An Improved Expeditious Meta-Heuristic Clustering Method for Classifying Student Psychological Issues with Homogeneous Characteristics

Abstract

Nowadays, cluster analyses are widely used in mental health research to categorize student stress levels. However, conventional clustering methods experience challenges with large datasets and complex issues, such as converging to local optima and sensitivity to initial random states. To address these limitations, this research work introduces an Improved Grey Wolf Clustering Algorithm (iGWCA). This improved approach aims to adjust the convergence rate and mitigate the risk of being trapped in local optima. The iGWCA algorithm provides a balanced technique for exploration and exploitation phases, alongside a local search mechanism around the optimal solution. To assess its efficiency, the proposed algorithm is verified on two different datasets. The dataset-I comprises 1100 individuals obtained from the Kaggle database, while dataset-II is based on 824 individuals obtained from the Mendeley database. The results demonstrate the competence of iGWCA in classifying student stress levels. The algorithm outperforms other methods in terms of lower intra-cluster distances, obtaining a reduction rate of 1.48% compared to Grey Wolf Optimization (GWO), 8.69% compared to Mayfly Optimization (MOA), 8.45% compared to the Firefly Algorithm (FFO), 2.45% Particle Swarm Optimization (PSO), 3.65%, Hybrid Sine Cosine with Cuckoo search (HSCCS), 8.20%, Hybrid Firefly and Genetic Algorithm (FAGA) and 8.68% Gravitational Search Algorithm (GSA). This demonstrates the effectiveness of the proposed algorithm in minimizing intra-cluster distances, making it a better choice for student stress classification. This research contributes to the advancement of understanding and managing student well-being within academic communities by providing a robust tool for stress level classification.

1. Introduction

Nowadays, student mental health is a significant problem in higher education institutes that requires intense attention and contribution [1]. Mental health disorders are common among students, with different challenges such as anxiety, depression, and stress significantly impacting academic performance [2,3]. Psychological factors like increasing workloads, disruptions in eating and sleeping patterns, poor time management, as well as changes in social and academic environments contribute to increased mental health problems among students [4,5]. According to a survey of 126,000 students, it is found that 20.3% of students experience depression and abnormal psychological problems [6]. In 2023, students faced 76% psychological problems, with 36% suffering anxiety, 28% depression, and 99% facing academic challenges [7]. Students faced considerable variations in mental health, influencing their perspectives on daily life and academic success [8]. Avoiding mental health can further lead to decreased academic performance, increased suspension rates, and social isolation, as well as contributing to long-term consequences on individuals’ lives. Efforts to address mental health problems can be significantly enhanced through the collection and analysis of data from university mental health databases. Early and precise detection of data is important for effective decision-making and optimizing psychological therapy services in higher education institutes [9,10].

Student mental health includes various emotional, cognitive, and behavioral factors affecting academic performance and well-being. These factors involve emotional distress, behavioral changes, cognitive difficulties, physical symptoms, social challenges, and academic effects. However, accurately classifying student mental health problems is challenging because these problems are subjective, overlap across different disorders, and some of them are homogeneous. Moreover, the homogeneity of certain mental health issues further increases the classification complexity [11,12,13]. Efforts to improve classification consider these complexities and adopt a comprehensive approach that effectively addresses them.

Recently, machine learning and computational techniques have been analyzed for classifying mental stress and the evaluation of anxiety and depression among students [14,15]. The deep learning methods are analyzed, particularly for analyzing expression demonstrations of mental illness and integrating diverse data sources for developing more precise evaluation methods for mental health conditions. Moreover, these methods are based on mental health [16] monitoring schemes customized for students, utilizing convolutional neural networks to accurately classify students’ mental health problems and address various mental health indicators, such as sleeping disorders, depression, suicidal ideation, personality development, and self-esteem. Different computational techniques, including the AdaBoost algorithm and the Mamdani fuzzy rule-based method, are being explored to determine students’ mental health issues. The aim is to identify an effective algorithm for classifying students and understand various factors that influence mental health among students [17,18].

Machine learning and computational techniques have made significant developments in various fields, including healthcare and psychology. However, when it comes to categorizing the mental health of students, they face certain challenges. These challenges arise due to the homogeneity of symptoms and the complexity of disorders [19,20,21]. Additionally, iteration in specific datasets [22], limited data availability, and accuracy issues in assessing psychological problems contribute to these challenges [23,24,25]. By addressing these challenges in the existing methods, the proposed research work aims to classify the mental health of students.

In response to these challenges in existing research work, researchers have introduced metaheuristic clustering algorithms (MCA) in recent years [26,27]. Traditional MCAs often struggle with a large amount of data and the complexity of the problems, encouraging the adoption of meta-framework algorithms. Nature-inspired algorithms are now commonly employed across different fields to address these issues effectively [28,29,30]. Clustering techniques, along with other data mining methods, have advanced significantly by using collective computational algorithms. The use of metaheuristic algorithms for clustering is gaining attraction as a choice to traditional methods, signifying significant progress in data analysis techniques [31,32]. However different MCAs have been suggested for addressing clustering issues [33] but their effectiveness remains challenging in accurately classifying student psychological problems. Therefore, there is still a need for further advancements in metaheuristic algorithms to effectively address clustering problems.

However, due to some challenges in the existing methods, this research developed a classification system for student psychological problems and proposed an iGWCA clustering algorithm. The iGWCA is a derivative of the original GWO algorithm, which is inspired by the social behavior and hunting strategies of grey wolves.

The original GWO can rapidly achieve better solutions due to its hierarchical structure. However, the GWO algorithm struggles with effectively classifying mental health problems characterized by homogeneous symptoms due to certain limitations. For example, GWO may ignore small fluctuations in data, resulting in misclassification, and GWO may not completely explore the solution space, due to prematurely converging to local optima, limiting its ability to find accurate classification solutions when symptoms are similar across different disorders. Due to these challenges, a modification in the GWO using parameter “a” is adjusted dynamically, which means its value changes during the optimization process. Dynamic adjustment of “a” is important because it allows the algorithm to adapt more effectively to different stages of optimization. This modification is important as it improved the algorithm’s adaptability, enabling it to make a better balance between exploration and exploitation. By dynamically adjusting “a” the iGWCA can potentially improve convergence speed, solution quality, and robustness across student classification problems. The main contribution of this proposed work is as follows.

• Improvement of GWO. This research analyzes an iGWCA to classify student psychological problems. The improvement focuses on two key issues such as slow convergence and the tendency to local optima.
• Performance Analysis. The effectiveness of the iGWCA algorithm has been comprehensively evaluated and confirmed to have better performance than other competing algorithms, including GWO, MOA [34], FFO [35], PSO [36], HSCCS [37], FAGA [38], and GSA [39,40].
• Validation of iGWCA. Statistical analysis and comparison of obtained results from iGWCA with various established algorithms consistently demonstrate superior results in both accuracy and efficiency.

