Search

In this article, we elaborate on a design and realization of fuzzy rule-based model in the horizontal federated learning framework. Traditional machine learning in distributed environment often involves sharing sensitive information with other sites or transferring data to a central server on which a global model is trained. These situations increase the communication overhead and pose serious threats to the privacy of sensitive data. Federated learning opens up the possibility for collaboratively training a global model on a basis of distributed on-site data without sacrificing data privacy. While fuzzy rule-based models have been used in system modeling due to their substantial modeling abilities and good interpretability, the implementation of fuzzy rule-based models in a distributed environment without compromising data privacy still requires careful consideration. This article proposes a two-step federated learning approach to train a global model on a basis of private data located across different sites without their centralization. The first step concerns the determination of the structure of the data through federated collaborative clustering. Subsequently, a shared global model is trained jointly by all the participating clients. An advantage of the proposed method is that it achieves high accuracy without violating data privacy. A series of experimental studies are conducted to gain a detailed insight into the realization steps and demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 article, we propose a two-stage time-series clustering approach to cluster time series with different shapes. The first step is to represent the time series by a suite of information granules following the principle of justifiable granularity to perform dimensionality reduction, while the second step is to realize the fuzzy clustering of the time series in the transformed representation space (viz., the space of information granules). In the dimensionality reduction process, the numerical data are granulated using a collection of information granules forming a new sequence that can well describe the original time series. Then, when clustering the time series, dynamic time warping (DTW) is employed to measure the similarity between time series and DTW barycenter averaging (DBA) is generalized to weighted DBA to be involved in the fuzzy ${C}$ -means (FCMs) algorithm. Finally, the experiments are conducted on the datasets coming from UCR time-series database and Chinese stocks to demonstrate the effectiveness and advantages of the proposed fuzzy clustering approach. [ABSTRACT FROM AUTHOR]

As an augmentation of classic fuzzy models, granular fuzzy models (GFMs) have been applied to many fields being in rapport with experimental data, models, and users. However, most of the existing methods used to construct GFMs are based on the principle of optimal allocation of information granularity, which requires that a numeric model be provided in advance. In this paper, a straightforward and convincing modeling method is proposed to directly construct GFM on a basis of experimental data. The method first granulates the output space to form some interval information granules with distinct semantics and then uses them to partition the entire input space into a series of input subspaces. Subsequently, an initial GFM is emerged by using “If-Then” rules to relate with those interval information granules positioned in the output space and structures expressed in prototypes that are produced by clustering individual input subspaces. Further, the initial GFM is also refined by continuously migrating prototypes in individual input subspaces. The experimental studies using the synthetic dataset and several real-world datasets are reported. They offer a useful insight into the feasibility and effectiveness of the proposed modeling method and reveal the impact of parameters on the performance of the ensuing GFMs. An application example is also presented to exhibit the advantages of the resulting GFM. [ABSTRACT FROM AUTHOR]

Type-2 fuzzy sets provide an efficient vehicle for handling uncertainties of real-world problems, including noisy observations. Bringing type-2 fuzzy sets to clustering algorithms offers more flexibility to handle uncertainties associated with membership concepts caused by a noisy environment. However, the existing type-2 fuzzy clustering algorithms suffer from a time-consuming type-reduction process, which not only hampers the clustering performance but also increases the burden of understanding the clustering results. In order to alleviate the problem, this article introduces a set of typical characteristics of type-2 fuzzy sets and establishes a characteristic-based type-2 fuzzy clustering algorithm. Being different from the objective function used in the fuzzy C-means (FCM) algorithm that produces cluster centers and type-1 memberships, the objective function in the proposed algorithm contains additional characteristics of type-2 membership grades, namely, centers of gravity and cardinalities of the secondary fuzzy sets. The derived iterative formulas used for these parameters are much more efficient than the interval type-2 FCM algorithm. The experiments carried out in this study show that the proposed typical characteristic-based type-2 FCM algorithm has an ability of detecting noise as well as assigning suitable membership degrees to the individual data. [ABSTRACT FROM AUTHOR]

