Forecasting Methods for Road Accidents in the Case of Bucharest City ^†

Abstract

This paper aims to emphasize the necessity for policy reform, improvements in vehicle design and enhanced public awareness through the projection of future trends in road accidents, injuries and fatalities. The statistical methods that are used in this study are the empirical laws of Smeed and Andreassen. The main gap that the researchers identify is the lack of a standardized methodology with the help of which the appropriate forecasting method can be chosen in the area of traffic accidents. In the present study, the authors propose such a methodology that can be generalized, being suitable for use for any urban agglomeration at the micro and macro level.

1. Introduction

The issue of road safety has become a major global priority, as evidenced by numerous studies investigating mortality trends on both a global and national level. These studies, alongside statistics compiled by organizations like the World Health Organization, the Romanian Police and Eurostat, consistently highlight road accidents as a significant contributor to the loss of young lives.

According to the World Health Organization [1],

• Each year, road traffic accidents result in the deaths of approximately 1.19 million people;
• For individuals aged 5–29 years, road traffic injuries are the primary cause of death;
• Although low- and middle-income countries account for about 60% of the world’s vehicles, they experience 92% of the global road fatalities;
• Vulnerable road users, such as pedestrians, cyclists and motorcyclists, represent more than half of all road traffic deaths;
• The economic impact of road traffic crashes amounts to 3% of the gross domestic product in most countries;
• The United Nations General Assembly aims to reduce the global number of deaths and injuries from road traffic accidents by 50% by 2030 (A/RES/74/299).

Investigating the factors contributing to accidents, their causes and their impacts is crucial in identifying effective strategies to enhance traffic safety. The EU is confronted with a range of interlinked demographic, public health and environmental challenges. These include the shifting climate, the persistent stagnation in reducing road fatalities, escalating urbanization rates, deteriorating air quality, upward trends in obesity rates and the demographic shift towards an aging population.

This paper aims to emphasize the necessity for policy reform, improvements in vehicle design and enhanced public awareness through the projection of future trends in road accidents, injuries and fatalities.

The statistical methods that are used in this study are the empirical laws of Smeed and Andreassen [2].

The Smeed law establishes that as the traffic volume increases (reflected by an increase in motor vehicle registrations), there is an increase in fatalities per capita and a decrease in fatalities per vehicle (as motor vehicle ownership increases, the death rates per vehicle decrease). The corrected Smeed law addresses the overestimation in the original Smeed law by considering that an increase in the vehicle fleet typically leads to investments in vehicle safety and infrastructure improvements, which subsequently reduce road fatalities. Andreassen’s law links the population and the number of vehicles to the fatalities on the roads.

The functional diagram of the used methodology is presented in Figure 1.

The use of a prediction model depends on the specifics of the input data, the parameters of the system for which the forecast is generated and the external factors that influence them. The primary gap that researchers identify is the lack of a standardized methodology with the help of which the appropriate forecasting method can be chosen. The present authors propose such a methodology that can be generalized, being suitable for use for any urban agglomeration at the micro and macro level.

Romania holds the highest position in the European ranking concerning the frequency of road accidents per capita [3]. A case study carried out in Bucharest sheds light on the seriousness of its road safety problems. Finally, the results of the case study serve as a crucial catalyst for policymakers, urban planners and community stakeholders to collaborate in developing effective strategies to reduce the incidence of accidents and create safer streets.

2. Literature Background

Research into the forecasting of road accidents and fatalities is currently a dynamic field. While individual accidents are inherently unpredictable, the aggregate annual number of accidents tends to exhibit discernible trends that can be modeled. Additionally, road accidents can be considered in terms of factors such as road conditions, driver awareness and vehicle safety. The literature background seeks to identify suitable methodologies for the modeling of road accidents, along with the parameters relevant to these models.

A pioneering and enduring principle in road safety research for the forecasting of fatalities is Smeed’s law [4], which elucidates the functional correlation between the fatality rate and the rate of motorization. Numerous authors have studied the applicability of Smeed’s law [5,6,7,8,9,10,11,12]. In 1980, John Adams re-evaluated Smeed’s law and determined that the total vehicle distance traveled (in vehicle-kilometers) provided a better indication of risk exposure [5]. A recent study discovered that Smeed’s formula accurately depicts the alteration in fatalities up to a motorization rate of 0.2–0.3 vehicles per person. However, beyond this threshold, the formula appears to overestimate the fatality rate [6].

Persia et al. (2015) used Smeed’s law for Italy. They reached the conclusion that calibrating the Smeed model yielded excellent outcomes, given that the experimental curve closely aligned (in terms of parameters) with the theoretical formulation proposed by Smeed [7].

Andreassen suggested another equation for the prediction of fatalities [13]. Studies have confirmed a notable trend whereby the fatality rate per person is declining faster than Smeed’s law predicts. Moreover, there has been a decline in fatalities per vehicle observed in numerous countries, especially in developed nations, contradicting the forecasts of Smeed’s law.

