HFSA: hybrid feature selection approach to improve medical diagnostic system

Rejection layer

Outlier rejection, also known as anomaly detection, is a data mining technique used to identify data points that deviate significantly from the norm (Rabie, Saleh & Mansour, 2022; Rabie et al., 2022; Saleh & Rabie, 2023b). In the context of medical diagnosis, these outliers can represent incorrectly recorded data due to equipment malfunction or human error, mistakes made during the input of patient information, or rare conditions or diseases that are not commonly observed. These outliers have a bad effect on the diagnostic model. Hence, outlier rejection is crucial for building accurate and reliable medical diagnosis models. Several methods are employed to detect outliers in medical data. These methods are categorized as statistical methods, clustering-based methods, and machine learning or optimization methods. while statistical and clustering-based methods are faster than optimization methods but optimization methods give more accurate results. In this work, GA is used as an optimization algorithm to give accurate results rather than its execution time because the outlier rejection process is performed only once and offline (Saleh & Rabie, 2023b).

When using GA for outlier rejection, the process typically involves the following steps. First, a population of candidate solutions is initialized. Each solution is a set of data points considered to be “normal” or representative of the inlier data. The remaining data points are implicitly treated as outliers. Next, an evaluation function (fitness function) assesses the quality of each solution. This function typically measures how well the selected data points fit a certain model or distribution, or how similar they are to each other, with better solutions receiving higher fitness scores. Then, selection operators choose solutions with higher fitness scores to become parents for the next generation, mimicking natural selection. Crossover operators combine parts of parent solutions to create new offspring solutions, exploring different combinations of data points. Mutation operators introduce small random changes to solutions, further diversifying the search. These steps of selection, crossover, and mutation are repeated for several generations, allowing the GA to evolve the population of solutions towards better representations of the inlier data. Finally, after a certain number of generations or when a satisfactory solution is reached, the algorithm outputs the best solution found, which represents the set of inlier data points. The data points not included in this final solution are then identified as outliers.

GA offers several advantages as an outlier rejection model, particularly when dealing with complex datasets where the nature of outliers might not be easily defined or where multiple types of outliers may exist. GAs can effectively search a high-dimensional feature space to identify optimal subsets of data points that represent the “normal” data distribution, implicitly flagging the remaining points as outliers. This flexibility is valuable compared to traditional statistical methods that often rely on assumptions about the data distribution, which may not hold in real-world scenarios. GA has been chosen for outlier detection because of its ability to adapt to complex data landscapes and potentially discover more nuanced outliers that other approaches might miss. While the computational complexity of GAs is a valid concern, especially with large datasets, it can be mitigated through various strategies. These include optimizing the GA parameters (population size, mutation rate, etc.), employing parallelization techniques to distribute the computational load, and incorporating domain-specific knowledge to constrain the search space. Furthermore, the offline nature of outlier rejection as a preprocessing step lessens the impact of GA’s execution time on real-time applications.

Selection layer

In this subsection, the proposed HFSA will be described in detail. In fact, medical diagnosis system is affected by irrelevant predictors (features) because these useless features mislead the model to provide accurate results and also cause over-fitting (Remeseiro & Bolon-Canedo, 2019; Ghaffar Nia, Kaplanoglu & Nasab, 2023). Hence, selecting the best predictors is a very important process before using the prediction model to give predication model the ability to provide correct predictions (Chae et al., 2024; Shivahare et al., 2024). Generally, there are two main categories of selection methods, namely; filter and wrapper (Rabie, Saleh & Mansour, 2022; Rabie et al., 2022; Saleh & Rabie, 2023b; Sarker, 2021). While filter type can speedily determine the most effective predictors, wrapper type can correctly determine the most effective predictors. Recently, wrapper methods based on bio-inspired algorithms have been used to correctly select the best predictors. During SL, HFSA is provided as a new method to select the best predictors that can correctly train the prediction model as illustrated in Fig. 3. Related to Fig. 3, HFSA includes two necessary stages; (i) FS and (ii) AS where FS is a filter stage that can quickly select a set of predictors and AS is a wrapper stage that can accurately select the best predictors. Consequently, all predictors are passed to FS to quickly select a set of predictors, and then the selected set of predictors from FS are passed to AS to accurately select the meaningful predictors.

