Improving Feature Selection for Infant Mortality Risk Assessment via Binary Multi-Objective Cheetah Optimization

1.1. Enhanced Binary Multi-objective Cheetah Optimization (BMOCO)

We propose an improved version of the Binary Multi-Objective Cheetah Optimization method, tailored for feature selection in large-scale, high-dimensional medical datasets. The enhancement incorporates dynamic control mechanisms that regulate the balance between exploration and exploitation to achieve better convergence.

1.2. Optimized Transfer Functions for Binary Conversion

To improve the transformation of continuous position vectors into binary format, eight S-shaped and V-shaped transfer functions are systematically evaluated and optimized. This optimization enhances the precision and robustness of the feature selection process, which is especially important when working with sensitive medical data.

1.3. Comprehensive Benchmarking Against State-of-the-art Algorithms

The proposed BMOCO method is extensively benchmarked against several leading multi-objective evolutionary algorithms, including MOGA [16], MOALO [23], NSGA-II [24], and MOQBHHO [17], demonstrating its competitive advantage.

1.4. Application to Real-world Infant Mortality Datasets

The proposed methodology is validated on real-world U.S. birth datasets. In addition to enhancing classification accuracy, the method identifies key predictors of infant mortality, such as birth weight, maternal education, and prenatal care frequency, providing actionable insights for healthcare practitioners.

1.5. Framework for Data-driven Public Health Decision-making

This work establishes a methodological foundation for leveraging advanced optimization and machine learning techniques to inform targeted public health interventions, particularly those aimed at reducing infant mortality rates in underserved populations.

1.6. Background and Related Work

Research into neonatal health through predictive analytics has become increasingly vital as data science and machine learning technologies advance. It provides a comprehensive review of significant contributions in the relevant literature, focusing on methodologies and findings that directly influence the understanding and prediction of infant health and mortality. In recent studies, various maternal characteristics were scrutinized using logistic regression, naive Bayes, and linear support vector machines on data from U.S. Territories in 2013 [25]. These analyses have provided insights into dangerous maternal factors significantly impacting neonatal survival rates. Similarly, Gaussian Process Classification was proposed to predict hospital mortality among neonates, showcasing the potential of advanced probabilistic models in healthcare settings [26]. The application of machine learning techniques to assess the cost-effectiveness of long-term newborn care was explored, demonstrating the utility of predictive models in managing healthcare costs effectively [27].

Research has consistently highlighted the strong correlation between neonatal deaths and premature births. Predictive models were created using data from Italian neonates and subsequently tested and validated across various temporal settings, demonstrating their robustness and adaptability [28]. Furthermore, machine learning techniques were employed in Neonatal Intensive Care Units (NICUs) to assess short-term mortality, which has crucial implications for clinical decision-making and treatment optimization [29]. Exploring maternal traits associated with Small Gestational Age (SGA) has been crucial for early identification of at-risk neonates, particularly in resource-limited settings [30]. Different socioeconomic characteristics, such as maternal education and birth order, were analyzed to predict infant mortality using a three-year comprehensive dataset, which provided deep insights into the social determinants of health [31].

Despite all these significant advancements, the evolution of methodologies for effectively selecting variables that predict infant mortality still needs to be improved. A study utilized a dataset of 275 neonates to demonstrate how random forest models could excel in predicting preterm neonatal death, emphasizing the need for high sensitivity and specificity in predictive models [32]. Adapting swarm-based optimization methods, inspired by the foraging and hunting behaviors of untamed creatures, introduces another novel approach to feature selection in predictive modeling [33-35]. Specifically, the cheetah’s unique hunting strategies have inspired algorithms that balance efficiency with the ability to cover extensive search areas [11]. Evolutionary strategies such as these can improve algorithmic performance by adapting search tactics based on the success of previous attempts, thereby showcasing flexible problem-solving in dynamic environments [36].

The feature selection task can be framed as a binary optimization problem [37] where the transfer function plays a pivotal role [38]. This function is essential for converting a numeric search space into a discrete domain, a step that is crucial for applying a binary optimization method effectively [39]. Optimizing transfer functions and feature subsets has been emphasized as a means to enhance the exploration and exploitation capabilities of algorithms, thereby avoiding local optima and improving overall model performance. Integrating machine learning with traditional epidemiological approaches has also opened new avenues for understanding complex interactions between genetic, environmental, and social factors contributing to neonatal outcomes [28]. Such a multidisciplinary approach could enable a more comprehensive understanding of the causes of infant health and mortality, which is crucial for developing more targeted interventions.

