Introduction

Clinical Decision Support Systems (CDSSs) have evolved from Expert Systems (ESs), which are knowledge-driven frameworks aimed at mimicking expert-level decision-making. ESs are composed of three main components: a knowledge/rule base, an inference engine, and a user interface1. In the healthcare sector, CDSSs leverage these components to support clinicians in diagnosing and treating patients effectively2. By integrating clinical guidelines with patient-specific data, CDSSs enhance the decision-making process, ultimately improving patient outcomes. Knowledge-driven CDSSs utilize structured knowledge bases containing clinical guidelines, protocols, and heuristics to guide healthcare professionals in their decision-making3. These systems rely on expert-defined rules and clinical evidence to provide recommendations, which are particularly useful in complex medical scenarios. Notable works in this area include the studies in4, which explored the integration of expert knowledge into clinical settings, and later advancements in5, highlighting the evolution and impact of knowledge-driven CDSSs in improving healthcare delivery. In contrast, data-driven CDSSs leverage machine learning techniques, neural networks, and swarm intelligence optimization algorithms to analyze large datasets and derive insights that inform clinical decisions. These systems focus on pattern recognition within patient data, identifying trends and correlations that may not be evident through traditional expert-based approaches. Pioneering research in6 illustrates the potential of data-driven methods to enhance predictive analytics in healthcare, while the work of7 emphasizes the effectiveness of combining neural networks with optimization algorithms to develop robust CDSSs capable of adapting to dynamic clinical environments.

At the core of CDSSs lie the knowledge/rule base and inference mechanism, both of which are crucial for generating accurate and interpretable clinical recommendations. The knowledge/rule base, often structured around fundamental IF-THEN rules, integrates clinical guidelines and decision protocols, guiding the system’s reasoning process8. In these systems, IF-THEN statements define conditions under which specific clinical actions or decisions should be recommended, such as If symptom X and test result Y are present, then consider diagnosis Z. This approach not only supports reliable recommendations but also enhances the interpretability of CDSSs, allowing clinicians to understand and verify the rationale behind each recommendation9. However, building a comprehensive and precise knowledge/rule base that balances accuracy with interpretability poses significant challenges. The vast complexity of medical knowledge and patient variability require adaptable rules that can account for nuanced clinical scenarios10, making rule base development a resource-intensive task. Additionally, as medical guidelines evolve, regularly updating the rule base is essential to ensure recommendations remain current, reliable, and transparent11. Research in12 emphasizes that well-curated and precise rule bases not only improve CDSS accuracy but also increase clinician trust by providing recommendations that are both evidence-based and easily understood.

Recent advancements have introduced Weighted Fuzzy Production Rules (WFPRs) to improve CDSSs by incorporating weights and fuzzy logic, offering a more nuanced representation of medical knowledge. WFPRs address clinical uncertainty by assigning degrees of membership to inputs, which captures the inherent ambiguity in clinical data. This approach has shown success in diabetes diagnosis13, enhancing prediction accuracy and supporting better decision-making by accounting for variability in patient responses. Furthermore, WFPRs contribute to the interpretability of clinical models, especially in managing complex, non-linear bioinformatics data14,15, thereby enhancing both classification accuracy and clinical insight.

To effectively extract WFPRs, Back Propagation Neural Networks (BPNNs) have proven valuable due to their ability to learn complex relationships. However, BPNNs are highly sensitive to initial weight settings, which significantly affect both convergence speed and model accuracy16. Inadequate initialization can lead to convergence issues or entrapment in local minima, thus impacting rule extraction quality17. To address this challenge, metaheuristic algorithms, such as the Harmony Search (HS) Algorithm, have been employed to optimize the initial weights of BPNNs18. HS iteratively refines solutions by balancing the exploitation of past information with random exploration, enabling a more thorough search across the solution space.

Nevertheless, standard HS faces limitations, particularly with large or complex datasets, where it can struggle to achieve an optimal balance between local and global search. To address these issues, several HS variants have been developed. For instance, Adaptive Global Optimal Harmony Search (AGOHS)19 and Harmony Search with Cuckoo Search (HSCS)20 Algorithms introduce enhancements that improve convergence and precision, especially when integrated with BPNNs for clinical rule extraction. These advanced variants offer refined optimization processes, allowing for more accurate and reliable rule extraction in complex clinical datasets. However, challenges persist, as these variants occasionally struggle to maintain an optimal balance between exploration and exploitation during high-dimensional weight optimization in BPNNs, which can lead to premature convergence or stagnation in local optima.

Recent advancements in metaheuristic optimization algorithms have further improved the performance of such methods in high-dimensional and complex optimization problems. For example, Adegboye and Deniz Ülker21 proposed a hybrid artificial electric field algorithm that integrates cuckoo search with refraction learning, demonstrating superior performance in engineering optimization problems. This approach leverages the strengths of both cuckoo search and refraction learning to enhance exploration and exploitation capabilities, particularly in high-dimensional spaces. The hybrid algorithm’s ability to dynamically adjust its search strategy based on the problem’s fitness landscape makes it highly effective for complex optimization tasks, such as those encountered in BPNN weight initialization. Similarly, Adegboye and Deniz Ülker22 introduced an improved artificial electric field algorithm that incorporates Gaussian mutation and specular reflection learning with a local escaping operator. This approach enhances the algorithm’s ability to escape local optima and explore diverse regions of the search space, leading to improved convergence rates and solution quality. The integration of Gaussian mutation and local escaping operators ensures that the algorithm maintains a balance between exploration and exploitation, making it particularly suitable for high-dimensional optimization problems. These advancements highlight the importance of hybrid and adaptive strategies in metaheuristic algorithms, which can be effectively applied to optimize BPNN weights and improve the accuracy of WFPR extraction in CDSSs.

HS is a metaheuristic optimization algorithm inspired by the process of musical improvisation. Since its introduction by Geem, et al.23, HS has been widely applied to various optimization problems due to its simplicity, flexibility, and effectiveness. Over the years, numerous variants of HS have been proposed to address its limitations and improve its performance. The standard HS algorithm balances exploration and exploitation through three key parameters: Harmony Memory Considering Rate \(\:\left(HMCR\right)\), Pitch Adjustment Rate \(\:\left(PAR\right)\), and Bandwidth \(\:\left(bw\right)\). While effective for many problems, the standard HS has limitations in high-dimensional and complex search spaces, where it may struggle with slow convergence and premature stagnation. Improved Harmony Search (IHS) Algorithm introduces dynamic adjustments to the \(\:PAR\) and \(\:bw\) parameters to enhance the algorithm’s convergence speed and solution accuracy. By adaptively tuning these parameters, IHS improves the balance between exploration and exploitation, making it more effective than the standard HS in many applications24. Global-Best Harmony Search (GHS) Algorithm incorporates information from the global best solution to guide the search process. This approach enhances the algorithm’s ability to exploit promising regions of the search space, leading to improved convergence and solution quality compared to the standard HS25. Self-Adaptive Harmony Search (SAHS) Algorithm automates the adjustment of HS parameters \(\:\left(HMCR,\:PAR,\:bw\right)\) based on the algorithm’s performance during the search process. This self-adaptive mechanism reduces the need for manual parameter tuning and improves the algorithm’s robustness across different problem domains26. Hybrid Harmony Search (HHS) Algorithm combines HS with other optimization techniques, such as genetic algorithms, particle swarm optimization, or simulated annealing, to leverage the strengths of multiple methods. These hybrid approaches often achieve better performance than the standard HS, particularly in complex and high-dimensional problems27.

Recent advancements in HS algorithm have focused on enhancing their performance through adaptive parameter control, hybridization with machine learning techniques, and applications to real-world problems. For instance, a novel intelligent global harmony search algorithm based on dynamic parameter tuning has been proposed, improving the algorithm’s ability to solve complex optimization problems28. The integration of HS with machine learning techniques has been explored to enhance optimization efficiency. For example, a hybrid HS-Support Vector Machine (SVM) model has been proposed for feature selection and hyperparameter tuning, demonstrating improved performance in credit scoring applications29. HS and its variants have also been successfully employed in energy optimization. To illustrate, an improved version of HS called Equilibrium Optimization-Based Harmony Search Algorithm with Nonlinear Dynamic Domains (EO-HS-NDD) has been applied to optimize energy systems, demonstrating its effectiveness in managing complex energy distribution networks30. These advancements highlight the versatility and robustness of HS algorithms in addressing diverse and complex optimization tasks. However, despite these improvements, several gaps remain in the current research on HS variants. These include challenges in high-dimensional optimization, exploration-exploitation balance, robustness to noise and imbalanced data, and the need for stronger theoretical foundations. These gaps highlight the necessity for more advanced HS variants, such as the Dynamic Dimension Adjustment Harmony Search (DDA-HS) Algorithm, which is specifically designed to address these challenges through a dynamic dimension adjustment mechanism and enhanced theoretical analysis.

Building on these challenges, there is a critical need for an HS variant specifically tailored to high-dimensional BPNN weight optimization, enhancing rule extraction accuracy in CDSSs. To address this, the DDA-HS algorithm is introduced, designed for specialized optimization tasks like BPNN weight initialization. Traditional methods often struggle in high-dimensional spaces, leading to inefficient searches and suboptimal convergence. DDA-HS employs a dynamic dimension adjustment mechanism that adapts the optimization scope based on the fitness landscape. Our approach first conducts a Markov analysis on the basic HS, identifying areas for enhancement in convergence behavior. Driven by this theoretical insight, DDA-HS refines HS’s balance of exploration and exploitation, improving convergence speed and solution quality in high-dimensional settings. Specifically designed for CDSS applications, DDA-HS, when integrated with BPNNs, achieves precise rule extraction and enhances the reliability of the knowledge base. Utilizing WFPRs, this approach strengthens diagnostic accuracy and interpretability, crucial for effective clinical support systems. The main contributions of this paper are:

  1. 1)

    Markov analysis is applied to the standard HS, enabling the identification of specific areas for enhancement. This theoretical foundation informs the subsequent refinement of HS, leading to more efficient search strategies and improved convergence properties.

