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K-medoids clustering is among the most popular methods for cluster analysis despite its use requiring several assumptions about the nature of the latent clusters. In this paper, we introduce the Convex Fuzzy k-Medoids (CFKM) model, which not only relaxes the assumption that objects must be assigned entirely to one and only one medoid, but also that medoids must be assigned entirely to one and only one cluster. The resulting model is convex, thus its resolution is completely robust to initialization. To illustrate the usefulness of the CFKM model, we compare it with two fuzzy k-medoids clustering models: the Fuzzy k-Medoids (FKM) and the Fuzzy Clustering with Multi-Medoids (FMMdd), both solved approximately by heuristics because of their hard computational complexity. Our experiments with both synthetic and real-world data as well as a user survey reveal that the model is not only more robust to the choice hyperparameters of the fuzzy clustering task, but also that it can uniquely discover important aspects of data inherently fuzzy in nature.

In this paper we propose different robust fuzzy clustering models for classifying heteroskedastic (volatility) time series, following the so-called model-based approach to time series clustering and using a partitioning around medoids procedure. The proposed models are based on a GARCH parametric modeling of the time series, i.e. the unconditional volatility and the time-varying volatility GARCH representation of the time series. We first suggest a timid robustification of the fuzzy clustering. Then, we propose three robust fuzzy clustering models belonging to the so-called metric, noise and trimmed approaches, respectively. Each model neutralizes the negative effects of the outliers in the clustering process in a different manner. In particular, the first robust model, based on the metric approach, achieves its robustness with respect to outliers by taking into account a robust distance measure; the second, based on the noise approach, achieves its robustness by introducing a noise cluster represented by a noise prototype; the third, based on the trimmed approach, achieves its robustness by trimming away a certain fraction of outlying time series. The usefulness and effectiveness of the proposed clustering models is illustrated by means of a simulation study and two applications in finance and economics.

Clustering high dimensional data and identifying central nodes in a graph are complex and computationally expensive tasks. We utilize k-nn graph of high dimensional data as efficient representation of the hidden structure of the clustering problem. Initial cluster centers are determined by graph centrality measures. Cluster centers are fine-tuned by minimizing fuzzy-weighted geodesic distances. The shortest-path based representation is parallel to the concept of transitive closure. Therefore, our algorithm is capable to cluster networks or even more complex and abstract objects based on their partially known pairwise similarities.The algorithm is proven to be effective to identify senior researchers in a co-author network, central cities in topographical data, and clusters of documents represented by high dimensional feature vectors.

Medoid-based fuzzy clustering generates clusters of objects based on relational data, which records pairwise similarities or dissimilarities among objects. Compared with single-medoid based approaches, our recently proposed fuzzy clustering with multiple-weighted medoids has shown superior performance in clustering via experimental study. In this paper, we present a new version of fuzzy relational clustering in this family called fuzzy clustering with multi-medoids (FMMdd). Based on the new objective function of FMMdd, update equations can be derived more conveniently. Moreover, a unified view of FMMdd and two existing fuzzy relational approaches fuzzy c-medoids (FCMdd) and assignment-prototype (A-P) can be established, which allows us to conduct further analytical study to investigate the effectiveness and feasibility of the proposed approach as well as the limitations of existing ones. The robustness of FMMdd is also investigated. Our theoretical and numerical studies show that the proposed approach produces good quality of clusters with rich cluster-based information and it is less sensitive to noise.