Using Neural Networks to Forecast the Amount of Traffic Accidents in Poland and Lithuania

Abstract

Globally, and specifically in Poland and Lithuania, the incidence of road accidents has been on a decline over the years. The overall figures remain significantly high. Thus, it is imperative to take substantial measures to further decrease these statistics. The objective of this article is to estimate the future frequency of traffic accidents in both countries. To achieve this, a comprehensive yearly analysis of traffic incidents in Poland and Lithuania was performed. Using police records, forecasts for the years from 2024 to 2030 were established. Various neural network models were employed to predict the number of accidents. The results suggest that there remains potential for stabilization in traffic accident rates. It is undeniable that the increasing volume of vehicles on the roads, along with the development of new highways and expressways, plays a crucial role in this scenario. The result obtained depends on the model parameters (testing, validation, and training phases). Sustainable development requires comprehensive solutions, which also include improving road safety. Our research contributes to this goal by creating a tool that provides insight into the number of road accidents in analyzed countries.

1. Introduction

Traffic accidents are regrettable occurrences that not only inflict injuries or fatalities on drivers but also cause significant property loss. The World Health Organization (WHO) reports that around 1.3 million individuals lose their lives in vehicle collisions annually. On a broader level, nations face an average decline of 3% in their Gross Domestic Product (GDP) due to these incidents. For the demographic aged five to twenty-nine, road traffic accidents rank as the primary cause of death. The United Nations (UN) General Assembly has set a target to achieve a 50% reduction in fatalities and injuries related to traffic by 2030 [1].

When assessing the gravity of a traffic incident, understanding the extent of the accident is essential. It is crucial for relevant authorities to evaluate the seriousness of incidents to create effective road safety regulations that can prevent accidents and reduce injuries, deaths, and property damage [2,3]. Before introducing strategies to alleviate or avert the severity of accidents, it is important to identify the main factors that contribute to those severities [4]. Yang et al. [5] suggest a multi-node Deep Neural Network (DNN) framework aimed at predicting different levels of injuries, fatalities, and property damage, facilitating a thorough assessment of traffic accident severity.

Accident statistics are sourced from a variety of channels, primarily compiled and analyzed by government entities through designated agencies. Data collection occurs through multiple means, such as police documentation, insurance claims, and hospital records. As a result, the transportation sector is becoming more involved in in-depth analysis of traffic accident data [6].

At present, intelligent transportation systems serve as the most crucial resource for the analysis and forecasting of traffic events. Information from GPS devices installed in vehicles can be leveraged for analytical purposes [7]. Additionally, roadside microwave detection systems can continuously gather data regarding moving vehicles, including their type, speed, and overall traffic volume [8]. License plate recognition systems can also accumulate substantial amounts of traffic data over designated timeframes [9]. Moreover, social media may represent an alternative source of information about traffic incidents, although the reliability of such reports may be affected by the inexperience of those reporting [10].

Working with a variety of data sources and properly validating this information are crucial to ensuring that accident data are valuable. The analytical results can be considerably more accurate by combining several data sources and combining disparate traffic accident facts [11].

In order to assess the gravity of the issue and establish a link between traffic participants and accidents, Vilaca et al. [12] carried out a statistical study. The study’s conclusions recommend raising the bar for road safety laws and putting in place more safety precautions.

Bak et al. [13] conducted a statistical analysis of traffic safety in a particular Polish region in another study, utilizing the frequency of traffic accidents as a primary metric to investigate their causes. Multivariate statistical analysis was used in this study to examine the safety parameters linked to accident-causing persons.

The particular traffic problems under investigation will determine which accident data sources are used for this study. The efficiency of accident prediction and accident prevention can be greatly increased by combining statistical models with additional data from actual driving situations or information obtained from intelligent traffic systems [14].

The literature includes a variety of techniques for forecasting the frequency of traffic accidents. One of the most often used methods for assessing accident frequency is time series approaches [15,16]. However, they have drawbacks, such as the frequent occurrence of autocorrelation in residuals and the inability to assess forecast accuracy based on prior predictions [17]. For example, Sunny et al. [18] used the Holt–Winters exponential smoothing approach, whereas Procházka et al. [19] used a multi-seasonality model for their forecasts. The inability to incorporate exogenous variables into the model is a significant limitation of current approaches [20].

Another key statistic to consider is the incidence of accidents per 10,000 individuals. In Poland, there were 20,936 reported road accidents in 2023, with a total population of 37.6 million. Based on these data, we can calculate an accident rate of 5.57 per 10,000 people in Poland for that year. In comparison, Lithuania, with a population of 2.8 million, recorded 2863 road incidents during the same timeframe, leading to a notably higher rate of 10.23 accidents per 10,000 inhabitants in 2023.

$$NRA={\displaystyle \frac{NR}{NI}}\ast 10000,$$

(1)

where

• NR—number of road accidents;
• NI—number of inhabitants.

