A Probabilistic Model Development to Predict Traffic Accident Tendency
under Snowy Conditions for Expressway
Bui Tien Manh
Supervisor: Prof. Kazushi SANO
Urban Transport Laboratory
1. Introduction
1.1. Background
Road traffic accidents occurred with the change between seasons. The frequency of traffic accidents inthe winter season is three times higher than in summer because drivers are affected by adverse factors such as
heavy snowfall, blowing snow, high winds, and fog, which lead to snow pavement, poor visibility, and rust isincreasing. According to statistics from the US, Canada, and Japan and recent studies have shown that thefrequency of traffic accidents tends to increase under snowy conditions. The study aims to develop a
probabilistic model to analyse the factors and predict traffic accident under snowy conditions on expressways.
According to statistics about accidents on expressways in Japan, over a 10-year period (2010-2020), thenumber of accidents per 100,000 persons on expressways decreased by 57% but the number of accidents
remained at a high level (309,178 in 2020). According to the analysis of the accident dataset on fiveexpressways in Niigata prefecture (2012-2020), this study has shown that accidents occurring on expressways
during the winter season (from December to March) are higher compared to other seasons, in which accidents
are concentrated in January and tend to decrease gradually until the end of March and are shown in Figure 1.

Figure 1. Number and percentage of accidents occurring on expressways in Niigata prefecture, Japan(2012-2020)1

At present, there have not been many studies on developing models to analyse influencing factors andpredict traffic accident trends for expressways that consider snowy conditions in the winter season. Besides,
these studies have not considered factors such as average flow speed, and segment type, and identified thefactors affecting accidents in snowy conditions. In addition, the influencing factors are aggregated over a long
period of time (years or days), which leads to decrease accuracy in forecasting and analysing the impact offactors on accidents under winter conditions.
1.2. Novelty of Research
The novelty of this study will develop a probabilistic model based on traditional statistic model andmachine learning to analyse factors effecting the tendency of predict the tendency of hourly accident andpredict accident frequency occurring on expressways in short-term (hour) that consider snow conditions.
Besides, the study also considers factors affecting accidents in a short time (hour and minutes) to ensure theaccuracy of the probabilistic model.
1.3. Research objective
This study includes some objectives as follows:- Determines multicollinearity in the model based on correlation analysis and analyses factors affecting
the traffic accident tendency based on NB models.
- Predicts the number of accidents occurring in the short-term (hour) on expressways under snowy
conditions based on the NB model and ANN model.
- Finally, compare the performance between the NB model and ANN model to propose a suitable model.
2. Methodology
The study used correlation analysis to determine the correlation between factors used in the model. Thathelps to detect the multicollinearity in the regression model. Then, the study develops the probabilistic modelthat includes Negative Binomial Regression Model (NB) in the traditional statistic model and Artificial Neural
Network (ANN) in Machine Learning algorithms to analyse factors effecting the accident tendency and predict
the accident frequency occurring on expressways in short-term (hour) under snowy conditions. Finally, thestudy evaluates and compares the fit of two models based on the mean absolute error (MAE), the mean squared
error (MSE), and the root mean square error (RMSE).
2.1. Correlation analysis
This study uses correlation analysis to determine the correlations between the variables that used theprobabilistic model to avoid multicollinearity. The sample correlation coefficient of X and Y, denoted r, iscalculated by the following formula:2 ( X − X )(Y − Y )nr=ii =1i ( X − X )  (Y − Y )ni =1n2

ii =1=
2

XY − X YS X SYiCorrelation coefficient values range -1 to +1. The closer to 1 the correlation coefficient gets the“stronger” the correlation. If the absolute value of the correlation coefficient is close to 0.8, collinearity islikely to exit.
2.2. Negative binomial regression model (NB)
The negative binomial regression model with probability density function is as follows: ( yi +  −1 )

 −1

 1 Pr (Y = yi |ij ,  ) = ( −1 )  ( yi + 1)  1 + i  i 
 1 + i yiWhere:- Pr (Y = yi |i ,  ) is the probability of accidents yi occurring segment ith over the hour jth.
- yi: is the number of accidents occurring to a given segment ith during the hour j, yi = 0,1, 2.
- μi is the expected number of accidents of segment ith during the hour j. In this study, μi follows thebelow function:

ij = e  Li  TVi   e(  X +.........+  X + )0122ikkii- Li: is the length of segment ith.
- TVi and Xki: is the hourly traffic volume and other explanatory variables at given segment ith during
the hour jth, respectively.
- βk and α: is coefficients of models and the dispersion parameter, respectively. These parameters areestimated by maximizing the log-likelihood function.
2.3. Artificial Neural Network (ANN)
ANN model is designed with an input layer, an output layer and two hidden layers. The input layer
consists of 8 neurons that are factors effecting traffic accidents. The output layer consists of one neuron whichis the hourly accident frequency. There are 2 hidden layers with 5 neurons and 3 neurons, respectively.
Between layers in ANN model are connected full connections by weights, bias, and activation functions (theleaky rectified linear activation unit). The structure of the ANN model is shown in Figure 2 below.