According to Fig. 3, FS can quickly select a set of predictors based on the chi-square method (Saleh & Rabie, 2023a). After that, the selected predictors are passed to HOA as a new optimization algorithm in AS to accurately select the best predictors. In fact, HFSA is proposed to enhance the efficiency and accuracy of the diagnostic model. This HFSA employs a two-stage process. First, FS method utilizing chi-square analysis rapidly eliminates the least relevant features from the initial dataset. This pre-filtering step significantly reduces the dimensionality of the data before it is passed to the second stage (AS). The AS stage, which typically involves a computationally intensive optimization algorithm (HOA), benefits greatly from this reduction in feature space. By receiving a smaller, more promising subset of features from the FS stage, the AS process can more quickly and effectively identify the most significant features for accurate diagnosis, leading to improved model performance. Chi-square is chosen as the filtering methodology in the FS of the HFSA due to its computational efficiency and suitability for handling high-dimensional medical datasets. In such datasets, where the number of features can far exceed the number of samples, quick reduction of dimensionality is crucial. Chi-square excels at this by independently evaluating the relationship between each feature and the target variable (diagnosis). It assesses the statistical independence between them, effectively identifying features that are highly dependent on the diagnosis. This dependence suggests that the feature carries valuable information for distinguishing between different diagnostic outcomes.

Compared to other statistical methods, chi-square offers several advantages in this context. Methods like ANOVA, while powerful, assume the normality of data, an assumption often violated in medical datasets. Non-parametric alternatives like Kruskal-Wallis, while robust to non-normality, can be computationally more expensive than chi-square, especially with a large number of features. Furthermore, chi-square is specifically designed for categorical data, which is common in medical datasets (e.g., presence/absence of symptoms, genetic markers). While continuous features can be discretized, the method is inherently well-suited for many medical applications. Its computational simplicity allows for rapid filtering of features, making it ideal for the FS of the HFSA. While more sophisticated filtering methods exist, the goal of the FS stage is speed and sufficient reduction, not necessarily finding the absolute best subset. Chi-square offers a good balance between these competing priorities in the context of high-dimensional medical data.

The HOA in AS is indeed the basic method for accurate feature selection. However, its computational intensity makes it unsuitable for direct application to the original, high-dimensional dataset. Therefore, the chi-square-based filtering in the upper stage called FS serves as a crucial pre-processing step. By drastically reducing the number of features before they are passed to the HOA, the FS significantly curtails the HOA’s execution time. This allows the HOA to focus its computational resources on a smaller, more promising subset of features, ultimately leading to a more efficient and accurate feature selection process.

According to HOA in AS, it consists of two main algorithms, which are GA in a binary form (Saleh & Rabie, 2023b) and $T^{2}A$ in a binary form (Ab. Rashid, 2021). In a signal cycle of executing HOA, GA will be implemented on a set of predictors selected by FS and then new chromosomes in GA’s population will be passed as players in $T^{2}A$’s population for implementation before starting to execute a new cycle. In other words, the output of GA passes into the input of $T^{2}A$ and then the output of $T^{2}A$ passes into the input of GA for the next cycle. To classify bio-inspired algorithms, there are two categories which are swarm intelligence and evolutionary (Davahli, Shamsi & Abaei, 2020). HOA includes the characteristics of both bio-inspired categories by using GA as an evolutionary algorithm and $T^{2}A$ as a swarm intelligence algorithm that mimics the football play behavior. GA can give the optimal global search that is related to its powerful in the exploration operation (Davahli, Shamsi & Abaei, 2020). GA can share the main information between chromosomes. At the same time, it can analyze many solutions. Also, it does not depend on mathematical derivation.

Although the advantages of GA, it cannot provide the best local search and it cannot give the optimal solution in the case if the probability of crossover and mutation are not suitable (Davahli, Shamsi & Abaei, 2020). Accordingly, GA can reach a local optimal solution but cannot reach a global one. To overcome the issues of GA and improve its performance, $T^{2}A$ is used because of its flexibility and scalability. Additionally, $T^{2}A$ that simulates the football play behavior can provide a high-speed convergence level (Ab. Rashid, 2021). Although the benefits of $T^{2}A$, the performance of $T^{2}A$ has been affected by the change in the coefficient and also it relies on mathematical derivation. To overcome the issues of GA and $T^{2}A$ and utilize their benefits at the same time, HOA that includes both algorithms is provided to select the best predictors accurately.

