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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]

Financial time series are generally high-dimensional, nonstationary, and exhibit heteroscedasticity. To derive a suitable way to cluster financial time series, these characteristics have to be taken into consideration. With this aim, in this article, the financial time series is firstly modeled using generalized autoregressive conditional heteroscedasticity (GARCH) models, where the parameters of GARCH models can represent the dynamic feature of the volatility in each time series. Therefore, the following clustering is realized based on the GARCH model parameters, which can help reduce the dimensionality of the original time series at the same time. Then, to produce semantically sound clustering results, we granulate the parameters based on the axiomatic fuzzy set (AFS) theory and structure them into a collection of meaningful and semantically sound entities, i.e., AFS information granules. Furthermore, the hierarchical structure of AFS information granules is built to realize time series clustering under the framework of granular computing. In the proposed approach, the characteristics of financial time series is fully considered to proceed dimensionality reduction, and the semantic clustering results obtained for different numbers of clusters are guaranteed to be the most informative. In the experiments, an application for clustering the time series coming from Chinese Yuan exchange rates against international currencies is presented to demonstrate the performance of the proposed clustering method. The results of clustering of the proposed method are the same as those of the fuzzy C-means algorithm and the hierarchical clustering with ward linkage, where the clustering results produced by the AFS hierarchical clustering exhibit well-articulated semantics at each level of the hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we propose reinforced fuzzy clustering-based ensemble neural networks (FCENNs) classifier. The objective of this paper is focused on the development of the design methodologies of ensemble neural networks classifier for constructing the network structure and enhancing the learning methods of fuzzy clustering-based neural networks through the combination of the probabilistic model and its learning mechanism. The proposed FCENNs classifier takes into consideration a cross-entropy error function to improve learning while $L_2$ -norm regularization is used to reduce overfitting as well as enhance generalization abilities. The essential points of the proposed reinforced FCENNs classifier can be enumerated as follows: First, in the proposed classifier, the cross-entropy error function is used as a cost function; to do this, a softmax function is applied to represent a categorical distribution located at the nodes of the output layer. Second, the learning mechanism is composed of two parts. First, fuzzy C-means clustering forms the connections (weights) of the hidden layer while the connections of the output layer are adjusted with the aid of the nonlinear least squares method using Newton's method-based learning. Third, $L_2$ norm-regularization is considered to avoid the degradation of generalization ability caused by overfitting. The learning mechanism similar to ridge regression is realized by adding $L_2$ penalty term to the cross-entropy error function. From the viewpoint of performance improvement achieved through the proposed novel learning method, the design methodology for the ensemble neural networks classifier is discussed and analyzed with the aid of a diversity of two-dimensional synthetic data and machine learning datasets. [ABSTRACT FROM AUTHOR]

Abstract Fuzzy rule-based models have attracted attention because of their modular architectures, well-developed design methodologies and practices as well as interpretability aspects. Methods exploiting factors of randomness offer significant efficiency and implementation simplicity that are essential in numerous application areas. In this study, we propose an original development of fuzzy rule-based models established with the aid of concepts of randomization algorithms. Several design strategies involving different random prototypes generation and basis functions approximation are studied. We investigate performance aspects of randomized rule-base and look at the performance versus the key components of the models such as the number of rules and the use of the randomized algorithms in the development. Furthermore, a comparative study is offered to quantify the efficiency of randomized algorithms. Experimental studies are reported for a series of publicly available data sets to illustrate the effectiveness of the proposed method and discuss its main features. [ABSTRACT FROM AUTHOR]

The Fuzzy C-Means (FCM) algorithm is a widely used objective function-based clustering method exploited in numerous applications. In order to improve the quality of clustering algorithms, this study develops a novel approach, in which a transformed data-based FCM is developed. Two data transformation methods are proposed, using which the original data are projected in a nonlinear fashion onto a new space of the same dimensionality as the original one. Next, clustering is carried out on the transformed data. Two optimization criteria, namely a classification error and a reconstruction error, are introduced and utilized to guide the optimization of the performance of the new clustering algorithm and a transformation of the original data space. Unlike other data transformation methods that require some prior knowledge, in this study, Particle Swarm Optimization (PSO) is used to determine the optimal transformation realized on a basis of a certain performance index. Experimental studies completed for a synthetic data set and a number of data sets coming from the Machine Learning Repository demonstrate the performance of the FCM with transformed data. The experiments show that the proposed fuzzy clustering method achieves better performance (in terms of the clustering accuracy and the reconstruction error) in comparison with the outcomes produced by the generic version of the FCM algorithm. [ABSTRACT FROM AUTHOR]