In summary, the reviewed literature highlights the critical role of feature selection and predictive modeling in advancing neonatal health analytics. Despite notable progress, there remains a pressing need for more adaptive, scalable, and interpretable methods that can effectively extract meaningful predictors from high-dimensional datasets. This study addresses this gap by proposing an optimized feature selection framework grounded in evolutionary computation.

2.1. Foundations of Cheetah-Based Multi-objective Feature Selection

We delve into the advanced optimization techniques underpinning the current research study, focusing on integrating Cheetah Optimization and Multi-Objective Optimization to address the complex challenges inherent in predicting infant mortality. These approaches capitalize on the inherent patterns and behaviours observed in nature and adapt them to address the challenges of intricate, high-dimensional spaces, which are often typical of medical data analytics. The proposed Binary Multi-Objective Cheetah Optimization (BMOCO) method represents an approach specifically tailored to manage the discrete complexities of high-dimensional datasets effectively, particularly for feature selection. By balancing the accuracy of feature selection with the overarching goal of model simplicity, BMOCO aims to enhance predictive performance while minimizing computational demands. It outlines a detailed description of the Cheetah Optimization Algorithm and the theoretical foundations of multi-objective optimization. The Cheetah Optimization Algorithm (COA) [11] is a novel, population-based optimization approach inspired by cheetahs’ dynamic and strategic hunting behaviour. Before discussing the specifics of the COA, it is important to visualize how a cheetah approaches its hunt. Figure 1 illustrates the hunting strategies employed by cheetahs, highlighting their strategic and responsive nature, which the algorithm seeks to emulate computationally. The cheetah's hunting strategy often begins with a period of vigilance. When a cheetah sights a target (prey) in its surroundings, it may choose to remain stationary and wait for the prey to move closer before launching an attack. This attack mode typically involves stages of stirring and entrapping. Conversely, the cheetah may abandon the hunt for several reasons, such as fatigue or the prey's rapid movement. In such cases, the cheetah may take a break and subsequently return to hunting after a pause. The cheetah selects the optimal tactic by assessing the target’s (prey) state, location, and proximity (Fig. 1a-d) [11]. The COA mimics these natural tactics to address optimization problems, integrating biologically inspired strategies to enhance algorithmic efficiency and adaptability.

Specifically, the COA model adapts to dynamic target (prey) availability and the cheetah’s physical state, employing a blend of passive and active hunting tactics. These strategies are designed to optimize energy expenditure and increase the probability of successful hunts, directly analogous to optimizing computational resources and solution accuracy in complex optimization scenarios.

2.1.1. Search Strategy

The COA models the cheetah’s prey-search behaviour by employing either an active patrol or a passive wait approach, depending on the density and behaviour of the target (prey).

In scenarios where the target (prey) is abundant and stationary, the algorithm adopts a scanning mode from a fixed position, thereby maximizing energy efficiency. Conversely, when the target (prey) is dispersed, it necessitates an active, energy-intensive search strategy. This adaptability is mathematically modelled as follows: (Eq. 1)

where X_i,j^t+1 is the cheetah's next position, X_i,j^t is the current position, t is the current time, and α_i,j^t represent the randomization parameter and step length, respectively. This dual-mode hunting strategy reflects the algorithm’s ability to switch between exploration and exploitation, optimizing the search for optimal solutions.

2.1.2. Sit-and-wait Strategy

This energy-conserving strategy involves the cheetah lying in wait, minimizing its movements to avoid alerting nearby targets (prey). The algorithm mirrors this behaviour by maintaining the position until optimal conditions for an attack arise, thus saving computational resources and focusing efforts only when high-quality solutions are within reach: (Eq. 2)

2.1.3. Attack Strategy

Mirroring the cheetah’s rapid chase, this strategy is activated when the target (prey) is within optimal range. The algorithm calculates the trajectory for an effective intercept, utilizing speed and tactical repositioning, which represents a rapid convergence towards the best-known solution: (Eq. 3)

where X_B,j^t indicates the best current position of the target (prey), and rˇ_i,j, and β_i,j^t are the turning and interaction factors, respectively.