The main reason for selecting GA and $T^{2}A$ is that GAs excel at exploring a wide range of potential feature subsets, effectively navigating complex search spaces to identify near-optimal solutions. Their strength lies in their ability to handle high dimensionality and non-linearity, crucial for medical datasets where feature interactions can be complex and the relationship between features and outcomes is often non-linear. However, GAs can be computationally expensive and may converge to local optima. $T^{2}A$, inspired by the fluid passing style of football, offers a different approach. It emphasizes exploitation, efficiently refining promising solutions by iteratively improving upon them. $T^{2}A$ is generally faster than GAs and less prone to getting trapped in local optima due to its focused search strategy. The hybridization of these two algorithms capitalizes on their complementary strengths. The GA provides broad exploration, casting a wide net to identify potentially good feature subsets.

Then, the $T^{2}A$ algorithm takes these promising subsets and exploits them, fine-tuning the feature selection to achieve higher accuracy and efficiency. This combined approach addresses the limitations of each algorithm individually. The GA’s exploration helps $T^{2}A$ avoid premature convergence, while $T^{2}A$’s exploitation accelerates the search process and improves the quality of the final feature subset. This synergistic combination is particularly beneficial for medical datasets, where identifying the most relevant features can significantly impact diagnostic accuracy and treatment effectiveness. The hybridization is needed because neither algorithm alone guarantees optimal performance. Specifically, these two methods were chosen because their exploration and exploitation strategies are highly complementary, offering a balanced approach that is well-suited for the challenges of feature selection in complex medical datasets.

There are many steps to execute HOA as illustrated in Fig. 3. Initially, it is proposed that the medical dataset consists of ‘u’ predictors. Based on this dataset, the chi-square method will be executed in FS to quickly select a set of predictors that consists of ‘v’ predictors, where v¡u. After that, the dataset that includes ‘v’ predictors will be transferred to HOA in AS to accurately select the best set of predictors that includes ‘m’ predictors, where m¡v. Hence, the dataset that includes ‘v’ predictors in FS will be passed to generate GA’s population to be executed. After that, a new population will be passed to $T^{2}A$ to be executed before starting the next cycle.

Initially, the execution of GA needs to create a population $(Po{p}_{GA})$ that includes ‘x’ chromosomes. The dimension of each chromosome is ‘v’ which represents the same number of predictors in the dataset (Rabie, Saleh & Mansour, 2022). Accordingly, the representation of each chromosome will be in a binary form including zero or one value. One value refers to the selected feature (predictor) but zero value refers to the unselected feature (predictor). Hence, a binary string representation (encoding) has been used where each bit corresponds to a feature in the medical dataset. A ‘1’ in the string indicates that the corresponding feature is selected, while a ‘0’ indicates that it is excluded. For example, if the dataset has 10 features, the string “1010011001” would represent a feature subset containing features 1, 3, 6, 7, and 10. This binary representation is simple, efficient, and commonly used in feature selection with GAs. After creating the GA’s initial population, NB’s accuracy will be calculated as an evaluation value to evaluate chromosomes using Eq. (1) (Rabie, Saleh & Mansour, 2022; Saleh & Rabie, 2023b).

(1) $$F(x_i(t))=Accuracy\mathrm{\_}NB(x_i(t))$$where $F(x_i(t))$ is a fitness value of $i^{th}$ chromosome at cycle $t$ and $Accuracy\mathrm{\_}NB(x_i(t))$ is an accuracy of NB related to $i^{th}$ chromosome at cycle $t$. In fact, this fitness function directly leverages the predictive power of a NB classifier. Each chromosome, representing a candidate feature subset, is evaluated by training NB model using only the features indicated by the ‘1’s in the chromosome’s binary string. The trained NB model’s performance, typically measured by its accuracy through cross-validation, directly forms the basis of the fitness score. Specifically, the fitness function can be defined as the NB model’s accuracy. This direct relationship between fitness and predictive performance ensures that the GA searches for feature subsets that maximize the NB classifier’s accuracy. In the next step, three operators called selection using the roulette wheel method, crossover using probability equals 0.9 and 1 bit method, and mutation using probability equals 0.01 and 1 flip bit will be used to produce new chromosomes in the population. In this study, two fit parents (chromosomes) are chosen at random using the roulette wheel approach, based on their probability of selection $((P_{sel})$ falling within the interval [0,1]; $(P_{sel})\in [0,1]$. Based on the probability of crossover $(P_{cross})$ pertaining to the interval [0,1]; $(P_{cross})\in [0,1]$, the crossover operator between the chosen parents is carried out using the one bit crossover method. Since crossover is a desired process, $(P_{cross})$ should be 0.9 or close to it. However, the mutation operator is carried out by flipping one bit according to the probability of mutation $(P_{mut})$ that belongs to the interval [0,1]; $((P_{mut})\in [0,1]$. Since mutation is actually a bad process, $(P_{mut})$ ought to be 0.01. Until the size of the new population matches the size of the original population, these three operators will be repeated.

