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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

This study is concerned with clustering carried out in presence of labeled patterns. An objective of this optimization is to reconcile between the structure residing in data (and being primarily discovered by the underlying clustering mechanism) and the labels of the patterns forming such structure. In this sense, one can consider the supervised fuzzy clustering to be a framework of preliminary data analysis providing with a thorough insight into the structure of the data and supporting the ensuing design of detailed classifiers. The proposed method augments the standard fuzzy C-means algorithm by extending the original objective function by the supervision component (labeled patterns). Experimental results illustrate the approach and discuss the use of this type of clustering in vector quantization. [Copyright &y& Elsevier]

Fuzzy clustering becomes a broadly accepted synonym of fundamental endeavors aimed at finding structures in multidimensional data. In essence, these methods operate in unsupervised mode. This means that they act upon data while being directed by some predefined objective function (criterion) for which they “discover” a structure (clusters) that yields a minimal value of this criterion. In this study, we discuss an issue of exploiting and effectively incorporating auxiliary problem dependent hints being available as a part of the domain knowledge associated with the pattern recognition problem at hand. As such hints are usually expressed by experts/data analysts at the level of clusters (information granules) rather than individual data (patterns), we refer to them as knowledge-based indicators and allude to a set of them as a knowledge-based guidance available to fuzzy clustering. The proposed paradigm shift in which fuzzy clustering incorporates this type of knowledge-based supervision is discussed and contrasted with the “pure” (that is data-driven) version of fuzzy clustering. Several fundamental categories of the guidance mechanisms are introduced and discussed, namely partial supervision, proximity-based guidance and uncertainty driven knowledge hints. The details on how the guidance machinery translates into updates of the partition matrices are presented. We also present a number of practical scenarios in which the role of knowledge hints becomes evident and highly justifiable. This concerns Web exploration, exploitation of labeled patterns, issues of incomplete feature spaces, and constraints of typicality of patterns, to name a few representative applications. [Copyright &y& Elsevier]

In this study, we introduce a new clustering architecture in which several subsets of patterns can be processed together with an objective of finding a structure that is common to all of them. To reveal this structure, the clustering algorithms operating on the separate subsets of data collaborate by exchanging information about local partition matrices. In this sense, the required communication links are established at the level of information granules (more specifically, fuzzy sets forming the partition matrices) rather than patterns that are directly available in the databases. We discuss how this form of collaboration helps meet requirements of data confidentiality. A detailed clustering algorithm is developed on a basis of the standard FCM method and illustrated by means of numeric examples. [Copyright &y& Elsevier]

The paper is concerned with a logic-based expansion of the standard FCM clustering. The proposed algorithm captures the logic fabric of the structure in a dataset by describing it in the form of a union of the clusters (that is fuzzy relations) determined by the clustering algorithm. In contrast to the standard FCM, the elements (clusters) are combined together as a union of such fuzzy relations—clusters and this form of combination arises as a constraint in the clustering method. In this sense, the introduced clustering environment gives rise to the clustering that is regarded as a logic-driven data decomposition. A detailed algorithm is presented along with some illustrative examples. [Copyright &y& Elsevier]

Granular Computing (GrC) is a human-centric way to discover the fundamental structure of data sets. The resulting information granules can be efficiently exploited to organize knowledge and reveal data descriptions, which can play a pivotal role in the classification problems. Furthermore, information granules are abstract collections of data entities and exhibit flexibility and tolerance when it comes to the representation of incomplete data. However, most of the existing methods focused on the data imputation and classification separately. They also require better interpretability. The crux of this study is to develop a novel information granule-based classification method for incomplete data and a way of representing missing entities and regarding them as information granules in a unified framework. The first aspect focuses on revealing the structural backbone of multiple labeled subspaces of data by fuzzy clustering of missing values. It emerges a classifier with interpretable "IF-THEN" rules by the refinement of fuzzy prototypes in a supervised mode to capture the critical relationship of the multi-class incomplete data. The second aspect concerns the construction of some information granules to impute and represent missing values according to the refined prototypes and classification findings. The experimental studies involved synthetic and publicly available datasets in quantifying the advantages of the classification and representation abilities of the proposed methods on incomplete data. • We proposed a novel information granule-based classifier to reveal the structural of the subspaces of data, which is easier to be interpreted. • We considered incomplete data among classification modeling, so that this classifier can deal with the incomplete data straightforwardly. • We present a refinement mechanism for the information granule-based classifier to optimize the prototypes of the classification rules. • As a byproduct of the classification, the imputed information granules are distinguished from present data and have more tolerance to the imputation error. [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]