The ongoing growth of the Internet of Things and machine learning technology have provided increased motivation for the development of smart healthcare. In this study, a disease diagnosis system is proposed for remote identification and early prediction in smart healthcare environments. The originality of this study resides in the innovative implementation of ensuing modules to improve diagnostic accuracy of the system. First, fuzzy clustering based on the forest optimization algorithm is employed to detect outliers and a self-organizing fuzzy logic classifier is applied to supplement missing data in electronic medical records (EMRs). A feature selection technique using the battle royale optimization algorithm is then developed to remove redundant information and identify optimal EMR features. The refined and fused data are further classified using an eigenvalue-based machine learning algorithm to determine whether a patient exhibits a certain disease. Simulation experiments are conducted with widely used heart disease and diabetes datasets to evaluate the performance of the proposed system, using accuracy, precision, recall, and F-measure as evaluation metrics. • A diagnostic system is proposed for early disease prediction in smart healthcare. • Fuzzy clustering is applied to remove outliers from electronic medical records. • A self-organizing fuzzy logic classifier is developed to supplement missing data. • A feature selection scheme is included to identify optimal features from the data. • Eigenvalue classification is used to ascertain whether a patient exhibits a disease. [ABSTRACT FROM AUTHOR]

Neural networks (NNs) with least square error (LSE) estimation form a certain type of single hidden layer feed-forward NNs. In this class of networks, the input connections (weights) and the biases of hidden neurons are generated randomly and fixed after being generated. The output connections are estimated by the LSE method rather than the back-propagation method. The random generation of the input connection weights and the hidden biases results in the larger number of hidden neurons to assure the quality of classification performance. To reduce the number of neurons in the hidden layer while maintaining the classification performance, we apply a “divide and conquer” strategy in this article. In other words, we divide an overall input space into several subspaces by using information granulation technique (Fuzzy C-Means clustering algorithm) and determine the local decision boundaries among related subspaces. A decision boundary defined in the input space can be considered as being composed of several decision boundaries defined in subspaces that form the entire input space. For the decision boundaries defined in the subspaces, their nonlinearity becomes lower in comparison with the one being encountered when considering the entire input space. Through the weighted LSE estimation instead of using the LSE estimation method, the connections of several NNs can be estimated without interfering with each other. After estimating the weights, the decision boundaries defined in the related subspaces are merged to a single decision boundary by using fuzzy ensemble technique. Several machine learning datasets and one real world application dataset are used to evaluate and validate the proposed fuzzy ensemble classifier. Based on the experimental results, the proposed classifier shows better classification performance when compared with the performance of some selected classifiers. [ABSTRACT FROM AUTHOR]

In this paper, an integrated model combining interval deep belief network (IDBN) and neural network with nonlinear weights, called IDBN-NN, is proposed for interval-valued data modeling. Firstly, the IDBN with variable learning rate is designed to initialize parameters of each sub-model. Based on a modified contrastive divergence algorithm the least square method is adopted to identify the coefficients of nonlinear weights in the output layer. Then, to improve the modeling accuracy, the Fuzzy C-Means (FCM) method and the Particle Swarm Optimization (PSO) algorithm are applied to tune the weights of sub-models. Though each sub-model can capture the nonlinear feature of the original system, by intersecting cut sets the synthesizing modeling scheme can further improve the performance of the proposed model. Some numerical examples show that the IDBN-NN with nonlinear output structure can achieve higher accuracy than some interval-valued data modeling methods. [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]

Fuzzy classifiers have been studied in the area of fuzzy sets for a long time resulting in a number of architectures. In this study, we thoroughly investigate and critically assess fuzzy rule-based classifiers. A topology of the classifier is discussed along with a discussion of the role of fuzzy set technology in the construction of condition and conclusion parts of the classification rules. Some optimization mechanisms utilized in the adjustment of information granules forming the rules are presented. Performance of the fuzzy classifiers is quantified in terms of their accuracy and an area under curve ( AUC ) determined for the receiver operating characteristics ( ROC ). The performance of the classifier is evaluated vis-à-vis a collection of triangular norms used in the construction of the fuzzy classifiers. Experimental studies involve synthetic and publicly available data. Furthermore, comparative studies include the experiments with the commonly used non-fuzzy classifiers. [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]

