The increasing effect of Internet of Things (IoT) unlocks the massive volume of the availability of Big Data in many fields. Generally, these Big Data may be in a non-independently and identically distributed fashion (non-IID). In this paper, we have contributions in such a way enable multi-view k-means (MVKM) clustering to maintain the privacy of each database by allowing MVKM to be operated on the local principle of clients' multi-view data. This work integrates the exponential distance to transform the weighted Euclidean distance on MVKM so that it can make full use of development in federated learning via the MVKM clustering algorithm. The proposed algorithm, called a federated MVKM (Fed-MVKM), can provide a whole new level adding a lot of new ideas to produce a much better output. The proposed Fed-MVKM is highly suitable for clustering large data sets. To demonstrate its efficient and applicable, we implement a synthetic and six real multi-view data sets and then perform Federated Peter-Clark in Huang et al. 2023 for causal inference setting to split the data instances over multiple clients, efficiently. The results show that shared-models based local cluster centers with data-driven in the federated environment can generate a satisfying final pattern of one multi-view data that simultaneously improve the clustering performance of (non-federated) MVKM clustering algorithms.

The current data acquisition techniques enable the gathering and storage of extensive datasets, encompassing multidimensional arrays. Recent researchers focus on the analysis of large datasets having diverse data points. These multidimensional datasets comprise diverse data points and can be represented as tensors or multidimensional arrays. Clustering, a data analysis technique, can be used to discover and reveal latent data patterns from these datasets. The traditional clustering algorithms such as k-means, fuzzy c-means (FCM), and possibilistic c-means (PCM) pose some drawbacks in efficiently delivering high-quality clustering results for tensor or multidimensional array data. This may stem from the fact that these algorithms are primarily designed for single-view or low-array datasets, rendering them less suitable for the complexities of multidimensional arrays. In response to this challenge, this article introduces the tensor-based PCM (TPCM) algorithm. TPCM utilizes a tensor distance (TD) function as the distance metric, different from the usual Euclidean distance. The TD function evaluates the distance between data points and cluster centers by considering relationships among different coordinates. To further enhance the analysis, the canonical polyadic decomposition (CPD) method and PARAFAC2 decomposition techniques are used to restructure heterogeneous data into low-order tensors. Our experiments consider two types of datasets: multiview datasets and tensor datasets. CPD is applied for tensor data decomposition, while PARAFAC2, a CPD variant, addresses multiview data with varying feature space sizes in each view. Through comprehensive illustrations and evaluations using synthetic and real datasets, we demonstrate the superior performance of TPCM. Experimental results reveal that TPCM consistently achieves higher clustering performance compared to most existing clustering algorithms.

Krishnapuram and Keller first proposed possibilistic c-means (PCM) clustering in 1993. Afterward, PCM was widely studied with various extensions. The PCM algorithm and its extensions always treat feature components under equal importance, but, in real applications, different features may better have different weights. Recently, Yang and Benjamin in 2021 proposed a feature-weighted PCM clustering with feature reduction. Although Yang and Benjamin (2021) can reduce feature dimensions, it still encounters the curse of dimensionality for high dimensional data. One possible way to address this problem is to conduct a sparse clustering technique. In this paper, we further study the PCM clustering by incorporating the idea of sparsity with different feature weights. We propose two approaches that use the PCM clustering with the least absolute shrinkage and selection operator (Lasso). The first one is the sparse PCM subject to a Lasso constraint of feature weights, called S-PCM1. The second is the sparse PCM by adding a Lasso penalty term of feature weights in the objective function, called S-PCM2. We show that S-PCM1 and S-PCM2 are theoretically the same, and both can induce sparsity in features, but they use different procedures in algorithms. Synthetic and real data sets are used to compare S-PCM1 and S-PCM2 with some existing sparsity clustering algorithms. Experimental results and comparisons demonstrate the effectiveness and usefulness of the proposed S-PCM1 and S-PCM2 clustering algorithms.