Deoxyribonucleic acid (DNA) microarray is an important technology, which supports a simultaneous measurement of thousands of genes for biological analysis. With the rapid development of the gene expression data characterized by uncertainty and being of high dimensionality, there is a genuine need for advanced processing techniques. With this regard, Fuzzy Possibilistic C-Means Clustering (FPCM) and Granular Computing (GrC) are introduced with the aim to solve problems of feature selection and outlier detection. In this study, by taking advantage of the FPCM and GrC, an Advanced Fuzzy Possibilistic C-Means Clustering based on Granular Computing (GrFPCM) is proposed to select features as a preprocessing phase for clustering problems while the developed granular space is used to cope with uncertainty. Experiments were completed for various gene expression datasets and a comparative analysis is reported. [ABSTRACT FROM AUTHOR]

Data clustering is aimed at discovering a structure in data. The revealed structure is usually represented in terms of prototypes and partition matrices. In some cases, the prototypes are simultaneously formed using data and features by running a co-clustering (bi-clustering) algorithm. Interval valued fuzzy clustering exhibits advantages when handling uncertainty. This study introduces a novel clustering technique by combining fuzzy co-clustering approach and interval-valued fuzzy sets in which two values of the fuzzifier of the fuzzy clustering algorithm are used to form the footprint of uncertainty (FOU). The study demonstrates the performance of the proposed method through a series of experiments completed for various datasets (including color segmentation, multi-spectral image classification, and document categorization). The experiments quantify the quality of results with the aid of validity indices and visual inspection. Some comparative analysis is also covered. [ABSTRACT FROM AUTHOR]

Abstract: In this article, a proximity fuzzy framework for clustering relational data is presented, where the relationships between the entities of the data are given in terms of proximity values. We offer a comprehensive and in-depth comparison of our clustering framework with proximity relational knowledge to clustering with distance relational knowledge, such as the well known relational Fuzzy C-Means (FCM). We conclude that proximity can provide a richer description of the relationships among the data and this offers a significant advantage when realizing clustering. We further motivate clustering relational proximity data and provide both synthetic and real-world experiments to demonstrate both the usefulness and advantage offered by clustering proximity data. Finally, a case study of relational clustering is introduced where we apply proximity fuzzy clustering to the problem of clustering a set of trees derived from software requirements engineering. The relationships between trees are based on the degree of closeness in both the location of the nodes in the trees and the semantics associated with the type of connections between the nodes. [Copyright &y& Elsevier]

Abstract: In this study, we design polynomial-based radial basis function neural networks (P-RBF NNs) based on a fuzzy inference mechanism. The essential design parameters (including learning rate, momentum coefficient and fuzzification coefficient of the underlying clustering method) are optimized by means of the particle swarm optimization. The proposed P-RBF NNs dwell upon structural findings about training data that are expressed in terms of a partition matrix resulting from fuzzy clustering in this case being the fuzzy C-means (FCM). The network is of functional nature as the weights between the hidden layer and the output are some polynomials. The use of the polynomial weights becomes essential in capturing the nonlinear nature of data encountered in regression or classification problems. From the perspective of linguistic interpretation, the proposed network can be expressed as a collection of “if–then” fuzzy rules. The architecture of the networks discussed here embraces three functional modules reflecting the three phases of input–output mapping realized in rule-based architectures, namely condition formation, conclusion creation, and aggregation. The proposed classifier is applied to some synthetic and machine learning datasets, and its results are compared with those reported in the previous studies. [Copyright &y& Elsevier]