  2. 2)

    DDA-HS is introduced, specifically designed with considerations from the No-Free-Lunch (NFL) Theorem. This design ensures that DDA-HS is particularly effective for high-dimensional BPNN weight optimization, addressing the unique challenges presented by complex clinical datasets. By integrating DDA-HS with BPNNs, precise rule extraction is achieved, which is critical for the reliability of CDSSs.

  3. 3)

    WFPRs are employed within the framework to enhance diagnostic accuracy and interpretability. This integration ensures that the generated knowledge/rule base is not only robust but also aligned with clinical reasoning, ultimately improving the effectiveness of decision-making in medical settings.

Materials and methods

HS

HS is a nature-inspired optimization algorithm modeled after the improvisational process in music. Known for its simplicity, flexibility, and effectiveness, HS has gained widespread adoption in solving optimization problems. The algorithm mimics the process of musicians improvising to find the best harmony, which is analogous to finding the optimal solution in an optimization problem. The key components and main steps of HS are introduced.

Key components of HS

  1. 1)

    Harmony Memory (HM): The HM stores a set of candidate solutions (harmonies), each representing a potential solution to the optimization problem. The size of the HM is defined by the Harmony Memory Size \(\:\left(HMS\right)\).

  1. 2)

    \(\:HMCR\): The \(\:HMCR\) determines the probability of selecting a value from the HM for a new solution. A high \(\:HMCR\) value encourages exploitation of existing solutions, while a low \(\:HMCR\) value promotes exploration of new solutions.

  2. 3)

    \(\:PAR\): The \(\:PAR\) controls the probability of adjusting the selected value from the HM. This adjustment is analogous to fine-tuning a musical pitch and helps the algorithm explore the local search space around existing solutions.

  3. 4)

    \(\:bw\): The \(\:bw\) defines the range within which the pitch adjustment occurs. A larger \(\:bw\) allows for broader exploration, while a smaller \(\:bw\) focuses on fine-tuning the solution.

Main steps of HS

The main steps of the standard HS are provided as follows:

  1. 1)

    initialization: the HM is initialized with random solutions within the search space

  2. 2)

    Improvisation: A new solution is generated by selecting values from the HM (based on \(\:HMCR\)) or randomly generating new values. The selected values may be adjusted based on \(\:PAR\) and \(\:bw\).

  3. 3)

    Update: If the new solution is better than the worst solution in the HM, it replaces the worst solution.

  4. 4)

    Termination: The algorithm repeats the improvisation and update steps until a stopping criterion (e.g., maximum iterations or convergence) is met.

The flowchart of the standard HS is provided, elucidating its operation in detail shown in Fig. 1.

Fig. 1
figure 1

Flowchart of HS.

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Strengths and limitations of HS

  1. 1)

    Strengths: HS is easy to implement and requires fewer parameters compared to other metaheuristic algorithms, and it effectively balances exploration and exploitation, making it suitable for a wide range of optimization problems31,32,33.

  2. 2)

    Limitations: In high-dimensional problems34, HS may struggle with slow convergence due to its fixed parameter settings. Moreover, the algorithm may get trapped in local optima, especially in complex and multimodal search spaces.

Motivation for DDA-HS

The limitations of standard HS, particularly in high-dimensional and complex optimization problems, motivated the development of the DDA-HS algorithm. By introducing a dynamic dimension adjustment mechanism, DDA-HS addresses these limitations and enhances the algorithm’s performance in challenging optimization tasks. Following the background on HS, the DDA-HS algorithm is introduced, which builds upon the strengths of HS while addressing its limitations.

DDA-HS

In the context of optimization algorithms, particularly metaheuristic approaches like HS, understanding the underlying behavior and convergence properties is crucial for enhancing both performance and reliability. One effective way to achieve this is through a theoretical analysis based on Markov processes35. By modeling the iterative steps of HS as a Markov chain, it becomes possible to assess the algorithm’s stochastic behavior, convergence probability, and overall dynamics in a structured mathematical framework. This analysis not only helps to understand how the algorithm explores the search space, but also provides insights into its development mechanism to evaluate its convergence properties and provide a deeper theoretical understanding, ultimately guiding the development of improved variants.

HS theoretical analysis

HS’s optimization process can be modeled as an absorbing Markov chain. At each iteration, HS updates the population of solutions, where the next state \(\:{X}_{t+1}\) depends only on the current state \(\:{X}_{t}\), following the Markov property. Once the global optimal solution \(\:{x}_{opt}\) appears in the population, it remains in all subsequent populations, making it an absorbing state, which can be formally expressed as:

$$\:P\left\{{x}_{opt}\notin\:{X}_{t+1}|{x}_{opt}\in\:{X}_{t}\right\}=0$$
(1)

Therefore, an absorbing Markov chain model with \(\:N\) states is proposed based on36, with state \(\:N\) as the absorbing state representing optimal fitness, as illustrated in Fig. 2. The states are arranged from left to right according to their fitness values. In a simplified scenario with a single individual in a one-dimensional problem, the transition probability \(\:{P}_{i\to\:j}\) indicates the likelihood of moving from state \(\:i\) to state \(\:j\). Notably, each perturbation operation either maintains the current state or advances it to the right, toward improved fitness.

Fig. 2
figure 2

Absorbing Markov chain model.

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The transition probabilities can be categorized:

  1. 1)

    Self-transition probability: The probability of remaining in state \(\:i\) is \(\:{P}_{i\to\:i}=HMCR\cdot\:PAR\cdot\:1/2+HMCR\cdot\:\left(1-PAR\right)+\left(1-HMCR\right)\cdot\:i/N\). It increases as the algorithm approaches the absorbing state \(\:N\), reflecting a risk of getting stuck in local optima due to incremental adjustments in HS.

  2. 2)

    Forward-transition probability: For \(\:i<j\le\:N\), \(\:{P}_{i\to\:j}=HMCR\cdot\:PAR\cdot\:1/2+\left(1-HMCR\right)\cdot\:(N-i)/N\), indicating movement towards higher fitness states.

To improve global optimization, reducing \(\:{P}_{i\to\:i}\) and increasing \(\:{P}_{i\to\:j}\) helps the algorithm escape local optima and explore better solutions. Additionally, the likelihood of finding the global optimal solution is crucial and can be illustrated through probability analysis. In an \(\:N\)-dimensional problem with \(\:n\) possible values per dimension, there are \(\:{n}^{N}\) possible solutions, only one of which is optimal. To quantify the probability of discovering this optimal solution, the probability \(\:\epsilon\) is introduced, representing the minimum chance that a newly generated individual, derived from existing individuals, will be optimal. The equation for \(\:\epsilon\) is:

$$\:\epsilon={\sum\:}_{k=0}^{N}{C}_{N}^{k}{\left[(1-HMCR)\cdot\:1/n\right]}^{N-k}{\left[HMCR\cdot\:PAR\cdot\:1/n\right]}^{k}>0$$
(2)

Here, \(\:\epsilon=P\left\{{x}_{opt}\in\:{X}_{t+1}|{x}_{opt}\notin\:{X}_{t}\right\}\) indicates the probability that the optimal solution \(\:{x}_{opt}\) enters the new population \(\:{X}_{t+1}\) given its absence in \(\:{X}_{t}\). The equation captures the worst-case scenario, combining scenarios of randomly selecting values in \(\:N-k\) dimensions and adjusting \(\:k\) dimensions from existing solutions. A non-zero \(\:\epsilon\) indicates a persistent, albeit small, chance of generating a new optimal solution in each iteration. Therefore, HS theoretically avoids getting trapped in local optima, maintaining a non-zero probability of discovering the global optimal solution.

Furthermore, the convergence of HS can be established by showing that, with sufficient iterations, the algorithm will almost surely identify the optimal solution. Let \(\:P\left\{{x}_{opt}\in\:{X}_{t}\right\}\) denote the probability that the optimal solution \(\:{x}_{opt}\) is present in the population \(\:{X}_{t}\) at iteration \(\:t\). The goal is to demonstrate that:

$$\:\underset{t\to\:\infty\:}{\text{lim}}P\left\{{x}_{opt}{\notin\:X}_{t}\right\}=0$$
(3)

This indicates that as \(\:t\) increases, the probability of \(\:{x}_{opt}\) not being in \(\:{X}_{t}\) approaches \(\:0\). To prove this, the law of total probability is applied:

$$\:P\left(A\right)=P\left(A|B\right)\cdot\:P\left(B\right)+\left(A|\neg\:B\right)\cdot\:P\left(\neg\:B\right)$$
(4)

Where \(\:A\) is the event \(\:{x}_{opt}{\notin\:X}_{t+1}\), \(\:B\) is \(\:{x}_{opt}\in\:{X}_{t}\), and \(\:\neg\:B\) is \(\:{x}_{opt}\notin\:{X}_{t}\). Thus, the probability \(\:P\left\{{x}_{opt}\notin\:{X}_{t+1}\right\}\) can be expressed in two cases:

  1. 1)

    \(\:{x}_{opt}\) is in \(\:{X}_{t}\) but not in \(\:{X}_{t+1}\). The probability for this scenario is \(\:P\left\{{x}_{opt}\notin\:{X}_{t+1}|{x}_{opt}\in\:{X}_{t}\right\}\), with prior probability of \(\:P\left\{{x}_{opt}\in\:{X}_{t}\right\}\).