Since social media, virtual communities and networks rapidly grow, multiview data become more popular. In general, multiview data always contain different feature components in different views. Although these data are extracted in different ways (views) from diverse settings and domains, they are used to describe the same samples, which make them highly related. Hence, applying (single-view) clustering methods for multiview data poses difficulty in achieving desirable clustering results. Thus, multiview clustering methods should be developed that will utilize available multiview information. Most of multiview clustering techniques currently use k-means due to its conceptual simplicity, and use fuzzy c-means (FCM) that the datapoints can belong to more than one cluster based on their membership degrees from 0 to 1. However, the use of k-means or FCM may degrade its performance due to the presence of noise and outliers, especially on large or high-dimensional datasets. The constraint imposed on the membership degrees of k-means and FCM tends to assign a corresponding high membership value to an outlier or a noisy data point. To address these drawbacks, possibilistic c-means (PCM) relaxes the membership constraint of k-means and FCM so that outliers and noisy datapoints can be properly identified. On the other hand, there are various extensions of k-means and FCM for multiview data, but no extension of PCM for multiview data was made in the literature. Thus, we use PCM in our proposed multiview clustering model. In this article, we propose novel weighted multiview PCM algorithms designed for clustering multiview data as well as view and feature weights on PCM approaches, called W-MV-PCM and W-MV-PCM with L2 regularization (W-MV-PCM-L2). In multiview clustering, different views may vary with respect to its importance and each view may contain some irrelevant features. In the proposed algorithms, a learning scheme is constructed to compute for the view weights, and feature weights within each view. This scheme will be able to identify the importance of each view and, at the same time, it will also identify and select relevant features in each view. Comparisons of W-MV-PCM-L2 with existing multiview clustering algorithms are made on both synthetic and real datasets. The experimental results are evaluated using accuracy rate (AR) and external validity indexes, such as Rand index (RI) and normalized mutual information (NMI). The proposed W-MV-PCM-L2 algorithm with comparisons of existing algorithms under criteria of AR, RI, and NMI shows that it is a feasible and effective multiview clustering algorithm.

In 1993, Krishnapuram and Keller proposed possibilistic c-means (PCM) clustering, where the PCM had various extensions in the literature. However, the PCM algorithm with its extensions treats data points under equal importance for features. In real applications, different features in a dataset should take different importance with different weights. In this article, we first propose a feature-weighted PCM (FW-PCM). We then construct a feature-reduction framework. Therefore, we give a feature-weighted possibilistic c-means clustering with a feature-reduction framework, termed as a feature-weighted reduction PCM (FW-R-PCM) algorithm. The proposed FW-R-PCM can improve the clustering performance of PCM by calculating feature weights to identify important features, and so it can consequently eliminate these irrelevant features to reduce feature dimension. Its theoretical behavior and computational complexity are also analyzed. The effectiveness and usefulness of FW-R-PCM are demonstrated through experimental results using synthetic and real datasets, where comparisons of FW-R-PCM with PCM, FW-PCM, and some existing feature-weighted clustering algorithms are also made.

Pythagorean fuzzy sets (PFSs) were proposed by Yager in 2013 to treat imprecise and vague information in daily life more rigorously and efficiently with higher precision than intuitionistic fuzzy sets. In this paper, we construct new distance and similarity measures of PFSs based on the Hausdorff metric. We first develop a method to calculate a distance between PFSs based on the Hasudorff metric, along with proving several properties and theorems. We then consider a generalization of other distance measures, such as the Hamming distance, the Euclidean distance, and their normalized versions. On the basis of the proposed distances for PFSs, we give new similarity measures to compute the similarity degree of PFSs. Some examples related to pattern recognition and linguistic variables are used to validate the proposed distance and similarity measures. Finally, we apply the proposed methods to multicriteria decision-making by constructing a Pythagorean fuzzy Technique for Order Preference by Similarity to an Ideal Solution and then present a practical example to address an important issue related to social sector. Numerical results indicate that the proposed methods are reasonable and applicable and also that they are well suited in pattern recognition, linguistic variables, and multicriteria decision-making with PFSs.

A similarity measure is a useful tool for determining the similarity between two objects. Although there are many different similarity measures among the intuitionistic fuzzy sets (IFSs) proposed in the literature, the Jaccard index has yet to be considered as way to define them. The Jaccard index is a statistic used for comparing the similarity and diversity of sample sets. In this study, we propose a new similarity measure for IFSs induced by the Jaccard index. According to our results, proposed similarity measures between IFSs based on the Jaccard index present better properties. Several examples are used to compare the proposed approach with several existing methods. Numerical results show that the proposed measures are more reasonable than these existing measures. On the other hand, measuring the similarity between IFSs is also important in clustering. Thus, we also propose a clustering procedure by combining the proposed similarity measure with a robust clustering method for analyzing IFS data sets. We also compare the proposed clustering procedure with two clustering methods for IFS data sets.

