Road Infrastructure Investment Limits Based on Minimal Accidents Using Artificial Neural Network

Abstract

Road traffic accidents are still among the top major global causes of death, injury, and disability. Despite this cause for alarm and several preventive initiatives, global road accident statistics are not improving. This study modeled annual road accidents (ARAs) as a function of demographic, economic, passenger movement, freight movement, and road capital investment indicators. The research is based on 22 years of data from more than 36 Organization for Economic Co-operation and Development (OECD) member and partner countries. Artificial neural network (ANN), multiple linear regression (MLR), and Poisson regression (PR) analysis were employed for this purpose. The ANN model outperformed the regression models by far, thus making it possible for reliable new insights and accurate results to be obtained. The ANN’s superior performance was shown to be a result of the non-linear relationship between ARA and some of the predicting variables. The average relative contribution of each variable in describing the ARA models was estimated using connection weight analysis (from the ANN model) and relative weight analysis for the regression model. The profile method was used to perform sensitivity analysis and to establish the partial variation trend of the ARA with each of the variables. The Existing Road Maintenance Investment (ERMI) and New Road Infrastructural Investment (NRII) showed a nonlinear concave-up relationship with ARA for given demography, economy, freight, and passenger movements. A combination of per capita NRII and ERMI corresponding to the minimum ARA exists. These sets of NRII and ERMI were considered safe road investment limits. The ANN-ARA model was utilized to estimate these limits with their relative proportion for diverse combinations of demography, economy, freight level, and passenger movement.

1. Introduction

Road traffic accidents are one of the major causes of death, injury, and disability globally [1,2,3,4,5]. Up to 1.3 million people lose their lives, and another 50 million sustain serious injuries due to road accidents each year, and the numbers keep increasing [6]. Annual road accidents (ARAs), which cause serious and irreparable economic and social harm to nations, are thus one of the main concerns of transportation managers around the world [7]. Successful implementation of road infrastructural safety management requires sufficient funding in addition to regulations, institutional capacity, and the availability of safety data [8]. Road management authorities are faced with the difficult task of identifying and prioritizing new projects and maintenance actions. Several optimization tools for resource allocation in road asset management have been adopted for efficient and well-informed decision-making processes [7,9,10,11,12,13]. Due to budget constraints, most of these optimization tools focus on the best alternative solutions within available budget constraints [14]. The question of how much is enough to tackle a single objective is usually given little attention effectively. The multi-criteria solution orientation of these decision frameworks also contributes to this lack of focus [9]. Safety, political importance, economic impact, and functional and structural performance are among the key priority functions for road asset management. However, among these factors, only the safety aspect affects human lives directly. This study focuses on safe limits for the annual maintenance of the road network and new infrastructural development investments in terms of minimum system-wide road accidents for diverse demographic, economic, and transport systems.

Numerous studies were conducted towards modeling the frequency of road accidents using artificial neural network (ANN) [15,16,17,18,19]. These studies focused on predicting accident frequency based on road features, environmental conditions, driver attributes, and specific types of road facilities. Traffic characteristics such as volume and vehicle mix proportions were also used as predicting variables [20]. Accident prediction models such as these are often useful in developing crash modification factors [21] and understanding and prioritizing accident risk indicators [16]. Some crash prediction models (CPMs) are incredibly helpful tools for quantitative road safety analysis and can be applied to road network screening to identify the network’s most important portions and to more effectively guide in-depth studies [22]. Other road accident predictors, such as vehicle ownership, per capita gross domestic product (GDP), and the relative usage of transport modes, have also been investigated [23]. Moreover, integrating the safety of VRUs—pedestrians, cyclists, and motorcyclists, has also resulted in significant consideration when making judgments about the management of road infrastructure [24]. Previously, ANN models were applied successfully to forecast road traffic [25] and travel time [26]; to optimize signal timing at road intersections [27], road vehicle collision avoidance systems, and road sign detection and classification [28]; and to evaluate pedestrian road crossing behavior [29]. The major aim of this study is to model and critically analyze the relationship between the frequency of road accidents and investments to maintain existing roads and for new road construction using ANN and regression analysis.

Programs to improve infrastructure-related safety are typically capital-intensive and call for management and planning. Despite being constructed to engineering standards, the road network, which consists of sections and junctions, will typically need renovations and adjustments over its lifespan due to changes, most notably in traffic volumes and additional safety requirements. Different countermeasures will typically be needed as a result of these adjustments to increase safety requirements and to repair problematic areas of the road [30]. The countermeasures can be divided into capital and maintenance work categories. However, road maintenance and new construction investments (in EUR/km) were found to result in both positive and negative effects on the severity of road accidents in Spain [31]. The work suggested further study on this observed anomaly. In an earlier study on data from the same country, the maintenance investment was found to achieve consistently lower accident fatality results [32], while investments in new construction were believed to only increase the risk and fatality of road accidents. Contrarily, a study conducted in the United States showed highway capital investment to have a negative effect on road fatalities [33]. The effects of infrastructural and demographic changes on accident frequency and fatality were studied from 8-year accident data from the State of Illinois [34]. Increasing the number and width of lanes were found to result in more crashes, while changes in demography were found to have an insignificant effect on the observed results. Reports from these studies on the effect of road investment on accidents cannot be generalized due to inconsistent findings. These contradictions could be due to several different factors related to the study area, such as demography, economy, road transport system characteristics, etc. However, it can be concluded that there has been no clear general explanation as to why road investment, whether for maintenance or new construction, can result in either a decrease or increase in the crash rate or in the number of fatalities. In this study, the inconsistent effect of road infrastructure investment on traffic accidents was shown to be due to a concave nonlinear relationship between the parameters. A simple method for estimating safe limits for both maintenance and new road infrastructure investments based on minimum Annual Road Accident (ARA) using the developed ANN-ARA model was also illustrated.

2. Data and Methodology

The data utilized in this study were obtained from the Organization for Economic Co-operation and Development (OECD) database. The OECD has about 36 member countries and several partner countries. Some of these countries include the United States of America (USA), Germany, France, Canada, Spain, Australia, Japan, Italy, etc. Annual data from member and partner countries for 22 years (1994–2016) were employed in this study. Historical data that were analyzed included annual road accidents, freight and passenger transport indicators, changes in the number of new vehicles registered, investments in new road infrastructure and existing road maintenance, and demographic and economic indicators (see

Table 1. Basic statistics of data.

