Assessing environmental influences on radon levels: analysis of independent variables

3.1. Unsupervised Bayesian multiple regression analysis

Unsupervised Bayesian Multiple Regression Analysis is a novel approach which introduced for the predictive analysis problem to address the issue of radon concentration due to the complex interactions between several factors, including different house surroundings, ventilation systems, etc. to determine the precise impacts of each predictor variable separately, therefore Unsupervised Bayesian Multiple Regression Analysis has been implemented to solve the predictive analysis problem. The proposed approach was the combination of two existing techniques namely Unsupervised Bayesian analysis and Universal multiple regression analysis which will be explained detailed in below.

Unsupervised Bayesian analysis is a statistical technique used to analyze data without labeled outcomes. It involves incorporating existing information or assumptions about relationships among variables, such as correlations between factors. The goal of unsupervised learning is to identify hidden patterns or structures in the data without predefined labels or instructions. Bayesian analysis updates probabilities as new data is observed, refining the understanding of the data over time. It also manages uncertainty by providing a range of probable values, allowing for better understanding of predictions and conclusions. The Bayes Theorem is the core of Bayesian analysis, which mathematically combines prior information with new data to update the probability of a hypothesis. Bayesian methods are well-suited for managing complex data interactions, as they model these interactions more accurately than traditional methods. Furthermore, by incorporating prior knowledge and updating probabilities with new data, Bayesian analysis becomes more robust, providing reliable results even in complex and noisy data. It connects the posterior distribution of the parameters $\left(P\right(\theta \left|X\right))$ to the prior distribution of the parameters $\left(P\right(\theta \left)\right)$ and the probabilities of the data given the parameters $\left(P\right(X\left|\theta \right))$, which shown in Eq. (1):

The combination of the probabilities and parameter priors yields the posterior for a specific model, which implemented in both batch and online versions, which shown in Eq. (2) and Eq. (3):

Eq. (2) refers to batch processing which handling data in large sets, often all at once, and is well-suited for offline analysis scenarios where the complete dataset is accessible for processing:

Eq. (3) refers to online processing which involves the incremental handling of data, either in real-time or in small batches, which is ideal for scenarios involving streaming or continuous data streams.

Batch processing and Online processing aims to compute the predictive posterior distribution by combining prior knowledge and updating probabilities given in Eq. (4):

where: $P\left(\theta \right|X)$ is the posterior distribution, representing the updated beliefs about the parameters given the observed data $X$; $P\left(X\right|\theta )$ is the probability, indicating the probability of observing the data $X$ given a particular set of parameters $\theta$; $P\left(\theta \right)$ is the prior distribution, representing the prior beliefs about the parameters before observing the data; $P\left(X\right)$ is the marginal probability, serving as a normalizing constant ensuring that the posterior distribution integrates to 1; data set $D$, and a model m with parameters $\theta$; prior over model parameters: $p\left(\theta \right|m)$; probability of model parameters for data set $D:p\left(D\right|\theta ,m)$; prior over model class: $p\left(m\right)$.

A posterior distribution $\left(P\right(\theta \left|X\right))$ representing updated views about the parameters, given the observed data $X$ is the output of Unsupervised Bayesian analysis. According to Bayes’ Theorem, this posterior distribution captures the integration of prior beliefs $\left(P\right(\theta \left)\right)$ with the probabilities of the data given the parameters $\left(P\right(X\left|\theta \right)).$ The posterior distribution offers a thorough and current understanding of the underlying parameters by combining the information from previous studies with the new signal found in the data. The foundation of Bayesian inference is this integration process, which allows for a methodical approach to revealing hidden structures, patterns, or parameters in unsupervised learning settings. Moreover, this output provides better to Universal multiple regression analysis which is explained below.