Other studies regarding forecasting methods for road accidents include Gorzelanczyk et al. In their paper, they outline an approach to assessing the impact of various factors on road accidents in Poland. They use multi-criteria optimization methods [14]. Hossain and Faruque applied an additive time series model approach, incorporating trend estimation, seasonal variation and random variation through the use of the triple exponential smoothing method [15]. Imran and Nasir utilized a series of eleven curve fitting models, including linear, cubic, quadratic, inverse, logarithmic, exponential growth model, logistic curve and compound models, to forecast road traffic accidents in Pakistan [16].

Borsos [17] used the methodology for 26 countries. The conclusions of the study were the following:

• A country-level analysis indicated that the model could be effectively applied to most countries;
• Road safety is increasingly becoming a priority, occupying a prominent position on the agenda of both developed and developing nations.

3. Methodology

In this study, the statistical analysis relies on the empirical laws formulated by Smeed and Andreassen and on parametric methods.

3.1. The Smeed Law

Smeed’s law describes the functional connection between the annual number of deaths, the number of vehicles and the population [4].

$$D=\alpha ·{(N·P^2)}^{\beta }$$

(1)

where

D = annual deaths; N = number of vehicles; P = population.

If we log the equation, we will have

$$\mathrm{l}\mathrm{n}\left(D\right)=\mathrm{l}\mathrm{n}\left(\alpha \right)+\beta ·ln\left(N·P^2\right)$$

(2)

If $y=\mathrm{l}\mathrm{n}\left(D\right)$, b$=\ln \left(\alpha \right),$ a = β and $x=\mathrm{l}\mathrm{n}(N·P^2)$, then

$$y=ax+b$$

(3)

3.2. The Corrected Smeed Law

Koren and Borsos contemplated whether Smeed’s law remains applicable in the present context [5]. They modified Smeed’s formula by introducing a negative exponential term to accommodate the reduction in fatalities with the increase in the motorization rate. Their rationale was that, with an increase in the number of vehicles, there is often an associated improvement in infrastructure, which contributes to a reduction in road accident deaths. The corrected Smeed’s law according to them is

$$\frac{D}{P}=a·\frac{N}{P}·e^{-b·\frac{N}{P}}$$

(4)

$a·\frac{N}{P}$ represents the increasing exposure as the vehicle numbers increase. When $\frac{N}{P}$ is relatively low, then $e^{-b·\frac{N}{P}}$ is approximately 1, making the first part of the formula predominant.

$e^{-b·\frac{N}{P}}$ describes a negative exponential function. It signifies that an increase in vehicle ownership tends to coincide with improvements in vehicle safety and the quality of road infrastructure, as well as improvements in education and law enforcement measures.

If we log the equation, we will have

$$\ln \left(D/P\right)=\mathrm{l}\mathrm{n}\left(a\right)+ln\left(N/P\right)-b·N/P$$

(5)

We consider the following equation as the linear model:

$$y=x_1-b{x}_2+c$$

(6)

where $y=\ln \left(\frac{D}{P}\right)$; $x_1=ln\left(\frac{N}{P}\right)$, $x_2=\frac{N}{P}$ and c = ln(a).

The parameters for both Smeed’s law and the corrected Smeed’s law model are estimated using linear regression.

3.3. The Andreassen Law

Andreassen revised Smeed’s law. He argued that the values of the parameters α and β provided by Smeed are inaccurate because they arise from an erroneous relationship between the variables N/P and D/P. Both variables include the population P in the denominator [18]. Andreassen suggested taking into account not only the number of deaths but also the number and severity of accidents, particularly when generating results for accident reduction purposes. Andreassen’s law [19] links the population and the number of vehicles to the fatalities as follows [2]:

$$D=k·N^{B_1}·P^{B_2}$$

(7)

where

D represents the number of deaths per year, N is the number of vehicles and P is the population.

If we log the equation, we will have

$$ln\left(D\right)=ln\left(k\right)+B_{1}ln\left(N\right)+B_{2}ln\left(P\right)$$

(8)

We consider the following linear model:

$$y=a{x}_1+b{x}_2+c$$

(9)

where

$y=\ln \left(D\right)$; $x_1=ln\left(N\right),$ $x_2=\mathrm{l}\mathrm{n}\left(P\right)$, $c=ln\left(k\right)$, $a=B_1$ and $b=B_2$.

Linear regression is used to estimate the parameters of the model.

4. Case Study: Bucharest City

The case study conducted in Bucharest illuminates the gravity of road safety issues. To implement the forecasting methods outlined, the following data from

Table 1. Number of accidents in Bucharest from 2010 to 2023.

