An extension of global fuzzy c-means using kernel methods

Abstract

Fuzzy c-means (FCM) is a simple but powerful clustering method using the concept of fuzzy sets that has been proved to be useful in many areas. There are, however, several well known problems with FCM, such as sensitivity to initialization, sensitivity to outliers, and limitation to convex clusters. In this paper, global fuzzy c-means (G-FCM) and kernel fuzzy c-means (K-FCM) are combined and extended to form a non-linear variant of G-FCM, called kernelized global fuzzy c-means (KG-FCM). G-FCM is a variant of FCM that uses an incremental seed selection method and is effective in alleviating sensitivity to initialization. There are several approaches to reduce the influence of noise and properly partition non-convex clusters, and K-FCM is one. K-FCM is used in this paper because it can easily be extended with different kernels, which provide sufficient flexibility to allow for resolution of the shortcomings of FCM. By combining G-FCM and K-FCM, KG-FCM can resolve the shortcomings mentioned above. The usefulness of the proposed method is demonstrated by experiments using artificial and real world data sets.

Kernelized Fuzzy C-Means Method and Gaussian Mixture Model in Unsupervised Cascade Clustering

“…Moreover, the Euclidean norm in the objective function and the resulting hyperspherical cluster shapes -may cause inappropriate data separation. The FCM method is also sensitive to the starting conditions -partition matrix and the number of clusters [8].…”

“…The kernelised clustering methods constitute the development of conventional FCM clustering algorithm, discussed in detail in [7,8,9].…”

“…The improvement of clusters shape in the feature space can be implemented using KFCM clustering. Two different approaches are used [7,12,13,8]: (1) the cluster centers (prototypes) are calculated in the original feature space and the clustering process is transformed to the kernel space, (2) the whole clustering process is transformed to the multidimensional space -described as clustering in the kernel space with implicit prototypes [7].…”

“…The transition from feature space F to kernel space H is obtained through mapping φ(x) and the corresponding kernel function q is described by formula in [8,7,5]. In the proposed classification algorithm the Radial Basis Function (RBF) given as…”

Objective function‐based clustering

“…Practically, the kernel function K(x i , x j ) can be integrated into the distance function of a clustering algorithm which changes the update equations. 45,46 The most general approach is to construct the cluster center prototypes in kernel space 46 because it allows for more kernel functions to be used. Here, we will take a look at hard and FKM approaches to objective function-based clustering with kernels.…”