  2. 2)

    \(\:{x}_{opt}\) is not in \(\:{X}_{t}\) and remains absent in \(\:{X}_{t+1}\). The probability for this scenario is \(\:P\left\{{x}_{opt}\notin\:{X}_{t+1}|{x}_{opt}\notin\:{X}_{t}\right\}\), with the prior probability \(\:P\left\{{x}_{opt}\notin\:{X}_{t}\right\}\).

Combining these cases yields:

$$\:P\left\{{x}_{opt}\notin\:{X}_{t+1}\right\}=P\left\{{x}_{opt}\notin\:{X}_{t+1}|{x}_{opt}\in\:{X}_{t}\right\}\cdot\:P\left\{{x}_{opt}\in\:{X}_{t}\right\}+P\left\{{x}_{opt}\notin\:{X}_{t+1}|{x}_{opt}\notin\:{X}_{t}\right\}\cdot\:P\left\{{x}_{opt}\notin\:{X}_{t}\right\}$$
(5)

Given that Eq. (1) and Eq. (2), it follows that \(\:1-\epsilon=P\left\{{x}_{opt}\notin\:{X}_{t+1}|{x}_{opt}\notin\:{X}_{t}\right\}\). Thus, Eq. (5) simplifies to:

$$\:P\left\{{x}_{opt}\notin\:{X}_{t+1}\right\}=0\cdot\:P\left\{{x}_{opt}\in\:{X}_{t}\right\}+(1-\epsilon)\cdot\:P\left\{{x}_{opt}\notin\:{X}_{t}\right\}=(1-\epsilon)\cdot\:P\left\{{x}_{opt}\notin\:{X}_{t}\right\}$$
(6)

For \(\:t+n\) iterations, iterating this relationship yields:

$$\:P\left\{{x}_{opt}\notin\:{X}_{t+n}\right\}={\left(1-\epsilon\right)}^{n}\cdot\:P\left\{{x}_{opt}\notin\:{X}_{t}\right\}$$
(7)

\(\:As\:n\to\:\infty\:,\:{\left(1-\epsilon\right)}^{n}\to\:0,\:leading\:to:\)

$$\:\underset{t\to\:\infty\:}{\text{lim}}P\left\{{x}_{opt}{\notin\:X}_{t}\right\}=0$$
(8)

Consequently, with increasing iterations, the optimal solution \(\:{x}_{opt}\) will almost certainly be present in \(\:{X}_{t}\), proving the convergence of HS.

In summary, the Markov-based fundamental analysis of HS demonstrated that HS can theoretically discover the optimal solution at any iteration. Furthermore, it was shown that given a sufficient number of iterations, HS is guaranteed to converge to the optimal solution. This theoretical analysis reinforces the robustness and convergence properties of HS, confirming its ability to effectively navigate the search space. In addition, these findings lay the groundwork for introducing an enhanced HS variant. Building on the guaranteed convergence of HS, the proposed improvements aim to optimize performance by refining its efficiency and adaptability in practical applications.

Inspiration of HS

The feasible values for each decision variable are assumed to already exist within the HM, removing the need for fine-tuning. After several iterations, only one decision variable per harmony remains suboptimal. Thus, the HM is represented as:

$$\:\text{H}\text{M}=\left[\begin{array}{c}{X}^{1}\\\:{X}^{2}\\\:\vdots\\\:{X}^{HMS}\end{array}\right]=\left[\begin{array}{cccc}{x}_{1}^{1}&\:{x}_{2}^{opt}&\:\cdots\:&\:{x}_{n}^{opt}\\\:{x}_{1}^{opt}&\:{x}_{2}^{2}&\:\cdots\:&\:{x}_{n}^{opt}\\\:\vdots&\:\vdots&\:\ddots\:&\:\vdots\\\:{x}_{1}^{opt}&\:{x}_{2}^{opt}&\:\cdots\:&\:{x}_{n}^{HMS}\end{array}\right]$$
(9)

HS generates a new harmony \(\:{X}^{new}\) by selecting values from the HM, where the probability that a given dimension equals its optimal value is \(\:(HMS-1)/HMS\). The probability of achieving optimal values across all dimensions diminishes rapidly as the number of dimensions increases, denoted as \(\:{\left(\right(HMS-1)/HMS)}^{n}\). In contrast, a proposed method randomly selects a harmony from the HM and adjusts one dimension, giving a probability of \(\:1/n\cdot\:(HMS-1)/HMS\) for achieving optimal values across all dimensions. As shown in Fig. 3, the success rate of the HS method sharply declines as dimensionality increases, whereas the proposed method’s performance remains more stable, outperforming HS in high-dimensional problems. This analysis highlights that HS struggles with high-dimensional optimization, while the proposed method shows better potential. Therefore, DDA-HS is proposed to leverage this insight.

Fig. 3
figure 3

Success rate comparison.

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According to the NFL theorem, no algorithm is universally superior across all problem domains37. This limitation highlights the necessity for a novel HS variant, specifically tailored to optimize BPNN initial weights and improve the rule extraction in CDSSs. As a result, DDA-HS is introduced, specifically designed for high-dimensional optimization challenges namely the initialization of weights in BPNNs. To be specific, high-dimensional spaces often pose significant difficulties for traditional algorithms, leading to inefficient search strategies and suboptimal convergence. DDA-HS addresses this by incorporating a dynamic dimension adjustment mechanism, enabling the algorithm to adaptively adjust the number of dimensions optimized based on the fitness landscape. The motivation for this advancement stems from the inherent strengths and weaknesses of swarm intelligence methods38, such as HS. While these approaches balance exploration and exploitation effectively in moderate-dimensional spaces, their performance degrades as dimensionality increases. DDA-HS, through its adaptive adjustment mechanism, refines this balance, enhancing both convergence speed and solution quality in high-dimensional settings. This improvement is particularly critical for tasks like BPNN weight optimization, where dimensionality directly impacts the model’s training effectiveness. Once optimized, BPNN is leveraged to extract WFPRs, constructing a reliable knowledge/rule base critical for CDSSs. The introduction of DDA-HS thus represents a significant enhancement particularly for complex, high-dimensional problems, positioning it as a more efficient and robust alternative for such challenges.

Main steps of DDA-HS

The theoretical analysis of HS using Markov chains provided critical insights into the algorithm’s convergence behavior and exploration-exploitation balance. HS was modeled as a Markov chain, and the transition probabilities between states, which represent potential solutions in the search space, were analyzed39. This analysis revealed two key aspects that directly influenced the design of DDA-HS. First, the self-transition probability, which increases as the algorithm approaches the optimal solution, indicates a risk of getting stuck in local optima due to incremental adjustments in HS. To address this, DDA-HS incorporates a dynamic dimension adjustment mechanism that reduces the likelihood of self-transition by adaptively adjusting the number of dimensions optimized based on the fitness landscape. Second, the forward-transition probability, which governs the movement towards improved fitness states, was enhanced in DDA-HS to increase the chances of escaping local optima and discovering the global optimum. This was achieved by refining the balance between exploration and exploitation, allowing DDA-HS to focus on dimensions more likely to lead to improved solutions. These insights from the Markov chain analysis directly influenced key design aspects of DDA-HS, including its dynamic dimension adjustment mechanism, fitness-based dimension selection, and cyclic dimension adjustment strategy, ensuring robust performance in high-dimensional and complex search spaces. The specific steps of DDA-HS are as follows:

  1. 1)

    fitness standardization: Z-score standardization is applied to normalize the fitness values

$$\:{\widehat{f}}_{i}=\left({f}_{i}-{\mu\:}_{f}\right)/{\sigma\:}_{f}$$
(10)

Where \(\:{f}_{i}\) represents the fitness of individual \(\:i\), and \(\:{\mu\:}_{f}\) and \(\:{\sigma\:}_{f}\) denote the mean and standard deviation of the population’s fitness values, respectively.

  1. 2)

    Fitness normalization: Standardized fitness is normalized to \(\:\left[0,\:1\right]\).

$$\:{\widehat{f}}_{i}=\frac{{\widehat{f}}_{i}-\text{m}\text{i}\text{n}\left(\widehat{f}\right)}{\text{ma}x\left(\widehat{f}\right)-\text{m}\text{i}\text{n}\left(\widehat{f}\right)}$$
(11)
  1. 3)

    Dimension adjustment count calculation: Each individual’s dimension adjustment count \(\:{D}_{i}\) is determined based on fitness.

$$\:{D}_{i}={D}_{min}+{\widehat{f}}_{i}\cdot\:\left({{D}_{max}-D}_{min}\right)$$
(12)

Where \(\:{D}_{min}\) and \(\:{D}_{max}\) are the predefined minimum and maximum number of dimensions to be adjusted. The resulting \(\:{D}_{i}\) value is rounded to the nearest integer.

