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In this article, K-meansclustering-based Kernel canonical correlation analysis algorithm is proposed for multimodal emotion recognition in human–robot interaction (HRI). The multimodal features (gray pixels; time and frequency domain) extracted from facial expression and speech are fused based on Kernel canonical correlation analysis. K-means clustering is used to select features from multiple modalities and reduce dimensionality. The proposed approach can improve the heterogenicity among different modalities and make multiple modalities complementary to promote multimodal emotion recognition. Experiments on two datasets, namely SAVEE and eNTERFACE‘05, are conducted to evaluate the accuracy of the proposed method. The results show that the proposed method produces good recognition rates that are higher than the ones produced by the methods without K-means clustering; more specifically, they are 2.77% higher in SAVEE and 4.7% higher in eNTERFACE‘05. [ABSTRACT FROM AUTHOR]

The generation of collective preference assessments occupies a critical position in deriving accurate and reliable alternative rankings in the context of large-scale group decision making (LSGDM). In general, the collective opinion generation framework entails the following three phases, which are clustering analysis, weighting clusters, and preference aggregation. However, the clustering of experts has been frequently based on preference similarities among them without taking into account individual opinions in which knowledge elicitation plays a crucial role. The traditional collective opinion generation framework suffering from this drawback may result in unreliable decision outcomes. To this end, we propose an expertise-structure and risk-appetite-integrated two-tiered collective opinion generation framework to address this concern. The first tier of the two-tiered collective opinion generation framework divides the entire expert group into several subgroups based on individual expertise structures, which are extracted from hesitant fuzzy linguistic term set (HFLTS)-based preference assessments, and it then weighs the resulting clusters in accordance with the overall expertise levels. The second-tier clusters the first-tier subgroups conditioned on the indicator of individual assessment similarities and gathers the generated subgroup preference constructs in the use of the risk appetite-oriented power average operator. In addition, the notion of proportional HFLTSs was introduced to manifest collective evaluations in second-tier subgroups to eliminate information loss and distortion. The effectiveness and flexibility of the proposed collective opinion generation algorithm are eventually illustrated by a case study and a comparison analysis. [ABSTRACT FROM AUTHOR]

In time-series forecasting, it is an important task to make an accurate and interpretable long-term prediction. In this article, we present a novel approach developed from the perspective of granular computing (GrC) to realize the long-term prediction of time series. The proposed method first employs a sliding window strategy to smooth on the raw time series. Subsequently, the smoothed time series is transformed into the corresponding granular time series that is depicted by evolving shape with the aid of the clustering algorithm based on the dynamic time warping (DTW) distance. Finally, a Takagi–Sugeno (TS) architecture-like granular model (GrM) is formed by deriving the relations implying in the granular time series and offers the granular output in the numeric vector format. The GrM adopts the pattern-to-pattern inference mechanism to realize the long-term prediction of time series at the vector level. Experiments on several datasets demonstrate that the proposed method not only has the ability to circumvent the cumulative error but also makes the resulting GrM equip better interpretability. [ABSTRACT FROM AUTHOR]

In this article, a reinforced iterative fuzzy radial basis function neural networks (IFRBFNN) is introduced as an augmented FRBFNN architecture generated through the iterative refinement process of weighted fuzzy C-means (WFCM) clustering and weighted least square error estimation. It is well known that the location of the clusters have an effect on the classification performance of the FRBFNN. The underlying idea behind this article is how to define the center points of clusters based on the data distribution analysis as well as the improvement of classification performance. In a nutshell, while the fuzzy C-means clustering to define the positions of radial basis functions (RBFs) based in the unsupervised learning manner is usually used in a FRBFNN, the parameters (cluster centers and their ensuing polynomial coefficients) refinement of FRBFNN through the proposed iterative method lead to superb classification performance by relocating the positions of RBFs over the input space implied by supervised learning. The idea of the proposed approach is to relocate the centers (prototypes) of the fuzzy clusters by using WFCM clustering algorithm and to re-estimate the coefficients by weighted least square estimation with the aid of the cross-entropy loss values of data so that the classification performance is improved. The weight related to each data is determined by some auxiliary information (i.e., the modified version of the cross-entropy loss value of each data). When it comes to the estimation of the coefficients of the consequent polynomial in IFRBFNN, they are estimated by the weighted least square error estimation technique, where the weights of the coefficients are defined based on the cross-entropy loss values. The auxiliary information such as the weights for relocation of the centers of RBFs and the weights for the re-estimation of the polynomial coefficients should be defined from the viewpoint of enhancement of the classification performance. The cross-entropy loss value of each data is able to meet the necessity of the required supervision signal. Several numerical experiments are provided to demonstrate the usefulness of the proposed design method for the FRBFNN for classification problems. [ABSTRACT FROM AUTHOR]

