Unbalanced data is one of the fundamental challenges in clustering algorithms. In these datasets, the data distribution is such that the number of samples belonging to one or more classes (minority class) is significantly lower compared to other classes (majority class) [9], [10]. This imbalance causes clustering objective functions, which are adjusted based on the general density of the data, to bias towards densely populated clusters and ignore clusters belonging to minority classes [10]. Mathematically, the degree of imbalance in a dataset can be defined by the Imbalance Ratio (IR):

Nmajority

IR = , (1)

Nminority where ∣ Nmajority| and ∣ Nminority| represent the number of samples of the majority class and the minority class, respectively. When the IR value is significantly larger than one, traditional algorithms such as FCM are unable to detect the structure of minority clusters due to their focus on reducing the mean error, and thus classify them as noise or part of the majority classes [11], [12].

Clustering is a fundamental part of unsupervised pattern recognition. Its goal is to group unlabeled samples into several categories so that the data within each category is most similar and the data in different categories is distinct from each other [1]. The similarity criterion varies depending on the application. But usually the distance, density, and distribution pattern of the data play a major role. The FCM algorithm is the most common fuzzy method in which each sample belongs to a cluster with a certain membership degree. Simplicity and understandability are the strengths of this method [13]. However, this method is sensitive to the initialization of cluster centers and, in many cases, tends to a local optimal response. Especially when the problem dimensions are large, deterministic search methods such as the gradient method usually suffer from these weaknesses. For this reason, the clustering problem is often considered as a multivariate optimization problem [1].

The FCM algorithm performs data clustering by minimizing the following objective function:

J(U, V) = ∑. ∑ u ⋅ ‖xk − vi‖ , (2)

ik i=1 k=1

j is the main objective function to be minimized. c is the number of clusters, and n is the number of data samples. U is the membership degree of the k-th sample in the ith cluster, and m is the fuzzification parameter, which usually has a value greater than 1. v is the center of the ith cluster. x represents the feature vector of the k-th sample (input dataset). ‖xk − vi‖ represents the Euclidean distance between xk and the cluster center vi. Then, the cluster centers are updated at each iteration using the following equation:

∑ uik ⋅ xk

vi = . (3)

∑ uik

The FCM algorithm is one of the most widely used fuzzy clustering methods due to its simplicity and efficiency. However, in unbalanced and noisy datasets, its performance suffers from reduced accuracy. In this algorithm, using constant parameters causes the clustering process to be unable to adapt to variations in cluster density and unbalanced data distribution. For this purpose, the AFCM has been proposed with the aim of increasing the flexibility of the clustering process. In AFCM, algorithm parameters are adjusted adaptively to the data characteristics during algorithm execution. As a result, samples belonging to minority classes and low-density areas play a more effective role in determining the centers of clusters. This feature increases the accuracy in determining cluster centers, reduces the impact of outliers, and prevents the majority classes from dominating the clustering process. Finally, AFCM provides more accurate and robust clustering than FCM on unbalanced datasets [14]. To deal with the challenge of non-uniform data distribution, the AFCM algorithm modifies the standard FCM objective function as follows:

c n C

J = ∑. ∑ u ⋅ ‖x − v ‖ + λ ∑ φ (v ). (4)

AFCM ik k i i i=1 k=1 i=1

Uik is the membership degree of the k-th sample in the i-th cluster, and m is the fuzzification parameter. vi is the center of the i-th cluster and xk represents the feature vector of the k-th sample (input dataset). φ(v is a penalty function that is adjusted based on the local density of the data and allows the algorithm to increase the weighting of minority classes using the adaptive parameter λ. The cluster centers are updated at each iteration using the following equation:

∑ uik ⋅ xk

vi = . (5)

∑ uik This mathematical structure causes the centers of clusters to also tend towards less dense areas (minority classes) rather than just focusing on densely populated clusters.

Population-based metaheuristic algorithms are inspired by natural processes and have become very popular in solving complex problems due to their simplicity of implementation and independence from gradient information [15]. These methods usually have a good ability to escape from local optima. The WOA is a metaheuristic algorithm inspired by the hunting behavior of humpback whales. These whales use a unique method called "bubble net feeding." The hunting process consists of two main phases. During the exploitation phase, whales surround the prey and move towards it in a spiral motion. In the exploration phase, they seek new paths to increase the likelihood of finding better answers. What increases the efficiency of the WOA algorithm is its flexible way of switching between broad search and precise search modes. This displacement is set by a control parameter (vector A). In the mathematical model of the algorithm, when the absolute value of this vector is greater than one (|A| ≥1), the algorithm agents move away from the reference point and perform an extensive search in space [16]. On the other hand, if its value is less than one, the movements are focused towards the reference point, and an accurate search is formed. This flexible behavior helps WOA algorithm avoid premature halting at undesirable solutions in complex, multidimensional spaces, such as many clustering problems [5], [16].