Furthermore, Universal multiple regression analysis aims to represent the connection between one independent variable and one dependent variable. It is an extension of the notion of basic linear regression. It utilizes multiple explanatory factors to predict the value of a response variable, thereby enhancing predictive power, especially crucial in navigating the complexities of air pollution studies where numerous variables influence indoor radon levels. By modeling the linear connections between multiple explanatory factors and the response variable, it captures intricate real-world relationships essential for accurate predictions. This method extends ordinary least-squares regression to incorporate multiple predictors, providing flexibility in analyzing air pollution data and adapting to intricate complexities. Additionally, it offers quantitative assessments of each variable's contribution to the variance in indoor radon levels, facilitating informed decision-making and prioritizing interventions. Its robustness to multicollinearity ensures reliable coefficient estimates even with highly correlated predictor variables, enhancing model reliability. An equation for multiple linear regression takes the following generic form in Eq (5):

where: $Y$ is the dependent variable (e.g., indoor radon concentration); ${\beta }_0$ is the intercept; ${\beta }_1$, ${\beta }_2$,…, ${\beta }_n$ are the coefficients for the independent variables $X_1$, $X_2$,…, $X_n$; $\epsilon$ is the error term.

The output of universal multiple regression analysis is a predictive model represented by Eq. (5), where the coefficients $({\beta }_0,{\beta }_1,\dots ,{\beta }_n)$ are determined based on the observed data. This model provides predictions $\left(Y\right)$ for the dependent variable (e.g., indoor radon concentration) using the values of the independent variables $(X_1,X_2,\dots ,X_n)$. The output of universal multiple regression analysis improves predictive analysis by providing a robust model that considers multiple factors, offers insights into variable contributions.

Unsupervised Bayesian Multiple Regression Analysis combines the strengths of unsupervised Bayesian analysis and universal multiple regression. It enhances predictive analysis by providing insights into the contributions of each independent variable, reducing uncertainty in parameter estimate. This approach combines the modeling capabilities of universal multiple regression with the probabilistic output of unsupervised Bayesian analysis, addressing challenges in predictive analysis. The outcome is a sophisticated prediction model that considers uncertainties, combines prior information, and raises an intricate understanding of data correlations. This comprehensive approach produces a powerful tool for predictive modeling, offering robust solutions to complex analytical problems by computing the posterior distribution of the regression coefficients $\beta$ and the error term $ϵ$given the observed data and the model given as Eq. (6):

where, $p(\beta ,ϵ\mid Y,X,D,m)$ is the posterior distribution of the regression coefficients $\beta$ and the error term $ϵ$ given the observed data $Y$, the independent variables $X$, the observed data $D$, and the model $m$; $p\left(Y\right|\beta ,ϵ,X,D,m)$is the probability function representing the probability of observing the dependent variable $Y$; $p(\beta ,ϵ|D,m)$ is the prior distribution representing the regression coefficients $\beta$ and the error term $ϵ$ before observing the data $D$ and the model $m$; $p\left(Y\right|X,D,m)$ is the marginal probabilities representing the probability of observing the data $Y$ given the independent variables $X$, the observed data $D$, and the model $m$.

This combines the prior knowledge and updating probabilities using Bayes theorem to obtain the posterior distribution, which represents the updated beliefs about the regression coefficients and the error term after observing the data.

Algorithm 1 displays the pseudocode of Unsupervised Bayesian Multiple Regression Analysis.

Algorithm 1: Unsupervised Bayesian Multiple Regression Analysis.

Input:

A dataset with observations of the dependent variable $\left(Y\right)$ and multiple independent variables $(X_1,X_2,\dots ,X_n)$.

Output:

Coefficients $({\beta }_0,{\beta }_1,\dots ,{\beta }_n)$ representing the linear relationship between the dependent variable and independent variables.

Procedure:

Step 1-Initialize Prior for Bayesian Analysis:

Set initial beliefs about the distribution of parameters based on prior knowledge (prior distribution).

Step 2-Universal Multiple Regression Analysis:

Utilize the provided algorithm for Universal Multiple Regression Analysis using the training data to obtain coefficients $({\beta }_0,{\beta }_1,\dots ,{\beta }_n)$.

Step 3- Probability Calculation:

Assess the probability of observing the data given the parameters using the obtained coefficients from the regression analysis.

Step 4-Prior and Probability Combination:

Combine the prior distribution and probabilities to obtain the unnormalized posterior distribution.

Step 5-Normalization:

Normalize the unnormalized posterior distribution to obtain the posterior distribution.

Step 6-Update Beliefs:

Use the posterior distribution as the updated beliefs about the parameters.

