Optimization of Artificial Neural Networks for Predicting the Radiological Risks of Thermal Waters in Türkiye

Abstract

In this study, the prediction of four radiological risk parameters of thermal waters in Türkiye (dose contribution (D_E) from radon release in thermal water to air for workers and visitors, the annual effective dose from radon ingestion (D_ing) and the annual effective dose to the stomach from radon ingestion (D_sto)) from three physicochemical properties of thermal waters (electrical conductivity (EC), pH and temperature (T)) was investigated using multilayer perceptron (MLP) and radial basis function (RBF) artificial neural networks (ANNs). To achieve this, two separate MLPANN and RBFANN models were constructed using data from the literature. The MLPANN and RBFANN models were verified using performance metrics (relative absolute error (RAE), root mean square error (RMSE), mean absolute error (MAE), and ratio of RMSE to data standard deviation (RSR)). The comparison of performance metrics shows that MLPANN models achieved approximately 54% lower error metrics than RBF models. The performance of the developed models was further examined using rank analysis, Taylor and Scaled Percentage Error (SPE) plots. Rank analysis and Taylor and SPE graphs showed that MLPANN models predicted the values of four radiological risk parameters of thermal waters more correctly than RBFANN models. This study demonstrates that MLPANNs significantly outperformed RBFANNs in predicting the radiological risks of thermal waters in Türkiye.

1. Introduction

Radon attracts the most attention among the radioisotopes contributing to natural background radiation because it poses the greatest risk to human health [1]. Radon, a naturally occurring radioactive inert gas, is typically the most harmful contributor to radiation exposure [2]. Radon occurs naturally in three isotopes: ²¹⁹Rn, ²²⁰Rn, and ²²²Rn. ²²²Rn, produced by the decay of uranium (²³⁸U), which makes up 99.3% of all uranium in the earth’s crust, is the most stable, abundant, dominant, and dangerous radionuclide, with a half-life of 3.8 days compared with the other isotopes [3]. Consequently, the term “radon” corresponds to ²²²Rn in most cases, including this study. Due to its toxicity and longer half-life, ²²²Rn continues to be a focus of scientific research and has received attention for its health risks to both humans and other living organisms [4,5]. Among the natural sources of radiation, ²²²Rn accounts for 50 per cent of the effective dose received by humans [4].

²²²Rn can be found in varying concentrations in the environment. Soils and rocks produce different amounts of ²²²Rn depending on their uranium content. However, the most important natural resources of ²²²Rn concentrations (C_Rn) are air, soil, and water [6,7]. Decaying ²²⁶Ra continuously forms and releases ²²²Rn into small air or water containing pores between soil and rock particles [4]. The water dissolves the radon in the saturated pores and carries it rapidly and far away beyond its initial position [8]. When pressure and temperature are sufficiently high, part of the carried water flows through cracks and fissures, generating thermal waters [9]. Thermal waters flowing through the Earth’s crust come into contact with large areas of igneous rock containing radium, such as granite, quartz, and basalt [10]. Thus, the thermal water used for treatment in spas typically has a very high C_Rn value. The majority of the ²²²Rn in thermal waters (about 70%) is frequently released into indoor air [11]. Consequently, especially in spas, ²²²Rn and short-lived decay products of it can rise to intolerable levels, leading to health problems such as lung cancer [2,12]. Consuming water with a high level of C_Rn can significantly increase the risk of developing cancer of the stomach [13]. It is thus vital to monitor and analyze the levels of C_Rn in thermal waters in order to protect the general public against potential hazards associated with excessive radiation exposure. C_Rn levels in thermal waters have been reported for many years in many locations around the world, and the associated health hazards have been studied by researchers such as Bozkurt and Erturk [14], Çetinkaya and Biçer [15], Hofmann et al. [16], Silva and Dinis [17], and Tabar and Yakut [18].

Artificial neural networks (ANNs) are machine learning techniques based on the human nervous system, simulating the primary learning function of the brain. The first research on ANNs was performed by McCulloch and Pitts [19], who created a simple ANN model for computing. ANNs have grown rapidly and are now widely employed in a number of disciplines for creating and mapping non-linear relationships between inputs and outputs, owing to their superior predictive ability since Hebb [20] devised a learning algorithm for them [21]. The popularity of ANN is due to its ability to learn directly from situations, eliminating the need to analyze parameters using statistical methods [22]. As a result, ANN models are valuable tools for identifying trends in large datasets from a series of experiments [23]. Depending on where the neurons are located and how the layers are connected, a variety of ANNs can be created. The most common and commonly utilized ANNs for problem solving are radial basis function (RBF) and multilayer perceptron (MLP) ANNs. RBF and MLP ANNs are thus considered to be universal approximations for non-linear input-output mappings [24].

Taking into account MLP and RBF ANNs’ advantage for creating and mapping non-linear relationships between outputs and inputs within different experimental datasets, in this study, MLP and RBF ANNs were used to estimate four radiological risk parameters of thermal water: the dose contribution (D_E) values from radon release in thermal water to air for workers and visitors, the annual effective dose from radon ingestion (D_ing) and the annual effective dose to the stomach from radon ingestion (D_sto). As mentioned earlier, radon is soluble in water, and the degree of solubility of radon depends on various factors such as the temperature (T) and the pH of the water. For many years, researchers all over the world have been studying radon levels in thermal waters and how they change with physicochemical properties (like electrical conductivity (EC), pH, and T), as well as the health risks associated with them. However, no study has yet been carried out to predict the radiological risk parameters of thermal water using ANNs from physicochemical properties of thermal waters. As a result, this study is the first to explore the prediction of four radiological risk parameters by MLP and RBF ANNs. Two separate MLPANN models (MLPANN-1 and MLPANN-2) and RBFANN models (RBFANN-1 and RBFANN-2) were constructed for this purpose: (i) MLPANN-1 and RBFANN-1 models for predicting the D_E values for workers and visitors, and (ii) MLPANN-2 and RBFANN-2 models for predicting the D_ing and D_sto values. For this purpose, a total of 189 datasets, including the three physicochemical properties (EC, pH, and T) and C_Rn values of thermal waters, were collected from available published sources (Baştan [25]; Çetinkaya and Biçer [15]; Revan [26]; Tabar [27]; Tabar and Yakut [18]). Four radiological risk parameters (D_E values for workers and visitors, and D_ing and D_sto values) predicted by MLPANN and RBFANN models were compared with experimentally determined values to assess their accuracy in predicting radiological risk parameters. In addition, the predictive ability of MLPANN and RBFANN models was evaluated using various performance measures.

2. Research Methodology

The current study applies and evaluates models MLPANN and RBFANN to find optimal ones for estimating four radiological risk parameters (D_E values for workers and visitors, D_ing and D_sto values) of thermal waters. A total of 189 datasets were collected from available published sources for the creation, learning, and analysis of the models. The models were trained on randomly selected 80% of the data and then tested on the remaining 20%. Four performance indices: relative absolute error (RAE), root mean square error (RMSE), the ratio of RMSE to the standard deviation of the data (RSR), and mean absolute error (MAE) were used to evaluate the training and testing performance of the MLPANN and RBFANN models. The performance was further validated using rank analysis, and visual interpretation was provided using Taylor and scaled percentage error (SPE) graphs. The methodology of this study is shown in Figure 1.

2.1. Data Analysis

In the present work, MLP and RBF ANNs were implemented to predict four radiological risk parameters of thermal waters using their three physicochemical properties (T, EC, and pH). To do so, two different MLPANN and RBFANN models were constructed. A total of 189 datasets collected from available published sources were used to build, learn, and analyze the models. The variables taken into account in this study were T, EC, pH, and D_E for workers and visitors, D_ing and D_sto. In all MLPANN and RBFANN models, T, EC, and pH values of thermal waters were considered as input variables. In models MLPANN-1 and RBFANN-1, D_E values for workers and visitors were used as output parameters, while in models MLPANN-2 and RBFANN-2, D_ing and D_sto values were used as output parameters. The descriptive statistics for each of the parameters used in models MLPANN and RBFANN are presented in

Table 1. The descriptive statistical values of the parameters used in the MLPANN and RBFANN models.

	pH	T	EC	DE for Workers	DE for Visitors	Ding	Dsto
Parameters		(°C)	(mS/cm)	(μSv Year−1)	(μSv Year−1)	(μSv Year−1)	(μSv Year−1)
Category	Input	Input	Input	Output	Output	Output	Output
N total	189	189	189	189	189	189	189
Minimum	5.90	0.129	20.00	0.07992	0.00400	0.02331	0.00280
Maximum	9.00	16.290	98.00	22.32000	1.11600	6.51000	0.78120
Mean	7.25	3.562	44.70	3.11779	0.15589	0.90936	0.10912
Median	7.07	2.260	41.00	1.87200	0.09360	0.54600	0.06552
Standard deviation	0.72	3.45	20.20	3.90	0.19	1.14	0.14
Skewness	0.60	1.97	1.11	2.61	2.61	2.61	2.61
Kurtosis	−0.57	3.38	0.53	7.55	7.55	7.55	7.55

. Figure 2 illustrates the visual representation of the input and output variables in the models, including histograms.