2. Related Work

In recent years, there has been a growing interest in utilizing various methods to assess students’ mental health. Data mining techniques are also explored to predict psychological issues in students. This aims to improve psychological management systems through early warning systems and continuous monitoring of at-risk students and increase problem alert efficiency [41]. To analyze burnout measures in students, highlighting the direct correlation between emotional tiredness and the onset of illnesses among participants is explored using partial least squares, and structural equation modeling is presented in [42]. The author classifies identical mental health classification in adolescents using latent profile analysis [43], however, it is bound to limited data size.

The authors proposed a neuro-fuzzy-based guidance and counseling system customized for college students, aiming to address the lack of guidance and counseling services in career path prediction [44]. An aspect-oriented convolutional neural network is introduced in [45], along with long short-term memory equipped with an attention mechanism, for academic emotion classification and recognition. These approaches offer valuable insights into students’ emotional experiences within academic environments. In [46], a clustering algorithm for intuitionistic fuzzy graphs is analyzed, focusing on edge density and practical applications, and introducing algorithms addressing clustering and minimum spanning tree problems, emphasizing parameterization for optimal data processing. There are different data sources, including social media activity, smartphone sensor data, electronic health records, and self-reported surveys, to gain insights into students’ psychological well-being. By applying machine learning algorithms to analyze these diverse data streams, researchers and mental health professionals aim to identify patterns, predict mental health results, and provide timely interventions to support students’ mental health [47,48,49]. The machine learning methods offer different advantages in evaluating the mental health of students, including early detection, objectivity, personalization, scalability, and resource efficiency, however, they have several challenges such as quality, interpretability, slow performance with a large number of datasets [50], limitations, such as their need for a large amount of data [51], reproducing model solution [52], and stochastic graph problems [53]. Additionally, the heuristic fuzzy c-means clustering algorithm struggles with limited dataset sizes [54]. Addressing these challenges involves careful data collection, interdisciplinary collaboration, robust evaluation techniques, and validation to maximize the benefits and minimize the risks of using machine learning in student mental health assessment.

Cluster analysis, an unsupervised data mining technique, is gaining attention for categorizing student psychological levels. With proper focus, these methods can offer valuable insights for guiding policy and implementation [55]. Clustering techniques are categorized into hierarchical clustering and partitional clustering. In hierarchical clustering, data members are initially assigned to individual clusters and then merged gradually, as one cluster. Partitional clustering, which is the focus of this paper, aims to divide the dataset into non-overlapping clusters without a nested structure [56,57,58]. Each cluster in this method is represented by its centroid, and initially, each student data object is assigned to the nearest centroid. The centroids are then updated based on the current problem and by optimizing criteria [59]. Traditional partitional clustering algorithms, such as the k-means algorithm are generally used for their speed and simplicity, aiming to minimize the average squared distance [60]. However, k-means’ efficiency heavily depends on the initial selection of cluster centers and is inclined to converge toward local optima. To address these challenges, the k-means++ algorithm is introduced to enhance the initial centroid selection [61]. Despite this improvement, k-means++ still faces difficulties in converging to local optima [62].

Therefore, in recent years, MCAs have gained attention for solving NP-hard problems like data clustering, aiming to minimize convergence issues and reduce the risk of being trapped in local optima. The literature review shows that there are multiple challenges associated with existing techniques to achieving better student classification, prediction accuracy, and student performance. Therefore, in this research work, iGWCA is proposed for effectively classifying student stress levels. There are the following main reasons for selecting iGWCA as a metaheuristic technique for clustering.

• iGWCA provides simplicity and ease of implementation.
• Only a few control parameters for tuning.
• The iGWCA offers better convergence speed and an improved balance between exploration and exploitation processes.
• It shows robustness and applicability in classifying psychological problems.

3. Problem Model Construction

Davies–Bouldin Index (DBI) [63] is a measure used in cluster analysis to evaluate the quality of clustering results. The objective function for optimizing DBI in the area of student clustering problems using iGWCA can be formulated as follows.

Assume that

• X is the set of the student data set;
• C is the set of clusters;
• d_ij is the distance between data point i and data point j;
• μ_k is the centroid of cluster k;
• σ_i is the average distance from the centroid of cluster k to the student data point within the cluster.

$$f\left(x\right)=\frac{1}{\left|C\right|}{\sum }_{k=1}^{\left|C\right|}{max}_{i\ne k}\left(\frac{{\sigma }_i+{\sigma }_k}{d_{\mu i\mu k}}\right)$$

(1)

where

• $\left|C\right|$ is the set of clusters;
• μ_i and μ_k are the centroids of clusters i and k, respectively;
• d_μiμk is the distance between μ_i and μ_k;
• σ_i and σ_k are the average distances from centroids μ_i and μ_k to the data points within clusters i and k, respectively.

The constraints for this problem are that each student data point is assigned to only one cluster and possibly setting boundaries on the minimum and maximum number of student data points assigned to each cluster. These constraints can be formulated mathematically as follows.

Each student data point must belong to exactly one cluster

$${\sum }_{k=1}^{\left|C\right|}{x}_{ij}=1 {\forall }_j\in X$$

(2)

Number of student data points allocated to each cluster

$${\sum }_{j=1}^{\left|X\right|}{x}_{ij}\ge \mathrm{minimum} \mathrm{cluster} \mathrm{size} {\forall }_j\in C$$

(3)

$${\sum }_{j=1}^{\left|X\right|}{x}_{ij}\ge \mathrm{maximum} \mathrm{cluster} \mathrm{size} {\forall }_j\in C$$

(4)

where

• X_ij is the binary variable indicating whether data point i belongs to cluster j;
• min_cluster_size and max_cluster size are the minimum and maximum allowed number of data points in a cluster, respectively.

The number of clusters k must be predefined

$$k=\mathrm{predefine} \mathrm{value}$$

(5)

The DBI should be minimized to achieve more homogeneous clusters

$$\mathrm{Minimized} f\left(x\right)$$

(6)

4. Proposed Method

4.1. Introduction to the Principle and Basic Process of GWO Algorithm

The GWO algorithm is initially suggested by Mirjalili in [64] and it is based on the social structure and hunting activities of grey wolves. GWO has gained recognition as a dependable and efficient optimization method in different applications. Its efficiency and ability to make a balance between exploration and exploitation have established it as a preferred approach for resolving complex optimization challenges [65]. The original GWO can rapidly achieve better solutions due to its structure. Furthermore, its low time complexity enables it to fulfill a significant role in addressing optimization problems.

Additionally, the innovative features and working principles of the GWO are explained as follows.

• Initially, the algorithm randomly places a pack of grey wolves in the search space, with each wolf representing a potential solution corresponding to a set of cluster centroids.
• Secondly, a predefined DBI objective function computes the current positions of the wolves (i.e., cluster centroids) and the student psychological problem data.
• Then, during each iteration, the algorithm evaluates the fitness of each wolf using the objective function and updates their positions accordingly, aiming to minimize the DBI and improve clustering quality.
• Finally, once the optimization reaches the maximum number of iterations, the final solution, consisting of optimized cluster centroids, is obtained. These clusters can be used to classify student psychological problems based on their similarities, with each student assigned to a specific cluster.

However, there are certain limitations, such as the GWO being susceptible to becoming trapped in local optima during the classification of psychological problems.