Given the above, it is clear that the authors do not use neural networks to predict the number of road accidents. For this reason, the authors’ research hypothesis is that it is possible to forecast the number of road accidents using neural networks. The research will help to predict the number of accidents in the near future, which in turn will improve road safety.

The researchers estimated the number of accidents occurring on the roads of Poland and Lithuania based on the previously provided data. To forecast the incidence of accidents in both countries, they utilized neural networks. The conducted research will help reveal and assess the symmetry and asymmetry of the traffic accident situation in neighboring countries Poland and Lithuania. In the future, this would also help in selecting appropriate measures to improve traffic safety in both countries and in properly coordinating them as well as coordinating the investigation of the circumstances of traffic accidents.

Sustainable development requires comprehensive solutions, which also include improving road safety. Our research contributes to this goal by creating a tool that provides insight into the number of road accidents in the countries analyzed. Reducing the number of road accidents is not only a social aspect but also an economic one. Thanks to our research, it is possible to reduce the costs associated with treating the injured, repairing vehicles, and loss of productivity, which translates into economic growth.

2. Materials and Methods

Each year, roadways witness a substantial number of traffic accidents. While there has been a reduction in these incidents in recent years attributed to the epidemic, this trend has impacted the reliability of predictive models. Despite this decline, traffic accidents remain prevalent [21,22]. Therefore, it is essential to recognize the types of roads that are more susceptible to accidents and to enact all necessary measures to minimize their frequency (refer to Figure 1 and Figure 2). The number of accidents on Poland’s and Lithuania’s roads is decreasing, but is still very high (

Table 1. Numbers of road accidents in Poland and Lithuania in 1990–2023 [21,22].

Year	Poland	Lithuania	Year	Poland	Lithuania
1990	50,432	5135	2007	49,536	6448
1991	54,038	6067	2008	49,054	4795
1992	50,990	4049	2009	44,196	3805
1993	48,901	4319	2010	38,832	3530
1994	53,647	3902	2011	40,065	3266
1995	56,904	4144	2012	37,046	3173
1996	57,911	4579	2013	35,847	3391
1997	66,586	5319	2014	34,970	3255
1998	61,855	6445	2015	32,967	3033
1999	55,106	6356	2016	33,664	3201
2000	57,331	5807	2017	32,760	3051
2001	53,799	5972	2018	31,674	2925
2002	53,559	6090	2019	30,288	3190
2003	51,078	5963	2020	23,540	2826
2004	51,069	6372	2021	22,816	2808
2005	48,100	6771	2022	21,322	2878
2006	46,876	6658	2023	20,936	2863

).

In Poland and Lithuania, particular neural network models were utilized to forecast the likelihood of traffic accidents. A significant benefit of this approach is its capacity to mimic the functions of the human brain. Neural networks are composed of nodes, which encompass inputs, weights, variances, and outputs. The optimal weights for the analysis were calculated using Statistica software 13.3. The results of the predictions produced by this method are affected by the models and parameters employed.

A neural network can be conceptualized as a mathematical framework that functions similarly to the nervous system. These networks are generally structured in multiple layers, which contribute to their overall design. The first layer is responsible for processing various forms of data, such as text, images, numbers, and sounds, through a training phase. During this stage, the network may evaluate hundreds of inputs before making specific decisions. The core elements of neural networks are artificial neurons, which mathematically replicate the behavior of biological neurons. Like their natural counterparts, artificial neurons take in multiple inputs but yield a single output value, similar to the dendrites found in real neurons (Figure 3) [23].

The progress of artificial intelligence is significantly centered around neural networks. This domain seeks to develop models that demonstrate intelligent behavior, including the formation of hierarchical knowledge structures.

Neural networks have diverse applications across multiple fields. For example, streaming platforms leverage them to examine user viewing habits and recommend films that align with individual preferences. They can also personalize product offerings for bidders in online auctions or improve the efficiency of text translations in applications like Google Translate. Additionally, forecasts generated by neural networks are used to estimate the frequency of traffic incidents.

To assess the number of traffic accidents in the analyzed counties, a neural network model is employed. A primary advantage of this technology is its ability to emulate the functionality of the human brain. The architecture of a neural network consists of nodes that encompass inputs, weights, variations, and outputs.

Additionally, one of the benefits of a neural network is that, similar to the human brain’s ability to function even after “conditioning” from substances like alcohol, it may continue to function up to a certain extent even in the event of significant data destruction. The brain does not refuse to comply until a certain level of damage is reached. This results in drug-induced psychosis, Alzheimer’s disease, and later stages of alcoholism, among other conditions. The capacity of neural networks to generalize learned information is another significant benefit. This implies that the network will detect pink and light yellow, which are colors that are close to those it already knows, if it learns to distinguish, for instance, red and yellow.