End

Fig. 2Flowchart of unsupervised Bayesian multiple regression analysis

The benefits of both universal multiple regression analysis and unsupervised Bayesian analysis are combined in the Unsupervised Bayesian Multiple Regression Analysis method. It accepts previous knowledge about parameter distributions and training data with observations of dependent and independent variables as input. The approach first initializes the regression coefficients, then iteratively optimizes them by adding observed data and use Bayesian analysis to modify them. A probabilistic representation of the parameters is the result, which offers a more complex understanding of the correlations between the variables. Through the integration of Bayesian ideas into the multiple regression framework, this hybrid technique improves predictive modeling. Fig. 2 describes the flowchart of Unsupervised Bayesian Multiple Regression Analysis.

The flowchart illustrates the Unsupervised Bayesian Multiple Regression Analysis algorithm. It begins by inputting training data and prior beliefs. The algorithm initializes prior beliefs for Bayesian analysis and applies the Universal Multiple Regression Analysis to obtain initial coefficients. probability is calculated, and the prior and probabilities are combined to create the unnormalized posterior distribution, which is then normalized. The updated beliefs are derived, and the algorithm outputs both the posterior distribution and the final regression coefficients, offering a comprehensive probabilistic model. In addition, the output of the Unsupervised Bayesian Multiple Regression Analysis method solves the variable identification problem by utilizing "Quadratic Discriminant Extreme Learning Machine". Further details are provided in the next section.

3.2. Quadratic discriminant extreme learning machine

Quadratic Discriminant Extreme Learning Machine is a novel approach which implemented for the variable identification problem due to the intricacy of the residual distribution makes it more difficult to identify the variables affecting indoor radon levels. The variable identification problem has therefore been addressed with the introduction of the Quadratic Discriminant Extreme Learning Machine. The proposed approach combined two approaches quadratic discriminant analysis and the Extreme Learning Machine, which are described in more depth as follows.

Extreme Learning Machines (ELM) is a type of feedforward neural network used for tasks such as classifications and regression. The unique aspect of ELM is its ability to compute the output layer’s weights using a generalized inverse of the hidden layers output matrix. In this approach, ELM effectively handles Non-Gaussian residual distributions and provide more accurate predictions of indoor radon levels. Their strength lies in random input weight initialization and analytical output weight solutions. This approach enhances prediction accuracy, enabling ELMs to adapt to complex relationships within the data, especially when dealing with residuals that deviate from Gaussian distributions in indoor radon level predictions.

The output of ELM is calculated as follows given in Eq. (7):

where: $y$ is the output layer vector; $\theta$ is the weight vector between the hidden layer and the output layer; $H$ is the hidden layer output matrix.

The Hidden Layer Output Matrix ($H$) is calculated as follows given in Eq. (8):

where: $g$ is the activation function; $w$ is the weight vector between input and hidden layer; $x$ is the input vector; $b$ is the bias vector.

The following provides a thorough argument for the existing method:

A learning framework for single hidden layer feed forward neural networks (SLFN) is called Extreme Learning Machines. Extreme Learning Machines work on the general principle of generating connection weights at random between the input and hidden layers, then analytically computing the weights connecting the hidden layer to the output layer.

The architecture of Extreme Learning Machines is seen in Fig. 3. The hidden layer, which has a variable number of nodes depending on the issue, comes after the input layer, which is the signal source node. The transformation function of the hidden layer is based on the attenuation of a nonnegative linear function and its radial basis. In response to the input pattern, the hidden layer modifies activation function parameters and learning speed more slowly, whereas the output layer modifies linear weight and learning speed more quickly. Eq. (9) is a mathematical representation of $N$ neurons in the buried layer:

where ${\omega }_i=[{{\omega }_i}_1,{{\omega }_i}_2,\dots ,{{\omega }_i}_n{]}^T$ is the threshold of the $i$th hidden neuron, and $y_j$ is its value. The weight vector is what links the input neurons with the $i$th hidden neuron. Let ${\beta }_x=[{{\beta }_x}_1,{{\beta }_x}_2,\dots ,{{\beta }_x}_m{]}^T$ is the weight vector that joins the output neurons to the $i$th hidden neuron.