  1. 4)

    Dimension selection and adjustment: A cyclic approach selects dimensions based on \(\:{D}_{i}\), from \(\:start\), initially set to \(\:0\), to \(\:end\), calculated as:

$$\:end=\left(start+{D}_{i}\right)\%V$$
(13)

Where \(\:V\) is the total number of dimensions. If \(\:end\) is less than \(\:start\), the indices wrap around, and the selected dimensions are given by:

$$\:indices=\left\{\begin{array}{c}\left\{start,\dots\:,V-1\right\}\cup\:\left\{0,\dots\:,end-1\right\},\:if\:end<start\\\:\left\{start,\dots\:,end-1\right\},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:otherwise\end{array}\right.$$
(14)

Once the dimensions are selected, the adjustment process begins. For \(\:HMCR\) of the cases, the adjustment follows this rule:

$$\:{x}_{i}^{(t+1)}={x}_{i}^{\left(t\right)}+0.7\cdot\:{r}_{1}\cdot\:\left({x}_{best}-{x}_{i}^{\left(t\right)}\right)+0.3\cdot\:{r}_{2}\cdot\:({x}_{i}^{\left(t\right)}-{x}_{worst})$$
(15)

Where \(\:{r}_{1}\) and \(\:{r}_{2}\) are random numbers in \(\:\left[0,\:1\right]\), and \(\:{x}_{best}\) and \(\:{x}_{worst}\) represent the best and worst individuals, respectively. The traditional \(\:PAR\) mechanism is removed, relying solely on \(\:HMCR\). For the remaining \(\:1-HMCR\) cases, Levy flight is used to explore new regions of the solution space with the update rule:

$$\:{x}_{i}^{\left(t+1\right)}={r}_{3}\cdot\:\left({x}_{best}+{x}_{worst}\right)-{x}_{best}+\vartriangle$$
(16)

Where \(\:{r}_{3}\) is a random number in \(\:\left[0,\:1\right]\), and \(\:\vartriangle\) represents the Levy flight step. \(\:start\) for the next individual is updated after the dimension adjustments:

$$\:start\leftarrow\:end$$
(17)
  1. 5)

    Fitness evaluation and update: If the new fitness improves:

$$\:{f}_{i}^{\left(t+1\right)}<{f}_{i}^{\left(t\right)}$$
(18)

the individual is updated. This process iteratively drives convergence by dynamically adjusting dimensions based on fitness, improving solution diversity and performance.

PIMA dataset preprocessing

The PIMA dataset, containing \(\:768\) records with \(\:8\) feature attributes and a binary classification label, underwent key preprocessing steps:

  1. 1)

    Data import and formatting: The dataset was divided into feature data (first \(\:8\) columns) and classification labels (last column).

  2. 2)

    Fuzzy k-means clustering: Each feature column was clustered into \(\:3\) groups. This process resulted in a membership matrix \(\:U\) of size \(\:768\times\:3\) for each feature. Each element \(\:{u}_{i,j}\) in \(\:U\) represents the degree of membership of data point \(\:{x}_{i}\) to cluster \(\:j\), with normalization ensuring that the sum of memberships for each data point equals \(\:1\):

$$\:\left\{\begin{array}{c}U=\left[\begin{array}{ccc}{u}_{\text{1,1}}&\:{u}_{\text{1,2}}&\:{u}_{\text{1,3}}\\\:{u}_{\text{2,1}}&\:{u}_{\text{2,2}}&\:{u}_{\text{2,3}}\\\:\vdots&\:\vdots&\:\vdots\\\:{u}_{\text{768,1}}&\:{u}_{\text{768,2}}&\:{u}_{\text{768,3}}\end{array}\right]\:\:\:\:\:\:\:\:\:\:\\\:{\sum\:}_{j=1}^{3}{u}_{i,j}=1,\:for\:i=\text{1,2},\dots\:,768\end{array}\right.$$
(19)
  1. 3)

    Cluster center update: Cluster centers \(\:C=\left[{c}_{1},{c}_{2},{c}_{3}\right]\) were iteratively updated using the weighted average equation:

$$\:{c}_{j}={\sum\:}_{i=1}^{768}({{u}_{i,j})}^{m}\cdot\:{x}_{i}/{\sum\:}_{i=1}^{768}({{u}_{i,j})}^{m},\:for\:j=\text{1,2},3$$
(20)

Here, \(\:m\) is the fuzzification factor (typically \(\:m=2\)), and \(\:{x}_{i}\) is the feature value of data point \(\:i\). This ensures that cluster centers are adjusted toward data points with higher membership degrees.

  1. 4)

    Distance calculation and membership update: The Euclidean distance \(\:{d}_{i,j}\) between each data point \(\:{x}_{i}\) and cluster center \(\:{c}_{j}\) was calculated:

$$\:{d}_{i,j}=\Vert{x}_{i}-{c}_{j}\Vert$$
(21)

A distance matrix \(\:D\) of size \(\:768\times\:3\) was formed:

$$\:D=\left[\begin{array}{ccc}{d}_{\text{1,1}}&\:{d}_{\text{1,2}}&\:{d}_{\text{1,3}}\\\:{d}_{\text{2,1}}&\:{d}_{\text{2,2}}&\:{d}_{\text{2,3}}\\\:\vdots&\:\vdots&\:\vdots\\\:{d}_{\text{768,1}}&\:{d}_{\text{768,2}}&\:{d}_{\text{768,3}}\end{array}\right]$$
(22)

The membership values were updated using:

$$\:{u}_{i,j}=1/{\sum\:}_{k=1}^{3}{\left({d}_{i,j}/{d}_{i,k}\right)}^{\left(2/\left(m-1\right)\right)},\:for\:i=\text{1,2},\dots\:,768;\:j=\text{1,2},3$$
(23)

This process was repeated until convergence, defined by:

$$\:\underset{i,j}{\text{max}}\left|{u}_{i,j}^{current}-{u}_{i,j}^{previous}\right|<\epsilon$$
(24)

with \(\:\epsilon=0.0000001\) as the convergence threshold.

  1. 5)

    One-hot encoding of labels: The binary classification labels (\(\:1\) for diabetic and \(\:0\) for non-diabetic) were converted into one-hot encoded vectors. For diabetic samples, the one-hot encoding was \(\:\left[1,\:0\right]\), and for non-diabetic samples, it was \(\:\left[0,\:1\right]\).

After fuzzy k-means clustering, the resulting membership matrices from each feature were combined into a final matrix of size \(\:768\times\:24\), reflecting the membership values for \(\:3\) clusters across the \(\:8\) features. The classification labels were transformed into a one-hot encoded format \(\:(768\times\:2)\).

BPNN workflow

  1. 1)

    Data partitioning and normalization: The preprocessed PIMA dataset is randomly partitioned into training \(\:\left(75\%\right)\) and testing \(\:\left(25\%\right)\) subsets, ensuring representativeness and reducing bias in the evaluation. This results in \(\:576\) training samples and \(\:192\) testing samples. Subsequently, normalization is applied to both sets, transforming each \(\:24\)-dimensional vector \(\:x\) into a unit norm. For a feature vector \(\:x=[{x}_{1},{x}_{2},\dots\:,{x}_{24}]\), the Euclidean norm (\(\:L2\) norm) is computed as:

$$\:{\Vert x \Vert}_{2}=\sqrt{{\sum\:}_{i=1}^{24}{x}_{i}^{2}}$$
(25)

The normalized vector \(\:\stackrel{\sim}{x}\) is obtained by:

$$\:\stackrel{\sim}{x}=x/{\Vert x\Vert}_{2}$$
(26)

Thus, each normalized vector satisfies:

$$\:{\Vert\stackrel{\sim}{x}\Vert}_{2}=1$$
(27)

The normalized input matrix \(\:\stackrel{\sim}{X}\), containing \(\:768\) samples characterized by \(\:24\) fuzzy attributes from fuzzy k-means clustering, is represented as:

$$\:\stackrel{\sim}{X}=\left[\begin{array}{cccc}{\stackrel{\sim}{x}}_{\text{1,1}}&\:{\stackrel{\sim}{x}}_{\text{1,2}}&\:\cdots\:&\:{\stackrel{\sim}{x}}_{\text{1,24}}\\\:{\stackrel{\sim}{x}}_{\text{2,1}}&\:{\stackrel{\sim}{x}}_{\text{2,2}}&\:\cdots\:&\:{\stackrel{\sim}{x}}_{\text{2,24}}\\\:\vdots&\:\vdots&\:\ddots\:&\:\vdots\\\:{\stackrel{\sim}{x}}_{\text{768,1}}&\:{\stackrel{\sim}{x}}_{\text{768,2}}&\:\cdots\:&\:{\stackrel{\sim}{x}}_{\text{768,24}}\end{array}\right],\:\stackrel{\sim}{X}\in\:{\mathbb{R}}^{768\times\:24}$$
(28)

Here, \(\:{\stackrel{\sim}{x}}_{i,j}\) represents the normalized value of the \(\:j\)-th fuzzy attribute for the \(\:i\)-th sample.

  1. 2)

    Hyperparameter configuration: BPNN is configured with essential hyperparameters to optimize performance. The input layer contains \(\:24\) neurons for the \(\:24\)-dimensional feature vectors, while the output layer has \(\:2\) neurons for binary classification. An empirical choice of \(\:4\) neurons is made for the hidden layer to balance complexity and computational efficiency. Moreover, the performance of BPNN is highly dependent on the choice of hyperparameters, such as the learning rate, regularization coefficient, batch size, and the number of training epochs. To optimize the model, extensive experiments were conducted to determine the most effective hyperparameter settings. The learning rate \(\:\left(\eta\:\right)\) was set to \(\:0.008\) after testing values ranging from \(\:0.005\) to \(\:0.1\), as this value provided the best balance between convergence speed and stability. A regularization coefficient \(\:\left(\lambda\:\right)\) of \(\:0.001\) was chosen to prevent overfitting without overly restricting the model’s ability to learn complex patterns. For efficient training, a batch size of \(\:128\) was selected, as it offered a good trade-off between computational efficiency and training stability. The model was trained for \(\:100\) epochs, with each epoch comprising \(\:5000\) iterations, ensuring sufficient convergence without overfitting. These hyperparameter settings were determined through systematic experimentation, demonstrating the robustness of the proposed method within a reasonable range of values. For instance, higher learning rates (e.g., \(\:0.05\)) led to unstable training, while smaller batch sizes (e.g., \(\:16\)) introduced noise into the gradient updates. The final configuration, namely \(\:\eta\:=0.008,\:\lambda\:=0.001,\:batch\:size=128,\:epoch=100\), and \(\:iteration=5000\), provided the best balance between model performance and computational efficiency.

