A Chaos-Enhanced Binary Newton-Raphson Optimizer for High-Dimensional Sensor Data Feature Selection

1. Introduction

One of the pre-processing steps in machine learning is feature selection (FS). For complex or huge datasets, selecting the most relevant feature subset is one of the most difficult challenges [1,2]. The search for hidden patterns or important information in massive amounts of data has become a critical issue. It has been demonstrated that feature selection efficiently eliminates superfluous and irrelevant features [3]. FS is a preprocessing method that selects relevant features to reduce overfitting, enhance model accuracy and interpretability, speed up learning, and reduce dataset memory needs [4]. It can also decrease the amount of storage needed, lower the computational cost, and enhance classifier performance. Both the number of attributes/features and the number of instances of data have increased in recent years [5]. Computational procedures known as feature selection algorithms are used to choose a collection of characteristics that maximizes an assessment metric that represents the quality of the characteristics [6]. There are two important factors to consider when choosing the optimum feature subsets, according to the literature: maximizing classification accuracy and minimizing the attributes found in the datasets [7,8].

An important consideration in dimension reduction is the evaluation metrics, which establish the quality of the chosen features. Dimension reduction techniques are typically separated into two major classes based on the evaluation measures: wrapper approaches and filter approaches, a learning/ classification method is used in wrapper approaches to determine the selected properties [9].

As a result, wrappers typically outperform filter approaches in classification. However, they suffer from high processing costs and a loss of generality, which makes them specific to a single classification algorithm [10]. Any learning algorithm has no bearing on filter methods. Therefore, appropriate evaluation metrics are typically required for filter approaches. Wrappers typically outperform filter approaches in terms of classification performance, but may incur significant processing costs and loss of generality due to algorithm specificity [11].

The size of the search space in dimension reduction issues increases exponentially with the number of attributes. In most situations, a thorough search is not necessary. Heuristic search techniques for dimension reduction have limitations, such as lengthy computation times and being stuck in local optima. To handle dimension reduction challenges, it is recommended to use the cheapest global search technique [12,13].

FS is typically performed as a pre-processing stage before model construction utilizing specialized algorithms [14,15]. Data mining and machine learning researchers are actively researching the creation of feature selection algorithms (FSAs). FSAs are computational methods that pick a set of features to optimize an assessment measure of feature quality [16,17].

In recent years, various new metaheuristic algorithms have been introduced to overcome the disadvantages of exact algorithms and to improve the performance of other metaheuristics [18]. Exact algorithms are not able to solve complex high-dimensional problems in a reasonable time because the search space increases exponentially with the problem size [3]. Some of the metaheuristic algorithms have provided outstanding performance in the problems. Most of them have been implemented for real search spaces [19].

Hence, different techniques have been proposed to use meta-heuristic algorithms in the binary search, such as Whale Optimization Algorithm (WOA) [20], Ant Colony Optimization (ACO) [21], Bat Algorithm (BA) [22], Artificial Bee Colony (ABC) [23], Particle Swarm Optimization (PSO) [24], Biogeography Based Optimization (BBO) [25], Genetic Algorithm (GA) [26], Harmony Search Algorithm (HSA) [27], Flower Pollination (FP) algorithm [28] Grasshopper Optimization Algorithm (GOA) [29], and Binary Dragonfly (BDF) algorithm [30].

Chaotic maps have been increasingly integrated into metaheuristic algorithms to improve exploration, maintain diversity, and avoid premature convergence. By replacing random components with deterministic chaotic sequences, these algorithms can better navigate complex search spaces and escape local optima. Recent studies have shown the effectiveness of chaos-enhanced metaheuristics: a chaotic transient search optimization algorithm improves global search performance [31], the Binary Chaotic Crow Search algorithm (BCCSA) [32], while a chaotic dwarf mongoose optimizer enhances feature selection accuracy [33].

For continuous optimization problems, Newton Raphson Based optimizer is among the most recently used metaheuristic algorithms [34]. It is straightforward and simple to put into practice. It also has a few control parameters that need to be modified. It has been widely applied to resolve a variety of optimization issues in the real world. Recent studies have focused on enhancing the NRBO algorithm across a range of application domains. For example, a multi-strategy improved version of NRBO has been proposed for engineering optimization problems [35], while a population-based variant has been developed for continuous optimization tasks [36]. In another application, an enhanced NRBO-XGBoost model was introduced for remote sensing of shallow water depth in the nearshore regions of the Beibu Gulf [37]. NRBO has also been applied to multi-level threshold image segmentation for tomato plant disease detection [38]. Given that neuroimaging and wearable rehabilitation data are typically high-dimensional, noisy, and redundant, feature selection becomes a crucial step for extracting stable digital biomarkers, reducing computational complexity, and improving clinical interpretability.

In this work, a novel binary chaotic variant of the Newton–Raphson-Based Optimizer, referred to as BCNRBO, is introduced. Different chaotic maps are examined to identify the most effective one for integration into the proposed algorithm, with the goal of improving search efficiency and enhancing feature selection performance. The selected chaotic map is utilized through a chaotic enforcement operator to dynamically adjust and improve the step size during the optimization process. Moreover, BCNRBO incorporates a new mechanism called Dynamical Potential, which is formulated as a time-varying potential function that adapts throughout the search process to regulate the algorithm’s behavior. The DP mechanism is based on the Lennard–Jones potential [39], in which the Hamming distance [40] is embedded to quantify the interaction force between candidate solutions and promote population diversity. In addition, a novel transfer function is proposed to transform continuous solutions into binary representations. This transfer function is driven by chaotic values and follows an S-shaped form, enabling an effective balance between exploration and exploitation in the binary search space.

In order to maximize classification accuracy, choose the most advantageous features that minimize dataset size, and enhance algorithm stability; The performance of the proposed BCNRBO method was evaluated using 26 different benchmark datasets, including medical datasets with different sizes (low, medium, and large scale). The results show that the proposed BCNRBO achieves better results than other similar algorithms in the literature, such as: Binary Arithmetic Optimization Algorithm (BAOA) [41], Binary Bat Algorithm (BBA) [42], Binary Flower Pollination Algorithm (BFPA) [43], Binary Particle Swarm Optimization (BPSO) [44], BCCSA [32], Binary Atom Search Optimization (jBASO) [45], and Binary Dwarf Mongoose (BDA) [46].

The rest of the paper is organized as follows: Section 2.2 gives an outline of the standard Newton Raphson Based Optimizer (NRBO), chaotic maps,and the potential technique. Section 3 presents the proposed binary chaotic newton-raphson based optimizer. Section 4 describes the Computational experiments design for the feature selection problem. Section 5 presents the experimental results and discussions. Section 6 concludes the study and provides insight into future trends.

2. Preliminaries

2.1. Background

Recent advances in metaheuristic optimization algorithms have demonstrated their effectiveness in solving feature selection and classification problems, particularly in high-dimensional search spaces. For example, human-inspired and nature-inspired optimization techniques have attracted increasing attention due to their strong exploration and exploitation capabilities [47]. The cultural history optimization algorithm has been proposed as an effective human-inspired metaheuristic for engineering optimization tasks [9]. In addition, several hybrid frameworks integrating swarm intelligence with machine learning models, such as PSO-LSTM and Whale Optimization Algorithm combined with LSTM, have shown improved convergence behavior and classification accuracy.

Furthermore, comprehensive studies on Elephant Optimization Algorithms [48] and Mountain Gazelle Optimizers [47] have provided valuable insights into algorithm classification, application domains, and future research directions. Recently, adaptive optimization techniques combined with deep learning have also been successfully applied to classification and cyber security problems, such as phishing email detection [3]. These studies highlight the importance of population diversity enhancement, adaptive search mechanisms, and hybridization strategies, which motivate the development of the proposed BCNRBO for feature selection.

Recent advances in learning-based feature selection and optimization techniques have shown promising results across various domains. Zhou et al. [49] proposed an attention-based interaction-aware spatio-temporal graph neural network (AST-GNN) for trajectory prediction, demonstrating the effectiveness of modeling complex dependencies through deep neural architectures. Fan et al. [50] introduced an adaptive data structure regularized multiclass discriminative feature selection method, which effectively captures the underlying structure of high-dimensional datasets for improved classification performance. Furthermore, Wu et al. [51] presented ReffaceNet, a reference-based framework for face image generation from line art drawings, highlighting recent developments in neural network-based representation learning.

In contrast, metaheuristic optimization methods, such as the proposed BCNRBO, provide flexible and computationally efficient alternatives capable of handling high-dimensional binary search spaces while maintaining competitive classification accuracy. This motivates the integration of chaotic dynamics into NRBO to enhance the exploration–exploitation balance in binary feature selection tasks.

On the other hand, chaotic maps have been widely adopted in metaheuristic optimization to improve exploration capabilities, enhance diversity, and mitigate premature convergence by replacing or augmenting traditional random components with deterministic yet unpredictable sequences. Instead of relying solely on pseudo-random number generators, sequences produced by chaotic maps allow algorithms to explore the search space more thoroughly and avoid stagnation in local optima, leading to better overall search performance in many applications. Recent studies have systematically analyzed the role of chaos in metaheuristics, showing significant improvements for variants of many well-known algorithms, such as chaotic particle swarm optimization, chaotic whale and sine cosine algorithms, and other chaos-enhanced population-based methods across benchmark and engineering problems [31,52,53,54]. Moreover, chaos-based feature selection approaches have demonstrated enhanced convergence and accuracy compared with their non-chaotic counterparts by integrating chaotic maps directly into initialization, parameter control, and transfer functions [32,33,55]. These results highlight the increasing effectiveness of chaos integration as a strategy for improving modern metaheuristic algorithms.