4.2. The Proposed iGWCA Algorithm

In this research, iGWCA is proposed, focusing on optimizing the balance between exploration and exploitation, these two processes are the fundamental aspects of optimization algorithms. Exploration involves extensively searching the solution space to discover better solutions and avoid local optima. As iterations increase, the position shifts towards exploitation, where solutions are developed to converge towards the optimal solution. To keep a balance between these processes is important for optimizing algorithm performance and motivating the exploration of new approaches. In the original GWO, the parameter “a” serves as a regulator for these elements, emphasizing exploration to entirely explore the solution space [26]. However, as iterations increase, “a” gradually transitions towards exploitation, concentrating on promising regions identified during the exploration phase. This transition is important for the algorithm to converge to an optimal solution. The proposed modification introduces variations in the linear behavior of parameter “a” throughout the optimization process. By adjusting “a” dynamically, the algorithm can effectively adapt its exploration and exploitation levels. This dynamic adjustment enables a more balanced approach, finally improving the convergence speed and solution quality.

$$\overrightarrow{a}=2-2{\left(\frac{t}{t_{max}}\right)}^k$$

(7)

where

• $t_{max}$ is the total number of iterations, t is the current iteration, k is a constant

• Step 1: Initialization of the student datasets
  • Initialize student stress datasets for dataset-I and dataset-II, then set the search agents and maximum number of iterations. The grey wolf at the location moves and updates its location according to the prey location and the algorithm defines the position of the best agent based on the positions of the grey wolf and prey by controlling parameters $\overrightarrow{A}$ and $\overrightarrow{C}$ using Equations (4) and (5). The value of the parameter $\overrightarrow{A}$ decreases during the algorithm’s simulation, and the fluctuation rate also decreases to maintain a balanced process.

• Step 2: DBI Objective Function
  • Next, initialize the population of grey wolves within the search space and assess the quality of each grey wolf’s solution using the DBI objective function f(x).

• Step 3: Hierarchy Formation
  • Grey wolves in the population are categorized into alpha (α), beta (β), and delta (δ) types of wolves. The α wolves are the best solutions, followed by β and δ wolves. Within the algorithm loop, continuously update the positions of the α, β, and δ wolves within the search space, ensuring all wolf positions remain feasible.

• Step 4: Update α, β, and δ Positions:
  • The algorithm selects the α, β, and δ wolves based on their performance. These wolves estimate the prey’s location and update the positions of wolves around the prey according to the following Equations.

$$\overrightarrow{D}=\left|\overrightarrow{C} {\overrightarrow{X}}_p-\overrightarrow{X_t}\right|$$

(8)

$${\overrightarrow{X}}_{(iter+1)}={\overrightarrow{X}}_p-\overrightarrow{A} \overrightarrow{D}$$

(9)

$$\overrightarrow{A}=2\overrightarrow{a}·\overrightarrow{r_1}-\overrightarrow{a}$$

(10)

$$\overrightarrow{C}=2\overrightarrow{r_2}$$

(11)

$$\left.\begin{array}{c}{\overrightarrow{D}}_{\alpha }=\left|\left({\overrightarrow{C}}_1-{\overrightarrow{X}}_{\alpha }\left(t\right)\right)\overrightarrow{X}\right|\\ {\overrightarrow{D}}_{\beta }=\left|\left({\overrightarrow{C}}_2-{\overrightarrow{X}}_{\beta }\left(t\right)\right)\overrightarrow{X}\right|\\ {\overrightarrow{D}}_{\delta }=\left|\left({\overrightarrow{C}}_3-{\overrightarrow{X}}_{\delta }\left(t\right)\right)\overrightarrow{X}\right|\end{array}\right\}$$

(12)

$$\left.\begin{array}{c}{\overrightarrow{X}}_1=\left|{\overrightarrow{X}}_{\alpha }\left(t\right)-\left({\overrightarrow{A}}_1{\overrightarrow{D}}_{\alpha }\right)\right|\\ {\overrightarrow{X}}_2=\left|{\overrightarrow{X}}_{\beta }\left(t\right)-\left({\overrightarrow{A}}_2{\overrightarrow{D}}_{\beta }\right)\right|\\ {\overrightarrow{X}}_3=\left|{\overrightarrow{X}}_{\delta }\left(t\right)-\left({\overrightarrow{A}}_3{\overrightarrow{D}}_{\delta }\right)\right|\end{array}\right\}$$

(13)

where

• T is the current iteration, $\overrightarrow{A}$ and $\overrightarrow{C}$ are the co-efficient, ${\overrightarrow{X}}_p$ is the position of prey, $\overrightarrow{X}$ is the position of a wolf.
• $D_{\alpha },{ D}_{\beta },{ D}_{\mathsf{\delta }}$ are the distances, ${\overrightarrow{r}}_1$, ${\overrightarrow{r}}_2$ are random values and a is the value from 2 to 0.

• Step 5: Update the Position of the prey

Update the positions of the prey using the following Equation (14).

$${\overrightarrow{X}}_{(iter+1)}=\frac{{\overrightarrow{X}}_1+{\overrightarrow{X}}_2+{\overrightarrow{X}}_3}{3}$$

(14)

In Equation (1), the parameter “a” is adjusted in a non-linear manner, gradually decreasing from 2 to 0. When “k” varied from 0 to 1, the algorithm focused more on exploitation, potentially leading to a compromise in search quality. However, for “k” values > 1, the algorithm thoroughly explores the search space before transitioning to exploitation. To improve the performance of GWO, it is important to experiment with various trials to determine the most suitable value for “k”.

To address the reduced number of iterations during exploitation phases, a mapping method is proposed to conduct local searches around the best solution. This method requires mapping the position of the best wolf to a new position and evaluating whether it results in improved fitness. If the fitness improves, the best wolf is moved to this new position using Equation (15)

$${\overrightarrow{X}}_n={\overrightarrow{X}}_{\alpha }\left(t\right)+r(U_b-L_b)\left(m-0.5\right)$$

(15)

where

• ${ U}_b$ and ${ L}_b$ are the upper and lower boundaries, r is the center, and m is the mapping parameter, which is updated at each iteration

$${\overrightarrow{m}}_{(t+1)}=4\times \overrightarrow{m}\times (1-{\overrightarrow{m}}_t)$$

(16)

• Step 6: Boundary Constraints
  • The DBI function evaluates student clustering quality based on cluster separation and constraints such as ensuring each student data point belongs to exactly one cluster and allocating the correct number of student data points to each cluster.

• Step 7: Update Fitness Values

Analyze the fitness of each wolf’s updated position using the DBI objective function.

• Step 8: Update Hierarchy of α, β, and δ wolves
  • Update the hierarchy of wolves according to their fitness values.

• Step 9: Termination Criteria
  • Update the convergence curve to monitor optimization progress until a stopping criterion is satisfied. When the stopping criterion is satisfied, return the best solution obtained from the optimization process otherwise, start the process.

The iGWCA iteratively searches the solution space, with the α, β, and δ wolves guiding the exploration and exploitation process. The flowchart and pseudo-code of the proposed algorithm are presented in Algorithm 1 and Figure 1, respectively.