In the proposed method, each "whale" or search agent represents a complete set of cluster centers. Assuming the dataset has K clusters and D features, each whale is encoded as a continuous vector, as shown in Eq. (5):

X = [c{11}, c{12}, … , c{KD}], (6)

where X is the position vector of a whale, cij represents the coordinates of the center of the ith cluster in the jth dimension (feature), K is the total number of clusters, and D represents the dimensions of the data.

In the proposed algorithm, each whale provides a possible solution (a set of cluster centers). To measure the quality of these solutions, a criterion called a "fitness function" is needed. Here, the fitness function is exactly the same as the cost function of the AFCM algorithm, denoted by the symbol jm. This function calculates the sum of weighted distances of all data to the proposed centers. Therefore, the goal of the whale algorithm is to minimize the value of this function, which is defined in Eq. (6), by moving the centers:

Jm = ∑ ∑ u{ } ∥ Xi − Cj ∥ . (7)

ij i=1 j=1

Uij is the membership degree of the ith sample in the jth cluster, and m is the fuzzification parameter, which usually has a value greater than 1. xi represents the feature vector of the ith sample (input dataset). C is the number of clusters and N is the number of data samples . cj is the center of the j-th cluster. ‖xi − Cj‖ represents the Euclidean distance between xi and the cluster center Cj. The lower the value of Jm, the more compact and accurate the clustering is.

At each iteration, whales adjust their position based on the current best solution (x ). If the whale is in the "prey encirclement" or "spiral movement" phase (i.e., when the probability value p ≥ 0.5), its new position is calculated based on Eq. (7):

bl ∗ (8)

X(t + 1) = D. e . cos(2πl) + x (t), where D represents the distance of the whale from the current best position, b is a logarithmic constant that determines the shape of the spiral of the movement, and l is a random number in [1-1]. If the whale is in the “search” phase (p < 0.5) and wants to explore a new region of the problem space (a state where ∣A∣≥1), the new position is calculated using Eq. (9):

X(t + 1) = xrand−A. |C . X{rand} − X(t)|. (9)

In the above equation, x_rand represents the position of a whale that is randomly selected from the population. Using this mechanism helps prevent the search path from becoming monotonous and the algorithm from getting stuck in local optima that are very common in unbalanced data.

In order to measure the efficiency of the proposed method and compare it with other algorithms, three standard metrics, including accuracy, Normalized Mutual Information (NMI), and Adjusted Rand Index (ARI), have been used.

Accuracy: The simplest metric for evaluating clustering performance and represents the ratio of the number of correctly labeled samples to the total samples:

∑i=1 ai

Accuracy = × 100, (10)

where k is the number of classes, n is the total number of data points, and ai is the number of samples correctly identified in class i.

NMI: This metric is used to measure the amount of shared information between two partitions (real clustering and algorithm clustering) and is independent of the number of clusters:

2 × I(X; Y)

NMI(X, Y) = , (11)

H(X) + (H)Y where I(X; Y) is the mutual information between the actual labels X and predicted Y, and H(X) and H(Y) are the related entropies. An NMI value close to 1 indicates a perfect match.

ARI index: This index eliminates the chance of clustering being random and measures the similarity between two clustering methods.

ai bj

(∑i( )) (∑j( ))

nij 2 2

∑ij( ) −

ARI = , (12)

ai bj

(∑i( )) (∑j( )) 1 a b 2 2 [∑i( ) + ∑j( )] − 2 2 2 n

where n is the total number of samples, nij is the number of samples in common between the actual class i and the predicted cluster j, and ai and bj are the sum of samples of class i and cluster j, respectively. An ARI value close to 1 indicates perfect matching and desired clustering.

In order to evaluate the quality of clustering, the following evaluation criteria were used: Accuracy, NMI, and ARI. Also, to evaluate the performance of algorithms fairly and accurately on minority classes in an unbalanced dataset, metrics extracted from the confusion matrix, such as the F1-Score and BA metrics, have been used.

Table 1 shows the results of applying different algorithms to the studied dataset. As can be seen from Table 1, the proposed WOA-AFCM method outperforms other methods in all criteria. In the Iris dataset, the FCM algorithm has provided an accuracy of 85.71% due to being stuck in local optima. The Reptile Search Algorithm (RSA)-FCM method, by improving the initial search, was able to reach 89.33% accuracy, but it was still unable to completely separate overlapping data.

On the other hand, the proposed method using the whale spiral search mechanism has identified the best primary centers and achieved an accuracy of 97.14% and an ARI index equal to 0.9320. These values indicate the high matching of the formed clusters with the real classes.