Fig. 3Architecture of extreme learning machines

The output of Extreme Learning Machines is a trained neural network model with optimized connection weights between the input and hidden layers, as well as the analytically computed weights connecting the hidden layer to the output layer. In the context of the provided mathematical representation (Eq. (9)), the output involves the modified activation function parameters and learning speeds in the hidden layer, and the linear weights and learning speeds in the output layer. This output improves variable identification in indoor radon levels by efficiently capturing nonlinear patterns, adapting to data complexity, and providing optimized connections. The output of Extreme Learning Machines is given to quadratic discriminant analysis which explained as follows.

Moreover, each class in quadratic discriminant analysis is given a quadratic decision boundary, which allows for more flexibility when modeling complex interactions between predictors and response variables. Quadratic Discriminant Analysis is appropriate in scenarios where linear bounds, as in Linear Discriminant Analysis, are insufficient to appropriately reflect the underlying relationships because it permits a more intricate depiction of class distributions, which allows for the accommodation of complicated data patterns. The following provides a thorough argument of Quadratic Discriminant Analysis method:

Compared to Linear Discriminant Analysis, Quadratic Discriminant Analysis handles different covariance matrices for each class, which makes it flexible enough to handle situations when there are different variances across classes. Because of its adaptability, Quadratic Discriminant Analysis more effectively represents the intricate and non-linear decision boundaries that are observed in datasets with a variety of class features. The benefit of using Quadratic Discriminant Analysis is that, as opposed to linear decision boundaries, it produces a first class decision boundary by being less stringent and accepting varying covariance matrix characteristics for different classes. A single-variable statistical technique called quadratic discriminant analysis has been used to build an algorithm based on the groups that reveal the agents or influencing layers that are visible. For class $k$, the discriminant function is represented as follows as in Eq. (10):

where: $x$ is the vector of environmental parameters, ${\sum }_k$is the covariance matrix for class; ${\mu }_k$ is the mean vector for class; ${\pi }_k$ is the prior probability of class.

The output of the discriminant function in quadratic discriminant analysis is ${\delta }_k\left(x\right)$. It is a metric that utilized for classifying observations into several groups. More specifically, the above Eq. (8) is used to construct the discriminant function ${\delta }_k\left(x\right)$ for each class $k$. Next, the most likely class for a particular input vector $x$ is projected to be the one with the greatest discriminant function value. This output contributes to solving the variable identification problem by the discriminant function in Quantitative Discriminant Analysis involves coefficients that are associated with each environmental parameter (variable). These coefficients are derived from the mean vectors, covariance matrices, and prior probabilities of the different classes during the training phase.

The Quadratic Discriminant Extreme Learning Machine provides a useful method for identifying variables related to indoor radon levels. The combined result is a trained neural network that recognizes complex nonlinear patterns. To further improve variable identification, the Quadratic Discriminant Extreme Learning Machine also makes use of discriminant function coefficients from Quadratic Discriminant Analysis. By effectively addressing the intricacies of the data and maximizing the relationships between variables in the model, this all-encompassing method raises the accuracy of indoor radon level estimates. The pseudocode for the Quadratic Discriminant Extreme Learning Machine is shown in Algorithm 2.

Algorithm 2: Quadratic Discriminant Extreme Learning Machine.

Input:

$X:$ Input data matrix of size $(n\times m)$, where $n$ is the number of samples and $m$ is the number of features.

$Y:$ Target class labels or output matrix of size $(n\times p),$ where $p$ is the number of output nodes.

Number of classes, $K.$

Number of hidden nodes, $H.$

Output:

Output Weights $W$ of size $(H\times K).$

Bias vector $b$ of size $(1\times K).$

Class-specific mean vectors ${\mu }_k$ ($k=$ 1 to $K$).

Class-specific covariance matrices ${\mathrm{\Sigma }}_k$($k=$ 1 to $K$).

Class priors ${\pi }_k$ ($k=$ 1 to $K$).