  2. 3)

    BPNN construction: BPNN is constructed with an input layer, hidden layer, and output layer. It is trained exclusively on the training set \(\:{\stackrel{\sim}{X}}_{training}\), learning relationships between input fuzzy attributes and output classifications.

$$\:{\stackrel{\sim}{X}}_{training}=\left[\begin{array}{cccc}{\stackrel{\sim}{x}}_{\text{1,1}}&\:{\stackrel{\sim}{x}}_{\text{1,2}}&\:\cdots\:&\:{\stackrel{\sim}{x}}_{\text{1,24}}\\\:{\stackrel{\sim}{x}}_{\text{2,1}}&\:{\stackrel{\sim}{x}}_{\text{2,2}}&\:\cdots\:&\:{\stackrel{\sim}{x}}_{\text{2,24}}\\\:\vdots&\:\vdots&\:\ddots\:&\:\vdots\\\:{\stackrel{\sim}{x}}_{\text{576,1}}&\:{\stackrel{\sim}{x}}_{\text{576,2}}&\:\cdots\:&\:{\stackrel{\sim}{x}}_{\text{576,24}}\end{array}\right],\:\stackrel{\sim}{X}\in\:{\mathbb{R}}^{576\times\:24}$$
(29)

The network’s weights are initialized using DDA-HS, generating a weight vector \(\:w\) divided into input-to-hidden weights \(\:{W}_{ih}\) and the hidden-to-output weights \(\:{W}_{ho}\). The weight vector contains \(\:104\) elements structured as:

$$\:w=\left[{w}_{\text{1,1}}^{\left(ih\right)},{w}_{\text{1,2}}^{\left(ih\right)},\dots\:,{w}_{\text{24,4}}^{\left(ih\right)},{w}_{\text{1,1}}^{\left(ho\right)},{w}_{\text{1,2}}^{\left(ho\right)},\dots\:,{w}_{\text{4,2}}^{\left(ho\right)}\right],\:w\in\:{\mathbb{R}}^{24\times\:4+4\times\:2}$$
(30)

The input-to-hidden weight matrix \(\:{W}_{ih}\) is defined as:

$$\:{W}_{ih}=\left[\begin{array}{cccc}{w}_{\text{1,1}}^{\left(ih\right)}&\:{w}_{\text{1,2}}^{\left(ih\right)}&\:\cdots\:&\:{w}_{\text{1,4}}^{\left(ih\right)}\\\:{w}_{\text{2,1}}^{\left(ih\right)}&\:{w}_{\text{2,2}}^{\left(ih\right)}&\:\cdots\:&\:{w}_{\text{2,4}}^{\left(ih\right)}\\\:\vdots&\:\vdots&\:\ddots\:&\:\vdots\\\:{w}_{\text{24,1}}^{\left(ih\right)}&\:{w}_{\text{24,2}}^{\left(ih\right)}&\:\cdots\:&\:{w}_{\text{24,4}}^{\left(ih\right)}\end{array}\right],\:{W}_{ih}\in\:{\mathbb{R}}^{24\times\:4}$$
(31)

The hidden-to-output weight matrix \(\:{W}_{ho}\) is defined as:

$$\:{W}_{ho}=\left[\begin{array}{cc}{w}_{\text{1,1}}^{\left(ho\right)}&\:{w}_{\text{1,2}}^{\left(ho\right)}\\\:{w}_{\text{2,1}}^{\left(ho\right)}&\:{w}_{\text{2,2}}^{\left(ho\right)}\\\:{w}_{\text{3,1}}^{\left(ho\right)}&\:{w}_{\text{3,2}}^{\left(ho\right)}\\\:{w}_{\text{4,1}}^{\left(ho\right)}&\:{w}_{\text{4,2}}^{\left(ho\right)}\end{array}\right],\:{W}_{ho}\in\:{\mathbb{R}}^{4\times\:2}$$
(32)
  1. 4)

    BPNN training: The training employs mini-batch gradient descent, processing each batch in a single iteration. Forward propagation computes predicted outputs, while backpropagation updates weights based on the loss function gradient. In forward propagation, the hidden layer’s output is computed using the sigmoid activation function:

$$\:H=\sigma\:({\stackrel{\sim}{X}}_{training}\cdot\:{W}_{ih})$$
(33)

Where\(\:\:H\) denotes the hidden layer output, and \(\:\sigma\:\left(z\right)=1/(1+\text{e}\text{x}\text{p}(-z\left)\right)\) is applied elementwise. The output layer activation \(\:O\) is calculated as:

$$\:O=\sigma\:(H\cdot\:{W}_{ho})$$
(34)

In backpropagation, the error at the output layer is computed and propagated back to adjust weights. The Mean Squared Error \(\:\left(MSE\right)\) measures the difference between true labels \(\:{y}_{i}\) and predicted outputs \(\:{\widehat{y}}_{i}\):

$$\:MSE=1/N\cdot\:{\sum\:}_{i=1}^{N}{({y}_{i}-{\widehat{y}}_{i})}^{2}$$
(35)

Where \(\:N=576\) for the training set. The total loss function incorporates \(\:MSE\) and \(\:L1\) regularization:

$$\:Loss=MSE+\lambda\:\cdot\:{\Vert w\Vert}_{1}$$
(36)

The \(\:L1\) norm of \(\:{\Vert w\Vert}_{1}\) is defined as:

$$\:{\Vert w\Vert}_{1}={\Vert{W}_{ih}\Vert}_{1}+{\Vert{W}_{ho}\Vert}_{1}={\sum\:}_{i=1}^{104}\left|{w}_{i}\right|$$
(37)

Once \(\:Loss\) is calculated, weights are updated to minimize it:

$$\:\left\{\begin{array}{c}{W}_{ih}\leftarrow\:{W}_{ih}-\eta\:\cdot\:\partial\:Loss/\partial\:{W}_{ih}\:\:\\\:{W}_{ho}\leftarrow\:{W}_{ho}-\eta\:\cdot\:\partial\:Loss/\partial\:{W}_{ho}\end{array}\right.$$
(38)
  1. 5)

    BPNN evaluation: BPNN is evaluated on both the training and testing datasets to assess its generalization and learning effectiveness, which produces output matrices \(\:{O}_{training}\in\:{\mathbb{R}}^{576\times\:2}\) and \(\:{O}_{testing}\in\:{\mathbb{R}}^{192\times\:2}\), where each element \(\:{o}_{i,j}\) represents the raw output for sample \(\:i\) and class \(\:j\). These outputs are converted to class probabilities using the softmax function:

$$\:{\stackrel{\sim}{o}}_{i,j}=\text{e}\text{x}\text{p}\left({o}_{i,j}\right)/{\sum\:}_{k=1}^{2}\text{e}\text{x}\text{p}\left({o}_{i,k}\right),\:for\:i=\text{1,2},\dots\:,N;\:j=\text{1,2}$$
(39)

Here \(\:N=576\) for training and \(\:N=192\) for testing. This ensures the sum of probabilities for each sample equals \(\:1\). The normalized probability matrices are:

$$\:\left\{\begin{array}{c}{\stackrel{\sim}{O}}_{training}=\left[\begin{array}{cc}{\stackrel{\sim}{o}}_{\text{1,1}}&\:{\stackrel{\sim}{o}}_{\text{1,2}}\\\:{\stackrel{\sim}{o}}_{\text{2,1}}&\:{\stackrel{\sim}{o}}_{\text{2,2}}\\\:\vdots&\:\vdots\\\:{\stackrel{\sim}{o}}_{\text{576,1}}&\:{\stackrel{\sim}{o}}_{\text{576,2}}\end{array}\right]\\\:{\stackrel{\sim}{O}}_{testing}=\left[\begin{array}{cc}{\stackrel{\sim}{o}}_{\text{1,1}}&\:{\stackrel{\sim}{o}}_{\text{1,2}}\\\:{\stackrel{\sim}{o}}_{\text{2,1}}&\:{\stackrel{\sim}{o}}_{\text{2,2}}\\\:\vdots&\:\vdots\\\:{\stackrel{\sim}{o}}_{\text{192,1}}&\:{\stackrel{\sim}{o}}_{\text{192,2}}\end{array}\right]\:\:\:\end{array}\right.$$
(40)

True labels are stored in one-hot encoded matrices \(\:{Y}_{training}\in\:{\mathbb{R}}^{576\times\:2}\) and \(\:{Y}_{testing}\in\:{\mathbb{R}}^{192\times\:2}\), where each row represents the class label:

$$\:{Y}_{i}=\left\{\begin{array}{c}\left[\begin{array}{cc}1&\:0\end{array}\right],\:if\:j=1\:\left(diabetic\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\\\:\left[\begin{array}{cc}0&\:1\end{array}\right],\:if\:j=2\:(non-diabetic)\end{array}\right.$$
(41)

For each sample \(\:i\), the predicted class \(\:{\widehat{y}}_{i}\) is the index of the highest probability:

$$\:{\widehat{y}}_{i}=\underset{j}{\text{a}{rg}\text{max}}{\stackrel{\sim}{o}}_{i,j},\:for\:i=\text{1,2},\dots\:,N$$
(42)

The true class \(\:{y}_{i}\) is similarly determined from the one-hot encoded labels. Accuracy is computed by comparing the predicted and true class indices:

$$\:\left\{\begin{array}{c}{Accuracy}_{training}=1/576\cdot\:{\sum\:}_{i=1}^{576}1\:\left({\widehat{y}}_{i}={y}_{i}\right)\\\:{\:Accuracy}_{testing}=1/192\cdot\:{\sum\:}_{i=1}^{192}1\:\left({\widehat{y}}_{i}={y}_{i}\right)\:\:\:\:\end{array}\right.$$
(43)

Here \(\:1\:\left({\widehat{y}}_{i}={y}_{i}\right)\) is an indicator function that equals \(\:1\) if the predicted class matches the true class.