The Pearson correlation coefficient (r) measures the strength of a linear correlation between the variables in question. Hair [28] suggested the following guide, showing the correlation between the variables according to the ranges of the r values:

$0.81\le \left|r\right|\le 1.00$	Very strong correlation between the variables;
$0.61\le \left|r\right|\le 0.80$	Strong correlation between the variables;
$0.41\le \left|r\right|\le 0.60$	Moderate correlation between the variables;
$0.21\le \left|r\right|\le 0.40$	Weak correlation between the variables;
$0.00\le \left|r\right|\le 0.20$	No correlation between the variables.

The r value computed for each input and output parameter used in this study is also shown in Figure 3. The r values of −0.43 for the input parameter pH in Figure 3 indicate a moderate relationship between pH and the four output variables, according to the guide suggested by Hair [28]. Furthermore, the r values of 0.22 for the input parameter EC in Figure 3 indicate a weak relationship between EC and the four output parameters, according to the guide suggested by Hair [28]. Moreover, r values of 0.027 for the input parameter T in Figure 3 indicate no relationship between T and the four output parameters, according to the guide suggested by Hair [28].

Preparation of the data is crucial for ensuring that all variables receive equal attention during training, resulting in faster learning outcomes [29]. In this study, as in previous ones (Erzin [30,31,32], Erzin and Yaprak [33], and Rabbani et al. [29,34,35]), a randomly selected 80% of the data was used to build and train the MLPANN and RBFANN models, and the remaining 20% was used to test the models. Normalizing both input and output variables is critical for improving the performance of soft computing (SC) models. According to the ranges shown in Table 1, all variables in the models were normalized between −0.9 and +0.9 by utilizing Equation (1), as employed by previous studies (Erzin [31,32] and Reddy et al. [36]). The minimum and maximum values are denoted by x_min and x_max in Equation (1), whereas the normalized and actual values are represented by x_norm and x.

x_norm = 1.8 × ((x − x_min)/(x_max − x_min)) − 0.9

(1)

2.2. Radiological Risk Assessment of Thermal Waters

Radiological risk assessment is the process of evaluating the dose and risk to humans from radioactive substances in the environment. Radon risk assessment is especially important because of its established health concerns, particularly as a cause of stomach and lung cancer [37,38]. For example, radon in water, if ingested or inhaled directly, delivers a radioactive dose to the stomach, which can cause a serious risk to the population.

The dose contribution (D_E) value from radon release in thermal water to air for workers and visitors is computed using Equation (2) [39].

D_E = C_Rn × F × R × DCF × T

(2)

where C_Rn is in Bq m⁻³; T is the exposure time in hours, estimated to be 2000 h year⁻¹ and 100 h year⁻¹ for workers and visitors, respectively; DCF is dose conversion factor for radon exposure, equal to 9 nSv h⁻¹ Bq⁻¹ m³; R is the ratio of radon in air to the radon in water, equal to 10⁻⁴; and F is the equilibrium factor between radon and its progenies, equal to 0.4 [39].

The annual effective dose from radon ingestion (D_ing) value is computed using Equation (3) [39].

D_ing = C_Rn × C_TWC × DCF

(3)

where C_Rn is in Bq L⁻¹, $C_{TWC}$ is the estimated thermal water consumption for a year (60 L year⁻¹), and $DCF$ is 3.5 × 10⁻⁹ Sv Bq⁻¹.

The annual effective dose to the stomach from radon ingestion (D_sto) has been assessed by the US National Research Council, and D_sto was found to be higher than in other tissues and organs [40]. The D_sto value in thermal water is computed using Equation (4) in this study.

D_sto = D_ing × W_T

(4)

where W_T is the weighting function for the stomach (0.12) [41].

2.3. Artificial Neural Networks (ANNs)

An ANN is a powerful computing system inspired by biological neural networks. Like the human cerebrum, ANNs can learn from examples, generalize, tolerate error, and respond intelligently to novel stimuli [42]. ANNs consist of two basic components: processing elements (neurons) and connections (weights). The former are used to process data, whereas the latter are employed to link neurons [43]. ANNs are now widely used for classification, clustering, pattern recognition, and prediction in a variety of disciplines [44]. Depending on variables such as architecture, function complexity, number of training instances, and training methods, there are numerous types of ANNs applicable to a particular problem [45]. The main difference between ANNs is the type of basis function of each ANN model [46]. The most commonly used ANN types are MLP and RBF ANNs, with the main difference being the way the neurons process the data [46]. MLP and RBF ANNs are universally applicable approximation methods to solve non-linear mapping problems [24], which are briefly described below.

2.3.1. Multilayer Perceptron Artificial Neural Networks (MLPANNs)

MLPANNs are widely employed in predictive applications and are characterized by their ability to differentiate data that is not easily separated by linearity. An MLPANN is composed of completely linked neurons with a non-linear activation function and is structured in at least three layers (an input layer, one or more hidden layers, and an output layer). Each layer is composed of many processing neurons, each of which is completely interconnected through weight links. Each input neuron represents a characteristic and its value. Each of the hidden neurons in a hidden layer takes the input from all the neurons in the preceding layer, performs a weighted sum, applies a bias, and then sends the result through a transfer function. The output layer, or the final layer in an ANN, generates the ANN’s final output based on the processed input from the hidden layers.

MLPANN’s efficiency is strongly affected by both the number of hidden layers and the number of hidden neurons in these layers [47]. The transfer function (TF) provides the primary benefit of enabling system analysis and design to be performed using straightforward algebraic equations as opposed to intricate differential equations. The choice of which type of TF to use is therefore crucial, as it generally affects the aim of the MLPANN model. The widely used TFs are pure linear, tan-sigmoid, and log-sigmoid [48].

MLPANN models learn from data and make predictions. Backpropagation (BP) learning algorithm is a widely applied learning technique for training MLPANNs [49]. BP offers various advantages, including ease of implementation, simplicity, and low computational complexity, etc. [50], making it a popular option for a wide range of machine learning applications. The training procedure begins with providing input data to MLPANNs. Neurons process input data and produce an output value. The error is calculated by subtracting the predicted value from the target value and is used to reweight the links. Until the error is reduced to an acceptable or minimal level, this process is repeated for many epochs. BP modifies the connection weights and bias values of the hidden and output layer neurons using a gradient descent technique, attempting to decrease errors. Following training, the relevant hidden and output layers’ neuron connection weights and bias values are stored in the memory of MLPANN. MLPANNs provide predictions for certain datasets that are not used during training, based on the weights recorded during the testing stage. The performance of the MLPANN can be evaluated by the difference between actual and predicted values.

2.3.2. Radial Basis Function Artificial Neural Networks (RBFANNs)

An RBFANN was a preferred alternative to an MLPANN because of its ability to handle arbitrarily distributed data, its rapid generalization to many spatial dimensions, and its spectral accuracy [51]. The RBFANN consists of input, hidden, and output layers. The input layer, which has the same number of neurons as the input variables, transmits data to the hidden layer. The hidden layer neurons compute the distances between the center and the inputs given [52]. RBF does non-linear mappings in the hidden layer using the provided data, and the results of the hidden neurons are sent to the output layer [52]. The basic formulation of the RBFANN is shown in the following [53].

$$y_i=\sum _{k=1}^N{W}_{ik}\times {\varnothing }_k\left(x,c_k\right)=\sum _{k=1}^N{W}_{ik}\times {\varnothing }_k\left({‖x-c_k‖}_2\right)\hspace{1em}\hspace{1em}i=1, 2, 3,..m$$

(5)

where N is the neurons’ number in the hidden layer; $x$ and y are the input and output; $W_{ik}$ represents the output layer’s weights; ${\mathrm{\varnothing }}_{\mathrm{k}}\left(.\right)$ is the RBF; and $c_k$ is the RB centers selected from the input vector space’s subset.

The RBFANN model’s major features include cell centers, output layer weights, and RBF form. Gaussian, cubic, linear, and multi-quadratic functions can be used in RBFANN models. The Gaussian function’s mathematical structure is represented by the following equation:

$${\mathrm{\varnothing }}_{\mathrm{k}}\left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{{‖\mathrm{x}-{\mathrm{c}}_{\mathrm{k}}‖}_2}^2/2{\mathsf{\sigma }}^2\right)$$

(6)

where σ is the standard deviation and ${‖.‖}_2$ is the Euclidean norm. The σ value, also known as the spread parameter, has a substantial impact on the RBFANN model’s performance and can be found via approximate equivalents from the literature or manually by trial and error [54].