However, the drawbacks of neural networks include the inability to provide exact and unambiguous outputs due to a variety of intricate computations. The reason for this is that neural networks are a mirror of the human brain, which is not designed to handle numbers precisely. Furthermore, fuzzy concepts—high, low, huge, tiny, medium, and bright—are used by neural networks. The network will frequently respond with “rather yes” or “probably not” if we anticipate it to say “yes” or “no.” Additionally, if multi-step reasoning is needed to solve the problem, we will not employ neural networks [24].

The integrated artificial neural network modules within the Statistica program were responsible for adjusting the optimal weights during the testing phase. For prediction purposes, a multilayer perceptron (MLP) neural network was employed, which consists of layers that include hidden neurons. In the analyzed cases, the number of neurons in the hidden layer ranged from two to eight. The output layer comprised a single neuron that represented the time series output values of traffic events. The number of neurons in the hidden layer in the analyzed case varied from 2 to 8 neutrons, and the output layer was a single neutron, representing the output values of the time series of the number of traffic accidents. The result of the forecast by the method discussed depends on the choice of the model and its parameters. The analyzed data were post-validated [25,26,27,28,29].

The outcomes generated by these methods are contingent upon the model and its associated parameters. To evaluate the quality of the predictions, the predictive accuracy was calculated using the prediction errors derived from specific Equations (2)–(7) with the following formula:

ME—mean error

$$ME={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left(Y_i-Y_p\right),$$

(2)

MAE—mean absolute error

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|Y_i-Y_p\right|,$$

(3)

MPE—mean percentage error

$$MPE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{Y_i-Y_p}{Y_i}},$$

(4)

MAPE—mean absolute percentage error

$$MAPE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\displaystyle \frac{\left|Y_i-Y_p\right|}{Y_i}},$$

(5)

SSE—mean square error

$$SSE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(Y_i-Y_p\right)}^2},$$

(6)

$M^2$—Theila measure

$$M^2={\displaystyle \frac{{\sum }_{i=1}^N{(Y_i-Y_p)}^2}{{\sum }_{i=1}^N{Y_i}^2}},$$

(7)

where

• n—length of forecast horizon;
• Y—observed value of road accidents;
• Yp—projected value of road accidents.

The frequency of traffic accidents was predicted using neural network models with the lowest average percentage error and average absolute percentage error.

3. Results

The Polish Police’s 1990–2023 data [21] were used to anticipate the country’s yearly traffic accident count; in Lithuania, the Transport Competence Agency’s data [22] were utilized. In both cases, the studies were conducted in Statistica software 13.3, assuming two random sample sizes:

• Teaching 70%, testing 15% and validation 15%;
• Teaching 80%, testing 10% and validation 10%;

with the following number of teaching networks: 20, 40, 60, 80, 100, 200, for which the MP error value was minimal (Appendix A,

Table A1. Summary of neural network learning for the case of random sample size teaching 70%, testing 15% and validation 15% for Poland.