Granular computing (GrC) has emerged as a unified conceptual and processing framework. Information granules are fundamental constructs that permeate concepts and models of GrC. This paper is concerned with a design of a collection of meaningful, easily interpretable ellipsoidal information granules with the use of the principle of justifiable granularity by taking into consideration reconstruction abilities of the designed information granules. The principle of justifiable granularity supports designing of information granules based on numeric or granular evidence, and aims to achieve a compromise between justifiability and specificity of the information granules to be constructed. A two-stage development strategy behind the construction of justifiable information granules is considered. First, a collection of numeric prototypes is determined with the use of fuzzy clustering. Second, the lengths of the semi-axes of ellipsoidal information granules to be formed around such prototypes are optimized. Two optimization criteria are introduced and studied. Experimental studies involving synthetic data set and data sets coming from the machine learning repository are reported. [ABSTRACT FROM PUBLISHER]

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]

Fuzzy rule-based models have been widely used due to their interpretability and effectiveness. However, they still encounter challenges when dealing with multivariable and large-scale data. In this study, we first propose a novel approach to establish a selective sampling and mapping data reduction method. The method focuses on reducing data variables while decreasing the number of samples, and an appropriate scaling size can be chosen for different situations. Then, a multivariable data-driven fuzzy rule-based model is developed based on the processed data. Moreover, the data projection approach using the distance metric helps to preserve the structural characteristics of the original data. The results are visually presented to facilitate an interpretable description of the subsequent rule-based modeling. Furthermore, due to the inevitable inaccuracy in the projection process of numeric modeling, we introduce the allocation of information granularity to extend the model to a granular form at a more abstract level. Experimental studies on both synthetic and publicly available datasets demonstrate that the proposed method has superior effectiveness and efficiency compared to the existing state-of-the-art regression algorithms. • We propose a novel selective sampling and mapping data reduction (SSMDR) method to exploit the data structure relationship. • We exploit the advantages of the SSMDR method to handle multivariable data-driven fuzzy modeling problems. • We extend the numerical fuzzy rule-based model to a higher-level granular model that can handle potential information loss. • We develop a comprehensive visual interpretable framework to achieve interpretable analysis in terms of three aspects: visualization structure, interpretable reasoning process, and granular structure. [ABSTRACT FROM AUTHOR]

This study elaborates on a comprehensive design methodology of fuzzy cognitive maps (FCMs). Here, the maps are regarded as a modeling vehicle of time series. It is apparent that whereas time series are predominantly numeric, FCMs are abstract constructs operating at the level of abstract entities referred to as concepts and represented by the individual nodes of the map. We introduce a mechanism to represent a numeric time series in terms of information granules constructed in the space of amplitude and change of amplitude of the time series, which, in turn, gives rise to a collection of concepts forming the corresponding nodes of the FCMs. Each information granule is mapped onto a node (concept) of the map. We identify two fundamental design phases of FCMs, namely 1) formation of information granules mapping numeric data (time series) into activation levels of information granules (viz., the nodes of the map), and 2) optimization of information granules at the parametric level, viz., learning (estimating) the weights between the nodes of the map. The learning is typically realized in a supervised mode on a basis of some experimental data. A construction of information granules is realized with the aid of fuzzy clustering, namely fuzzy C-means. The optimization is realized with the use of particle swarm optimization. The proposed approach is illustrated in detail by a series of experiments using a collection of publicly available data. [ABSTRACT FROM PUBLISHER]

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 concerned with a granular data description in which we propose a characterization of numeric data by a collection of information granules so that the key structure of the data, their topology and essential relationships are described in the form of a family of fuzzy sets – information granules. A comprehensive design process is introduced in which we show a two-phase development strategy: first, numeric prototypes are built with the use of Fuzzy C-Means (FCM) that is followed by their augmentation resulting in a collection of information granules. In the design of information granules we engage the fundamental ideas of Granular Computing, especially the principle of justifiable granularity. A series of experiments is presented to visualize the key steps of the construction of information granules. [ABSTRACT FROM AUTHOR]