Algorithm 1: Pseudo Code of the iGWCA. Algorithm 1. PSEUDO code of proposed method.
1	Input number students stress dataset
2	Initialize the population of grey wolves randomly within the search space using iGWCA
3	Initialize$\overrightarrow{a}$, $\overrightarrow{A}$ and $\overrightarrow{C}$
4	Evaluate the fitness value of each grey wolf solution using the Davies-Bouldin Index f(x)
5	While the stopping criterion is not met
6	for each search agent
7	Update the positions of $X_{\alpha },X_{\beta } \mathrm{a}\mathrm{n}\mathrm{d} X_{\mathsf{\delta }}$ wolves
8	end for
9	Update$\overrightarrow{a}$, $\overrightarrow{A}$ and $\overrightarrow{C}$
10	Evaluate the fitness value of each grey wolf Davies-Bouldin Index f(x)
11	Update the new position of $X_{\alpha },X_{\beta } \mathrm{a}\mathrm{n}\mathrm{d} X_{\mathsf{\delta }}$ wolves
12	t = t + 1
13	End while
14	Return $X_{\alpha }$ as the best obtained solution for iGWCA

4.3. Time Complexity Analysis of iGWCA Algorithm

In the iGWCA, if the number of population size is N, D is the dimension of the student dataset, and Max: Iter is the number of maximum iterations. The time complexity of the iGWCA is presented in

Table 1. Time complexity of iGWCA.

Phase	Generation of the Initial Phase	Control Parameters Calculation	Update the Search Agent Position	Evaluate the Fitness Value of Each Search Agent	iGWCA
Time Complexity	O(N  ×  D)	O(N  ×  D)	O(N  ×  D)	O(N  ×  D)	O(N  ×  D  × Max: Iter )

using big O notation. The algorithm’s time complexity plays a significant role in evaluating the abilities and limitations of the algorithm.

The actual time achieved on the computer for the best-case scenario is 8.162354 s, for the worst-case scenario is 20.817287 s, and for the average-case scenario is 15.979641 s.

5. Experimental Results

This section presents the results of the proposed iGWCA algorithm, with a detailed description provided below. The results are then discussed, analyzed, and compared to those of other algorithms.

5.1. Experimental Method Description

The proposed algorithm is implemented in MATLAB R2019a on a computer equipped with an Intel(R) Core(TM) i5-7200U CPU @ 2.50 GHz, 16 GB of memory, and running Windows 10 with a 64-bit operating system. The iGWCA algorithm performance is evaluated with competent algorithms such as GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. The best optimal solution for our datasets based on the average and standard deviation (SD) are determined. Each algorithm simulates 30 independent runs for dataset-I and dataset-II, with each consisting of 100 maximum number of iterations.

Table 2. Parameter setting.

Types	Parameter
iGWCA	Search agents = 30, m = 0.15, k = 2
GWO	Search agents = 30, r1, r2 = [0, 1]
MOA	Particle number = 40, α1 = 1, α2 = 1.5, β = 2, d = 0.1, fl = 0.1
FFO	population size = 20, a1, pa = 0.92
PSO	Particle number = 30; c1 = 1.5, c2 = 2; ω = 0.9
HSCCS	Alpha = 0.7, A and a = 2
FAGA	Population size = 30, mutation rate = 0.05
GSA	Alpha = 20, gravitational constants = 100

demonstrates the parameter configurations for all algorithms. In iGWCA the search agents are 30, however, in PSO, the acceleration coefficients c₁ and c₂ are set to 1.5 and 2, respectively. Meanwhile, the inertia weight (w) dynamically changes from 0.9 to 0.5 during the optimization process [66].

The statistical metrics, including mean (avg), standard deviation (SD), and efficiency, are mentioned in Equations (17) and (18). These metrics provide insights into the overall performance and variability of the proposed method. Moreover, to ensure robustness, each algorithm is run in more >30 trials, and the results are rounded to five decimal places to minimize statistical errors and ensure statistically significant outputs. The mean computes the average of the best-obtained values by the algorithm across a predetermined number of runs as expressed in Equation (17).

$$\mathrm{A}\mathrm{v}\mathrm{g}=\frac{1}{n}=\sum _{i=1}^{n}FF{V}_i$$

(17)

where

• n is the number of runs, and the FFV_i represents the best value obtained in the ith iteration.

The SD measure is explored to analyze whether the algorithm under evaluation consistently achieves the same best value across multiple trials, therefore it serves as an accuracy test for the algorithm results. It is expressed as follows.

$$SD=\sqrt{\frac{1}{\left(n-1\right)}\sum _{i=1}^n{\left(FF{V}_i-Avg\right)}^2}$$

(18)

$$\mathrm{E}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}=\frac{FF{V}_{min}}{FF{V}_i}\times 100\%$$

(19)

where

• FFV_min is the minimum value of the fitness function.

5.2. Discussion on Statistical, Convergence Performance

Nowadays, statistical analysis has gained more attention as an important tool for evaluating the performance of computational methods. These analyses are commonly employed to compare the efficiency of different algorithms in the proposed work as mentioned in

Table 3. Statistical analysis of optimization techniques.

Statistical Analysis	iGWCA [Proposed]	GWO [Studied]	MOA [Studied]	FFO [Studied]	PSO [Studied]	HSCCS [Studied]	FAGA [Studied]	GSA [Studied]
Best Value	0.52461	0.53248	0.57456	0.57301	0.53781	0.54452	0.57150	0.57452
Worst Value	0.58452	0.58231	0.65322	0.64799	0.60992	0.58443	0.67141	0.66444
Avg	0.5453	0.5497	0.6017	0.5989	0.5627	0.5844	0.6060	0.6055
SD	0.0290	0.0241	0.0380	0.0363	0.0349	0.0193	0.0483	0.0435
Number of Hits	31	32	28	32	30	33	30	33
Computational Time (s)	8.162354	12.454219	18.791092	26.140449	29.559220	15.777735	14.877140	11.791202
% Decrease		1.48%	8.69%	8.45%	2.45%	3.65%	8.20%	8.68%

using dataset-I. In this research, the iGWCA demonstrated better accuracy compared to GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. Moreover, the iGWCA algorithm’s ability to avoid becoming stuck in local optima, unlike traditional methods, is beneficial because it integrates social behaviors and hunting strategies observed in grey wolves.

5.2.1. Efficiency Experiments

The effectiveness of the iGWCA method is further validated through statistical analysis, confirming it shows better accuracy. This comparison is important for determining whether a proposed method offers a significant improvement over existing methods for a given problem. Researchers often prefer this analysis due to their simplicity and efficiency in computational processing. The statistical analysis includes the best, worst, and average value, SD, number of hits, and computational time, analyzing dataset-I with parameters as mentioned in Table 3.