To provide a clear overview of the existing research and highlight the key differences among prior studies, a summary table of the reviewed papers is presented below (Table 1). This table compares the algorithms, datasets, and main contributions, which helps to identify the remaining research gaps and justify the proposed approach.

2.2. Newton Raphson Based Optimizer

The Newton Raphson Based Optimizer is an advanced optimization algorithm inspired by the Newton-Raphson method, which is commonly used to find solutions to non-linear equations. NRBO leverages iterative approximation to efficiently converge to the optimal solution by adjusting the search parameters based on the gradient and curvature of the objective function [65]. Newton’s approach uses gradient information to iteratively improve initial estimates and guide the search for solutions [34]. The NRBO introduces three key principles for exploring solution spaces: using gradients to reach optimal points, iterative adjustments with random variations to maintain diversity and avoid premature convergence, and using a trap avoidance strategy helps to navigate around local optimum points. Further details are provided in the flowchart shown in Figure 1.

2.2.1. Initialization and Fitness Evaluation

NRBO is a population optimization system that produces and updates several agents $N_P$ during each iteration ($Iter$) until the maximum number of iterations ($MaxIter$) is reached. Each agent is addressed by a decision vector $X_n^j$ of dimensionality Dim. The solution vectors for agents are randomly assigned within the specified boundaries ($Uppe{r}^j\mathrm{and}Lowe{r}^j$), and calculated as follows:

$$X_i^j(Iter=1)=Lowe{r}^j+\left([Uppe{r}^j-Lowe{r}^j]\times rand\right),i=1:N_P,j=1:Dim$$

(1)

where rand denotes the random number. For each agent $X_i^j$ in the population, the objective function is evaluated. The agents are ranked based on their fitness values.

The parameter Draft factor (DF) is responsible about determining the strategy compared between the following two phases. The value of DF is set to be equal to 0.6 as shown in the original paper.

2.2.2. Newton-Raphson Search Rule (NRSR)

The NRSR determines the agents, allowing for improved exploration of the feasible zone and better positioning. The NRSR used to update each agents position in the population using gradient based information for optimization. This rule uses a formula to update agents locations based on relative fitness values, iteratively. The update method incorporates a random perturbation factor ($\Delta X$) that is proportionate to the difference between the best-performing and current agents. This increases the diversity of solutions. The NRSR formula calculates a weighted difference ($\Delta X$) between two solution vectors. This method changes agents’ positions instead of their fitness values, resulting in more efficient computation and fewer recalculations in the following steps.

Solutions are generated by combining three components, each perturbed randomly to encourage diverse exploration. The primary components ($X_1,X_2,$ and $X_3$) for each solution are calculated iteratively using factors that account for previous best solutions and random elements. At each iteration, the location of the agent ($X_i^j$) is updated based on weighted averages of ($X_1,X_2,$ and $X_3$), as shown below.

$$\begin{array}{cc}\hfill X_i^j(Iter+1)=r{d}_1^j\times \left(r{d}_1^j\times X_{1i}^j\left(Iter\right)+(1-r{d}_1^j)\times X_{2i}^j\left(Iter\right)\right)& +\left((1-r{d}_1^j)\times X_{3i}^j\left(Iter\right)\right),\hfill \\ \hfill & i=1:N_p,j=1:Dim,\hfill \end{array}$$

(2)

where $X_{1i}^j\left(Iter\right)$, $X_{2i}^j\left(Iter\right)$, and $X_{3i}^j\left(Iter\right)$ are formulated with additional random variations and factors that incorporate differences between the best and worst agents, as well as the mean of current positions as follow:

$$X_{1i}^j\left(Iter\right)=X_i\left(Iter\right)-(randn\times \left({\displaystyle \frac{(Y_W-Y_V)\times \Delta X}{2\times (Y_W+Y_V-2\times X_i)}}\right)))+\rho .$$

(3)

$$X_{2i}^j\left(Iter\right)=X_{best}^j-randn\times \left({\displaystyle \frac{(Y_W-Y_V)\times \Delta X}{2\times (Y_W+Y_V-2\times X_i)}}\right)+\rho ,$$

(4)

$$X_{3i}^j\left(Iter\right)=X_i\left(Iter\right)+\delta \times \left(X_{2i}^j\left(Iter\right)-X_{1i}^j\left(\mathrm{Iter}\right)\right),$$

(5)

where $randn$ denotes the normal-distributed random number with mean 0 and variance 1, $r{d}_1^j$ is a random value between 0 and 1. The parameter $\delta$ is calculated from the following equation:

$$\delta ={\left(1-{\displaystyle \frac{2·Iter}{MaxIter}}\right)}^5.$$

(6)

The exploitation of the NRBO is improved by including another parameter called $\rho$, which directs the population in the right direction. The expression for $\rho$ is given as follows.

$$\rho =a\times \left(X_{best}-X_i\left(Iter\right)\right)+b\times \left(X_{r1}\left(Iter\right)-X_{r2}\left(Iter\right)\right),$$

(7)

where $X_{r1}\left(Iter\right)$ and $X_{r2}\left(Iter\right)$ correspond to two different randomly chosen vector agents among the population, a and b are arbitrary values that range from 0 to 1. The components of $Y_V$ and $Y_W$ can be derived using the following equations:

$$Y_V=r{d}_2^j\times \left(Mean(X_i-randn\times \left({\displaystyle \frac{(X_{worst}-X_{best})\times \Delta X}{2\times (X_{worst}+X_{best}-2\times X_i)}}\right),X_i)\right)-r{d}_2^j\times \Delta X,$$

(8)

$$Y_W=r{d}_2^j\times \left(Mean(X_i-randn\times \left({\displaystyle \frac{(X_{worst}-X_{best})\times \Delta X}{2\times (X_{worst}+X_{best}-2\times X_i)}}\right),X_i)\right)+r{d}_2^j\times \Delta X,$$

(9)

where $X_{worst}$ denotes the best position and $X_{best}$ denotes the worst position, $r{d}_2^j$ is a random value between 0 and 1 and $\Delta X$ indicates its weighted differences as described as follows:

$$\Delta X=rand\times |X_{best}-X_i|,rand\in [0,1].$$

(10)

2.2.3. Trap Avoidance Strategy (TAS)

In the NRBO, the TAS enhances the suggested NRBO capability to address real-world problems and is used to help agents escape local optima and avoid premature convergence.

This technique creates new positions by using the best performing agent ($X_{best}$) and the current agent’s position ($X_i\left(Iter\right)$).

$$X_{TAS}\left(Iter\right)=\left\{\begin{array}{cc}{X}_i(Iter+1)+X_A\left(Iter\right)+X_B\left(Iter\right),\hfill & \mathrm{if}{\mu }_a<0.5\hfill \\ X_{best}+X_A\left(Iter\right)+X_B\left(Iter\right),\hfill & otherwise,\hfill \end{array}\right.$$

(11)

$$X_A\left(Iter\right)={\theta }_1\times \left(({\mu }_a\times X_{best})-({\mu }_b\times X_i\left(Iter\right))\right),$$

(12)

$$X_B\left(Iter\right)=\left({\theta }_2\times \delta \right)\times \left(({\mu }_a\times Mean\left(X_i\right))-({\mu }_b\times X_i\left(Iter\right))\right),$$

(13)

where ${\theta }_1$ and ${\theta }_2$ are uniform random numbers between $(-1,1)$ and $(-0.5,0.5)$, respectively, DF denotes the deciding factor that controls the NRBO performance, and ${\mu }_a$ and ${\mu }_b$ are random numbers, and are generated using Equation (14) and Equation (15), respectively.

$${\mu }_a=(1-\beta )+3\times \beta \times rand,{\mu }_b=(1-\beta )+\beta \times rand,$$

(14)

$$\beta =\left\{\begin{array}{cc}0,\hfill & \mathrm{if}rand>0.5\hfill \\ 1,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(15)

where rand denotes the uniform random number between (0,1) and $\beta$ denotes a binary number, either 1 or 0. If the value of $rand$ is greater than or equal to 0.5, then the value of $\beta$ is 0; otherwise, the value is 1. Because of the randomness in the selection of the parameters ${\mu }_a$ and ${\mu }_b$, the population becomes more diverse and escapes from the local optimum solutions, which helps to improve its diversification.

For each agent $X_i$ in the population, the objective function is evaluated. The agents are ranked according to their fitness values to identify the best agent $X_{best}$ with the lowest fitness value and the worst agent $X_{worst}$ with the highest fitness value.

This following steps summarize the NRBO procedure.

Step 1.

Specify the NRBO parameters: $Dim,Lower,Upper,N_p,MaxIter,a,b$.

Step 2.

Initialize the solution vectors using Equation (1).

Step 3.

For each solution vector, evaluate the fitness function to assess the performance of each solution.

Step 4.

