with LX safety are devoted to developing qualitative approaches, in order to understand the potential reasons causing accidents at LXs, such as surveys [START_REF] Wigglesworth | A human factors commentary on innovations at railroad-highway grade crossings in Australia[END_REF], interviews [START_REF] Read | Walking the line: Understanding pedestrian behaviour and risk at rail level crossings with cognitive work analysis[END_REF], focus group methods [START_REF] Stefanova | Systems-based approach to investigate unsafe pedestrian behaviour at level crossings[END_REF] or driving simulators [START_REF] Larue | Assessing driver acceptance of Intelligent Transport Systems in the context of railway level crossings[END_REF], rather than collecting real field data. In recent years, a systems analysis framework [START_REF] Leveson | Engineering a Safer World: Systems Thinking Applied to Safety[END_REF][START_REF] Read | Walking the line: Understanding pedestrian behaviour and risk at rail level crossings with cognitive work analysis[END_REF][START_REF] Wilson | Fundamentals of systems ergonomics/human factors[END_REF] and a psychological schema theory [START_REF] Salmon | The crash at Kerang: Investigating systemic and psychological factors leading to unintentional non-compliance at rail level crossings[END_REF][START_REF] Stanton | Exploring the psychological factors involved in the Ladbroke Grove rail accident[END_REF]) have been used to analyze the contributory factors underlying the accidents occurring at LXs. A study presented by [START_REF] Salmon | The crash at Kerang: Investigating systemic and psychological factors leading to unintentional non-compliance at rail level crossings[END_REF] described a collision between a loaded semi-trailer truck and a train, which occurred in North Victoria, Australia, when the truck crossed the LX while the LX is occupied by railways without lights flashing. According to the investigation of the Office of the Chief Investigator (OCI), the truck driver in this study was not aware of the train and the activated state of the level crossing until it was too late to stop the truck. A study conducted by [START_REF] Davey | The experiences and perceptions of heavy vehicle drivers and train drivers of dangers at railway level crossings[END_REF] discussed the intentional violation of vehicle drivers crossing LXs, particularly focusing on vehicle driver's complacency due to the high level of familiarity. [START_REF] Tey | Measuring driver responses at railway level crossings[END_REF] conducted an experiment to measure vehicle drivers' responses to LXs equipped with stop signs (passive), flashing lights and half barriers with flashing lights (active) respectively. In this study, the vehicle drivers' responses result from both the field survey and a driving simulator. Although these available qualitative approaches are beneficial to understand factors causing LX accidents, they do not allow for predicting the number or the probability of accident occurrence, or quantifying the contribution degree of the various impacting factors. Thereby, quantitative safety analysis approaches are crucial to thoroughly understand the impacting factors and enable the identification of practical design and improvement recommendations to prevent accidents at LXs.

One can notice that a number of quantitative studies on statistical models to predict LX accident frequency open a significant vista on understanding the risk related to LX accidents. In 1941, L. E. Peabody and T. B. Dimmick of the U.S. Bureau of Public Roads developed one of the earliest railway-highway crossing accident prediction models to estimate the number of accidents at railway-highway crossings in 5 years, named Peabody-Dimmick Formula [START_REF] Ogden | Railroad-Highway Grade Crossing Handbook[END_REF]. This formula was developed based on the accident data of rural railway-highway crossings in 29 states in the U.S. and was utilized through the 1950s. As shown in Eq. ( 1), the parameters considered in this formula are the average daily road traffic V, the average daily railway traffic T , and the protection coefficient indicative of warning devices adopted P. K is an additional parameter.

A 5 = 1.28 × (V 0.170 × T 0.151 ) P 0.171 + K (1)
However, advances in both warning device technologies and LX design features quickly led to an unavailability of the predefined formula form and coefficients that reflected the conditions pertaining to LX accidents in 1941.

The next evolutionary step in LX accident prediction was the New Hampshire Index [START_REF] Oh | Accident prediction model for railway-highway interfaces[END_REF] which is given as follows:

HI = V × T × P f (2)
where HI represents the hazard index; V is the average daily road traffic; T is the average daily railway traffic and P f is the protection factor indicative of the warning devices adopted.

The New Hampshire model is a relative formula which can be used to rank the importance of crossing upgrades.

Due to its simplicity, it has been widely used across the U.S. However, it is limited in that it does not predict the expected number of collisions, but only gives some indications about the priorities in terms of LX safety.

The accident prediction formula developed by the U.S. Department of Transportation (USDOT) in the early 1980s sought to overcome the limitations of earlier models [START_REF] Chadwick | Highway-rail grade crossing safety challenges for shared operations of high-speed passenger and heavy freight rail in the US[END_REF]. This comprehensive formula comprises three primary equations:

a = K × EI × MT × DT × HP × MS × HT × HL (3) B = T 0 T 0 + T × a + T T 0 + T × ( N T ), T 0 = 1 0.05 + a (4) A =         
0.7159 × B, for passive devices; 0.5292 × B, for flashing lights; 0.4921 × B, for gates;

(5) where a is the initial collision prediction (collisions per year at a given LX); K is the formula constant; EI is the exposure index (a variant of traffic moment) based on the product of highway and railway traffic; MT is the index for the number of main tracks; DT is the index for daily through trains during daylight; HP is the index for highway paved; MS is the index for maximum train speed; HT is the index for highway type; HL is the index for highway lanes. B is the adjusted accident frequency; T 0 is the weighting factor and N is the number of accidents observed in T years at a given LX. Finally, A is the normalized accident frequency.

The USDOT formula is the most commonly used model in the U.S. today. A specified table of USDOT provides each of the indexes for LXs equipped with passive controls, flashing lights and gates [START_REF] Austin | An alternative accident prediction model for highway-rail interfaces[END_REF].

Although the formula is comprehensive, its current definition makes it difficult to identify or prioritize design or improvement activities that will most effectively address LX safety-related problems, since it does not provide the magnitude of the characteristics' contribution to the LX safety.

The Australian Level Crossing Assessment Model (ALCAM) is a location specific and parameterized risk model which provides a method for assessing risks to LX users, train passengers and train staff [START_REF] Woods | Safer European Level Crossing Appraisal and Technology (SELCAT)-D3 Report on Risk Modeling Techniques for level crossing risk and system safety evaluation[END_REF]. The ALCAM model is given as follows:

ALCAM Risk Score = Infrastructure Factor × Exposure Factor × Consequence Factor (6)
where the Infrastructure Factor is the output of a complex scoring algorithm that assesses how the physical properties at each LX site will affect human behavior; the Exposure Factor is a function of the LX control type, vehicle (or pedestrian) volumes and train volumes (i.e., the Peabody-Dimmick Formula is used as the Exposure Factor function)

to address the combined exposure of trains and road vehicles (or pedestrians) pertaining to various LX control types;

the Consequence Factor is the expected consequence of a collision which includes deaths and injuries involving both railway and roadway. The Infrastructure Factor adjusts the accident probability per year to reflect the actual LX site conditions. Multiplying the Infrastructure Factor by the Exposure Factor will give the actual annual likelihood of an accident occurring at a particular LX (National ALCAM Committee, 2012). The Consequence Factor is expressed in terms of an expected number of equivalent fatalities per year. An equivalent fatality is a combination of all types of harm using the ratio: 1 fatality = 10 serious injuries = 200 minor injuries. The ALCAM has been applied across all Australian states and in New Zealand since 2003, and overseen by a committee of representatives from the various jurisdictions of these countries to ensure its consistency in terms of development and application.

However, the ALCAM does not cover all kinds of LX accidents, since its main focus is deliberate and accidental collisions involving user errors but excluding vandalism and suicide. It should be noticed that some LX physical properties considered in ALCAM show a high correlation between each other, which implies the existence of a kind of redundancy between the model inputs, and consequently a bias in terms of the outputs.