Information granules can discover interpretable and meaningful relationships offering a full description for time series. This article presents an information granulation method with rectangular information granules and applies it to time-series similarity measurement. First, the fuzzy $c$ -means clustering algorithm transforms the time series and its first-order difference time series to data clusters. With the maximum volume of rectangular information granules viewed as the criterion, the optimal rectangular information granules are formed using the data cluster by the principle of justifiable granularity and the gravitational search algorithm. The time-series similarity is measured by calculating the similarity between the upper and lower bounds of the optimal rectangular information granules built from the time series. Finally, an experiment is performed on a public dataset to verify the feasibility of the proposed method. The result shows that the rectangular information granulation method can capture the change characteristics of time series. The similarity measurement method can effectively evaluate the similarity of the time series belonging to different classes. [ABSTRACT FROM AUTHOR]

In this article, we elaborate on a Kullback–Leibler (KL) divergence-based Fuzzy ${C}$ -Means (FCM) algorithm by incorporating a tight wavelet frame transform and morphological reconstruction (MR). To make membership degrees of each image pixel closer to those of its neighbors, a KL divergence term on the partition matrix is introduced as a part of FCM, thus resulting in KL divergence-based FCM. To make the proposed FCM robust, a filtered term is augmented in its objective function, where MR is used for image filtering. Since tight wavelet frames provide redundant representations of images, the proposed FCM is performed in a feature space constructed by tight wavelet frame decomposition. To further improve its segmentation accuracy (SA), a segmented feature set is reconstructed by minimizing the inverse process of its objective function. Each reconstructed feature is reassigned to the closest prototype, thus modifying abnormal features produced in the reconstruction process. Moreover, a segmented image is reconstructed by using tight wavelet frame reconstruction. Finally, supporting experiments coping with synthetic, medical, and real-world images are reported. The experimental results exhibit that the proposed algorithm works well and comes with better segmentation performance than other peers. In a quantitative fashion, its average SA improvements over its peers are 4.06%, 3.94%, and 4.41%, respectively, when segmenting synthetic, medical, and real-world images. Moreover, the proposed algorithm requires less time than most of the FCM-related algorithms. [ABSTRACT FROM AUTHOR]

Information granulation and degranulation play a fundamental role in granular computing (GrC). Given a collection of information granules (referred to as reference information granules), the essence of the granulation process (encoding) is to represent each data (either numeric or granular) in terms of these reference information granules. The degranulation process (decoding) that realizes the reconstruction of original data is associated with a certain level of reconstruction error. An important issue is how to reduce the reconstruction error such that the data could be reconstructed more accurately. In this study, the granulation process is realized by involving fuzzy clustering. A novel neural network is leveraged in the consecutive degranulation process, which could help significantly reduce the reconstruction error. We show that the proposed degranulation architecture exhibits improved capabilities in reconstructing original data in comparison with other methods. A series of experiments with the use of synthetic data and publicly available datasets coming from the machine-learning repository demonstrates the superiority of the proposed method over some existing alternatives. [ABSTRACT FROM AUTHOR]