Rank the solutions based on their fitness values. Identify the agent with the lowest fitness as the best, and the one with the highest fitness as the worst.

Step 5.

Generate a random number $RN$ to determine the strategy compaired with Draft factor (DF=0.6) to apply:

• Apply the NRSR using Equation (2) to update agent positions using gradient-based information for local refinement.
• Apply the TAS using Equation (11) to enhance exploration and help agents escape local optima.

Step 6.

Re-evaluate the fitness function for the updated design variables.

Step 7.

Repeat steps 3–6 until the maximum number of iterations is reached. Once complete, the final position of the best agent is considered the optimal solution.

2.3. Chaotic Maps

Chaotic maps are deterministic systems that respond to beginning conditions and exhibit disorganized or random behavior. Chaos theory investigates both deterministic and dynamic system behaviors. Depending on the initial conditions, chaotic variables can cycle through all states in specific periods without repetition. Creating chaotic number sequences is quick and easy. Chaos search outperforms random search when it comes to escaping the local optimum due to its unique tendencies [66,67].

Chaos refers to dynamical processes that appear random, but are actually generated by regular systems. This behavior is typical of stochastic processes, such as the movement of molecules in a gas filled vessel. However, chaotic systems are inherently nonlinear; typical statistical approaches that are frequently linear are insufficient for their analysis. Chaotic system characteristics are similar to random like systems, making them readily confused with random noise.

In this study, the chaotic maps will be incorporated into the binary NRBO in order to enhance its performance.

Ten chaotic maps are used in this study, namely Chebyshev, Circle, Gauss/Mouse, Iterative, Logistic, Piecewise, Sine, Singer, Sinusoidal and Tent map. Table 2 contains additional specific information about the common mathematical equations of the shared chaotic maps. These chaotic maps are visualized in Figure 2 for 10 different chaotic map values. Most chaotic maps used in integrating chaos theory into metaheuristic algorithms are either one or two dimensional. In the proposed algorithm, one-dimensional chaotic maps are used as random number generators at the beginning of the search process to improve the performance of BCNRBO, find effective solutions in the search space, and increase the stability of the algorithm for solving FS problem. The starting value was assigned randomly within the range of [0, 1] for all chaotic maps used in the proposed algorithm.

As shown in Figure 2, Map10 exhibits significantly different chaotic behavior compared to the other maps. While several maps demonstrate periodic tendencies or limited oscillation ranges, Map10 maintains a highly irregular and non-periodic pattern throughout the iterations. Moreover, the amplitude of Map10 varies dynamically, indicating stronger chaotic properties and better randomness, which is particularly beneficial for binary optimization and feature selection problems [68].

2.4. Lennard Potential

The Lennard potential is a mathematical model that describes how atoms or molecules interact with each other based on their distance [39,69,70]. In this technique, the solution maintains a relative distance that varies within a certain range at all times due to repulsion or attraction. The change in amplitude of the repulsion relative to the equilibrium distance is significantly greater than that of the attraction.

The Lennard-Jones potential is used as the interaction force acting on the current atom from the other atom in the jth dimension at $Iter$ time, which can be written as:

$$\mathrm{P}=n\left(Iter\right)\times \left(12{(-h_i^j)}^{-13}-6{(-h_i^j)}^{-7}\right),$$

(16)

The function behaviors of P with different values of $n\left(Iter\right)$ corresponding to the values of $h_i^j$ ranging from 0.9 to 2. The $n\left(Iter\right)$ is the depth function of the current iteration to adjust the repulsion region or the attraction region based on the maximum number of iterations ($MaxIter$), which can be defined as:

$$n\left(Iter\right)={\left(1-{\displaystyle \frac{Iter-1}{MaxIter}}\right)}^3,$$

(17)

The behavior of the function $n\left(\mathrm{Iter}\right)$ varies with different values of h, as defined in Equation (16), within the range of 0.9 to 2. For varying $n\left(\mathrm{Iter}\right)$ values, repulsion occurs when $h_i^j\in [0.9,1.12)$, while attraction takes place when $h_i^j\in (1.12,2]$. Equilibration is observed precisely at $h_i^j=1.12$. Notably, the attraction effect diminishes to nearly zero when $h_i^j\ge 2$. To enhance exploration capabilities, $h_i^j$ is defined as:

$$h_i^j\left(Iter\right)=\left\{\begin{array}{cc}{h}_{min}\hfill & {\displaystyle \frac{r_i^j\left(Iter\right)}{\sigma \left(Iter\right)}}<h_{min}\hfill \\ {\displaystyle \frac{r_i^j\left(Iter\right)}{\sigma \left(Iter\right)}}\hfill & h_{min}\le {\displaystyle \frac{r_i^j\left(Iter\right)}{\sigma \left(Iter\right)}}\le h_{max}\hfill \\ h_{max}\hfill & {\displaystyle \frac{r_i^j\left(Iter\right)}{\sigma \left(Iter\right)}}>h_{max},\hfill \end{array}\right.$$

(18)

where $h_{min}$ and $h_{max}$ are the lower and upper limits of $h_i^j$, respectively. The main equations of the function help maintain diversity among candidate solutions while refining their positions over iterations.

The amplitude of the repulsion change with respect to the equilibration distance is significantly larger than the attraction change, which is equal to $r=1.12\sigma$, where the length scale $\sigma \left(Iter\right)$ is defined as

$$\sigma \left(Iter\right)={∥x_y\left(Iter\right),{\displaystyle \frac{1}{\left|K\right(Iter\left)\right|}}\sum _{j\in K_{\mathrm{best}}}{x}_y\left(Iter\right)∥}_2$$

(19)

where $K_{\mathrm{best}}$ is a subset of an agent population, consisting of the first K neighbors agents with the best fitness values. The system boundaries are defined as [69]:

$$\left\{\begin{array}{c}{h}_{min}=g_0+g\left(Iter\right)\hfill \\ h_{max}=u,\hfill \end{array}\right.$$

(20)

where $g_0$ and u are values of h corresponding to a values between the minimum value and maximum value of the attraction. The drift factor $g\left(Iter\right)$, which transitions the algorithm from exploration to exploitation, is given by

$$g\left(Iter\right)=0.1\times sin\left({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{Iter}{MaxIter}}\right)$$

(21)

3. BCNRBO: Binary Chaotic Newton Raphson Based Optimizer

Metaheuristics were often created to address ongoing optimization problems. But a lot of optimization issues (like FS problem) have a binary search space. It is necessary to modify an algorithm in order to solve binary problems. So, in this study, we introduce the binary version of the improved NRBO.

In order to help agents avoid early convergence and escape local optima, the following steps improve the proposed BCNRBO capability to handle real-world problems such feature selection problem. The main contributions of this study are summarized as follows: First, to prevent being located in the local search, we omit the TAS step. Second, we improve the new population in the NRSR phase by changing the random numbers (a and b) into a new parameter called “dynamic potential”. Third, since chaotic maps have been shown to increase the effectiveness of optimization algorithms, we incorporate them in certain places throughout the proposed method. Lastly, we utilize a modification of the NRBO’s parameter $\delta$ to propose a new transfer function that depends on both the s-shape and v-shape functions, thereby transforming the continuous solutions into binary ones.

In the beginning, the initial positions are initialized in binary form as follows:

$$X_i^j(Iter=1)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}rand\le 0.5,\hfill \\ 1,\hfill & \mathrm{if}rand>0.5.\hfill \end{array}\right.$$

(22)

Where $X_i^j$ is the jth dimension of the ith agent, $Iter$ is the current iteration, and $rand$ is a random number from the distributions uniformly distributed over the interval $[0,1]$. The objective function is assessed, and the fitness values of the agents are stored for every initial binary agent in the population.

The next subsections will provide a detailed explanation of all the proposed main steps, which are utilized to enhance the quality of the updated solutions.

3.1. Dynamic Potential (DP)

Dynamic Potential is a time-dependent potential function that evolves during the search process to adaptively influence algorithm behavior, rather than remaining fixed. A dynamic potential adapts according to the iteration or time step, allowing the algorithm to guide the search process more effectively while preserving optimal behavior [71].

In this study, to enhance the exploration of the proposed algorithm in the first stage of iterations, each agent needs to interact with as many agents as possible with a better fitness value than its $K_{best}$ neighbors. The $K_{best}$ is calculated from the following equation:

$$K_{best}=⌈N_p-(N_p-2)\times \sqrt{{\displaystyle \frac{Iter}{MaxIter}}}⌉$$

(23)

where `$\lceil ·\rceil$’ is the ceil value, $N_p$ is total number of agents (population size), $Iter$ is the current iteration, and $MaxIter$ is the maximum number of iterations. It is worth mentioning that $K_{best}$ is an integer number that starts close to $N_p$ and then gradually decreases to approximately the value 2 during the search progress (Equation (23)).