Abstract: Several fuzzy c-means based clustering techniques have been developed to tackle many problems in a number of areas such as pattern recognition, image analysis, communication, data mining. Among all, a common use of such a class of clustering algorithms is in the training of radial basis function neural networks (RBFNNs). In this paper, we describe a novel approach to fuzzy clustering, which organizes the data in clusters on the basis of the input data and a ‘prototype’ regression function built, in the output space, as a summation of a number of linear local regression models. This methodology is shown to be effective in the training of RBFNNs leading to improved performance with respect to other clustering algorithms. [Copyright &y& Elsevier]

Missing values are a common phenomenon when dealing with real-world data sets. Analysis of incomplete data sets has become an active area of research. In this paper, we focus on the problem of clustering incomplete data, which is intended to introduce some prior distribution information of the missing values into the algorithm of fuzzy clustering. First, non-parametric hypothesis testing is employed to describe the missing values adhering to a certain Gaussian distribution as probabilistic information granules based on the nearest neighbors of incomplete data. Second, we propose a novel clustering model, in which probabilistic information granules of missing values are incorporated into the Fuzzy C-Means clustering of incomplete data by involving the maximum likelihood criterion. Third, the clustering model is optimized by using a tri-level alternating optimization utilizing the method of Lagrange multipliers. The convergence and the time complexity of the clustering algorithm are also discussed. The experiments reported both on synthetic and real-world data sets demonstrate that the proposed approach can effectively realize clustering of incomplete data. [ABSTRACT FROM AUTHOR]

In this study, kernel interval-valued Fuzzy C-Means clustering (KIFCM) and multiple kernel interval-valued Fuzzy C-Means clustering (MKIFCM) are proposed. The KIFCM algorithm is built on a basis of the kernel learning method and the interval-valued fuzzy sets with intent to overcome some drawbacks existing in the “conventional” Fuzzy C-Means (FCM) algorithm. The development of the method is motivated by two factors. First, uncertainty is inherent in clustering problems due to some information deficiency, which might be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, etc. With this regard, interval-valued fuzzy sets exhibit advantages when handling such aspects of uncertainty. Second, kernel methods form a new class of pattern analysis algorithms which can cope with general types of data and detect general types of relations (geometric properties) by embedding input data in a vector space based on the inner products and looking for linear relations in the space. However, as the clustering problems may involve various input features exhibiting different impacts on the obtained results, we introduce a new MKIFCM algorithm, which uses a combination of different kernels (giving rise to a concept of a composite kernel). The composite kernel was built by mapping each input feature onto individual kernel space and linearly combining these kernels with the optimized weights of the corresponding kernel. The experiments were completed for several well-known datasets, land cover classification from multi-spectral satellite image and Multiplex Fluorescent In Situ Hybridization (MFISH) classification problem. The obtained results demonstrate the advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The Social Web offers increasingly simple ways to publish and disseminate personal or opinionated information, which can rapidly exhibit a disastrous influence on the online reputation of organizations. Based on social Web data, this study describes the building of an ontology based on fuzzy sets. At the end of a recurring harvesting of folksonomies by Web agents, the aggregated tags are purified, linked, and transformed to a so-called fuzzy grassroots ontology by means of a fuzzy clustering algorithm. This self-updating ontology is used for online reputation analysis, a crucial task of reputation management, with the goal to follow the online conversation going on around an organization to discover and monitor its reputation. In addition, an application of the Fuzzy Online Reputation Analysis (FORA) framework, lessons learned, and potential extensions are discussed in this article. [ABSTRACT FROM AUTHOR]