Year	Population 2	Accidents 1 (Total)	Fatalities 1	Seriously Injured 1	Easily Injured 1	Registered Vehicles 3
2011	1,883,425	920	86	899	149	1131807
2012	1,886,866	960	79	946	248	1118125
2013	1,875,389	783	62	762	102	1131694
2014	1,865,563	614	61	583	91	1152796
2015	1,853,638	670	68	628	110	1193775
2016	1,843,962	480	66	441	87	1253692
2017	1,826,579	471	74	408	73	1320230
2018	1,828,869	569	58	521	109	1381620
2019	1,832,802	690	58	657	123	1457889
2020	1,841,052	535	52	498	58	1502169
2021	1,828,781	378	63	323	66	1535310
2022	1,722,865	441	47	415	69	1570965
2023	1,725,271	400	45	361	45	1598284

Source: ¹ [20], ² [21], ³ [22,23].

were used.

The results for Smeed’s law, Andreassen’s law and the corrected Smeed’s law (Koren–Borsos) using linear regression are presented in

Table 2. Smeed’s law with linear regression: (a) model fit measures; (b) model coefficients—ln(D).

(a) Model Fit Measures
Model			R			R2
1			0.623			0.388
(b) Model Coefficients—ln(D)
						95% Confidence Interval
Predictor	Estimate	SE	t	p	Stand. Estimate	Lower	Upper
Intercept	56.84	19.960	2.85	0.016
ln(NP2)	−1.23	0.465	−2.64	0.023	−0.623	−1.14	−0.104

,

Table 3. Results for corrected Smeed’s law (Koren–Borsos) with linear regression: (a) model fit measures; (b) model coefficients—ln(D/P).

(a) Model Fit Measures
Model			R			R2
1			0.763			0.583
(b) Model Coefficients—ln(D/P)
						95% Confidence Interval
Predictor	Estimate	SE	t	p	Stand. Estimate	Lower	Upper
Intercept	−9.51	0.202	−47.01	<0.001
N/P	−1.07	0.273	−3.92	0.002	−0.763	−1.19	−0.335

and

Table 4. Results for Andreassen’s law with linear regression: (a) model fit measures; (b) model coefficients—ln(D).

(a) Model Fit Measures

Model

R

R2

F

df1

df2

p

1

0.825

0.681

10.7

2

10

0.003

(b) Model Coefficients—ln(D)

95% Confidence Interval

95% Confidence Interval

Predictor

Estimate

SE

Lower

Upper

t

p

Stand. Estimate

Lower

Upper

Intercept

−34.066

33.604

−108.94

40.808

−1.01

0.335

ln(N)

−0.527

0.421

−1.47

0.411

−1.25

0.239

−0.381

−1.060

0.297

ln(P)

3.164

1.984

−1.26

7.585

1.59

0.142

0.485

−0.193

1.164

.

The R² coefficient is 0.388; therefore, 38.8% of the variance of $ln\left(D\right)$ is due to $ln(N·P^2)$’s variance. N·P² is a significant predictor of the number of fatalities (P = 0.023).

The resulting Smeed’s law for Bucharest is

$$D=e^{56.84}·{(N·P^2)}^{-1.23}$$

(10)

The R² coefficient is 0.583; therefore, 58.3% of the variance of $ln\left(D/P\right)$ is predicted by the variance of $N/P$. $N/P$ is a significant predictor of the number of fatalities (P = 0.002). The model is verified for the linear regression.

The resulting corrected Smeed’s law for Bucharest is

$$\frac{D}{P}=e^{9.51}·\frac{N}{P}·e^{1.07·\frac{N}{P}}$$

(11)

$$\frac{D}{P}=\frac{N}{P}·e^{1.07·\frac{N}{P}+9.51}$$

(12)

Because the R² in the corrected Smeed’s law model is greater than the R² in the Smeed’s law model, we can conclude that the first one is more appropriate to describe the evolution of fatalities relative to the population and vehicle number.

The corrected Smeed’s law can be characterized as a strong model for the prediction of the number of fatalities.

The R² coefficient is 0.681, so 68.1% of the variance in fatalities is due to the predictors, namely the population and number of vehicles. Still, in the linear regression, ln(N) and ln(P) are weak predictors of ln(D).

The resulting Andreassen’s law for Bucharest is

$$D=e^{-34.066}·N^{-0.527}·P^{3.164}$$

(13)

5. Conclusions

The increased number of severe accidents in Bucharest could be due to the following.

• Romania has not made significant progress in utilizing technology in the field of road traffic or in modernizing the road infrastructure. This could contribute to maintaining an old vehicle fleet and poor traffic management in cities, including Bucharest.
• The vehicle fleet is old. Older vehicles may be less fuel-efficient and emit more pollutants, and the safety technologies may be less advanced. Additionally, these vehicles may require more maintenance and repairs, leading to a higher frequency of road incidents.
• The levels of road education among the population and adherence to traffic rules in Romania are not at the levels of other European countries. Poor road education can contribute to unsafe behaviors on roads and a higher frequency of road accidents.

The results from the case study are explicable as follows. Smeed’s law and the corrected Smeed’s law applied to Bucharest’s accident statistics are robust models. Given the high traffic density in Bucharest [24], the applicability of both Smeed’s law and the corrected Smeed’s law is verified. Smeed’s law states that the higher the traffic density, the lower the probability of road accidents. This is due to drivers becoming more attentive and cautious in heavy traffic conditions.