To update the new guiding position, the following steps describe how to compute the probability ${\mathcal{P}}_i$ for each agent based on the new parameter called mass ($M_i$) of the ith agent, which is described as a probability distribution or fitness-based ranking of agents. It can be calculated as follows:

$${\mathcal{P}}_i={\displaystyle \frac{M}{\sum M_i}},M_i=exp\left({\displaystyle \frac{{\mathrm{Fitness}}_{best}-{\mathrm{Fitness}}_{1,\dots ,N_p}}{{\mathrm{Fitness}}_{worst}-{\mathrm{Fitness}}_{best}}}\right),$$

(24)

where the best and worst fitness values for a minimization problem at iteration $Iter$ are calculated for the current population as follows,

$${\mathrm{Fitness}}_{best}\left(Iter\right)=\underset{i\in [1,\dots ,N_p]}{min}{\mathrm{Fitness}}_i\left(Iter\right),{\mathrm{Fitness}}_{worst}\left(Iter\right)=\underset{i\in [1,\dots ,N_p]}{max}{\mathrm{Fitness}}_i\left(Iter\right),$$

(25)

where ${\mathrm{Fitness}}_i\left(Iter\right)$ is the value of the objective function for the ith individual in iteration $Iter$ and $N_p$ is the size of the population.

A new guiding position $XK\left(Iter\right)$ is determined as the mean values of the selected $K_{best}$ agents based on the mass $M_i$ as follows:

$$X{K}_i\left(Iter\right)={\displaystyle \frac{{\sum }_{m=1}^{K_{best}}{X}_m}{K_{best}}},$$

(26)

where $m=1,2,\dots ,K_{\mathrm{best}}$ is the index of the sorting ${\mathcal{P}}_i$ (Equation (24)) for each agent. It is sorted descendingly from highest to lowest values ($M_i^{\left(1\right)}\ge M_i^{\left(2\right)}\ge \dots \ge M_i^{\left(N_P\right)}$).

In the next step, for binary or discrete optimization problems, we employ the Hamming distance ($hd$) [40,72,73], which measures the dissimilarity between two position vectors ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ by counting differing elements as follows:

$$hd({\mathbf{X}}_1,{\mathbf{X}}_2)=\sum _{j=1}^{Dim}(X_1^j\ne X_2^j).$$

(27)

The Hamming distance calculation served as a technique to accommodate the discrete nature of the search space. By incorporating the Hamming distance, the algorithm can effectively measure and navigate the differences between potential solutions, thereby improving its ability to find optimal or near optimal solutions in optimization problems. From the Lennard-Jones potential function (Section 2.4), it can be seen that solution keep a relative distance varying in a certain range all the time due to the repulsion or the attraction, and the change amplitude of the repulsion relative to the equilibration distance ($r=1.12\sigma$) is much greater than that of the attraction.

Specifically, the type of interaction force—whether it is attractive or repulsive—between an agent and its neighbors depends on the ratio between their distance and the characteristic length scale $r_i$. This length scale, ${\sigma }_i$, indicates how far each agent is from the average position of its top $K_{best}$ nearest neighbors.

In this step, for each agent, the values of $r_i$ and ${\sigma }_i$ (Equation 19) will be replaced by new values that are calculated based on the Hamming distance (Equation (27)) as follows:

$${\sigma }_i\left(Iter\right)=hd(X_i,X{K}_i)=\sum _{i=1}^n(X_i\left(Iter\right)\ne X{K}_i\left(Iter\right)),$$

(28)

$$r_i\left(Iter\right)=hd(X_i,X_{best})=\sum _{i=1}^n(X_i\left(Iter\right)\ne X_{best}\left(Iter\right)),$$

(29)

The new values of $h_i$ (Equation (18)) are computed after updating the parameters ${\sigma }_i$ and $r_i$, as follows:

$$h_i=\left\{\begin{array}{cc}{h}_{min},\hfill & \mathrm{if}{\displaystyle \frac{hd(X_i,X_{best})}{hd(X_i,X{K}_i)}}<h_{min}\hfill \\ h_{max},\hfill & \mathrm{if}{\displaystyle \frac{hd(X_i,X_{best})}{hd(X_i,X{K}_i)}}>h_{max}\hfill \\ {\displaystyle \frac{hd(X_i,X_{best})}{hd(X_i,X{K}_i)}},\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(30)

where $h_{min}$ and $h_{max}$ are calculated from Equation (20), where $g_0$ in this study is equal to ($h=1.12$) when the attraction reaches the minimum value and u is equal to ($h=1.24$) when the attraction reaches the maximum value. The value of g is calculated by Equation (21).

The ultimate value of potential (Equation (16)) represents the interaction force between two distinct agents and is updated based on the new modified $h_i$ as follows:

$$\mathrm{DP}=n\left(Iter\right)\left(12{(-h_i)}^{-13}-6{(-h_i)}^{-7}\right)$$

(31)

where $n\left(Iter\right)$ is calculated from Equation (17). See Algorithm 1 for further information and the steps involved in the dynamic potential.

Algorithm 1 Pseudocode for Dynamic Potential.

Require: $Iter$ (Iteration), $MaxIter$ (Maximum iterations), $H_0=1.12$ and $u=1.24$ Ensure: Updated Dynamic Potential 1: Generate $K_{best}$ from Equation (23) 2: Calculate mass of agent (M) from Equation (24) 3: Extract new guiding position $XK\left(Iter\right)$ based on $K_{best}$ from Equation (26) 4: Calculate the Hamming distances $\sigma =h{d}_1$ between the current solution and the current $XK$ based on Equation (28) 5: Calculate the Hamming distances $r_i=h{d}_2$ between the current solution and the best one based on Equation (29) 6: Calculate $n\left(Iter\right)$ from Equation (17) and $g\left(Iter\right)$ from Equation (21) 7: Calculate $h_{min}$ and $h_{max}$ using Equation (20) 8: Calculate $h_i$ from Equation (30) 9: Calculate Dynamic Potential $DP$ using Equation (31) 10: return$D{P}_i$ value

3.2. Chaotic Enforcement (CE)

To avoid premature convergence to local optima and effectively guide solutions toward promising regions of the search space, we introduce a new chaotic map-based component named Chaotic Enforcement ($CE$). $C{E}_i^j$ represents the aggregated influence of randomly weighted components based on the chaotic map w in the jth dimension acting upon the ith agent and the best agent, formulated as follows:

$$C{E}_i^j=w\times (X_{best}^j-X_i^j),\forall j\in [1,\mathrm{Dim}].$$

(32)

where $X_{best}^j$ is the position of the best agent and the parameter w is the chaotic value in each iteration. The value of $CE$ depends on both the current chaotic value and the inconstant bond length between the current agent and the best agent. Hence, the constraint force $CE$ can be obtained for each dimension jth of the agent ith by the effect of the chaotic map.

Chaotic enforcement helps determine the step size for the search area, as the step size in the proposed method impacts the stability and accuracy of the approximated solution, thereby influencing how closely the optimal solution aligns with the true solution. $CE$ will be incorporated into the main rule of the proposed algorithms, which will be presented in the next subsection.

3.3. Chaotic Newton-Raphson Search Rule (CNRSR)

The basic NRSR controls the solutions, which allows for a more accurate exploration of viable regions and better positioning. For more accurate results during the binary search, we aim to enhance both exploration and exploitation techniques during this phase by refining the exploitation of this phase through mathematical equation adaptations. The CNRSR basically depends on the dynamical potential and chaotic enforcement, as shown in Equations (31) and (32) respectively.

In the proposed BCNRBO, a new parameter called $c\rho$ is driven from Equation (7). This new parameter is responsible for the exploration and exploitation during the search process, and it’s considered the main step of moving the new solution to the promising region. The fundamental factor that guides the population in the right direction is the dynamical potential and chaotic enforcement. From Equation (7), the parameters a and b which is a random numbers between (0,1) have been replaced by the proposed parameters DP and $CE$ respectively. The new chaotic $\rho$ parameter can be referred as the step size for each solution, and is given as follows.

$$c{\rho }^i=\mathrm{DP}\times (X_{best}-X_i)+\mathrm{DP}\times (C{E}_i-X_{RS}),$$

(33)

where DP is the dynamical potential from Equation (31) and $CE$ parameter is calculated from Equation (32). $RS$ is a unique random integers from the range 1 to $N_p$. Each agent’s $c{\rho }_i$ value is checked to ensure it remains within the boundary of 0 and 1, preparing it for the transfer function in the next step.

Algorithm 2 Pseudocode of Binary Chaotic Newton Raphson Based Optimizer (BCNRBO)

Require: $N_p$ (Number of solutions), $MaxIter$ (Maximum iterations), Dim (Dimension/number of features), w (Chaotic value) Ensure: $X_{best}$ and ${\mathrm{Fitness}}_{best}$ 1: Randomly initialize binary population (Equation 22) 2: $Iter\leftarrow 1$ 3: Calculate fitness for each binary solution $X_i$ ($i=1,2,\dots ,N_p$) 4: while$Iter\le MaxIter$do 5:     for $i=1$ to $N_p$ do 6:         Calculate the dynamic potential by applying Algorithm 1 7:         Calculate Chaotic Enforcement term ($C{E}_i^j$) using chaotic map (Equation 32) 8:         Calculate the agent chaotic size $c\rho$ (Equation 33) 9:         Update the current binary agent using transfer function (Equation 37) 10:         Evaluate the new binary solution using the objective function 11:         Update the current best $X_{best}$ and the best fitness ${\mathrm{Fitness}}_{best}$ 12:     end for 13:     $Iter\leftarrow Iter+1$ 14: end while 15: return$X_{best}$ (position of best binary solution)

Figure 3. Flowchart of BCNRBO.

