3.3.1 Radial Basis Function Networks (RBFN).

Organizations use Radial Basis Function Network (RBFN) as a three-layer feedforward neural architecture because it provides complex nonlinear approximation ability through a straightforward and understandable structure. The network architecture of RBFN comprises features including an input layer and then a single hidden layer, which includes radial basis activation functions and finishes with a linear output layer [36]. The architecture of the RBFN proves helpful for environment datasets because it functions best when data does not divide linearly.

Let denote the input vector representing a sample instance consisting of n normalized water quality features, such as pH, Hardness, Solids, Sulfate, Organic Carbon, Chloramines, Trihalomethanes, and Turbidity. The RBFN transforms this input via M radial basis functions, each centered at , resulting in a hidden layer activation given by:

Where is the width (standard deviation) parameter of the i-th Gaussian kernel, controlling each neuron’s receptive field. The output of the network is a linear combination of the hidden layer responses:

with w_i representing the weights connecting the i-th hidden unit to the output node. In binary classification tasks, the sign or thresholded value of determines the predicted class label.

The primary advantage of RBFNs is that they perform localized approximations that yield more precise decision boundaries by needing fewer training samples than multilayer perceptrons. Radial basis functions within RBFNs operate with high significance only in particular regions of the input space, thus offering both interpretability and modularity to the network. The training process of RBFNs is divided into independent steps involving unsupervised cluster center and spread calculation before supervised least squares or metaheuristics output weight optimization.

The choice of Radial Basis Function Network (RBFN) for this study, while other deep learning models like Long Short-Term Memory (LSTM) and Convolutional Neural Networks (CNN) are popular in agri-data contexts, is deliberate and justified by several key advantages for the specific problem of water quality classification. Firstly, RBFNs are particularly well-suited for nonlinear function approximation and pattern recognition in multidimensional spaces, which aligns perfectly with the complex and often non-linearly separable nature of water quality parameters. Unlike LSTMs, which excel in sequential data processing, and CNNs, which are powerful for spatial feature extraction, RBFNs offer a strong capability for static, tabular data classification without the need for complex temporal or spatial dependencies. Secondly, RBFNs possess inherent interpretability due to their clear three-layer structure and the localized nature of their radial basis functions. Each hidden neuron can be viewed as representing a cluster in the input space, making it easier to understand which input features are most influential for a given classification outcome. This interpretability is highly valuable in precision agriculture, where understanding the ‘why’ behind a prediction can inform better decision-making for farmers. Finally, RBFNs often require less training data and computational resources compared to more complex deep learning architectures like LSTMs or CNNs, especially when dealing with data that doesn’t inherently have strong temporal or spatial patterns. Their faster training process and simpler architecture make them a more practical choice for deployment in scenarios where computational resources might be constrained, as is often the case in agricultural settings. This efficiency, combined with their robust performance on nonlinear classification tasks, makes RBFN a highly effective and justifiable choice for this water quality prediction application.

The RBFN performs better than other state-of-the-art (SOTA) ML models in various evaluation metrics. The superior RBFN performance derives from its natural capability to handle complex nonlinear water quality parameter interactions while maintaining compatibility with Dipper Throated Optimizer (DTO) and its hybrid version DTOPRS. The implemented optimization strategies allow researchers to achieve accurate performance combinations between feature selection approaches and network parameters, which enhances their ability to generalize predictions for unknown data points. RBFN is the top-performing model among experimental setups because it provides an excellent combination of predictive power, computational efficiency, and transparent algorithmic behavior.

3.3.2 Benchmark machine learning models.

A wide range of methodological categories was covered by selecting benchmark machine learning models for complete performance evaluation. These models served as choices since they demonstrate historical importance in classification work while maintaining interpretability, scalability, and direct applicability to agricultural and environmental data analytics.

