Multi-view clustering has been received considerable attention due to the widespread collection of multi-view data from diverse domains and sources. However, storing multi-view data across multiple devices in many real scenarios poses significant challenges for efficient data analysis. Federated Learning framework enables collaborative machine learning on distributed devices while preserving privacy constraints. Even though there have been intensive algorithms on multi-view fuzzy clustering, federated multi-view fuzzy clustering has not been adequately investigated so far. In this study, we first develop the federated learning mode into multi-view fuzzy clustering and realize the federated optimization procedure, called Federated Multiview Fuzzy C-Means clustering (FedMVFCM). Then, we design an original strategy of consensus prototype learning during federated multi-view fuzzy clustering. It is termed as Federated Multi-view Fuzzy c-means consensus Prototypes Clustering (FedMVFPC). We also further develop the federated alternative optimization algorithm with proven convergence. This study also introduces the notion of clustering prototype communication within the federated learning framework, and integrates the clustering prototypes of different views into a unified optimization formulation. The experimental studies on various benchmark datasets demonstrate that the proposed FedMVFPC method improves the federated clustering performance and efficiency. It achieves comparable or better clustering performance against the existing state-of-the-art multi-view clustering algorithms

Ensemble methods are widely used to tackle class imbalance problem. However, for existing imbalanced ensemble (IE) methods, the samples in each subset are resampled from the same dataset, and are directly input to the classifier for training, so the quality (diversity and separability) of the subsets is unsatisfactory usually. To solve the problem, a deep fuzzy envelope sample generation mechanism is proposed. First, the fuzzy C-means clustering based deep sample envelope prenetwork (DSEN) is designed to mine correlation information among samples, thereby increasing the quality of the subsets. Second, the local manifold structure metric and global structure distribution metric are designed to construct local-global structure consistency mechanism (LGSCM) to enhance distribution consistency of interlayer samples of DSEN. Third, the DSEN and LGSCM are combined to form the final deep sample envelope network–DSENLG to refresh the existing subsets. Finally, base classifiers are applied on the new subsets generated by the DSENLG and then fused, thereby realizing a new IE algorithm. The experimental results show that the proposed algorithm is significantly better than existing representative IE algorithms and it achieves the highest improvement of 10.64%, 19.5%, 18.67% and 22.33% on four criteria over the state-of-the-art methods. The originality of the article is threefold: proposing the concept of “deep fuzzy samples” or “envelope samples”, which comprehensively considers the correlation information among original samples; proposing the LGSCM to resolve the distribution inconsistency of interlayer samples; and forming an fuzzy envelope sample based IE algorithm.

Multi-view learning becomes increasingly attractive and promising because multimodal or multi-view data are commonly encountered in real-world applications. In this study, we develop a novel multi-view Takagi–Sugeno–Kang (TSK) fuzzy system framework to handle classification problems for such data. We propose an anchor and graph subspace clustering strategy to discover and represent the actual latent data distribution for each view separately. In this way, the discriminate anchors (landmarks) are learned to capture the main structure of the multi-view data. This strategy also provides a computationally efficient clustering algorithm with respect to the number of instances. These resulting anchors are formed as the prototypes of information granules (IGs) for fuzzy modeling. Then we construct an information-granule-based multi-view TSK fuzzy classification model inherited from the natural interpretability of fuzzy rule-based systems. Concretely, the relationship between the multi-view input and label output spaces is depicted by IGs-oriented fuzzy rules. The experimental studies involve various commonly used benchmark datasets, which indicate that our proposed method achieves comparable or better performance compared to the state-of-the-art algorithms.

Incomplete data are frequently encountered and bring difficulties when it comes to further processing. The concepts of granular computing (GrC) help deliver a higher level of abstraction to address this problem. Most of the existing data imputation and related modeling methods are of numeric nature and require prior numeric models to be provided. The underlying objective of this study is to introduce a novel and straightforward approach that uses information granules as a vehicle to effectively represent missing data and build granular fuzzy models directly from resulting hybrid granular and numeric data. The evaluation and optimization of this method are guided by the principle of justifiable granularity engaging the coverage and specificity criteria and carried out with the help of particle swarm optimization. We provide a collection of experimental studies using a synthetic dataset and several publicly available real-world datasets to demonstrate the feasibility and analyze the main features of this method.

Fuzzy rule-based models (FRBMs) are sound constructs to describe complex systems. However, in reality, we may encounter situations, where the user or owner of a system only owns either the input or output data of that system (the other part could be owned by another user); and due to the consideration of data privacy, he/she could not obtain all the needed data to build the FRBMs. Since this type of situation has not been fully realized (noticed) and studied before, our objective is to come up with some strategy to address this challenge to meet the specific privacy consideration during the modeling process. In this study, the concept and algorithm of the collaborative fuzzy clustering (CFC) are applied to the identification of FRBMs, describing either multiple-input-single-output (MISO) or multiple-input-multiple-output (MIMO) systems. The collaboration between input and output spaces based on their structural information (conveyed in terms of the corresponding partition matrices) makes it possible to build FRBMs when input and output data could not be collected and used in unison. Surprisingly, on top of this primary pursuit, with the collaboration mechanism the input and output spaces of a system are endowed with an innovative way to comprehensively share, exchange, and utilize the structural information between each other, which results in their more relevant structures that guarantee better model performance compared with performance produced by some state-of-the-art modeling strategies. The effectiveness of the proposed approach is demonstrated by experiments on a series of synthetic and publicly available datasets.