3. Results and Discussions

3.1. Development of MLPANN Models

In this study, two MLPANN models (MLPANN-1 and MLPANN-2) were developed: the MLPANN-1 model for predicting the D_E values for workers and visitors of thermal waters, and the MLPANN-2 model for predicting the D_ing and D_sto values of thermal waters. For this purpose, data on three physicochemical parameters (T, EC, and pH) and C_Rn values of thermal waters, obtained from available published sources, were used. The C_Rn values of the thermal waters obtained were found to vary between 0.111 Bq L⁻¹ to 31 Bq L⁻¹. All of the C_Rn values of the thermal waters were found to be below the limit level of 100 Bq L⁻¹ suggested by both the World Health Organization (WHO) [55] and the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) [4]. Additionally, approximately 91% of the C_Rn values of thermal waters were found to be below the suggested limit value of 11 Bq L⁻¹ by the United States Environmental Protection Agency (USEPA) [56]. The D_E values for workers and visitors were then calculated using Equation (2), while the D_ing and D_sto values were calculated using Equations (3) and (4), respectively. The D_E values range from 0.080 μSv year⁻¹ to 22.320 μSv year⁻¹ for spa workers and from 0.004 μSv year⁻¹ to 1.116 μSv year⁻¹ for visitors. The D_ing and D_sto values range from 0.023 μSv year⁻¹ to 6.510 μSv year⁻¹ and from 0.003 μSv year⁻¹ to 0.781 μSv year⁻¹, respectively. The D_E values for both workers and visitors, as well as the D_ing and D_sto values, are found to be below the safe limit of 100 μSv year⁻¹ defined by WHO for all age groups. These results suggest that the thermal waters examined in this research do not pose significant radiological health risks due to radon.

The T, EC, and pH values were used as input parameters in both MLPANN models. The D_E values for workers and visitors were used as output parameters for the MLPANN-1 model, while the D_ing and D_sto values were used as output parameters for the MLPANN-2 model. There are an infinite number of ways to construct an MLPANN architecture. The search for the MLPANN architecture and its creation is not based on theory [57]. MLPANN design is the process of determining the MLPANN architecture, providing optimal results in any given task. A trial-and-error approach was used in this work to investigate the optimal design of each MLPANN model. The MLPANN models were trained using the Levenberg–Marquardt backpropagation algorithm.

When choosing the optimal architecture for MLPANNs, three important hyperparameters to consider are the number of hidden layers and of hidden neurons per hidden layer, and the type of TF [58]. The hyperparameters have a significant impact on the accuracy of MLPANN predictions [59]. The hidden layer number varies depending on the problem type and its level of difficulty [59]. The ANN model begins to overfit as the number of hidden layers increases [60]. As a result, this work developed one- and two-hidden-layer MLPANN models. It is also difficult to determine how many hidden neurons are in the hidden layer(s). The number of hidden neurons in each hidden layer (or layers) ranged from 1 to 8 in this study. To obtain the best structure for the MLPANN models during training and testing phases, the commonly used hyperbolic tangent sigmoid TF, called tansig, was used.

The performance of SC models is analyzed using various performance metrics [61]. The reliability of the constructed MLPANN models was evaluated and verified using four performance measures presented by Equations (7)–(10).

$$MAE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|y_i-{\widehat{y}}_i\right|$$

(7)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(8)

$$RSR={\displaystyle \frac{RMSE}{\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(y_i-\overline{y_i}\right)}^2}}}$$

(9)

$$RAE={\displaystyle \frac{{\sum }_{i=1}^N\left|y_i-{\widehat{y}}_i\right|}{{\sum }_{i=1}^N\left|y_i-\overline{y_i}\right|}}$$

(10)

where N is the number of samples; $y_i$ is a determined value; $\overline{y_i}$ is $y_i$ values’ mean value; ${\widehat{y}}_i$ is the predicted value. A SC model is deemed to be reliable if the values of MAE, RMSE, RSR, and RAE are approaching zero.

Details of the optimum performance are provided in

Table A1. Details of the optimal performance of the MLPANN-1 model in predicting the D_E values for workers of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
MLPANN-1_A	2.1457	3.4516	0.8867	0.8692	16	2.1934	3.2304	0.8514	0.6176	9
MLPANN-1_B	1.9668	3.1488	0.8089	0.7967	13	2.0849	3.1910	0.8410	0.5870	6
MLPANN-1_C	1.8891	3.1571	0.8111	0.7652	14	1.9682	3.3024	0.8704	0.5542	7
MLPANN-1_D	1.5873	2.8972	0.7443	0.6430	11	1.9065	3.1562	0.8319	0.5368	5
MLPANN-1_E	1.2688	1.9780	0.5082	0.5140	10	2.4096	3.6274	0.9561	0.6784	15
MLPANN-1_F	0.8838	1.2392	0.3183	0.3580	8	2.3257	3.5097	0.9250	0.6548	13
MLPANN-1_G	0.8326	1.1953	0.3071	0.3373	7	2.3707	3.6539	0.9630	0.6675	16
MLPANN-1_H	0.6772	1.0375	0.2665	0.2743	5	2.1858	3.7108	0.9780	0.6154	13
MLPANN-1_I	2.1414	3.4390	0.8835	0.8675	15	2.2112	3.2407	0.8541	0.6226	11
MLPANN-1_J	1.7540	2.8363	0.7286	0.7105	12	1.7831	2.7057	0.7131	0.5020	3
MLPANN-1_K	1.2035	1.7864	0.4589	0.4875	9	1.7522	3.1317	0.8254	0.4933	3
MLPANN-1_L	0.7353	1.0576	0.2717	0.2979	6	1.9565	3.4788	0.9169	0.5509	8
MLPANN-1_M	0.5214	0.8933	0.2295	0.2112	3	2.1756	3.4702	0.9146	0.6126	10
MLPANN-1_N	0.6135	0.9051	0.2325	0.2485	4	2.1340	3.7495	0.9882	0.6008	12
MLPANN-1_0	0.4502	0.7139	0.1834	0.1824	1	1.5654	2.2512	0.5933	0.4407	2
MLPANN-1_P	0.4191	0.7792	0.2002	0.1698	1	1.4653	2.0797	0.5481	0.4126	1

Bold values correspond to the best architectural model.

and

Table A2. Details of the optimal performance of the MLPANN-1 model in predicting the D_E values for visitors of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
MLPANN-1_A	0.1073	0.1726	0.8867	0.8692	16	0.1256	0.2050	0.9204	0.7546	15
MLPANN-1_B	0.0983	0.1574	0.8089	0.7966	13	0.1201	0.1917	0.8605	0.7219	12
MLPANN-1_C	0.0945	0.1579	0.8111	0.7651	13	0.1143	0.1949	0.8750	0.6869	12
MLPANN-1_D	0.0795	0.1425	0.7322	0.6440	11	0.1086	0.1935	0.8685	0.6526	11
MLPANN-1_E	0.0634	0.0989	0.5082	0.5139	10	0.1207	0.1818	0.8160	0.7256	10
MLPANN-1_F	0.0442	0.0620	0.3183	0.3580	8	0.1252	0.1902	0.8537	0.7523	14
MLPANN-1_G	0.0416	0.0598	0.3071	0.3372	7	0.1175	0.1810	0.8124	0.7061	9
MLPANN-1_H	0.0339	0.0519	0.2665	0.2743	5	0.0934	0.1372	0.6160	0.5614	4
MLPANN-1_I	0.1070	0.1720	0.8835	0.8668	15	0.1264	0.2048	0.9196	0.7597	16
MLPANN-1_J	0.0877	0.1418	0.7286	0.7104	12	0.1050	0.1826	0.8196	0.6313	8
MLPANN-1_K	0.0602	0.0893	0.4589	0.4875	9	0.1035	0.1848	0.8296	0.6220	7
MLPANN-1_L	0.0368	0.0529	0.2717	0.2978	6	0.0819	0.1211	0.5436	0.4925	2
MLPANN-1_M	0.0261	0.0447	0.2295	0.2112	3	0.1045	0.1659	0.7448	0.6281	6
MLPANN-1_N	0.0307	0.0453	0.2325	0.2485	4	0.0908	0.1579	0.7090	0.5458	5
MLPANN-1_0	0.0225	0.0357	0.1834	0.1824	1	0.0891	0.1387	0.6225	0.5357	3
MLPANN-1_P	0.0210	0.0390	0.2002	0.1698	1	0.0748	0.1082	0.4858	0.4492	1

Bold values correspond to the best architectural model.

for the MLPANN-1 model in predicting D_E values for workers and visitors of thermal waters, respectively, and in

Table A3. Details of the optimal performance of the MLPANN-2 model in predicting the D_ing values of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
MLPANN-2_A	0.6243	1.0012	0.8819	0.8671	16	0.6558	0.9564	0.8643	0.9343	15
MLPANN-2_B	0.5533	0.9023	0.7948	0.7684	14	0.5920	0.9186	0.8301	0.8435	6
MLPANN-2_C	0.5257	0.8707	0.7669	0.7301	12	0.5773	0.8971	0.8107	0.8224	3
MLPANN-2_D	0.3267	0.4705	0.4144	0.4538	10	0.5153	0.9839	0.8891	0.7341	7
MLPANN-2_E	0.3176	0.4313	0.3799	0.4411	8	0.6113	1.0081	0.9109	0.8709	16
MLPANN-2_F	0.2766	0.4050	0.3568	0.3841	7	0.6105	0.9989	0.9026	0.8698	13
MLPANN-2_G	0.2370	0.3366	0.2964	0.3292	6	0.5185	0.9815	0.8869	0.7387	8
MLPANN-2_H	0.2266	0.3270	0.2880	0.3147	5	0.5617	1.0445	0.9438	0.8003	11
MLPANN-2_I	0.6068	0.9979	0.8790	0.8427	15	0.6554	0.9553	0.8632	0.9337	12
MLPANN-2_J	0.5492	0.8909	0.7847	0.7628	13	0.5708	0.9279	0.8385	0.8132	4
MLPANN-2_K	0.3801	0.5856	0.5158	0.5280	11	0.5509	0.8967	0.8103	0.7849	2
MLPANN-2_L	0.3240	0.4578	0.4032	0.4501	9	0.5575	0.9420	0.8513	0.7943	5
MLPANN-2_M	0.1766	0.2934	0.2584	0.2452	4	0.6125	0.8325	0.7523	0.8727	9
MLPANN-2_N	0.1142	0.1803	0.1588	0.1586	1	0.4730	0.6418	0.5799	0.6738	1
MLPANN-2_0	0.1139	0.1890	0.1664	0.1581	1	0.5922	0.9379	0.8475	0.8438	10
MLPANN-2_P	0.1452	0.2401	0.2115	0.2017	3	0.6506	0.9640	0.8711	0.9269	14