Network Number	Network Name	Quality (Learning)	Quality (Testing)	Quality (Validation)	Learning Algorithm	Error Function	Activation (Hidden)	Activation (Output)	Errors
									ME	MAE	MPE	MAPE	SSE	Theil
20	MLP 1-7-1	0.97	0.98	0.99	BFGS 12	SOS	Tanh	Logistic	932.03	2374.97	2.85%	6.69%	2747.76	4.56 × 10−³
20	MLP 1-8-1	0.97	0.98	0.99	BFGS 4	SOS	Linear	Logistic	664.14	2092.80	3.19%	6.46%	2667.94	4.30 × 10−3
20	MLP 1-2-1	0.97	0.98	0.99	BFGS 4	SOS	Exponential	Exponential	1168.74	2119.69	3.23%	5.69%	2657.54	4.26 × 10−3
20	MLP 1-4-1	0.96	0.97	0.99	BFGS 4	SOS	Exponential	Linear	1815.06	2891.74	2.84%	7.52%	3551.21	7.61 × 10−3
20	MLP 1-3-1	0.97	0.97	0.99	BFGS 42	SOS	Tanh	Logistic	1113.77	2387.70	3.25%	6.64%	2802.32	4.74 × 10−3
40	MLP 1-7-1	0.97	0.98	0.99	BFGS 8	SOS	Exponential	Exponential	837.43	2001.24	2.77%	5.68%	2484.15	3.72 × 10−3
40	MLP 1-3-1	0.97	0.98	0.99	BFGS 8	SOS	Exponential	Exponential	909.41	2052.58	2.80%	5.71%	2526.15	3.85 × 10−3
40	MLP 1-2-1	0.97	0.96	0.99	BFGS 7	SOS	Logistic	Logistic	1110.51	2279.94	3.35%	6.40%	2764.05	4.61 × 10−3
40	MLP 1-8-1	0.97	0.96	0.99	BFGS 5	SOS	Logistic	Exponential	1483.15	2363.70	4.57%	6.80%	2927.92	5.17 × 10−3
40	MLP 1-3-1	0.96	0.95	0.99	BFGS 5	SOS	Logistic	Exponential	1035.60	2590.23	3.60%	7.73%	3048.65	5.61 × 10−3
60	MLP 1-4-1	0.97	0.97	0.99	BFGS 7	SOS	Tanh	Logistic	1031.48	2377.94	3.10%	6.68%	2772.29	4.64 × 10−3
60	MLP 1-7-1	0.97	0.97	0.99	BFGS 5	SOS	Tanh	Logistic	777.04	2415.30	2.17%	6.63%	2763.52	4.61 × 10−3
60	MLP 1-2-1	0.97	0.97	0.99	BFGS 9	SOS	Exponential	Logistic	1109.93	2233.82	3.17%	6.12%	2715.75	4.45 × 10−3
60	MLP 1-5-1	0.97	0.97	0.99	BFGS 12	SOS	Tanh	Exponential	1090.27	2283.83	3.28%	6.40%	2721.35	4.47 × 10−3
60	MLP 1-6-1	0.97	0.98	0.99	BFGS 5	SOS	Exponential	Exponential	1071.39	1987.71	3.83%	5.94%	2590.07	4.05 × 10−3
80	MLP 1-5-1	0.97	0.97	0.99	BFGS 7	SOS	Tanh	Logistic	1040.10	2404.10	3.15%	6.78%	2796.23	4.72 × 10−3
80	MLP 1-8-1	0.97	0.98	0.99	BFGS 14	SOS	Exponential	Logistic	1023.90	2217.06	3.03%	6.15%	2661.45	4.27 × 10−3
80	MLP 1-3-1	0.97	0.98	0.99	BFGS 7	SOS	Exponential	Logistic	801.65	2239.47	2.48%	6.27%	2623.54	4.15 × 10−3
80	MLP 1-7-1	0.97	0.96	0.99	BFGS 7	SOS	Logistic	Logistic	978.70	2426.95	2.68%	6.63%	2851.96	4.91 × 10−3
80	MLP 1-7-1	0.97	0.98	0.99	BFGS 13	SOS	Exponential	Logistic	873.08	2237.49	2.54%	6.16%	2638.93	4.20 × 10−3
100	MLP 1-8-1	0.97	0.98	0.99	BFGS 18	SOS	Exponential	Logistic	1021.62	2260.14	2.99%	6.25%	2688.97	4.36 × 10−3
100	MLP 1-5-1	0.97	0.97	0.99	BFGS 6	SOS	Logistic	Logistic	1108.50	2402.26	3.24%	6.69%	2819.42	4.80 × 10−3
100	MLP 1-4-1	0.97	0.98	0.99	BFGS 11	SOS	Logistic	Exponential	909.58	2320.20	2.93%	6.65%	2707.86	4.43 × 10−3
100	MLP 1-2-1	0.96	0.95	0.99	BFGS 7	SOS	Tanh	Logistic	1114.15	2426.31	4.15%	7.43%	3005.79	5.45 × 10−3
100	MLP 1-2-1	0.97	0.97	0.99	BFGS 8	SOS	Tanh	Logistic	894.03	2347.67	2.83%	6.67%	2720.34	4.47 × 10−3
200	MLP 1-6-1	0.96	0.96	0.99	BFGS 8	SOS	Tanh	Logistic	644.88	2480.32	2.22%	7.15%	2814.19	4.78 × 10−3
200	MLP 1-6-1	0.97	0.96	0.99	BFGS 7	SOS	Tanh	Logistic	770.95	2330.09	2.51%	6.64%	2702.56	4.41 × 10−3
200	MLP 1-3-1	0.97	0.97	0.99	BFGS 10	SOS	Logistic	Logistic	970.77	2347.01	2.97%	6.61%	2750.08	4.56 × 10−3
200	MLP 1-2-1	0.95	0.92	0.99	BFGS 7	SOS	Logistic	Exponential	319.55	2657.15	0.32%	7.61%	3035.78	5.56 × 10−3
200	MLP 1-4-1	0.97	0.97	0.99	BFGS 6	SOS	Tanh	Logistic	1200.19	2356.37	3.63%	6.66%	2816.40	4.79 × 10−3
								Minimal	319.55	1987.71	0.32%	5.68%	2484.15	3.72 × 10−3

,

Table A2. Summary of neural network learning for the case of random sample size teaching 80%. testing 10% and validation 10% for Poland.