Statistics	Mean	Standard Deviation	Median	Maximum	Minimum	25th Percentile	50th Percentile	75th Percentile
Total Accidents/yr. (1000)	202	449	41	2275	1	9	41	157
Freight (million ton-km)	269,976	776,473	37,199	4,018,805	1864	17,344	37,199	156,986
Passenger (million passenger-km)	543,779	1,162,300	124,355	5,846,312	2143	67,088	124,355	680,863
Change in Number of New Vehicles Registered (%)	2.45	18.97	1.29	152.44	−72.94	−4.82	1.29	7.47
Existing Road Maintenance Investment “ERMI” (EUR/yr.)	3,509,432,464	6,486,214,266	755,494,568	35,926,399,759	16,694,491	204,500,000	755,494,568	3,085,019,372
New Road Infrastructure Investment “NRII” (EUR/yr.)	8,118,209,606	16,109,624,922	1,893,299,904	79,312,039,934	46,000,000	653,000,000	1,893,299,904	6,034,830,090
ERMI Per Capita (EUR/yr./Capita)	84	71	73	387	0	29	73	117
NRII Per Capita (EUR/yr./Capita)	163	126	128	757	11	85	128	205
Population (Million)	44.30	68.62	10.55	318.86	1.32	5.40	10.55	58.81
Proportion of Work Group (%)	67.12	2.13	66.81	73.10	62.41	65.60	66.81	68.07
Population of Work Group (Million)	29.60	45.72	7.03	211.55	0.87	3.63	7.03	38.65
Gross Domestic Product (USD/Capita)	30,005	10,974	28,707	67,051	8608	22,784	28,707	36,519

). Some previous studies that have employed the OECD database include the statistical modeling of road mortality [35,36], comparisons of death from road accidents with international terror [37], modeling traffic fatalities using autoregressive nonlinear time series [38], the risk of severe fatal accidents using Bayesian data analysis [39], etc. In this study, annual road accidents (ARAs) were modeled as a function of all the other listed variables using multiple linear regression (MLR), Poisson regression (PR) analysis, and artificial neural network (ANN). Because the ANN model is more accurate, it was utilized to closely examine how these variables collectively affect annual road accidents. Safe investment limits for new road infrastructure and for the maintenance of existing roads based on the minimum ARAs were estimated for a wide range of demographic, freight transport, passenger transport, and economic systems.

2.1. Description and Basic Statistics of Data

The basic statistics of the variables, including the average, standard deviation, median, maximum, and minimum, are presented in Table 1. A detailed description of the dataset is given in the following paragraphs. Rows with missing data were excluded from the analysis, leaving about 435 rows.

Annual Road Accidents (ARAs) [30]: The ARAs are defined as the number of annual road accidents involving a road motor vehicle (excluding suicide using a road motor vehicle) recorded that might have resulted in death (s), injury, and or loss of property. Road motor vehicles include buses, tramways or streetcars, coaches, trolleys, and road vehicles used to transport goods and passengers.

Freight Transport [31]: This is defined as the total movement of goods using road transport on a given network expressed in million ton-km.

Passenger Transport [32] is defined as the total movement of passengers using road transport on a given network expressed in million passenger-km.

Change in Amount of New Vehicle Registrations (CNRV) [33] is defined as the annual increase or decrease in the amount of newly registered vehicles (private and commercial) expressed as a percentage.

Existing Road Maintenance Investment (ERMI) [34]: This is defined as the total annual monetary spending towards the preservation of the existing road infrastructure expressed in EUR (EUR).

New Road Infrastructure Investment (NRII) [35]: This covers the total annual monetary spending on new road construction to improve the existing road network expressed in EUR (EUR).

Population [36] is defined as all nationals present in, or temporarily absent from, a country, and aliens who have permanently settled in a country, and is expressed in millions.

The proportion of Working Group [37] is the percentage of the population between 15 and 64 years of age who are employed. The Population of Work Group was estimated from the total population and the proportion of work group and is represented in millions. The Proportion of Work Group is the percentage of the working population with respect to the total population. Different countries can have similar proportions of groups but can also have different populations.

Gross Domestic Product (GDP) Per Capita [38]: All OECD countries estimate their econometric data according to the 2008 System of National Accounts and are thus comparable for any single year. GDP per capita was measured in US dollars at current prices and purchasing power parity exchange rates.

2.2. Regression Modeling

Stepwise multiple linear regression (MLR) and Poisson regression (PR) analysis were utilized to develop simple regression models for annual road accidents as a function of other variables. Stepwise regression analysis starts with a single variable and then continuously adds and or removes the potential predicting variables to fit the model until a combination of predicting variables that yield the best model performance was achieved. A value of 0.15 for α-to-include and α-to-exclude was used, where α is the level of statistical significance. MiniTab16^TM statistical software was employed for this task. General model equations for the MLR and the PR are given by Equations (1) and (2), respectively. MLR is the basic statistical modeling tool, while the PR and similar discrete count models were considered more appropriate for modeling accident frequency [40].

$$Y={{\displaystyle \sum }}_{i=1}^N{\beta }_i{x}_i$$

(1)

where $Y$ represents the dependent variable ARA, $x_i$ is the $i^{th}$ predicting variable, ${\beta }_i$ is the coefficients of the $i^{th}$ variable, ${\beta }_1=1$, and $N$ is the total number of terms in the final model.

$$Y=e^{\left({\displaystyle {\sum }_{i=1}^N}{\beta }_i{x}_i\right)}$$

(2)

where $e$ is natural log constant (2.718281), and all other terms are described as in Equation (1).