Bold values correspond to the best architectural model.

and

Table A4. Details of the optimal performance of the MLPANN-2 model in predicting the D_sto values of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
MLPANN-2_A	0.0749	0.1201	0.8819	0.8670	16	0.0787	0.1148	0.8643	0.7718	15
MLPANN-2_B	0.0664	0.1083	0.7948	0.7683	14	0.0710	0.1102	0.8301	0.6968	6
MLPANN-2_C	0.0631	0.1045	0.7669	0.7302	12	0.0693	0.1077	0.8107	0.6794	4
MLPANN-2_D	0.0392	0.0565	0.4145	0.4537	10	0.0618	0.1181	0.8892	0.6065	7
MLPANN-2_E	0.0381	0.0518	0.3800	0.4413	8	0.0733	0.1210	0.9109	0.7194	16
MLPANN-2_F	0.0332	0.0486	0.3568	0.3842	7	0.0733	0.1199	0.9027	0.7186	13
MLPANN-2_G	0.0284	0.0404	0.2965	0.3293	6	0.0622	0.1178	0.8871	0.6104	8
MLPANN-2_H	0.0272	0.0392	0.2881	0.3147	5	0.0674	0.1253	0.9439	0.6612	11
MLPANN-2_I	0.0728	0.1198	0.8790	0.8426	15	0.0786	0.1146	0.8633	0.7713	12
MLPANN-2_J	0.0654	0.1065	0.7817	0.7568	13	0.0661	0.1084	0.8164	0.6488	3
MLPANN-2_K	0.0456	0.0703	0.5158	0.5278	11	0.0661	0.1076	0.8103	0.6483	2
MLPANN-2_L	0.0389	0.0549	0.4032	0.4500	9	0.0669	0.1130	0.8513	0.6562	5
MLPANN-2_M	0.0212	0.0352	0.2584	0.2452	4	0.0735	0.0999	0.7523	0.7209	9
MLPANN-2_N	0.0137	0.0216	0.1588	0.1586	1	0.0568	0.0770	0.5800	0.5568	1
MLPANN-2_0	0.0137	0.0227	0.1665	0.1581	1	0.0711	0.1126	0.8476	0.6971	10
MLPANN-2_P	0.0174	0.0288	0.2115	0.2016	3	0.0781	0.1157	0.8711	0.7657	13

Bold values correspond to the best architectural model.

for the MLPANN-2 model in predicting D_ing and D_sto values of thermal waters, respectively. The MLPANN-1_P model was chosen as the optimal MLPANN-1 model by analyzing each model’s rank values for both training and test datasets in Table A1 and Table A2, with two hidden layers and eight hidden neurons in the hidden layers (3-8-8-1 structure), a momentum factor of 0.001, and 99 epochs. The MLPANN-2_N model was chosen as the optimal MLPANN-2 model by analyzing each model’s rank values for both training and test datasets in Table A3 and Table A4, with two hidden layers and six hidden neurons in the hidden layers (3-6-6-1 structure), a momentum factor of 0.001, and 173 epochs.

As previously mentioned, the training data were randomly selected, and the remainder was used as the test data when developing the optimal MLPANN models. The 5-fold cross-validation technique was also implemented to enhance the robustness and credibility of the optimal MLPANN models. The predictive capability of the MLPANN models was therefore investigated using four folds of different training/testing datasets, and the errors are presented in

Table A5. Details of the performance of the MLPANN-1 model for different train/test samples in predicting the D_E values for workers of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.6399	0.9017	0.2343	0.2391	1.5725	2.2922	0.5778	0.5770
Fold 2	0.6434	0.9912	0.2566	0.2333	1.4576	2.2516	0.5759	0.6070
Fold 3	0.5806	0.9554	0.2293	0.1995	1.5379	2.9217	1.2344	0.8550
Fold 4	0.5075	0.7519	0.2105	0.1937	1.5239	3.0460	0.6189	0.5145
Optimal model	0.4191	0.7792	0.2002	0.1698	1.4653	2.0797	0.5481	0.4126

and

Table A6. Details of the performance of the MLPANN-1 model for different train/test samples in predicting the D_E values for visitors of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.0314	0.0445	0.2210	0.2397	0.0754	0.1079	0.5442	0.5533
Fold 2	0.0313	0.0484	0.2400	0.2315	0.0717	0.1120	0.5728	0.5974
Fold 3	0.0290	0.0478	0.2207	0.2031	0.0945	0.1685	1.4241	1.0500
Fold 4	0.0248	0.0376	0.2001	0.1932	0.0880	0.1631	0.6629	0.5941
Optimal model	0.0210	0.0390	0.2002	0.1698	0.0748	0.1082	0.4858	0.4492

for the MLPANN-1 model predicting D_E values for thermal water workers and visitors, respectively, and in

Table A7. Details of the performance of the MLPANN-2 model for different train/test samples in predicting the D_ing values of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.1184	0.1872	0.1659	0.1646	0.4805	0.6982	0.6159	0.5610
Fold 2	0.1237	0.1986	0.1754	0.1626	0.4875	0.8387	0.7377	0.6655
Fold 3	0.1315	0.2070	0.1705	0.1568	0.4168	0.7332	1.1648	1.2387
Fold 4	0.1164	0.1806	0.1727	0.1596	0.4762	0.8770	0.6108	0.5441
Optimal model	0.1142	0.1803	0.1588	0.1586	0.4730	0.6418	0.5799	0.6738

and

Table A8. Details of the performance of the MLPANN-2 model for different train/test samples in predicting the D_sto values of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.0142	0.0225	0.1659	0.1643	0.0576	0.0837	0.6154	0.5604
Fold 2	0.0148	0.0238	0.1754	0.1625	0.0584	0.1006	0.7371	0.6649
Fold 3	0.0158	0.0248	0.1706	0.1569	0.0500	0.0880	1.1648	1.2389
Fold 4	0.0140	0.0217	0.1727	0.1597	0.0571	0.1052	0.6105	0.5439
Optimal model	0.0137	0.0216	0.1588	0.1586	0.0568	0.0770	0.5800	0.5568

for the MLPANN-2 model predicting D_ing and D_sto values for thermal waters, respectively. It can be seen from Table A5, Table A6, Table A7 and Table A8 that all the models from fold 1 to fold 4 exhibit good prediction performance. Table A5, Table A6, Table A7 and Table A8 also indicate that both optimal MLPANN-1 and MLPANN-2 models achieve the best performance in predicting the radiological risks of thermal waters in Türkiye.

3.2. Development of RBFANN Models

Two different RBFANN models (RBFANN-1 and RBFANN-2) were also constructed in this paper: (i) RBFANN-1 for predicting D_E values for workers and visitors, and (ii) RBFANN-2 for predicting D_ing and D_sto values. To do this, the same datasets used to create MLPANN models were utilized to create RBFANN models. Both RBFANN models used T, EC, and pH values as input parameters, as did the MLPANN models. The models RBFANN-1 and RBFANN-2 have the same output parameters as the MLPANN-1 and MLPANN-2. The RBFANN-1 and RBFANN-2 models were built using the same training and test datasets as the MLPANN-1 and MLPANN-2 models. The neural network toolbox in MATLAB Version 7.5.0.342 (R2007b) was used to train and test both RBFANN models. The Gaussian function, a well-known RBF [62], was used to build each RBFANN model. As mentioned earlier, σ has a major impact on RBFANN models’ performance. The optimum σ value was found by increasing the value by 0.5 each time from 0.5 to 15 for each RBFANN model. Then, the RBFANN models’ performance was evaluated using the same four performance measures employed in the MLPANN models to determine the optimal RBFANN structure.