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

In this article, we are concerned with the formation of type-2 information granules in a two-stage approach. We present a comprehensive algorithmic framework which gives rise to information granules of a higher type (type-2, to be specific) such that the key structure of the local granular data, their topologies, and their diversities become fully reflected and quantified. In contrast to traditional collaborative clustering where local structures (information granules) are obtained by running algorithms on the local datasets and communicating findings across sites, we propose a way of characterizing granular data (formed) by forming a suite of higher type information granules to reveal an overall structure of a collection of locally available datasets. Information granules built at the lower level on a basis of local sources of data are weighted by the number of data they represent while the information granules formed at the higher level of hierarchy are more abstract and general, thus facilitating a formation of a hierarchical description of data realized at different levels of detail. The construction of information granules is completed by resorting to fuzzy clustering algorithms (more specifically, the well-known Fuzzy C-Means). In the formation of information granules, we follow the fundamental principle of granular computing, viz., the principle of justifiable granularity. Experimental studies concerning selected publicly available machine-learning datasets are reported. [ABSTRACT FROM AUTHOR]

This article proposes a customized kernel-based fuzzy C-means (CKFCM) clustering, which provides an original framework for data interpretation and data analysis and delivers an effective solution for an accurate division of actual run data under multiple operating conditions in an iron ore sintering process (IOSP). The improvement of CKFCM clustering is achieved by including an adjustment factor introduced into the kernel-based fuzzy C-means clustering. The adjustment factor only needs to consider a small amount of labeled data that are determined based on expert experience. Subsequently, the CKFCM clustering is applied to the modeling of the IOSP, and the Spearman’s rank correlation coefficient method is utilized to determine input variables of the model under different operating conditions. Furthermore, the broad learning model is employed to build the prediction model for each operating condition. Finally, we conducted an in-depth analysis of the presented clustering method. Its performance has been experimentally demonstrated by many publicly available datasets. Meanwhile, simulation results involving actual run data of the IOSP demonstrate the superiority and effectiveness of the developed model in carbon efficiency prediction. We show that the developed model outperforms the state-of-the-art prediction models in achieving a good balance between simplicity and accuracy. [ABSTRACT FROM AUTHOR]

While granular computing has experienced rapid growth in the past decades and some milestones have been reached, a comprehensive study of the representation capabilities delivered by numeric prototypes and granular prototypes produced by different techniques still calls for comprehensive research and a comparative analysis. Well-constructed information granules are reflective of the nature of the numeric evidence and serve as backbones of granular classifiers and granular models. The objective of this study is to review a number of clustering paradigms aimed at the construction of information granules, discuss the development of granular prototypes, and conduct a comprehensive evaluation of quality of numeric prototypes and their corresponding augmentations coming in the form of granular prototypes. We have been witnessing many studies devoted to the construction of information granules, but a comparative analysis of the quality of information granules constructed on a basis of prototypes produced by different clustering algorithms is still lacking. In this regard, the review of the clustering algorithms supporting the formation of information granules and the comprehensive comparative study of their usefulness in classification and modeling tasks offered in this study make sense. This will promote the usage of information granules in various future works, especially classification problem and system modeling. [ABSTRACT FROM AUTHOR]

Information granulation is an effective vehicle to explore data structure and has become a timely research topic. In this article, a top-down granulation model (T-GrM) adhering to the two fundamental requirements (coverage and specificity) in the principle of justifiable granularity is designed. Here, this principle is employed as a fundamental method for constructing information granules in each layer, and as an indicator to determine whether the obtained partitions need to be further divided in the next layer. For capturing the correlation among features and reflecting the data structure, an improved version of a T-GrM is offered by incorporating the principal component analysis. The proposed granulation models are evaluated on synthetic datasets, where the obtained experimental results offer some insights into the feasibility of the designed models and describe the effect of parameter values on the constructed information granules. The improved version of the T-GrM has the advantage of being transparent by generating information granules on several principal components, and thereby, the description of information granules is simpler and descriptive. As application examples, two real-world datasets are analyzed to exhibit the advantage of the constructed information granules. [ABSTRACT FROM AUTHOR]

