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Fuzzy clustering techniques, especially fuzzy C-means (FCM) and its weighted variants, are typical partitive clustering models that are widely used for revealing possible hidden structures in data. Although they can quantitatively depict the overlapping areas with a partition matrix, their performances deteriorate when dealing with high-dimensional data because the distance computations may be negatively impacted by the irrelevant features, and then the concentration effect may arise. Moreover, they are sensitive to noisy environments. To tackle these obstacles, a robust jointly sparse fuzzy clustering method (RJSFC) is proposed in this study. The representative prototypes, sparse membership grades, and an orthogonal projection matrix are simultaneously learnt when optimizing RJSFC. The obtained low-dimensional embeddings can preserve the local neighborhood structure, and the clustering is conducted in the transformed lower dimensional space rather than the original space, which improves the capability of fuzzy clustering for dealing with high-dimensional scenarios. Furthermore, ${L_{2,1}}$ -norm is exploited as the basic metric for both loss and regularization parts in RJSFC, the robustness of the model and the interpretability of the extracted features are enhanced. The notions of fuzzy clustering, neighborhood structure preservation, and feature extraction are seamlessly integrated into a unified model. The limitation of the previous two-stage clustering framework when dealing with high-dimensional data entailing dimensionality reduction and clustering procedures separately can be effectively addressed. Extensive experimental results on various well-known datasets demonstrate the usefulness of RJSFC when comparing with some state-of-the-art methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Clustering offers a general methodology and comes with a remarkably rich conceptual and algorithmic framework for data analysis and data interpretation. As one of the most representative algorithms of fuzzy clustering, fuzzy C-means (FCM) is a widely used objective function-based clustering method exploited in various applications. In this study, a virtual-based fuzzy clustering algorithm is proposed to improve the classification performance coming as a result of using fuzzy clustering. This improvement is achieved by forming a virtual space based on the original data space. First, we construct a piecewise linear transformation function to modify the similarity matrix of the original data and build the so-called virtual similarity matrix (VSM). Considering the VSM, the effect of closeness becomes amplified; in other words, high similarity values (say, larger than α which is a cutoff value of the large and small similarity in this paper) present in the original similarity matrix are made higher, whereas lower similarity levels (say, smaller than α) are further reduced. In addition, data with high similarity (say, larger than a certain threshold value) observed in the original space will overlap (the attributes of the samples are exactly the same) significantly in the virtual space; the overlapping samples can be treated as one sample. This modification makes possible easier to identify clusters. Second, we build a relationship matrix between the original dataset and the determined similarity values and present two closed-form solutions to the problem of building the relationship matrix. Subsequently, a virtual space of the original data space is derived through the modified similarity matrix and the introduced relationship matrix. We offer a thorough analysis behind the developed clustering algorithm. The experimental results are in agreement with the underlying conceptual basis. Furthermore, the resulting classification performance is significantly improved compared with the results produced by the FCM and the kernel-based fuzzy C-means. [ABSTRACT FROM AUTHOR]

Fuzzy clustering has emerged as one of the fundamental conceptual and algorithmic frameworks supporting the development of information granules. Generic fuzzy clustering such as fuzzy C-means (FCM) has been utilized in a broad range of applications. However, the constructs resulting from fuzzy clustering, namely a partition matrix and prototypes, are numeric and as such are not capable of fully capturing the essence of the overall data. In this study, we propose an alternative augmented way of building information granules by generating hypercube-like information granules. A collection of hypercubes is referred to as a family of ϵ-information granules. This family is constructed around numeric prototypes generated through a modified version of the FCM algorithm whose running time is linear with respect to the number of clusters. By admitting a certain level of information granularity $(\varepsilon)$ , a collection of hypercubes is formed around the prototypes. The quality of information granules realized in this way is assessed by involving them in the granulation—degranulation process as well as determining a value of the coverage criterion. The level of information granularity and the number of the granular prototypes in the family of $\varepsilon $ -information granules form an important design asset directly impacting the obtained coverage level of the data. The computational facet of the approach is stressed. It has been demonstrated that the granular enhancements of the description of data come with a very limited computing overhead. Experimental studies involve synthetic data as well as data coming from the UCI Machine Learning repository. The granular reconstruction capabilities delivered by the family of $\varepsilon $ -information granules are discussed. [ABSTRACT FROM AUTHOR]