WFPR extraction

  1. 1)

    Weight pruning: Elements in the weight matrices \(\:{W}_{ih}^{*}\) and \(\:{W}_{ho}^{*}\) are set to \(\:0\) if their absolute values are less than or equal to \(\:0.5\) after BPNN training. The pruned matrix \(\:{\stackrel{\sim}{W}}_{ih}^{*}\) is:

$$\:{\stackrel{\sim}{W}}_{ih}^{*}=\left[\begin{array}{cccc}{\stackrel{\sim}{w}}_{\text{1,1}}^{(ih,*)}&\:{\stackrel{\sim}{w}}_{\text{1,2}}^{(ih,*)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{\text{1,4}}^{(ih,*)}\\\:{\stackrel{\sim}{w}}_{\text{2,1}}^{(ih,*)}&\:{\stackrel{\sim}{w}}_{\text{2,2}}^{(ih,*)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{\text{2,4}}^{(ih,*)}\\\:\vdots&\:\vdots&\:\ddots\:&\:\vdots\\\:{\stackrel{\sim}{w}}_{\text{24,1}}^{(ih,*)}&\:{\stackrel{\sim}{w}}_{\text{24,2}}^{(ih,*)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{\text{24,4}}^{(ih,*)}\end{array}\right]$$
(44)

Where each element \(\:{\stackrel{\sim}{w}}_{i,j}^{(ih,*)}=0,\:if\:\left|{w}_{i,j}^{\left(ih,*\right)}\right|\le\:0.5\); otherwise, it remains its value. Similarly, the pruned matrix \(\:{\stackrel{\sim}{W}}_{ho}^{*}\) is:

$$\:{\stackrel{\sim}{W}}_{ho}^{*}=\left[\begin{array}{cc}{\stackrel{\sim}{w}}_{\text{1,1}}^{(ho,*)}&\:{\stackrel{\sim}{w}}_{\text{1,2}}^{(ho,*)}\\\:{\stackrel{\sim}{w}}_{\text{2,1}}^{(ho,*)}&\:{\stackrel{\sim}{w}}_{\text{2,2}}^{(ho,*)}\\\:{\stackrel{\sim}{w}}_{\text{3,1}}^{(ho,*)}&\:{\stackrel{\sim}{w}}_{\text{3,2}}^{(ho,*)}\\\:{\stackrel{\sim}{w}}_{\text{4,1}}^{(ho,*)}&\:{\stackrel{\sim}{w}}_{\text{4,2}}^{(ho,*)}\end{array}\right]$$
(45)

Where \(\:{\stackrel{\sim}{w}}_{i,j}^{(ho,*)}=0\:if\:\left|{w}_{i,j}^{\left(ho,*\right)}\right|\le\:0.5\); otherwise, it retains its value. These pruned matrices represent the simplified layer connections, removing smaller, less significant weights to streamline BPNN.

  1. 2)

    Acquisition of the weighted matrix: The new weighted matrix \(\:{W}_{io}\), representing the relationship between input and output layers, is obtained by taking the dot product of the pruned weight matrices \(\:{\stackrel{\sim}{W}}_{ih}^{*}\) and \(\:{\stackrel{\sim}{W}}_{ho}^{*}\):

$$\:{W}_{io}={\left({\stackrel{\sim}{W}}_{ih}^{*}\cdot\:{\stackrel{\sim}{W}}_{ho}^{*}\right)}^{T}$$
(46)

The dot product results in a \(\:{\mathbb{R}}^{24\times\:2}\) matrix, which is transposed to yield \(\:{W}_{io}\in\:{\mathbb{R}}^{2\times\:24}\), aligning it for subsequent steps.

  1. 3)

    Preprocessing of the weighted matrix: The weighted matrix \(\:{W}_{io}\:(2\times\:24)\) represents the contribution of each input fuzzy attribute to the output classes. To facilitate rule extraction, each element is rounded to \(\:2\) decimal places:

$$\:{\stackrel{\sim}{w}}_{i,j}^{\left(io\right)}=\text{r}\text{o}\text{u}\text{n}\text{d}\left({w}_{i,j}^{\left(io\right)},2\right)$$
(47)

The preprocessed matrix \(\:{\stackrel{\sim}{W}}_{io}\) is:

$$\:{\stackrel{\sim}{W}}_{io}=\left[\begin{array}{cccc}{\stackrel{\sim}{w}}_{\text{1,1}}^{\left(io\right)}&\:{\stackrel{\sim}{w}}_{\text{1,2}}^{\left(io\right)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{\text{1,24}}^{\left(io\right)}\\\:{\stackrel{\sim}{w}}_{\text{2,1}}^{\left(io\right)}&\:{\stackrel{\sim}{w}}_{\text{2,2}}^{\left(io\right)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{\text{2,24}}^{\left(io\right)}\end{array}\right]$$
(48)

The input features \(\:{A}_{1},\:{A}_{2},\dots\:,{A}_{8}\) each have \(\:3\) fuzzy attributes \(\:{A}_{n}:\left\{{A}_{n,1},{A}_{n,2},{A}_{n,3}\right\}\), and the output classes \(\:{C}_{1}\) \(\:\left(diabetic\right)\) and \(\:{C}_{2}\) \(\:(non-diabetic)\) are linked to the fuzzy attributes. Thus, the matrix can be expressed as:

$$\:{\stackrel{\sim}{W}}_{io}=\left[\begin{array}{ccccc}{\stackrel{\sim}{w}}_{{C}_{1},{A}_{\text{1,1}}}^{\left(io\right)}&\:{\stackrel{\sim}{w}}_{{C}_{1},{A}_{\text{1,2}}}^{\left(io\right)}&\:{\stackrel{\sim}{w}}_{{C}_{1},{A}_{\text{1,3}}}^{\left(io\right)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{{C}_{1},{A}_{\text{8,3}}}^{\left(io\right)}\\\:{\stackrel{\sim}{w}}_{{C}_{2},{A}_{\text{1,1}}}^{\left(io\right)}&\:{\stackrel{\sim}{w}}_{{C}_{2},{A}_{\text{1,2}}}^{\left(io\right)}&\:{\stackrel{\sim}{w}}_{{C}_{2},{A}_{\text{1,3}}}^{\left(io\right)}&\:\cdots\:&\:{\stackrel{\sim}{w}}_{{C}_{2},{A}_{\text{8,3}}}^{\left(io\right)}\end{array}\right]$$
(49)
  1. 4)

    WFPR construction: A general WFPR for \(\:Class\:i\) is:

$$\:\mathbf{I}\mathbf{F}\:{Condition}_{1}\:\mathbf{A}\mathbf{N}\mathbf{D}\:{Condition}_{2}\:\mathbf{A}\mathbf{N}\mathbf{D}\cdots\:\mathbf{A}\mathbf{N}\mathbf{D}{\:Condition}_{8}\:\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}\:Class\:i$$
(50)

Each condition links an input feature to its fuzzy attributes, reflecting their contribution to the output class. For example, a specific WFPR for \(\:Class\:1\) might be:

$$\:\mathbf{I}\mathbf{F}\:{A}_{1}\:is\:{A}_{\text{1,1}}\:\left[\left|{\stackrel{\sim}{w}}_{{C}_{1},{A}_{\text{1,1}}}^{\left(io\right)}\right|\right]\:\mathbf{A}\mathbf{N}\mathbf{D}\:{A}_{4}\:is\:\mathbf{N}\mathbf{O}\mathbf{T}\:{A}_{\text{4,2}}\:\left[\left|{\stackrel{\sim}{w}}_{{C}_{1},{A}_{\text{4,2}}}^{\left(io\right)}\right|\right]\:\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}\:Class\:1$$
(51)

Here, \(\:{A}_{n}\) represent input features with \(\:3\) fuzzy attributes \(\:{A}_{n,k}\). Positive contributions to a class are written as \(\:{A}_{n}\) is \(\:{A}_{n,k}\) \(\:\left[\left|{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\right|\right]\), while negative contributions are expressed as \(\:{A}_{n}\) is NOT \(\:{A}_{n,k}\) \(\:\left[\left|{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\right|\right]\), where \(\:{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\) is the rounded weight value. In constructing WFPRs, only one non-zero fuzzy attribute per feature is selected. If a feature has no non-zero attributes, it is skipped. The number of possible rules is based on the non-zero fuzzy attributes of each feature, with a maximum of \(\:{3}^{8}=6561\) rules if all fuzzy attributes are non-zero. If all fuzzy attributes are \(\:0\), no rules are generated. The total number of rules \(\:{R}_{i}\) for each output class \(\:Class\:i\) is:

$$\:{R}_{i}={\prod\:}_{n=1}^{8}nz\left({A}_{n}\right)$$
(52)

Where \(\:nz\left({A}_{n}\right)\) is the number of non-zero fuzzy attributes for feature \(\:{A}_{n}\). The total number of WFPRs is:

$$\:R={\sum\:}_{i=1}^{2}{R}_{i}$$
(53)
  1. 5)