Algorithm Steps:

Step 1 –Apply Quadratic Discriminant Analysis (QDA):

a) For each class $k$ ($k=$ 1 to $K$):

i. Calculate the class-specific mean vector ${\mu }_k.$

ii. Calculate the class-specific covariance matrix ${\mathrm{\Sigma }}_k.$

iii. Calculate the class prior probability ${\pi }_k.$

Step 2 –Initialize Extreme Learning Machine (ELM):

a) Randomly initialize input weights $w_i$ ($i=$ 1 to $m$) and biases $b_i$ ($i=$ 1 to $H$) connecting input layer to hidden layer.

Step 3 –Calculate ELM hidden layer output matrix $\mathit{H}\_\mathit{o}\mathit{u}\mathit{t}$ using QDA parameters:

a) For each hidden node $h$ ($h=$ 1 to $H$):

i. Calculate activation using QDA mean vectors and covariance matrices: $activatio{n}_h=activation\_function(X{w}_i+b_i,{\mu }_h,{\mathrm{\Sigma }}_h)$ for each hidden node.

Step 4 –Compute the ELM output weights $\mathit{W}$ and bias $\mathit{b}$ using the following equation:

a) $W=pinv\left(H_{out}\right)Y,$ where pinv denotes the pseudo-inverse.

b) $b=0.5(Y-H_{out}W)$$\#$ Bias can be calculated for better fitting.

End

The Quadratic Discriminant Extreme Learning Machine algorithm combines the discriminative power of Quadratic Discriminant Analysis with the flexibility of Extreme Learning Machine to optimize weights and biases. It utilizes Quadratic Discriminant Analysis’s class-specific statistics to enhance ELM’s ability to capture nonlinear patterns, resulting in improved variable identification and accurate predictions for multi-class classification tasks. Fig. 4 represents the flowchart of Quadratic Discriminant Extreme Learning Machine for Radon Prediction.

Fig. 4Flowchart of quadratic discriminant extreme learning machine for radon prediction

The data processing process depicted in the flowchart starts with the input data, initializes Quadratic Discriminant Analysis and ELM components, looks for objective update criteria, computes output from a hidden layer, determines when to stop, computes output weight and biases, integrates QDA and ELM components, stresses variable identification, and outputs the intended outcome in the end. Because the process is data-driven, each step's goal must be determined within a certain context or domain. The “END” label marks the end of the flowchart.

Overall, the proposed approach integrates Unsupervised Bayesian Multiple Regression Analysis with a Quadratic Discriminant Extreme Learning Machine for comprehensive indoor radon level predictions. It utilizes Unsupervised Bayesian Multiple Regression Analysis for predictive analysis in radon concentration, which combines universal multiple regression and unsupervised Bayesian analysis for linear relationships. Then, it makes use of a Quadratic Discriminant Extreme Learning Machine for variable identification in indoor radon levels, which combines the Quadratic Discriminant Analysis and Extreme Learning Machines for efficient handling of nonlinear patterns. This synergistic method enhances variable identification, providing a sophisticated predictive model that considers uncertainties and complex correlations in the data. The algorithm combines statistical and machine learning techniques for robust indoor radon level predictions, addressing multifaceted interactions. Section 4 will describe the performance and comparison of the proposed technique.

4.1. Experimental setup

Every home’s radon gas concentration is determined using a dosimeter that was created by BARC Mumbai. The dosimeter is maintained one meter above the ground, and measurements are taken in the summer, winter, and rainy seasons.

Tool used: Tableau Desktop; OS: Windows 10 home; Processor: Intel (R) Core (TM) i5-8265U CPU@1.60 GHZ 1.80 GHZ; RAM: 16 .00 GB.

4.2. Dataset description

Three different seasons were observed at the Dadar, Mumbai location: the rainy season (June to September), the winter season (October to January), and the summer season (February to May). The measured values of radon concentrations were used to calculate the Pearson Correlation Coefficient for three environmental parameters: temperature, humidity, and atmospheric pressure. Table 2 summarizes the Pearson correlation matrix for radon and climatic parameter.

There is no discernible relationship between environmental parameters and radon concentration at the Dadar, Mumbai, Maharashtra location. However, Table 2 shows that during the summer, humidity had a recorded influence of $r=$0.402, while for other parameters, the correlation coefficient in absolute value is less than 0.25 during the rainy, winter, and summer seasons.