Support Vector Machine (SVM): SVM is the first selection because it utilizes kernel-based methods to develop optimal hyperplanes for class separation. SVM’s main advantage is its ability to maximize margins because it improves generalization, particularly in spaces with high dimensions [37]. The radial basis function kernel was the RBF implementation in SVM to transform non-linearly separable inputs into a space with linear separation and feasible boundaries. Synergism exists between SVM performance and regularized parameter C and kernel width gamma, which require precise optimization procedures during application.

Decision Tree (DT): DT classifier operates as a non-parametric algorithm that divides the input space through feature value choices that produce maximum information gain or impurity reduction measurements (computing Gini index or entropy), Decision Trees provide both great interpretability and simple implementation, but they lead to overfitting training data unless proper pruning strategies are integrated [38].

Random Forest (RF): RF model creates an ensemble learning approach by joining various decision trees that process bootstrapped data sets using randomly chosen features. The wisdom-of-crowds mechanism in RFs enables improved predictive performance and overfitting control [39]. Random Forest models increase interpretability difficulties while needing adjustment of meta-settings that determine the number of trees and maximum tree depth.

Recurrent Neural Network (RNN): RNN processes sequential or temporal data dependencies as a dynamic model. Despite the time-invariant nature of the water quality dataset, RNNs help identify complex inter-feature relationships because of their feedback mechanism and previous input activation retention ability [40]. The standard RNN shows performance constraints because the gradient vanishes when performing backpropagation over time.

Long Short-Term Memory (LSTM): LSTM network served as the gateway solution because it stands as a gated variant of RNN. The LSTM network utilizes memory cells with input, forget, and output gate functions to control information transit through time steps. Compared to standard RNNs, the LSTM network uses a memory mechanism that enables better detection of sustained dependencies [41]. The high computational demands of LSTMs make them an excellent choice for analyzing data containing hidden temporal relationships in information and multiple feature interactions.

The three predictive models provide independent computational views about predicting water quality outcomes. Through comprehensive benchmarking, the RBFN can receive rigorous performance evaluation, creating an effectiveness hierarchy dedicated to sustainable agricultural resource management.

3.3.3 Hyperparameter tuning.

To ensure optimal performance and generalization capability, a meticulous hyperparameter tuning process was conducted for all machine learning models employed in this study. Hyperparameter tuning is crucial as it directly impacts a model’s ability to learn from the training data and perform accurately on unseen data, preventing both underfitting and overfitting. The goal was to identify the best combination of hyperparameters that yielded the highest classification accuracy on the validation set, while maintaining model stability.

The tuning strategy involved a systematic search across predefined ranges for each hyperparameter. For complex models like RNN, LSTM, and RBFN, a combination of grid search and random search approaches, guided by cross-validation, was utilized to efficiently explore the hyperparameter space. For traditional machine learning models (SVM, Decision Tree, Random Forest), similar cross-validation-based search techniques were applied. Each model’s performance was evaluated using metrics such as accuracy, and the set of hyperparameters that resulted in the best performance on the validation folds was selected as the optimal configuration.

Below, we describe the key hyperparameters for each model, followed by a table (Table 2) summarizing their applied search ranges and the optimal values obtained after the tuning process.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Hyperparameter ranges and optimal values for all classification models.

https://doi.org/10.1371/journal.pone.0327230.t002

Support Vector Machine (SVM) hyperparameters.

For SVM, critical hyperparameters tuned included C (Regularization parameter), which controls the trade-off between achieving a low training error and a large margin; kernel, defining the type of kernel function used (e.g., ‘rbf’ for non-linear separation); gamma, the kernel coefficient for ‘rbf’, ‘poly’ and ‘sigmoid’ kernels, influencing the reach of a single training example; and degree for the ‘poly’ kernel. Other parameters like ‘shrinking’, ‘probability’, ‘tol’, ‘cache_size’, ‘class_weight’, ‘verbose’, ‘max_iter’, ‘decision_function_shape’, and ‘break_ties’ were also considered to fine-tune the model’s behavior and efficiency.