Learning a proper distance for clustering from prior knowledge falls into the realm of semisupervised fuzzy clustering. Although most existing learning methods take prior knowledge (e.g., pairwise constraints) into account, they pay little attention to local knowledge of data, which, however, can be utilized to optimize the distance. In this article, we propose a novel distance learning method, which learns from the Group-level information, for semisupervised fuzzing clustering. We first present a new format of constraint information, called Group-level constraints, by elevating the pairwise constraints (must-links and cannot-links) from point level to Group level. The Groups, generated around data points contained in the pairwise constraints, carry not only the local information of data (the relation between close data points) but also more background information under some given limited prior knowledge. Then, we propose a novel method to learn a distance by using the Group-level constraints, namely, Group-based distance learning, in order to optimize the performance of fuzzy clustering. The distance learning process aims to pull must-link Groups as close as possible while pushing cannot-link Groups as far as possible. We formulate the learning process with the weights of constraints by invoking some linear and nonlinear transformations. The linear Group-based distance learning method is realized by means of semidefinite programming, and the nonlinear learning method is realized by using the neural network, which can explicitly provide nonlinear mappings. Experimental results based on both synthetic and real-world datasets show that the proposed methods yield much better performance compared to other distance learning methods using pairwise constraints.

In this article, we advocate that a knowledge tidbit residing in the output space could be helpful in improving the performance (accuracy) of the fuzzy rule-based model. It states thatif two outputs are far apart from each other,it is advisable to place their corresponding inputs in different clusters when forming subspaces of the input space. Considering this knowledge guidance mechanism, we propose two different methods to partition the input space. In the first method, input data are first partitioned with the use of the standard clustering algorithm, say fuzzy C-means; here, a constructed partition matrix is reflective of the structure present in the input space. Then, the knowledge tidbit is used to adjust the entries of the original partition matrix in such a way that those input data whose corresponding output data are far apart from each other are assigned with low values of proximity. In the second method, we propose two strategies to modify the distance between input data and a prototype (cluster center) identified in the input space. The crux of this method is that if there are many input data (which, in virtue of the knowledge tidbit, are regarded as being far-apart from the input data of interest) around a certain prototype, the distance between the input data of interest and this prototype should be penalized. Thus, the membership of these input data to the prototype is reduced. The comprehensive experimental studies carried out on both synthetic and publicly available data are used to examine the usefulness of the proposed methods.

In this paper, we propose a gradient-based method to approximate a fuzzy set through a trapezoidal fuzzy set (TFS). By adding some constraints in the formulated optimization problem, the major characteristics of the fuzzy set such as the core, the major part of the support, and the shape of the membership function could be preserved; also the form of the optimized result as a TFS is guaranteed. We regard the optimized TFS as the "skeleton" (blueprint) of the original fuzzy set. Based on this skeleton, we further extend the TFS to a higher type, that is, an interval type-2 TFS (IT2 TFS), so that more information about the original fuzzy set could be captured but the number of the parameters used to describe the original fuzzy set is still maintained low (nine parameters are required for an IT2 TFS). The principle of justifiable granularity is used to ensure that the formed type-2 information granule exhibits a sound interpretation. Both synthetic fuzzy sets and those constructed by the fuzzy C -means algorithm applied to the publicly available data have been used to demonstrate the usefulness of the proposed approximation methods.

Big data with a large number of observations (samples) have posed genuine challenges for fuzzy clustering algorithms and fuzzy C-means (FCM), in particular. In this article, we propose an original algorithm referred to as a hyperplane division method to split the entire data set into disjoint subsets. By disjoint subsets, we mean that the data subspaces (parts of the entire data space), each of which is supported or spanned by the data points in the corresponding subset, do not overlap each other. The disjoint subsets turned out to be beneficial to the improvement of the quality of the clusters formed by the clustering algorithms. Moreover, considering that either a large number (say, thousands) or a small number (say, a few) of clusters may be pursued in the clustering task, we propose corresponding strategies (based on the hyperplane division method) to make clustering processes feasible, efficient, and effective. By validating the proposed strategies on both synthetic and publicly available data, we show their superiority (in terms of both efficiency and effectiveness) manifested in a visible way over the method of clustering the entire data and over some representative big data clustering methods.

Conceptually and algorithmically, hotspots could be regarded as information granules. In this study, we propose an aggregation of Fuzzy C-Means (FCM) algorithm and the principle of justifiable granularity (PJG) as a new approach to forming hotspots. With the proposed method, the quality of the hotspots formed in this manner could also be provided as an additional information to the decision makers. Moreover, a weighted granular clustering method is presented to further abstract the constructed hotspots, and this delivers a higher level of abstraction of the phenomenon of interest. A collection of synthetic data is used to show the proposed process of identifying the hotspots, and to demonstrate its differences with some other representative hotspot identification methods. Besides, real-world data are also used to illustrate the performance of the proposed method.

In this paper, we develop a comprehensive conceptual and algorithmic framework to cope with a problem of clustering homogeneous information granules. While there have been several approaches to coping with granular (viz. non-numeric) data, the origin of granular data themselves considered there is somewhat unclear and, as a consequence, the results of clustering start lacking some full-fledged interpretation. In this paper, we offer a holistic view at clustering information granules and an evaluation of the results of clustering. We start with a process of forming information granules with the use of the principle of justifiable granularity (PJG). With this regard, we discuss a number of parameters used in this development of information granules as well as quantify the quality of the granules produced in this manner. In the sequel, Fuzzy C -Means is applied to cluster the derived information granules, which are represented in a parametric manner and associated with weights resulting from the usage of the PJG. The quality of clustering results is evaluated through the use of the reconstruction criterion (quantifying the concept of information granulation and degranulation). A suite of experiments using synthetic and publicly available datasets is reported to quantify the performance of the proposed approach and highlight its key features.