3Figure 2. The structure of ANN model
The goal of the training processing is to find the weight and bias such that make the loss function (meansquared error) as small as possible. To achieve this goal, this study used stochastic gradient descent and thebackpropagation algorithm in training process. However, before conducting training processing for ANN
model, this study normalizes the input and output data using the minimax algorithm.
2.4. Model evaluation
To evaluate the predictive ability of the model, this study used main metrics which are the mean absoluteerror (MAE), the mean squared error (MSE), and the root mean square error (RMSE). The desirable model
should fit the data as closely as possible, which is presented by the smaller value of MAE, MSE, and RMSE.
Their formula is shown as follows:nMAE = |y − y |iini

( y − y )ni, MSE = ii

2, RMSE = MSEnWhere: yi, y i are the observed accident frequency, the predicted accident frequency, respectively. n thesize of the set test.
3. Study scope and dataThe scope of this study focuses on traffic accidents and stacks occurring on five major expressways which
covers the entire Niigata Prefecture. The dataset used in this study includes subdivided datasets which are anaccident and stack dataset, a traffic condition dataset, a roadway dataset, and a weather condition dataset. After
collecting the datasets, this study synthesized and matched the datasets into a common dataset based on spatial(segments) and temporal (hour). The data processing is performed by R software. Descriptive statistics ofexplanatory variables used in model are shown in Table 1 below.

4Table 1. Descriptive statistics about explanatory variables in model
Explanatory variables

MinMaxMeanSD

Accident frequency (per hour)

0.00

2.00

0.00030.019

Segment Length (Km)

0.21

3.91

0.95

0.56

Hourly traffic volume

1

4,435

426

369.48

Truck percentage (%)

050

2513

Average flow speed (km/h)

0.17

150.42

84.54

11.18

Average Snowfall (cm/10minutes)

0.00

4.93

0.03

0.10

Temperature (oC)

-11.42

31.28

4.53

4.36

Vertical gradient (%)

-4.5

4.50

-0.02

1.50

Explanatory variables
Segment type

CategoryNumber of Segments

Un-divided

174 (19.98%)

Divided697 (80.02%)

4. Results and Discussions
4.1. Results of correlation analysis
The results of the correlation analysis show that the correlation coefficient between the explanatory
variables in the model which are shown in heat map (Figure 4) below ranges from -0.35 to 0.376. This indicatesthat between explanatory variables have a weak correlation and do not cause multicollinearity for the model.

Figure 3. Correlation between factors in the model5

4.2. Results of Negative binomial regression model (NB)
After analysing the correlation between the variables in the model, the study trains NB model that used thetraining dataset (80%). The result of training this model is presented in Table 2. In additional, this table displays
the goodness of fit test of the fitted model for the hourly accident frequency occurring on segments of theexpressway under snowy conditions. In this model, Akaike's information criterion (AIC) is 97,819, the residual
deviance is 8,423, and the ratio of the log-likelihood index (ρ2) is 0.19.
Table 2. Parameter estimates of the NB model.
Explanatory variables

Estimate

Std. Errorz value

Pr(>|z|)Intercept-7.0410.271-26.025< 2e-16

Ln (Hourly traffic volume)

0.734

0.044

16.811< 2e-16

Truck percentage-2.281

0.247-9.242< 2e-16

Average snowfall

1.382

0.272

5.083

3.72e-08

Temperature-0.0730.008-8.952< 2e-16

Average flow speed-0.6520.003-21.770< 2e-16

Segment type

0.787

0.101

7.783

7.07e-15-0.0860.020-4.405

1.06e-06

Divided segments

Vertical gradient2 x log-likelihood-97,819

Standard error

1.10e-06

AIC

97,837Null deviance

6,856

Residual deviance

8,423

Ratio of log-likelihood index

0.19

Observations

16,221,504

The results of NB model show that all the factors in the model are statistically significant at the significancelevel of 1%. Besides, the results of this model have shown that the hourly traffic volume and average snowfall
have a positive effect on the expected number of traffic accidents occurring in short-term (hour). Meanwhile,
factors including truck percentage, average flow speed, temperature, and vertical gradients have a negative
effect on the expected number of traffic accidents occurring in short-term. In addition, the expected number oftraffic accidents occurring in short-term on the divided segments has higher than those on the undivided
segments on expressways under snowy conditions. Finally, the average snowfall is one of risk factors that has6

the highest effect on the expected number of traffic accidents occurs in short term on each segment undersnowy conditions.
The percentage change of the expected number of traffic accidents occurring in short-term on each segmentof expressways under snowy conditions when the risk factors (variables) except variables, hourly trafficvolume, in the NB model change one unit is summarized in Table 3.
Table 3. The percentage change of the expected number of traffic accidents

Explanatory variables

Coefficients(βi)

Percentage
Exp(βi)Change(%)

Lower

Upper

Std.