Decision tree hyperparameters.

Decision Trees were optimized by tuning criterion (e.g., ‘gini’ or ‘entropy’ for measuring split quality); splitter (e.g., ‘best’ or ‘random’ for split strategy); max_depth, controlling the maximum depth of the tree; min_samples_split and min_samples_leaf, which regulate the minimum number of samples required to split a node or be at a leaf node, respectively, helping prevent overfitting. Additional parameters like ‘max_features’, ‘ccp_alpha’, ‘min_weight_fraction_leaf’, ‘max_leaf_nodes’, and ‘min_impurity_decrease’ were also adjusted.

Random forest hyperparameters.

Random Forest, an ensemble of Decision Trees, had its performance optimized through tuning n_estimators (the number of trees in the forest); criterion for individual trees; max_depth, min_samples_split, and min_samples_leaf (similar to Decision Tree); max_features (the number of features to consider when looking for the best split); and bootstrap (whether bootstrap samples are used when building trees). Other parameters like ‘oob_score’, ‘ccp_alpha’, ‘min_impurity_decrease’, ‘max_leaf_nodes’, ‘n_jobs’, and ‘warm_start’ were also evaluated.

Recurrent Neural Network (RNN) hyperparameters.

For RNNs, crucial hyperparameters included units (or hidden_size), defining the dimensionality of the output space of the hidden layer; activation_function for the recurrent layers; learning_rate for the optimizer; optimizer (e.g., Adam, RMSprop); batch_size for training; and epochs (with early stopping to prevent overfitting). dropout_rate and recurrent_dropout_rate were applied to mitigate overfitting, and number_of_layers determined the network’s depth.

Long Short-Term Memory Networks (LSTM) hyperparameters.

Similar to RNNs, LSTM networks were tuned using units (hidden_size); activation_function for the internal gates; recurrent_activation for the recurrent steps; learning_rate and optimizer; batch_size and epochs. dropout_rate and recurrent_dropout_rate were also critical for regularization, as was the number_of_layers to control model complexity.

Radial Basis Function Network (RBFN) hyperparameters.

For RBFN, key hyperparameters optimized included number_of_centers (or hidden_units), which dictates the complexity and approximation capability of the network; gamma (or width), controlling the spread of the radial basis functions; learning_rate for weight updates; and a regularization_parameter if explicit regularization was applied. The center_selection_method (e.g., k-means, random_subset) was also a critical choice influencing the initial setup of the hidden layer.

3.4.1 Role of metaheuristics in feature selection.

The fundamental preprocessing method in machine learning involves selecting the most critical subset of features that strongly impacts predictions through variable reduction of unhelpful features. Model generalization and efficiency calculations suffer for the water quality dataset because the attributes including pH, Solids, Chloramines, and Trihalomethanes exhibit high levels of dimensionality and inter-feature correlations.

Traditional approaches to feature selection through wrapper and filter methods (including correlation coefficients and mutual information and forward selection and backward elimination) struggle to identify the best feature subsets due to non-linear feature relationships. Analysis methods that employ simplistic statistical rules and approaches that are not practicable for big data sets present limitations. The embedded techniques LASSO and decision tree importance scores depend on specific learning algorithms for their operation, and their results do not transfer well between different algorithms.

A different option exists in metaheuristic algorithms, which demonstrate high flexibility and power throughout the process. These search methods gain inspiration from nature’s processes of evolution and swarming and thermodynamics, so they explore extensive domains through randomly guided yet purposeful steps. The Dipper Throated Optimizer (DTO) and the binary Grey Wolf Optimizer (bGWO) along with the binary Particle Swarm Optimization (bPSO) and hybrid variants effectively explore the feature subset combinatorial space through their optimized searching algorithms.

A metaheuristic framework uses candidate solutions that contain binary vectors that show whether each feature gets included (1) or left out (0). Two optimization goals are balanced through a fitness function that maximizes valid classifications and minimizes the choice of features for selection. The two-objective optimization format of the selection process guarantees both predictive effectiveness while improving comprehensibility and lowering model fitting errors.