Figure 3. Flowchart of BCNRBO.

The iterative pseudocode steps of the pseudocode for the binary chaotic Newton-Raphson-based optimizer can be summarized as shown in Algorithm 2.

The updating of the binary agents can be found in details in the following subsection.

3.4. The Proposed Transfer Function

Utilizing transfer functions (TFs) is among the most straightforward techniques for converting a continuous to a binary form [74,75]. With the exception of a few minor adjustments to certain operators, they maintain the original continuous algorithm’s structure, which contributes to its simplicity. The probability that a continuous value in a solution will be mapped to a binary value is determined using TFs. Each solution is selected from the population and modified by incorporating information as explained previously.

In BCNRBO, the FS is represented as a binary vector with values of 0 or 1. A binary bit string of length Dim for a Dim-dimensional dataset represents the position of the NRBO. While each bit represents a feature, a value of ‘‘1” means that feature is selected, and ‘‘0” means that the feature is not selected. Each position vector generated is a subset of features generated from the original dataset.

In order to convert the current continuous solution to binary values bounded in the interval $[0,1]$, the following new transfer function is suggested:

$$S_i^j\left(Iter\right)=n\delta +\left|tanh\left({\displaystyle \frac{\pi }{4}}c{\rho }_i^j\left(Iter\right)\right)\right|,S\in [0,1],$$

(34)

where $c\rho$ is the new agent chaotic size and is calculated using Equation (33). In the proposed transfer function, we introduce a new parameter $n\delta$ which is derived from Equation (6) as follows:

$$n\delta =1-exp\left({\displaystyle \frac{-1}{MaxIter}}\right).$$

(35)

The parameter $n\delta$ is a small positive constant (in range [0,0.2]) introduced to avoid zero-probability stagnation in the transfer function. Since $|tanh(·\left)\right|$ may approach zero when the input value is small, adding $n\delta$ ensures a minimum activation level for $S_i^j\left(\mathrm{Iter}\right)$. This maintains a non-zero probability of bit flipping in the binary search space, thereby enhancing exploration and preventing premature convergence without significantly affecting the overall shape or boundedness of the S-shaped transfer function.

The obtained values of S are compared to random values as shown in Equation (36) to update the new binary population.

The update of the binary solution $X_i^{\prime j}$ would be as follows:

$$X_i^{\prime j}\left(Iter\right)=\left\{\begin{array}{cc}\mathrm{Complement}\left(X_i^j\left(Iter\right)\right),\hfill & \mathrm{if}rand<S_i^j\left(Iter\right)\hfill \\ X_i^j\left(Iter\right),\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(36)

After calculating the probabilities using the transfer function S and updating the binary solutions based on Equation (36), the final position updating equation is presented based on the current chaotic map value as follows:

$$X_i^{\prime j}\left(Iter\right)=\left\{\begin{array}{cc}{X}_{best}^j,\hfill & \mathrm{if}w<rand\hfill \\ X_i^j\left(Iter\right),\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(37)

where w is the current chaotic value in the current iteration and $rand$ is the random value between 0 and 1.

Evaluate the current solution and update the best-known solution if an improvement is found; then assess the current solution and revise the best solution accordingly.

3.5. Computational Complexity

The computational complexity of the proposed BCNRBO is analyzed based on its core operations and compared to the original NRBO. For the original NRBO, the complexity is calculated as $O(MaxIter\times N\times (N\times Dim+T_{fobj}))$. This result is obtained because the population mean is calculated inside the agent loop in the trap avoidance operator, requiring $O(N\times Dim)$ operations for each of the N agents in every iteration.

In the proposed BCNRBO, the computational complexity is derived based on the actual implementation steps. Unlike the original NRBO, the sorting operation is performed inside the main agent loop.

Specifically, for each agent in every iteration, the probability vector P is sorted in descending order. Since sorting a population of size N requires $O(NlogN)$ operations, and this process is repeated for all N agents, the total sorting cost per iteration becomes $O(N^{2}logN)$.

Furthermore, the centroid of the $K_{best}$ individuals is recalculated inside the loop for each agent, which costs $O(N\times Dim)$ per agent, leading to $O(N^2\times Dim)$ per iteration.

The Hamming distance computation between binary vectors of length $Dim$ costs $O\left(Dim\right)$ per agent. However, this term is dominated by the quadratic terms above.

The evaluation of the objective function for the entire population costs $O(N\times T_{fobj})$, where $T_{fobj}$ denotes the computational cost of evaluating the objective (fitness) function for a single agent.

By combining these steps, the total complexity of BCNRBO becomes

$$O\left(MaxIter\times \left(N^{2}logN+N^2\times Dim+N\times T_{fobj}\right)\right).$$

By grouping the dominant quadratic terms, it can be expressed as

$$O\left(MaxIter\times \left(N^2(logN+Dim)+N\times T_{fobj}\right)\right).$$

.

This analysis shows that the proposed BCNRBO preserves the same quadratic order with respect to the population size as the original NRBO.

Although the inclusion of sorting operations introduces an additional $O(N^{2}logN)$ term, the overall complexity remains dominated by the quadratic components $O(N^2\times Dim)$.

Moreover, the incorporation of Hamming distance and chaotic mapping mechanisms does not significantly increase the computational burden, since these operations scale linearly with respect to the population size and dimensionality.

3.6. Code Availability

The complete MATLAB implementation of BCNRBO and the custom scripts used to generate the results are publicly available at Zenodo: https://doi.org/10.5281/zenodo.18602047.

4. Computational Experiments

The suggested feature selection approach aims to determine the best feature subset that delivers optimal classification performance with the least number of features. Wrapper-based feature selection is used in this experiment with an objective function to evaluate the fitness of each solution [76]. This framework makes use of the fitness function known as classification error which considers the following aspects:

(a)

This study focuses on low to medium dimensional large scale datasets, including the Pd Speech dataset with 753 features, to demonstrate the scalability of the proposed algorithm.

(b)

This study simultaneously optimizes classification accuracy (minimum misclassifications error) while minimizing selected features.

The BCNRBO technique has been tested on many datasets in the UCI repository [77], including low, medium, and large scales, employing three different classifiers.

■

KNN classifier [78]: The k-nearest neighbor method is a popular classification method in data mining and statistics due to its simplicity and significant classification performance. It uses k nearest neighbors to determine the class of examples, making it a memory-based classification, lazy learning technique. The data were classified with k = 5 for KNN classification, and the training set results were obtained by divide data into training and testing. KNN is still utilized today, proving its accuracy and professionalism [79,80,81].

■

DT classifier [82]: Decision trees is a tree-based technique used in data mining, where a path starts at the root and ends at the leaf node [82]. They are hierarchical exemplifications of knowledge relationships, with nodes representing purposes. Decision trees are widely used in fields like machine learning, image processing, and pattern identification. They unify basic tests efficiently and cohesively, comparing numeric features to threshold values [83]. They are commonly used for grouping purposes and classification models in data mining. Decision trees have found many implementation fields due to their simple analysis and precision on multiple data forms.

■

NB classifier [84]: The Naive Bayes Classifier is a simple yet effective approach based on the Bayesian theorem that is especially well-suited for high-dimensional inputs [84]. It is capable of outperforming more advanced classification algorithms. The prior probability, based on the percentage of Green and Red objects, is used to predict outcomes before they occur, ensuring that new cases are classified accordingly. It was derived from Bayesian Classification is a supervised learning and statistical method that uses probabilistic models to capture uncertainty and solve diagnostic and predictive problems.

4.1. Algorithms, Parameters and Experimental Setup

Various datasets are used to assess the efficiency of the proposed algorithms [7]. The proposed algorithm, along with other algorithms, was implemented using MATLAB R2022b on a PC with an Intel(R) Pentium(R) CPU running at 3.2GHz and 32 GB of RAM.

The parameter values of 20 and 50 have been utilized for the population size and maximum iterations, respectively.

The proposed algorithm is compared with state-of-the-art feature selection algorithms using the following criteria:

• Fitness values (using wrapper approach): They are obtained from each approach as reported (classification error: it is obtained by using the selected features on the test dataset). The mean, min, and max fitness values are compared [85]. The average is calculated over 20 independent runs.
• Average selection features: It is the other comparison that has been presented in here.
• $p-$value from Wilcoxon’s rank sum test and the mean rank value from Friedman test (Wilcoxon’s rank sum test and Friedman test are non-parametric statistical tests with 5% significance level [86]).

The statistical test is necessary to show that the proposed algorithm provides a significant improvement compared to other algorithms. Here we are using two non-parametric tests: Wilcoxon and Friedman rank tests [86]. Generally speaking, the best values of p are when p value $<0.05$. Therefore, it can be considered sufficient evidence against the null hypothesis.

The values for the Wilcoxon’s test indicate that the null hypothesis R is rejected for `1’ and the proposed algorithm outperforms the other one.

`$-1$’ means the null hypothesis is rejected and the proposed algorithm is inferior to the other one. `0’ reveals that the null hypothesis is accepted, and the proposed algorithm binds the other one. The tables report on the results obtained from both the Wilcoxon and Friedman rank tests, respectively, with a confidence level of 0.95 ($\alpha$ = 0.05).