This article proposes a new network approach toward the implementation of Takagi–Sugeno (T–S) fuzzy models referred to as disjunctive fuzzy neural networks (DJFNNs). The proposed DJFNN involves a novel network architecture and a greedy learning algorithm. Being different from the existing grid-based and clustering-based network architectures, the proposed architecture adds an OR neural layer positioned between the fuzzification layer and the rule layer. In this way, the implied constraint between the number of rules and the number of fuzzy labels is excluded so that a curse of dimensionality can be overcome and more interpretable models are formed. Furthermore, inspired by the core algorithm for building a decision tree, a top–down, nonbacktracking, and greedy algorithm is proposed to learn the unknown parameters of the networks. The input space splits into smaller and smaller subspace along the predefined fuzzy grids in a supervised manner meanwhile the associated conditions of the T–S fuzzy model are identified. The greedy algorithm is applicable to high-dimensional problems since there is no exponential growth in time or space as the dimensionality increases. The new network architecture and greedy learning algorithm make the proposed DJFNN a regression model of high interpretability and good prediction capability, particularly suitable for solving the high-dimensional problems. The DJFNN was experimented with using a synthetic dataset and 28 real-world datasets and compared with classical and state-of-the-art methods through nonparametric statistical tests. The results confirmed the effectiveness of the DJFNN in terms of accuracy, interpretability, and computational cost. [ABSTRACT FROM AUTHOR]

In this article, we develop a residual-sparse Fuzzy C-Means (FCM) algorithm for image segmentation, which furthers FCM's robustness by realizing the favorable estimation of the residual (e.g., unknown noise) between an observed image and its ideal version (noise-free image). To achieve a sound tradeoff between detail preservation and noise suppression, morphological reconstruction is used to filter the observed image. By combining the observed and filtered images, a weighted sum image is generated. Tight wavelet frame decomposition is used to transform the weighted sum image into its corresponding feature set. Taking such feature set as data for clustering, we impose an $\ell _0$ regularization term on residual to FCM's objective function, thus resulting in residual-sparse FCM, where spatial information is introduced for improving its robustness and making residual estimation more reliable. To further enhance segmentation accuracy of the proposed FCM, we employ morphological reconstruction to smoothen the labels generated by clustering. Finally, based on the prototypes and smoothed labels, a segmented image is reconstructed by using tight wavelet frame reconstruction. Experimental results regarding synthetic, medical, and real-world images show that the proposed algorithm is effective and efficient, and outperforms its peers. [ABSTRACT FROM AUTHOR]

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

The analytic hierarchy process (AHP) is widely employed to guide the decision-maker to rank or evaluate the alternatives in decision activities. Its fuzzy set-based version, i.e., the fuzzy AHP, has also been widely studied and applied since its inception. The essential distinction between the AHP and fuzzy AHP comes from the diverse transformation methods between the linguistic and numeric judgments. In this article, we conduct a thorough comparative study between the AHP and fuzzy AHP methods in the framework of two linguistic models, i.e., the linguistic model based on the membership functions and two-tuple linguistic model. First, four AHP and three fuzzy AHP methods are revisited with the involvement of two linguistic models. Then, the comparison criteria are involved by calculating the cardinal or ordinal deviation between the original information and decision solutions, and the effects of the transitivity of the reciprocal matrix are also discussed in the comparative study. Finally, the detailed experiments along with a thorough comparative analysis are conducted based on the random and publicly available data to show the difference between the AHP and fuzzy AHP methods. [ABSTRACT FROM AUTHOR]