In recent studies, authors tended to adopt the Poisson regression model, the NB regression model or variants of the Poisson regression model (e.g., zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB)) combined with the estimated λ = e m j=1 β j x j +σ (x j is the independent variable considered and β j is the estimated coefficient of x j ) [START_REF] Cameron | Econometric models based on count data: Comparisons and applications of some estimators and tests[END_REF][START_REF] Lawless | Negative binomial and mixed Poisson regression[END_REF][START_REF] Cameron | Regression-based tests for overdispersion in the Poisson model[END_REF][START_REF] Miaou | The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions[END_REF][START_REF] Austin | An alternative accident prediction model for highway-rail interfaces[END_REF][START_REF] Chang | Analysis of freeway accident frequencies: Negative binomial regression versus artificial neural network[END_REF][START_REF] Lu | Accident prediction model for public highway-rail grade crossings[END_REF] to deal with accident statistics. However, this form of estimated λ is not appropriate in our case. According to the constraints between the LX accident frequency and impacting variables, presented in section 3.2, some variables (e.g., the average daily railway traffic, the average daily road traffic and the road traffic accidents) should not be used in an exponential form, due to the logical assumption that the case where these variables are equal to 0, would directly lead to 0 accident occurrence. Therefore, these aforementioned approaches combined with traditional λ will introduce high bias when predicting the LX accident frequency and the probability of accident occurrence. As shown in Table 1, SAL2 (more than 10,000) is the most widely used type of LX in France. Moreover, more than 4,000 accidents at SAL2 LXs contributed most to the total number of accidents at LXs from 1974 to 2014. LX accidents are caused by the following transport modes: 1) motorized vehicle (MV), 2) pedestrian and bicycle (PB). As illustrated in Fig. 2, the motorized vehicle is the main transport mode causing LX accidents in the 21 geographical regions in France. Moreover, as the LX accident frequency caused by motorized vehicles increases, the entire LX accident frequency increases accordingly. On the contrary, pedestrians and cyclists contribute very little to the overall risk1 related to LX accidents [START_REF] Liang | Risk analysis on level crossings using a causal Bayesian network based approach[END_REF]. Considering the train/motorized vehicle (train-MV) collisions, SAL2 LXs also have the most part of LX accidents according to the statistics shown in Fig. 3. For all these reasons, we will limit the scope of our analysis to train-MV accidents occurring at SAL2 LXs; in fact, from the aforementioned observation, these accidents can be considered as the most representative for LX accidents in general.

Methodology

Data sources and coding

The data to support our investigation come from a dedicated accident/incident database provided by SNCF Réseau.

SNCF Réseau investigated and recorded various attributes of LX accidents/incidents, railway and roadway traffic characteristics, surrounding characteristics of LXs. Therefore, the accident/incident data and involved LX information that cover SAL2 LXs in 21 geographical administrative regions in mainland France from 1990 to 2013 are obtained.

In the present study, an adequate sample is selected to include the data in the decade from 2004 to 2013, which provides reliable and sufficient information about both LX accidents and railway, roadway and LX characteristics.

Namely, the selected LX inventory presents the LX identification number, the LX location, the LX accident timestamp, the railway traffic volume, the road traffic volume, the LX dimension, the profile and alignment of the entered road and so on. There are 8,332 public SAL2 LXs involved in our investigation. The total number of SAL2 LXs in

France is about 10,000. However, in our investigation we considered only those about which we had enough information (8,332). Using the LX identification number and the LX accident timestamp in the accident/incident database, the annual accident frequency at a given SAL2 is obtained. Then, a new database containing 10 years of data is created, using again the LX identification number as a common data element, which includes annual LX accident frequency, railway, roadway and LX characteristics at a given SAL2 and annual roadway accident statistics. Impacting parameters pertaining to LX accidents considered in our investigation should be thought to be: (1) important in determining accident frequency, (2) more permanent in nature (e.g., sight obstruction noted as a problematic factor due to involved alterable construction topography, vegetation and other environmental elements) and ( 3) not accidentdependent [START_REF] Austin | An alternative accident prediction model for highway-rail interfaces[END_REF]. This combined database formed the basis of our investigation. The parameters considered in this investigation are shown in Table 2. As shown in Table 2, some minor data transformations in the combined database were necessary. Variables that have multiple non-numeric choices (e.g., profile, alignment) are encoded as singular indicator variables. Numerical variables, such as the average daily road traffic, the average daily railway traffic, the railway speed limit, the LX width and the crossing length are used as they are without transformation. The region risk factor is determined by the general accident frequency per SAL2 in the region. The road accident factor is determined by the ratio of the annual number of road accidents in a given year to the average number of road accidents per year over the period of 10 years considered. The statistical characterization of the variables considered is given in Table 3.

It is worth noticing that by using the Spearman correlation checking [START_REF] Borkowf | Computing the nonnull asymptotic variance and the asymptotic relative efficiency of Spearman's rank correlation[END_REF], we found that some other parameters tested were not significant (e.g., the road-rail track angle at a given LX) or highly correlated with the parameters considered in our analysis (e.g., the number of lanes at a given LX is highly correlated with the LX width).

Preliminary accident prediction model

Based on some preliminary analyses, it is worth noticing that five constraints need to be considered so as to develop the model for predicting annual accident frequency at a given SAL2:

-The predicted accident frequency should always be non-negative;

-It should be 0 if the average daily railway traffic is 0;

-It should be 0 if the average daily road traffic is 0;

-It should be 0 if the annual road traffic accidents are 0;

-The model should be time-dependent, i.e., it should reflect the variation of accident frequency as time advances.

For the preliminary accident prediction model, we considered only three parameters in Table 2, which are the average daily railway traffic, the average daily road traffic and the annual road accidents. The preliminary model is developed as follows:

λ 10P = K × F RAcc × V a × T b (7)
where λ 10P represents the annual accident frequency at a given SAL2 during the period of 10 years considered; K is the constant coefficient; F RAcc is the road accident factor; V is the average daily road traffic and T is the average daily railway traffic. Here, F RAcc is a time-dependent variable which can reflect the variation of annual road accidents as time advances.

The conventional formula of the traffic moment is given by: Traffic moment = Road traffic frequency × Railway traffic frequency [START_REF] Liang | Risk analysis on level crossings using a causal Bayesian network based approach[END_REF]. However, based on some previous analyses, we adopt a variant called "corrected moment", or CM for short. CM = V a × T b , where b = 1a and the best value of a in terms of fitting is computed to be a = 0.354 based on the previous statistical analysis performed by SNCF Réseau (SNCF Réseau, 2010). Therefore, we consider (V 0.354 × T 0.646 ) as an integrated parameter that reflects the combined exposure frequency of both railway and road traffic. One can notice that Eq. ( 7) can be rewritten as

λ 10P = K × RM, where RM = F RAcc × V 0.354 × T 0.646 .
Thus, this model can be regarded as a linear model with respect to the composite parameter RM. The Ordinary Least-Squares (OLS) method is employed to estimate coefficient K. As shown in Fig. 4, K is estimated as 1.319e-04 (tstatistic = 33.72 > 1.96 corresponding to a 95% confidence level).

Fig. 4 indicates that this preliminary model shows that, for high values of corrected moment, there is a significant deviation between observed accident frequencies and predicted accident frequencies at SAL2 LXs. Therefore, further statistical analysis is carried out to evaluate the quality of the transformed linear model. In this case, we make group classification, which means that the data set is divided into 100 groups with the same number of samples in each group. Then, the mean value of λ 10P and RM of each group are computed respectively to generate a linear relationship between the group-mean λ 10P and the group-mean RM. Hence, we can adopt "Residuals vs. Fitted" graph, "Normal Q-Q" graph, "Scale-Location" graph and "Residual vs. Leverage" graph [START_REF] Anscombe | Graphs in statistical analysis[END_REF][START_REF] Chatterjee | Regression analysis by example[END_REF] to check the linearity, the normal distribution of residuals, the homoscedasticity [START_REF] Jarque | Efficient tests for normality, homoscedasticity and serial independence of regression residuals[END_REF] and the abnormal values of the model, respectively. These four graphs pertaining to our group classification analysis are shown in Fig. 5. For an idealized linear model: 1) the red line in Fig. 5a significant. These residuals correspond to SAL2 LXs with high corrected moment in our data set. Therefore, the statistical test results attest the significant deviation between the observed accident frequencies and the predicted accident frequencies at SAL2 LXs having high corrected moment. Through checking the accident/incident data, these SAL2 LXs with high corrected moment are correspondingly accident-prone LXs in general. We conjecture that this preliminary accident prediction model is not appropriate to predict the annual accident frequency at SAL2 LXs with high corrected moment. A thorough analysis needs to be performed to develop an improved model which can predict the annual accident frequency at a given SAL2 more accurately. Meanwhile, the model should consider more impacting variables.