The performance of the binary algorithm BCNRBO is compared with other optimization algorithms. These algorithms are proposed in the literature for binary optimization or for solving the feature selection problem. These algorithms are BAOA [41], BBA [42], BFPA [43], BPSO [44], BCCSA [32], jBASO [45] and BDA [46]. We added to the experimental test two more algorithms, the bee colony algorithm [87] and differential evolution [88]. These two algorithms were converted to binary versions (BABC and BDE, respectively) using the sigmoid function to demonstrate the effect of two powerful continuous algorithms on the binary problem. The parameter settings for all adopted optimization algorithms are taken from the original papers, as shown in Table 3.

4.2. Datasets Description

The BCNRBO technique has been tested on many datasets in the UCI repository, including low, medium, and large scales. The performance of BCNRBO algorithms was tested on 26 benchmark medical and normal datasets from UCI [77], KEEL, Kaggle, and Arizona State University [89]. The details of the shared datasets, which vary in size, features, and sample counts, are displayed in Table 4.

These datasets cover a wide range of application domains, including medical diagnosis, biological analysis, and general classification problems, with varying numbers of samples and feature dimensions. As summarized in Table 4, the datasets range from low-dimensional problems such as Olive and Wisconsin to high-dimensional datasets such as LSVT and Pd Speech, which makes them suitable for assessing the robustness and scalability of feature selection algorithms.

For each dataset, the proposed algorithm was applied to select an optimal subset of features, which was then evaluated using three different classifiers: K-Nearest Neighbors, Decision Tree, and Naive Bayes. To ensure fair comparison and reduce randomness effects, each experiment was independently repeated multiple times, and the average classification accuracy was reported.

All datasets were preprocessed by normalizing feature values to a common scale before the optimization process. The same experimental settings, including population size and maximum number of iterations, were used for all algorithms to guarantee a fair and consistent comparison.

4.3. Evaluation and Analysis of the Proposed Algorithm Using Chaotic Maps

In the preliminary testing phase, ten various chaotic maps were tested to show which one of them would improve the convergence speed, local optima avoidance, and scalability of the BCNRBO algorithm. Ten different datastes with different sizes are used in this test. The shared datasets in this test are: Diabetes, Parkinsons, Diagnostic, Dermatology, Chess, Lung Cancer, Sonar, Hillvalley, LSVT, and Pd Speech. See Table 4. It is noteworthy that the KNN is used in this test with a population size of 20 over 20 runs and a maximum iteration of 50.

To demonstrate the effect of chaotic maps on the proposed method, ten different maps are integrated into the proposed algorithm. Furthermore, BNRBO represents an enhanced version of the proposed algorithm’s phases, but without the integration of chaotic maps.

Table 5 and Table 6 present the performance results of BCNRBO utilizing various chaotic maps, alongside the enhanced binary BNRBO without chaotic map integration. Table 5 specifically summarizes the mean and standard deviation values for both variants.

The statistical analysis highlights Map10 (Tent map) as the most effective configuration, consistently outperforming others across multiple evaluation metrics. Notably, Map10 and Map7 each achieved top mean fitness values rankings on four datasets of differing dimensions. However, Map10 exhibited superior stability, recording the lowest standard deviation in eight out of ten benchmark datasets.

Furthermore, Map10 delivered the highest precision means for four specific datasets (Diabetes, Dermatology, Lung Cancer, and Sonar) and the most stable results (measured by standard deviation) across eight datasets, including Diabetes, Parkinsons, Diagnostic, Dermatology, Lung Cancer, Sonar, LSVT, and Pd Speech. Comparatively, Map8 and the original BNRBO algorithm ranked second in stability, achieving optimal standard deviation values in five out of the ten evaluated datasets.

Similarly, BNRBO outperforms other algorithms when selecting the more significant features from the enormous available feature set.

To improve the reliability of the results, the Friedman test was used. The obtained results of this test are represented in Table 6, which confirms the previous analysis, demonstrating that the chaotic map Map10 outperforms its counterparts.

Based on the results of the Friedman test, Table 7 presents the p-values from the Wilcoxon signed-rank test comparing Map10 against its counterparts, along with their corresponding rank values (R).

The experimental findings demonstrate that integrating Map10 with the BNRBO algorithm significantly enhances its effectiveness in solving feature selection problems. Additionally, computational times across 10 different datasets were analyzed.

Overall, the results confirm that incorporating chaotic maps into the binary optimizer improves performance, with the original BNRBO (without chaotic maps) showing comparatively lower effectiveness. Among all tested chaotic map variants, the Tent map emerged as the most effective, consistently delivering superior results.

5. Experimental Results and Discussion

The effectiveness of the proposed algorithms is evaluated across 26 different datasets, with details provided in Table 4. To obtain statistically meaningful results, every algorithm is run independently 20 times. Moreover, for all proposed algorithms, the size of the population and the number of iterations are set to 20 and 50, respectively, over 20 independent runs.

For the classification process, each dataset was randomly partitioned into two subsets: 80% of the data was used for training the classifier, while the remaining 20% was reserved for testing. This split ensures that the model is trained on a substantial portion of the data while still being evaluated on unseen instances to assess its generalization performance. The random partitioning was repeated in each run to minimize bias and ensure robust evaluation of the proposed algorithm.

In the results presented in Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18 and Table 19, we conducted rigorous statistical analyses, including: (1) descriptive statistics (best, worst, mean, standard deviation of fitness values, and the average number of selected features); (2) the non-parametric Friedman test for overall performance ranking across all datasets; and (3) pairwise Wilcoxon signed-rank tests for detailed comparisons with state-of-the-art algorithms. The best results across all tables are highlighted in bold to underscore their significance relative to the other values.

5.1. Evaluation of BCNRBO with KNN Classifier

In this section, we present the results obtained by the proposed algorithm alongside those from other benchmark algorithms using the KNN classifier, serving as the first experiment in our evaluation. The purpose of this experiment is to assess and compare the classification performance of each algorithm based on selected features.

The primary objective of optimization algorithms is to minimize the fitness function, thereby identifying the most optimal solution to a given problem. As illustrated in Table 8Table A2, the proposed BCNRBO algorithm consistently achieved better solutions across most evaluated datasets in terms of min, mean, worst, and std, fitness values, respectively.

These results underscore the robustness and effectiveness of BCNRBO in tackling feature selection challenges, as it repeatedly outperformed alternative methods in minimizing the objective function.

Table 8 presents the best fitness values obtained using the KNN classifier across 26 datasets. Bold values indicate the best (lowest) result for each dataset.

The BCNRBO algorithm shows the best overall performance, achieving the top result on 10 datasets. The BDA algorithm ranks second with 8 best results, followed by BBA with 6. The BABC and jBASO algorithms each performed best on 7 datasets.

Table 9 presents the average fitness values obtained using the KNN classifier.

The BCNRBO algorithm demonstrates superior consistency, achieving the best average performance on 10 out of 26 datasets. Both BDE and BABC algorithms follow with 7 best average results each. The BCCSA algorithm secured 5 best averages, while BAOA, jBASO, BFPA, BBA, and BPSO each obtained 4.

As is evident from the Table A1 presents the worst (maximum) fitness values obtained using the KNN classifier. Bold values indicate the best (lowest worst-case) result for each dataset.

The BCNRBO algorithm demonstrates the best robustness, achieving the lowest worst-case error on 9 datasets. BABC, BPSO, and BDA follow with strong performance, each securing the best worst-case result on 8 datasets. BAOA and BDE each performed best on 6 datasets.

Higher standard deviation values observed in some algorithms indicate unstable performance and a wider spread of outcomes, reflecting sensitivity to stochastic variations.

Table A2 presents the standard deviation of fitness values obtained using the KNN classifier. Bold values indicate the most consistent (lowest standard deviation) result for each dataset.

The BPSO algorithm shows the highest consistency, achieving the lowest standard deviation on 12 datasets. BCNRBO and BABC also demonstrate strong stability, each securing the most consistent results on 10 and 8 datasets respectively. This indicates that BPSO, BCNRBO, and BABC produce the most reliable and repeatable performance when used with the KNN classifier.

In addition, Table A3 reports the average number of selected features obtained by the proposed binary algorithm BCNRBO and other algorithms. Similarly to the classification error and fitness values, the number of selected features is a significant metric to evaluate the effectiveness of feature selection methods tested by KNN classifier. The classification model was implemented using KNN classifier in MATLAB (fitcknn) function with 5 nearest neighbors (k=5).

In Table A3 presents the average number of features selected using the KNN classifier. Bold values indicate the most compact (lowest average feature count) result for each dataset.

The BCNRBO algorithm achieves the most compact feature subsets, selecting the fewest features on 15 datasets. The jBASO algorithm follows, performing best on 6 datasets. This indicates that BCNRBO is the most effective at identifying minimal feature subsets while maintaining classification performance with KNN.

5.1.1. The Statistical Test for KNN Classifier

The superior performance of BCNRBO comes from its dynamic switching mechanism between 0s and 1s (or vice versa), guided by the proposed dynamic potential, chaotic enforcement and proposed transfer function techniques. This approach effectively compensates for the lack of prior experiential knowledge in the solutions, ensuring robust and adaptive performance.

The Friedman test results presented in Table 10 provide statistically rigorous evidence for performance comparisons among algorithms using the K-Nearest Neighbors classifier across 26 datasets.