Table 3 compares the performance of various optimization algorithms. The average best values obtained for iGWCA, GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA are 0.52461, 0.53248, 0.57456, 0.57301, 0.53781, 0.54452, 0.57150, and 0.57452, respectively, with iGWCA demonstrating the best performance. Considering computational time, iGWCA achieves the lowest computational time at 8.162354 s, followed by GWO at 12.4542199 s, MOA at 18.791092 s, FFO at 26.140449 s, PSO at 29.559220 s, HSCCS at 15.777735 s, FAGA at 14.877140 s and GSA at11.791202 s. Additionally, FFO shows lower computational time than PSO, while GSA ranks second in this analysis. However, iGWCA has the lowest computational time among competent algorithms, emphasizing its efficiency. On average, the proposed algorithm shows a 5.175% reduction in optimal value compared to other techniques. These results indicate that the iGWCA algorithm effectively obtains the best optimal solution, making it a better choice for student stress classification tasks due to its superior performance. Therefore, it is a better choice than other methods.

The iGWCA convergence is marked by a better choice, aiming to make a balance between exploration and exploitation while improving solutions toward the global optimum solution obtained using dataset-I. It consistently exhibits smooth convergence curves and faster rates compared to GWO, particularly for optimization problems. However, GWO may show slower convergence rates as compared to iGWCA as mentioned in Figure 2. In WOA, convergence involves iteratively improving solutions through exploration and exploitation phases, resembling whale hunting behavior. PSO convergence curve typically shows a gradual decrease in the objective function with fluctuations due to particle movement. HSCCS combines elements of the sine cosine and cuckoo search algorithms to enhance optimization, measured by reducing the objective function. In FAGA, solutions are refined through a combination of evolutionary operators from the Firefly and Genetic Algorithms. GSA attracts solutions towards better ones, mimicking gravitational attraction.

In addition to this, convergence in deep learning models is assessed by monitoring loss function decrease over epochs, while data mining technology convergence depends on specific algorithm strategies. AdaBoost convergence is followed by observing classification accuracy improvements, and the gradient boosting machine algorithm minimizes a differentiable loss function similar to AdaBoost.

Table 4. Performance analysis of the proposed method.

S.No	Method	Performance (%)
1	Machine Learning Algorithms	0.8571 [15]
2	Data Mining Technology	0.90 [67]
3	AdaBoost Algorithm	0.8175 [18]
4	Aspect-Oriented Convolutional Neural Network (A-CNN)	0.89 [45]
5	Machine Learning Technology (CatBoost)	0.826 [14]
6	Gradient Boosting Machine Algorithm (Dataset-I, Class H)	0.78 [68]
7	iGWCA	0.96

presents a comparative analysis of the efficiency of various algorithms. The machine learning algorithm outperforms the AdaBoost algorithm with a performance ratio of 0.8571. Furthermore, the CatBoost model achieves a performance ratio of 0.826, exceeding the gradient boosting machine algorithm. However, the data mining method’s performance ratio is slightly lower compared to the proposed method. These results demonstrate the efficiency and capability of the proposed method for further applications.

5.2.2. Exploring Data Credibility

The iGWCA shows multiple advantages over GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA, such as smooth convergence, lowest optimal value, a balance between exploration and exploitation as mentioned in Figure 3, ease of implementation, and a smaller number of parameters. These attributes make iGWCA a better choice for addressing the student clustering problem analyzing dataset-I. The proposed method is further analyzed as presented in

Table 5. Confusion matrix for different optimization techniques.

Types	iGWCA [Proposed]	GWO [Studied]	MOA [Studied]	FFO [Studied]	PSO [Studied]	HSCCS [Studied]	FAGA [Studied]	GSA [Studied]
TPR	0.9666	0.9333	0.8	0.8333	0.9	0.866	0.8	0.7666
TNR	0.35	0.5	0.65	0.6	0.4	0.13	0.2333	0.2666
FPR	0.75	0.6	0.2	0.5	0.6	0.2	0.06	0.8
FNR	0.20687	0.214286	0.24	0.215156	0.2413	0.2650	0.213	0.2234

using common evaluation metrics such as False Negative Ratio (FNR), False Positive Ratio (FPR), True Negative Ratio (TNR), and True Positive Ratio (TPR) conducted [69,70].

Table 5 demonstrates the confusion matrix of GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA, indicating that iGWCA has the best values for TPR, TNR, FPR, and FNR, with 0.9666, 0.35, 0.75, and 0.20687, respectively. To prove the ability of iGWCA in the classification of psychological problems, it is compared with different competent algorithms. The obtained result shows that iGWCA achieved better performance than others. In addition to this, it is observed that the results of GWO and PSO show better performance than MOA, FFO, HSCCS, FAGA, and GSA.

Figure 4 shows the dispersion of student data points across different trials, serving as a measure of consistency compared to other methods. Analyzing convergence patterns and box plots allows for a detailed analysis of the proposed method’s performance in student psychological problems. The highest and lowest values are represented by cross symbols positioned at the top and bottom of the box, respectively. The rectangular box shows the interquartile range, including half of the data (50%). This visualization effectively shows the distribution of optimal values achieved across multiple runs, providing insights into strategies to reduce the risk of local optima. Furthermore, the probability of attaining the minimum fitness function value is particularly higher, given that the median for the proposed method is closer to the lower quartile.

Figure 3 compares the box plots of the proposed method with GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. The minimum and maximum data points are presented as points at the ends of the whiskers extending from the box. Box plots show the distribution of quantitative data, facilitating comparisons among fitness functions across iGWCA and other algorithms. The averages of iGWCA outperform GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. Specifically, PSO outperforms FFO, HSCS, FAGA, and GSA, while GWO achieves a lower average value than GWO. These box plots indicate better chances of obtaining the minimum fitness function value, as the median from iGWCA is closer to the lower quartile.

5.3. Comparative Analysis of Classification Results Based on Dataset-I

5.3.1. Dataset Description

The study involved analyzing student stress data from dataset-I, which consisted of 1100 students obtained from the Kaggle database, each characterized by three attributes. In this work, these attributes are selected as academic performance as test case 1, future career concerns as test case 2, and headaches as test case 3. Below is an explanation of these attributes.

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Test Case 1: Academic Performance Stress

Academic performance stress among students is a universal problem characterized by psychological and emotional stress arising from various academic pressures. These stressors include the pressure to succeed, fear of failure, heavy workload, time pressure, performance expectations, competition, perfectionism, uncertainty about the future, and challenges with time management and study skills. Such stress can lead to physical symptoms like headaches and fatigue, as well as mental health concerns such as anxiety and depression. Identifying and addressing academic stress is significant for promoting student well-being and academic success. Time management techniques, seeking support from teachers or counselors, practicing self-care, and developing coping skills can help students manage academic stress more effectively.

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Test Case 2: Future Career Concern Stress

Future career concern stress among students during their studies is a significant issue characterized by anxiety and fear about their professional prospects after graduation. This stress arises from various factors, including uncertainty about job opportunities, fear of career expectations, pressure to secure employment in a competitive market, and concerns about academic choices with future career goals. Additionally, students may experience stress related to the perceived importance of obtaining internships, networking, and building relevant skills for their desired career paths. Future career concern stress can be determined by physical symptoms such as tension and fatigue, as well as psychological symptoms like worry and self-doubt. Addressing these stress problems is important for supporting students’ well-being and career development. Strategies such as career counseling, networking opportunities, skill-building workshops, and fostering resilience can help students navigate future career concerns and stress more effectively, enabling them to make informed decisions about their professional trajectories.