4.3. Experimental results

Tableau was used for data analysis and simulation, with both independent and dependent variables (such as temperature, humidity, and air pressure) as input. A differential equation was used to compute the integrated radon concentration, which was then compared to the measured value. There were not many variations between the calculated and observed values, according to the study. As a result, it concluded that temperature, humidity, and air pressure have a negligible effect on the integrated radon concentration that was computed because of the differential equation that was used in the simulation.

4.4. Performance analysis of the proposed method

This integrated predictive analysis and variable identification approach is evaluated using key indicators such as air pressure and radon content. It also evaluates how well Unsupervised Bayesian Multiple Regression Analysis enhances predictive analysis and detects diseases caused by radon concentrations. It is evaluated how well Quadratic Discriminant Extreme Learning Machine identify specific variables identification.

The link between temperature and radon concentration is seen in Fig. 5. The graph shows that the value of the radon concentration is roughly proportionate to the floor-by-floor temperature. With a correlation coefficient of 0.159, the temperature during the rainy season is positively and inversely correlated. The multiple regression equation for the measurement at the Dadar Mumbai site during the rainy season was computed using the following parameters: temp and correlation coefficient, $r=$ 0.159 in Eq. (11):

Fig. 5Plotting a regression path between temperature and radon levels during the rainy season in a scattering diagram

Fig. 6A scattering diagram showing the direction of the plotted relationship between the wintertime temperature and radon concentration

Fig. 6 indicates that there is a negative relationship between the two variables. The presented graph demonstrates that the value of the indoor radon concentration is directly correlated with the floor-wise temperature. The winter temperature has a slight correlation with the correlation coefficient of 0.208. Eq. (12) presents the multiple regression equation that was developed for the measurement at Dadar Mumbai location during the winter season, taking into account the parameters temp and $r=$ 0.208:

Fig. 7 indicates a marginally favorable relationship between the two variables. With a correlation value of $r=$ 0.233, it shows the direct proportionality of indoor radon concentration with temperature variations floor by floor during the summer. The multiple regression equation that was computed for the measurement at the Dadar Mumbai site during the summer season took into account the temperature and the correlation coefficient, $r=$ 0.233, as indicated in Eq. (13):

Fig. 7A scattering diagram showing the direction of the plotted relationship between summertime temperature and radon concentration

Fig. 8 indicates a marginally favorable relationship between the two variables. The plotted graph demonstrates that the relationship between the value of indoor radon concentration and humidity is direct. The correlation coefficient, which is 0.128, shows a modest relationship between humidity and the rainy season. The multiple regression equation for the measurement in Dadar, Mumbai, during the rainy season, taking into account the humidity and $r=$ 0.128, is represented in Eq. (14):

Fig. 8Plotted regression direction between radon concentration and humidity during the rainy season is displayed in a scattering diagram

Fig. 9Plotted regression direction between wintertime humidity and radon levels on a scattering diagram

Examining the association between radon concentration and humidity levels is shown in Fig. 9. The above graph illustrates how indoor radon levels and humidity are directly correlated throughout the winter. Eq. (15) displays the multiple regression equation with humidity and $r=$ 0.189 for the Dadar Mumbai area during the winter season:

There is a somewhat favorable association between humidity and radon concentration in the summer according to the scatter plot in Fig. 10. The graph’s red trend line indicates that radon concentration grows in tandem with increased humidity. The somewhat favorable association between greater humidity levels and enhanced radon concentrations is indicated by the correlation coefficient, $r=$ 0.402 which shows in Eq. (16):

The link between air pressure and radon levels during the rainy season is shown in Fig. 11. This graph shows the straight proportionality between the independent variable, the atmospheric pressure during the rainy season, and the radon concentration. The regression Eq. (17) for pressure and radon concentration parameters in Dadar, Mumbai, during the rainy season has a correlation value of $r=$ 0.107:

Fig. 10A scattering diagram showing the direction of the observed relationship between summertime humidity and radon concentration

Fig. 11Plotted regression direction between pressure and radon concentration during the rainy season is displayed in a scattering diagram.