This capability is beneficial for the water quality dataset researchers are currently examining. The selected features demonstrate statistical significance and environmental interpretability through metaheuristic search because they represent crucial chemical properties that affect plant health. The process reduces input parameters to lower dimensions, speeding up training time and streamlining deployment when applied to agricultural decision systems.

3.4.2 Role of metaheuristics in ML and DL optimization.

Metaheuristic algorithms play a pivotal role in enhancing the performance of machine learning (ML) and deep learning (DL) models, particularly when dealing with complex, high-dimensional, and nonlinear data typical of precision agriculture. Unlike traditional exhaustive or grid search methods, metaheuristics efficiently navigate large and multimodal search spaces to identify near-optimal solutions for both feature selection and optimization tasks.

In the context of this study, metaheuristics enable two critical functions. First, feature selection through binary-coded search strategies reduces input dimensionality by selecting the most informative and non-redundant variables. This reduction not only improves computational efficiency but also mitigates overfitting and enhances model interpretability. Second, metaheuristics optimize; such as network architecture, kernel widths, and regularization weights—which are crucial for balancing bias-variance trade-offs and achieving robust generalization in both ML and DL frameworks.

The proposed hybrid metaheuristic algorithm combining Dipper Throated Optimization (DTO) with Polar Rose Search (PRS) uniquely leverages the exploitation strengths of DTO and the systematic global exploration properties of PRS. This synergy facilitates more effective searches over the hyperparameter and feature subset spaces, thereby improving convergence speed and solution quality.

Moreover, metaheuristic-based optimization integrates seamlessly with the Radial Basis Function Network (RBFN) used in this study, enhancing its capacity to model complex nonlinear relationships inherent in irrigation water quality data. The metaheuristic framework’s adaptability to diverse learning architectures—from classical ML classifiers to advanced DL models like LSTM and RNN—highlights its broad applicability and potential as a key enabler in precision agriculture analytics.

Overall, the utilization of metaheuristics transforms the model development process from a manual, trial-and-error approach to a systematic, data-driven optimization paradigm, which is essential for building scalable, interpretable, and high-performing predictive systems in agricultural domains.

3.4.3 The proposed metaheuristic optimization algorithm: DTO + PRS.

The research design utilizes two sophisticated metaheuristic approaches to create a new DTO+PRS hybrid algorithm that combines the biological DTO method with the mathematical PRS approach. The combined optimizer incorporates DTO and PRS approaches that utilize their exploitation capabilities for DTO and the exploratory properties of polar rose curves to address feature selection and optimization problems. The exposition then explains theoretical frameworks and algorithmic cascades, which form the basis for both components and their subsequent integration as a unified search operation.

3.4.4 The DTO algorithm: Theoretical foundation and dynamics.

As a recent swarm-based algorithm, the Dipper Throated Optimization (DTO) derives concepts from the behavioral patterns of dipper birds (Cinclus spp.) during their water foraging and bowing movements in turbulent water environments [42]. DTO reproduces the behaviors of these birds through two primary mechanisms: swimming-based exploitation and flight-based exploration.

Mathematically, let the search space be d-dimensional, with a population of n candidate solutions (referred to as birds), each represented by a position matrix and a velocity matrix . The birds iteratively update their positions based on probabilistic decisions governed by a parameter . For R < 0.5, birds perform a dipper-inspired swimming update, modeled as:

Where C₁ and C₂ are dynamic control coefficients computed via:

with and t denoting the current iteration.

For , birds engage in flight-based updates, where their position is governed by classical velocity adjustment:

Where , and C₃, C₄, C₅ are fixed coefficients controlling inertia and acceleration toward both local and global optima.