2.3. Artificial Neural Network (ANN) Modeling

A feed-forward 2-layer ANN with 7 neurons was developed in MatLab^TM (R2017a) to train the annual road accident model. Based on the balanced performance between the training and testing results, the number of neurons was optimized. This was achieved by varying the number of neurons, randomly re-splitting the training/testing data, and retraining with random initial conditions. A schematic of the final ANN structure is shown in Figure 1. The terms W and b represent the weight and bias matrices of the model, respectively. Each of the variables from the input matrix $X$ is connected to each neuron through the weight matrix $IW$. In this case, $a^1$ is a 7-element column vector formed by “$F^1$” from the weighted sum of the input variables $x_i$ and bias $b_i$ of the neurons’ outputs. The neurons’ outputs serve as inputs, which are transformed to fall between [−1, 1] using the hyperbolic tangent sigmoid equivalent transfer function “F¹” given by Equation (3). This quickens the learning process. These normalized outputs from “F¹” serve as input to the output layer. “$F^2$” is a purelin transfer function that again normalizes the weighted sum and biases of the outputs from the hidden layer between [−1, 1] to yield $a^2$. Equation (4) is a post-processing function that reverses $a^2$ to yield a scalar $y_p$ between [$y_{max}$, $y_{min}$] that is matched and compared with the actual target (ARA) $y_a$. The training cycles follow this sequence while the weight and bias matrices are optimized for better model performance. Bayesian-regularized (BR) Levenberg–Marquardt optimization was selected as the training algorithm [41,42]. The BR-ANNs were found to be robust and difficult to over-train or over-fit, and the validation process is unnecessary [43]. Data partitioning of 75/25 for the training and testing data was adopted, respectively. The model performance was measured based on the mean square error (MSE) and coefficient of correlation between the actual and predicted ARA. Equation (5) represents the root mean square error. The correlation between anticipated/predicted and actual/observed ARA was estimated using Equation (6).

$$F^1\left(r\right)=2/\left(1+e^{-2r}\right)-1$$

(3)

where $F^1$ is a tangent sigmoid equivalent transfer function; $r$ is the independent variable, while $e$ denotes the natural log constant (2.718281).

$$y_p\left(u\right)=\frac{\left(y_{max}-y_{min}\right)\left(u-u_{min}\right)}{\left(u_{max}-u_{min}\right)}+y_{min}$$

(4)

where $u$ denotes a set of finite real numbers between [−1, 1], while the minimum and maximum values of the original target dataset is represented by $y_{min}$ and $y_{max}$, respectively.

$$RMSE=\sqrt{MSE}=\sqrt{\frac{{{\displaystyle \sum }}_i^{n_t}{\left(y_a{}_i-y_p\left(u_i\right)\right)}^2}{n_t-n_p}}$$

(5)

$$R^2=1-\frac{{{\displaystyle \sum }}_i^{n_t}{\left(y_a{}_i-y_p\left(u_i\right)\right)}^2}{{{\displaystyle \sum }}_i^{n_t}{\left(y_a{}_i-\overline{y_a}\right)}^2}$$

(6)

where $RMSE$: root mean square error, $y_a{}_i$: actual observed ARAs, $y_p\left(u_i\right)$: modeled or predicted ARAs, $n_t$: total number of observed/actual ARAs, $n_p$: number of model parameters, $\overline{y_a}$: mean of observed/actual ARAs.

2.4. Relative Importance Analysis and Ranking

Using the connection weight matrix method, the relative significance of the predictive variables for the ANN model was calculated [44]. The concept of using the weight connection to determine the relevance of predicting variables in an ANN model was initially proposed by Garson [45]. The overall absolute weight contribution fraction of the variables with regard to each neuron was calculated using the Garson algorithm to determine the relative importance of the predicting variable, as indicated in Equation (7). Olden and colleagues later claimed that the net sum of the weight contribution for each predicting variable across the neurons yielded more accurate rankings [44], as given in Equation (8). Both approaches were utilized in this study to rank the relative relevance of the predicting variables from the ANN model.

$$Garson Relative Importance\left(x_i\right)=\frac{{{\displaystyle \sum }}_{j=1}^n\left|W_{ij}^h\ast W_j^o\right| }{{{\displaystyle \sum }}_{i=1}^m\left\{{{\displaystyle \sum }}_{j=1}^n\left|W_{ij}^h\ast W_j^o\right| \right\}}$$

(7)

$$Olden Relative Importance\left(x_i\right)=\frac{\left|{{\displaystyle \sum }}_{j=1}^n{W}_{ij}^h\ast W_j^o\right|}{{{\displaystyle \sum }}_{i=1}^m\left\{\left|{{\displaystyle \sum }}_{j=1}^N{W}_{ij}^h\ast W_j^o\right|\right\}}$$

(8)

where $x_i$ is the $i^{th}$ predicting variable, $W_{ij}^h$ is the weight matrix of the hidden layer of the ANN model, $W_j^o$ is the weight matrix of the output layer, $n$ is the maximum number of neurons in the hidden layer, and $m$ is the number of predicting variables.

The relative importance of the variables from the regression models was also estimated using relative weight analysis [46,47]. The traditional use of normalized coefficients was found to yield misleading results due to some level of correlation between the predicting variables [47]. The relative weight analysis eliminates the effect of variable inter-correlation by transforming the predictors into sets of orthogonal variables. The end result is a decomposed contribution of each variable to the R² of the model in explaining the dependent variable. The analysis was run using R code web-based software with all variables at the 5% significant level and with 10,000 bootstrapping iterations [48].

2.5. Sensitivity Analysis Using ANN Model

The profile method (Lek’s profile) was employed for the sensitivity analysis of the ANN model (Gevrey et al., 2003). Lek and colleagues introduced this technique to show the general partial variation trends in the dependent variable of ANN models with each predicting variable (Lek et al., 1996). The dependent variable’s profile with respect to a certain predicting variable is derived by changing the predicting variable only partially across its range while holding the minimum, mode/median, quartiles, and maximum values of all other variables constant. The various results are averaged to obtain a single curve. The disadvantage of the method is the high computational tasks, especially when dealing with numerous predicting variables. However, comparative studies using other similar ANN sensitivity analysis methods found the profile method to yield useful and reliable insight [49]. Outputs from 12 places along the range of the predicting variable were advised to reduce the amount of calculation needed (Lek et al., 1996). In this study, the other variables were held constant at their quartiles (25th, 50th, and 75th percentiles), mean, and median, while the ARAs were estimated at 50 equal intervals along the range of each predictive variable. These curves were averaged to obtain the overall profile for a predictive variable. Because some outcomes from these extreme constants produce negative ARA values that could create mistakes in the trends seen, the minimum and maximum are removed.