Improved accident prediction model

Based on the previous analysis in section 3.2, we consequently developed an improved version of the prediction model, as shown below:

λ 10Y = K × F RAcc × (V 0.354 × T 0.646 ) × e (C Pro f ile ×I Pro f ile +C Align ×I Align +C Wid ×Wid+C Leng ×Leng+C RS L ×RS L+C Reg ×F Reg ) (8)
where λ 10Y represents the annual accident frequency at a given SAL2 for a period of 10 years; K is the constant coefficient; F RAcc is the road accident factor; V is the average daily road traffic; T is the average daily railway traffic and V 0.354 × T 0.646 is regarded as the corrected moment; I Pro f ile and C Pro f ile are respectively the profile indicator and its corresponding coefficient; I Align and C Align are respectively the alignment indicator and its corresponding coefficient;

Wid and C Wid are respectively the LX width and its corresponding coefficient; Leng and C Leng are respectively the crossing length and its corresponding coefficient; RS L and C RS L are respectively the railway speed limit and its corresponding coefficient; F Reg and C Reg are respectively the region factor and its corresponding coefficient (see Table 2).

Note that appropriate higher orders and interaction terms of covariates can be included in Eq. ( 8) without difficulty, due to the use of exponential form [START_REF] Miaou | The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions[END_REF]. The Nonlinear Least-Squares (NLS) method and Gauss-Newton algorithm [START_REF] Madsen | Methods for non-linear least squares problems (2nd Edition)[END_REF]) are adopted to estimate the coefficients of variables. Estimated coefficients are provided in Table 4. A |t -statistic| > 1.96 is introduced to identify significant parameters corresponding to a 95% confidence level. The average daily railway traffic, the railway speed limit, the average daily road traffic, the annual road accidents, the road alignment, the LX width, the crossing length and the region LX-accident-prone factor have significant and positive influence on SAL2 accident frequency. However, the test shows that the road profile does not have a significant impact. The tstatistic for road profile is not significant which indicates that the coefficient of road profile C Pro f ile should be 0 and the impact of road profile could be neglected (The interpretation of this fact is discussed in section 5.1.3). Moreover, the coefficients of the considered variables with the exponential form can reflect their respective contribution degrees to the SAL2 accident frequency. According to these contribution degrees (ranked in brackets), the region LX-accident-prone factor is the most important contributor among these variables.

Model validation and distribution identification

In this section, we will validate the quality of the two prediction models and identify an appropriate statistical distribution combined with the prediction model of accident frequency, in such a way as to make a more accurate estimation of the probability of accidents occurring at a given SAL2 in a given year.

Statistical test evaluation

We applied the two prediction models to estimate the annual accident frequency based on the 10-year accident data of SAL2 LXs. The Monte-Carlo test for randomly sampling annual accident frequencies which meet the condition that the predicted annual accident frequency at a given SAL2 is equal to or more than the observed annual accident frequency at the SAL2 considered is performed (Considering a safety strict principle, the predicted annual accident frequency should not be lower than the observed annual accident frequency). Then, the percentages of randomly sampled annual accident frequencies that meet this condition are computed to compare with the actual percentages of specified entire sampled annual accident frequencies (e.g., as for the entire 80,000 annual accident frequencies sampled out of 83,320, the actual entire percentage is computed as 80,000/83,320; while k annual accident frequencies within the 80,000 frequencies sampled meet the above condition, thus, the percentage of randomly sampled annual accident frequencies meeting this condition is computed as k/83,320). Table 5 shows the Monte-Carlo test results.

One can notice that, for the specified entire random sampling size 80,000, 40,000, 10,000, 5,000 and 500, the percentages of randomly sampled annual accident frequencies meeting the aforementioned condition computed using λ 10Y are all closer to the actual percentages of specified entire sampled annual accident frequencies, compared with the percentages of randomly sampled annual accident frequencies computed using λ 10P . Moreover, the similarity between the percentages of randomly sampled annual accident frequencies meeting the aforementioned condition, which are computed using λ 10Y , and the actual specified entire percentages is relatively high.

Although the Monte-Carlo test results indicate that the λ 10Y model seems more appropriate, the tested percentages of annual accident frequencies sampled according to the aforementioned condition closer to the actual percentages are not able to thoroughly attest to the fact that the quality of λ 10Y model is definitely better, since the predicted accident frequency may be much higher than the accident frequency observed. Therefore, further statistical tests are required to comprehensively evaluate the model quality. Akaike's information criterion (AIC) [START_REF] Bozdogan | Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions[END_REF], the Bayesian information criterion (BIC) [START_REF] Weakliem | A critique of the Bayesian information criterion for model selection[END_REF], the Pearson chi-square statistic test (PCS) [START_REF] Dahiya | Pearson chi-squared test of fit with random intervals[END_REF] and the degree of freedom (DF) are computed to evaluate the goodness of fit (GOF) of the model. They can be expressed as follows:

AIC = n + n × ln(2π) + n × ln(RSS/n) + 2(l + 1) (9) BIC = n + n × ln(2π) + n × ln(RSS/n) + (l + 1)ln(n) (10) PCS = n i=1 (O i -λ i ) 2 λ i (11) DF = n -(l + 1) (12)
where n is the sample size; RSS is the sum of the squares of residuals between the annual accident frequencies observed and the annual accident frequencies estimated; l is the number of independent exponential parameters; O i is the annual accident frequency observed and λ i is the annual accident frequency expected.

The AIC and BIC are two statistical measures to test the relative quality of models for a given set of data. Smaller AIC and BIC values indicate a better model fitting. The PCS test is used to determine whether there is a significant difference between the values expected and the values observed. The PCS is roughly equal to the model DF if the model fits the data perfectly without any dispersion. In other words, the closer the PCS is to the DF, the better the model fits the data [START_REF] Lu | Accident prediction model for public highway-rail grade crossings[END_REF]. These statistical test results are shown in Table 6 with the goodness ranked in brackets. Some findings can be noticed: 1) considering AIC and BIC, the λ 10Y model gives better results, since the AIC and BIC values corresponding to the λ 10Y model are much smaller than those for the λ 10P model; 2) as for PCS, the λ 10Y model is also the preferred one, since the PCS of the λ 10Y model is closer to DF (DFs of the λ 10Y and the λ 10P are considerably approximative).

Statistical distribution identification

In this section, further analysis is performed to identify a more appropriate statistic distribution combined with the accident frequency prediction model. The Poisson distribution shown as Eq. ( 13) is a natural first choice for modeling such accident occurrence. [START_REF] Chang | Analysis of freeway accident frequencies: Negative binomial regression versus artificial neural network[END_REF] indicates that accident frequency is likely to be over-dispersed (cf. Eq. ( 14))

and suggests using the negative binomial (NB) distribution as an alternative.

Poi(X = k) = λ k e -λ k! , k = 0, 1, 2, . . . ( 13
)
where Poi(X = k) is the probability of k accidents occurring, k ∈ IN and λ is the expectation value of the number of accidents. In our study, λ is expressed by Eq. ( 7) or Eq. ( 8).

V AR(X)

         = E(X) > E(X), over-dispersed < E(X), under-dispersed (14)
The NB model as a special case of Poisson-Gamma mixture model is a variant of the Poisson model designed to deal with over-dispersed data [START_REF] Lord | The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives[END_REF][START_REF] Buddhavarapu | Modeling unobserved heterogeneity using finite mixture random parameters for spatially correlated discrete count data[END_REF]. The over-dispersion could come from several possible sources, e.g., omitted variables, uncertainty in exposure data, covariates or nonhomogeneous LX environment [START_REF] Miaou | The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions[END_REF]. The NB model considered in this study has the following form:

P NB (X = k) = Γ k + 1 α Γ (k + 1) Γ 1 α 1 1 + αλ 1/α αλ 1 + αλ k , k = 0, 1, 2, . . . ( 15
)
where P NB (X = k) is the probability of k accidents occurring, k ∈ IN; λ is the expectation value of the number of accidents and α is the dispersion parameter.

The relationship between the mean value and the variance in the NB model is given as follows:

V AR(X) = E(X) + αE(X) 2 ( 16)

If α > 0, there is an over-dispersion; if α < 0, there is an under-dispersion and, in the case where α = 0, the NB model reduces to the Poisson model.