Owning to their abilities to reveal structural relationships in data, fuzzy clustering plays a pivotal role in fuzzy modeling, pattern recognition, and data analysis. As supporting an unsupervised mode of learning, fuzzy clustering, brings about unique opportunities to build a structural backbone of numerous constructs in the areas identified above. A follow-up phase is required when the structural findings developed in the form of fuzzy clusters need some refinements, usually when the clustering results are used afterwards in the model being developed in a supervised mode. Following this general line of thought, in this study we propose a novel approach to optimize the clustering and classification performance of the Fuzzy C-Means (FCM) algorithm. Proceeding with a collection of clusters (information granules) produced by the FCM, we carry out the refinements of the results (in order to improve the representation or classification capabilities of fuzzy clusters) by adjusting a location of the prototypes so that a certain performance index becomes optimized. At this phase, the optimization is carried out in a supervised mode with the aid of Differential Evolution (DE). We propose five different strategies to adjust a location of the prototypes. Experimental studies completed on synthetic data and publicly available real-world data quantify the improvement of the representation and classification abilities of the clustering method. [ABSTRACT FROM AUTHOR]

Incomplete data clustering is often encountered in practice. Here the treatment of missing attribute value and the optimization procedure of clustering are the important factors impacting the clustering performance. In this study, a missing attribute value becomes an information granule and is represented as a certain interval. To avoid intervals determined by different cluster information, we propose a congeneric nearest-neighbor rule-based architecture of the preclassification result, which can improve the effectiveness of estimation of missing attribute interval. Furthermore, a global fuzzy clustering approach using particle swarm optimization assisted by the Fuzzy C-Means is proposed. A novel encoding scheme where particles are composed of the cluster prototypes and the missing attribute values is considered in the optimization procedure. The proposed approach improves the accuracy of clustering results, moreover, the missing attribute imputation can be implemented at the same time. The experimental results of several UCI data sets show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this study, we discuss a design of fuzzy associative structures (memories) realized within the framework of fuzzy clustering. Associative memories are inherently direction-free structures (and the recall of objects can be realized for any variable or a subset of variables). Fuzzy clustering being direction-free comes here as a sound design alternative. Two recall proposals are studied: one involves prototypes (being the key descriptors of the structure of the data) and their activation in the presence of partially available data to be recalled whereas the second proposal involves fuzzy correlation matrices and in principle exhibits some resemblance with a standard correlation associative memories. In the setting of associative memories, Fuzzy C-Means (FCM) is studied. The recall error is discussed with regard to the essential parameters of the FCM (the number of clusters and the fuzzification coefficient). Furthermore we discuss an optimization of the distance function used in the clustering algorithm realized with regard to the recall error. Experimental results are provided along with a comparative study involving correlation-based associative memories. [ABSTRACT FROM AUTHOR]

In this study, we introduce a concept of hierarchical granular clustering and establish its algorithmic framework. We show that the proposed model naturally gives rise to information granules that are both of higher order and higher type, offering a compelling justification behind their emergence. In a concise way, we can capture the overall architecture of information granules as a hierarchy exhibiting conceptual layers of increasing abstraction: numeric data → information granules → information granules of type-2, order-2 → … information granules of higher type/order. The elevated type of information granules is reflective of the visible hierarchical facet of processing and the inherent diversity of the individual locally revealed structures in data. While the concept and the methodology deliver some general settings, the detailed algorithmic aspects are discussed in detail when using fuzzy clustering realized by means of fuzzy c-means. Furthermore, for illustrative purposes, we mainly focus on interval-valued fuzzy sets and granular interval fuzzy sets arising at the higher level of the hierarchy. Higher type fuzzy sets are formed with the help of the principle of justifiable granularity. The conceptually sound hierarchy is established in a general way, which makes it equally applicable to various formalisms of representation of information granules. Experiments are reported for synthetic and publicly available datasets. [ABSTRACT FROM AUTHOR]