2.6. Annual NRII and ERMI Limits

Among all of the independent variables of the ARA model, annual NRII and EMRI are the only variables that can be easily controlled to influence the ARAs. Demographic and economic variables depend on several other factors and require a long period of time to control. The freight and passenger movement of a given country depend on the demography and economy of that country. Due to these reasons (more reasons will follow in the Results section), the annual per capita safe limits of the NRII and EMRI were considered important parameters influencing ARAs in this study. As a result, the per capita NRII and EMRI corresponding to the minimum ARAs for several sets of all other variables were estimated. This was carried out by estimating the various ARAs within the space [Minimum NRII, Maximum NRII] and [Minimum ERMI, Maximum ERMI]. The results for each run were exported to an Excel sheet, where simple algorithm searches were conducted for the minimum ARAs and the corresponding per capita NRII and ERMI. Figure 2 below shows a simplified flow chart for the safe annual per capita NRII and EMRI limits estimation process.

3. Results and Discussion

3.1. MLR, PR and ANN ARA Models

This section discusses the fitting results of the ARA results obtained using ANN, PR, and MLR. It also highlights the need for sophisticated model fitting techniques such as ANN (for this study) instead of conventional methods such as MLR and PR.

The parameters of the ARA model from the stepwise regression analysis are summarized in

Table 2. ARA regression model parameters.

		MLR		PR
Terms	Unit	Coefficient	p-value	Coefficient	p-value
Constant		−1,625,191	0.0000	1.77931	0.0000
Freight	MTkm	3.56900 × 10−01	0.0000	−1.28461 × 10−06	0.0000
Passenger	Million passenger km	−1.07521×10−01	0.0000	1.62813 × 10−07	0.0000
CNRV	(%)	--	--	−2.14206 × 10−03	0.0000
Total ERMI	EUR/yr.	--	--	−1.50190× 10−10	0.0000
Total NRII	EUR/yr.	1.12648 × 10−05	0.0000	4.42200 × 10−11	0.0000
ERMI Per Capita	EUR/yr./Capital	5.40957 × 1002	0.0000	1.08639 × 10−02	0.0000
NRII Per Capita	EUR/yr./Capital	−2.65564 × 1002	0.0001	−3.40675 × 10−03	0.0000
Population	Million	4.62782 × 1004	0.0000	1.07097 × 10−01	0.0000
Proportion of Work Group	(%)	2.39945 × 1004	0.0000	1.31254 × 10−01	0.0000
Population of Work Group	Million	−6.63550 × 1004	0.0000	−1.12287× 10−01	0.0000
GDP	(USD/capita)	--	--	−2.45351 ×10−05	0.0000

. All of the included parameters showed significant influence on the ARA (p-value < 0.05). Some of the variables were excluded from the MLR models, which means that based on this model, these variables did not have a significant effect on the independent variable (ARA), with p-values greater than 0.15. The excluded variables include GDP/capita, CNVR, and total ERMI/yr. On the other hand, all of the variables were found to significantly affect the ARA according to the PR model. The PR model is considered more appropriate for discrete count variable statistics such as ARA [40]. It is worth mentioning that the inclusion of similar variables was obtained with either standardized or non-standardized data for both models. A comparison of the coefficients between the MLR and PR models reveals similar results, as previously reported by some studies in the literature. For example, increasing NRII/Capita will result in lower ARAs, while increasing EMRI/Capita will lead to more ARAs according to both the MLR and PR models. Based on both model structures and intercepts, any variable with a negative sign has a negative effect on the independent variable (i.e., ARA), and vice versa. This characteristic of attributing only a single effect to a variable with a single regression coefficient limits their performance when dealing with highly non-linear problems. The total NRII, ERMI/Capita, NRII/Capita, and Proportion and Population of Work Group values showed consistently similar effects on the ARAs in both the MLR and PR models. However, the freight and passenger variables alternated signs between the two models. This can be attributed to the different nature of the models. Whatever the sign or observed effect of the variables within a model, the degree of acceptability of the results depends on the overall accuracy of that model. The accuracies of the MLR and PR models are discussed in the next paragraph.

Plots of the actual versus predicted ARAs for the MLR, PR, and ANN models are presented in Figure 3 and Figure 4, respectively. Although the MLR and PR models showed good coefficients of correlation (R²) between the actual and the predicted ARA, these models are inconsistent and low in accuracy. The inconsistency can be seen from their deviation from the Y = X line. On the other hand, the ANN model showed a higher R² value as well as a high degree of accuracy. The difference in the level of accuracy between these models can be clearly seen from the comparison plots of their error histograms shown in Figure 5. The MLR and PR models only have a few accurate predictions, with most of the remaining estimated ARAs being far different from the actual values. The high number of observations with deviation errors of 25 or more attest to this fact. However, the ANN model showed that most of the ARA values were within a small margin of error with the actual values. This was deduced from the higher number of observations with deviation errors closer to zero for the ANN model.

Table 3. ANN model vs. MLR model performance.

	MLR	PR	ANN
	All	All	Training	Testing	All
RMSE	96.07	127.53	7.34	9.42	8.64
R2	0.9554	0.9233	0.9999	0.9991	0.9996

shows the summary of the various model performances in terms of root mean square error (RMSE) and R². Comparing the MLR, PR, and ANN ARA models in terms of RMSE, the ANN model is at least 11 times more accurate than the other models. In accordance with these observations, only the ANN model was considered reliable for the sensitivity analysis of the ARA and the model application example.

Table 4. Trained ANN model weight and bias matrices.