Network Number	Network Name	Quality (Learning)	Quality (Testing)	Quality (Validation)	Learning Algorithm	Error Function	Activation (Hidden)	Activation (Output)	Errors
									ME	MAE	MPE	MAPE	SSE	Theil
20	MLP 1-5-1	0.96	0.99	1.00	BFGS 8	SOS	Logistic	Linear	422.40	1830.32	0.90%	5.12%	2362.11	3.37 × 10−3
20	MLP 1-5-1	0.96	0.99	1.00	BFGS 5	SOS	Linear	Tanh	420.05	2152.64	0.39%	6.51%	2773.07	4.64 × 10−3
20	MLP 1-3-1	0.96	0.99	1.00	BFGS 63	SOS	Tanh	Logistic	702.37	1986.10	2.31%	5.57%	2455.03	3.64 × 10−3
20	MLP 1-8-1	0.96	0.99	1.00	BFGS 6	SOS	Linear	Tanh	326.74	2130.77	0.17%	6.45%	2734.11	4.51 × 10−3
20	MLP 1-8-1	0.96	0.99	1.00	BFGS 6	SOS	Logistic	Tanh	265.62	1759.88	0.80%	4.63%	2294.49	3.18 × 10−3
40	MLP 1-5-1	0.96	0.99	1.00	BFGS 5	SOS	Tanh	Exponential	1544.71	2539.20	6.27%	8.23%	3300.44	6.57 × 10−3
40	MLP 1-5-1	0.96	0.99	1.00	BFGS 6	SOS	Linear	Tanh	180.47	2355.28	0.72%	7.31%	2994.43	5.41 × 10−3
40	MLP 1-2-1	0.96	0.99	1.00	BFGS 6	SOS	Linear	Tanh	184.52	2325.08	1.67%	7.28%	2934.77	5.20 × 10−3
40	MLP 1-6-1	0.96	0.98	1.00	BFGS 4	SOS	Logistic	Logistic	725.12	2046.17	3.35%	6.03%	2699.40	4.40 × 10−3
40	MLP 1-2-1	0.96	0.99	1.00	BFGS 10	SOS	Logistic	Tanh	397.20	1761.51	1.05%	4.76%	2339.08	3.30 × 10−3
60	MLP 1-2-1	0.95	0.98	1.00	BFGS 5	SOS	Logistic	Exponential	46.12	2638.20	0.89%	7.75%	3021.27	5.51 × 10−3
60	MLP 1-6-1	0.96	0.99	1.00	BFGS 5	SOS	Linear	Tanh	381.15	2625.38	2.79%	8.41%	3359.27	6.81 × 10−3
60	MLP 1-6-1	0.95	0.98	1.00	BFGS 5	SOS	Logistic	Logistic	1436.54	2605.98	2.62%	6.58%	3107.29	5.83 × 10−3
60	MLP 1-3-1	0.95	0.98	1.00	BFGS 7	SOS	Tanh	Tanh	225.51	2181.93	1.10%	6.66%	2827.63	4.83 × 10−3
60	MLP 1-6-1	0.95	0.99	1.00	BFGS 7	SOS	Exponential	Logistic	231.31	2206.35	0.69%	5.98%	2657.70	4.26 × 10−3
80	MLP 1-2-1	0.96	0.99	1.00	BFGS 11	SOS	Logistic	Tanh	63.00	2068.87	0.35%	6.24%	2669.34	4.30 × 10−3
80	MLP 1-3-1	0.96	0.99	1.00	BFGS 4	SOS	Linear	Tanh	261.75	2325.06	0.42%	7.18%	2957.41	5.28 × 10−3
80	MLP 1-2-1	0.96	0.98	1.00	BFGS 7	SOS	Logistic	Linear	553.25	2205.41	2.23%	6.74%	2759.02	4.59 × 10−3
80	MLP 1-2-1	0.95	0.98	1.00	BFGS 6	SOS	Tanh	Logistic	81.51	2328.80	0.41%	6.55%	2719.89	4.46 × 10−3
80	MLP 1-7-1	0.96	0.99	1.00	BFGS 5	SOS	Linear	Tanh	159.97	2374.17	0.82%	7.38%	3018.42	5.50 × 10−3
100	MLP 1-7-1	0.96	0.99	1.00	BFGS 7	SOS	Linear	Tanh	573.15	2175.01	0.84%	6.54%	2792.33	4.71 × 10−3
100	MLP 1-2-1	0.95	0.99	1.00	BFGS 9	SOS	Tanh	Logistic	334.46	2310.29	1.71%	6.79%	2726.36	4.49 × 10−3
100	MLP 1-5-1	0.96	0.99	1.00	BFGS 5	SOS	Linear	Tanh	180.96	2441.21	1.90%	7.72%	3101.83	5.81 × 10−3
100	MLP 1-2-1	0.96	0.99	1.00	BFGS 7	SOS	Linear	Tanh	573.25	2174.78	0.84%	6.54%	2791.98	4.70 × 10−3
100	MLP 1-4-1	0.96	0.99	1.00	BFGS 5	SOS	Linear	Tanh	100.84	2331.11	0.91%	7.25%	2967.66	5.32 × 10−3
200	MLP 1-8-1	0.96	0.99	1.00	BFGS 6	SOS	Tanh	Tanh	380.18	2350.66	2.18%	7.47%	3034.06	5.56 × 10−3
200	MLP 1-2-1	0.96	0.98	1.00	BFGS 7	SOS	Tanh	Linear	265.66	2300.27	1.76%	7.12%	2877.84	5.00 × 10−3
200	MLP 1-8-1	0.96	0.99	1.00	BFGS 2	SOS	Tanh	Tanh	1932.17	2744.39	4.08%	6.86%	3486.86	7.34 × 10−3
200	MLP 1-3-1	0.96	0.98	1.00	BFGS 7	SOS	Logistic	Tanh	38.54	1969.49	0.44%	5.51%	2441.32	3.60 × 10−3
200	MLP 1-4-1	0.95	0.98	1.00	BFGS 5	SOS	Logistic	Logistic	704.60	2296.70	1.35%	6.17%	2731.82	4.50 × 10−3
								Minimal	38.54	1759.88	0.17%	4.63%	2294.49	3.18 × 10−3