In this article, we introduce novel reinforced hybrid fuzzy rule-based neural networks (RHFNNs). This article is concerned with the development of the design methodologies of hybrid fuzzy rule-based model for constructing the network structure and enhancing its predictive abilities through the combination of inhomogeneous neurons [i.e., clustering-based polynomial neurons (CPNs) and polynomial neurons (PNs)] and tournament selection. The key points of the proposed RHFNN are enumerated as follows: The first layer of the proposed network consists of CPNs. CPN can effectively reflect the complex nonlinear structure encountered in the data space, and refine (granulate) it with the help of the clustering algorithm. Two types of CPNs including hard C-means (HCM) clustering-based polynomial neuron (HCPN) and fuzzy C-means (FCM) clustering-based polynomial neuron (FCPN) are designed. According to the type of CPN used in the first layer, RHFNN can be categorized into two types, namely, RHFNN based on HCPN (HRHFNN) and RHFNN based on FCPN (FRHFNN). We use PNs to construct the second and consecutive layers. PN can identify and approximate the nonlinear relationship among system's inputs and outputs. A tournament-based performance selection (TPS) algorithm stemming from evolutionary computation is used for selection of neuron. TPS not only ensures that the candidate nodes have sufficient fitting ability but also enhances the individual diversity in the node set and provides the abilities to generate better prediction nodes. In addition, L2-norm regularization is considered to reduce the deviation between coefficients and ameliorate overfitting as well as boost generalization ability. The performance of RHFNN is discussed through a variety of publicly available machine learning datasets. From the experimental results, we conclude that RHFNN achieves the best prediction accuracy on 13 of 15 datasets; the statistical analysis also confirms the superiority of RHFNN. [ABSTRACT FROM AUTHOR]

The current societal demands and technological developments have resulted in the participation of a large number of experts in making decisions as a group. Conflicts are imminent in groups and conflict management is complex and necessary especially in a large group. However, there are few studies that quantitatively research the conflict detection and resolution in the large-group context, especially in the multicriteria large-group decision making (GDM) context. This article proposes a dynamic adaptive subgroup-to-subgroup conflict model to solve multicriteria large-scale GDM problems. A compatibility index is proposed based on two kinds of conflicts among experts: 1) cognitive conflict and 2) interest conflict. Then, the fuzzy $c$ -means clustering algorithm is used to classify experts into several subgroups. A subgroup-to-subgroup conflict detection method and a weight-determination approach are developed based on the clustering results. Afterward, a conflict resolution model, which can dynamically generate feedback suggestion, is introduced. Finally, an illustrative example is provided to demonstrate the effectiveness and applicability of the proposed model. [ABSTRACT FROM AUTHOR]

In this article, a reinforced fuzzy radial basis function neural network (R-FRBFNN) classifier is proposed. It focuses on the development of methodologies of reinforced architecture to improve classification accuracy and enhance the robust capability based on two learning strategies. The two learning strategies are summarized: 1) R-FRBFNN designed via support vector (SV)-based fuzzy C-means (FCM) clustering and softmax-based iterative reweighted least square (IRLS), which concentrate on improving the classification performance of R-FRBFNN; and 2) R-FRBFNN designed via SV-based FCM and softmax-based iterative quadratic programming (IQP), which focus on improving the robust abilities of the R-FRBFNN and reducing the effects of noise and outliers. The essential points of the proposed R-FRBFNN classifier are summarized as follows. a) The proposed R-FRBFNN consists of three phases: condition, conclusion, and inference. b) An SV-based FCM is considered for prioritizing the classification boundary and improving the classification performance of the proposed classifier. c) Three types of polynomials construct the conclusion phase. Two learning techniques are designed to update the coefficients of the polynomials. Softmax-based IRLS is a type of iterative learning technique based on Newton's method. Softmax-based IQP is more robust and avoids the degradation of generalization capabilities caused by outliers and noisy data. d) In the concept of reinforced architecture, SV-based FCM imposes compensation (membership degrees) on learning techniques according to the data characteristics encountered in the inference phase. Experimental results reported for benchmark data and outliers/noisy datasets demonstrate that the proposed classifier shows improved classification performance compared with other previously studied methods. [ABSTRACT FROM AUTHOR]

Rule-based models are applicable to model the behavior of complex and nonlinear systems. Due to limited experience and randomness involving constructing information granules, an insufficient credible rules division could reduce the model’s accuracy. This paper proposes a new rule-based modeling approach, which utilizes density-based spatial clustering of applications with noise (DBSCAN)-based information granules to construct the rules. First, bear in mind the advantages of density-based clustering, DBSCAN is proposed to generate data structures. Based on these data structures, two rule-based models are constructed: 1) models using DBSCAN clusters to construct granules and rules directly and 2) models generating subgranules in each DBSCAN cluster for rule formation. Experiments involving these two models are completed, and obtained results are compared with those generated with a traditional model involving fuzzy C-means-based granules. Numerical results show that the rule-based model, which builds rules from subgranules of DBSCAN structures, performs the best in analyzing system behaviors. [ABSTRACT FROM AUTHOR]