These new chromosomes will be transferred to $T^{2}A$ as an input population $(Po{p}_T)$ where each player includes one chromosome in binary form. After that, evaluating players will be performed by using the same fitness function of GA presented in Eq. (1). The best (key) player (q) will be assigned according to the evaluation values of players (Ab. Rashid, 2021). $q$ will be modified in every cycle and also will be used as a global position to modify every player’s position through population. The ball’s position will be updated before the players’ position are modified using Eq. (2) (Ab. Rashid, 2021).

(2) $$p_i(t+1)=\{\begin{array}{cc}r(p_i(t)-p_{i+1}(t))+p_i(t),\hfill & \mathrm{i}\mathrm{f}{rand}_p>\epsilon \hfill \\ p_i(t)-(C_i+r)(p_i(t)-p_{i+1}(t)),\hfill & \mathrm{i}\mathrm{f}ran{d}_p\le \epsilon \hfill \end{array}$$where a new ball’s position is $p_i(t+1)$ in the next cycle $t+1$ while $ran{d}_p$ is a random value in (0,1). The basic idea of $T^{2}A$ depends on short passing which refers to moving the ball from the player to the nearby player. Losing the ball to the opponent may have occurred although the increase in the successful passing rate. For this reason, the probability of losing the ball is $\epsilon$ which has a percentage in [0.1–0.3]. In $T^{2}A$, $ran{d}_p>\epsilon$ represents the successful pass while $ran{d}_p\le \epsilon$ represents the unsuccessful pass. $C_1$ represents the ball’s reflection magnitude in the unsuccessful pass that takes values in [0.5,1.5]. $(p_i(t)-p_{(i+1)}(t))$ is the distance between $i^{th}$ position and $i^{th}+1$ position of ball. In the case if $p_i(t)$ equals $(p_v(t))$ that represents the last position of ball, $p_{(i+1)}(t)$ will be $p_1(t)$. After updating the ball’s position, every player’s position in $Po{p}_T$ will be updated related to the ball’s position and the key player archive $(q)$ using Eq. (3) (Ab. Rashid, 2021).

(3) $$p{l}_i(t+1)=p{l}_i(t)+r\ast C_2\ast (p_i(t)-p{l}_i(t))+r\ast C_3\ast (q-p{l}_i(t))$$where $C_2$ is the coefficient that has value in [1,2.5] while $C_3$ is the coefficient that has value in [0.5,1.5]. $C_2$ and $C_3$ are used to balance the player’s position (pl) between p and q. The representation of the new ball’s position and each player’s position includes continuous values. Hence, these continuous values should be converted to binary ones using a sigmoid function. Hence, the sigmoid function will be applied on the ball’s position; $p_i=(p_i^1,p_i^2,\dots ..,p_i^v)$ and every player’s position; $p{l}_i=(p{l}_i^1,p{l}_i^2,\dots ..,p{l}_i^v)$ to convert them into binary form; $p_b{\mathrm{\_}}_i=(p_b^1{\mathrm{\_}}_i,p_b^2{\mathrm{\_}}_i,\dots ..,p_b^v{\mathrm{\_}}_i)$ and $p{l}_b{\mathrm{\_}}_i=(p{l}_b^1{\mathrm{\_}}_i,p{l}_b^2{\mathrm{\_}}_i,\dots ..,p{l}^{v}b{\mathrm{\_}}_i)$ respectively using Eqs. (4) and (5) (Rabie, Saleh & Mansour, 2022; Rabie et al., 2022).

