# Fuzzy C-Means Clustering Algorithm with Multiple Fuzzification Coefficients
## 2 Preliminaries
### Interval Type-2 Fuzzy C-Means (FCMT21) Clustering  
FCM is a breakthrough algorithm compared to other well-known clustering algorithms because FCM has a more general representation and more accurate clustering results in many cases. However, the FCM algorithm has some disadvantages:  
* The algorithm is sensitive to noise and foreign elements;
* Clustering is not accurate and valid for elements located at the boundary between clusters;
* There is no specific criterion to select the value for the parameter \(m\), which is often selected after testing multiple times.  
With these disadvantages, the FCM algorithm requires more research and development to be further improved.  
A research [6] on the extension or modification of the distance metrics \(D(i,k)\) uses different fuzzy metrics for \(U(i,k)\) or discusses the exponential parameter \(m\). The clustering decisions made in FCM is based on Equation (2), which presents the fuzzy membership of element \(X_{i}\) that is assigned by the relative distance between \(X_{i}\) and \(V_{k}\). This formula has a characteristic that is, the smaller \(m\), the higher the absolute membership \(U\), as shown in Figure 1 by the case of \(C=2\).  
For the elements on the cluster boundary, the absolute degree of \(U\) must be lower than the elements near the center of the cluster. The reason is that we cannot make a clustering decision with elements at the immediate boundary, so we need the \(U\) graph to be more flexible in terms of cluster selection for these elements. Meanwhile, the elements near the center of the cluster should have a steeper \(U\) graph to ensure that the clustering control is always in place in the nearest cluster. Since FCM only uses a single \(m\) value, the absolute level of \(U\) is the same for the elements near the center of the cluster and the elements at the boundary, which is not reasonable. [PERSON] et al. (2016) studied the elements on the boundary of two clusters. It was found that when choosing a large \(m\) for the degree of \(U\) to cluster center \(V_{1}\) and choosing a small \(m\) for the degree of \(U\) to cluster center \(V_{2}\), the cluster boundary would be distributed with a wide left region and narrow right region, as shown in Figure 2.  
As seen from the above analysis, different exponential parameters \(m\) can be determined depending on the distance between the element and the cluster center. The interval type-2 fuzzy C-means clustering (FCMT2I) method, proposed in [PERSON] et al. (2016), uses two parameters \(m_{L}\) and \(m_{U}\). These calculations are derived from Equations (2) and (3). However, the membership of \(X_{i}\) in the cluster \(k\) becomes a range of values, hence, the calculation of the cluster center in FCMT2I requires additional steps from the type-reduction algorithm to obtain a single final value. Therefore, even though the performance of the FCMT2I algorithm is, in general, better than that of the FCM with one  
Figure 1: Membership of an element to a cluster for different values of m.  
Figure 2: The cluster boundary between two clusters for different values of m.  
the FCMT2I algorithm requires more calculations because the exponential parameters depend on the distance to the cluster center and the cluster center changes after every iteration.  
The proposed idea in this study is to assign each element \(X_{i}\) an exponential parameter \(m_{i}\) which is constant throughout the entire clustering process. The determination of \(m_{i}\) depends on the characteristics of the corresponding element \(X_{i}\) in the dataset.