To generalize the Rand index (RI) from crisp partitions to fuzzy partitions, we first propose a graph method in which color edges in the graph for crisp partitions are used to determine the relation matrix between objects such that the matrix trace can be employed to calculate the RI. This approach is then introduced into fuzzy partitions to generalize the RI to the fuzzy RI (FRI). Compared with previous fuzzy generalizations, the most unique aspect of our method has the following important characteristics that for any two partition matrices M(1) and M(2), the result with is the necessary and sufficient condition for the result that the FRI is equal to 1. This important characteristic renders our fuzzy generalization of the RI is not only able to determine the similarities between fuzzy partitions and crisp reference partitions, but also to identify the similarity between fuzzy partitions and fuzzy reference partitions. The method can even be used to explore and compare the similarities between various data sets and the same fuzzy reference partition. Finally, we use synthetic data and real data to give more demonstrations, and further perform comparisons of our method with those existing fuzzy extensions of the RI.

Fuzzy clustering algorithms generally treat data points with feature components under equal importance. However, there are various datasets with irrelevant features involved in clustering process that may cause bad performance for fuzzy clustering algorithms. That is, different feature components should take different importance. In this paper, we present a novel method for improving fuzzy clustering algorithms that can automatically compute individual feature weight, and simultaneously reduce these irrelevant feature components. In fuzzy clustering, the fuzzy c-means (FCM) algorithm is the best known. We first consider the FCM objective function with feature-weighted entropy, and construct a learning schema for parameters, and then reduce these irrelevant feature components. We call it a feature-reduction FCM (FRFCM). During FRFCM processes, a new procedure for eliminating irrelevant feature(s) with small weight(s) is created for feature reduction. The computational complexity of FRFCM is also analyzed. Some numerical and real datasets are used to compare FRFCM with various feature-weighted FCM methods in the literature. Experimental results and comparisons actually demonstrate these good aspects of FRFCM with its effectiveness and usefulness in practice.

Belief and plausibility functions based on Dempster–Shafer theory have been used to measure uncertainty. They are also widely studied and applied in diverse areas. Numerous studies in the literature have presented various generalizations of belief and plausibility functions to fuzzy sets. However, there are still less generalizations of belief and plausibility functions to intuitionistic fuzzy sets. Because intuitionistic fuzzy sets can present the degrees of both membership and nonmembership with a degree of hesitancy, the knowledge and semantic representation becomes more general and applicable than fuzzy sets. In this paper, we propose a generalization of belief and plausibility functions to intuitionistic fuzzy sets based on fuzzy integral. Some numerical examples show the effectiveness of the proposed generalization. Furthermore, this generalization of belief and plausibility functions to intuitionistic fuzzy sets is able to catch more information about the change of intuitionistic fuzzy focal elements.

In this paper, we consider cluster analysis based on T-transitive interval-valued fuzzy relations. A fuzzy relation with its partitional tree for obtaining an agglomerative hierarchical clustering has been studied and applied. In general, these fuzzy-relation-based clustering approaches are based on real-valued memberships of fuzzy relations. Since interval-valued memberships may be better than real-valued memberships to represent higher order imprecision and vagueness for human perception, in this paper we first extend fuzzy relations to interval-valued fuzzy relations and then construct a clustering algorithm based on the proposed T-transitive interval-valued fuzzy relations. We use two examples to demonstrate the efficiency and usefulness of the proposed method. In practical application, we apply the proposed clustering method to performance evaluations for academic departments of higher education by using actual engineering school data in Taiwan.

In this paper we propose clustering methods based on weighted quasiarithmetic means of T-transitive fuzzy relations. We first generate a T-transitive closure RT from a proximity relation R based on a max-T composition and produce a T-transitive lower approximation or opening RT from the proximity relation R through the residuation operator. We then aggregate a new T-indistinguishability fuzzy relation by using a weighted quasiarithmetic mean of RT and RT. A clustering algorithm based on the proposed T-indistinguishability is thus created. We compare clustering results from three critical ti-indistinguishabilities: minimum (t3), product (t2), and Łukasiewicz (t1). A weighted quasiarithmetic mean of a t1-transitive closure [Formula: see text] and a t1-transitive lower approximation or opening [Formula: see text] with the weight [Formula: see text], demonstrates the superiority and usefulness of clustering begun by using a proximity relation R based on the proposed clustering algorithm. The algorithm is then applied to the practical evaluation of the performance of higher education in Taiwan.