DTO offers rapid convergence and efficient local search due to its dynamic modulation of position and velocity parameters. However, like many population-based methods, it may exhibit premature convergence in highly multimodal landscapes. This limitation motivates its hybridization with the PRS strategy to enhance global search diversity.

3.4.5 The PRS algorithm: Polar rose-inspired global search.

The Polar Rose Search (PRS) component is inspired by rose curves, a family of sinusoidal curves expressed in polar coordinates by the equation:

where r is the radial coordinate, is the polar angle, a is the amplitude (controlling the radius of the petals), and k is the angular frequency (determining the number and symmetry of petals). These curves create deterministic, symmetrical trajectories with high coverage of the angular domain, which is ideal for injecting structured diversity into population-based searches.

In PRS, candidate solutions are probabilistically perturbed along rose curves defined over polar projections of the search space. Each solution vector is first transformed to a 2D polar representation by selecting a principal plane of variation (e.g., ) and computing the corresponding polar coordinates . The perturbed position is then obtained by modifying the radius:

Where is a rotational offset introduced to maintain temporal diversity, the updated Cartesian coordinates are retrieved using:

The rose curves guarantee uniform angular distribution and controlled radial expansion/contraction, effectively balancing exploration and exploitation. The choice of k controls the complexity of the search pattern: odd values yield k petals, even values produce 2k petals, and irrational k leads to a non-repeating quasi-chaotic trajectory.

3.4.6 DTO + PRS: Integrated hybrid framework.

The hybrid DTO+PRS algorithm synergistically combines the focused convergence dynamics of DTO with the global symmetry-driven diversification of PRS. The integration is orchestrated through an adaptive control mechanism that switches between DTO’s local update schemes and PRS-based perturbations based on an annealing-like schedule or entropy-based diversity metrics.

Specifically, the hybrid operates in two alternating phases:

1. DTO Phase (Exploitation): Population updates follow the DTO’s swimming or flying equations, rapidly converging toward local optima. Feature selection is encoded as binary vectors, where continuous outputs are transformed using a sigmoid activation:
2. followed by thresholding at 0.5 to yield binary decisions.
3. PRS Phase (Exploration): A subset of elite solutions is perturbed along polar rose trajectories. The polar parameters (a,k) are varied adaptively across iterations to avoid pattern saturation. This phase injects new candidate regions into the search process, improving escape from local minima and enabling re-diversification when population entropy drops below a critical threshold.

Algorithmically, the hybrid optimizer is formally defined as follows in Algorithm 1:

Algorithm 1. Refined hybrid DTO+PRS algorithm.

Theoretical justification and computational rationale.

Theoretical grounds demonstrate that DTO and PRS work together because their search-exploration methods support one another. The population distribution under DTO moves toward promising regions, using adaptive velocity and social interaction models for guidance. During its operation, PRS creates geometric diversity structures that help prevent premature convergence and allow the search to discover new attraction basins. Luxurious symmetrical sampling patterns within the polar curve system offer organized exhaustive search capabilities that reduce uncertainty in basic stochastic optimization processes.

The hybrid system lets users control the convergence behavior through angular frequency k and amplitude parameters a that can undergo real-time tuning or evolution. The adjustable parameters give the optimizer the power to solve problems with different shapes and requirement specifications.

As shown in Table 3, all metaheuristic algorithms were configured using standardized population sizes and iteration limits to ensure a fair and consistent benchmarking environment. This uniformity guarantees that any observed performance differences stem from algorithmic capabilities rather than configuration biases. While the proposed hybrid metaheuristic-deep learning framework demonstrates superior performance, it is crucial to acknowledge its computational complexity and resource requirements. The training of deep learning models, especially when integrated with metaheuristic optimization for feature selection and hyperparameter tuning, can be computationally intensive. This demands significant processing power (CPU), memory (RAM), and often specialized hardware like Graphics Processing Units (GPUs) for efficient execution, as detailed in the Materials and Methods section. This computational demand can be a limiting factor in real-time applications or in resource-constrained agricultural environments where access to high-performance computing infrastructure may be limited for individual farmers. Future work will focus on optimizing the framework for deployment in edge computing devices or cloud-based solutions to mitigate these computational barriers and enhance practical applicability in diverse farming settings.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Initial parameter values for all algorithms and algorithm-specific settings.