(4) $$p_{b_i}^y(t+1)=\{\begin{array}{cc}1\hfill & \mathrm{i}\mathrm{f}r(0,1)\ge \mathrm{Sig}(p_{b_i}^y)\hfill \\ 0\hfill & \mathrm{E}\mathrm{l}\mathrm{s}\mathrm{e}\hfill \end{array}$$

(5) $$p{l}_{b_i}^y(t+1)=\{\begin{array}{cc}1\hfill & \mathrm{i}\mathrm{f}r(0,1)\ge \mathrm{Sig}({pl}_{b_{\mathrm{i}}}^y)\hfill \\ 0\hfill & \mathrm{E}\mathrm{l}\mathrm{s}\mathrm{e}\hfill \end{array}$$where the next cycle is $t+1$, the binary value of $i^{th}$ player for the next cycle at $y^{th}$ position is $p{l}_b^y{\mathrm{\_}}_i(t+1)$, and the ball’s binary value for the next cycle at $y^{th}$ position is $p_b^y{\mathrm{\_}}_i(t+1)$. $y$ represents a pointer to the current position (predictor); $y=1,2,3,\dots .,v$. A random value that is belonging to [0,1] is r(0,1) while sigmoid functions for each player and the ball are $Sig(p{l}_b^y{\mathrm{\_}}_i)$ and $Sig(p_b^y{\mathrm{\_}}_i)$ respectively which can be calculated by using Eqs. (6) and (7) (Rabie, Saleh & Mansour, 2022; Rabie et al., 2022).

(6) $$\mathrm{Sig}(p_{b_i}^y)=\frac{1}{1+e^{-p_{b_i}^y}}$$

(7) $$\mathrm{Sig}(p_{b_i}^y)=\frac{1}{1+e^{-p_{b_i}^y}}$$where $e$ indicates the base of the natural logarithm. Related to $p{l}_b^y{\mathrm{\_}}_i(t+1)$ as a new player’s position in $Po{p}_T$, new players in $Po{p}_T$ are produced. Then, the new $Po{p}_T$ is passed to $Po{p}_{GA}$ as an input of GA. New chromosomes in $Po{p}_{GA}$ will be evaluated using Eq. (1). These steps will be continued in the following cycles until the stopping criteria are satisfied. In the end, the optimal solution is represented in the chromosome that gives the maximum evaluation value where the best predictors are represented in all ones in this solution. Algorithm 1 presents the main steps of the HFSA algorithm.

Diagnostic layer

After implementing outlier rejection and feature selection processes, the medical diagnostic model uses NB classifier to analyze patient data and predict disease outcomes (Sarker, 2021). By removing outliers and irrelevant features, NB classifier significantly improves accuracy and reliability. Outliers, which are data points that deviate significantly from the norm, can distort the model’s learning process. Irrelevant features, on the other hand, can introduce noise and hinder the model’s ability to identify meaningful patterns. By eliminating these hindrances, NB classifiers can provide more precise and confident diagnoses, aiding healthcare professionals in making informed treatment decisions and ultimately improving patient outcomes.

Dataset description

A medical dataset is a collection of historical data collected from patients with various illnesses (Rabie & Saleh, 2024). Based on 132 symptoms (features), 4,920 cases were identified in this dataset as belonging to 41 distinct illness categories. In other words, 41 distinct illness types can be identified using the 132 features in this dataset. Each characteristic in this dataset has a value of either “1” or “0,” which indicates whether or not the feature (symptom) has an impact on the illness. As a result, “1” denotes that the trait influences the disease, whereas “0” denotes that it has no effect. A total of 30% of the dataset’s cases are utilized for testing, while 70% of the cases are used for training. A training dataset of 3,444 cases and a testing dataset of 1,476 cases were therefore created from the 4,920 cases.

The complexity of DS

The DS’s complexity, or Big O, is primarily determined by two factors: the size of the training dataset (T) and the number of features (F). The size of the training dataset factor correlates with the training procedure’s $O(T)$ complexity. The complexity of DS in respect to T and F can be calculated using Eq. (8) (Rabie & Saleh, 2024; Saleh et al., 2025).