This study is concerned with a fundamental problem of expanding (refining) information granules being treated as functional entities playing a pivotal role in Granular Computing and ensuing constructs such as granular models, granular classifiers, and granular predictors. We formulate a problem of refinement of information granules as a certain optimization task in which a selected information granule is refined into a family of more detailed (precise, viz. more specific) information granules so that a general partition requirement becomes satisfied. As the ensuing information granules are directly linked with the more general information granule positioned at the higher level of hierarchy, the partition criterion is conditional by being implied (conditioned) by the description of the granule positioned one level up in the hierarchy. A criterion guiding a refinement of information granules is formulated and made fully reflective of the nature of the problem (being of regression-like or of classification character), which leads to a distinct way in which the diversity of information granules is articulated and quantified. With regard to the detailed algorithmic setting, we discuss the use of a so-called conditional Fuzzy C-Means and show how information granules (fuzzy sets) are formed in a successive manner. The method helps highlight the ensuing calculations of the resulting membership functions and reveal how the detailed structure of the data is captured. A number of numeric studies in the realm of system modeling are provided to demonstrate the performance of the approach and highlight the nature of the resulting information granules along with the performance of the fuzzy models in which these information granules are used. [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]

In spatiotemporal data commonly encountered in geographical systems, biomedical signals, and the like, each datum is composed of features comprising a spatial component and a temporal part. Clustering of data of this nature poses challenges, especially in terms of a suitable treatment of the spatial and temporal components of the data. In this study, proceeding with the objective function-based clustering (such as, e.g., fuzzy C-means), we revisit and augment the algorithm to make it applicable to spatiotemporal data. An augmented distance function is discussed, and the resulting clustering algorithm is provided. Two optimization criteria, i.e., a reconstruction error and a prediction error, are introduced and used as a vehicle to optimize the performance of the clustering method. Experimental results obtained for synthetic and real-world data are reported. [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]

Abstract: Cognitive maps and fuzzy cognitive maps offer interesting and transparent modeling capabilities by functioning at a level of conceptual entities (nodes) and their relationships expressed either at the qualitative level of excitatory/inhibitory relationships or being further numerically quantified as encountered in fuzzy cognitive maps. While there has been a vast array of conceptual enhancements, a relatively less attention has been paid to the design of the maps especially when dealing with an algorithmic way of forming the map. The objective of this study is to offer a design strategy in which starting with experimental evidence in the form of numeric data, those data are transformed into a finite and small number of concepts (nodes) of the map and afterwards the connections of the map are estimated. We show that techniques of Granular Computing, especially fuzzy clustering are effectively used to form concepts (nodes) of well-articulated semantics. In the sequel, we show the use of global optimization in the form of Particle Swarm Optimization (PSO) to carry out calibration of the connections of the interrelationships between the nodes of the map. Numeric examples are concerned with the representation of time series and their visualization in the form of fuzzy cognitive maps. Further interpretation of the maps is also discussed. [Copyright &y& Elsevier]

Abstract: Fuzzy clustering being focused on the discovery of structure in multivariable data is of relational nature in the sense of not distinguishing between the natures of the individual variables (features) encountered in the problem. In this study, we revisit the generic approach to clustering by studying situations in which there are families of features of descriptive and functional nature whose semantics needs to be incorporated into the clustering algorithm. While the structure is determined on the basis of all features taken en-block, it is anticipated that the topology revealed in this manner would aid the effectiveness of determining values of functional features given the vector of the corresponding descriptive features. We propose an augmented distance in which the families of descriptive and predictive features are distinguished through some weighted version of the distance between patterns. The optimization of this distance is guided by a reconstruction criterion, which helps minimize the reconstruction error between the original vector of functional features and their reconstruction realized by means of descriptive features. Experimental results are offered to demonstrate the performance of the clustering and quantify the effect of reaching balance between semantically distinct families of features. [ABSTRACT FROM AUTHOR]