However, the NB model is limited to handling under-dispersed data (α < 0) [START_REF] Lord | The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives[END_REF]. If the dispersion parameter α is set as a negative value to try to handle under-dispersion issue, it would no longer be an NB model and would lead to unreliable estimation, especially when the sample mean is low and the sample size is small Fig. 6. Constraint between the group variance and the group mean of annual accidents for all SAL2 LXs. [START_REF] Lu | Accident prediction model for public highway-rail grade crossings[END_REF]. [START_REF] Oh | Accident prediction model for railway-highway interfaces[END_REF] proposed the Gamma model to handle under-dispersed samples. The Gamma model is given as follows:

VAR (n) = 1.0171E(n), R^2 = 0.
P G (X = k) = Gamma(βk, λ) -Gamma(β(k + 1), λ) (17)
where P G (X = k) is the probability of k accidents occurring, k ∈ IN; λ is the expectation value of the number of accidents and β is the dispersion parameter. If β < 1, there is an over-dispersion; if β > 1, there is an under-dispersion and if β = 1, the Gamma model reduces to the Poisson model. However, the Gamma model shown as Eq. ( 18) is limited to the time-dependent observation assumption and zero observations [START_REF] Lord | The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives[END_REF], since general Γ(x) restricts discrete responses to positive values.

Gamma(βk, λ) =            1, if k = 0 1 Γ(βk) λ 0 u βk-1 e -u du , if k > 0 (18)
Therefore, the restriction between variance and mean value is significant to identify an appropriate statistical distribution. Firstly, we adopted group classification to make preliminary variance analysis, which is that the annual accidents at a given SAL2 during the 10 years were divided into 100 groups with the same number of samples in each group. Then, the mean value and variance of accidents in each group were computed respectively to analyze the relationship between the group variance and the group mean value. The variance analysis is shown in Fig. 6. It seems that there is a slight over-dispersion of the data set, since the variance V AR(n) is a bit bigger than the mean

E(n) (V AR(n) = 1.0171E(n)).
Since the mean value and the variance are very close to each other, we performed meticulous analyses to assess both the Poisson and the NB models with regard to all SAL2 LXs in our accident database so as to identify which model is more effective. When applying the NB distribution, we adopt the Maximum Likelihood Estimation (MLE) method to estimate the dispersion parameter α of the data set [START_REF] Dai | Maximum likelihood estimate for the dispersion parameter of the negative binomial distribution[END_REF]. Using R language to perform the MLE method, α is estimated at 1.9594.

The log-likelihood statistic (LL) is adopted to assess the GOF of the accident frequency prediction model combined with a statistical distribution and identify the more appropriate distribution for estimating the probabilities of accident frequency observed. The larger the LL, the more preferred the model [START_REF] Lu | Accident prediction model for public highway-rail grade crossings[END_REF]. The mathematical description of the LL is given as:

LL = n i=1 ln( Pi ) ( 19
)
where n is the sample size and Pi is the estimated probability of accident frequency observed. Pi is computed respectively according to the accident frequency prediction model combined with the Poisson or the NB distribution.

LL test results are shown in Table 6. One can notice that for λ 10Y model combined with either the Poisson or NB distribution, its GOFs are significantly better than λ 10P model's GOFs according to LL results. Furthermore, the GOF of λ 10Y combined with the NB distribution is better than when combined with the Poisson distribution. Based on the predicted probability of the accident frequency observed, further Cumulative Distribution Function (CDF) analysis with regard to the Poisson and the NB distributions is performed to evaluate the quality of the accident frequency prediction model combined with these two statistical distributions. As shown in Fig. 7, the relationship between the CDF and the corresponding probability of a given event is depicted. P(•) denotes the predicted probability of a given event obtained through the Poisson or NB distribution; O i is the observed accident frequency and λ i is the estimated accident frequency. The blue curve "CDF NB λ 10P , O i > λ i " represents the CDF of event " O i > λ i " obtained through the NB distribution combined with the λ 10P ; the red curve "CDF NB λ 10P , O i <= λ i " represents the CDF of event " O i <= λ i " obtained through the NB distribution combined with the λ 10P ; the green curve "CDF POI λ 10P , O i > λ i " represents the CDF of event " O i > λ i " obtained through the Poisson distribution combined with the λ 10P ; the violet curve "CDF POI λ 10P , O i <= λ i " represents the CDF of event " O i <= λ i " obtained through the Poisson distribution combined with the λ 10P . The interpretation of the remaining curves involving the λ 10Y can be similarly obtained. Given that some curves are almost covered by some others in Fig. 7, the extracted results of CDF analysis shown in Table 7 become clearer for discussion. At a later stage in this paper, we carry out a comparison between the predicted probabilities of annual accident frequency observed at a given SAL2 according to the λ 10Y and the λ 10P combined with the Poisson and the NB distribution. In this context, group classification analysis is adopted to compute the mean predicted probability and the mean observed accident frequency of each group with the same sample size. The relationship between the mean predicted probability and the mean observed accident frequency is shown in Fig. 8. The blue and red scatters respectively represent the λ 10Y model combined with the Poisson and NB distributions. The interpretation of the remaining scatters involving the λ 10P can be similarly obtained. One can notice that the λ 10Y model combined with the Poisson and NB distributions respectively predict higher probabilities of observed accident frequencies than the λ 10P model combined with them. This difference is particularly noticeable in the case where the two prediction models are combined with the NB distribution (i.e., the red and green scatters). Moreover, one can also notice that the probability of accident occurrence estimated by the Poisson distribution combined with the λ 10Y model is higher than that estimated by the NB distribution combined with the λ 10Y model.

However, it should be recalled that the higher probability predicted does not necessarily indicate a higher predictive accuracy, since the probability of accident occurrence in reality would not be high. Therefore, further analysis to assess the predictive accuracy of the Poisson and the NB distributions should be carried out. As shown in on average is computed as: fk = n i=1 P(X i = k)/n, where P(X i = k) is the estimated probability of k accidents occurring at a given SAL2 in a given year. According to the goodness of predictive accuracy ranked in brackets, the NB distribution shows a higher predictive accuracy with regard to various annual numbers of accidents occurring at a given SAL2 during the 10-year period, particularly, when combining with the λ 10Y . In the cases of 0, 1 and 2 accidents occurring at a given SAL2 in a given year, the predictive accuracy of the NB distribution combined with the λ 10Y takes the first place in all the cases, which means that the probabilities of accident occurrence predicted by the NB distribution combined with the λ 10Y are closest to the actual frequencies of accident occurrence. In the case of more than 2 accidents occurring at a given SAL2 in a given year, the predictive accuracy of the NB distribution combined with the λ 10Y takes the second place, with the deviation of only 0.0002% compared with f k , the actual percentage of observed annual accident frequency samples. In fact, there are no SAL2 LXs showing more than 2 accidents in the same year during the 10-year period considered.

In fact, the prediction performance of ZIP and ZINB regression methods were also examined but resulted in no higher goodness-of-fit values and a quite small number of significant parameters (only 4 and 3 significant parameters corresponding to ZIP and ZINB, respectively) compared with the Poisson and NB models and, hence, were not reported in the comparison.

Discussion

Accident frequency prediction model evaluation

Based on the aforementioned analyses using the Monte-Carlo, AIC, BIC and PCS tests, one can notice that λ 10Y model for accident frequency prediction has better GOF and considers impacting variables more comprehensively (cf.

Table 6, 9 characteristics considered in λ 10Y vs. 3 characteristics considered in λ 10P ). In this section, we will scrutinize the impact of variables considered in λ 10Y model on SAL2 accident frequency.

Railway traffic characteristics

Two different characteristics regarding railway traffic are proved to be significant in terms of affecting SAL2 accident frequency, namely the average daily railway traffic and the railway speed limit. They positively influence the annual accident frequency at a given SAL2. The average daily railway traffic with a power of 0.646 has a more decisive impact on the accident frequency than the average daily road traffic with a power of 0.354, since the higher the railway traffic frequency appearing at SAL2 LXs, the much higher the SAL2 accident risk. Furthermore, the higher the railway speed limit, the higher the SAL2 accident risk. A tentative to explain this finding is that a high railway speed limit corresponds to a high actual train speed, therefore, the risk for train-MV collisions is higher.

Roadway traffic characteristics

There are two roadway traffic characteristics which have a significant impact on SAL2 accident frequency, namely the average daily road traffic and the annual road accidents. The average daily road traffic with a power of 0.354 has a positive impact on the annual accident frequency at a given SAL2. If there is no road traffic at a given SAL2, there would be no accidents caused by motorized vehicles at this SAL2. Moreover, the higher the combined exposure of railway and roadway traffic, the higher the likelihood of an accident occurring.

One of the most important characteristics to estimate the annual accident frequency at a given SAL2 is the annual road accidents. The impact of road accidents was likely to be ignored in the previous studies pertaining to LX safety analysis. The present study has clearly shown that the accidents at LXs are above all road accidents and they highly depend on the road safety level. Moreover, the road accident factor in λ 10Y model is time-dependent, since road accidents have an annual variation (vary every year). Correspondingly, the risk related to LX accidents has an annual variation as well.