2.1.4. Fundamentals of Multi-Objective Optimization (MOO)

Multi-Objective Optimization (MOO) involves optimizing multiple conflicting objectives simultaneously, a common challenge in complex decision-making scenarios [40]. The goal is to find a solution that achieves the best possible balance among competing objectives. Mathematically, an MOO problem can be formulated as follows: (Eq. 4)

In the above formulation, f₁, f₂,..., f_m and g₁, g₂,..., g_n are the objective functions to be minimized and maximized, respectively. e_j and h_k represent the inequality and equality constraints the solutions must satisfy, with y_iLB and y_iUB denoting the lower and upper bounds of the decision variables y_i. The concept of dominance is crucial to this approach. A solution p1 is said to dominate another solution p2 if it performs better in at least one objective without being worse in any of the others. Optimal solutions form a Pareto front, a benchmark for evaluating trade-offs among competing objectives [41]. This approach enables a holistic view of potential solutions, often leading to innovative outcomes that traditional single-objective optimization might overlook [42-44].

Despite its potential, the literature suggests the need for further research on multi-objective evolutionary approaches to feature selection in health data. Exploring this gap can lead to significant advancements in the analysis and understanding of complex medical datasets, offering a balanced approach to predictive accuracy and computational efficiency and ultimately enhancing patient outcomes. The exploration of advanced optimization techniques sets a solid foundation for the methodologies employed in this study, aiming to leverage these sophisticated algorithms to improve predictive models of infant health outcomes. The integration of these techniques represents a convergence of biological inspiration and mathematical precision, poised to make significant contributions to healthcare analytics. Such an evolutionary-based advanced optimization approach is Cheetah Optimization for feature selection.

2.2. Proposed Method

The study presents the proposed Binary Multi-Objective Cheetah Optimization (BMOCO) model in detail, outlining its algorithmic components and the workflow for feature selection in high-dimensional medical datasets. Inspired by the adaptive hunting behavior of cheetahs, BMOCO incorporates multiobjective optimization principles and a dynamic transfer function mechanism to identify the most relevant features while maintaining classification accuracy. The method is structured around several key components, including position encoding, fitness evaluation, archive management, and strategy-driven position updates. Each component contributes to a flexible and efficient search process capable of handling the complexities of large-scale health data.

2.2.5. Balance between Exploration and Exploitation

CO’s hunting strategies prevent premature convergence, a common issue in optimization tasks. By enabling a balanced approach to exploration (searching through diverse areas of the search space) and exploitation (intensifying the search around promising areas), CO ensures a thorough investigation of potential solutions.

2.2.7. Feature Selection Using Binary Multi-Objective Cheetah Optimization (BMOCO)

Figure 2 displays the conceptual framework of the proposed feature selection methodology using the binary version of Multi-Objective Cheetah Optimization (BMOCO). This adaptation addresses the discrete nature of the feature selection problem, where features are either selected or not, making the original continuous-domain CO algorithm unsuitable without modifications [11].

The adjustments necessary to adapt CO for feature selection are outlined as follows:

2.2.7.1. Position Encoding

In BMOCO, each cheetah is represented by a binary string of length L + 3 (where L is the number of features in the dataset). This encoding is crucial for the discrete nature of feature selection, where each bit in the string (0 or 1) signifies the exclusion or inclusion of a feature, and the additional three bits determine the choice of transfer function used during optimization. Figure 3 illustrates this binary position encoding.

2.2.7.2. Fitness Computation

The fitness of each cheetah’s configuration in BMOCO is evaluated based on two primary objectives: (Eq. 5)

The first objective, Obj1, aims to minimize the number of features selected to simplify the model and reduce overfitting. This reduction is crucial in enhancing model generalizability and computational efficiency. (Eq. 6)

Where X represents the position vector of a cheetah. To compute Obj2 features corresponding to ones in X, compress the dataset, and evaluate classification performance using a KNN classifier and 10-fold cross-validation.