Abstract: In this study, we present a comprehensive comparative analysis of kernel-based fuzzy clustering and fuzzy clustering. Kernel based clustering has emerged as an interesting and quite visible alternative in fuzzy clustering, however, the effectiveness of this extension vis-à-vis some generic methods of fuzzy clustering has neither been discussed in a complete manner nor the performance of clustering quantified through a convincing comparative analysis. Our focal objective is to understand the performance gains and the importance of parameter selection for kernelized fuzzy clustering. Generic Fuzzy C-Means (FCM) and Gustafson–Kessel (GK) FCM are compared with two typical generalizations of kernel-based fuzzy clustering: one with prototypes located in the feature space (KFCM-F) and the other where the prototypes are distributed in the kernel space (KFCM-K). Both generalizations are studied when dealing with the Gaussian kernel while KFCM-K is also studied with the polynomial kernel. Two criteria are used in evaluating the performance of the clustering method and the resulting clusters, namely classification rate and reconstruction error. Through carefully selected experiments involving synthetic and Machine Learning repository (http://archive.ics.uci.edu/beta/) data sets, we demonstrate that the kernel-based FCM algorithms produce a marginal improvement over standard FCM and GK for most of the analyzed data sets. It has been observed that the kernel-based FCM algorithms are in a number of cases highly sensitive to the selection of specific values of the kernel parameters. [Copyright &y& Elsevier]

Abstract: Ordinal feature values are totally ordered labels that can be considered as fuzzy sets. The formulation of proper fuzzy sets for ordinal labels is important for the systems that deal with the objects of mixed feature types. When a proper ordinal–numerical mapping for an ordinal feature of interest is given, proper fuzzy sets for the labels of the ordinal feature can be easily formulated. In this paper, we propose an adaptive method to learn proper ordinal–numerical mappings for ordinal features of interest from a given objects of mixed features including the ordinal features. The method starts with uniform ordinal–numerical mappings, and performs two steps iteratively. The first step computes a fuzzy partition over the given object set with the ordinal–numerical mappings. The second step learns new ordinal–numerical mappings from the new fuzzy partition in the way that the new mappings make the similarity between two ordinal labels be similar to the average similarity between the objects having the two labels, respectively. Through the alternate repetition of the two steps, both of the ordinal–numerical mappings and the clustering quality become gradually improved. The validity of the proposed method is strongly supported through the experiments with a modified fuzzy C-means clustering algorithm in which the proposed method is implemented. [Copyright &y& Elsevier]

Polynomial neural networks have been known to exhibit useful properties as classifiers and universal approximators. In this study, we introduce a concept of polynomialbased radial basis function neural networks (P-RBF NNs), present a design methodology and show the use of the networks in classification problems. From the conceptual standpoint, the classifiers of this form can be expressed as a collection of "if-then" rules. The proposed architecture uses two essential development mechanisms. Fuzzy clustering (Fuzzy C-Means, FCM) is aimed at the development of condition parts of the rules while the corresponding conclusions of the rules are formed by some polynomials. A detailed learning algorithm for the P-RBF NNs is developed. The proposed classifier is applied to two-class pattern classification problems. The performance of this classifier is contrasted with the results produced by the "standard" RBF neural networks. In addition, the experimental application covers a comparative analysis including several previous commonly encountered methods such as standard neural networks, SVM, SOM, PCA, LDA, C4.5, and decision trees. The experimental results reveal that the proposed approach comes with a simpler structure of the classifier and better prediction capabilities. [ABSTRACT FROM AUTHOR]

Abstract: Frequency-based cluster prototypes have been used to cluster categorical objects, based on the simple matching dissimilarity measure. This paper introduces a new generalization called fuzzy p-mode prototype, of frequency-based prototypes. A fuzzy p-mode cluster prototype at a categorical feature is expressed as a list of p labels that have larger frequencies than others in the cluster. This paper also presents a new generalization of the fuzzy C-means clustering algorithm for the objects of mixed features. In the general fuzzy C-means clustering algorithm, any dissimilarity measures at the categorical feature level are assumed, not like other clustering algorithms that use the simple matching dissimilarity. The convergence of the general fuzzy C-means clustering algorithm under the optimization framework is proved. It is also explained through experiments over real object sets that the size of fuzzy p-mode prototypes and the fuzzification coefficients affect clustering performance. [Copyright &y& Elsevier]