,

Table A3. Summary of neural network learning for the case of random sample sizes teaching 70%. testing 15% and validation 15% for Lithuania.

Network Number	Network Name	Quality (Learning)	Quality (Testing)	Quality (Validation)	Learning Algorithm	Error Function	Activation (Hidden)	Activation (Output)	Errors
									ME	MAE	MPE	MAPE	SSE	Theil
20	MLP 1-7-1	0.95	0.88	0.79	BFGS 5	SOS	Exponential	Tanh	116.84	293.52	2.65%	7.20%	472.03	1.10 × 10−2
20	MLP 1-5-1	0.96	0.80	0.79	BFGS 11	SOS	Exponential	Logistic	50.81	287.67	1.24%	6.68%	486.55	1.17 × 10−2
20	MLP 1-7-1	0.95	0.87	0.80	BFGS 14	SOS	Tanh	Tanh	129.74	300.58	3.23%	7.40%	474.81	1.11 × 10−2
20	MLP 1-4-1	0.95	0.86	0.80	BFGS 17	SOS	Tanh	Tanh	118.34	309.49	3.08%	7.64%	477.01	1.12 × 10−2
20	MLP 1-8-1	0.95	0.86	0.80	BFGS 6	SOS	Linear	Tanh	101.48	346.30	2.87%	8.73%	492.44	1.20 × 10−2
40	MLP 1-7-1	0.95	0.86	0.80	BFGS 5	SOS	Linear	Tanh	56.14	342.71	1.59%	8.56%	484.27	1.16 × 10−2
40	MLP 1-2-1	0.95	0.85	0.80	BFGS 9	SOS	Logistic	Linear	103.22	333.19	2.90%	8.31%	486.14	1.16 × 10−2
40	MLP 1-3-1	0.94	0.88	0.80	BFGS 2	SOS	Linear	Tanh	280.75	432.36	6.18%	9.80%	530.19	1.39 × 10−2
40	MLP 1-5-1	0.95	0.86	0.80	BFGS 14	SOS	Tanh	Tanh	72.85	324.85	2.20%	7.97%	469.73	1.09 × 10−2
40	MLP 1-4-1	0.95	0.84	0.81	BFGS 10	SOS	Logistic	Tanh	68.48	338.43	2.04%	8.37%	486.85	1.17 × 10−2
60	MLP 1-5-1	0.95	0.85	0.80	BFGS 5	SOS	Linear	Tanh	15.31	363.05	1.03%	9.38%	512.18	1.29 × 10−2
60	MLP 1-2-1	0.95	0.85	0.80	BFGS 13	SOS	Logistic	Tanh	97.04	327.47	2.72%	8.11%	485.12	1.16 × 10−2
60	MLP 1-2-1	0.95	0.86	0.80	BFGS 7	SOS	Linear	Tanh	99.28	346.01	2.85%	8.70%	490.43	1.19 × 10−2
60	MLP 1-6-1	0.95	0.86	0.80	BFGS 3	SOS	Logistic	Linear	47.65	355.01	2.64%	8.34%	473.70	1.11 × 10−2
60	MLP 1-3-1	0.95	0.85	0.80	BFGS 8	SOS	Logistic	Linear	98.13	332.87	2.77%	8.28%	486.39	1.17 × 10−2
80	MLP 1-4-1	0.95	0.86	0.80	BFGS 5	SOS	Linear	Tanh	57.41	342.51	1.64%	8.55%	483.87	1.15 × 10−2
80	MLP 1-7-1	0.94	0.86	0.81	BFGS 2	SOS	Tanh	Tanh	193.51	447.86	2.57%	9.52%	566.17	1.58 × 10−2
80	MLP 1-5-1	0.95	0.86	0.80	BFGS 5	SOS	Linear	Tanh	21.46	344.49	0.49%	8.64%	486.20	1.16 × 10−2
80	MLP 1-6-1	0.94	0.85	0.81	BFGS 2	SOS	Tanh	Tanh	143.23	402.10	2.54%	9.28%	509.69	1.28 × 10−2
80	MLP 1-3-1	0.95	0.86	0.80	BFGS 5	SOS	Linear	Tanh	59.53	342.41	1.72%	8.55%	483.56	1.15 × 10−2
100	MLP 1-7-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	97.95	346.00	2.76%	8.72%	492.12	1.19 × 10−2
100	MLP 1-8-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	79.26	344.72	2.07%	8.71%	494.49	1.21 × 10−2
100	MLP 1-5-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	100.76	346.25	2.85%	8.72%	492.49	1.20 × 10−2
100	MLP 1-4-1	0.95	0.85	0.80	BFGS 13	SOS	Tanh	Tanh	91.68	346.05	2.62%	8.69%	491.42	1.19 × 10−2
100	MLP 1-6-1	0.95	0.86	0.80	BFGS 5	SOS	Linear	Tanh	4.80	352.90	0.26%	8.99%	499.52	1.23 × 10−2
200	MLP 1-7-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	101.52	346.31	2.87%	8.73%	492.47	1.20 × 10−2
200	MLP 1-4-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	101.01	346.26	2.86%	8.72%	492.37	1.19 × 10−2
200	MLP 1-3-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	75.43	345.50	1.91%	8.75%	496.30	1.21 × 10−2
200	MLP 1-8-1	0.95	0.86	0.80	BFGS 5	SOS	Linear	Tanh	101.59	346.31	2.88%	8.73%	492.42	1.20 × 10−2
200	MLP 1-8-1	0.95	0.86	0.80	BFGS 4	SOS	Linear	Tanh	78.13	345.02	2.02%	8.73%	495.22	1.21 × 10−2
								Minimal	4.80	287.67	0.26%	6.68%	469.73	1.09 × 10−2