	$\mathbf{Hidden} \mathbf{Layer} \mathbf{Weight} \mathbf{Matrix} \left({\mathit{W}}_{\mathit{i}\mathit{j}}\right)$
Inputs ($x_i)$/Neurons ($n$)	$n_1^1$	$n_2^1$	$n_3^1$	$n_4^1$	$n_5^1$	$n_6^1$	$n_7^1$
Freight	−0.01957	−0.03012	−0.87492	−0.39241	−0.74901	−1.96745	−0.59826
Passenger	0.50547	1.344694	0.85682	−1.09923	0.954988	0.425543	−0.31455
CNRV	1.266489	0.042526	0.137132	−0.58993	0.032595	0.035055	0.112455
Total ERMI	−1.98258	0.067577	−1.65205	2.627753	−0.38092	−0.0193	−0.99349
Total NRII	0.270451	0.579433	0.359455	1.800905	1.374441	1.101101	0.150742
ERMI Per Capita	1.404668	−1.05879	−0.25969	−2.63473	−0.83877	−1.47092	−0.68379
NRII Per Capita	−0.05638	0.63253	0.185425	−2.40709	0.636397	1.087314	0.397435
Population	−0.58079	−0.80266	0.564306	1.295473	−0.3578	0.179384	−0.0249
Proportion of Work Group	0.446989	0.523725	−0.33594	−0.08935	0.866879	1.350303	−0.04303
Population of Work Group	−1.42903	0.59404	1.841841	1.313592	1.691216	2.568981	1.323976
GDP	−2.28676	0.178019	−0.53137	2.542531	0.362622	0.493147	−0.27178
Hidden Layer Bias ‘$b_i$’	0.763762	1.411427	0.530619	−1.21196	1.324628	0.295899	−0.81256
	Output Layer Weight Matrix
Out Put	2.942207	1.177333	1.196882	2.90994	−1.60138	1.148312	−1.75404
Output Layer Bias ‘b’	−0.889019679

presents the weight and bias matrices of the ANN model’s hidden and output layers. These results were used to estimate the relative importance of the predicting variables of the ANN model.

3.2. Relative Influence of Predicting Variables

The relative importance and ranking of the predicting variables of the ANN model were estimated according to Equations (7) and (8) using the data in Table 4. Similarly, the relative influence for the regression model predictors was also determined. These relative importance results for the predicting variables in the ARA model are presented in

Table 5. Relative influence/importance ranking of variables.

$\mathbf{Input}/\mathbf{Variable} \left({\mathit{x}}_{\mathit{i}}\right)$	Regression (%)	ANN-Garson (%)	ANN-Olden (%)	Average (%)	Overall Rank/11
Freight	18.21	7.74	7.88	11.28	4
Passenger	16.48	9.67	1.40	9.18	8
CNRV	0.08	3.06	6.93	3.36	11
Total ERMI	14.64	11.84	7.72	11.40	3
Total NRII	14.63	8.29	20.45	14.46	1
ERMI Per Capita	0.42	12.68	14.57	9.22	7
NRII Per Capita	1.14	7.73	22.94	10.60	5
Population	16.87	5.86	8.99	10.57	6
Proportion of Work Group	0.08	6.04	5.18	3.77	10
Population of Work Group	16.77	18.23	1.52	12.17	2
GDP	0.68	8.86	2.43	3.99	9

. The results obtained using the relative weight analysis for the regression showed that CNVR, ERMI per Capita, and Proportion of Work Group contributed to explaining R² the least in the ARA. However, these variables were found to have a certain level of relevance according to the weight contribution analysis of the ANN model. This can be associated with the different structure, accuracy, theory, method of relative importance estimation, etc., of the models. This also highlights how almost all the predicting variables included in the ANN model played some role in the model performance. Although the relative importance does not represent the level of statistical significance, it certainly can reveal high-priority variables. The relative influence results of the variables were utilized to rank them, as presented in the last column of Table 5. The #1 rank represents the variable with the highest relevance, while the rank of #11 represents the variable with the least relative influence. The overall ranking was obtained based on averages of relative importance of each variable across the different methods. This overall ranking was necessary because the various models and relative importance estimation methods did not yield similar results. Variation in the results was anticipated, but since Garson’s method is still widely utilized and Olden’s method was supported with compelling evidence, both results were included. The relative weight analysis of the regression has strong fundamental theoretical bases but is still affected by all the limitations of multiple regression [46]. As such, the average or the overall ranking is considered to be much closer to the fact. Overall, the variables in decreasing order of influence on ARA are total NRII (#1), population of work group (#2), total ERMI (#3), freight movement (#4), NRII/capita (#5), population (#6), ERMI/capita (#7), passengers (#8), GDP (#9), proportion of work group (#10), and CNRV (#11). The set of road infrastructure investment variables can be seen to be in the top levels with regard to the influence on ARA. Being the lowest in the ranking does imply insignificance to the dependent variable. It only means lower relative linear, nonlinear, or both influence the dependent variable with respect to the other variables.

3.3. Sensitivity Analysis of ANN Model

The average variation trend of ARAs with the different predicting variables is presented in Figure 6. The figure is scaled by normalizing the ranges of each input variable ($X_i)$ between [0, 100] and their corresponding ARA outputs (Y) between [0, 100]. Each curve is an average of various curves, as previously described in the methodology. Starting from Figure 6a, it can be observed that the ARA exhibits a positive exponential growth relationship as Freight Movement increases. This trend is in line with previous findings that the number of accidents and fatalities increase as the number of heavy goods vehicles and trucks per capita increase [50,51]. It should be noted that none of these studies reported the specific nature of the ARA–Freight relationship. In addition, it can be deduced that eliminating the Freight Movement from roads (even if possible) can only enable the lowering of the ARA to a certain cap (not zero). On the other hand, the ARA increases linearly as passenger movement increases (positive linear relationship), but there is a non-zero intercept on the passenger movement axis. This observation is also in agreement with reports on the effect of passenger exposure expressed as vehicle kilometer travel [52] and gasoline sales [53] on the accident rate. A less significant logarithmic growth relationship between ARA and CNRV was also observed. These first three observations (for freight, passenger, and CNRV) are very clear and can be easily understood. The same cannot be said regarding the next four variables related to new road infrastructural investments and existing road maintenance investments (total NRII, total ERMI, NRII/capita, and EMRI/capita). Generally, the ARA decreases to a certain extent with the increase in these predicting variables; then, the ARAs begin to rise as the variables increase further. This means that monetary investment in new road infrastructure or existing road maintenance could result in either a decrease or increase in ARAs, depending on the amount invested. It also explains some of the anomalies on the effects of road investments on ARA variables, as reported by previous studies such as [31,52,53].