LX characteristics

Four LX characteristics, namely the road alignment, the LX width, the crossing length and the LX-accident-prone factor of the region, have a significantly positive impact on the annual accident frequency at a given SAL2. It is worth recalling that the vertical road profile has no significant impact (|t -statistic| of profile < 1.96). A tentative is discussed with SNCF experts to explain this fact as follows: on the one hand, the "hump or cavity" profile would cause an increasing risk of accidents involving long/heavy vehicles (trucks, buses, etc.), with relatively low population; on the other hand, for most of ordinary cars, such a profile obliges ordinary cars to cross the LX with low speeds, thus helps reduce the risk of LX accident occurrence. Besides, for LXs lacking the road profile information, they are treated as normal situations (no hump or cavity) according to SNCF experts' advice. Namely, 0 is used as the profile indicator for them when performing data coding. These conjectural reasons need to be further investigated in future works. In addition, although the above reasons may be suitable for French LX situation, they would depend on case-by-case scenarios and the policy implications in other countries.

For the other significant LX characteristics, a higher region LX-accident-prone factor, a poorer road alignment trace, a larger LX width and a larger crossing length correspond to a higher risk of LX accidents. According to the contribution degrees of variables ranked in Table 4, the contribution of the region LX-accident-prone factor to the risk of LX accidents takes the first place followed by the contribution of the road alignment, the LX width and the crossing length successively.

Statistic distribution evaluation

According to LL test results shown in Table 6, the NB distribution combined with the λ 10Y shows a better quality for predicting accident frequency with over-dispersion. Considering the CDF analysis shown in Fig. 7 and Table 7, the NB distribution combined with the λ 10Y also shows higher percentages of cases meeting the two conditions that the probability of the event "the accident frequency estimated no less than the actual accident frequency observed" is higher than 0.95 and 0.995, than those of the Poisson distribution combined with the λ 10Y . On the other hand, the NB distribution combined with the λ 10Y shows a lower percentage of cases meeting the condition that the probability of the event "the accident frequency estimated less than the actual accident frequency observed" is higher than 0.05, than those of the Poisson distribution combined with the λ 10Y .

Moreover, in terms of estimated probabilities of actual accidents occurring at a given SAL2 (cf. Table 8), the NB distribution combined with the λ 10Y shows a higher predictive accuracy with regard to various annual numbers of accidents occurring at a given SAL2 during the 10 years.

It is worth noticing that, although the NB distribution is more effective than the Poisson distribution when dealing with over-dispersed accident count, it requires more extensive computations to estimate the model parameters as well as the dispersion parameter and, to generate inferential statistics, compared with the Poisson model.

Conclusions

In the present study, we have developed an accident frequency prediction model based on LX accident statistics and various impacting factors. This model allows for predicting accident occurrence with a considerably high accuracy and has a more appropriate form compared with the existing models pertaining to LX accident prediction. Although the developed prediction model is tailored to SAL2 LX accidents in France, the general formula form of the model and the methodology adopted to set up the model and validate its quality can be applied to different contexts. The scientific selection process in our study ensures that the main impacting variables are considered and redundant variables are excluded. In fact, impacting variables pertaining to LX risk should be thought to be important in determining accident frequency, more permanent in nature and not accident-dependent.

The region LX-accident-prone factor which can indicate the impact of regional LX safety level on LX accident frequency is originally utilized in the improved model. CM, a more effective factor, is proposed in this study to replace the conventional traffic moment, single average daily railway traffic or single average daily road traffic when explaining the likelihood of LX collisions. The significant impact of road accidents, almost ignored in the past studies, is well considered in our study.

A validation of the model quality is performed by means of a set of comprehensive statistical approaches, namely the Monte-Carlo, AIC, BIC and PCS tests, which all indicate that the improved accident prediction model involving various influential factors is trustworthy and sound. Moreover, the LL test, CDF analysis and predictive accuracy analysis are conjunctively employed to identify a more appropriate statistical distribution for predicting the probability of LX accident occurrence. The results obtained attest that the improved model combined with NB distribution has relatively high predictive accuracy of the probability of accident occurrence. To the best of our knowledge, such a thorough validation process is rarely achieved in similar existing works.

To sum up, the above contributions of the present study offer an in-depth perspective on potential parameters causing LX accidents. Moreover, these contributions pave the way for identifying practical design measures and improvement recommendations to prevent accidents at LXs. In future works, we will establish Bayesian risk models to quantify the causal relationships between the impacting parameters and the risk related to LX accidents, and assess their respective impact on the LX safety level. In addition, the effectiveness of various technical solutions will be investigated based on some experiments that we have carried out at several LXs.

and Networks (IFSTTAR).

Fig. 1 .

 1 Fig. 1. Four types of LXs in France.

Fig. 2 .

 2 Fig. 2. Comprehensive accident frequency in different regions.

Fig. 3 .

 3 Fig. 3. The number of collisions (train-MV) at different types of LX in France from 1978 to 2013.

F

  Fig. 4. The preliminary accident frequency prediction model λ 10P vs. (F RAcc × V 0.354 × T 0.646 ).

  should be a horizontal line at 0 and residuals should randomly distribute around this line; 2) the standardized residuals shown in Fig. 5b should fall in the 45 • direct line, which can attest the normal distribution of residuals; 3) the red line shown in Fig. 5c should be a horizontal line at a certain value and square roots of standardized residuals should randomly distribute around this line; 4) Fig. 5d can identify abnormal values and significant values which have an important impact on the model fitting, through Cook's distance. One can notice that residuals in the groups with big ID (i.e., 95, 99 and 100) are

Fig. 5 .

 5 Fig. 5. Statistical evaluation of the model quality: (a) Residuals vs. Fitted, (b) Normal Q-Q, (c) Scale-Location, (d) Residual vs. Leverage.

  Fig. 7. CDF of the Poisson and the NB distributions combined with the λ 10P and λ 10Y models according to the estimated probability.

Table 1 .

 1 Accidents at different types of LXs in France from 1974 to 2014.

	Type of LX	Number	# Accident
	SAL4	> 600	> 600
	SAL2	> 10,000	> 4,000
	SAL0	> 60	> 50
	Crossbuck LX	> 3,000	> 700

Table 2 .

 2 Parameters considered and data coding.

	Parameter	Explanation	Data coding
	Railway traffic characteristics
	Average daily railway traffic	The average number of trains	Numerical, used directly;
		crossing the LX daily
	Railway speed limit	The maximum permission	Numerical, used directly;
		speed of train within the LX
		section
	Roadway traffic characteristics
	Average daily road traffic	The average number of road ve-	Numerical, used directly;
		hicles crossing the LX daily
	Annual road accidents	The number of road accidents in	Road accident factor:
		a given year	Annual road accidents in a given year / Average
			road accidents per year over the period observed;
	LX characteristics
	Alignment	Horizontal road alignment	Alignment indicator; 0, 1 and 2 represent
		shape: "straight", "curve" or	"straight", "curve" and "S", respectively;
		"S"
	Profile	Vertical road profile shape:	Profile indicator; 0 and 1 represent "normal" and
		"normal" or "hump or cavity"	"hump or cavity", respectively;
	LX width	The entered road width	Numerical, used directly;
	Crossing length	The length of LX that road ve-	Numerical, used directly;
		hicles need to cross
	Region risk	The region of the LX considered	Region risk factor, highlighting the general
			LX-accident-prone region:

T he number o f S AL2 accidents over the observation period in the region considered / T he number o f S AL2 LXs in the region considered;

Table 3 .

 3 Statistical characterization of variables considered.

	Variable	Mean	Variance	StdDev	Min	Max
	Annual LX accident	0.0057	0.0060	0.0776	0	2
	Average daily railway traffic	26.0636	914.5413	30.2413	0.5000	330
	Railway speed limit	92.4599	1.7963e+03	42.3829	5	160
	Average daily road traffic	826.8022	3.1718e+06	1.7810e+03	0.5700	2.5570e+04
	Corrected moment	51.4744	3.7377e+03	61.1367	1.2781	938.5449
	Road accident factor	1.0001	0.0189	0.1378	0.8058	1.1988
	Alignment	0.2587	0.3209	0.5665	0	2
	Profile	0.1488	0.1266	0.3559	0	1
	Length	9.6766	14.9545	3.8671	3	59
	Width	5.4504	1.8414	1.3569	2	24
	Region risk factor	0.3487	0.0142	0.1194	0.1739	0.7747

Table 4 .