In this study, we propose a cluster-oriented development of fuzzy models. An overall design process is focused on an efficient usage of fuzzy clustering, Fuzzy C-Means (FCM), in particular, to form information granules—clusters that are used in the construction of the fuzzy model. Fuzzy models are regarded as mappings from information granules expressed in the input and output spaces. This position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules. This study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages. The augmented FCM method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and this way the performance of the resulting fuzzy model is optimized. It is shown that the cluster-oriented modeling gives rise to the Mamdani-like fuzzy rules and a zero-order Takagi–Sugeno model (under a certain decoding scheme). We identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding–decoding (or granulation–degranulation) encountered in processing fuzzy sets and Granular Computing, in general. Furthermore, refinements of zero-order fuzzy models are investigated leading to first-order fuzzy models with linear functions standing in the conclusions of the rules. A series of experiments is reported where we used synthetic and real-world data using which an issue of generalization capabilities is elaborated in detail. [ABSTRACT FROM AUTHOR]

Clustering forms one of the most visible conceptual and algorithmic framework of developing information granules. In spite of the algorithm being used, the representation of information granules–clusters is predominantly numeric (coming in the form of prototypes, partition matrices, dendrograms, etc.). In this paper, we consider a concept of granular prototypes that generalizes the numeric representation of the clusters and, in this way, helps capture more details about the data structure. By invoking the granulation–degranulation scheme, we design granular prototypes being reflective of the structure of data to a higher extent than the representation that is provided by their numeric counterparts (prototypes). The design is formulated as an optimization problem, which is guided by the coverage criterion, meaning that we maximize the number of data for which their granular realization includes the original data. The granularity of the prototypes themselves is treated as an important design asset; hence, its allocation to the individual prototypes is optimized so that the coverage criterion becomes maximized. With this regard, several schemes of optimal allocation of information granularity are investigated, where interval-valued prototypes are formed around the already produced numeric representatives. Experimental studies are provided in which the design of granular prototypes of interval format is discussed and characterized. [ABSTRACT FROM AUTHOR]

There are some variants of the widely used Fuzzy C-Means (FCM) algorithm that support clustering data distributed across different sites. Those methods have been studied under different names, like collaborative and parallel fuzzy clustering. In this study, we offer some augmentation of the two FCM-based clustering algorithms used to cluster distributed data by arriving at some constructive ways of determining essential parameters of the algorithms (including the number of clusters) and forming a set of systematically structured guidelines such as a selection of the specific algorithm depending on the nature of the data environment and the assumptions being made about the number of clusters. A thorough complexity analysis, including space, time, and communication aspects, is reported. A series of detailed numeric experiments is used to illustrate the main ideas discussed in the study. [ABSTRACT FROM AUTHOR]

Abstract Rule-based modeling has been one of the key directions in fuzzy modeling since its inception. Nowadays it exhibits a number of conceptual development and algorithmic pursuits. Fuzzy clustering, especially Fuzzy C-Means (FCM), is a commonly used algorithmic tool to construct rules, in particular building fuzzy sets forming conditions of the rules. While being efficient with this regard and transforming data into a collection of fuzzy sets, one should note that the agenda of fuzzy clustering (and clustering, in general) does not fully align with the agenda of system modeling and because of this, it requires some attention and calls for further refinements. Clustering is a direction-free (relational) process and developed constructs (clusters) are optimized in light of the direction-free criterion (say, a commonly used objective function). In contrast, in fuzzy models the rules are direction-sensitive artifacts, which implies that the clusters themselves need to be reflective of this directionality requirement. This paper contributes to this direction of studies by bringing a collection of augmentations of the generic FCM algorithm along this line. There are three original enhancements considered in the study: (i) an accommodation of extreme (minimal and maximal) values encountered in the output variable, (ii) a reduction of spurious impact of rules being the result of variable overlap existing among fuzzy sets forming the condition parts of the rules, and (iii) a development of the core (granular) structure of rules and analysis of their features. The motivation behind these augmentations is presented followed by the detailed algorithms along with a series of illustrative examples. In the sequel, a number of numeric studies are conducted demonstrating in a quantitative manner the contributions delivered by the refinements. [ABSTRACT FROM AUTHOR]