(8) $$D{S}^{\mathrm{\prime }}scomplexity=O(T\ast F)$$

Testing the proposed hybrid feature selection approach

The proposed HFSA will be evaluated to prove its efficiency against other feature selection methodologies after performing GA method as an outlier rejection method (Rabie, Saleh & Mansour, 2022). Then, three different classifiers, called NB, KNN, and ANN, will be used to validate HFSA compared to these previous methodologies (Rabie, Saleh & Mansour, 2022; Mahesh et al., 2022;). These selection techniques are FSGA (Abdollahi & Nouri-Moghaddam, 2022), IBGWO (Rabie, Saleh & Mansour, 2022), EST (Bashir et al., 2022), and HSM (Rabie et al., 2022). Figures 4–8 illustrate the evaluation of accuracy, error, precision, recall, and implementation time. In fact, HFSA outperforms other recent methods according to NB, KNN, and ANN.

Figures 4–7 proved that the proposed HFSA is better than FSGA, IBGWO, EST, and HSM. HFSA introduces the best results at training patient number 3,444 according to all classifiers. Figures 4A–4C proved that HFSA is more accurate than other feature selection methods because it provides accuracy values for NB, KNN, and ANN that equal 89.91%, 85.85%, and 82.02% respectively. On the other hand, the error values of HFSA according to the same classifiers at the same order are 10.09%, and 17.98% as showed in Figs. 5A–5C. Hence, Fig. 4A showed that the accuracy of HFSA, FSGA, IBGWO, EST, and HSM according to NB are 89.91%, 62.21%, 73.92%, 75.48%, and 81.02%, respectively. Figure 4B showed that the accuracy of the same methods according to KNN are 85.85%, 61.01%, 71.82%, 73.40%, and 79.00%, respectively. According to ANN, Fig. 4C showed that the accuracy of the same methods is 85.00%, 61.00%, 71.02%, 72.42%, and 79.90%, respectively.

Figure 5A showed that the error of HFSA, FSGA, IBGWO, EST, and HSM according to NB is 10.09%, 37.79%, 26.08%, 24.52%, and 18.98%, respectively. Figure 5B showed that the error of the same methods according to KNN are 14.15%, 38.99%, 28.18%, 26.60%, and 21.00%, respectively. According to ANN, Fig. 5C showed that the accuracy of the same methods is 15.00%, 39.00%, 28.98%, 72.42%, and 20.10%, respectively.

Figures 6A–6C showed the precision of the proposed HFSA against other feature selection methods according to NB, KNN, and ANN. Figure 6A illustrated that the precision of HFSA, FSGA, IBGWO, EST, and HSM according to NB are 73.05%, 55.12%, 61.16%, 65.99%, and 71.05%, respectively. Figure 6B illustrated that the precision of the same techniques according to KNN are 72.00%, 54.01%, 59.18%, 64.00%, and 69.54%, respectively. Figure 6C illustrated that the precision of the same techniques according to ANN are 72.15%, 67.50%, 52.11%, 59.00%, and 64.02%, respectively.

Figures 7A–7C showed the recall of the proposed HFSA against other feature selection methods according to NB, KNN, and ANN. Figure 7A illustrated that the recall of HFSA, FSGA, IBGWO, EST, and HSM according to NB are 75.14%, 55.1%, 58.88%, 63.39%, and 69.47% respectively. The recall of the same methods in the same order according to KNN are 74.11%, 54.01%, 57.88%, 62.00%, and 68.40%, respectively as shown in Fig. 7B. Figure 7C showed that the recall of the same methods in the same order according to ANN are 73.00%, 54.55%, 57.01%, 62.55%, and 67.20%, respectively. According to these results, it is noted that HFSA represents the best feature selection method while FSGA represents the worst one according to NB, KNN, and ANN.

Experimental results demonstrate the superiority of the proposed HFSA, comprising Chi-square in FS followed by AS employing a hybrid algorithm (HOA) combining GA and $T^{2}A$, over other feature selection methods, particularly in terms of diagnostic accuracy and precision. The HFSA consistently achieves higher performance metrics compared to methods like FSGA (which uses only GA for feature selection). This performance gap can be attributed to several factors. First, the initial filtering step in HFSA, by quickly eliminating irrelevant features, reduces the search space for the subsequent optimization stage, leading to more efficient exploration of the feature subset space. Second, the HOA within the AS stage likely provides a more robust search strategy compared to GA alone. GA excels at exploring the broader search space, while $T^{2}A$ helps refine the search around promising solutions, preventing premature convergence to local optima and improving the chances of finding the globally optimal feature subset. This combination of efficient pre-filtering and enhanced optimization contributes to the improved accuracy and precision of the HFSA.