Table A9. Details of the optimal performance of the RBFANN-1 model in predicting the D_E values for workers of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
RBFANN-1_A	0.0001	0.0001	0.0000	0.0000	1	4036.448	21,032.545	5543.446	1385.7277	30
RBFANN-1_B	0.3489	0.7415	0.1905	0.1413	2	249.9901	1053.9425	277.7825	85.8225	29
RBFANN-1_C	0.6460	1.1675	0.2999	0.2617	3	71.1450	256.7383	67.6673	24.4243	28
RBFANN-1_D	0.8001	1.5304	0.3932	0.3241	4	16.3937	57.3411	15.1131	5.6280	27
RBFANN-1_E	0.9399	1.6806	0.4318	0.3807	5	11.0034	38.0199	10.0207	3.7775	26
RBFANN-1_F	0.9693	1.7734	0.4556	0.3927	6	4.5359	9.4355	2.4869	1.5572	25
RBFANN-1_G	1.0044	1.8285	0.4697	0.4069	7	3.5734	6.3450	1.6723	1.2268	24
RBFANN-1_H	0.9928	1.8465	0.4744	0.4022	8	2.9897	5.0147	1.3217	1.0264	23
RBFANN-1_I	1.0828	1.9178	0.4927	0.4386	9	2.8972	4.5579	1.2013	0.9946	20
RBFANN-1_J	1.0976	1.9247	0.4945	0.4446	10	3.0529	4.7836	1.2608	1.0481	22
RBFANN-1_K	1.1263	1.9504	0.5011	0.4562	11	2.8651	4.5511	1.1995	0.9836	19
RBFANN-1_L	1.1446	1.9791	0.5084	0.4637	12	2.6054	4.3277	1.1406	0.8944	9
RBFANN-1_M	1.1928	2.0181	0.5184	0.4832	13	2.8856	4.9392	1.3018	0.9907	21
RBFANN-1_N	1.2214	2.0357	0.5230	0.4948	14	2.6072	4.4726	1.1788	0.8951	16
RBFANN-1_O	1.2416	2.0857	0.5358	0.5029	15	2.5916	4.3297	1.1412	0.8897	8
RBFANN-1_P	1.2683	2.1013	0.5398	0.5138	16	2.7630	4.5019	1.1866	0.9485	18
RBFANN-1_R	1.3009	2.1467	0.5515	0.5270	17	2.5204	4.2968	1.1325	0.8652	6
RBFANN-1_S	1.3005	2.1476	0.5517	0.5268	18	2.5930	4.3365	1.1430	0.8902	10
RBFANN-1_T	1.3012	2.1477	0.5517	0.5271	19	2.5966	4.3385	1.1435	0.8914	12
RBFANN-1_U	1.3409	2.1575	0.5542	0.5432	21	2.5084	4.2725	1.1261	0.8612	2
RBFANN-1_V	1.3380	2.1580	0.5544	0.5420	20	2.5177	4.2775	1.1274	0.8643	3
RBFANN-1_W	1.3382	2.1582	0.5544	0.5421	21	2.5199	4.2784	1.1276	0.8651	4
RBFANN-1_X	1.3381	2.1584	0.5545	0.5420	23	2.5225	4.2794	1.1279	0.8660	5
RBFANN-1_Y	1.3387	2.1584	0.5545	0.5423	24	2.5267	4.2806	1.1282	0.8674	7
RBFANN-1_Z	1.3428	2.1652	0.5562	0.5440	27	2.6267	4.3331	1.1421	0.9018	13
RBFANN-1_AA	1.3423	2.1651	0.5562	0.5438	25	2.6289	4.3351	1.1426	0.9025	15
RBFANN-1_AB	1.3412	2.1654	0.5563	0.5433	26	2.6250	4.3338	1.1422	0.9012	14
RBFANN-1_AC	1.3444	2.1675	0.5568	0.5446	28	2.6629	4.3585	1.1488	0.9142	17
RBFANN-1_AD	1.3906	2.2051	0.5665	0.5633	29	2.5003	4.2499	1.1201	0.8584	1
RBFANN-1_AE	1.3771	2.2325	0.5735	0.5578	29	2.5897	4.3928	1.1578	0.8891	11

Bold values correspond to the best architectural model.

and

Table A10. Details of the optimal performance of the RBFANN-1 model in predicting the D_E values for visitors of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
RBFANN-1_A	0.0000	0.0000	0.0000	0.0000	1	201.838	1051.628	4720.877	1213.125	30
RBFANN-1_B	0.0174	0.0371	0.1905	0.1413	2	12.4836	52.6968	236.5618	75.0314	29
RBFANN-1_C	0.0323	0.0584	0.3000	0.2618	3	3.5731	12.8385	57.6337	21.4756	28
RBFANN-1_D	0.0401	0.0766	0.3934	0.3245	4	0.8275	2.8678	12.8741	4.9738	27
RBFANN-1_E	0.0470	0.0840	0.4317	0.3807	5	0.5648	1.9035	8.5448	3.3948	26
RBFANN-1_F	0.0485	0.0887	0.4555	0.3927	6	0.2427	0.4823	2.1649	1.4585	25
RBFANN-1_G	0.0502	0.0914	0.4698	0.4069	7	0.1939	0.3314	1.4877	1.1652	24
RBFANN-1_H	0.0496	0.0923	0.4743	0.4021	7	0.1644	0.2681	1.2033	0.9881	23
RBFANN-1_I	0.0541	0.0959	0.4927	0.4386	9	0.1608	0.2501	1.1227	0.9663	20
RBFANN-1_J	0.0549	0.0962	0.4945	0.4444	10	0.1685	0.2591	1.1632	1.0125	22
RBFANN-1_K	0.0563	0.0975	0.5011	0.4561	11	0.1591	0.2507	1.1253	0.9562	19
RBFANN-1_L	0.0572	0.0990	0.5085	0.4637	12	0.1461	0.2406	1.0802	0.8780	6
RBFANN-1_M	0.0596	0.1009	0.5185	0.4831	13	0.1601	0.2672	1.1993	0.9625	21
RBFANN-1_N	0.0611	0.1018	0.5230	0.4949	14	0.1462	0.2441	1.0959	0.8789	10
RBFANN-1_O	0.0621	0.1043	0.5357	0.5028	15	0.1454	0.2411	1.0824	0.8738	4
RBFANN-1_P	0.0634	0.1051	0.5398	0.5138	16	0.1540	0.2498	1.1213	0.9256	18
RBFANN-1_R	0.0650	0.1073	0.5515	0.5270	17	0.1419	0.2448	1.0991	0.8528	8
RBFANN-1_S	0.0650	0.1074	0.5517	0.5270	17	0.1456	0.2466	1.1069	0.8751	11
RBFANN-1_T	0.0650	0.1074	0.5517	0.5268	17	0.1456	0.2466	1.1069	0.8749	11
RBFANN-1_U	0.0670	0.1079	0.5543	0.5432	23	0.1413	0.2442	1.0962	0.8493	2
RBFANN-1_V	0.0669	0.1079	0.5544	0.5420	20	0.1417	0.2445	1.0977	0.8519	3
RBFANN-1_W	0.0669	0.1079	0.5544	0.5420	22	0.1419	0.2446	1.0979	0.8526	5
RBFANN-1_X	0.0669	0.1079	0.5545	0.5420	21	0.1420	0.2446	1.0982	0.8535	7
RBFANN-1_Y	0.0669	0.1079	0.5545	0.5424	24	0.1422	0.2447	1.0984	0.8545	9
RBFANN-1_Z	0.0671	0.1083	0.5562	0.5439	25	0.1472	0.2468	1.1077	0.8846	15
RBFANN-1_AA	0.0672	0.1083	0.5563	0.5441	27	0.1473	0.2468	1.1080	0.8854	16
RBFANN-1_AB	0.0670	0.1083	0.5564	0.5431	25	0.1470	0.2467	1.1076	0.8836	14
RBFANN-1_AC	0.0672	0.1084	0.5568	0.5444	28	0.1489	0.2477	1.1118	0.8952	17
RBFANN-1_AD	0.0695	0.1103	0.5665	0.5633	29	0.1409	0.2418	1.0857	0.8468	1
RBFANN-1_AE	0.0689	0.1116	0.5734	0.5579	29	0.1454	0.2487	1.1164	0.8738	13

Bold values correspond to the best architectural model.

give the RBFANN-1 model’s optimal performance in the prediction of D_E values for workers and visitors of thermal waters, respectively. Table A9 and Table A10 show that after comparing the rank values for the training and test data, the RBFANN-1_U model with a σ value of 10.0 was selected as the optimal RBFANN-1 model.