https://doi.org/10.1371/journal.pone.0327230.t003

The convergence behavior of the classification models is shown in Fig 6, where DTOPRS-RBFN demonstrates a rapid and consistent drop in fitness over iterations, indicating superior convergence efficiency compared to DTO-RBFN, RBFN, LSTM, and RNN.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Classification convergence time showing fitness reduction over 100 iterations for DTOPRS-RBFN, DTO-RBFN, and benchmark models.

https://doi.org/10.1371/journal.pone.0327230.g006

For feature selection, Fig 7 displays the convergence behavior across all tested metaheuristics. The bDTOPRS algorithm reaches optimal fitness values significantly faster than other methods, confirming its efficiency in identifying minimal and high-quality feature subsets.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Feature selection convergence time comparison of various binary metaheuristic algorithms.

bDTOPRS achieves rapid and stable convergence.

https://doi.org/10.1371/journal.pone.0327230.g007

The DTO+PRS hybrid metaheuristic is a strong, mathematically sound, performance-driven optimization solution. The ability of DTO + PRS to combine feature selection with optimization throughout a unified system makes it ideal for identifying optimal complex prediction models with high dimensions like water quality assessment in irrigation systems. The subsequent sections demonstrate the method’s superior empirical performance, which outcomes from the techniques explained in this section.

3.4.7 SOTA benchmark optimization algorithms.

Evolutionary computation and nature-inspired search techniques have fully validated many metaheuristic algorithms for their competence in resolving complex optimization problems. This study employs six benchmark optimization algorithms as comparative baselines against the proposed hybrid approach (DTO + PRS). These algorithms are state-of-the-art (SOTA) in swarm intelligence and bio-inspired computation. This paper describes the operational behavior of multiple algorithms rather than showing their mathematical compositions because full computational details are omitted.

Grey Wolf Optimizer (GWO): Population-based optimization algorithm Grey Wolf Optimizer (GWO) operates through simulated natural behavior of grey wolves, including their hierarchical organization and hunting activities. The algorithm divides solutions into four hierarchical levels, from alpha down to beta and delta before omega, which dictates how the search direction will be managed. Position updates in the GWO follow wolf cooperative hunting methods by considering the influence of the three leading wolves in the pack [43]. GWO dynamically controls its exploration-exploitation balance, so it performs global searches initially until it changes to local refinement through progressive iterations. GWO finds widespread application because it offers basic implementation, stable convergence characteristics, and strong solution capabilities for continuous and discrete optimization.

Particle Swarm Optimization (PSO): The Particle Swarm Optimization (PSO) algorithm ranks among the first and most applicable swarm intelligence methods that came into existence due to observations of bird flocking and fish swarm behaviors. The population contains particles that function as potential solutions that modify their position and velocity through personal historical achievement and group-swarm-located excellence [44]. The independent forces of self and group components guide particles across the search space using exploratory and exploitative behavior. The prominent features of PSO include swift convergence speed with straightforward programming and overall flexibility; however, its performance deteriorates when dealing with highly multimodal functions.

Bat Algorithm (BA): The Bat Algorithm (BA) based its operational method on microbat echolocation techniques, where these flying mammals use sonar pulses to locate prey while navigating in darkness. When investigating the search environment, virtual bats in the population modify their locations, movement rates, emitted sound frequencies [45], and volume characteristics. Bats with echolocation adaptations utilize this system to manage the merger between exploration and exploitation activities based on their proximity to finding optimal solutions. The probabilistic frequency and loudness modification method fits BA exceptionally well when searching complicated non-convex search areas.