BCNRBO demonstrates statistical superiority by achieving the lowest average rank of 4.227 in Table 10, followed by BDA with an average rank of 4.575, and BBA with 5.083. In contrast, BCCSA shows the highest average rank of 7.048, indicating statistically weaker performance.

Examining the specific datasets where BCNRBO achieved the top rank reveals its consistent performance across diverse data characteristics. BCNRBO secured the first position in datasets including Chess (D1), Lung Cancer (D5), Diabetes (D6), Sonar (D9), Hillvalley (D11), BreastEW (D17), Diabetes (D20), Parkinsons (D23), Dermatology (D24), and Heart Failure Clinical (D26). This distribution across 10 out of 26 datasets represents the highest success rate among all evaluated algorithms.

To assess the statistical significance of BCNRBO’s performance, additional comprehensive comparisons were conducted against several state-of-the-art algorithms.

In this context, the Wilcoxon signed-rank test results presented in Table 11 provide strong pairwise statistical confirmation of the Friedman test findings for the KNN classifier. The test demonstrates that BCNRBO significantly outperformed most competing algorithms.

Specifically, BFPA and BCCSA recorded the highest number of losses (19 and 18 datasets, respectively) against BCNRBO, followed by BDE (16 losses) and jBASO (15 losses). In contrast, BBA and BPSO exhibited the most competitive performance, with the fewest losses (4 and 3 datasets, respectively) and the highest number of ties (10 each) when compared with the proposed algorithm.

5.1.2. Convergence Graphic for Different Dataset Using KNN Classifier

The convergence characteristic curves of the basic BCNRBO on the different dimensions of 15 datasets studied using KNN classifier. The four weakest performing algorithms were removed from the graph to improve explanation, and their weak performance was statistically confirmed based on the Friedman test results (Table 10).

According to Figure 4 illustrate the convergence behavior of the competing algorithms over 15 benchmark functions with different dimensionalities. For clarity and better visualization, the four weakest-performing algorithms were excluded from the comparison.

Figure 4. Convergence history for different datasets using KNN test.

Figure 4. Convergence history for different datasets using KNN test.

Figure 4. Convergence history for different datasets using KNN test (Continued).

Figure 4. Convergence history for different datasets using KNN test (Continued).

As shown in the figures, the proposed BCNRBO algorithm achieves the best fitness values on functions $F1$, $F9$, $F11$, $F17$, $F20$, $F23$, and $F26$, demonstrating its strong capability to balance exploration and exploitation during the search process. In addition, BCNRBO exhibits competitive performance by attaining comparable results on functions $F5$, $F6$, and $F24$, indicating stable and consistent convergence behavior across different problem landscapes.

It is also observed that some competing algorithms, including JBASO, BBA, BFPA, and BABC, achieve lower fitness values on a limited number of functions. This outcome highlights the problem-dependent nature of metaheuristic optimization algorithms. Nevertheless, the overall results confirm the superiority and robustness of the proposed BCNRBO algorithm when compared with the other binary optimization methods.

5.2. Evaluation of BCNRBO with DT Classifier

This section presents the performance of the proposed algorithm against benchmark methods using the DT classifier in our second experiment. The goal is to evaluate and compare the classification error of each algorithm on selected feature subsets. The experimental results demonstrate the superior capability of the proposed BCNRBO algorithm, as evidenced by the comprehensive evaluation in Table 12 and Table A5.

The statistical analysis further validates that the proposed algorithm consistently outperforms competing methods in terms of reducing classification error. Table 13 and Table A5 present the mean and standard deviation of the fitness values achieved by the proposed binary chaotic NRBO algorithm. Since lower fitness values reflect better performance, these metrics serve as key indicators of algorithmic effectiveness.

Table 12 presents the best fitness values obtained using the Decision Tree classifier across 26 datasets. The bolded values indicate the best performance (lowest fitness value) for each dataset.

From the analysis of the results, the BCNRBO algorithm demonstrates the most superior performance, achieving the best results on 10 datasets (D1, D2, D3, D6, D8, D12, D13, D14, D20, D24). It is followed by the BBA algorithm, which excelled on 8 datasets. The BABC algorithm also performed competitively, each securing the best fitness value on 7 datasets.

As illustrated in Table 13, the average fitness values obtained using the DT classifier across 26 datasets provide insights into the consistent performance of each algorithm. BCNRBO maintains its leading position, achieving the lowest average fitness values in 8 datasets. BAOA follows with the best average performance in 5 datasets, while BDE and BCCSA each show superior average results in 4 datasets. The remaining algorithms demonstrate more variable performance: BABC excels in 2 datasets, and BDA, BBA, BFPA, jBASO, and BPSO each lead in only 1 dataset based on average values.

Table A4 presents the worst fitness values obtained using the Decision Tree classifier across the 26 datasets. This analysis of worst-case performance provides valuable insights into algorithm robustness and stability. BCNRBO demonstrates exceptional reliability by achieving the best (lowest) worst-case performance in 13 datasets, substantially more than any other algorithm. Following distantly, BDE shows the best worst-case performance in 9 datasets, with BABC achieving this in 6 datasets, and BAOA in 4 datasets.

These results reveal an important dimension of algorithm performance. While several algorithms can achieve good results under optimal conditions, BCNRBO maintains superior performance even in worst-case scenarios.

Table A5 presents the standard deviation of fitness values obtained using the Decision Tree classifier across 26 datasets. Standard deviation measures the consistency and stability of algorithm performance, with lower values indicating more reliable results.

In this analysis, BDE and BABC algorithms demonstrate the highest consistency, each achieving the lowest standard deviation in 10 datasets. Following closely, BAOA and jBASO show strong stability with the lowest deviation in 6 datasets each, while BCNRBO and BDA also achieve this in 6 and 5 datasets respectively.

These results provide an important perspective on algorithm reliability. While BCNRBO showed superior performance in terms of best, average, and worst fitness values, the standard deviation analysis reveals that BDE and BABC offer greater consistency in their results.

Table A6 presents the average number of selected features obtained using the Decision Tree classifier across 26 datasets. This metric is crucial for evaluating algorithm efficiency in feature selection, where fewer selected features generally indicate better dimensionality reduction while maintaining classification performance.

BCNRBO demonstrates remarkable efficiency by selecting the smallest number of features in 15 datasets, significantly more than any other algorithm. Following distantly, jBASO achieves the minimal feature count in 6 datasets.

The examination of average feature selection efficiency using the Naive Bayes classifier, presented in Table A9, reveals important patterns in dimensionality reduction capability. BCNRBO demonstrates exceptional efficiency by selecting the smallest number of features in 14 datasets, continuing its strong performance in feature reduction observed with the DT classifier. Following this, jBASO shows competitive efficiency with the minimal feature count in 8 datasets. This substantial lead in feature reduction efficiency, combined with previously demonstrated strong fitness performance, positions BCNRBO as particularly effective for applications where both classification performance and model simplicity are important.

5.2.1. The Statistical Test for DT Classifier

The experimental results revealed statistically significant performance improvements ($\alpha <0.05$) over all comparative methods, as quantified through classification error. These findings validate BCNRBO’s enhanced exploration exploitation balance and its capability to overcome local optima traps common in high dimensional search spaces. However, the proposed BCNRBO is statistically significant and ranked first compared with the other algorithms. From Table 14, and Table 15 the p value and mean ranks R of the proposed algorithm versus six algorithms BCNRBO, BAOA, jBASO, BFPA, BBA, BCCSA, BPSO and BDA.

The Friedman test results presented in Table 14 provide robust statistical evidence for performance comparisons using the Decision Tree classifier across 26 datasets. The analysis shows BCNRBO in first place with an average rank of 4.206. It achieved the top rank in 8 datasets: D2 (Wisconsin), D3 (Breast), D6 (Diabetes), D11 (Hillvalley), D13 (Zoo), D14 (Hepatitis), D15 (Diagnostic), D24 (Dermatology), and D25 (Pd Speech). BBA followed in second place with an average rank of 4.966, and BABC came third with 4.654. BCCSA showed the weakest performance with an average rank of 8.150.

To further validate the statistical significance of BCNRBO’s performance, comprehensive pairwise comparisons were extended to the Decision Tree classifier.

In this context, the Wilcoxon signed-rank test results presented in Table 15 provide robust pairwise statistical confirmation of the Friedman test findings for the DT classifier. The test demonstrates that BCNRBO significantly outperformed all competing algorithms. Specifically, BCCSA recorded the highest number of losses (22 datasets) against BCNRBO, followed by BAOA and BFPA (17 losses each). In contrast, BBA exhibited the most competitive performance, with the fewest losses (5 datasets) and the highest number of ties (8) when compared with the proposed algorithm.

5.2.2. Convergence Graphic for Different Dataset Using DT Classifier

The convergence characteristics of the baseline BCNRBO algorithm were evaluated across varying dimensions of 15 benchmark datasets, with performance analysis conducted using the DT classifier.

Figure 5 present the convergence behavior of the competing algorithms using the Decision Tree classifier. As illustrated in the figures, the proposed BCNRBO algorithm achieves the lowest fitness values in nine benchmark functions, namely $F2$, $F3$, $F8$, $F9$, $F14$, $F15$, $F20$, $F23$, and $F24$, demonstrating its strong optimization capability when coupled with the DT classifier.