Table A11. Details of the optimal performance of the RBFANN-2 model in predicting the D_ing values for workers of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
RBFANN-2_A	0.0000	0.0000	0.0000	0.0000	1	1177.297	6134.492	5543.447	1385.728	30
RBFANN-2_B	0.1018	0.2163	0.1866	0.1413	2	72.9138	307.3999	277.7825	85.8225	29
RBFANN-2_C	0.1884	0.3405	0.2938	0.2617	3	20.7506	74.8820	67.6673	24.4243	28
RBFANN-2_D	0.2334	0.4464	0.3852	0.3241	4	4.7815	16.7245	15.1131	5.6280	27
RBFANN-2_E	0.2741	0.4902	0.4230	0.3807	5	3.2093	11.0891	10.0207	3.7775	26
RBFANN-2_F	0.2827	0.5172	0.4463	0.3927	6	1.3230	2.7520	2.4869	1.5572	25
RBFANN-2_G	0.2930	0.5333	0.4602	0.4069	7	1.0422	1.8506	1.6723	1.2268	24
RBFANN-2_H	0.2896	0.5386	0.4647	0.4022	8	0.8720	1.4626	1.3217	1.0264	23
RBFANN-2_I	0.3158	0.5594	0.4826	0.4386	9	0.8450	1.3294	1.2013	0.9946	20
RBFANN-2_J	0.3201	0.5614	0.4844	0.4446	10	0.8904	1.3952	1.2608	1.0481	22
RBFANN-2_K	0.3285	0.5689	0.4908	0.4562	11	0.8357	1.3274	1.1995	0.9836	19
RBFANN-2_L	0.3338	0.5772	0.4981	0.4637	12	0.7599	1.2622	1.1406	0.8944	9
RBFANN-2_M	0.3479	0.5886	0.5079	0.4832	13	0.8416	1.4406	1.3018	0.9907	21
RBFANN-2_N	0.3562	0.5937	0.5123	0.4948	14	0.7604	1.3045	1.1788	0.8951	16
RBFANN-2_O	0.3621	0.6083	0.5249	0.5029	15	0.7559	1.2628	1.1412	0.8897	8
RBFANN-2_P	0.3699	0.6129	0.5288	0.5138	16	0.8059	1.3131	1.1866	0.9485	18
RBFANN-2_R	0.3794	0.6261	0.5402	0.5270	17	0.7351	1.2532	1.1325	0.8652	6
RBFANN-2_S	0.3793	0.6264	0.5405	0.5268	17	0.7563	1.2648	1.1430	0.8902	10
RBFANN-2_T	0.3795	0.6264	0.5405	0.5271	19	0.7573	1.2654	1.1435	0.8914	12
RBFANN-2_U	0.3911	0.6293	0.5430	0.5432	21	0.7316	1.2461	1.1261	0.8612	2
RBFANN-2_V	0.3903	0.6294	0.5431	0.5420	20	0.7343	1.2476	1.1274	0.8643	3
RBFANN-2_W	0.3903	0.6295	0.5431	0.5421	21	0.7350	1.2479	1.1276	0.8651	4
RBFANN-2_X	0.3903	0.6295	0.5432	0.5420	23	0.7357	1.2482	1.1279	0.8660	5
RBFANN-2_Y	0.3904	0.6295	0.5432	0.5423	24	0.7370	1.2485	1.1282	0.8674	7
RBFANN-2_Z	0.3917	0.6315	0.5449	0.5440	27	0.7661	1.2638	1.1421	0.9018	13
RBFANN-2_AA	0.3915	0.6315	0.5449	0.5438	25	0.7668	1.2644	1.1426	0.9025	15
RBFANN-2_AB	0.3912	0.6316	0.5450	0.5433	26	0.7656	1.2640	1.1422	0.9012	14
RBFANN-2_AC	0.3921	0.6322	0.5455	0.5446	28	0.7767	1.2712	1.1488	0.9142	17
RBFANN-2_AD	0.4056	0.6432	0.5549	0.5633	29	0.7292	1.2396	1.1201	0.8584	1
RBFANN-2_AE	0.4016	0.6511	0.5618	0.5578	30	0.7553	1.2812	1.1578	0.8891	11

Bold values correspond to the best architectural model.

and

Table A12. Details of the optimal performance of the RBFANN-2 model in predicting the D_sto values for workers of thermal waters.

	Training					Testing
Model ID	MAE	RMSE	RSR	RAE	Rank	MAE	RMSE	RSR	RAE	Rank
RBFANN-2_A	0.0001	0.0001	0.0004	0.0007	1	141.288	736.203	5543.927	1385.848	30
RBFANN-2_B	0.0122	0.0260	0.1866	0.1413	2	8.7504	36.8912	277.8067	85.8298	29
RBFANN-2_C	0.0226	0.0409	0.2938	0.2617	3	2.4903	8.9866	67.6732	24.4264	28
RBFANN-2_D	0.0280	0.0536	0.3852	0.3241	4	0.5738	2.0071	15.1143	5.6284	27
RBFANN-2_E	0.0329	0.0588	0.4230	0.3807	5	0.3851	1.3308	10.0217	3.7777	26
RBFANN-2_F	0.0339	0.0621	0.4463	0.3927	6	0.1588	0.3303	2.4871	1.5572	25
RBFANN-2_G	0.0352	0.0640	0.4602	0.4069	7	0.1251	0.2221	1.6724	1.2268	24
RBFANN-2_H	0.0348	0.0646	0.4647	0.4022	7	0.1046	0.1755	1.3217	1.0264	23
RBFANN-2_I	0.0379	0.0671	0.4826	0.4386	9	0.1014	0.1595	1.2013	0.9946	20
RBFANN-2_J	0.0384	0.0674	0.4844	0.4446	10	0.1068	0.1674	1.2608	1.0480	22
RBFANN-2_K	0.0394	0.0683	0.4908	0.4562	11	0.1003	0.1593	1.1995	0.9835	19
RBFANN-2_L	0.0401	0.0693	0.4981	0.4637	12	0.0912	0.1515	1.1407	0.8944	9
RBFANN-2_M	0.0417	0.0706	0.5079	0.4832	13	0.1010	0.1729	1.3018	0.9906	21
RBFANN-2_N	0.0427	0.0712	0.5123	0.4947	14	0.0912	0.1565	1.1788	0.8949	16
RBFANN-2_O	0.0435	0.0730	0.5249	0.5029	15	0.0907	0.1515	1.1412	0.8895	8
RBFANN-2_P	0.0444	0.0735	0.5288	0.5138	16	0.0967	0.1576	1.1866	0.9484	18
RBFANN-2_R	0.0455	0.0751	0.5402	0.5269	17	0.0882	0.1504	1.1325	0.8651	6
RBFANN-2_S	0.0455	0.0752	0.5405	0.5268	18	0.0907	0.1518	1.1429	0.8900	10
RBFANN-2_T	0.0455	0.0752	0.5405	0.5270	19	0.0909	0.1518	1.1435	0.8913	12
RBFANN-2_U	0.0469	0.0755	0.5430	0.5431	21	0.0878	0.1495	1.1261	0.8610	2
RBFANN-2_V	0.0468	0.0755	0.5431	0.5420	20	0.0881	0.1497	1.1274	0.8642	3
RBFANN-2_W	0.0468	0.0755	0.5431	0.5420	21	0.0882	0.1497	1.1276	0.8650	4
RBFANN-2_X	0.0468	0.0755	0.5432	0.5420	23	0.0883	0.1498	1.1279	0.8659	5
RBFANN-2_Y	0.0468	0.0755	0.5432	0.5422	24	0.0884	0.1498	1.1282	0.8673	7
RBFANN-2_Z	0.0470	0.0758	0.5449	0.5439	27	0.0919	0.1517	1.1421	0.9017	13
RBFANN-2_AA	0.0470	0.0758	0.5449	0.5437	25	0.0920	0.1517	1.1426	0.9024	15
RBFANN-2_AB	0.0469	0.0758	0.5450	0.5432	26	0.0919	0.1517	1.1422	0.9011	14
RBFANN-2_AC	0.0470	0.0759	0.5455	0.5445	28	0.0932	0.1525	1.1488	0.9141	17
RBFANN-2_AD	0.0487	0.0772	0.5549	0.5632	29	0.0875	0.1487	1.1201	0.8582	1
RBFANN-2_AE	0.0482	0.0781	0.5618	0.5578	30	0.0906	0.1537	1.1578	0.8889	11

Bold values correspond to the best architectural model.

give the RBFANN-2 model’s optimal performance in predicting D_ing and D_sto values of thermal waters, respectively. Table A11 and Table A12 show that after comparing the rank values for the training and test data, the RBFANN-2_U model with a σ value of 10.0 was selected as the optimal RBFANN-2 model. Table A9, Table A10, Table A11 and Table A12 show that some RBFANN models have extremely low training errors but high testing errors (e.g., RBFANN-1_A model developed with a σ value of 0.5 (see Table A9): MAE = 0.0001, RMSE = 0.0001, RSR = 0.0000, and RAE = 0.0000 for training samples and MAE = 4036.448, RMSE = 21,032.545, RSR = 5543.446, and RAE = 1385.7277 for testing samples), indicating severe overfitting. These results demonstrate that the sensitivity of RBFANNs to the σ value and their tendency to create overly complex models that memorize noise in the training data, instead of learning the underlying pattern.