The conventional fuzzy C-means (FCM) algorithm is not robust to noise and its rate of convergence is generally impacted by data distribution. Consequently, it is challenging to develop FCM-related algorithms that have good performance and require less computing time. In this article, we elaborate on a comprehensive FCM-related algorithm for image segmentation. To make FCM robust, we first utilize a morphological grayscale reconstruction (MGR) operation to filter observed images before clustering, which guarantees noise-immunity and image detail-preservation. Since real images can generally be approximated by sparse coefficients in a tight wavelet frame system, feature spaces of observed and filtered images can be obtained. Taking such features to be clustered, we investigate an improved FCM model in which a sparse regularization term is introduced into the objective function of FCM. We design a three-step iterative algorithm to solve the sparse regularization-based FCM model, which is constructed by the Lagrangian multiplier method, hard-threshold operator, and normalization operator, respectively. Such an algorithm can not only perform well for image segmentation, but also come with high computational efficiency. To further enhance the segmentation accuracy, we use MGR to filter the label set generated by clustering. Finally, a large number of supporting experiments and comparative studies with other FCM-related algorithms available in the literature are provided. The obtained results for synthetic, medical and color images indicate that the proposed algorithm has good ability for multiphase image segmentation, and performs better than other alternative FCM-related algorithms. Moreover, the proposed algorithm requires less time than most of the existing algorithms. [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 study is concerned with a description of large numeric data with the aid of building a limited collection of representative information granules with the objective of capturing the structure of the original data. The proposed development scheme consists of two steps. First, a clustering algorithm characterized by high flexibility of coping with the diverse geometry of data structure and efficient computational overhead is invoked. At the second step, a clustering algorithm applied to the clusters already formed during the first phase, yielding a collection of numeric prototypes is involved and the numeric prototypes produced there are then generalized into their granular prototypes. The quality of granular prototypes is quantified while their build-up is supported by the mechanisms of granular computing such as the principle of justifiable granularity. In this paper, the clustering algorithms of DBSCAN and fuzzy ${C}$ -means were used in successive phases of the processed approach. The experimental studies concerning synthetic data and publicly available data are covered and the performance of the developed approach is assessed along with a comparative analysis. [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]

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

In recent years, image processing in a Euclidean domain has been well studied. Practical problems in computer vision and geometric modeling involve image data defined in irregular domains, which can be modeled by huge graphs. In this paper, a wavelet frame-based fuzzy $C$ -means (FCM) algorithm for segmenting images on graphs is presented. To enhance its robustness, images on graphs are first filtered by using spatial information. Since a real image usually exhibits sparse approximation under a tight wavelet frame system, feature spaces of images on graphs can be obtained. Combining the original and filtered feature sets, this paper uses the FCM algorithm for segmentation of images on graphs contaminated by noise of different intensities. Finally, some supporting numerical experiments and comparison with other FCM-related algorithms are provided. Experimental results reported for synthetic and real images on graphs demonstrate that the proposed algorithm is effective and efficient, and has a better ability for segmentation of images on graphs than other improved FCM algorithms existing in the literature. The approach can effectively remove noise and retain feature details of images on graphs. It offers a new avenue for segmenting images in irregular domains. [ABSTRACT FROM AUTHOR]