Change-point (CP) regression models have been widely applied in various fields, where detecting CPs is an important problem. Detecting the location of CPs in regression models could be equivalent to partitioning data points into clusters of similar individuals. In the literature, fuzzy clustering has been widely applied in various fields, but it is less used in locating CPs in CP regression models. In this paper, a new method, called fuzzy CP (FCP) algorithm, is proposed to detect the CPs and simultaneously estimate the parameters of regression models. The fuzzy c -partitions concept is first embedded into the CP regression models. Any possible collection of all CPs is considered as a partitioning of data with a fuzzy membership. We then transfer these memberships into the pseudomemberships of data points belonging to each individual cluster, and therefore, we can obtain the estimates for model parameters by the fuzzy c-regressions method. Subsequently, we use the fuzzy c -means clustering to obtain the new iterates of the CP collection memberships by minimizing an objective function concerning the deviations between the predicted response values and data values. We illustrate the new approach with several numerical examples and real datasets. Experimental results actually show that the proposed FCP is an effective and useful CP detection algorithm for CP regression models and can be applied to various fields, such as econometrics, medicine, quality control, and signal processing.

In fuzzy clustering, the fuzzy c-means (FCM) is the most known algorithm. Several extensions and variations of FCM had been proposed in the literature. The first important extension to FCM was proposed by Gustafson and Kessel (GK). In the GK fuzzy clustering, they considered the effect of different cluster shapes except for spherical shapes by replacing the Euclidean distance of the FCM objective function with the Mahalanobis distance. The GK algorithm has become one of the most frequently used clustering algorithms. Just like FCM, the fuzziness index m is a parameter in which the value will greatly influence the performance of the GK algorithm. However, there is no theoretical work on the parameter selection for the fuzziness index m of GK. In this paper, we reveal the relation between the stable fixed points of the GK algorithm and the datasets using Jacobian matrix analysis, and then provide a theoretical base for selecting the fuzziness index m in the GK algorithm. Some experimental results verify the effectiveness of our theoretical results.

Similarity measures between generalized trapezoidal fuzzy numbers (GTFNs) are employed to indicate the degrees of similarity between GTFNs. Although several similarity measures of GTFNs have been proposed in the literature, none has considered using the Jaccard index. In general, the Jaccard index is a statistic used for comparing the similarity and diversity of sample sets. This paper presents a new similarity measure between GTFNs, which involves the Jaccard index. The proposed similarity measure is found to have better properties. Several examples are employed to compare the proposed measure with some existing methods. An experiment is performed using 15 sets of GTFNs to compare the proposed similarity measure with existing ones. Numerical results show that the proposed measure is more reasonable than those existing methods.

In 1993, Yang first extended the classification maximum likelihood (CML) to a so-called fuzzy CML, by combining fuzzy c-partitions with the CML function. Fuzzy c-partitions are generally an extension of hard c-partitions. It was claimed that this was more robust. However, the fuzzy CML still lacks some robustness as a clustering algorithm, such as its in-ability to detect different volumes of clusters, its heavy dependence on parameter initializations and the necessity to provide an a priori cluster number. In this paper, we construct a robust fuzzy CML clustering framework that has a robust clustering method. The eigenvalue decomposition of a covariance matrix is firstly considered using the fuzzy CML model. The Bayesian information criterion (BIC) is then used for model selection, in order to choose the best model with the optimal number of clusters. Therefore, the proposed robust fuzzy CML clustering framework exhibits clustering characteristics that are robust in terms of the parameter initialization, robust in terms of the cluster number and also in terms of its capability to detect different volumes of clusters. Numerical examples and real data applications with comparisons are provided, which demonstrate the effectiveness and superiority of the proposed method.

In this paper we propose a robust clustering algorithm for interval data. The proposed method is based on similarity measure that is not necessary to specify a cluster number and initials. Several numerical examples demonstrate the effectiveness of the proposed robust clustering algorithm. We then apply this algorithm to the real data set with cities temperature interval data. The proposed clustering algorithm actually presents its robustness.

In this paper, new similarity, inclusion measure and entropy between type-2 fuzzy sets corresponding to grades of memberships are proposed. We also create the relationships among these measures between type-2 fuzzy sets. Several examples are used to present the calculation of these similarity, inclusion measure and entropy between type-2 fuzzy sets. The comparison results show that the proposed similarity measure presents better than those of Hung and Yang (2004) and Yang and Lin (2009). Moreover, measuring the similarity between type-2 fuzzy sets is important in clustering. We also use the proposed similarity measure as a clustering method for type-2 fuzzy sets.