The $X$-axis represents atmospheric pressure, while the $Y$-axis represents radon concentration. Fig. 12 illustrates the link between radon concentration and atmospheric pressure. This graph shows the straight proportionality between the independent variable, the atmospheric pressure during the winter, and the radon concentration. Eq. (18) displays the results of a multiple regression analysis using pressure and $r=$ 0.152 for the Dadar Mumbai area during the winter season:

Fig. 12Plotted regression path between wintertime pressure and radon concentration on a scattering diagram

Fig. 13 illustrates the relationship between summertime air pressure and radon levels. The data points show that as air pressure increases from 90 to 95, there is a modest increase in radon concentration. The multiple regression equation for the parameters Pressure and Indoor Radon Concentration at location Dadar Mumbai in the summer, with $r=$ 0.233, is displayed in Eq. (19):

Fig. 13Plotting a regression path between summertime pressure and radon concentration in a scattering diagram

4.5. Comparison method

This section, as noted in [29], contrasts the proposed approach's Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), Root Mean Squared Logarithmic Error (RMSLE), Percentage Bias, and Coefficient of relation with the existing Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging approaches. It illustrates how the proposed approach outperforms or is on trend with current methods in the variables identification and predictive analysis domains. A thorough analysis of these measures is used to achieve this. The comparison goes beyond performance metrics to include Percentage Bias and Coefficient of Relation to confirm the benefit or efficiency of the proposed strategy in managing diagnostics that enable a thorough comprehension of the radon concentration of predictive analysis and variable identification.

Fig. 14Comparison of MAE with existing methods

The comparison of the proposed model's MAE with that of other current methods is shown in Fig. 14. The MAE of the proposed method is contrasted with that of already available methods like Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging approaches. The proposed model's MAE comes in at 5.1 %, whereas the MAE of Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging approaches are, in that order, 10.5 %, 8.3 %, and 6.8 %. Thus, variable identification has improved due to the Quadratic Discriminant Extreme Learning Machine with low MAE.

Fig. 15Comparison of RMSE with existing methods

Fig. 15 displays a comparison of the RMSE of the proposed approach with those of other available techniques. The RMSE of the proposed approach is compared to those of existing techniques such as Ordinary Kriging approaches, Empirical Bayesian Kriging, and Inverse distance weighting. The RMSE of the proposed model is 5.5 %, while the RMSE of the techniques such as inverse distance weighting, empirical Bayesian kriging, and ordinary kriging are, respectively, 9.6 %, 8.2 %, and 7.0 %. Thus, the Unsupervised Bayesian Multiple Regression Analysis has enhanced predictive analysis with low RMSE.

Fig. 16Comparison of RMSLE with existing methods

A comparison of the proposed approach’s RMSLE with other existing methodologies is shown in Fig. 16. The proposed method’s RMSLE is contrasted with that of other methods already in use, including inverse distance weighting, empirical Bayesian kriging, and ordinary kriging approaches. The proposed model's RMSLE is 0.08 %, whereas the RMSLE of methods utilizing inverse distance weighting, empirical Bayesian kriging, and conventional kriging are 0.12 %, 0.18 %, and 0.15 %, respectively. Thus, the use of machine learning in Quadratic Discriminant Extreme Learning Machine has enhanced variable identification with low RMSLE.

Fig. 17Comparison of percentage Bias with existing methods

Fig. 17 displays a comparison of the Percentage Bias of the proposed methodology with various current approaches. The Percentage Bias of the proposed approach is compared with that of existing techniques, such as Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging. The percentage bias of the proposed model is 92 %, while the percentage biases of approaches that use Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging are, respectively, 20 %, 42 %, and 76 %. Thus, the Unsupervised Bayesian Multiple Regression Analysis has enhanced predictive analysis with high Percentage Bias.

A comparison of the proposed methodology’s coefficient of relation with many other existing methods is shown in Fig 18. The proposed method’s coefficient of relation is contrasted with that of other methods now in use, including Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging approaches. The proposed model’s correlation coefficient is 81, compared to the correlation coefficients of models utilizing Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging, which are 24, 48, and 66, respectively. Thus, the Unsupervised Bayesian Multiple Regression Analysis has enhanced predictive analysis with high Coefficient of relation.