Information granules form an abstract and efficient characterization of large volumes of numeric data. Fuzzy clustering is a commonly encountered information granulation approach. A reconstruction (degranulation) is about decoding information granules into numeric data. In this study, to enhance quality of reconstruction, we augment the generic data reconstruction approach by introducing a transformation mapping of the originally produced partition matrix and setting up an adjustment mechanism modifying a localization of the prototypes. We engage several population-based search algorithms to optimize interaction matrices and prototypes. A series of experimental results dealing with both synthetic and publicly available data sets are reported to show the enhancement of the data reconstruction performance provided by the proposed method. [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]

The study is focused on a development of a global structure in a family of distributed data realized on a basis of locally discovered structures. The local structures are revealed by running fuzzy clustering (Fuzzy C-Means), whereas building a global view is realized by forming global proximity matrices on a basis of the local proximity matrices implied by the partition matrices formed for the individual data sets. To capture the diversity of local structures, a global perspective at the structure of the data is captured in terms of a granular proximity matrix, which is built by invoking a principle of justifiable granularity with regard to the aggregation of individual proximity matrices. The three main scenarios are investigated: (a) designing a global structure among the data through building a granular proximity matrix, (b) refining a local structure (expressed in the form of a partition matrix) by engaging structural knowledge conveyed at the higher level of the hierarchy and provided in the form of the granular proximity matrix, (c) forming a consensus-building scheme and updating all local structures with the aid of the proximity dependences available at the upper layer of the hierarchy. While the first scenario delivers a passive approach to the development of the global structure, the two others are of an active nature by facilitating a structural feedback between the local and global level of the hierarchy of the developed structures. The study is illustrated through a series of experiments carried out for synthetic and publicly available data sets. [ABSTRACT FROM AUTHOR]

The study is concerned with a concept and a design of granular time series and granular classifiers. In contrast to the plethora of various models of time series, which are predominantly numeric, we propose to effectively exploit the idea of information granules in the description and classification of time series. The numeric (optimization-oriented) and interpretation abilities of granular time series and their classifiers are highlighted and quantified. A general topology of the granular classifier involving a formation of a granular feature space and the usage of the framework of relational structures (relational equations) in the realization of the classifiers is presented. A detailed design process is elaborated on along with a discussion of the pertinent optimization mechanisms. A series of experiments is covered leading to a quantitative assessment of the granular classifiers and their parametric analysis. [ABSTRACT FROM AUTHOR]

Clustering is a powerful vehicle to reveal and visualize structure of data. When dealing with time series, selecting a suitable measure to evaluate the similarities/dissimilarities within the data becomes necessary and subsequently it exhibits a significant impact on the results of clustering. This selection should be based upon the nature of time series and the application itself. When grouping time series based on their shape information is of interest (shape-based clustering), using a Dynamic Time Warping (DTW) distance is a desirable choice. Using stretching or compressing segments of temporal data, DTW determines an optimal match between any two time series. In this way, time series exhibiting similar patterns occurring at different time periods, are considered as being similar. Although DTW is a suitable choice for comparing data with respect to their shape information, calculating the average of a collection of time series (which is required in clustering methods) based on this distance becomes a challenging problem. As the result, employing clustering techniques like K-Means and Fuzzy C-Means (where the cluster centers – prototypes are calculated through averaging the data) along with the DTW distance is a challenging task and may produce unsatisfactory results. In this study, three alternatives for fuzzy clustering of time series using DTW distance are proposed. In the first method, a DTW-based averaging technique proposed in the literature, has been applied to the Fuzzy C-Means clustering. The second method considers a Fuzzy C-Medoids clustering, while the third alternative comes as a hybrid technique, which exploits the advantages of both the Fuzzy C-Means and Fuzzy C-Medoids when clustering time series. Experimental studies are reported over a set of time series coming from the UCR time series database. [ABSTRACT FROM AUTHOR]

In the framework of Axiomatic Fuzzy Set (AFS) theory, we propose a new approach to data clustering. The objective of this clustering is to adhere to some principles of grouping exercised by humans when determining a structure in data. Compared with other clustering approaches, the proposed approach offers more detailed insight into the cluster's structure and the underlying decision making process. This contributes to the enhanced interpretability of the results via the representation capabilities of AFS theory. The effectiveness of the proposed approach is demonstrated by using real-world data, and the obtained results show that the performance of the clustering is comparable with other fuzzy rule-based clustering methods, and benchmark fuzzy clustering methods FCM and K -means. Experimental studies have shown that the proposed fuzzy clustering method can discover the clusters in the data and help specify them in terms of some comprehensive fuzzy rules. [ABSTRACT FROM AUTHOR]

Highlights: [•] Interval attribute weights are proposed and introduced for fuzzy clustering. [•] Genetic mechanism and gradient-based iteration constitute optimization strategy. [•] Data partition and weights can be obtained by minimizing the objective function. [•] Reasonable clustering results can be achieved more easily. [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]