Cluster analysis is a tool for data analysis. It is a method for finding clusters of a data set with most similarity in the same group and most dissimilarity between different groups. In general, there are two ways, mixture distributions and classifi

In this paper, we construct fuzzy renewal processes involving fuzzy random variables. We first extend the renewal processes to the fuzzy renewal processes where interarrival times, rewards, and stopping times are all fuzzy random variables. According to these fuzzy renewal processes, we then extend some theorems of renewal processes to those in fuzzy renewal processes. These are elementary renewal theorem, asymptotic expected average reward, and Wald's equation. In each part, we also give examples for applications. © 2010 Wiley Periodicals, Inc.

Krishnapuram and Keller [IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp. 98-110, May 1993] proposed a possibilisticc-means (PCM) clustering by relaxing the constraint in fuzzyc-means (FCMs) that the memberships of a data point across classes sum to 1. The PCM algorithm has a tendency to produce coincident clusters. This can be a merit of PCM as a good mode-seeking algorithm if initials and parameters are suitably chosen. However, the performance of PCM heavily depends on initializations and parameter selection. In this paper, we propose a mechanism of robust automatic merging. We then create an automatic merging possibilistic clustering method (AM-PCM), where the proposed algorithm does not only solve these parameter-selection and initialization problems but also obtains an optimal cluster number. The proposed AM-PCM algorithm first uses all data points as initial cluster centers and then automatically merges these surrounding points around each cluster mode such that it can self-organize data groups according to the original data structure. The AM-PCM can exhibit the robustness to parameter, noise, cluster number, different volumes, and initializations. The computational complexity of AM-PCM is also analyzed. Comparisons between AM-PCM and other clustering methods are made. Some numerical data and real datasets are used to show these good aspects of AM-PCM. Experimental results and comparisons actually demonstrate that the proposed AM-PCM is an effective and parameter-free robust clustering algorithm.

In this paper we propose a similarity-based clustering algorithm for handling LR-type fuzzy numbers. The proposed method does not need to specify a cluster number and initial values in which it is robust to initial values, cluster number, cluster shapes, noise and outliers for clustering LR-type fuzzy data. Numerical examples and real data demonstrate the effectiveness of the proposed clustering algorithm.

This paper considers competitive learning networks using three types of hard, soft, and fuzzy learning schemes. The hard competitive learning algorithm is with the winner-take-all. The soft competition learning algorithm is with a stochastic relaxation technique using the Gibbs distribution as a dynamic neighborhood function. The fuzzy competition learning algorithm is with a fuzzy relaxation technique using fuzzy membership functions as kernel type neighborhood interaction functions. Some numerical examples are made for these three types of competitive learning schemes. The numerical results show that the fuzzy learning has better performance than hard and soft learning under the normal mixture data. We then present an application to magnetic resonance image segmentation. A real case of ophthalmology recommended by a neurologist with MR image data is examined in this paper. These competitive learning algorithms are used in segmenting the ophthalmological MRI data for reducing medical image noise effects with a learning mechanism. Based on the segmentation results, the fuzzy learning gives better performance than hard and soft learning so that the fuzzy competitive learning algorithm is recommended for use in MRI segmentation as an aid for support diagnoses. © 2010 Wiley Periodicals, Inc.

In this paper we propose a robust-type clustering algorithm that has more accuracy than the alternative FCM (AFCM) proposed by Wu and Yang. Moreover, to speed up the proposed algorithm, we use the suppressed idea to modify it. The modified robust-type clustering algorithm presents fast convergence speed and also robustness. Finally, this algorithm is applied in the segmentation of the magnetic resonance image (MRI) of an ophthalmic patient. In our comparisons of the proposed algorithm with the AFCM for these MRI segmentation results, we find that the proposed algorithm provides better detection of abnormal tissue than AFCM.

Most clustering algorithms, such as k-means and fuzzy c-means (FCM), are used to cluster a set of objects based on a function of dissimilarities between objects. However, clustering on attribute variables of objects may give more cluster information. Thus, to have a clustering algorithm that can be designated to construct simultaneously an optimal partition of objects and also attribute variables into homogeneous block is important. This kind of clustering was called block clustering (see Duffy and Quiroz, 1991). Recently, Govaert and Nadif (2003) proposed a block classification EM (block CEM) algorithm and then proposed block fuzzy c-methods (block FCM) in 2006. In this paper, based on Huang and Ng's (1999) fuzzy k-modes (FKM) method, we propose a block FKM clustering algorithm. Several examples are used to make the comparisons between block FCM and the proposed block FKM.