According to the trends seen here, an average nonlinear concave-up relationship exists, between ARA and new road infrastructure and existing road maintenance investment. The drop and rise in the ARA due to increased maintenance investment is not as pronounced as in the case of new road capital investments. The NRII curves are basic cases of constant demand (fixed freight and passenger movement) and increasing supply (additional road infrastructure). Moderate to low NRII levels will improve the system efficiency by decongesting the overall traffic. This means lesser volume, which leads to a decrease in the level of crash frequency [54]. However, high NRII will not only eliminate congestion, but will also result in a significant increase in the system-wide average speed. The accident rate was found to increase with a higher mean speed for all types of roads [55]. As for the ERMI curves: low to moderate maintenance investments are low-cost, but significant safety improvements such as signage fixing, road marking tracing, porthole patching, drainage clearing, lighting improvements, and even black spot rectification are costly. However, major road rehabilitation projects such as resurfacing and overlays require high capital expenditure. Unlike low-cost road maintenance projects such as patching, drainage clearing, etc., pavement overlays and resurfacing restore the ratings of the pavement condition and parameters such as the roughness index by 100%. Previous studies have shown that roads with better roughness indices (or semi-new roads) witness higher crash frequencies [56,57,58]. Thus, although major road rehabilitation results in high ride comfort and user satisfaction, it also results in the possibility of higher crash tendencies.

Both Population and Population of Work Group showed a positive S-curve relationship with the ARA. On average, the Proportion of Work Group showed a general non-linearly increasing association with ARAs. The decline in ARAs due to changes in the Proportion of Work Group prior to the increasing trend could not be fully explained. The ARA decreases linearly with an increase in the per capita GDP. The decreasing effect of GDP per capita on the accident rate was also reported in [59].

3.4. Example and Application of the ARA ANN Model

The variation in the ARA with NRII/capita along with ERMI/capita is illustrated for a medium population of 10 million. An annual freight and passenger movement of 50,000 million ton-km and 200,000 million passenger-km was assumed, respectively. A Working Population of 70 percent, no change in the number of new vehicles registered annually, and a per capita GDP of USD 40,000 was assumed. These conditions are rounded values typical of a medium population similar to that of Belgium. As of 2014, the annual number of road accidents in Belgium was around 42,000, with annual ERMI/capita of EUR 18 and NRII/capita of approximately EUR 37.

Figure 7 shows the ARA variation with NRII per capita for different per capita ERMI. Each curve represents constant ERMI allocation and varying NRII values. There is a limit to the extent by which the NRII reduces or increases the ARA for any given EMRI/capita. The ARA variation trends are similar to those previously observed from the sensitivity analysis plots for both NRII and ERMI.

3.5. NRII and ERMI per Capita Limits Estimation

The ANN model represents an accurate possible outcome of road investment interventions on ARA, for a given economy, demography, and transport system. It could serve as a useful tool for exploring effective and potential road investment limits that will result in the minimum accident rate. The best NRII/capita and ERMI/capita are the combination of NRII/capita and ERMI/capita that correspond to the lowest ARAs for a given set of demographic, economic, passenger, and freight conditions. Sample NRII/capita and ERMI/capita values corresponding to the minimum ARAs for the population of 10 million illustrated in the previous section are estimated and shown in

Table 6. Sample-minimized ARA and the corresponding NRII/capita and EMRII/capita for population of 10 million.

Freight (Million ton-km)	Passenger (Million Passenger-km)	CNVR (%)	Proportion of Working Group (%)	GDP (US $/Capita)	Minimum ARA (1000)	Opt. ERMI (EUR/Capita)	Opt. NRII (EUR/Capita)	NRII/ERMI	ERMI (%)	NRII (%)
50,000	200,000	0	60	20,000	14.922	20	2	0.1000	91%	9%
50,000	200,000	0	60	30,000	9.454	32	17	0.5313	65%	35%
50,000	200,000	0	60	40,000	4.873	47	62	1.3191	43%	57%
50,000	200,000	0	70	20,000	36.353	20	132	6.6000	13%	87%
50,000	200,000	0	70	30,000	15.937	17	142	8.3529	11%	89%
50,000	200,000	0	70	40,000	6.629	20	162	8.1000	11%	89%
50,000	200,000	20	60	20,000	15.098	20	2	0.1000	91%	9%
50,000	200,000	20	60	30,000	9.891	32	27	0.8438	54%	46%
50,000	200,000	20	60	40,000	5.164	41	72	1.7561	36%	64%
50,000	200,000	20	70	20,000	38.412	17	132	7.7647	11%	89%
50,000	200,000	20	70	30,000	18.827	17	142	8.3529	11%	89%
50,000	200,000	20	70	40,000	8.513	17	167	9.8235	9%	91%

. The per capita NRII and ERMI ratio and their relative proportions were also estimated. The NRII/ERMI ratio and their relative proportion are better when the budget is limited to below the best estimates. This is because the ratio or relative proportions are also better than random values. Safer NRII/capita and ERMI/capita limit results for populations of 5, 15, and 30 million with different demographic, economic, freight, and passenger conditions were also estimated. These results are presented in the Appendix A in

Table A1. Optimized annual ERMI and NRII per capita for population of 5 million.

