# Fuzzy C-Means Clustering Algorithm with Multiple Fuzzification Coefficients
## 1 Introduction  
Clustering is a machine learning technique that involves the grouping of data points into different clusters, where data points in the same cluster have a higher degree of similarity and any two data points in two different clusters have a lower degree of similarity. This technique improves the effectiveness of data mining, especially in big-data problems, as the data points are grouped into clusters with distinctive characteristics [1]. There are many different clustering techniques, such as centroid- based, hierarchical, density-based, distribution-based and grid-based clustering. In practical problems, some data elements may be missing information or containing uncertain information. The fuzzy set theory provides an appropriate method of representing and processing these types of data elements, along with the concept of membership function defined in the range [0,1]. In this concept, each element can belong to more than one cluster. The fuzzy C-means (FCM) algorithm [2], a method of fuzzy clustering, is an efficient algorithm for extracting rules and mining data from a dataset in which the fuzzy properties are highly common [3]. Many new methods based on the FCM algorithm were introduced, in order to overcome the limitations and improve the clustering ability of this algorithm in different cases. The performance of these new methods is summarized in [4]. The new and improved FCM-based methods extend or modify the distance metric betweenelements and cluster center, use different fuzzy measures for the membership of an element to a cluster or modify the exponential parameter for fuzzifying [3, 5]. An improvement direction is given by [PERSON] et al. in [6], which is the interval type-2 fuzzy C-means clustering (FCMT2I) algorithm. The study gives a fuzzy extension of the fuzzification coefficient \(m\), thereby applying the type-2 fuzzy set into the membership matrix \(U\). The parameter \(m\) in the FCMT2I algorithm can be any value in the range \([m_{L},m_{U}]\). A few articles [7, 8, 9, 10] discuss this approach further; after every iteration, an element belonging to the clusters is represented by an interval type-2 fuzzy set and further calculations are needed for type-reduction when calculating the cluster centers for the next steps. According to the type-reduction algorithm [6, 10], the cluster centers are calculated separately with the exponential parameters \(m_{L}\), \(m_{U}\), and then combined through the defuzzification step. In essence, after the type-reduction algorithm is executed, the resulting cluster center value is the same as the value calculated using an exponential parameter \(m\) in the range \([m_{L},m_{U}]\). The exponential parameter \(m\) is used to adjust the membership level of the elements in the clusters and its value can vary with clusters and steps. Thus, the FCMT2I algorithm allows the use of different exponential parameters. This study presents an approach that utilizes multiple different fuzzification coefficients as opposed to only one coefficient in the FCM algorithm. While in other past papers, the determination of fuzzification coefficients corresponds to the type-reduction calculations for the cluster centers, in this study, the fuzzification coefficients are determined for each element prior to calculations, depending on the characteristics of such element in the entire dataset. In a dataset, the density of each element, represented by its distance from the \"neighboring\" elements, is a useful measurement [6, 11, 12]. This study uses that measurement to determine the exponential parameter for each element during the clustering process. The main contribution of this study is the proposition of the use and manipulation of multiple fuzzification coefficients to enhance the performance of the original FCM algorithm. The experimental evaluation of the efficiency of the proposed algorithm is conducted using several common datasets from the University of California, Irvine (UCI) machine learning repository.