In statistics field, variation plays an important role. This is because greater variations in some features of data can provide more important information. Therefore, in this paper, we use this idea to select feature-weights in data. The proposed approach is simple to compute and interpret for feature-weights selection. Compared with the feature-weights proposed by Wang et al., Modha and Spangler, Pal et al. & Basak et al., we find that the proposed method provides a better clustering performance for the Iris data and color image segmentation and also has lower computational complexity..

In this paper, we first review several popular similarity measures between fuzzy sets and then extend those similarity measures to intuitionistic fuzzy sets. We also propose two new similarity measures between intuitionistic fuzzy sets. These similarity measures have been found to satisfy some similarity measure axioms. Several numerical experiments are performed to assess the performance of these measures. Numerical results clearly indicate these new measures to be superior in performance to the others. Finally, we apply the new measures to evaluate students' answerscripts. The experimental results show the superiority of the proposed measures for students' evaluation. © 2008 Wiley Periodicals, Inc.

In this paper we deal with the entropy of fuzzy sets. We first review several defined entropies of fuzzy sets and then propose a new one. Some comparisons are made with some existing entropies to show the effectiveness of the proposed one.

In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Moreover, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regressions (FCRs), still has the same drawbacks as FCM. In this paper, we propose the alpha-cut implemented fuzzy clustering algorithms, referred to as FCMalpha, which allow the data points being able to completely belong to one cluster. The proposed FCMalpha algorithms can form a cluster core for each cluster, where data points inside a cluster core will have a membership value of 1 so that it can resolve the drawbacks of FCM. On the other hand, the fuzziness index m plays different roles for FCM and FCMalpha. We find that the clustering results obtained by FCMalpha are more robust to noise and outliers than FCM when a larger m is used. Moreover, the cluster cores generated by FCMalpha are workable for various data shape clusters, so that FCMalpha is very suitable for embedding into switching regressions. The embedding of FCMalpha into switching regressions is called FCRalpha. The proposed FCRalpha provides better results than FCR for environments with noise or outliers. Numerical examples show the robustness and the superiority of our proposed methods.

In fuzzy clustering, the fuzzy c-mean (FCM) is a most used algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regression (FCR), was proposed by Hathaway and Bezdek in 1993. However, these FCRs always heavily depend on the initial values. In this paper, we propose mountain c-regressions (MCR) to solve the initial-value problem where the MCR is based on the modified mountain clustering method. We first transform the data set into a parameter space and then take a random sample from the transformed data set. We implement the modified mountain clustering on the random samples for switching regressions. The proposed MCR can form c estimated regression models for switching regression data sets without giving initials. Several examples show the accuracy and effectiveness of the proposed MCR method.

Uncertainty has been treated in science for several decades. It always exists in real systems. Probability has been traditionally used in modeling uncertainty. Belief and plausibility functions based on the Dempster–Shafer theory (DST) become another method of measuring uncertainty, as they have been widely studied and applied in diverse areas. Conversely, a fuzzy set has been successfully used as the idea of partial memberships of multiple classes for the presentation of unsharp boundaries. It is well used as the representation of human knowledge in complex systems. Nowadays, there exist several generalizations of belief and plausibility functions to fuzzy sets in the literature. In this article, we propose a new generalization of belief and plausibility functions to fuzzy sets based on the Sugeno integral. We then make comparisons of the proposed generalization with some existing methods. The results show the effectiveness of the proposed generalization, especially for being able to catch more information about the change of fuzzy focal elements. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1215–1228, 2007.

In this paper, we propose a generalized fuzzy clustering regularization (GFCR) model and then study its theoretical properties. GFCR unifies several fuzzy clustering algorithms, such as fuzzy c-means (FCM), maximum entropy clustering (MEC), fuzzy clustering based on Fermi-Dirac entropy, and fuzzy bidirectional associative clustering network, etc. The proposed GFCR becomes an alternative model of the generalized FCM (GFCM) that was recently proposed by Yu and Yang. To advance theoretical study, we have the following three considerations. 1) We give an optimality test to monitor if GFCR converges to a local minimum. 2) We relate the GFCR optimality tests to Occam's razor principle, and then analyze the model complexity for fuzzy clustering algorithms. 3) We offer a general theoretical method to evaluate the performance of fuzzy clustering algorithms. Finally, some numerical experiments are used to demonstrate the validity of our theoretical results and complexity analysis.