As in the MLPANN models, the training data was randomly selected, with the remainder is used as the testing data, when developing the optimal RBFANN models. The 5-fold cross-validation technique was also implemented to enhance the robustness and credibility of the optimal RBFANN models. The predictive capability of the RBFANN models was therefore investigated using four folds of different training/testing datasets, and the errors are presented in

Table A13. Details of the performance of the RBFANN-1 model for different train/test samples in predicting the D_E values for workers of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	1.3162	2.2875	0.5911	0.5335	3.8637	6.5414	1.6829	1.3155
Fold 2	1.4464	2.3694	0.6103	0.5546	2.6127	4.9820	1.2781	1.0404
Fold 3	1.5440	2.4824	0.5957	0.5306	2.7912	4.8087	2.0317	1.5517
Fold 4	1.5339	2.4437	0.6814	0.6132	2.7341	4.0143	0.8159	0.9522
Optimal model	1.3409	2.1575	0.5542	0.5432	2.5084	4.2725	1.1261	0.8612

and

Table A14. Details of the performance of the RBFANN-1 model for different train/test samples in predicting the D_E values for visitors of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.0745	0.1219	0.6002	0.5755	0.1988	0.3711	1.9096	1.3534
Fold 2	0.0787	0.1241	0.6095	0.5775	0.1484	0.2908	1.4875	1.1820
Fold 3	0.0814	0.1355	0.6259	0.5707	0.1417	0.2368	2.0011	1.5755
Fold 4	0.0815	0.1320	0.6971	0.6230	0.1522	0.2312	0.9399	1.0601
Optimal model	0.0670	0.1079	0.5543	0.5432	0.1413	0.2442	1.0962	0.8493

for the RBFANN-1 model in predicting D_E values for thermal water workers and visitors, respectively, and presented in

Table A15. Details of the performance of the RBFANN-2 model for different train/test samples in predicting the D_ing values of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.4026	0.6804	0.6029	0.5594	1.0701	1.7585	1.5511	1.2492
Fold 2	0.4198	0.6776	0.5984	0.5519	0.7502	1.4673	1.2905	1.0242
Fold 3	0.4503	0.7241	0.5964	0.5369	0.8129	1.3984	2.2215	2.4159
Fold 4	0.4474	0.7127	0.6814	0.6132	0.8751	1.2735	0.8868	0.9999
Optimal model	0.3911	0.6293	0.5430	0.5432	0.7316	1.2461	1.1261	0.8612

and

Table A16. Details of the performance of the RBFANN-2 model for different train/test samples in predicting the D_sto values of thermal waters.

	Training				Testing
Model ID	MAE	RMSE	RSR	RAE	MAE	RMSE	RSR	RAE
Fold 1	0.0483	0.0817	0.6029	0.5594	0.1284	0.2110	1.5512	1.2494
Fold 2	0.0504	0.0813	0.5984	0.5519	0.0900	0.1761	1.2907	1.0242
Fold 3	0.0540	0.0869	0.5964	0.5368	0.0975	0.1678	2.2215	2.4159
Fold 4	0.0537	0.0855	0.6814	0.6132	0.1050	0.1528	0.8868	1.0000
Optimal model	0.0469	0.0755	0.5430	0.5431	0.0878	0.1495	1.1261	0.8610

for the RBFANN-2 model in predicting D_ing and D_sto values of thermal waters, respectively. It can be seen from Table A13, Table A14, Table A15 and Table A16 that all the models from fold 1 to fold 4 exhibit good prediction performance. Table A13, Table A14, Table A15 and Table A16 further indicate that both optimal RBFANN-1 and RBFANN-2 models achieve the best performance in predicting the radiological risks of thermal waters in Türkiye.

3.3. Simulation of Results

Figure 4 and Figure 5 show a comparison of determined versus predicted D_E values from the MLPANN-1 model for workers and visitors, respectively. As shown in Figure 4, there was a remarkable similarity between the predicted D_E and the given D_E in both samples of training and test, with their r values of 0.987 and 0.945. Figure 5 indicates the high correspondence of predicted D_E values of visitors to determined values in both samples of training and test, with r values of 0.987 and 0.938. A comparison of determined versus predicted D_ing and D_sto values from the MLPANN-2 model is illustrated in Figure 6 and Figure 7. As illustrated in Figure 6 and Figure 7, the MLPANN-2 model demonstrated a high degree of accuracy in predicting the D_ing and D_sto values in both samples of training and test, with r values of 0.992 and 0.935.

Figure 8 and Figure 9 show a comparison of determined and predicted D_E values for workers and visitors of thermal waters using the RBFANN-1 model. These figures indicate that the determined and predicted D_E values for workers and visitors exhibited strong agreement for the samples of training, with an r value of 0.897. However, this agreement was not observed for workers and visitors in the samples tested, with r values of 0.724 and 0.672. Figure 10 and Figure 11 show a comparison of determined and predicted D_ing and D_sto values of thermal waters using the RBFANN-2 model. These figures show that the D_ing and D_sto values for the training samples are quite similar (r = 0.897), but not for the test samples (r = 0.724).

As illustrated in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, the MLPANN-1 and MLPANN-2 models yielded r values very close to 1 for both training and test samples, signifying an extremely strong correlation between the experimentally determined and predicted values of the radiological risk parameters of both developed MLPANN models. In other words, the determined and predicted values of four radiological risk parameters do not differ significantly.

In reality, the r value is a significant measure of the predictive ability of SC models. The performance of the models designed in this work was also assessed by four performance measures (MAE, RMSE, RSR, and RAE). The computed performance measures are presented in

Table 2. Performance indices of the best MLPANN and RBFANN models developed.

Model	Health Risk Parameter	Data	MAE	RMSE	RSR	RAE
MLPANN-1	DE for workers	Training set	0.4191	0.7792	0.2002	0.1698
		Testing set	1.4653	2.0797	0.5481	0.4126
	DE for visitors	Training set	0.0210	0.0390	0.2002	0.1698
		Testing set	0.0748	0.1082	0.4858	0.4492
MLPANN-2	Ding	Training set	0.1142	0.1803	0.1588	0.1586
		Testing set	0.4730	0.6418	0.5799	0.6738
	Dsto	Training set	0.0137	0.0216	0.1588	0.1586
		Testing set	0.0568	0.0770	0.5800	0.5568
RBFANN-1	DE for workers	Training set	1.3409	2.1575	0.5542	0.5432
		Testing set	2.5084	4.2725	1.1261	0.8612
	DE for visitors	Training set	0.0670	0.1079	0.5543	0.5432
		Testing set	0.1413	0.2442	1.0962	0.8493
RBFANN-2	Ding	Training set	0.3911	0.6293	0.5430	0.5432
		Testing set	0.7316	1.2461	1.1261	0.8612
	Dsto	Training set	0.0469	0.0755	0.5430	0.5431
		Testing set	0.0878	0.1495	1.1261	0.8610

and also illustrated in Figure 12 and Figure 13. As shown in Figure 3, there is little to no linear correlation between the input variables (EC, pH, and T) and the outputs used in the models, according to the guide suggested by Hair [28]. The results presented in Table 2 and illustrated in Figure 12 and Figure 13 justify the use of ANNs, as they are capable of modeling complex, non-linear relationships that linear correlation coefficients cannot capture.

As shown in Figure 12, the MLPANN-1 model achieved better predictive performance in predicting D_E values for both workers and visitors of thermal waters for both training and test samples compared to the RBFANN-1 model. Figure 13 indicates that the MLPANN-2 model outperformed the RBFANN-2 model in predicting both D_ing and D_sto values of thermal waters for training and test samples. These results suggest that MLPANNs are better suited to predicting four radiological risk parameters of thermal waters in Türkiye than RBFANNs. This is due to the hierarchical feature learning of MLPANNs being more effective at capturing the complex interactions between pH, T, and EC that govern radon release than the localized approximations of RBFANNs. The performance indices for the best MLPANN models also show that four radiological risk parameters of thermal waters in Türkiye can be predicted reliably and quickly using the developed MLPANN models, as long as the physicochemical properties (EC, pH, and T) of the thermal water are known.

3.4. Rank Analysis

Rank analysis is an easy and popular way to evaluate the performance of SC models [63,64]. SC models are scored on a variety of performance criteria during the training and testing phases [29]. The best-performing model is ranked highest; the worst-performing model is ranked lowest; and then the model with the highest score is selected as the best [65]. In the present study, various performance parameters were employed for determining scores, with each performance parameter’s optimal value serving as a reference (see

Table 3. Best values of performance parameters.

Performance Indices	Best Value
MAE	0
RMSE	0
RSR	0
RAE	0

).

Table 4. Rank analysis of the optimal MLPANN-1 and RBFANN-1 models.

		Training				Testing
		DE for Workers	DE for Workers	DE for Visitors	DE for Visitors	DE for Workers	DE for Workers	DE for Visitors	DE for Visitors
		MLPANN-1	RBFANN-1	MLPANN-1	RBFANN-1	MLPANN-1	RBFANN-1	MLPANN-1	RBFANN-1
MAE	Value	0.4191	1.3409	0.0210	0.0670	1.4653	2.5084	0.0748	0.1413
	Score	2	1	2	1	2	1	2	1
RMSE	Value	0.7792	2.1575	0.0390	0.1079	2.0797	4.2725	0.1082	0.2442
	Score	2	1	2	1	2	1	2	1
RSR	Value	0.2002	0.5542	0.2002	0.5543	0.5481	1.1261	0.4858	1.0962
	Score	2	1	2	1	2	1	2	1
RAE	Value	0.1698	0.5432	0.1698	0.5432	0.4126	0.8612	0.4492	0.8493
	Score	2	1	2	1	2	1	2	1
Total		8	4	8	4	8	4	8	4

and

Table 5. Rank analysis of the optimal MLPANN-2 and RBFANN-2 models.