 4 Estimated coefficients of the improved accident prediction model.

	Parameter	Coefficient	Estimated value	Standard error	t -statistic	Significant
		K	2.703e-05	5.078e-06	5.322	×
	I Pro f ile	C Pro f ile	3.626e-02	5.706e-02	0.635
	I Align	C Align	3.427e-01 (2)	2.942e-02	11.648	×
	Wid	C Wid	9.847e-02 (3)	1.494e-02	6.589	×
	Leng	C Leng	2.084e-02 (4)	4.284e-03	4.865	×
	RS L	C RS L	3.089e-03 (5)	7.586e-04	4.072	×
	F Reg	C Reg	4.962e-01 (1)	1.722e-01	2.882	×

Table 5 .

 5 Monte-Carlo test results.

	# Samples	Actual percentage of annual	λ 10Y -model estimated per-	λ 10P -model estimated per-
		accident frequencies sam-	centage of annual accident	centage of annual accident
		pled	frequencies sampled	frequencies sampled
	80,000	0.96015	0.95482	0.94191
	40,000	0.48008	0.47747	0.45463
	10,000	0.12002	0.11946	0.11416
	5,000	0.06001	0.05959	0.05665
	500	0.00600	0.00598	0.00576

Table 6 .

 6 Model GOF comparison.

	Parameter	λ 10Y Poisson	λ 10Y NB	λ 10P Poisson	λ 10P NB
	Railway traffic characteristics
	Average daily railway traffic	×	×	×	×
	Railway speed limit	×	×
	Roadway traffic characteristics
	Average daily road traffic	×	×	×	×
	Annual road accidents	×	×	×	×
	LX characteristics
	Alignment	×	×
	Profile	×	×
	LX width	×	×
	Crossing length	×	×
	Region	×	×
	AIC	-190,744 (1)	-190,744 (1)	-190,591 (2)	-190,591 (2)
	BIC	-190,670 (1)	-190,670 (1)	-190,573 (2)	-190,573 (2)
	PCS	65,796 (1)	65,796 (1)	53,108 (2)	53,108 (2)
	DF	83,313	83,313	83,319	83,319
	LL	-2,599 (2)	-2,596 (1)	-2,631 (4)	-2,629 (3)
	Goodness score (the lower, the better)	5	4	10	9

Table 7 .

 7 The extracted results of CDF analysis.

	Model CDF	P(O i > λ i ) > 0.005	P(O i > λ i ) > 0.05	P(O i <= λ i ) > 0.95	P(O i <= λ i ) > 0.995
		(CDF in percent)	(CDF in percent)	(CDF in percent)	(CDF in percent)
	CDF NB λ 10P	85.29 (3)	6.61 (1)	99.62 (1)	57.19 (3)
	CDF NB λ 10Y	79.10 (2)	7.68 (3)	99.36 (3)	66.07 (1)
	CDF POI λ 10P	85.29 (3)	6.82 (2)	99.61 (2)	57.15 (4)
	CDF POI λ 10Y	78.89 (1)	9.17 (4)	99.27 (4)	65.94 (2)

Table 7

 7 10P , O i > λ i :In 85.29% of cases, P(O i > λ i ) is more than 0.005; in 6.61% of cases, P(O i > λ i ) is more than 0.05;2) CDF POI λ 10P , O i > λ i :In 85.29% of cases, P(O i > λ i ) is more than 0.005; in 6.82% of cases, P(O i > λ i ) is more than 0.05;3) CDF NB λ 10Y , O i > λ i : In 79.10% of cases, P(O i > λ i ) is more than 0.005; in 7.68% of cases, P(O i > λ i ) is more than 0.05; 4) CDF POI λ 10Y , O i > λ i :In 78.89% of cases, P(O i > λ i ) is more than 0.005; in 9.17% of cases, P(O i > λ i ) is more than 0.05;5) CDF NB λ 10P , O i <= λAccording to the CDF analysis results shown in Table7, in the cases of " P(O i > λ i ) > 0.005" and " P(O i <= λ i ) > 0.995", for the λ 10Y model combined with either the Poisson or the NB distribution, its GOFs are significantly better than λ 10P model's GOFs. In the cases of " P(O i > λ i ) > 0.05" and " P(O i <= λ i ) > 0.95", the criteria of the two models combined with the Poisson and the NB distributions have no obvious distinction, in particular, for the criterion " P(O i <= λ i ) > 0.95". Furthermore, λ 10Y combined with the NB distribution shows a slightly better quality than when combined with the Poisson distribution.

	indicates that:
	1) CDF NB λ

i :

In 99.62% of cases, P(O i <= λ i ) is more than 0.95; in 57.19% of cases, P(O i <= λ i ) is more than 0.995;

6) CDF POI λ 10P , O i <= λ i :

In 99.61% of cases, P(O i <= λ i ) is more than 0.95; in 57.15% of cases, P(O i <= λ i ) is more than 0.995;

7) CDF NB λ 10Y , O i <= λ i :

In 99.36% of cases, P(O i <= λ i ) is more than 0.95; in 66.07% of cases, P(O i <= λ i ) is more than 0.995;

8) CDF POI λ 10Y , O i <= λ i :

In 99.27% of cases, P(O i <= λ i ) is more than 0.95; in 65.94% of cases, P(O i <= λ i ) is more than 0.995;

Table 8 ,

 8 f k denotes the percentage of samples of observed annual accident frequency with k accidents involved in a given year ( f k = the number of samples of observed annual accident frequency involving k accidents occurring in a given year / the total number of samples n). The estimated relative annual accident frequency reflected by estimated probabilities Fig. 8. Average predicted Probability of annual accidents observed at a given SAL2 using λ 10P and λ 10Y combined with the Poisson and the NB distribution.

		1
		0.98
	Average predicted Probability	0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96						Pro POI λ10Y Pro NB λ10Y Pro POI λ10P Pro NB λ10P
		0	0.01	0.02	0.03	0.04	0.05	0.06
				Average actual accidents per year at a given SAL2

Table 8 .

 8 The predictive accuracy comparison between the Poisson distribution and the NB distribution.

	# Annual ac-	Observed annual	NB-λ 10Y estimated	POI-λ 10Y estimated	NB-λ 10P estimated	POI-λ 10P estimated
	cidents consid-	accident frequency	relative	annual	relative	annual	relative	annual	relative	annual
	ered (k)	( f k in percent)	accident frequency	accident frequency	accident frequency	accident frequency
			( fk in percent)	( fk in percent)	( fk in percent)	( fk in percent)
	0	99.4371	99.3915 (1)		99.3903 (2)		99.3279 (3)		99.3255 (4)
	1	0.5485	0.5999 (1)		0.6033 (2)		0.6647 (3)		0.6673 (4)
	2	0.0144	0.0077 (1)		0.0062 (3)		0.0069 (2)		0.0055 (4)
	> 2	0	0.0002 (2)		0.0001 (1)		0.0001 (1)		0.0001 (1)
	Goodness score (the lower, the better) 5		8		9		13

In this paper, most of the time the term "risk" stands for the occurrence likelihood.

Acknowledgements

This work has been conducted in the framework of "MORIPAN project: MOdèle de RIsque pour les PAssages à Niveau" within the Railenium technological research institute, in partnership with the National Society of French

Railway Networks (SNCF Réseau) and the French Institute of Science and Technology for Transport, Development


Educational novels written by women in the 18 th century: a significant advance for the instruction of female readers -"Jamais fille chaste n'a lu de romans..." 1 -"A chaste girl has never read novels..." (Preface of La Nouvelle Héloise) -"Une femme bel esprit est le fléau de son mari, de ses enfants, de ses amis, de ses valets, de tout le monde." ( 2 Rousseau, Emile ou de l'éducation)

-"A quick-witted woman is the scourge of her husband, of her children, of her friends, of her servants, of everyone." (Rousseau, Emile ou de l'éducation) -"Toute fille lettrée restera fille toute sa vie tant qu'il n'y aura que des hommes sensés sur la terre" 3 -"Any learned girl will remain a girl all her life so long as there are only sensible men on earth." (Fifth book of Emile ou de l'éducation).

Rousseau's views on education profoundly marked readers at the end of the 18 th century. Among his numerous female readers, certain, however, did not hesitate to enter onto the path of educational writing, despite the great man's anathemas of "women authors". The writing of educational works still appears to be the only type of writing that a woman from polite society could justify at the time; she could put forward the reason that it was a useful step and in the very line of her maternal duties and thus avoid accusations of frivolous wit.