95.0%

95.0%

Error

Confidence

Confidence

Limit

Limit

Intercept-7.0410.001-99.9%

31.1%-160.9%-38.9%

Truck percentage-2.2810.102-89.8%

28.0%-144.7%-34.9%

Average snowfall

1.382

3.983

298.3%

31.3%

237.0%

359.6%

Temperature-0.073

0.930

-7.0%

0.8%

-8.6%

-5.5%

Average follow speed-0.6520.521-47.9%

0.3%-48.5%-47.3%

0.787

2.197

119.7%

10.6%

98.8%

140.5%-0.086

0.918

-8.2%

2.0%

-12.2%-4.3%

Segment

Dividedtypesegments

Vertical gradient4.3. Results of Artificial Neural Network (ANN)
ANN model is trained in training dataset and validation dataset (80%). This training process was conductedwith 100 epochs with loss function (mean squared error) and mean absolute error (MAE). The results of thetraining process are shown in Figure 4 below.

7Figure 4. The training processing for ANN model
The results of the training process are as follows:- The values of the loss function (MSE), and MAE in the training process are approximately zero. This
indicates that the predicted accident frequency closes with the actual accident frequency.
- The values of the loss function (MSE) and mean square error (MAE) in the validation process are closeto the value in the training process. Therefore, the training process of the ANN model is relatively good.
After the training process, ANN model was evaluated by a test dataset. The results show that the values ofMAE, MSE, and RMSE of the test process are relatively small and close to the value of the training processof the model. This indicates that the model is a good model or has high accuracy.
4.4. Comparison results between two models.
The study compared the predictive performance of two models based on MAE, MSE, and RMSE on a test
dataset (20%) are shown in Table 4 below.
Table 4. Comparison results between two models
MeasureNB ModelANN Model

The Mean Absolute Error (MAE)

0.00060.0004The Mean Squared Error (MSE)

0.00040.0003The Root Mean Squared Error (RMSE)

0.02

0.0173The study has shown some comparative results between the two models as follows:8- The values of MAE, MSE, and RMSE are small in both models. This indicates both models have the good
performance or high accuracy.
- The values of MAE, MSE, and RMSE in ANN model is less than NB model. Therefore, ANN model isbetter than the NB model.
5. Conclusions and Limitations
The purpose of this study was to analyse the affected factors and predict the accident frequency occurringin short-term (hour). This helps expressway operators make appropriate policies to reduce traffic accidents on
expressways under snowy conditions. The results of this study are as flows:- Results of correlation analysis show that the correlation coefficient between variables in the model ranges
from -0.35 to 0.376. This indicates between variables in the model have a weak correlation and do not cause
multicollinearity for the regression model.
- The results of the NB model have shown that all the factors are statistically significant at the significancelevel of 1%. Besides, the hourly traffic volume and average snowfall have a positive effect on the expected
number of traffic accidents occurring in short-term (hour) on each segment of expressways under snowy
conditions. Meanwhile, risk factors including truck percentage, average flow speed, temperature, and verticalgradients have a negative effect on the expected number of traffic accidents occurring in short-term on eachsegment of expressways. In addition, the expected number of traffic accidents occurring in short-term on eachdivided segment tends to have higher than those on each undivided segment of on expressways under snowy
conditions. Finally, the average snowfall is one of risk factors that has the highest effect on the expected
number of traffic accidents occurs in short term on each segment under snowy conditions.
- Besides, this study designed a structure of ANN model with 1 input layer, an output layer, and two hiddenlayers. The training process of the ANN model is relatively good. After training process, ANN model is tested
by MAE, MSE, and RMSE index on a test dataset. The results indicates that ANN model is a good model orhas high accuracy. Finally, this study compared the performance of both models. The results show that bothmodels have good performance or high accuracy. However, ANN model is better than NB model.
However, there are some limitations in this study related to considering other factors such as snow depthon the road, road surface condition, the driver's age, the driver's experience, and the daytime and night timeeffecting traffic accident under snowy conditions; solving an excess zero in dataset; and using different
optimization algorithms in training ANN model.

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