2.2.7.3. External Archive Maintenance

An external archive is crucial for storing non-dominated solutions identified during optimization. It maintains diverse solutions, reflecting the best trade-offs between minimizing feature count and maximizing classification accuracy. This approach is vital for handling multi-objective optimization’s inherent complexities and ensuring a broad solution for space exploration. To introduce the external archive’s management, Figure 4 depicts how solutions are evaluated and either retained or replaced based on their dominance relations.

2.2.7.4. Transfer Function Optimization

The adaptation of transfer functions in BMOCO is essential for effectively navigating the binary search space. These functions adjust the probabilities of bit flips in a cheetah’s position vector, influencing the exploration and exploitation dynamics within the algorithm. This nuanced handling helps significantly enhance the convergence behaviour, allowing the algorithm to effectively escape local optima and ensuring a comprehensive search across the feature space. Before diving into the specific functions used, it is beneficial to understand the variety and purpose of each transfer function employed. These functions are designed to modify the solution space navigation differently, optimizing the search and solution refinement processes as depicted in Table 1.

Figure 5a, b illustrates the visual appearance of the eight transfer functions. In each iteration, after assessing the fitness of the members of the cheetah population, the set of non-dominated solutions is identified and strategically arranged in reverse order based on Crowding Distance (CD) values. The solution with the highest CD value is then selected as the ’prey,’ and the corresponding transfer function is adopted for the subsequent step of the process. It is important to note that although the transfer function is selected dynamically during the run, it is not updated once chosen.

Table 1.

S-Shaped TFs			V-Shaped TFs
Name	Function	Bits	Name	Function	Bits
S1		000	V1		100
S2		001	V2		101
S3		010	V3		110
S4		011	V4		111

The selection of transfer functions within BMOCO is pivotal in effectively navigating the solution landscape. Each transfer function modifies the probability distribution for selecting features, thus influencing the algorithm’s balance between exploration and exploitation. By adopting these functions dynamically based on the optimization state, BMOCO can maintain diversity in the solution pool while efficiently converging to optimal solutions.

Three additional bits in each cheetah’s position string are used to select among these eight transfer functions, further tailoring the search process to the specific dynamics of the feature selection landscape. In each iteration, after assessing the fitness of each cheetah, the set of non-dominated solutions is strategically arranged in reverse order of crowding distance (CD) values. The highest CD value solution is then selected as the ’prey,’ guiding the subsequent search direction. The selected transfer function significantly impacts how the search progresses, ensuring adaptability and robustness in the search strategy.

2.2.7.5. Position Update

The position update mechanism in BMOCO reflects the adaptive strategies cheetahs employ in the wild, balancing energy conservation during the search phase and aggressive pursuit during attacks.

When hunting, either the seeking or the attacking strategy may be applied depending on the context, but as the cheetah’s energy declines, the likelihood of switching to the search strategy increases. During the early iterations, the algorithm emphasizes exploration through the search method, while later iterations favor exploitation by applying the attack strategy to refine candidate solutions. The transition between strategies is controlled using random thresholds. If r2 ≥ r3, the cheetah adopts a sit-and-wait approach, and its position remains unchanged as per Eq. (2). Otherwise, the value of r1—a random number in [0, 1]—is used to compute H = e^2(1−t/T)(2r1 − 1), which governs the strategy choice. If H ≥ r4, the attack strategy is employed using Eq. (3); otherwise, the search strategy in Eq. (1) is used. The prey’s location is defined as the solution with the highest crowding distance in the current archive, representing a high-quality, sparsely crowded solution. The selected transfer function from the prey is then used to probabilistically update the cheetah’s position via Eq. (7), allowing efficient exploitation of promising regions of the search space [38].

Where rnd is a random number between [0,1], X represents the cheetah’s position, L is the dimension, t is the current iteration, ¬ denotes negation, and T () is the chosen transfer function. Random thresholds govern the sit-and-wait approach and an active search-and-attack. This ensures that the strategy shifts dynamically based on the situation, reflecting the real-time decision-making processes observed in natural cheetah behaviour.