In this article we exploit the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets (IFSs). We then propose two families of entropy measures for IFSs and also construct the axiom definition and properties. Two definitions of entropy for IFSs proposed by Burillo and Bustince in 1996 and Szmidt and Kacprzyk in 2001 are used. The first one allows us to measure the degree of intuitionism of an IFS, whereas the second one is a nonprobabilistic-type entropy measure with a geometric interpretation of IFSs used in comparison with our proposed entropy of IFSs in the numerical comparisons. The results show that the proposed entropy measures seem to be more reliable for presenting the degree of fuzziness of an IFS. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 443–451, 2006.

This article presents a new similarity measure for LR-type fuzzy numbers. The proposed similarity measure is based on a defined metric between LR-type fuzzy numbers. It is known that an exponential operation is highly useful in dealing with the classical Shannon entropy and cluster analysis. We adopted, therefore, the exponential operation on this metric. Furthermore, we analyze its properties and make numerical comparisons to several similarity measures. The results show that the proposed similarity measure can overcome the drawbacks of the existing similarity measures. We then apply it to compound attributes for handling null queries to database systems. These applications can also be widely used in fuzzy queries to databases. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1001–1016, 2005.

In cluster analysis, the fuzzy c-means (FCM) clustering algorithm is the best known and most widely used method. It was proven to converge to either a local minimum or saddle points by Bezdek et al. Wei and Mendel produced efficient optimality tests for FCM fixed points. Recently, a weighting exponent selection for FCM was proposed by Yu et al. Inspired by these results, we unify several alternative FCM algorithms into one model, called the generalized fuzzy c-means (GFCM). This GFCM model presents a wide variation of FCM algorithms and can easily lead to new and interesting clustering algorithms. Moreover, we construct a general optimality test for GFCM fixed points. This is applied to theoretically choose the parameters in the GFCM model. The experimental results demonstrate the precision of the theoretical analysis.

In [1], Ozdemir and Akarun proposed an intercluster separation (ICS) fuzzy clustering algorithm. The ICS algorithm is useful in combined quantization and dithering. However, there are two errors in the update equations for the ICS algorithm. This correspondence first points out these errors and gives their corrections. Since the parameters m, c, and gamma are important factors in the performance of ICS, we also conduct a theoretical analysis of these ICS parameters. In order to analyze the parameters in ICS, we devise a theorem for the calculation of the Hessian matrix from the ICS objective function. We establish the fixed-point property of ICS based on the decomposition of the Hessian matrix and then analyze the effect of the parameters. Finally, we propose a numerical approach in choosing the appropriate parameters m and gamma for ICS. These experimental results give a better numerical perspective on the effect of parameters in ICS and have conclusions consistent with our theoretical analysis.

In this paper, we give similarity measures between type-2 fuzzy sets and provide the axiom definition and properties of these measures. For practical use, we show how to compute the similarities between Gaussian type-2 fuzzy sets. Yang and Shih's [22] algorithm, a clustering method based on fuzzy relations by beginning with a similarity matrix, is applied to these Gaussian type-2 fuzzy sets by beginning with these similarities. The clustering results are reasonable consisting of a hierarchical tree according to different levels.

Distribution mixtures are used as models to analyze grouped data. The estimation of parameters is an important step for mixture distributions. The latent class model is generally used as the analysis of mixture distributions for discrete data. In this paper, we consider the parameter estimation for a mixture of logistic regression models. We know that the expectation maximization (EM) algorithm was most used for estimating the parameters of logistic regression mixture models. In this paper, we propose a new type of fuzzy class model and then derive an algorithm for the parameter estimation of a fuzzy class logistic regression model. The effects of the explanatory variables on the response variables are described. The focus is on binary responses for the logistic regression mixture analysis with a fuzzy class model. An algorithm, called a fuzzy classification maximum likelihood (FCML), is then created. The mean squared error (MSE) based accuracy criterion for the FCML and EM algorithms to the parameter estimation of logistic regression mixture models are compared using the samples drawn from logistic regression mixtures of two classes. Numerical results show that the proposed FCML algorithm presents good accuracy and is recommended as a new tool for the parameter estimation of the logistic regression mixture models.