and

Table A4. Summary of neural network learning for the case of random sample size teaching 80%. testing 10% and validation 10% for Lithuania.

Network Number	Network Name	Quality (Learning)	Quality (Testing)	Quality (Validation)	Learning Algorithm	Error Function	Activation (Hidden)	Activation (Output)	Errors
									ME	MAE	MPE	MAPE	SSE	Theil
20	MLP 1-2-1	0.90	1.00	1.00	BFGS 6	SOS	Tanh	Exponential	134.74	393.58	2.22%	8.75%	591.36	1.72 × 10−2
20	MLP 1-2-1	0.90	1.00	1.00	BFGS 6	SOS	Tanh	Exponential	220.56	1410.13	17.04%	36.02%	1471.74	1.07 × 10−1
20	MLP 1-2-1	0.90	1.00	1.00	BFGS 6	SOS	Tanh	Exponential	107.06	293.38	2.98%	7.14%	452.54	1.01 × 10−2
20	MLP 1-2-1	0.90	1.00	1.00	BFGS 6	SOS	Tanh	Exponential	103.92	359.62	0.98%	7.64%	535.33	1.41 × 10−2
20	MLP 1-2-1	0.90	1.00	1.00	BFGS 6	SOS	Tanh	Exponential	54.42	341.90	1.87%	8.42%	474.11	1.11 × 10−2
40	MLP 1-2-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	79.02	330.92	2.38%	8.24%	465.18	1.07 × 10−2
40	MLP 1-8-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	94.95	332.66	2.86%	8.30%	468.35	1.08 × 10−2
40	MLP 1-8-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	92.36	327.48	2.79%	8.14%	463.80	1.06 × 10−2
40	MLP 1-5-1	0.93	1.00	1.00	BFGS 2	SOS	Tanh	Linear	86.38	769.14	4.16%	17.71%	875.68	3.78 × 10−2
40	MLP 1-2-1	0.93	1.00	1.00	BFGS 2	SOS	Exponential	Exponential	169.63	729.93	1.83%	16.07%	880.63	3.82 × 10−2
60	MLP 1-7-1	0.93	1.00	1.00	BFGS 0	SOS	Linear	Tanh	54.42	341.90	1.87%	8.42%	474.11	1.11 × 10−2
60	MLP 1-5-1	0.92	1.00	1.00	BFGS 5	SOS	Logistic	Logistic	141.96	386.07	1.79%	8.08%	570.73	1.61 × 10−2
60	MLP 1-5-1	0.93	1.00	1.00	BFGS 0	SOS	Linear	Tanh	54.42	341.90	1.87%	8.42%	474.11	1.11 × 10−2
60	MLP 1-5-1	0.93	1.00	1.00	BFGS 0	SOS	Linear	Tanh	54.42	341.90	1.87%	8.42%	474.11	1.11 × 10−2
60	MLP 1-6-1	0.94	1.00	1.00	BFGS 0	SOS	Logistic	Tanh	100.10	298.22	2.80%	7.27%	455.09	1.02 × 10−2
80	MLP 1-2-1	0.93	1.00	1.00	BFGS 4	SOS	Exponential	Exponential	6.39	317.71	1.23%	7.16%	483.83	1.15 × 10−2
80	MLP 1-2-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	97.24	327.26	2.96%	8.13%	463.60	1.06 × 10−2
80	MLP 1-3-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	93.10	326.55	2.78%	8.12%	463.80	1.06 × 10−2
80	MLP 1-8-1	0.93	1.00	1.00	BFGS 0	SOS	Linear	Tanh	54.42	341.90	1.87%	8.42%	474.11	1.11 × 10−2
80	MLP 1-4-1	0.93	1.00	1.00	BFGS 3	SOS	Exponential	Exponential	215.34	621.17	9.75%	16.10%	662.96	2.17 × 10−2
100	MLP 1-2-1	0.93	1.00	1.00	BFGS 0	SOS	Tanh	Linear	84.90	301.01	2.41%	7.37%	450.93	1.00 × 10−2
100	MLP 1-8-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	85.41	333.58	2.63%	8.26%	467.98	1.08 × 10−2
100	MLP 1-8-1	0.96	1.00	1.00	BFGS 166	SOS	Logistic	Exponential	3.37	227.91	0.80%	5.18%	363.52	6.51 × 10−3
100	MLP 1-4-1	0.93	1.00	1.00	BFGS 2	SOS	Exponential	Logistic	376.04	833.96	2.14%	17.19%	1089.97	5.85 × 10−2
100	MLP 1-2-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	92.59	326.49	2.81%	8.10%	462.81	1.06 × 10−2
200	MLP 1-6-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	95.16	324.12	2.86%	8.04%	461.95	1.05 × 10−2
200	MLP 1-4-1	0.93	1.00	1.00	BFGS 3	SOS	Exponential	Exponential	160.92	664.22	8.86%	16.78%	703.59	2.44 × 10−2
200	MLP 1-4-1	0.93	1.00	1.00	BFGS 4	SOS	Tanh	Logistic	66.82	446.60	4.59%	10.57%	552.80	1.51 × 10−2
200	MLP 1-5-1	0.94	1.00	1.00	BFGS 0	SOS	Tanh	Tanh	137.58	314.34	3.67%	7.79%	482.26	1.15 × 10−2
200	MLP 1-7-1	0.93	1.00	1.00	BFGS 0	SOS	Exponential	Linear	83.79	330.50	2.63%	8.13%	465.29	1.07 × 10−2
								Minimal	3.37	227.91	0.80%	5.18%	363.52	6.51 × 10−3