2.2.7.6. BMOCO Algorithm

The BMOCO Algorithm 1 encapsulates the entire feature selection process, from initialization to the final selection of optimal feature sets based on crowding distance. This process ensures the algorithm finds non-dominated solutions regarding feature count and classification accuracy, representing a balance between these competing objectives. BMOCO’s computational complexity primarily depends on the number of features and the size of the dataset. Each algorithm iteration evaluates all individuals in the population across the entire feature set, making the computational cost proportional to the product of these factors. This scaling is crucial in high-dimensional data, where efficient handling of large feature sets becomes imperative.

2.2.7.7. Selection of the Crowding Distance-Based Optimal Solution from the Repository

After a specified number of iterations, the external archive, filled with non-dominated solutions, is assessed to identify the optimal set of features. This selection uses the Crowding Distance (CD) metric, which measures density around each solution in the objective space, ensuring that the selected features are practical and diverse [17].

This comprehensive approach, embodied in the BMOCO algorithm, harnesses the dynamics of natural predation and adapts them for complex, high-dimensional feature spaces, driving towards solutions that offer a practical balance between minimal feature sets and maximum classification accuracy.

Algorithm 1 BMOCO-Based Feature Selection

Input: Population size N, Maximum iterations MaxIt, Dataset D

Output: Optimized feature set A in external archive 1: Initialize the population of cheetahs randomly.

2: Calculate initial objective values Obj1 and Obj2 for each cheetah.

3: Store the non-dominated solutions in an external archive. 4: fori = 1 to MaxIt do

5: Select K (where 2 ≤ K ≤ N) cheetahs randomly. 6: for all cheetahs in the selection do

7: Calculate rˆ, rˇ, α, β, and H based on current and local best positions. 8: Generate random numbers r2, r3, r4 uniformly distributed in [0, 1].

9: if r2 ≤ r3 then

10: if H ≥ r4 then

11: Update the position of the cheetah using the attack strategy equation. 12: else

13: Update position using the search strategy equation. 14: end if

15: else

16: Maintain position using a sit-and-wait strategy. 17: end if

18: Apply the appropriate transfer function.

19: Recalculate objective values Obj1 and Obj2.

20: Update the external archive with new non-dominated solutions. 21: end for

22: end for

23: Return the best solution from the archive based on Crowding Distance (CD).

2.3. Experimental Setup and Benchmarking

It outlines a comprehensive experimental framework designed to assess the effectiveness of the BMOCO method in feature selection tasks. It details the computational setup, the datasets employed, and the benchmarking algorithms against which BMOCO is compared. This framework facilitates a robust, reproducible, and fair evaluation by aligning computational settings, classifier choice, and metric definitions across all comparative algorithms. This rigorous evaluation not only underscores the adaptability and efficiency of BMOCO but also situates it within the broader context of existing multi-objective optimization methods. We further elucidate the dataset’s structure, the preprocessing steps to ensure data quality, and a suite of multi-objective performance indicators used to measure the efficacy of the proposed and existing methods. This methodical approach ensures a thorough understanding and validation of the optimization capabilities of BMOCO, aiming to establish new benchmarks in feature selection for complex datasets.

2.3.4. Generational Distance (GD)

This metric measures the Euclidean distance between the solutions obtained by the algorithm (computed front, A) and the true Pareto front (PF). It is defined as: (Eq. 8)

A smaller value of GD indicates that the computed solutions are closer to the Pareto front, reflecting higher solution quality. In this study, the actual Pareto front is estimated by pooling and filtering non-dominated solutions from multiple runs of all algorithms, as suggested by [47].

2.3.5. Inverted Generational Distance (IGD)

Complementary to GD, IGD measures how well a set of solutions approximates the true Pareto front: (Eq. 9)

An optimal algorithm would minimize IGD, indicating comprehensive coverage and proximity of its solutions to all members of the actual Pareto front.

2.3.7. Spread

This metric evaluates the diversity of solutions across the Pareto front by measuring the extent of spread among all objective function values: (Eq. 10)

A desirable property for an effective optimization is a broader spread, which indicates a diverse set of solutions spanning possible trade-offs between objectives.

Employing these metrics provides a robust framework for comparing the effectiveness and efficiency of various multi-objective feature selection methods, aiding in identifying the most suitable algorithm for handling the complexity and nuances of newborn dataset feature selection.