Multiagent systems are inherently associated with their distributivity, which enforces a great deal of communication mechanisms. To effectively arrive at meaningful solutions in a vast array of problem-solving tasks, it becomes imperative to establish a sound machinery of reconciling findings which might form partial solutions to an overall problem. In this paper, we focus on a broad category of problems of collaborative data analysis realized by a collection of agents having access to their individual data and exchanging findings through their collaboration activities. Such problems of data analysis arise in the context of building a global view at a certain phenomenon (process) by viewing it from different perspectives (and thus engaging various collections of attributes by various agents). Our goal is to develop some interaction between the agents so that they could form an overall perspective, where the knowledge available locally is shared and reconciled. The underlying format of knowledge built by the agents is that of information granules and fuzzy sets in particular. We develop a comprehensive optimization scheme and discuss its two-phase nature in which the communication phase of the granular findings intertwines with the local optimization being realized by the agents at the level of the individual datasite and exploits the evidence collected from other sites. We show how the mechanism of fuzzy granulation realized in the form of a well-known fuzzy c-means (FCM) clustering can be augmented to support collaborative activities required by the agents. For this purpose, we introduce augmented versions of the original objective function used in the FCM and derive algorithmic details. We also discuss an issue of optimizing the strength of collaborative linkages, so that the reconciled findings attain the highest level of consistency (agreement) The presented experimental studies include some synthetic data and selected data sets coming from the Machine Learning repository. [ABSTRACT FROM AUTHOR]

Abstract: In this study, we develop a concept of dynamic data granulation realized in presence of incoming data organized in the form of so-called data snapshots. For each of these snapshots we reveal a structure by running fuzzy clustering. The proposed algorithm of adjustable fuzzy C-means (FCM) exhibits a number of useful features which directly associate with the dynamic nature of the underlying data: (a) the number of clusters is adjusted from one data snapshot to another in order to capture the varying structure of patterns and its complexity, (b) continuity between the consecutively discovered structures is retained, viz the clusters formed for a certain data snapshot are constructed as a result of evolving the clusters discovered in the predeceasing snapshot. We present a detailed clustering algorithm in which the mechanisms of adjustment of information granularity (the number of clusters) become the result of solutions to well-defined optimization tasks. The cluster splitting is guided by conditional fuzzy C-means (FCM) while cluster merging involves two neighboring prototypes. The criterion used to control the level of information granularity throughout the process is guided by a reconstruction criterion which quantifies an error resulting from pattern granulation and de-granulation. Numeric experiments provide a suitable illustration of the approach. [Copyright &y& Elsevier]

Multiagent systems are inherently associated with their distributivity, which enforces a great deal of communication mechanisms. To effectively arrive at meaningful solutions in a vast array of problem-solving tasks, it becomes imperative to establish a sound machinery of reconciling findings which might form partial solutions to an overall problem. In this paper, we focus on a broad category of problems of collaborative data analysis realized by a collection of agents having access to their individual data and exchanging findings through their collaboration activities. Such problems of data analysis arise in the context of building a global view at a certain phenomenon (process) by viewing it from different perspectives (and thus engaging various collections of attributes by various agents). Our goal is to develop some interaction between the agents so that they could form an overall perspective, where the knowledge available locally is shared and reconciled. The underlying format of knowledge built by the agents is that of information granules and fuzzy sets in particular. We develop a comprehensive optimization scheme and discuss its two-phase nature in which the communication phase of the granular findings intertwines with the local optimization being realized by the agents at the level of the individual datasite and exploits the evidence collected from other sites. We show how the mechanism of fuzzy granulation realized in the form of a well-known fuzzy c-means (FCM) clustering can be augmented to support collaborative activities required by the agents. For this purpose, we introduce augmented versions of the original objective function used in the FCM and derive algorithmic details. We also discuss an issue of optimizing the strength of collaborative linkages, so that the reconciled findings attain the highest level of consistency (agreement). The presented experimental studies include some synthetic data and selected data sets coming from the Machine Learning repository. [ABSTRACT FROM AUTHOR]