).

4. Discussion

The research indicates that there might be a minor uptick in road accidents in Poland; however, the total traffic volume is projected to stabilize in the coming years. To arrive at these conclusions, a random sampling method was employed. It was found that the average percentage error tends to decline when the size of the learning group surpasses that of both the test and validation groups. In particular, the error rates recorded were 5.68% and 4.63% for the individuals in the learning group, with 15% allocated to both the test and validation groups during the first (80-10-10) and second (70-15-15) tests. In the near future, the number of road accidents can be expected to be around 20,000 (Figure 4).

The data available suggest that there may be a modest rise in road accidents across Lithuania; however, overall traffic is anticipated to stabilize in the near future. The process of selecting a random sample size is crucial to the resulting findings. It was observed that the average percentage error tends to decrease when the learning group’s size exceeds that of the test and validation groups. Specifically, for the learning group, the error percentage was 6.68% with a ratio of 70-15-15, while both the test group and the validation group recorded an error of 15%. In contrast, during the second test (80-10-10), the error rate was 5.18%. In the near future, the number of road accidents can be expected to be around 3000 (Figure 5).

This research helps to reveal the symmetry/asymmetry of the traffic accident situation in neighboring countries Poland and Lithuania. In the future, this would also help in selecting appropriate measures to improve traffic safety in both countries and in properly coordinating them as well as coordinate the investigation of the circumstances of traffic accidents [30].

5. Conclusions

Neural networks were employed to predict the occurrence of accidents in both Poland and Lithuania. This study was carried out within the Statistica environment, where the system evaluated the weights of this study to minimize both the mean absolute error and the mean absolute percentage error.

According to the gathered data, it is reasonable to anticipate a stable rate of traffic accidents in both nations, with only a slight rise expected. This situation should be considered in the context of the ongoing pandemic and the increasing number of vehicles on the roads. The estimated forecast errors offer valuable insights into the accuracy of the models used.

In response to the generated forecasts, measures should be taken to further reduce the incidence of traffic accidents. One potential strategy is to enhance penalties for traffic violations on Polish roads, which took effect on 1 January 2022. It is clear that the pandemic has impacted the research findings by significantly changing the frequency of road accidents.

For future studies, researchers plan to utilize additional statistical methods to more accurately determine the total number of traffic accidents. They also aim to explore various factors that could affect the accident rate, such as traffic volume, weather conditions, driver age, and the application of exponential techniques to estimate accident frequency in traffic scenarios.

The results of forecasting the number of road accidents are consistent with the authors’ other studies in this field using other methods of forecasting the number of road accidents, such as adaptive methods and trend methods [31,32].