This paper presents an alternating optimization clustering procedure called a similarity-based clustering method (SCM). It is an effective and robust approach to clustering on the basis of a total similarity objective function related to the approximate density shape estimation. We show that the data points in SCM can self-organize local optimal cluster number and volumes without using cluster validity functions or a variance-covariance matrix. The proposed clustering method is also robust to noise and outliers based on the influence function and gross error sensitivity analysis. Therefore, SCM exhibits three robust clustering characteristics: 1) robust to the initialization (cluster number and initial guesses), 2) robust to cluster volumes (ability to detect different volumes of clusters), and 3) robust to noise and outliers. Several numerical data sets and actual data are used in the SCM to show these good aspects. The computational complexity of SCM is also analyzed. Some experimental results of comparing the proposed SCM with the existing methods show the superiority of the SCM method.

Uncertainty always exists in nature and real systems. It is known that probability has been used traditionally in modeling uncertainty. Since a belief function was proposed as an another type of measuring uncertainty, Dempster-Shafer theory (DST) has been widely studied and applied in diverse areas. Because of the advent of computer technology, the representation of human knowledge can be processed by a computer in complex systems. The analysis of fuzzy data becomes increasingly important. Up to date, there are several generalizations of DST to fuzzy sets proposed in the literature. In this article, we propose another generalization of belief function, plausibility function, and Dempster's combinational rule to fuzzy sets. We then make the comparisons of the proposed extension with some existing generalizations and show its effectiveness. © 2003 Wiley Periodicals, Inc.

The alternative fuzzy c-means (AFCM) clustering algorithm proposed by Wu and Yang (2001) has shown more robustness than the fuzzy c-means (FCM) on the basis of the robust statistic and the influence function. We propose a possibilistic type of AFCM by relaxing the restriction /spl Sigma//sub i=1//sup c/ /spl mu//sub i/(x)=1 for all data points x. The resulting cluster memberships constitute a possibilistic partition which is different to a fuzzy partition from AFCM. The comparisons of the proposed method to FCM, AFCM and possibilistic c-means are made.

This paper presents a new validity index for fuzzy clustering called a partition separation (PS) index. It uses the factors of a normalized partition coefficient and an exponential separation measure. According to the numerical comparisons with the other five cluster-validity indexes, the proposed PS index shows its high ability to produce a valid clustering of the data set.

Classification of objects is an important area in a variety of fields and applications. In the presence of full knowledge of the underlying joint distributions, Bayes analysis yields an optimal decision procedure and produces optimal error rates. Many different methods are available to make a decision in those cases where information of the underlying joint distributions is not presented. The k nearest neighbor rule (k-NNR) is a well-known nonparametric decision procedure. Many classification rules based on the k-NNR have already been proposed and applied in diverse substantive areas. The edited k-NNR proposed by D.L. Wilson (1972) would be an important one. Fuzzy theory, originated by L.A. Zadeh (1965), is widely used to represent the uncertainty of class membership. The fuzzy k-NNR has been proposed by several investigators. In this paper an edited type of the fuzzy k-NNR is developed. Next, some asymptotic properties of the proposed edited fuzzy k-NNR are created. Moreover, numerical comparisons are made between the proposed edited fuzzy k-NNR and the other fuzzy k-NNR. Those results confirm that the edited fuzzy k-NNR has a better performance than the fuzzy k-NNR.

Since Tanaka et al. (1982) proposed a study of linear regression analysis with a fuzzy model, fuzzy regression analysis has been widely studied and applied in a variety of substantive areas. Regression analysis in the case of heterogeneity of observations is commonly presented in practice. The authors' main goal is to apply fuzzy clustering techniques to fuzzy regression analysis. Fuzzy clustering is used to overcome the heterogeneous problem in the fuzzy regression model. They present the cluster-wise fuzzy regression analysis in two approaches: the two-stage weighted fuzzy regression and the one-stage generalized fuzzy regression. The two-stage procedure extends the results of Jajuga (1986) and Diamond (1988). The one-stage approach is created by embedding fuzzy clusterings into the fuzzy regression model fitting at each step of procedure. This kind of embedding in the one-stage procedure is more effective since the structure of regression line shape encountered in the data set is taken into account at each iteration of the algorithm. Numerical results give evidence that the one-stage procedure can be highly recommended in cluster-wise fuzzy regression analysis.