    Confidence calculation: For each classification category \(\:{C}_{i}\), a fuzzy rule \(\:r\) is formed by selecting one non-zero fuzzy weight \(\:{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\) from the fuzzy sets of each feature \(\:n\) (\(\:8\) total). The confidence of a rule \(\:{CF}_{r}\) is calculated as:

$$\:{CF}_{r}={\sum\:}_{n=1}^{8}\left(\left\{\begin{array}{c}\left|{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\right|\cdot\:\left(1-{\stackrel{\sim}{x}}_{{A}_{n,k}}\right),\:if\:{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}<0\\\:{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\cdot\:{\stackrel{\sim}{x}}_{{A}_{n,k}},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:otherwise\end{array}\right.\right)$$
(54)

Where \(\:{\stackrel{\sim}{x}}_{{A}_{n,k}}\) is the fuzzy membership value. If the weight is negative, the contribution is adjusted by subtracting the membership value from \(\:1\). The total confidence \(\:{CF}_{{C}_{i}}\) for category \(\:{C}_{i}\) is:

$$\:{CF}_{{C}_{i}}={\sum\:}_{r\in\:{R}_{{C}_{i}}}{CF}_{r}$$
(55)

Only rules with \(\:{CF}_{r}\ge\:0.5\) are considered. For such rules:

$$\:{CF}_{{C}_{i}}={\sum\:}_{r\in\:{R}_{{C}_{i}}}{\sum\:}_{n=1}^{8}\left(\left\{\begin{array}{c}\left|{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\right|\cdot\:\left(1-{\stackrel{\sim}{x}}_{{A}_{n,k}}\right)-0.5,\:if\:{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}<0\\\:{\stackrel{\sim}{w}}_{{C}_{i},{A}_{n,k}}^{\left(io\right)}\cdot\:{\stackrel{\sim}{x}}_{{A}_{n,k}}-0.5,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:otherwise\end{array}\right.\right)$$
(56)
  1. 6)

    Effectiveness evaluation: Once the confidence for each data instance and category is calculated, the resulting matrix is represented as:

$$\:CF=\left[\begin{array}{cc}{CF}_{1}^{\left(1\right)}&\:{CF}_{2}^{\left(1\right)}\\\:{CF}_{1}^{\left(2\right)}&\:{CF}_{2}^{\left(2\right)}\\\:\vdots&\:\vdots\\\:{CF}_{1}^{\left(N\right)}&\:{CF}_{2}^{\left(N\right)}\end{array}\right]$$
(57)

Where \(\:N=576\) for the training set and \(\:N=192\) for the testing set. Each row represents the confidence values for \(\:2\) classification categories, derived from the respective fuzzy rules. The category with the highest confidence value determines the predicted class: If \(\:{CF}_{1}^{\left(N\right)}>{CF}_{2}^{\left(N\right)}\), the data instance is classified as \(\:Class\:1\); otherwise, it is classified as \(\:Class\:2\). The accuracy for both the training and testing datasets is computed:

$$\:\left\{\begin{array}{c}{Accuracy}_{training}=1/576\cdot\:{\sum\:}_{i=1}^{576}1\:\left({\widehat{y}}_{i}={y}_{i}\right)\\\:{\:Accuracy}_{testing}\:=1/192\cdot\:{\sum\:}_{i=1}^{192}1\:\left({\widehat{y}}_{i}={y}_{i}\right)\:\:\end{array}\right.$$
(58)

Here, \(\:{\widehat{y}}_{i}\) is the predicted class, \(\:{y}_{i}\) is the true class, and \(\:1\:\left({\widehat{y}}_{i}={y}_{i}\right)\) is an indicator function that equals \(\:1\) if they are the same.

Experimental setup

The experiment is designed to evaluate the performance of DDA-HS against four algorithms: HS, Cuckoo Search (CS), AGOHS, and HSCS algorithms. The parameter settings for each algorithm are listed in Table 1. Each algorithm runs \(\:30\) times to account for stochastic variability. The experiment was conducted on a \(\:2.3\) GHz quad-core Intel Core \(\:i5\) processor with \(\:8\) GB RAM, using Python \(\:3.7\).

Table 1 Parameter settings of the experimental algorithm.
Full size table

The target function \(\:E\left(w\right)\) represents the total error of BPNN for a given set of weights \(\:w\) on the PIMA training dataset. This objective function aims to minimize prediction error on the training set while incorporating a regularization term to prevent overfitting. Its specific equation is:

$$\:E\left(w\right)=1/2\cdot\:{\sum\:}_{i=1}^{N}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}+\lambda\:\cdot\:{\sum\:}_{j=1}^{M}\left|{w}_{j}\right|$$
(59)

Here, \(\:{y}_{i}\) and \(\:{\widehat{y}}_{i}\) are the actual and predicted outputs for instance \(\:i\), respectively. The regularization coefficient \(\:\lambda\:\) is set at \(\:0.001\), and \(\:{w}_{j}\) denotes the individual weight of BPNN. The number of training instances \(\:N\) is \(\:576\), and the total number of weights \(\:M\) is \(\:104\).

Results

Quantitative analysis

The convergence curve in Fig. 4 shows the average fitness values over \(\:100\) iterations for \(\:30\) runs of the algorithms. DDA-HS exhibits a consistently faster convergence rate and achieves a lower final fitness value, indicating its effective management of the exploration-exploitation trade-off for quick convergence to high-quality solutions. In contrast, other algorithms, particularly HS, display slower declines in fitness and higher final fitness values.

Fig. 4
figure 4

Error fitness mean convergence curves.

Full size image

Table 2 Provides quantitative results, revealing that DDA-HS has the best mean fitness of \(\:1.03e+02\), significantly outperforming HS’s mean of \(\:1.23e+02\). However, DDA-HS has a standard deviation of \(\:6.68e+00\), suggesting more variability across runs. The median fitness for DDA-HS also stands at \(\:1.03e+02\), confirming its superior performance. While its worst fitness value of \(\:1.16e+02\) is not the lowest as CS achieves \(\:1.14e+02\), DDA-HS excels with a best-case fitness of \(\:8.94e+01\). Furthermore, statistical significance of performance differences was assessed using the Wilcoxon signed-rank test, yielding P-values consistently below \(\:0.05\) (e.g., \(\:1.27e-10\) for HS, \(\:7.49e-04\) for CS), indicating that DDA-HS significantly outperforms the competing algorithms. All comparisons are marked with a \(\:{\prime\:}+{\prime\:}\) sign.

Table 2 Comparison of error fitness results.
Full size table

DDA-HS proves to be a more effective optimization technique than standard HS and CS, and hybrid variants namely AGOHS and HSCS. In contrast, slower convergence rates in HS, CS, AGOHS, and HSCS result in suboptimal solutions, as reflected in both the convergence curve and statistical analysis. Overall, DDA-HS demonstrates superior accuracy, convergence speed, and robustness, validating its efficacy for high-dimensional optimization tasks, such as in BPNN weight optimization.

In addition, the proposed DDA-HS algorithm demonstrates superior performance compared to other optimization methods, particularly in terms of convergence speed and solution accuracy. DDA-HS dynamically adjusts the number of dimensions optimized based on the fitness landscape, allowing it to focus on the most promising dimensions and significantly reduce the search space. This mechanism accelerates convergence, especially in high-dimensional problems where standard HS and its variants (e.g., AGOHS, HSCS) struggle due to fixed search strategies. Additionally, DDA-HS refines the balance between exploration and exploitation by adaptively adjusting the search scope, ensuring efficient exploration of the search space while exploiting promising regions. This adaptability enables DDA-HS to escape local optima and achieve higher solution accuracy, even in complex datasets with multiple local optima or non-linear relationships. Furthermore, DDA-HS excels in handling imbalanced and noisy datasets, such as medical datasets with high noise levels or imbalanced classes, by prioritizing relevant features and avoiding overfitting. In contrast, HS, CS, and their variants often rely on random or heuristic-based adjustments, which can lead to suboptimal solutions. Overall, DDA-HS’s robustness and adaptability make it highly effective across a wide range of problem complexities, from low-dimensional to high-dimensional datasets, and in scenarios involving non-linear relationships or noisy data.

BPNN and WFPR classification accuracy

Table 3 Presents the classification accuracy of BPNN and WFPRs across different optimization methods. For BPNN, the unoptimized BPNN achieved \(\:77.78\text{\%}\) training accuracy and \(\:71.88\text{\%}\) testing accuracy. Among the optimized models, AGOHS attained the highest training accuracy \(\:\left(79.17\text{\%}\right)\), yet its testing accuracy \(\:\left(71.35\text{\%}\right)\) showed limited improvement, suggesting overfitting. HSCS, on the other hand, exhibited more balanced performance with a testing accuracy of \(\:76.04\text{\%}\), outperforming other methods on unseen data. DDA-HS achieved \(\:74.48\text{\%}\) testing accuracy, surpassing HS, CS and AGOHS, while maintaining a solid \(\:78.30\text{\%}\) training accuracy, indicating effective training stabilization without overfitting. For WFPRs, the BPNN model optimized by DDA-HS achieved the highest performance, with a training accuracy of \(\:79.51\text{\%}\) and a testing accuracy of \(\:77.08\text{\%}\), indicating an effective balance between BPNN complexity and generalization. In contrast, HSCS showed robust performance with a testing accuracy of \(\:76.56\text{\%}\), although its training accuracy of \(\:77.95\text{\%}\) was lower than that of DDA-HS. The unoptimized BPNN exhibited a significant gap between training \(\:\left(76.22\text{\%}\right)\) and testing accuracy \(\:\left(72.40\text{\%}\right)\), underscoring its inefficiency in capturing complex data patterns. Both HS and CS yielded moderate improvements, with HS achieving a testing accuracy of \(\:73.96\text{\%}\) and CS slightly higher at \(\:75.00\text{\%}\), reflecting limited enhancements over the unoptimized BPNN.