Freight (Million ton-km)	Passenger (Million Passenger-km)	CNVR (%)	Proportion of Working Group (%)	GDP (US USD/Capita)	Opt. ERMI (EUR/Capita)	Opt. NRII (EUR/Capita)	NRII/ERMI	ERMI (%)	NRII (%)
20,000	200,000	0	60	10,000	11	2	0.1818	85%	15%
20,000	200,000	0	60	20,000	17	2	0.1176	89%	11%
20,000	200,000	0	60	30,000	32	27	0.8438	54%	46%
20,000	200,000	20	60	10,000	11	2	0.1818	85%	15%
20,000	200,000	20	60	20,000	17	2	0.1176	89%	11%
20,000	200,000	20	60	30,000	29	37	1.2759	44%	56%
20,000	200,000	0	70	10,000	125	2	0.0160	98%	2%
20,000	200,000	0	70	20,000	8	112	14.0000	7%	93%
20,000	200,000	0	70	30,000	8	132	16.5000	6%	94%
20,000	200,000	20	70	10,000	131	2	0.0153	98%	2%
20,000	200,000	20	70	20,000	5	117	23.4000	4%	96%
20,000	200,000	20	70	30,000	5	137	27.4000	4%	96%
50,000	200,000	0	60	10,000	11	2	0.1818	85%	15%
50,000	200,000	0	60	20,000	17	2	0.1176	89%	11%
50,000	200,000	0	60	30,000	32	29	0.9063	52%	48%
50,000	200,000	20	60	10,000	11	2	0.1818	85%	15%
50,000	200,000	20	60	20,000	17	2	0.1176	89%	11%
50,000	200,000	20	60	30,000	29	37	1.2759	44%	56%
50,000	200,000	0	70	10,000	131	2	0.0153	98%	2%
50,000	200,000	0	70	20,000	2	127	63.5000	2%	98%
50,000	200,000	0	70	30,000	2	147	73.5000	1%	99%
50,000	200,000	20	70	10,000	137	2	0.0146	99%	1%
50,000	200,000	20	70	20,000	2	132	66.0000	1%	99%
50,000	200,000	20	70	30,000	2	152	76.0000	1%	99%
50,000	100,000	0	70	10,000	113	2	0.0177	98%	2%
50,000	100,000	0	70	20,000	5	117	23.4000	4%	96%
50,000	100,000	0	70	30,000	5	142	28.4000	3%	97%
50,000	100,000	20	70	10,000	119	2	0.0168	98%	2%
50,000	100,000	20	70	20,000	2	122	61.0000	2%	98%
50,000	100,000	20	70	30,000	2	147	73.5000	1%	99%

,

Table A2. Optimized annual ERMI and NRII per capita for population of 15 million.

Freight (Million ton-km)	Passenger (Million Passenger-km)	CNVR (%)	Proportion of Working Group (%)	GDP (US USD/Capita)	Opt. ERMI (EUR/Capita)	Opt. NRII (EUR/Capita)	NRII/ERMI	ERMI (%)	NRII (%)
20,000	200,000	0	60	20,000	23	2	0.0870	92%	8%
20,000	200,000	0	60	30,000	35	7	0.2000	83%	17%
20,000	200,000	0	60	40,000	50	57	1.1400	47%	53%
20,000	200,000	20	60	20,000	20	2	0.1000	91%	9%
20,000	200,000	20	60	30,000	32	17	0.5313	65%	35%
20,000	200,000	20	60	40,000	44	67	1.5227	40%	60%
20,000	200,000	0	70	20,000	44	112	2.5455	28%	72%
20,000	200,000	0	70	30,000	38	122	3.2105	24%	76%
20,000	200,000	0	70	40,000	41	142	3.4634	22%	78%
20,000	200,000	20	70	20,000	44	112	2.5455	28%	72%
20,000	200,000	20	70	30,000	35	127	3.6286	22%	78%
20,000	200,000	20	70	40,000	41	147	3.5854	22%	78%
50,000	200,000	0	60	20,000	23	2	0.0870	92%	8%
50,000	200,000	0	60	30,000	35	7	0.2000	83%	17%
50,000	200,000	0	60	40,000	50	52	1.0400	49%	51%
50,000	200,000	20	60	20,000	23	2	0.0870	92%	8%
50,000	200,000	20	60	30,000	35	17	0.4857	67%	33%
50,000	200,000	20	60	40,000	44	67	1.5227	40%	60%
50,000	200,000	0	70	20,000	41	142	3.4634	22%	78%
50,000	200,000	0	70	30,000	32	137	4.2813	19%	81%
50,000	200,000	0	70	40,000	32	152	4.7500	17%	83%
50,000	200,000	20	70	20,000	41	142	3.4634	22%	78%
50,000	200,000	20	70	30,000	32	137	4.2813	19%	81%
50,000	200,000	20	70	40,000	32	157	4.9063	17%	83%
50,000	300,000	0	60	20,000	23	2	0.0870	92%	8%
50,000	300,000	0	60	30,000	35	2	0.0571	95%	5%
50,000	300,000	0	60	40,000	53	47	0.8868	53%	47%
50,000	300,000	20	60	20,000	23	2	0.0870	92%	8%
50,000	300,000	20	60	30,000	35	12	0.3429	74%	26%
50,000	300,000	20	60	40,000	47	62	1.3191	43%	57%
50,000	300,000	0	70	20,000	41 *	162 *	3.9512 *	20% *	80% *
50,000	300,000	0	70	30,000	29 *	142 *	4.8966 *	17% *	83% *
50,000	300,000	0	70	40,000	32 *	157 *	4.9063 *	17% *	83% *
20,000	300,000	0	60	20,000	23	2	0.0870	92%	8%
20,000	300,000	0	60	30,000	35	2	0.0571	95%	5%
20,000	300,000	0	60	40,000	50	47	0.9400	52%	48%
20,000	300,000	20	60	20,000	23	2	0.0870	92%	8%
20,000	300,000	20	60	30,000	35	12	0.3429	74%	26%
20,000	300,000	20	60	40,000	47	62	1.3191	43%	57%
20,000	300,000	0	70	20,000	35	122	3.4857	22%	78%
20,000	300,000	0	70	30,000	35	127	3.6286	22%	78%
20,000	300,000	0	70	40,000	38	147	3.8684	21%	79%
20,000	300,000	20	70	20,000	38	127	3.3421	23%	77%
20,000	300,000	20	70	30,000	32	132	4.1250	20%	80%
20,000	300,000	20	70	40,000	35	152	4.3429	19%	81%

* Similar results for 20% increase in new vehicles registered.

and

Table A3. Optimized annual ERMI and NRII for population of 30 million.