Whale Optimization Algorithm (WOA): The Whale Optimization Algorithm (WOA) uses inspiration from humpback whale bubble-net feeding behaviors to drive its operations. The bubble-net hunting method of whales generates trapping spirals that capture fish schools, and this approach uses logarithmic spiral mathematical models in the algorithm design. The significant components of WOA include prey-encircling behavior combined with spiraling movement through space and erratic prey-search movements [46]. WOA implements probabilistic mechanisms to balance its optimization techniques as it selects behaviors at random. This selection process produces a proper equilibrium between diversification and intensification. The adaptive movement capabilities of WOA demonstrate high performance in solving real-valued and combinatorial optimization problems.

Satin Bowerbird Optimizer (SBO): Through its mechanism, The Satin Bowerbird Optimizer (SBO) replicates the mating behaviors of satin bowerbirds that demonstrate bower construction coupled with intricate bower decoration to draw mates [47]. SBO features agents that model male birds that construct bower solutions before evaluation by visual appearance alongside novelty considerations. The optimization process combines exploiting elite solutions with exploration achieved through bower modifications. By modeling the competitive aspects of mating rituals, SBO enhances the convergence of its high-quality solutions when searching complex solution spaces.

Firefly Algorithm (FA): The Firefly Algorithm (FA) functions through a model derived from firefly communication using bioluminescence to create mating and prey attraction patterns. A key factor in FA’s operation includes using attractiveness based on objective function values (brightness) that result in fewer bright fireflies seeking out brighter ones [48]. The algorithm decreases in appeal as fireflies move farther from each other but maintain diverse searching capabilities at short and long distances. The simple rules of FA and its automatic organization capabilities allow it to solve rugged optimization problems effectively when dealing with high-dimensional features and multiple optimization scenarios.

The diverse analytic toolbox comprises six benchmark algorithms derived from different behavioral and biological foundations. A complete performance assessment of the proposed DTO+PRS hybrid method depends on its integration with these standard benchmark algorithms.

3.4.8 Algorithm’s parameters setup.

Establishing a standardized and well-defined parameter space ensures the fairness and reproducibility of comparative evaluations across different metaheuristics. The initialization parameters for each optimization algorithm used throughout this study are summarized in Table 3, including shared values and specific tuning constants as applicable. These parameters were selected based on established practices in the literature and calibrated to balance convergence speed, solution diversity, and computational efficiency. A detailed explanation of each parameter is provided below:

Common parameters: The population size refers to the number of individual candidate solutions (agents, particles, wolves, bats, etc.) that constitute the optimization algorithm’s population. Generally, a larger population increases exploration capabilities but also raises computational cost. The number of iterations defines the maximum number of generations or cycles the optimization algorithm will run, determining the total search effort. The number of runs indicates how many independent executions of each algorithm are performed; multiple runs are crucial for statistical analysis to accommodate the stochastic nature of metaheuristics and ensure result robustness. The search domain specifies the boundaries within which candidate solutions can exist; for binary feature selection, this is typically [0, 1], representing feature selection or non-selection.

Grey Wolf Optimizer (GWO) parameters: The control parameter a decreases linearly from 2 to 0 over the course of iterations. This parameter is vital for managing the balance between exploration and exploitation: a higher value promotes broad searching, while a lower value encourages local refinement of solutions.

Particle Swarm Optimization (PSO) parameters: The inertia weight W (defined by and ) controls the influence of a particle’s previous velocity on its current velocity, with higher values favoring exploration and lower values promoting exploitation. The acceleration constants C₁ and C₂ serve as cognitive and social coefficients: C₁ pulls particles toward their personal best positions, while C₂ draws them toward the global best position found by any particle, regulating the maximum step size toward these optima.