Table 3 Classification accuracy of BPNN and WFPRs.
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Generated WFPRs

WFPR extraction from the weighted matrix \(\:{\stackrel{\sim}{W}}_{io}\), is vital for understanding BPNN’s decision-making process and improving its interpretability. This matrix summarizes the influence of input features on output classes, with each weight reflecting a feature’s importance in determining the outcome:

$$\:{\stackrel{\sim}{W}}_{io}=\left[\begin{array}{cccccccccccccccccccccccc}-0.69&\:0&\:0&\:8.1&\:-22.54&\:0&\:0&\:0&\:0&\:0&\:2.59&\:0&\:0&\:0&\:0&\:8.55&\:0&\:0&\:-2.89&\:-4.68&\:0&\:0&\:3.39&\:-5.12\\\:0&\:0&\:0&\:-7.9&\:23&\:0&\:0&\:0&\:0&\:0&\:-2.52&\:0&\:0&\:0&\:0&\:-8.33&\:0&\:0&\:2.82&\:4.56&\:0&\:0&\:-3.31&\:4.98\end{array}\right]$$
(60)

WFPR extraction involves constructing logical rules from significant weights in the matrix. These rules follow an \(\:\mathbf{I}\mathbf{F}\:Condition\:\mathbf{T}\mathbf{H}\mathbf{E}\mathbf{N}\:Class\) format. From the matrix, \(\:8\) rules for each class are derived as WFPRs for \(\:Class\:1\:\left(diabetic\right)\) and \(\:Class\:2\:\left(non-diabetic\right)\) (\(\:16\) in total), as illustrated in Table 4.

Table 4 WFPRs for \(\:Class\:1\:\left(diabetic\right)\) and \(\:Class\:2\:(non-diabetic)\).
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By extracting interpretable rules, clinicians gain insights into the relationships between features and diabetes classification, fostering trust and facilitating informed decision-making in clinical contexts. Specifically, the interpretability of WFPRs is a key advantage in CDSSs, as it allows clinicians to understand and act upon the model’s predictions. For example, a WFPR extracted from the PIMA dataset might state: IF Glucose is High [Weight: 8.1] AND BMI is High [Weight: 8.55] AND Diabetes Pedigree Function is NOT Low [Weight: 2.89], THEN Class 1 (Diabetic). This rule highlights the importance of elevated glucose and BMI levels, along with a family history of diabetes, in predicting diabetes risk. Clinicians can use such rules to prioritize diagnostic testing and early interventions for high-risk patients. Conversely, another WFPR might classify a patient as non-diabetic based on normal glucose levels, such as IF Glucose is NOT High [Weight: 7.9] AND BMI is NOT High [Weight: 8.33] AND Diabetes Pedigree Function is Low [Weight: 2.82], THEN Class 2 (Non-Diabetic), helping to optimize screening resources by identifying low-risk individuals. The adaptability of fuzzy cluster thresholds (e.g., Low, Medium, High) ensures that WFPRs remain clinically relevant across different healthcare settings. For instance, glucose level categories can be adjusted according to updated diagnostic criteria. By providing transparent, interpretable rules, WFPRs enhance CDSSs by supporting personalized treatment decisions, reducing unnecessary testing, and improving overall diagnostic accuracy.

Discussion

The transition from IF-THEN rules to WFPRs marks a significant advancement in the development of CDSSs. IF-THEN rules have long served as the foundation, providing a straightforward framework for encoding medical expertise and guiding clinical decisions40. These rules rely on binary logic, where specific conditions lead to definitive outcomes. While such rules are effective in simple, well-defined cases, they often fail to capture the complexity and ambiguity inherent in medical data. For example, fever or pain often exist on a continuum, and the relationships between medical conditions are highly non-linear and context dependent. To overcome this issue, WFPRs offer a solution to these limitations by combining fuzzy logic with rule weighting. Fuzzy logic allows for a more flexible interpretation of clinical data, representing conditions along a spectrum rather than as binary states41. This nuanced representation more closely mirrors the clinical reasoning of healthcare professionals. Moreover, the integration of weights into fuzzy rules enables a dynamic prioritization of rules based on their relevance, reliability, or confidence level. This weighting mechanism ensures that rules of higher clinical significance exert a greater influence on decision-making processes.

WFPRs have proven to be effective in several CDSS applications, particularly in scenarios requiring the simultaneous evaluation of multiple diagnostic factors. Their probabilistic framework enables CDSSs to provide diagnoses accompanied by confidence levels, offering a significant advantage in complex clinical situations. This probabilistic approach allows clinicians to assess not only the likelihood of a diagnosis but also the level of certainty attached to the system’s recommendations42. This capability is particularly valuable in complex or ambiguous cases, where traditional rules might struggle to account for overlapping symptoms or uncertain data. By enhancing the flexibility and granularity of decision-making, WFPRs have demonstrated significant potential in improving diagnostic accuracy and reliability within CDSSs, ultimately contributing to better patient outcomes.

The quality of initial weights in BPNN is a crucial factor. Recent research emphasizes the importance of weight optimization strategies43, proposing various techniques such as metaheuristic algorithms, which enhance the initial weight distribution and improve overall convergence rates44. One such approach is the use of HS and its variants, which have proven effective in optimizing BPNN weights for improved classification accuracy in CDSSs45. As a result, BPNN is increasingly applied in WFPR extraction. The capacity to process and learn from large and complex datasets makes it well-suited for identifying subtle and intricate relationships in medical data. Unlike traditional rule extraction methods that rely heavily on expert knowledge, BPNN provides a data-driven, automated approach to rule learning, reducing human bias and enhancing the objectivity of the derived rules46. Therefore, it is possible to derive fuzzy rules that are interpretable and clinically relevant. These fuzzy rules, grounded in fuzzy logic principles, allow the system to account for the inherent uncertainty and ambiguity present in medical data, thus improving decision-making in uncertain environments.

HS is a metaheuristic optimization technique inspired by the improvisation process in musical performances. This analogy forms a framework, ideal for neural network weight optimization47. When applied to BPNN weight initialization, HS explores the weight space to identify near-optimal configurations that lead to improved convergence during the training phase. Meanwhile, HS iteratively searches the weight space and retains promising weight sets, which is particularly beneficial in BPNN, as better initial weights significantly enhance the network’s ability to quickly converge to an accurate solution. Early studies have shown that HS outperforms traditional random or heuristic-based initialization methods in neural network training, leading to faster learning and improved classification accuracy48. Moreover, recent innovations in HS have refined its effectiveness by incorporating dynamic parameter adjustments to balance exploration and exploitation throughout the optimization process, and the introduction of advanced strategies like mutation operations further increases diversity in the search space28,49. Therefore, these improvements accelerate convergence and enhance classification performance, highlighting HS’s potential to address weight initialization inefficiencies in machine learning.

The comprehensive evaluation highlights the effectiveness of DDA-HS in optimizing BPNN and extracting WFPRs. The method strikes a balance between enhancing training performance and generalizing to unseen data, as evidenced by its superior testing accuracy in both cases. This indicates that DDA-HS facilitates more flexible and adaptive weight initialization. Furthermore, the consistent improvements in classification accuracy underscore the potential of WFPRs to enhance the interpretability of machine learning models, particularly in critical areas like medical diagnostics, where transparent decision-making is essential. In summary, these findings validate DDA-HS as a robust optimization tool for BPNN, especially in extracting actionable insights from complex datasets namely the PIMA dataset.

Conclusions

This paper provided a thorough evaluation of DDA-HS for optimizing BPNN performance using the PIMA dataset. Key findings include:

  1. 1)

    Optimized performance: DDA-HS achieved a strong balance between complexity and generalization, with \(\:78.30\%\) training accuracy and \(\:74.48\%\) testing accuracy for BPNN. Additionally, the classification accuracy of WFPRs on the training set reached \(\:79.51\%\), and \(\:77.08\%\) on the testing set, outperforming traditional methods like HS, CS, AGOHS, HSCS, and unoptimized BPNN. This demonstrates the effectiveness of DDA-HS in both optimizing model performance and improving generalization.

  2. 2)

    Prediction interpretability: WFPR extraction from the weighted matrix offered transparency into BPNN’s decision-making, revealing the impact of input features on classification outcomes.

  3. 3)

    Clinical implications: The interpretability of WFPRs enhances trust in machine learning models for clinical diagnostics by providing clear reasoning for model predictions, aiding healthcare professionals.

In conclusion, DDA-HS proves to be a robust method for improving both accuracy and interpretability in the BPNN model. The primary goal of this study was to introduce and validate the DDA-HS algorithm for optimizing the initial weights of BPNNs in the context of WFPR extraction, using the PIMA dataset as a benchmark. While the results on the PIMA dataset are promising, future work will focus on extending this framework to larger and more complex datasets, such as MIMIC-III or eICU, to evaluate its scalability and robustness in high-dimensional and diverse clinical environments. Additionally, challenges related to imbalanced classes and noisy data are planned to be addressed, ensuring that DDA-HS remains effective in real-world scenarios where data quality and class distribution may vary. Potential strategies for improving scalability and robustness, such as adaptive hyperparameter tuning, will be explored to enhance the algorithm’s performance across various healthcare applications. Cross-dataset validation across multiple healthcare domains will further enhance the generalizability and robustness of the proposed method, paving the way for its broader application in CDSSs.