Freight (Million ton-km)	Passenger (Million Passenger-km)	CNVR (%)	Proportion of Working Group (%)	GDP (US USD/Capita)	Opt. ERMI (EUR/Capita)	Opt. NRII (EUR/Capita)	NRII/ERMI	ERMI (%)	NRII (%)
20,000	200,000	0	60	20,000	23	2	0.0870	92%	8%
20,000	200,000	0	60	30,000	38	2	0.0526	95%	5%
20,000	200,000	0	60	40,000	59	47	0.7966	56%	44%
20,000	200,000	20	60	20,000	23	2	0.0870	92%	8%
20,000	200,000	20	60	30,000	38	2	0.0526	95%	5%
20,000	200,000	20	60	40,000	56	62	1.1071	47%	53%
20,000	200,000	0	70	20,000	65 *	2 *	0.0308 *	97% *	3% *
20,000	200,000	0	70	30,000	65 *	72 *	1.1077 *	47% *	53% *
20,000	200,000	0	70	40,000	74 *	112 *	1.5135 *	40% *	60% *
50,000	200,000	0	60	20,000	23	2	0.0870	92%	8%
50,000	200,000	0	60	30,000	38	2	0.0526	95%	5%
50,000	200,000	0	60	40,000	59	47	0.7966	56%	44%
50,000	200,000	20	60	20,000	23	2	0.0870	92%	8%
50,000	200,000	20	60	30,000	38	2	0.0526	95%	5%
50,000	200,000	20	60	40,000	56	62	1.1071	47%	53%
50,000	200,000	0	70	20,000	65	2	0.0308	97%	3%
50,000	200,000	0	70	30,000	62	97	1.5645	39%	61%
50,000	200,000	0	70	40,000	68	122	1.7941	36%	64%
50,000	200,000	20	70	20,000	56	52	0.9286	52%	48%
50,000	200,000	20	70	30,000	62	102	1.6452	38%	62%
50,000	200,000	20	70	40,000	65	127	1.9538	34%	66%
50,000	300,000	0	60	20,000	23	2	0.0870	92%	8%
50,000	300,000	0	60	30,000	41	2	0.0488	95%	5%
50,000	300,000	0	60	40,000	62	37	0.5968	63%	37%
50,000	300,000	20	60	20,000	23	2	0.0870	92%	8%
50,000	300,000	20	60	30,000	41	2	0.0488	95%	5%
50,000	300,000	20	60	40,000	59	52	0.8814	53%	47%
50,000	300,000	0	70	20,000	74	132	1.7838	36%	64%
50,000	300,000	0	70	30,000	62	107	1.7258	37%	63%
50,000	300,000	0	70	40,000	65	122	1.8769	35%	65%
50,000	300,000	20	70	20,000	65	112	1.7231	37%	63%
50,000	300,000	20	70	30,000	62	112	1.8065	36%	64%
50,000	300,000	20	70	40,000	65	132	2.0308	33%	67%
20,000	300,000	0	60	20,000	23 *	2 *	0.0870 *	92% *	8% *
20,000	300,000	0	60	30,000	41 *	2 *	0.0488 *	95% *	5% *
20,000	300,000	0	60	40,000	62	42	0.6774	60%	40%
20,000	300,000	20	60	40,000	56	57	1.0179	50%	50%
20,000	300,000	0	70	20,000	62 *	2 *	0.0323 *	97% *	3% *
20,000	300,000	0	70	30,000	68 *	92 *	1.3529 *	43% *	58% *
20,000	300,000	0	70	40,000	71 *	112 *	1.5775 *	39% *	61% *

* Similar results for a 20% increase in new vehicles registered.

, respectively. It can be seen that the safe NRII/capita limit is only greater than the safe ERMI/capita limit for cases in which there is a higher working population (approximately 70%), GDP per capita (USD 30,000 and above), or a combination of a higher CNVR and GDP. Otherwise, the ERMI/capita should be prioritized over the NRII/capita for a lower ARA. Interpolation can be used to estimate in between results with some reasonable degree of accuracy.

4. Conclusions and Recommendations

The artificial neural network (ANN), MLR, and PR techniques were employed to model ARA based on 22 years of data from the OECD and its partner countries. These data include annual passenger and freight movement, demographic and economic indicators, and road network maintenance and new development investments. The ANN model was found to be more accurate and general in predicting the ARAs. The ANN model was used to illustrate the variation in ARAs with road infrastructure investments and the other variables mentioned. Safe limits per capita NRII and ERMI and their relative proportions for diverse combinations of demography, economy, freight level, and passenger movement were estimated and documented. Some other important observations are highlighted below:

• The ANN is, by far, more reliable than the MLR and PR in modeling the nonlinear relationship of ARAs with road investment variables, demographic indicators, GDP, etc.
• The results of the MLR model showed that all of the independent variables have a significant effect on the ARA, except per capita GDP and changes in the amount of newly registered vehicles.
• The PR model indicated that all predicting variables have a significant effect on the ARA.
• On average, the variables listed in decreasing order of influence on ARA are total NRII (#1), population of work group (#2), total ERMI (#3), freight movement (#4), NRII/capita (#5), population (#6), ERMI/capita (#7), passenger (#8), GDP (#9), proportion of work group (#10), and CNRV (#11).
• New road expansion and existing road maintenance investments were found to exhibit a nonlinear concave-up relationship with ARAs. This was considered to be an explanation for non-uniform conclusions on the influence of these parameters on road accident frequency, as reported by some past literature.
• A certain amount of NRII and ERMI that corresponds to minimum ARAs for a given demography, economy, freight level, and passenger movement exists.
• ARAs showed positive exponential growth with an increase in freight movement.
• ARAs exhibit a positive linear relationship as passenger movement increases but there is a non-zero intercept on the passenger movement axis.
• A positive logarithmic growth relationship between ARAs and CNRV was also observed.
• Population and population of work group showed a positive S-curve relationship with the ARAs. The proportion of the population work group showed a general non-linearly increasing association with ARAs.
• The ARAs decrease linearly as the per capita GDP increases.

Based on these observations, more and proper attention should be given to road safety as the target criteria for the capital funding of road network maintenance and infrastructural development. As a suggestion for further studies, different models for ARA prediction that will enable the estimations of safe road investment limits for specific road facilities could provide more effective alternatives for road infrastructure investments.