Figure 5. Convergence history for different datasets using DT test.

Figure 5. Convergence history for different datasets using DT test.

Figure 5. Convergence history for different datasets using DT test (Continued).

Figure 5. Convergence history for different datasets using DT test (Continued).

Furthermore, several competing algorithms, including BBA, BDA, BABC, and BPSO, also exhibit competitive performance on specific functions, confirming their effectiveness in certain problem scenarios. For clarity of visualization, the weakest-performing algorithms were excluded from the presented convergence plots.

Overall, the results indicate that BCNRBO maintains robust and consistent convergence behavior across the majority of the tested functions, outperforming the other binary optimization algorithms in terms of solution quality under the DT classification framework.

5.3. Evaluation of BCNRBO with NB Classifier

This section serves as the third and last experiment in our evaluation, comparing the results obtained from the suggested algorithm with those of other benchmark algorithms that employ the NB classifier. Examining and contrasting the performance of each algorithm’s classification accuracy using specific features is the aim of this study. The data sets were randomly split into training sets (80%) and testing sets (20%), and the quality of the selected subsets of characteristics was assessed using the NB classifier. To show how effective each approach is, performance measurements including fitness values, mean, standard deviation, and number of selected features are employed.

To rigorously assess the optimization performance of the proposed BCNRBO algorithm, we report the best, mean, worst, and standard deviation of fitness values obtained across all iterations. Table 16 and Table A8 illustrate the behavior of BCNRBO in various dimensional search spaces, highlighting its stability and effectiveness in finding optimal solutions.

The evaluation of algorithm performance using the NB classifier, as presented in Table 16, reveals a distinct pattern compared to the DT classifier results. BCNRBO maintains strong performance by achieving the best fitness values in 13 datasets, continuing its dominance across different classifier paradigms.

Following this, BAOA, BFPA, and BPSO each demonstrate competitive results with best fitness values in 8, 8, and 7 datasets respectively, while BABC and BDE achieve this in 6 datasets each.

The examination of average fitness values using the Naive Bayes classifier, displayed in Table 17, reveals important patterns in algorithm consistency. BCNRBO continues to demonstrate strong performance with the lowest average fitness in multiple datasets, though the distribution of top-performing algorithms shows greater diversity compared to best fitness results.

The jBASO algorithm achieves the best average performance in 9 datasets, indicating particular strength in maintaining consistent results across multiple runs. Following this, BCNRBO, BBA, and BFPA each secure the lowest average fitness in 5, 5, and 4 datasets respectively, with BDE and BABC showing competitive results in 4 and 6 datasets.

This pattern suggests that while different algorithms may achieve excellent peak performance with the NB classifier, their ability to maintain consistently good results across multiple runs varies. The jBASO algorithm’s strong showing in average performance metrics indicates it offers reliable consistency when paired with Naive Bayes classification.

This distinction between best-case and average-case performance highlights the importance of considering both peak optimization capability and result stability when evaluating feature selection algorithms for practical applications.

The evaluation of worst-case performance using the Naive Bayes classifier, presented in Table A7, provides insights into algorithm robustness under challenging conditions. BCNRBO demonstrates strong reliability by achieving the best worst-case performance in 10 datasets, maintaining its position as a consistently robust algorithm.

Following closely, BDE and BABC each secure the best worst-case results in 9 datasets, indicating these algorithms also offer substantial robustness when paired with the NB classifier. The BAOA, jBASO, and BFPA algorithms show competitive performance with best worst-case results in 6, 6, and 5 datasets respectively.

The standard deviation analysis of fitness values using the Naive Bayes classifier, shown in Table A8, provides critical insights into algorithm consistency and stability. BDE and BDA algorithms demonstrate the highest level of consistency, each achieving the lowest standard deviation in 11 datasets, indicating exceptional stability in their results. Following these, BABC shows strong consistency with the lowest standard deviation in 8 datasets, while BCNRBO, BAOA, and jBASO achieve this in 7, 6, and 6 datasets respectively.

In Table A9, BCNRBO has the most features in 14 of the 26 datasets, whereas jBASO has 8 of them. It displays the result of feature size (number of selected features) of 8 different algorithms on 26 datasets. In terms of feature size, BCNRBO can usually affirm the minimal number of selected features, followed by jBASO. Based on the result obtained, BCNRBO gave the smallest feature size on 8 datasets.

The results validate that BCNRBO was good at eliminating irrelevant and redundant features, thus providing high prediction accuracy. Among rivals, BCNRBO is highly capable of finding significant features, which can contribute to better classification performance.

5.3.1. The Statistical Test for NB Classifier

The Friedman test results presented in Table 18 provide robust statistical evidence for performance comparisons using the Naive Bayes classifier across 26 datasets. The analysis shows BCNRBO in first place with the lowest average rank of 3.949. It achieved the top rank in 9 datasets: D2 (Wisconsin), D6 (Diabetes), D7 (Heart), D11 (Hillvalley), D15 (Diagnostic), D18 (HeartEW), D19 (SPECT), D22 (ILPD), and D23 (Parkinsons). BBA followed in second place (avg rank 4.360), with BPSO third (4.671). BCCSA had the weakest performance with the highest average rank of 7.338.

The statistical significance analysis for the Naive Bayes classifier was further investigated using the Wilcoxon signed-rank test. As reported in Table 19, the majority of the obtained p-values are below the significance level of 0.05, confirming that the proposed BCNRBO algorithm performs significantly differently—and in most cases better—than the competing algorithms.

Although the BBA algorithm demonstrates favorable performance in certain datasets, BCNRBO outperformed it in 15 out of 26 datasets. Moreover, the Wilcoxon signed-rank test results provide strong pairwise statistical confirmation of the Friedman test findings. Specifically, BCCSA recorded the highest number of losses (21 datasets) when compared with BCNRBO, followed by BAOA and BDE, each with 19 losses.

In contrast, BDA exhibited the most competitive performance against the proposed algorithm, achieving the fewest losses (7 datasets) and the highest number of ties (8).

5.3.2. Convergence Graphic for Different Dataset Using NB Classifier

Figure 6 presents the convergence characteristics of the proposed BCNRBO algorithm across fifteen benchmark datasets with varying dimensionalities, where the NB classifier was employed for performance evaluation.

Figure 6. Convergence history for different datasets using NB test.

Figure 6. Convergence history for different datasets using NB test.

Figure 6. Convergence curves of BCNRBO and competing classifiers on different datasets using the NB test (continued).

Figure 6. Convergence curves of BCNRBO and competing classifiers on different datasets using the NB test (continued).

Figure 6 illustrate the convergence performance of the competing binary optimization algorithms when employing the NB classifier. As depicted in the figures, the proposed BCNRBO algorithm achieves the lowest fitness values in 10 out of the 15 benchmark functions, namely $F2$, $F4$, $F7$, $F11$, $F15$, $F18$, $F19$, $F20$, $F22$, and $F23$. This result demonstrates the strong effectiveness and robustness of BCNRBO under the NV classification framework across diverse problem landscapes.

In addition, it is observed that some competing algorithms, including BABC, BBA, and BPSO, are capable of reaching the global minimum in a limited number of functions, highlighting their competitive behavior in specific cases. Nevertheless, the overall convergence trends confirm that BCNRBO consistently outperforms the other algorithms in terms of solution quality and convergence stability when integrated with the NB classifier.

6. Conclusions and Future Work

In this study, a novel binary chaotic Newton-Raphson-based optimizer was proposed for solving the feature selection problem. The algorithm introduces a binary extension of NRBO enhanced with various chaotic maps to improve exploration in binary search spaces.

A dynamic potential mechanism inspired by the Lennard–Jones potential was incorporated, where the Hamming distance was integrated into DP to measure interactions between binary solutions. The Tent chaotic map is integrated into the proposed algorithm to enhance exploration and maintain solution diversity. Its chaotic sequences are used in the transfer function and to adaptively update the step size of each agent (chaotic enforcement), enabling a more dynamic and effective search in the binary feature selection space.

The performance of BCNRBO was evaluated using 26 benchmark datasets of varying sizes (low, medium, and large scale), including medical datasets. Three classifiers K-Nearest Neighbors, Decision Tree , and Naïve Bayes were used to assess classification accuracy. Experimental results demonstrated that BCNRBO consistently outperforms several state-of-the-art metaheuristic algorithms, including BAOA, BBA, BFPA, BPSO, BCCSA, jBASO, and BDA, in terms of classification accuracy, fitness value, and feature subset reduction. The superiority of the proposed method was further confirmed through statistical tests.

These results highlight several key contributions and developments:

• The introduction of a binary chaotic NRBO variant capable of effectively handling discrete feature selection problems.
• The incorporation of a dynamic potential mechanism to enhance population diversity and improve exploration–exploitation balance.
• The development of a transfer function mechanism to improve the conversion of continuous solutions into binary space.
• Demonstration of the superior performance of BCNRBO across multiple datasets, classifiers, and benchmark algorithms.

For future work, the proposed algorithm can be extended to multi-objective optimization settings and further enhanced through hybridization with other metaheuristic techniques, adaptive parameter control, and application to large-scale real-world feature selection tasks. Additionally, exploring other chaotic maps and dynamic transfer functions could provide further improvements in search efficiency and solution quality.