Bat Algorithm (BA) parameters: The frequency range defines the minimum and maximum echolocation frequencies used by bats, adjusted dynamically to control search behavior. Loudness A represents the intensity of emitted sounds, typically decreasing as a bat nears a promising solution to indicate exploitation; initial and minimum loudness values are denoted A₀ and , respectively. The pulse emission rate r governs how often bats emit pulses, generally increasing during convergence, signaling a local search shift. The constants a = c control the updating of loudness and pulse emission rates. A random walk component with a range [–1,1] supports local search by perturbing bat positions. The population size specifically for BA is given by the number of bats n.

Whale Optimization Algorithm (WOA) parameters: The spiral shape parameter b defines the logarithmic spiral’s tightness during the exploitation phase, influencing how whales encircle their prey.

Salp Swarm Algorithm (SBO) parameters: Parameters r₂, r₃, and r₄ are random numbers drawn from a uniform distribution [0,1]. These stochastic factors model the salps’ movement, affecting exploration and exploitation behaviors.

Firefly Algorithm (FA) parameters: The wormhole existence probability is a parameter that potentially enables global exploration or escape from local optima by allowing long-distance jumps via “wormholes." The step size regulates the magnitude of movement toward brighter fireflies, where larger values favor exploration and smaller values facilitate exploitation.

Dipper Throated Optimization (DTO) parameters: The exponential control parameter c typically decreases from a higher value (e.g., 2) to a lower value (e.g., 0) over iterations, balancing exploration and exploitation phases similarly to GWO’s parameter a. Parameters C₁ and C₂ guide dipper-throated birds’ movement toward the best solution, derived from c and a random variable , affecting step size and direction. Constants C₃, C₄, and C₅ model specific behavioral aspects during different search phases (such as diving or swimming), though their exact roles depend on the DTO formulation. Random factors r₁, r₂, and R, drawn uniformly from [0,1], introduce stochasticity, enhancing exploration and preventing premature convergence.

3.5.1 Machine learning prediction metrics.

As shown in Table 4, Multiple well-recognized metrics help evaluate the effectiveness of the ML models when used for irrigation water suitability classification. These assessment tools give a detailed picture of model discrimination power, the distribution of errors, and the capability to work with imbalanced classes. The set of metrics used for evaluation includes Accuracy combined with Sensitivity (also known as True Positive Rate, TPR) with Specificity (True Negative Rate, TNR), Positive Predictive Value (PPV) and Negative Predictive Value (NPV) along with the F-score.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Machine learning prediction metrics and their equations.

https://doi.org/10.1371/journal.pone.0327230.t004

Accuracy indicates the total model accuracy, while Sensitivity specifies correct predictions of positive cases, and Specificity measures correct negative classifications. PPV, along with NPV, strengthens the analysis of predictive class reliability. The F-score aligns precision and recall through an optimal balance between the two values. These metrics provide an all-encompassing assessment of model output behavior across data frequency patterns and prediction errors.

In the above equations, TP, TN, FP, and FN refer to the number of true positives, true negatives, false positives, and false negatives, respectively. These values are derived from the confusion matrix generated during model validation.

3.5.2 Feature selection metrics.

The efficient operation of the ML pipeline requires feature selection as a fundamental step, particularly in analyzing high-dimensional environmental datasets, including water quality measurements. Feature selection algorithm assessments require both an evaluation of predictive model accuracy and a measurement of the optimization procedure.

As shown in Table 5, Average Selection Size and the set of metrics, including Average Error, serve as evaluation elements for this specific purpose. These metrics help measure classification error, model compactness, and evaluate fitness quality using Average, Best, Worst, and Standard Deviation of Fitness. The multiple metrics measure the consistency and robustness alongside the efficiency performance of the metaheuristic algorithm across various runs.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Feature selection metrics and their definitions.

https://doi.org/10.1371/journal.pone.0327230.t005

The table includes M independent optimization runs while f_i stands for the fitness value of the i-th run. The utilized fitness function combines classification error measurement with feature count evaluation through appropriate weight distribution.

Such integrated approaches for evaluating final model performance and selecting the best features provide rigorous transparency for measuring evaluation success.