Figures 8A–8C provided the implementation time of HFSA, FSGA, IBGWO, EST, and HSM based on NB, KNN, and ANN. In Fig. 8A, the implementation time of HFSA, FSGA, IBGWO, EST, and HSM, are 8.2, 8.68, 6.99, 5.89, and 4.8 s, respectively. In Fig. 8B, the implementation time of HFSA, FSGA, IBGWO, EST, and HSM, are 8.3, 8.90, 7.25, 5.91, and 4.99 s, respectively. In Fig. 8C, the implementation time of HFSA, FSGA, IBGWO, EST, and HSM, are 8.5, 8.91, 7.27, 5.98, and 5.25 s, respectively. Thus, the implementation time of HFSA is long but the implementation time of HSM is small according to NB, KNN, and ANN.

It is important to note that both outlier rejection and feature selection, although potentially complex and time-consuming preprocessing steps are typically performed offline before the diagnostic model is trained. This means their computational burden is not a factor during real-time diagnosis. Once the data is cleaned and the optimal feature subset is identified, the resulting model can be deployed for real-time use, providing rapid diagnoses based on the pre-processed, filtered data. Therefore, while the development time for these preprocessing steps is a consideration, their offline nature means their execution time does not directly impact the speed of real-time diagnosis.

Related to these calculations in Figs. 4–7, HFSA is the first best technique and HSM is the second best one as they can give the best results according to all classifiers. On the contrary, FSGA is the worst technique as it introduces the worst results. Additionally, the results depending on NB are better than the results of KNN and ANN. Consequently, DS based on GA as an outlier rejection method, HFSA as a feature selection method, and NB as a classifier method outperformed other systems that include GA as an outlier rejection method, different feature selection methods, and NB as a classifier. In the next subsection, the proposed DS depending on GA as an outlier rejection, HFSA as a feature selection, and NB classifier will be tested and compared to other systems.

Testing the proposed diagnostic system against other systems

The proposed DS that consists of GA, HFSA, and NB classifier will be evaluated to prove its efficiency against other diagnostic systems. In this work, the proposed DS will be tested and compared to Statistical Naıve Bayes (SNB) (Rabie et al., 2022), Hybrid Diagnosis Model (HDM) (Rabie, Saleh & Mansour, 2022), Machine Learning Model (MLM) (Shivahare et al., 2024), and Ensemble Diagnosis Model (EDM) (Shokouhifar et al., 2024). Figures 9–15 illustrate the evaluation of accuracy, error, precision, recall, ROC-AUC, Boxplot, and implementation time. Additionally, Table 3 illustrate the 10-fold cross-validation results for accuracy, precision, and recall. DS outperforms other recent methods.

Figures 9–15 proved that the proposed DS is better than SNB, HDM, MLM, and EDM. DS introduces the best results at training patients number = 3,444. Figure 9 proved that DS is more accurate than other methods because it provides an accuracy of 89.91%. On the other hand, the error value of DS is 10.09% as shown in Fig. 10. Hence, Fig. 9 showed that the accuracy of DS, SNB, HDM, MLM, and EDM are 89.91%, 65.20%, 70.06%, 77%, and 83.85% respectively. Figure 10 showed that the error of DS, SNB, HDM, MLM, and EDM are 10.09%, 34.80%, 29.94%, 23%, and 16.15%, respectively. Figure 11 illustrated that the precision of DS, SNB, HDM, MLM, and EDM are 73.05%, 55.22%, 65%, 67%, and 72.19%, respectively. The recall of DS, SNB, HDM, MLM, and EDM are 75.14%, 56.08%, 60.98%, 67.29%, and 69.02% respectively as shown in Fig. 12. The ROC-AUC of DS, SNB, HDM, MLM, and EDM are 92.00%, 89.02%, 84.25%, 85.00%, and 87.25%, respectively as presented in Fig. 13. The Boxplot of the diagnosis models after performing 20 independent runs is presented in Fig. 14. According to these results, it is noted that DS represents the best diagnosis method while SNB represents the worst one. Figure 15, the implementation time of DS, SNB, HDM, MLM, and EDM is 8.2, 4.8, 6.89, 5.88, and 6.6 s, respectively. Table 3 demonstrates that the DS can produce the greatest results at each fold as well as average values. Consequently, the DS can obtain the best accuracy, precision, and recall values. Hence, DS can handle the changes in data distribution, class imbalance, or noisy features in real-world medical datasets.