In the Thyroid dataset, the performance of the algorithms has been evaluated based on two criteria, NMI and ARI; two valid indices that evaluate the degree of matching of the resulting clustering structure with the actual labels of the data, independent of the mere matching of the samples. In the Thyroid dataset, the classic FCM algorithm is unable to effectively separate minority samples, and as a result, it has obtained lower values of NMI and ARI indices. This performance degradation is due to the significant imbalance between the classes, as well as the dependency of the FCM algorithm on the cluster structures with relatively uniform distribution and sensitivity to the dominance of the majority classes. Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithm had the weakest performance with 86.51% accuracy. This issue is because this algorithm identifies data from lower-density classes (overactive/underactive patients) as noise and excludes them from the clustering process, leading to a drastic decrease in NMI and ARI metrics. In the proposed WOA-AFCM method, by using the adaptive weighting mechanism and optimizing the cluster centers with the WOA algorithm, it has been able to make the clustering structure more consistent with the real data distribution. This issue has led to a significant increase in the values of both NMI and ARI criteria. In particular, a higher value of ARI indicates a greater matching of the generated clusters with the real structure of the classes, after removing the effect of random matching. The results indicate a higher ability of the proposed method in identifying minority samples and maintaining the real boundaries between classes. These results show that WOA-AFCM is not only more stable in the face of unbalanced data, but also able to represent the inherent structure of the data more accurately and improve the clustering quality more significantly than the classic methods. By overcoming the imbalance problem, the proposed method has been able to increase the accuracy to 92.94%.

In order to visually evaluate the performance of the algorithm, the initial data distribution of the Iris dataset and the clustering results obtained from the proposed method are presented in Fig. 3. Fig. 3.a shows the initial distribution of data samples before clustering, while Fig. 3.b shows the final output of the WOA-AFCM algorithm after the clustering process. Comparing these two figures indicates that the proposed method was able to identify the data structure well and create clusters with appropriate resolution and desirable compliance with the actual data distribution. In the Thyroid dataset, the majority class tends to attract the cluster centers, which leads to the minority classes (patients) being ignored. Table 2 shows the performance of the proposed algorithm focusing on minority classes. As can be seen from Table 2, the performance of the algorithm with severe imbalance in the Thyroid dataset has been examined separately. The FCM algorithm fails to detect minority classes (patients) due to the high prevalence of the majority class (normal). Therefore, FCM records zero values for the evaluation of minority classes due to placing all samples in one cluster. In contrast, the results show that the proposed WOA-AFCM method has successfully overcome the attraction of the majority class and extracted the hidden structure of the minority classes. In minority class number 3, the algorithm has shown outstanding performance. An F1-score of 0.7273 and a BA of 0.8303 are evidence of the high ability of the hybrid algorithm to accurately locate cluster centers and distinguish the boundaries of the class, despite the limited number of samples. In minority class number 2, the model's behavior is extremely conservative. A prediction accuracy (precision) of 1.00 indicates that the model had no FP errors in this class and that all samples assigned to this cluster belonged to class 2 with complete certainty. Although the low recall rate in this class indicates a strong structural overlap between class 2 and majority class samples, the overall average shows that the whale-global-search (WOA) mechanism in fuzzy center optimization is an effective solution to prevent the phenomenon of "minority class swallowing" in complex medical data. In order to evaluate the performance of the algorithms in facing the imbalance challenge, accurately, a comparison of the evaluation criteria of the basic FCM algorithm and the proposed WOA-AFCM method on the Thyroid dataset is presented in Fig. 4. As can be seen from Fig. 4, the FCM algorithm has poor results in the F1-Score and BA metrics, which provide a more accurate assessment of the model's performance on unbalanced data. These undesirable results are due to the algorithm's sensitivity to class imbalance and the influence of the dominance of the majority classes. The result indicates that FCM faces limitations in correctly identifying minority samples and maintaining a balance of performance across classes. In contrast, the proposed WOA-AFCM method has been able to significantly increase the F1-Score and BA values while maintaining a high overall accuracy value. The successful performance of the proposed WOA-AFCM method is based on the utilization of cluster center optimization through the WOA and the adaptive clustering mechanism. The improvement in evaluation parameters indicates increased accuracy in identifying minority samples, reduced bias towards majority classes, and better balance in classification performance. Overall, the results show that the proposed method has a greater ability to handle unbalanced data compared to FCM and provides more stable and reliable performance in representing structure and identifying minority classes. Using the WOA to determine the initial cluster centers has increased the algorithm's ability to extract the true structure of the data, reduce the effect of majority class dominance, and create more balanced clustering. As a result, the increase in the overall performance of the proposed method is not only due to the improvement in general accuracy, but also reflects the improvement in the quality of clustering and proper separation of all classes, especially

In order to examine the behavior of the algorithms in reaching the optimal solution, the convergence graph for the three implemented algorithms (FCM, RSA-FCM, WOA-AFCM) is plotted in Fig. 5. The above graph shows that the FCM algorithm (blue line) stalls in the early iterations and is trapped in a local optimum with a high cost value. The RSA-FCM algorithm (green line) has better performance, but its convergence slope is slower than the proposed method. In contrast, the WOA-AFCM algorithm (red line) not only reduced the cost function value faster, but also managed to converge to the lowest final value. The result shown in the figure proves that the WOA has a high ability to escape local traps and guide cluster centers towards global