Domain knowledge can be introduced into fuzzy clustering with the aid of information granules, embodied by the concept of viewpoints. For such kind of fuzzy clustering methods, the strategy of acquisition of viewpoints has not been fully developed. Furthermore a way of determining the related information granules deserves more attention. Having these problems in mind, in this study, the density Viewpoint-based Weighted Kernel Fuzzy Clustering (VWKFC) algorithm is proposed. First, the kernel-based hypersphere density initialization (KHDI) algorithm is presented as a certain prerequisite, in which the kernel distance is utilized instead of the Euclidean distance. Besides, a novel density radius is put forward. Second, the concept of the weight information granule is established, which incorporates two parts. The feature weight matrix is provided, where different weights are assigned to different features to reduce the influence of unrelated features. Meanwhile a sample weight is assigned to each data point, thus the influence of noise and outliers on clustering can be reduced to a certain extent. Third, the data point with the highest local density obtained by KHDI is regarded as the density viewpoint. Then we combine kernel mechanism, density viewpoints, weight information granules and a maximum entropy regularization to design the VWKFC algorithm, and prove its convergence. Experimental results validate that VWKFC is superior over eight related clustering algorithms with regard to five evaluation indexes, especially when processing high-dimensional data. It has been shown that VWKFC makes the selection of initialized cluster centers and viewpoints more reasonable, and obtains better clustering results, and achieves higher convergence speed.

The introduction of domain knowledge opens new horizons to fuzzy clustering. Then knowledge-driven and data-driven fuzzy clustering methods come into being. To address the challenges of inadequate extraction mechanism and imperfect fusion mode in such class of methods, we propose the Knowledge-induced Multiple Kernel Fuzzy Clustering (KMKFC) algorithm. Firstly, to extract knowledge points better, the Relative Density-based Knowledge Extraction (RDKE) method is proposed to extract high-density knowledge points close to cluster centers of real data structure, and provide initialized cluster centers. Moreover, the multiple kernel mechanism is introduced to improve the adaptability of clustering algorithm and map data to high-dimensional space, so as to better discover the differences between the data and obtain superior clustering results. Secondly, knowledge points generated by RDKE are integrated into KMKFC through a knowledge-influence matrix to guide the iterative process of KMKFC. Thirdly, we also provide a strategy of automatically obtaining knowledge points, and thus propose the RDKE with Automatic knowledge acquisition (RDKE-A) method and the corresponding KMKFC-A algorithm. Then we prove the convergence of KMKFC and KMKFC-A. Finally, experimental studies demonstrate that the KMKFC and KMKFC-A algorithms perform better than thirteen comparison algorithms with regard to four evaluation indexes and the convergence speed.

The existing clustering validity indexes (CVIs) show some difficulties to produce the correct cluster number when some cluster centers are close to each other, and the separation processing mechanism appears simple. The results are imperfect in case of noisy data sets. For this reason, in this study, we come up with a novel CVI for fuzzy clustering, referred to as the triple center relation (TCR) index. The originality of this index is twofold. On the one hand, a new fuzzy cardinality is built on the strength of the maximum membership degree, and a novel compactness formula is constructed by combining it with the within-class weighted squared error sum. On the other hand, starting from the minimum distance between different cluster centers, the mean distance as well as the sample variance of cluster centers in the statistical sense are further integrated. These three factors are combined by means of product to form a triple characterization of the relationship between cluster centers, and hence a 3-D expression pattern of separability is formed. Subsequently, the TCR index is put forward by combining the compactness formula with the separability expression pattern. By virtue of the degenerate structure of hard clustering, we show an important property of the TCR index. Finally, based on the fuzzy $ C $ -means (FCMs) clustering algorithm, experimental studies were conducted on 36 data sets (incorporating artificial and UCI data sets, images, the Olivetti face database). For comparative purposes, 10 CVIs were also considered. It has been found that the proposed TCR index performs best in finding the correct cluster number, and has excellent stability.

The Bandler–Kohout subproduct (BKS) method is one of the two widely acknowledged fuzzy relational inference (FRI) schemes. The previous works related to its stability and robustness mainly concentrated on how the output values were changed with perturbation parameters of input values. However, the works on estimating oscillation bounds of output values with regard to varying limits of input, are lacking. In this study, we investigate the oscillation-bound estimation of perturbations for BKS. First, the BKS output variation scopes are acquired for interval perturbation, where the $R$ -implication, ( $S, N$ )-implication, QL-implication, and $t$ -norm implication are adopted. Second, in allusion to the more sophisticated problem of the fuzzy reasoning chain with BKS, the oscillation bounds of BKS output resulting from input interval perturbation are offered. Third, we construct the upper and lower bounds of BKS output deviation originated in the simple perturbation of the input fuzzy set, in which the situations of one rule and multiple rules are both dissected. Finally, the stable properties of all these BKS strategies are confirmed. It is emphasized that interval perturbation and simple perturbation are more general ways to give expression describing the robustness issue, and the obtained oscillation bounds also deliver more detailed characterization of the output deviation along with the input perturbation. This study further validates the sound properties of the BKS method.