Faced with the needs of women's education and the demand for training, cultivated women of high society followed the path opened up by Madame de Maintenon a century earlier and the writings of Madame de Lambert. Madame d'Epinay, with the first volume of Conversations d'Emilie (1774)(seventeen seventy-four) outlined the principles of an enlightened education founded on discussions between mothers and daughters; Madame de Genlis, after having written plays for the education of young girls , wrote a considerable educational work with her novel Adèle and Théodore in 1782 (seventeen eighty-two).

These two women authors knew perfectly Rousseau's theories, they read a lot of his work and both were also acquainted with him: what theories did they have on the education of girls? Where did they stand in relation to this dominantly masculine discourse? Madame d'Epinay and Madame de Genlis both wrote works of fiction filled with educational principles: however, their theories differed radically from those of their male contemporaries who were authors of fiction and who were interested in women's education.

After Rousseau, few men won renown at the end of the 18 th century through their theories on education. However, the question of women's education can be found in works of fiction written by men, in particular in licentious novels.

The famous novel by Choderlos de Laclos, Les Liaisons Dangereuses, was published in 1782 (seventeen eighty-two), the same year as Adèle et Théodore. In 1784 (seventeen eighty-four), Rétif de la Bretonne was also interested in the problem in his work La Paysanne perverti. Mirabeau also proposed a narrative of the education of young girls in his novel Le rideau levé ou l'éducation de Laure at the same period. The topic of an innocent young girl corrupted by a libertine (who uses books to seduce her) which had a strong foothold at the beginning of the century, is used again in these novels. Education always takes place to the detriment of virtue, whether for the young Cécile de Volanges or for the Marquise de Merteuil, a self-taught woman, but libertine. The collusion between reading and debauched sexuality is constantly appearing in these works of fiction, for the great pleasure of the libertine reader. 4 In contrast to these images of young girls corrupted by reading, Madame d'Epinay and Madame de Genlis proposed characters that were extremely well-behaved young girl readers, who educated themselves and acquired wisdom by their reading. These women's theories thus totally break with the topic of women's libertine reading found in a great number of 18 th century works of fiction; on a good many points they come closer to the precepts advocated by Rousseau for educating virtuous and chaste girls.

They replaced the "female reader and libertine seducer" duo by that of the "mother watching over her daughter". These authors, having read Rousseau, adapted into their educational principles lessons from this great man: both of these women depicted young girls guided by very devoted mothers; they were never put to the care of a nurse or in a convent but were raised, from the earliest age until marriage, by a loyal mother acting as private tutor, after the fashion of Sophie in Emile ou de l'éducation.

All of these mothers recommend an outdoor education and allow their girls to grow up and freely develop without stylish finery, as Rousseau strongly advises. The mother in Conversations d'Emilie invites the little girl to go and run in the park rather than stay in her room. Madame de Genlis' work Adèle et Théodore opens with a letter from the Baroness d'Almane announcing her departure from Paris for Languedoc with her husband and two small children; they move away to bring up their children in the country.

Both mothers also insist on the fact that above all they do not want to make scholars out of their daughters but rather well-mannered and virtuous young girls who will become devoted wives and attentive mothers. The Baroness d'Almane even writes in Madame de Genlis' work: "... on doit éviter avec soin d'enflammer l'imagination des femmes et d'exhalter leurs têtes, elles sont nées pour une vie monotone et dépendant." "one should carefully avoid kindling the imagination of women and letting it go to their heads; they are born to lead a monotonous and dependent life." 5 Through such discussion, they thereby defend themselves against all attacks against pedantic women scholars and distinguish their girls as those "of wit who will remain a girl all her life" according to Rousseau.

However, these women authors in their theories do not totally follow that of Rousseau; they in particular move away from him on the quality and quantity of instruction given to the young heroines. 5 Madame de Genlis, Adèle et Thédore (Paris, Michel Lambert, 1782, vol. 1, letter X, pg. 39).

They will not leave to the husband the care of educating their wives as they please, just as Rousseau wishes it in Emile. They judge education essential for a woman to be happy in her home 6 and defend the joy of learning. Madame d'Epinay reminds her readers that they must educate their children and that "to educate, one must be educated". 7 Her heroine complains about her lack of education and her daughter Emilie, who is not even ten years old yet in the first volume of Les Conversations, can read, make summaries, write abstracts and comment upon writers such as Plutarch with her mother. As for Madame de Genlis, she multiplies Adèle's activities and courses in her novel: an English nurse teaches her English at six months old, so that she can speak it as her mother tongue; a master of Italian drawing trains her at a very young age speaking Italian to her. She speaks and reads later in three languages, draws, plays music and is able to keep herself busy, learn and teach in multiple fields (history, geography, literature, geometry, etc.) without forgetting, of course, morals and religion which are a big part of her education. The images of women readers in these educational novels thus differ totally from images of "young girls corrupted by their reading". Wide-ranging fields of knowledge are open to young girls in the novels written by women authors, without danger. The declared objective is to train self-sufficient young women capable of taking the paths of knowledge on their own, while at the same time teaching others. There is a real difference with works by their male author contemporaries, who at best in their works of fiction deplore the unsuitable education of convents, or women reading (too scholarly for their understanding or too novelistic for them to be happy in their daily lives). The works by women authors offer veritable learning programs to their potential female readers.

These are works written to cause the feminine condition to advance. The authors are not content to show that young girls can read without any danger and learn with benefit; Madame d'Epinay and Madame de Genlis also include exercises and programs adapted to the age of their heroines in their works of fiction. Madame d'Epinay, in Les Conversations d'Emilie, explains how to accompany the child's readings: make the child copy and summarize passages difficult for his or her age, and show the child incoherencies of certain literary passages.

6 "Rousseau may well confine women to their homes, Madame d'Epinay gives them the great recipe for avoiding that. She reverses the principle of dependance established by Jean-Jacques. Therein lies the most obvious critique ever made at the time of Rousseau's doctrine. It was also of incomparable grandeur of Madame d'Epinay to understand two centuries before many others that women should themselves work for their happiness and not leave it up to men only" according to Elisabeth Badinter (Emile, Emilie ou l'ambition féminine au XVIIIe siècle, Paris, Flammarion, 1983, pg. 395).

Madame de Genlis at the end of the novel gives Adèle's reading program from the age of six up to twenty-two years old. She describes and explains certain exercises that aim to develop the critical mind of her two children: for example, the mother-teacher dreamed up imaginary correspondence between a brother and sister who were separated; he is in Paris, she in the provinces. He reads the wrong things and has dangerous liaisons, and speaks of them in his letters; she answers arguing so as to show him the dangers incurred and the errors in his reasoning. By giving the letters of this imaginary boy one by one to her daughter, she interests her in the individual and confers to her the task of guiding this young man down the good road by her reasoning. Together they correct Adèle's letters and the mother later reads her version of the letter to her; she thus proceeds with an assessment, correcting what the pupil writes. Furthermore, she emphasizes the flaws of certain books through this exercise and develops the young girl's critical ability.

Even if her role as an honest woman consists in obeying and keeping quiet, Adèle is trained to be like Julie, in charge of her estate, rather than Sophie. She has nothing to envy her brother in his education. She benefits just like he does from educational outings: factory and workshop visits accompanied by readings from the Encyclopedia. With her parents she goes on an initiatory voyage through Europe, traditionally reserved for boys. During these journeys, instruction does not stop for a moment and it is varied: the parents and children give an account of their readings to the group and then comment on and critique them together. Therefore these texts accord to women the same intellectual capacities as men and they explain to the real female readers how to develop the minds of new generations of women. Even more, they stand against Rousseau's allegations and all denigrators of women writers, and highlight women authors and their works. The comparison between the male author's work and that of Madame de Lambert aims at making the qualities of the woman's work shine. Feminine solidarity worked well; women authors sent each other compliments and laudatory references: Les Conversations d'Emilie are among Adèle's first readings. Madame de Genlis also gave her Madame de La Fite's Dialogues to read, Madame Riccoboni's books, Madame de Graffigny's Lettres Péruviennes in Italian, Madame Deshoulière's poetry; she calls to mind the theatre of Mademoiselle Barbier, and of course Madame de Sévigné's Lettres. Women authors are not the only ones mentioned: for example, Madame de Genlis cannot stop her praises for Fénelon and his Télémaque, and she praises Richardson's virtuous novels. However, they place women's works in their reading programs without making any difference with those by male authors, and this is a noteworthy development.