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Test Case 3: Headache Stress

Headache stress among students during their studies is a significant issue characterized by physical discomfort and tension in the head region. This stress is often triggered by academic pressures and demands. Factors contributing to headache stress include excessive studying, continuous screen time, insufficient rest, dehydration, and high levels of tension related to academic performance. Headache stress can significantly impact students’ ability to concentrate, focus, and retain information, thus affecting both their academic performance and overall well-being. Addressing headache stress involves implementing strategies such as taking regular breaks, staying hydrated, practicing relaxation techniques, managing time effectively, and seeking medical advice if headaches exist. By effectively managing headache stress, students can alleviate discomfort and optimize their study environment to improve their academic success. The iGWCA classifies the student data into two clusters, and statistical analysis is conducted using SPSS to evaluate the results obtained from the clustering process.

5.3.2. Classification Result and Analysis

Figure 4 shows the classification of student psychological problems on the dataset-I with three attributes, the number of clusters k = 2 is selected in this study. The concept of similarity is measured via Euclidean distance between student data points and centroids. Each small circle represents a student data point in cluster 1 and cluster 2, and they are connected with each data point. However, the limitation is set as each student data point must belong to exactly one cluster, as explained in Section 3. Then, the boundaries do not cross each other, and the data points become properly grouped.

According to Figure 4a, and in view of the obtained results, it is observed that the iGWCA classifies the student data points having similar characteristics into clusters 1 and 2 with the lowest average value. The graphical representation obtained by iGWCA facilitates effective exploration of the results for cluster 1 and cluster 2. The first one is a cluster with 824 population distribution where the clusters should be well separated. The second one is a 276 population distribution with separate clusters. According to Figure 4b, which shows the results obtained from GWO, the student dataset is grouped into cluster 1 and cluster 2 (n = 709, 391). Figure 4c shows the results obtained from MOA, showing a similar clustering pattern with cluster 1 containing n = 730 data points and cluster 2 containing n = 370 data points. Additionally, FFO resulted in a dataset of points of n = 748 and 352 for clusters 1 and 2 clusters, as mentioned in Figure 4d, respectively. According to Figure 4e, PSO obtained n = 746 and 354 dataset sizes for both clusters. FAGA obtained (n = 730, 370) sample size according to Figure 4f. According to Figure 4g,h HSCCS and GSA are obtained (n = 711, 389) for both clusters. However, iGWCA consistently demonstrates superior efficiency in classifying more accurate and compact student clusters across different cluster configurations. Moreover, from the graphical representation of the cluster, it is confirmed that iGWCA consistently exhibits the best performance and maintains a balanced distribution among all student data points inside the cluster.

The performance of iGWCA utilized in this study is presented in

Table 6. The result of clustering for dataset-I (N = 1100, cluster size = 2).

Algorithms	Test Case	Cluster	Size	Avg	Median	Rank	SD
iGWCA [Proposed]	Test Case 1	1	824	5.89	5.0	1	3.90
GWO [Studied]	Test Case 1	1	709	7.70	8.0	2	4.36
MOA [Studied]	Test Case 1	1	730	7.91	8.0	3	4.47
FFO [Studied]	Test Case 1	1	748	8.11	9.0	7	4.59
PSO [Studied]	Test Case 1	1	746	8.08	9.0	6	4.57
HSCCS [Studied]	Test Case 1	1	711	7.72	8.0	4	4.37
FAGA [Studied]	Test Case 1	1	730	7.91	8.0	5	4.47
GSA [Studied]	Test Case 1	1	711	7.72	8.0	4	4.37
iGWCA [Proposed]	Test Case 2	1	824	1.89	1.0	1	1.26
GWO [Studied]	Test Case 2	1	709	1.86	2.0	2	1.03
MOA [Studied]	Test Case 2	1	730	1.94	8.0	3	4.47
FFO [Studied]	Test Case 2	1	748	1.97	2.0	4	1.15
PSO [Studied]	Test Case 2	1	746	1.98	2.0	5	1.15
HSCCS [Studied]	Test Case 2	1	711	1.86	2.0	2	1.03
FAGA [Studied]	Test Case 2	1	730	1.94	2.0	3	1.11
GSA [Studied]	Test Case 2	1	711	1.87	2.0	3	1.04
iGWCA [Proposed]	Test Case 3	1	824	1.84	1.0	1	1.14
GWO [Studied]	Test Case 3	1	709	1.86	2.0	2	1.04
MOA [Studied]	Test Case 3	1	730	1.93	2.0	3	1.11
FFO [Studied]	Test Case 3	1	748	1.96	2.0	5	1.15
PSO [Studied]	Test Case 3	1	746	1.97	2.0	6	1.14
HSCCS [Studied]	Test Case 3	1	711	1.86	2.0	2	1.04
FAGA [Studied]	Test Case 3	1	730	1.93	2.0	4	1.11
GSA [Studied]	Test Case 3	1	711	1.87	2.0	3	1.05
iGWCA [Proposed]	Test Case 1	2	276	20.60	21.0	1	5.04
GWO [Studied]	Test Case 1	2	391	21.36	21.0	2	3.62
MOA [Studied]	Test Case 1	2	370	21.72	22.0	3	3.38
FFO [Studied]	Test Case 1	2	352	22.01	22.0	5	3.21
PSO [Studied]	Test Case 1	2	354	21.98	22.0	4	3.22
HSCCS [Studied]	Test Case 1	2	389	21.39	21.0	2	3.60
FAGA [Studied]	Test Case 1	2	370	21.72	22.0	3	3.38
GSA [Studied]	Test Case 1	2	389	21.39	21.0	2	3.60
iGWCA [Proposed]	Test Case 2	2	276	3.39	4.0	1	1.70
GWO [Studied]	Test Case 2	2	391	4.08	4.0	4	1.22
MOA [Studied]	Test Case 2	2	370	4.06	4.0	2	1.24
FFO [Studied]	Test Case 2	2	352	4.09	4.0	5	1.20
PSO [Studied]	Test Case 2	2	354	4.07	4.0	3	1.23
HSCCS [Studied]	Test Case 2	2	389	4.08	4	4	1.22
FAGA [Studied]	Test Case 2	2	370	4.06	4.0	2	1.24
GSA [Studied]	Test Case 2	2	389	4.08	4	4	1.22
iGWCA [Proposed]	Test Case 3	2	276	3.65	4.0	1	1.28
GWO [Studied]	Test Case 3	2	391	3.68	4.0	3	1.21
MOA [Studied]	Test Case 3	2	370	3.65	4.0	2	1.22
FFO [Studied]	Test Case 3	2	352	3.67	4.0	4	1.19
PSO [Studied]	Test Case 3	2	354	3.65	4.0	2	1.22
HSCCS [Studied]	Test Case 3	2	389	3.69	4.0	5	1.21
FAGA [Studied]	Test Case 3	2	370	3.65	4.0	2	1.22
GSA [Studied]	Test Case 3	2	389	3.68	4.0	4	1.22