In a Web-oriented society, organization, retrieval, and classification of digital images have become one of the major endeavors. In this paper, we study the mechanisms of fuzzy clustering and fuzzy clustering with partial supervision in the analysis and classification of images. It is demonstrated that the main features of fuzzy clustering become essential in revealing the structure in a collection of images and supporting their classification. The discussed operational framework of fuzzy clustering is realized by means of fuzzy c-means (FCM). When dealing with the mode of partial supervision, we augment an original objective function guiding the clustering process by an additional component expressing a level of coincidence between the membership degrees produced by the FCM and class allocation supplied by the user(s). The study also contrasts the use of the technology of fuzzy sets in image clustering with other approaches studied in this area. A suite of experiments deals with two collections of images, namely, Columbia object image library (COIL-20) and a database composed of 2000 outdoor images. [ABSTRACT FROM AUTHOR]

Abstract: Vector quantization (VQ) is a fundamental and omnipresent mechanism of data compression with various conceptual underpinnings and diversified algorithmic realizations. The objective of this study is to investigate the concept of VQ in the setting of fuzzy sets by forming a coherent algorithmic framework referred to as a fuzzy VQ (FVQ). Given the nature of the framework of VQ in which fuzzy sets are involved, we may refer to the discussed processes of FVQ as a fuzzy granulation and fuzzy degranulation. In comparison to the winner-takes-all strategy encountered in VQ where a result of decoding typically arises as a single element of the codebook, in the FVQ we exploit an efficient usage of all components of the codebook (fuzzy sets) in the reconstruction of the original data. In this study, we present a complete development scheme of the FVQ and elaborate on its essential features. Its main design phases involve: (a) an encoding in which we encode data in terms of the elements of the given codebook; (b) a decoding during which we reconstruct the original data; and (c) a development of the codebook. The mechanisms of encoding and decoding are created as a result of some well-formed optimization tasks. The buildup of the codebook is completed through a mechanism of global optimization realized in the form of the particle swarm optimization (PSO). We offer a collection of experiments using synthetic data by focusing on and quantifying the role of fuzzy sets in VQ. While FVQ outperforms VQ (which seems to be an intuitively appealing finding), we also show that this improvement could be achieved through a careful optimization of the elements of the granulation scheme. It is also shown that without optimization of the FVQ scheme, the enhancements could not be possible or may become very much limited. A series of experiments involving synthetic data and data sets coming from the Machine Learning repository is included as well. [Copyright &y& Elsevier]

Abstract: Information granules, such as e.g., fuzzy sets, capture essential knowledge about data and the key dependencies between them. Quite commonly, we may envision that information granules (fuzzy sets) have become a result of fuzzy clustering and therefore could be succinctly represented in the form of some fuzzy partition matrices. Interestingly, the same data set could be represented from various standpoints and this multifaceted view yields a collection of different partition matrices being reflective of the higher-order granular knowledge about the data. The levels of specificity of the clusters the data are organized into could be quite different—the larger the number of clusters, the more detailed insight into the structure of data becomes available. Given the granularity of the resulting constructs (rather than plain data themselves), one could view a collection of partition matrices as a certain type of a network of knowledge. Considering a variety of sources of knowledge encountered across the network, we are interested in forming consensus between them. In a nutshell, this leads to the construction of certain fuzzy partition matrices which “reconcile” the knowledge captured by the individual partition matrices. Given that the granularity of the sources of knowledge under consideration could vary quite substantially, we develop a unified optimization perspective by introducing fuzzy proximity matrices that are induced by the corresponding partition matrices. In the sequel, the optimization is realized on a basis of these proximity matrices. We offer a detailed algorithm and illustrate its performance using a series of numeric experiments. [Copyright &y& Elsevier]