Within these works, the very broad educational programs for young girls, as well as the references to works by famous women authors, aimed at erasing the differences between men and women, and pointed the way towards equality between the sexes. However, it is not only by discussion within their works that these women authors contributed to the advancement of women in the realm of knowledge. Their act of writing in itself consisted of an important progression for women in 18 th century society.

Louise d'Epinay, known for her relations with Rousseau, Duclos, Diderot, Voltaire and especially Grimm, raised her children with an attention unusual in her social class. She sent her first writings to Rousseau, Lettres à mon fils, and he criticized its style as being hardly appropriate for childhood. She pursued her duties as woman educator in bringing up her little girl, Emilie, and in writing Les Conversations d'Emilie, focusing this time on women's education. She presents herself as a simple woman, a mother having raised several children and who shares her experience with the public. This modest approach was well-received; she was the first to receive the Monthyon prize rewarding the book of the year "from which the greatest good for society could result" Madame de Genlis believed she would have the prize. She had campaigned with her partisans at the Academy and had denigrated the work of her rival: "A work full of language errors, without interest... which is only known because I praised it... which contains twenty pages of phrases which are not French and words only useful to chambermaids.(?)" However opposed to the Encyclopedists by her religious notions, she had three times less votes than that of her rival. See Gabriel de Broglie on this subject, Madame de Genlis (Paris, Librairie Académique Perrin, 1985, pg. 117 and following).

Even if she lost this battle with Madame d'Epinay, Caroline de Genlis was granted other honors: herself a mother of two daughters, she wrote plays for them which had success in polite society. She took under her charge the twin daughters of the Duchess of Chartres at the age of two years and withdrew herself from the height of her fame at the Palais Royal to raise them at a distance in a pavillion specially designed for her educational purposes. In 1781 (seventeen eighty-one), she published Les Annales de la vertu ou cours d'histoire à l'usage des jeunes gens under her own name. In 1782 (seventeen eighty-two), the twelve hundred pages of Adèle et Théodore introduced her educational theories at the very moment when high society was learning, with astonishment, that the Duke of Chartres was also entrusting to her the education of his two boys and for the first time a woman was named "tutor" of a prince and would ensure his education until adult age. At the Bellechasse pavillion, she thus directed a coeducational school and educated a future king: Louis Philippe. The author of 84 (eighty-four) books between 1779 (seventeen seventy-nine) and 1787 (seventeen eightyseven), she gained even more fame under the Consulate and the Empire. Strengthened by this position as a successful author, she published an essay in 1811 (eighteen eleven) entitled De l'influence des femmes sur la littérature française 12 , and she undertook nothing less than rewriting the Encyclopedia, removing all references contrary to Religion.

These women did not go as far in their works as they did in their lives... Coming from noble and well-to-do backgrounds, Madame de Genlis and Madame d'Epinay gradually moved forward towards the status of woman author while hiding behind the usefulness of their job as educator. In Madame de Genlis' work of fiction, Madame d'Almane, the mother of Adèle and Théodore, apologizes for always producing new educational novels; she explains that it is to compensate for the striking lack of work in that field, for the joy of her children, and to help other parents. Besides, she reserves the education of her son to her husband even though she also writes for her son. Not everyone was ready at the time to see a woman in charge of men's education: the conspiracy which followed her appointment as tutor illustrates that point. Lampoons slandering Madame de Genlis circulated in 1782 (seventeen eighty-two): they suspected La Harpe as being the man who wrote her texts and nicknamed him "deputy tutor" while the Duke of Luynes, who was very fat, was called the dauphin's nanny. A good many women criticized her for having taken this traditionally masculine role. Nevertheless, Madame de Genlis continued her work and her duties, insisting in all of her discussions on the importance of a woman being useful to society by her writings and her actions. Therefore, we can truly speak of writings that brought about changes in the women's condition of their time. Madame d'Epinay and Madame de Genlis broke new 12 De l'influence des femmes sur la littérature française comme protectrices des lettres et comme auteurs ; ou précis de l'histoire des femmes françaises les plus célèbres. By Madame de Genlis (Paris, chez Maradan, Libraire, rue des Grands-Augustins, no. 9, 1811). ground in presenting heroines that were lively, very learned and yet well-behaved. They showed in their works that when well-educated and taken care of, young girls could follow, without danger, an education equivalent to that of boys. Their mothers, loyal educators at their side, also brought forth a new image of woman and in particular, of women writing educational texts, of great use to society. These books were useful for the personal journeys of their authors and for their access to rare honors for women at that time. Despite the resonance of Rousseau's theories at the end of the century, Madame de Genlis and Madame d'Epinay were able pave their way into the only legitimate path for women : the education of children and departure from Rousseau precepts to develop open and enlightened women's education programs, giving concrete keys to their readers as to how to bring about changes in the condition of women in society... which enlightened men have never done.

  SANDRINE ARAGON, Aphra ben conférence 2000 Denver Colorado University of Lille III -Charles de Gaulle, France Madame d'Epinay, Madame de Genlis and Women's Education in Eighteenth Century France

  10 and was crowned by the Académie française in 1783 (seventeen eighty-three). She came ahead of a book by Daubenton on sheep, a work by Parmentier on potatoes and over Adèle et Théodore by Madame de Genlis. 11 10 Prize founded in 1782 by Mr. de Monthyon to award "the work in prose published during the year which was the most useful and the most well-written." Madame d'Epinay presented the second volume of Conversations d'Emilie, the first volume having great success seven years earlier. She was supported by all of her encyclopedist friends.

  11

  These charming comedies were the work of a lady of the court, famous for the graces of her spirit and the diversity of her talents. She shielded herself from the society which she charmed, to consecrate her time to an infinitely more interesting and more noble goal. It was quite natural that in Régentine's court there could be found a woman of such rare merit; but no one could conceive of why such an enlightened Princess had not conferred upon her the education of her daughters." She also gives the works of Madame de Lambert to her daughter to read, in parallel with the reading of Méditations by a male writer, and she predicts that she would very quickly see the text that she prefers: "Mère : (...) je vous donnerai moi les cahiers d'une femme d'un grand mérite de ma connaissance.

	"
	Madame d'Epinay praises Madame de Genlis' educational plays, her first works
	contemporary with Madame d'Epinay's Conversations d'Emilie. Within a little story
	inserted into the twelfth conversation:
	"Ces comédies charmantes étaient l'ouvrage d'une dame de la cour, célèbre par
	les grâces de son esprit et par la diversité de ses talents. Elle s'était soustraite à
	la société don't elle faisait le charme, pour consacrer son temps à un but
	infiniment plus intéressant et plus noble. Il était bien naturel qu'à la cour de
	Régentine il se trouvât une femme d'un mérite si rare; mais personne ne pouvait
	concevoir, comment une Princesse aussi éclairée ne lui avait pas confié
	l'éducation de ses filles." 8

Emilie : Qu'est-ce qu'il y a dans ces cahiers? Mère : Elle a fait l'extrait des vies des hommes illustres de Plutarque, à l'usage d'une jeune personne qui en a singulièrement profité. Vous savez ce que c'est qu'un extrait, et vous comprenez qu'elle a rapproché les traits les plus remarquables de tous les grands personnages de l'antiquité. Si cela vous convient, après chaque Méditation vous lirez un de ces extraits; nous verrons lequel de ces deux ouvrages vous aimerez le mieux à la longue." "Mother: ... I shall give you the notebooks of a woman of high esteem that I know of. Emilie: What is in these notebooks? Mother: She had made an extract of the lives of Plutarch's illustrious men, for use by a young person who particularly benefited from them. You know what an extract is all about, and you understand that she brought together the most remarkable traits of all the great individuals from antiquity. If you like, you should read one of these extracts after each Méditation; then we shall see which one of these two works you like the best in the end." 9

Even if Choderlos de Laclos distances himself from Rousseau's anathemas, his essay on women's education, published one year after Les Liaisons dangereuses, takes inspiration from the master but arrives at different conclusions. See Choderlos de Laclos, De l'éducation des femmes (1783),(Grenoble, Jérôme Millon, 1991).

 Madame d'Epinay, Les Conversations d'Emilie, Twelfth conversation (Oxford, Voltaire Foundation, 1996, pg. 238).

Madame d'Epinay, Les Conversations d'Emilie, Fourteenth conversation, pg. 286. Madame de Genlis' Théâtre à l'usage des jeunes personnes was published by Panckouke in 1779 and with Lambert et Baudoin in 1780.

Madamed'Epinay, Les Conversations d'Emilie, Sixteenth conversation,