Fig. 18Comparison of coefficient of relation with existing methods

Fig. 19 shows a comparison of the accuracy of the proposed technique with several different methods already in use. The accuracy of the proposed method is compared with existing approaches such as Ordinary Kriging, Empirical Bayesian Kriging, and Inverse distance weighting. The accuracy of the proposed model is 98.2 %, whereas the models that use Ordinary Kriging, Empirical Bayesian Kriging, and Inverse distance weighting have corresponding accuracy of 92.5 %, 89 %, and 85 %. Predictive analysis has therefore been improved by Unsupervised Bayesian Multiple Regression Analysis with accuracy rate.

Fig. 19Comparison of accuracy with existing methods

A comparison of the precision of the proposed approach with several existing methods is presented in Fig. 20. The precision of the proposed technique is contrasted with current methodologies like Inverse Distance Weighting, Ordinary Kriging, and Empirical Bayesian Kriging. The proposed model’s precision is 96.3 %, whereas the models that employ inverse distance weighting, ordinary kriging, and empirical Bayesian kriging have respective precisions of 91.7 %, 89.5 %, and 86.4 %. Thus, Unsupervised Bayesian Multiple Regression Analysis has enhanced predictive analysis with high precision rate.

Fig. 20Comparison of precision with existing methods

Fig. 21 compares the recall of the proposed approach with several other approaches that are currently in use. The proposed technique’s recall is compared to existing approaches such as Empirical Bayesian Kriging, Ordinary Kriging, and Inverse Distance Weighting. The proposed model has a recall of 97.8 %, whereas the models using conventional kriging, empirical Bayesian kriging, and inverse distance weighting had recalls of 84.7 %, 88.9 %, and 93.4 %, respectively. Thus, variable identification has improved due to the Quadratic Discriminant Extreme Learning Machine with high recall.

Fig. 21Comparison of recall with existing methods

The sensitivity of the proposed method is contrasted with many alternative ways that are currently in use in Fig. 22. The sensitivity of the proposed method is contrasted with that of other methods that are already in use, including Inverse Distance Weighting, Ordinary Kriging, and Empirical Bayesian Kriging. The models using conventional kriging, empirical Bayesian kriging, and inverse distance weighting have recalls of 95 %, 92 %, and 95 %, respectively, whereas the recommended model had a sensitivity of 96 %. Thus, Unsupervised Bayesian Multiple Regression Analysis has enhanced predictive analysis with high sensitivity rate.

Fig. 22Comparison of sensitivity with existing methods

Fig. 23 compares the specificity of the proposed approach with several other approaches that are currently in use. The proposed method’s specificity is compared to that of existing approaches, such as Empirical Bayesian Kriging, Ordinary Kriging, and Inverse Distance Weighting. The proposed approach has a sensitivity of 97 %, while the models utilizing inverse distance weighting, conventional kriging, and empirical Bayesian kriging have specificities of 96 %, 94 %, and 89 %, respectively. Thus, variable identification has improved due to the Quadratic Discriminant Extreme Learning Machine with high specificity.

Fig. 23Comparison of specificity with existing methods

Simple linear regression is a powerful mathematical algorithm that excels under certain conditions due to its simplicity, efficiency, and interpretability. It models the relationship between two variables with a straight line, making it easy to understand and communicate findings. It is computationally efficient, requiring only two parameters, making it suitable for large datasets. Its ease of implementation is enhanced by built-in functions in software packages and programming languages.

Overall the comparison method in predictive analysis and variable identification evaluates the proposed approach against existing methods using various performance metrics. These include MAE, RMSE, RMSLE, Percentage Bias, Coefficient of Relation, Accuracy, Precision, Recall, sensitivity and specificity. The proposed approach outperforms or aligns with current methods in these domains with a Percentage Bias of 92 %, and Coefficient of relation with 81. Additionally, the approach achieves an impressive 5.1 % of MAE, 5.5 % of RMSE, 0.08 % of RMSLE establishing it as a superior option for predictive analysis and variable identification in radon concentration. This advantage over Inverse distance weighting, Empirical Bayesian Kriging, and Ordinary Kriging techniques is particularly notable in real-world circumstances with complicated dynamics and interference difficulties.