		Training				Testing
		Ding	Ding	Dsto	Dsto	Ding	Ding	Dsto	Dsto
		MLPANN-2	RBFANN-2	MLPANN-2	RBFANN-2	MLPANN-2	RBFANN-2	MLPANN-2	RBFANN-2
MAE	Value	0.1142	0.3911	0.0137	0.0469	0.4730	0.7316	0.0568	0.0878
	Score	2	1	2	1	2	1	2	1
RMSE	Value	0.1803	0.6293	0.0216	0.0755	0.6418	1.2461	0.0770	0.1495
	Score	2	1	2	1	2	1	2	1
RSR	Value	0.1588	0.5430	0.1588	0.5430	0.5799	1.1261	0.5800	1.1261
	Score	2	1	2	1	2	1	2	1
RAE	Value	0.1586	0.5432	0.1586	0.5431	0.6738	0.8612	0.5568	0.8610
	Score	2	1	2	1	2	1	2	1
Total		8	4	8	4	8	4	8	4

show the MLPANN-1 and RBFANN-1 model rankings and the MLPANN-2 and RBFANN-2 model rankings, respectively, based on test and training performance metrics. As only two models were used, the highest score for a model in these tables was 2. Table 4 shows that the MLPANN-1 model scored higher (8 points) than the RBFANN-1 model (4 points) in predicting the D_E values for both workers and visitors of thermal waters in both the training and testing stages, which indicates that the MLPANN-1 model outperforms the RBFANN-1 model in terms of prediction accuracy. Table 5 shows that the MLPANN-2 model scored higher (8 points) than the RBFANN-2 model (4 points) in predicting both the D_ing and D_sto values of thermal waters in both the training and testing stages, which indicates that the MLPANN-2 model outperforms the RBFANN-2 model in prediction accuracy.

3.5. Taylor Graph

A simple visual representation of how SC model predictions relate to observations is also provided by Taylor graphs [64]. The graphs provide several statistical measures, such as r, centered RMS difference, and standard deviation ratio, to statistically compare the SC models’ performance [65]. The performance and competence of the proposed SC models are assessed against a reference value by using Taylor graphs [65]. The SC model whose position is most proximate to the reference point is selected as the optimum SC model [66]. Figure 14 and Figure 15 show Taylor graphs of the models MLPANN-1 and RBFANN-1 for predicting D_E values for workers and visitors of thermal waters throughout the training and testing stages, respectively. Figure 14 and Figure 15 also demonstrate that the MLPANN-1 model’s marker positions are nearer to the “reference” point than the RBFANN-1 model’s positions throughout the training and testing stages, indicating that the MLPANN-1 model predicts D_E values more accurately for both thermal water workers and visitors.

Figure 16 and Figure 17 illustrate Taylor graphs of models MLPANN-2 and RBFANN-2 for predicting the D_ing and D_sto values of thermal waters throughout the training and testing stages, respectively. Figure 16 and Figure 17 also show that the MLPANN-2 model’s marker positions are nearer to the “reference” point than the RBFANN-2 model’s positions throughout the training and test stages, implying that the MLPANN-2 model predicts both D_ing and D_sto values more accurately.

3.6. Scaled Percent Error Graph

The SPE values were calculated using Equation (11) to more comprehensively validate the predictive potential of the MLPANN and RBFANN models, used by Erzin [32] and Erzin and Ecemis [67].

$$\mathrm{SPE}={\displaystyle \frac{\left({\widehat{y}}_i-y_i\right)}{\left({y_i}_{max}-{y_i}_{min}\right)}}$$

(11)

where ${y_i}_{max}$ and ${y_i}_{min}$ are the measured values’ maximum and minimum values.

Figure 18 and Figure 19 show the SPE values computed for the D_E values of workers and visitors of thermal waters in the MLPANN-1 and RBFANN-1 models, respectively, plotted against cumulative frequency. Figure 18 illustrates that about 93% and 80% of the D_E values for workers of thermal water predicted by the MLPANN-1 and RBFANN-1 models, respectively, are within ±10% of the SPE. Figure 19 demonstrates that about 92% and 77% of the D_E values for visitors of thermal waters predicted by the MLPANN-1 and RBFANN-1 models, respectively, are within ±10% of the SPE. The SPE values of the MLPANN-1 and RBFANN-1 models (Figure 18 and Figure 19) indicate that the MLPANN-1 model performed better than the RBFANN-1 model in predicting D_E values for both workers and visitors of thermal waters.

Figure 20 and Figure 21 show the SPE values computed for the D_ing and D_sto values of thermal waters in the MLPANN-2 and RBFANN-2 models, respectively, plotted against cumulative frequency. Figure 20 illustrates that approximately 92% and 81% of the D_ing values of thermal waters predicted by the MLPANN-2 and RBFANN-2 models, respectively, are within ±10% of the SPE. Figure 21 demonstrates that about 92% and 80% of the D_sto values of thermal waters predicted by the MLPANN-2 and RBFANN-2 models, respectively, are within ±10% of the SPE. The SPE values for the MLPANN-2 and RBFANN-2 models (Figure 20 and Figure 21) indicate that the MLPANN-2 model outperformed the RBFANN-2 model in predicting both D_ing and D_sto values of thermal waters.

3.7. Sensitivity Analysis

A sensitivity analysis determines the effectiveness of input parameters in predicting output parameters. The parameters having the most significant impact on the prediction can be determined using this analysis [68]. Several researchers have used the cosine amplitude method for sensitivity analysis [68,69,70,71,72]. The mathematical formula for the cosine amplitude method is given by the equation below [73].

$$r_{ij}={\displaystyle \frac{{\sum }_{k=1}^n{X}_{ik}\times X_{jk}}{\sqrt{{\sum }_{k=1}^n{X}_{ik}^2}\times {\sum }_{k=1}^n{X}_{jk}^2}}$$

(12)

where X_jk and X_ik represent output and input parameters; n represents the data values’ number; and r_ij shows how input parameters impact the model output. The r_ij values vary from 0 to 1. If the output parameter is not related to the input parameter, the r_ij value will be zero. A larger r_ij value indicates that the input parameter has a greater effect on the output parameter.

The r_ij values for all input parameters are presented in Figure 22 for the MLPANN-1 and RBFANN-1 models and in Figure 23 for the MLPANN-2 and RBFANN-2 models. As shown in Figure 22 and Figure 23, the r_ij values for all input parameters are greater than 0.5. This suggests that all three input parameters influence the prediction of radiological risk for thermal waters in Türkiye, albeit to different extents, when using the MLP and RBF ANN models. Figure 22 shows that pH with r_ij values of 0.589 and 0.640 for the MLPANN-1 and RBFANN-1 models, respectively, ranks the highest, indicating that pH has the most significant input parameter affecting the prediction of the D_E values for workers and visitors of thermal waters from the MLPANN-1 and RBFANN-1 models. Figure 23 shows that pH with r_ij values of 0.578 and 0.640 for the MLPANN-2 and RBFANN-2 models, respectively, ranks the highest, indicating that pH has the most significant input parameter affecting the prediction of the D_ing and D_sto values of thermal waters from the MLPANN-2 and RBFANN-2 models.

4. Conclusions

The current research investigates the performance of MLPANN and RBFANN models in estimating four radiological risk parameters (D_E values for workers and visitors, D_ing and D_sto values) of thermal waters in Türkiye. To accomplish this, two distinct MLPANN models (MLPANN-1 and MLPANN-2) and RBFANN models (RBFANN-1 and RBFANN-2) were created using 151 training and 38 test datasets. The MAE, RMSE, RSR, and RAE statistics were then used to verify and evaluate the generated MLPANN and RBFANN models. Performance was also confirmed by rank analysis and visually using Taylor and SPE plots.

The research’s crisp conclusions are the following:

• During the training and testing phases, the MLPANN models produced a higher correlation (r values close to 1) between the determined and predicted four radiological risk parameters of thermal waters than the RBFANN models, indicating that the MLPANN models predicted four radiological risk parameters more accurately.
• MLPANN models had lower MAE, RMSE, RSR, and RAE values than RBFANN models, demonstrating that the MLPANN models outperformed the RBFANN models in predicting four radiological risk parameters of thermal waters.
• The MLPANN and RBFANN models′ rank analysis showed that MLPANN models had higher scores, signifying that MLPANN models achieved better prediction accuracy in the prediction of four radiological risk parameters of thermal waters than RBFANN models
• Taylor and SPE graphs showed that MLPANN models predicted four radiological risk parameters of thermal waters more precisely than RBFANN models.
• Four radiological risk parameters of thermal waters can be predicted reliably and quickly using the developed MLPANN models, as long as the physicochemical properties (EC, pH, and T) of thermal waters are known.

This study used MLPANN and RBFANN models to predict the radiological risks associated with thermal waters in Türkiye. As all the data used to develop the models in this study were collected from Turkish springs, the models’ generalizability to other geological settings may be limited. In future work, (a) geological data could be incorporated into the models, which could then be tested on international datasets using systematic hyperparameter optimization techniques such as grid search, random search, and Bayesian optimization, as well as cross-validation techniques. (b) If spatial data between springs were available, techniques such as Graph Neural Networks (GNNs) could be employed; and (c) ensemble methods (e.g., stacking MLP and RBF models) could potentially yield even better performance.