3.1. Performance Comparison based on Pareto Fronts

It delves into the comparative performance of the suggested Binary Multi-Objective Cheetah Optimization (BMOCO) method against established benchmarks using the representation of Pareto fronts. Pareto fronts illustrate the trade-offs between conflicting objectives, such as minimizing feature count while maximizing classification accuracy. By examining these fronts, we can evaluate each method’s effectiveness in balancing model simplicity with predictive performance.

The Pareto fronts depicted in Figure 6 show that those from BMOCO are consistently closer to the true Pareto fronts across all datasets. This proximity indicates that BMOCO efficiently generates solutions that reduce the number of features used and maintain or enhance classification accuracy. A detailed analysis reveals that BMOCO outperforms other methods, particularly in its ability to identify critical predictive features with fewer indicators, as evidenced by the high accuracy rates achieved with significantly reduced feature sets.

Further analysis of specific datasets reveals additional remarkable results. For instance, the 2016 dataset managed with only 18 features reached an accuracy of 96.1%, closely approximating the actual model performance. In 2017, both MOQBHHO and BMOCO achieved excellent results, but BMOCO’s 99.7% accuracy with 41 features stands out, showcasing its efficiency in feature reduction without compromising accuracy.

As discussed in the study, the Crowding Distance (CD) metric has been employed as the primary criterion for selecting the best non-dominated solutions at the end of an optimization process. This metric helps identify the most effective solutions that balance multiple objectives without bias, ensuring a fair comparison across all methods. Table 4 summarizes the best solutions selected based on the CD criterion, underscoring the superior performance of BMOCO in achieving high accuracy with fewer features across multiple datasets.

The detailed analysis of solutions presented in Table 4 highlights BMOCO’s exceptional ability to minimize the number of features while maximizing classification accuracy. For example, in the 2019 and 2020 datasets, BMOCO achieves nearly the highest accuracy with significantly fewer features than other methods, demonstrating its effectiveness in feature optimization and model efficiency. This performance indicates BMOCO’s robust strategy formulation and adaptation to the complexities of high-dimensional feature spaces. These results substantiate the potential of BMOCO in improving feature selection methodologies in complex, data-driven fields.

3.2. Performance Comparison based on Objective Function Metrics

It evaluates the performance of Binary Multi-Objective Cheetah Optimization (BMOCO) compared to other benchmark methods based on objective function metrics, average feature size, and classification accuracy. These metrics provide insights into each method’s efficiency in reducing feature set dimensionality while maintaining or improving the model’s predictive accuracy.

Table 5 presents a detailed analysis of the average feature sizes and classification accuracy values obtained after 30 iterations of feature selection. Notably, BMOCO consistently achieves lower average feature sizes while maintaining competitive classification accuracies, emphasizing its efficiency in simplifying models without significant loss in performance. For instance, in the 2016 and 2020 datasets, BMOCO significantly reduces the number of features while achieving the highest classification accuracy among the compared methods. Additionally, in 2017, NSGA-II identified a more significant feature subset and achieved slightly higher accuracy than BMOCO, highlighting a trade-off between feature reduction and accuracy. However, BMOCO achieves near-optimal accuracy with considerably fewer features across all datasets, illustrating a substantial improvement in balancing model complexity and performance.

3.3. Performance Comparison based on Multi-objective Performance Measures

It assesses the efficacy of Binary Multi-Objective Cheetah Optimization (BMOCO) relative to other established multi-objective feature selection algorithms. The evaluation is grounded on four critical performance metrics: Generational Distance (GD), Inverted Generational Distance (IGD), Hypervolume (HV), and Spread, which collectively offer a comprehensive view of each method’s capability to handle trade-offs between conflicting objectives effectively.

Analysis from Table 6 summarizes that BMOCO demonstrates superior or competitive performance across multiple datasets. Notably, BMOCO consistently exhibits the lowest GD and IGD values for the 2016, 2018, and 2019 datasets, suggesting a closer approximation to the actual Pareto fronts than other methods. Furthermore, BMOCO’s spread values are exceptionally high, indicating a comprehensive coverage across the objective space, which is essential for effectively exploring diverse solutions. For instance, in the 2022 dataset, BMOCO reached the highest spread of 1.03, significantly outperforming other methods for exploring extensive regions of the solution space. Additionally, the HV values of BMOCO are commendable, with consistently high marks, underscoring its ability to maintain a balance between convergence and diversity in solution quality.