, demonstrating its superiority over other methods employed. The clustering results are evaluated using six statistical indicators, including cluster size, average, median, rank, and SD, with mean rank serving as the primary metric for assessing performance. These results are tabulated in Table 6, which illustrates the distribution of the dataset from the simulation experiment on students’ psychological problems. The obtained result shows that iGWCA consistently obtained the lowest average value across test cases 1 and 2, indicating its effectiveness, particularly in test case 3, where iGWCA outperforms the competing algorithms, securing the top rank among them. Finally, to further validate the obtained results, comparisons are made with other competent algorithms. The iGWCA demonstrates not only lower computational time but also higher accuracy compared to others. These findings show the robustness of the iGWCA algorithm in the classification of psychological problems based on attributes, indicating its superiority over established techniques. The comprehensive evaluation, along with other algorithms, underscores the credibility of iGWCA as a better choice for student psychological problems. In Table 6 there is a variability in SD observed, and it can be attributed to several factors. Firstly, it is important to note that each algorithm undergoes >30 trials. During these trials, significant fluctuations in both minimum and maximum values are observed. Then, the SD values are obtained from these trials. This variability further shows the importance of robust statistical analysis to monitor the range of obtained results.

According to Table 6, the average values obtained using iGWCA, GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA for test case 1, considering cluster 1, are 5.89%, 7.70%, 7.91%, 8.11%, and 8.08%, respectively. For cluster 2, the average value for iGWCA is 20.60%, GWO is 21.36%, MOA is 21.72%, FFO is 22.01%, PSO is 21.98%, HSCCS, GSA is 7.72%, and FAGA is 7.91%. In this case, HSCCS outperforms PSO, however, iGWCA demonstrates the best performance overall. Therefore, it is suggested that the proposed method shows better performance compared to the other methods. In test case 2, considering cluster 1, the iGWCA achieves an average value of 1.89%, GWO 1.86%, MOA 1.94%, FFO 1.97%, PSO 1.98%, HSCCS 1.86%, FAGA is 1.94% and GSA is 1.87%. Particularly, iGWCA has better performance than PSO and FFO by achieving the lowest average value, indicating its efficacy in generating more precise and concise clusters. For cluster 2, the average values obtained for iGWCA are 3.50%, GWO is 3.39%, MOA is 4.06%, FFO is 4.09%, PSO is 4.07%, HSCCS, GSA is 4.08%, and FAGA is 4.06%.

In test case 3, considering cluster 1, the average values obtained are 1.84% for iGWCA, 1.86% for GWO, 1.93% for MOA, 1.96% for FFO, 1.97% for PSO, 1.86% for HSCCS, 1.93% for FAGA, and GSA is 1.87%. From these results, iGWCA, GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA show better performance. However, iGWCA consistently demonstrates superior efficiency in classifying more accurate and compact student clusters across different test cases and cluster configurations. Although iGWCA and GWO demonstrate comparable performance, PSO lags slightly in achieving lower average values, indicating its lesser effectiveness in this specific context. For cluster 2 in test case 3, the average values obtained are 3.31% for iGWCA, 3.68% for GWO, 3.65% for MOA, 3.67% for FFO, and 3.65% for PSO, however, HSCCS is 3.69%, FAGA is 3.65%, and GSA is 3.68%. The iGWCA significantly outperforms both FFO and MOA, demonstrating its superior ability to address the dataset-I more effectively. While the differences between FFO and PSO are relatively minor in this case, the proposed method maintains a slight edge by achieving a marginally lower average value.

Based on graphical representations of these two clusters, as mentioned in Figure 4, using iGWCA and other algorithms can help in the classification and monitoring of psychological problems more accurately. The graphical representation makes it easier to explore the results of cluster 1 and cluster 2, such as how these students in dataset-I are grouped and how they relate to one another within each cluster. Researchers can assess the consistency and separation of clusters visually, checking whether the two clusters are well-defined and different from one another. This can ensure the chosen number of clusters and the overall quality of the clustering solution. According to the obtained result mentioned in Table 6, iGWCA demonstrates the lowest average value between two clusters across different test cases, compared to GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. The analysis confirms iGWCA’s better performance in classifying student psychological characteristics, as mentioned in the DBI function.

5.4. Comparative Analysis of Classification Results Based on Dataset-II

5.4.1. Dataset Description

To explore the efficiency of the proposed algorithm, it is further verified using dataset-II comprising 824 individuals, including students, housewives, professionals, and others obtained from [71]. In this research work, the number of clusters k = 2, and two attributes are selected. Test Case 4 represents growing stress, and Test Case 5, represents social weakness. The original values in this dataset are “No/Maybe/Yes”, but for this work, they are mapped to 10/15/20, respectively. The following is the explanation of these attributes.

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Test Case 5: Growing Stress

Growing stress is an escalating level of psychological pressure experienced by individuals over time. It refers to the increasing burden of stressors that every individual perceives in their lives, which may stem from various sources such as work, school, relationships, financial concerns, or other life events. Growing stress can manifest in physical, emotional, and behavioral symptoms, including but not limited to fatigue, irritability, difficulty concentrating, changes in sleep patterns, and heightened anxiety. It is a significant aspect of mental health research and is often studied to understand its impact on individuals’ well-being and to develop strategies for stress management and prevention.

(2)

Test Case 6: Social Weakness

Social weakness refers to a condition where individuals experience difficulties in their social interactions, relationships, or support networks. It encompasses various challenges such as feelings of isolation, loneliness, inadequate social skills, a lack of social support, or difficulties in forming and maintaining meaningful connections with others. Social weakness can have negative impacts on individuals’ mental and emotional well-being, contributing to stress, depression, anxiety, and a sense of disconnection from society.

5.4.2. Classification Result and Analysis

Table 7. The result of clustering for dataset-II (N = 824, cluster size = 2).

Algorithms	Test Case	Cluster	Size	Avg	Median	Rank	SD
iGWCA [Proposed]	Test Case 4	1	229	11.0	15	1	4.62
GWO [Studied]	Test Case 4	1	572	16.9	20	4	3.67
MOA [Studied]	Test Case 4	1	195	16.1	15	3	4.25
FFO [Studied]	Test Case 4	1	129	18.6	20	6	2.25
PSO [Studied]	Test Case 4	1	692	15.9	15	2	3.99
HSCCS [Studied]	Test Case 4	1	131	18.6	20	6	2.24
FAGA [Studied]	Test Case 4	1	135	18.4	20	5	2.65
GSA [Studied]	Test Case 4	1	704	15.9	15	2	3.99
iGWCA [Proposed]	Test Case 5	1	229	16.0	15	3	2.06
GWO [Studied]	Test Case 5	1	572	13.4	15	1	2.41
MOA [Studied]	Test Case 5	1	195	17.9	20	7	2.91
FFO [Studied]	Test Case 5	1	129	17.8	20	6	2.79
PSO [Studied]	Test Case 5	1	692	14.3	15	3	3.98
HSCCS [Studied]	Test Case 5	1	131	17.7	20	5	4.02
FAGA [Studied]	Test Case 5	1	135	17.6	20	4	2.86
GSA [Studied]	Test Case 5	1	704	14.3	15	2	3.97
iGWCA [Proposed]	Test Case 4	2	595	11.5	15	1	5.44
GWO [Studied]	Test Case 4	2	252	11.6	10	2	2.35
MOA [Studied]	Test Case 4	2	629	15.0	15	6	4.03
FFO [Studied]	Test Case 4	2	695	14.7	15	4	4.07
PSO [Studied]	Test Case 4	2	132	11.8	10	5	2.78
HSCCS [Studied]	Test Case 4	2	693	14.6	15	3	4.07
FAGA [Studied]	Test Case 4	2	689	14.7	15	4	4.07
GSA [Studied]	Test Case 4	2	120	11.5	10	1	2.28
iGWCA [Proposed]	Test Case 5	2	595	13.7	15	1	2.56
GWO [Studied]	Test Case 5	2	252	18.2	20	5	2.41
MOA [Studied]	Test Case 5	2	629	14.0	15	2	3.88
FFO [Studied]	Test Case 5	2	695	14.4	15	3	4.01
PSO [Studied]	Test Case 5	2	132	17.8	20	4	2.91
HSCCS [Studied]	Test Case 5	2	693	14.4	15	3	4.02
FAGA [Studied]	Test Case 5	2	689	14.4	15	3	4.02
GSA [Studied]	Test Case 5	2	120	18.2	20	5	2.67