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

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

Fuzzy models have been commonly used in system modeling and model-based control. Among various fuzzy models, Takagi–Sugeno (TS) fuzzy models form one of the intensively studied and applied categories of models. In this study, we are concerned with a development of a granular TS fuzzy model realized on a basis of numerical evidence and completed through a combination of fuzzy subspace clustering and the principle of optimal allocation of information granularity. The TS fuzzy models are built with the use of the fuzzy subspace clustering algorithm. Information granularity is regarded as a crucial design asset whose optimal allocation gives rise to granular fuzzy models and makes the constructed models to become better in rapport with experimental data. In comparison with fuzzy models, granular fuzzy models produce results (outputs) that are information granules rather than numeric entities being encountered in fuzzy models. In contrast with the commonly used optimization criteria, which emphasize the highest accuracy encountered at the numeric level, the performance of the granular TS fuzzy model is quantified in terms of the coverage and specificity criteria where such criteria are of interest in the evaluation of quality of information granules vis-à-vis experimental (numeric) data. Experimental results are reported for both synthetic datasets and publicly available data sets coming from the UCI machine learning repository. [ABSTRACT FROM AUTHOR]

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

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

The study is devoted to the clustering of granular data and an evaluation of the results of such clustering. A comprehensive and systematic approach is developed, which is composed of three fundamental phases: 1) representation of granular data; 2) clustering carried out in the representation space of information granules; and 3) evaluation of quality of clusters following the reconstruction criterion. The reconstruction criterion formed originally for numeric data and leading to an idea of granular prototypes is revisited. We show here an emergence of granular information of higher type, which are used to implement granular interval prototypes. We discuss a way of forming granular data in the context of representation of time series and present clustering of granular time series. [ABSTRACT FROM AUTHOR]

In order to deal with the sequential decision problems with large or continuous state spaces, feature representation and function approximation have been a major research topic in reinforcement learning (RL). In this paper, a clustering-based graph Laplacian framework is presented for feature representation and value function approximation (VFA) in RL. By making use of clustering-based techniques, that is, $K$ -means clustering or fuzzy $C$ -means clustering, a graph Laplacian is constructed by subsampling in Markov decision processes (MDPs) with continuous state spaces. The basis functions for VFA can be automatically generated from spectral analysis of the graph Laplacian. The clustering-based graph Laplacian is integrated with a class of approximation policy iteration algorithms called representation policy iteration (RPI) for RL in MDPs with continuous state spaces. Simulation and experimental results show that, compared with previous RPI methods, the proposed approach needs fewer sample points to compute an efficient set of basis functions and the learning control performance can be improved for a variety of parameter settings. [ABSTRACT FROM PUBLISHER]

Principal curves arising as an essential construct in dimensionality reduction and data analysis have recently attracted much attention from theoretical as well as practical perspective. In many real-world situations, however, the efficiency of existing principal curves algorithms is often arguable, in particular when dealing with massive data owing to the associated high computational complexity. A certain drawback of these constructs stems from the fact that in several applications principal curves cannot fully capture some essential problem-oriented facets of the data dealing with width, aspect ratio, width change, etc. Information granulation is a powerful tool supporting processing and interpreting massive data. In this paper, invoking the underlying ideas of information granulation, we propose a granular principal curves approach, regarded as an extension of principal curves algorithms, to improve efficiency and achieve a sound accuracy–efficiency tradeoff. First, large amounts of numerical data are granulated into $C$ intervals—information granules developed with the use of fuzzy C-means clustering and the two criteria of information granulation, which significantly reduce the amount of data to be processed at the later phase of the overall design. Granular principal curves are then constructed by determining the upper and the lower bounds of the interval data. Finally, we develop an objective function using the criteria of information confidence and specificity to evaluate the granular output formed by the principal curves. We also optimize the granular principal curves by adjusting the level of information granularity (the number of clusters), which is realized with the aid of the particle swarm optimization. A number of numeric studies completed for synthetic and real-world datasets provide a useful quantifiable insight into the effectiveness of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

A class of clustering problems that is studied here is concerned with the development of a structure of a global nature given a collection of structures (clusters) constructed locally for data that are represented by several collections (blocks) of features. These blocks of features come with a well-defined semantics. For instance, in spatiotemporal data, a certain block of features concerns a spatial component of the data (say, x–y or x–y–z coordinates), while another one deals with the features that describe time series associated with the corresponding locations. The results of clustering that are being produced locally are reconciled by minimizing a distance between the proximity matrices that are formed at the higher conceptual level and induced by the individual partition matrices. The optimization problem is formulated and presented along with its iterative scheme. [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]

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