Fuzzy clustering forms a cornerstone of fuzzy (granular) modeling. The clusters (prototypes) are viewed as a blueprint of the model that is further refined through a number of detailed estimation techniques. In this study, we claim that while clustering is indisputable essential to fuzzy modeling, the essence of clustering mechanisms supporting this process of information granulation is not compatible with the character of the task at hand. In modeling, the required constructs are inherently direction-sensitive (that is we clearly distinguish between input and output variables). On the other hand, fuzzy clustering is direction neutral and during the formation of the clusters does not take this into consideration. We re-formulate the clustering so that the directionality aspect can be addressed in the optimization process. This leads to a new, augmented objective function to be minimized. A detailed algorithm is derived. As the directional sensitivity of the clustering method gives rise to different numbers of clusters in the input and output space, it becomes necessary to identify a mapping between these clusters which in turn gives rise to some allocation problem. Because of its inherently combinatorial character, the proposed solution is obtained through some genetic optimization. Comprehensive experiments demonstrate the performance of the approach and compare it with some of the generic version of the FCM clustering. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new category of logic neurons-unineurons that are based on the concept of uninorms. As uninorms form a certain generalization of the generic categories of fuzzy set operators such as t-norms and t-conorms, the proposed unineurons inherit their logic processing capabilities which make them flexible and logically appealing. We discuss several fundamental categories of uninorms (such as UNI̱OR, UNI̱AND, and alike). In particular, we focus on the interpretability of networks composed of unineurons leading to several categories of rules to be exploited in rule-based systems. The learning aspects of the unineurons are presented along with detailed optimization schemes. Experimental results tackle two categories of problems such as: (a) a logic approximation of fuzzy sets, and (b) a design of associations between information granules where the ensuing development schemes directly relate to the fundamentals of granular (fuzzy) modeling. [ABSTRACT FROM AUTHOR]

Abstract: Given the rapidly growing diversity of techniques and applications of fuzzy clustering, an interpretation of grouping results becomes of paramount relevance. Fuzzy clusters offer a lot of detailed information about the structure in data by allocating patterns to clusters with numeric degrees of membership. While this information could be highly beneficial, its level of detail could be too overwhelming and in some sense somewhat detrimental to the formation of the global view of the structure. To establish some sound compromise between the qualitative Boolean (two-valued) description of data and quantitative membership grades, we introduce an interpretation framework of shadowed sets. Shadowed sets are discussed as three-valued constructs induced by fuzzy sets assuming three values (that could be interpreted as full membership, full exclusion, and uncertain). The algorithm of converting membership functions into this quantification is a result of a certain optimization problem guided by the principle of uncertainty localization. With the shadowed sets of clusters in place, discussed is a taxonomy of patterns leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure. [Copyright &y& Elsevier]

The article introduces a concept and design of decision trees based on information granules, multivariable entities characterized by high homogeneity. The article shows how the tree is grown depending on some additional node expansion criteria such as cardinality at a given node and a level of structural dependencies of data existing there. A series of experiments is reported using both synthetic and machine learning data sets. The C-decision trees are classification constructs that are built on a basis of information granules, fuzzy clusters.

In our study we presented an effective method for clustering of Web pages. From flat HTML files we extracted keywords, formed feature vectors as representation of Web pages and applied them to a clustering method. We took advantage of the Fuzzy C-Means clustering algorithm (FCM), We demonstrated an organized and schematic manner of data collection. Various categories of Web pages were retrieved from ODP (Open Directory Project) in order to create our datasets. The results of clustering proved that the method performs well for all datasets. Finally, we presented a comprehensive experimental study examining: the behavior of the algorithm for different input parameters, internal structure of datasets and classification experiments. [ABSTRACT FROM AUTHOR]

In this study, we introduce and study a proximity-based fuzzy clustering. As the name stipulates, in this mode of clustering, a structure “discovery” in the data is realized in an unsupervised manner and becomes augmented by a certain auxiliary supervision mechanism. The supervision mechanism introduced in this algorithm is realized via a number of proximity “hints” (constraints) that specify an extent to which some pairs of patterns are regarded similar or different. They are provided externally to the clustering algorithm and help in the navigation of the search through the set of patterns and this gives rise to a two-phase optimization process. Its first phase is the standard FCM while the second step is concerned with the gradient-driven minimization of the differences between the provided proximity values and those computed on a basis of the partition matrix computed at the first phase of the algorithm. The proximity type of auxiliary information is discussed in the context of Web mining where clusters of Web pages are built in presence of some proximity information provided by a user who assesses (assigns) these degrees on a basis of some personal preferences. Numeric studies involve experiments with several synthetic data and Web data (pages). [Copyright &y& Elsevier]