3.4. Performance Comparison based on the Wilcoxon Signed-rank Test

Our study assesses the statistical significance of the performance differences between the proposed Binary Multi-Objective Cheetah Optimization (BMOCO) Feature Selection (FS) approach and other benchmark methods. To ensure robustness in the comparisons, we employ the Wilcoxon signed-rank test on Inverted Generational Distance (IGD) values collected from 20 distinct runs of each method. This nonparametric test helps determine whether two related paired samples come from the same distribution, an essential aspect of verifying the statistical superiority or equivalence of the proposed method relative to others.

As presented in Table 7, the Wilcoxon test results exhibit statistically significant superiority (’++’) of the BMOCO method over most of the compared methods across most datasets. Specifically, BMOCO consistently outperforms others in datasets ranging from 2016 to 2021, except for 2022, where MOGA-FS shows superior performance as indicated by the ’–’ in the significance column. These results provide substantial statistical evidence that BMOCO is generally more effective at optimizing the trade-off between feature reduction and classification accuracy. Table 7 also highlights instances (’==’) where the performance between methods is statistically equivalent, providing a nuanced view of the competitive landscape in multi-objective feature selection.

3.5. Performance Evaluation based on Execution Time

This evaluation assesses the computational efficiency of five multi-objective Feature Selection (FS) methods by analyzing the average execution times across seven datasets. The proposed Binary Multi-Objective Cheetah Optimization Algorithm (BMOCO) demonstrates superior performance due to its selective participation mechanism in the evolutionary process and dynamic adjustment between exploration and exploitation phases based on heuristic cues.

Table 8 presents that BMOCO often requires less time to complete its iterations than its counterparts. This is particularly notable in complex datasets where efficient execution is critical. The NSGA-II method, known for its rigorous merging of populations and generation of non-dominated fronts, consistently shows longer execution times. In contrast, BMOCO efficiently leverages its adaptive exploration and exploitation phases, reducing overall computational overhead. MOQBHHO also shows competitive execution times, benefiting from a similar mechanism that adjusts based on the situational needs of the algorithm. The results underscore BMOCO’s potential in applications where execution time and solution quality are paramount.

3.6. Computational Complexity Analysis

The computational complexity of the proposed Binary Multi-Objective Cheetah Optimization (BMOCO) used for feature selection in a K-Nearest Neighbours (KNN) classifier framework is determined by several parameters: number of training samples (X), maximum number of iterations (MaxIt), population size (N), number of objective functions (J), and feature dimension (L). The total computational complexity of the BMOCO algorithm is represented by the following equation: (Eq. 11)

The computational effort required for initialization and each fitness calculation, when X training samples are involved, primarily depends on these parameters: the dimension of the features and the number of bits for the transfer function selection (L + 3). Typically, the complexity of fitness calculation using a KNN classifier is O(X × L). Thus, the initialization complexity is approximated by the following equation: (Eq. 12)

Each iteration involves updating positions, calculating fitness, and rearranging the repository based on crowding distance, which is computed by O(J × N log N) [17], considering N potential solutions and J objective functions that must be sorted as represented by the following equation: (Eq. 13)

This analysis underlines the BMOCO’s efficiency, especially in handling high-dimensional data through evolutionary operations and intelligent exploitation of feature space, resulting in a time complexity of approximately O(N log N).

4.1. Analysis of Selected Features

The analysis of features selected from various datasets highlights several critical determinants of infant survival: maternal education, prenatal care, pre-existing maternal conditions such as diabetes, maternal smoking during pregnancy, maternal Body Mass Index (BMI), infant birth weight, and breastfeeding practices. These factors collectively contribute to the varying rates of infant mortality observed across different socio-economic and demographic groups.

4.2. Positive Aspects of the Proposed Feature-Selection Method

Evaluating the proposed Binary Multi-Objective Cheetah Optimization (BMOCO) across various metrics and tests has revealed several advantages that make it a robust feature selection method for medical data analytics, particularly in infant health. These advantages underscore BMOCO’s capability to efficiently navigate the complex landscape of feature selection by leveraging its unique algorithmic strategies.