shows the comparative analysis of iGWCA, GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. In test case 4, for cluster 1, the minimum values achieved are iGWCA 11.0%, GWO 16.9%, MOA 16.1%, FFO 18.6%, PSO 15.9%, HSCCS 18.6%, FAGA 18.4%, and GSA 15.9%. However, for cluster 2, the proposed method achieves the lowest average value of 11.5%. Therefore, it is suggested that the proposed method exhibits superior performance compared to the other methods. Similarly, in test case 5, for cluster 1, the average values obtained are iGWCA 16.0%, GWO 13.4%, MOA 17.9%, FFO 17.8%, PSO 14.3%, HSCCS 17.7%, FAGA 17.6%, and GSA 14.3%. Particularly, GWO outperforms others by achieving the lowest average value, however, iGWCA’s average value is slightly high in this case. For cluster 2, the average values are iGWCA 13.7%, GWO 18.2%, MOA 14.0%, FFO 14.4%, PSO 17.8%, HSCCS and GSA 14.4%, and FAGA 18.2%. The iGWCA demonstrates efficacy in generating more precise and concise clusters, while MOA and FFO yield better results than GWO and GSA.

The comparative analysis of iGWCA, GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA, based on the clustering experiments shows that iGWCA consistently demonstrates better efficiency in classifying more accurate and compact student clusters across test case 4, and test case 5 with different clusters configuration.

The combination of customized improvements in iGWCA and the utilization of the DBI function enables it to outperform GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA in classifying dataset-I and dataset-II student psychological problems. By optimizing cluster quality according to the DBI objective function, iGWCA ensures more accurate and meaningful classifications, making it a better choice for psychological classification problems in higher education institutes.

The iGWCA combines improvements to the original GWO algorithm with the help of dynamic parameter “a” making it more attractive at addressing the classification of student psychological problems. These improvements include improved exploration and exploitation and better handling of constraints (each student data point is assigned exclusively to one cluster, with the number of student data points allocated to each cluster specified) customized to the classification of dataset-I and dataset-II. The DBI objective function is well-suited for evaluating the quality of clusters formed during optimization. It measures both the inter-cluster separation and intra-cluster integration, providing a comprehensive assessment of clustering quality. By optimizing this objective function, iGWCA ensures the formation of clusters that are both different from each other and internally unified, leading to more meaningful and accurate classifications of student psychological problems. The iGWCA likely shows better convergence properties and exploration–exploitation balance compared to GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA. Through its improved optimization mechanisms, iGWCA can efficiently navigate the solution space, effectively balancing exploration to discover different cluster configurations and exploitation to refine better solutions. These improvements enable iGWCA to allow it to achieve more optimal solutions with improved convergence speed and greater consistency compared to other methods.

6. Conclusions

In this proposed work, the classifications of student stress levels are analyzed using iGWCA. The classification of student stress levels using the proposed method presents a capable approach to understanding and addressing the complexities of student psychological dataset-I and dataset-II. The experimental analysis demonstrates that the iGWCA achieves the lowest average values for test case 1 at 5.89% and 20.60% for test case 2. For test case 3, the iGWCA method obtained 1.62% and 3.5% for cluster 1 and cluster 2, respectively, compared to other methods. However, in test case 4 and test case 5, iGWCA achieved 11.0% and 11.5%, 16.0%, and 13.7% average values, respectively. The comparison between iGWCA, GWO, MOA, FFO, PSO, HSCCS, FAGA, and GSA indicates the significance of selecting an appropriate optimization technique for effectively classifying student stress levels. However, iGWCA shows better performance in this research work, offering more accurate and efficient classification results. The importance of classifying stress levels among students cannot be ignored. Understanding student stress levels is important for universities, administrators, and policymakers to provide effective help and support. By accurately classifying stress levels, educational institutions can modify resources and programs to meet students’ specific needs, ultimately fostering a healthier and more conducive learning environment. Moreover, the classification of stress levels enables early detection of potential issues and facilitates measures to minimize the adverse effects of stress on academic performance, mental health, and overall well-being. The classification of student stress levels using optimization techniques like iGWCA not only offers valuable insights into student well-being but also covers the way for evidence-based interventions that promote resilience and academic achievement. The experimental result shows that iGWCA achieves the lowest best and average values, at 0.52461 and 0.54532, respectively.

The iGWCA for classifying student psychological issues with homogeneous characteristics presents few limitations. Initially, student psychological issues are complex, sometimes influenced by various factors such as genetics, environment, and individual experiences. iGWCA, like many optimization algorithms, may struggle to capture the full complexity of these issues and adequately classify them based on limited input data. Secondly, the effectiveness of iGWCA relies heavily on the quality and availability of data. If the dataset used for classification is incomplete or contains errors, it may lead to inaccurate classifications of the results. Then the performance of the algorithm could be impacted by parameter configurations, necessitating careful tuning, which might present difficulties for users lacking extensive optimization skills. Addressing these limitations and improving the algorithm’s scalability, ease of implementation, and flexibility are important areas for future research and development, particularly for extensive applications in classification systems. This study establishes a robust foundation for future advancements in the classification of psychological problems. The success of the iGWCA method suggests following a few promising directions for further exploration.

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Evaluation of other optimization algorithms, such as whale optimization or moth flame optimization, for classifying student psychological issues is a promising area for future research. By refining the algorithm parameters and exploring adaptive mechanisms, its efficiency will further improve in accurately classifying different psychological issues.

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Additionally, integrating multimodal data sources, such as physiological measurements, behavioral patterns, and academic performance metrics, could further enhance the algorithm classification accuracy.

(3)

Furthermore, investigating the customization of the algorithm to address the unique challenges modeled by student psychological data, such as variability in symptom expression and the dynamic nature of mental health conditions, is significant.

Overall, the effectiveness of the iGWCA approach acts as an attractive tool for future research efforts aimed at further improving the accuracy and applicability of psychological problems.