Survey on granularity clustering
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paper_title: Fuzzy logic = computing with words

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As its name suggests, computing with words (CW) is a methodology in which words are used in place of numbers for computing and reasoning. The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa. Thus, as an approximation, fuzzy logic may be equated to CW. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW. In CW, a word is viewed as a label of a granule; that is, a fuzzy set of points drawn together by similarity, with the fuzzy set playing the role of a fuzzy constraint on a variable. The premises are assumed to be expressed as propositions in a natural language. In coming years, computing with words is likely to evolve into a basic methodology in its own right with wide-ranging ramifications on both basic and applied levels.
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paper_title: The Present Studying State of Granular Computing and Studying of Granular Computing Based on the Semantics of Rough Logic

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In this article, the proposed background, the present studying state and its developing direction of granular computing are summarized. Author′s view are also inserted in the article. The applied universality of granular computing is discussed, including the applications of image searching technology in AI, diagnosis and treatemnt systems in medicine, calculus in seriate mathematics and other approximate reasoning. Granular computing could hopefully be the theoretical tool and methodology on information processing and studying other subject. Granulations and related properties based on the semantics of Rough logic are also discussed. Deductive reasoning of the granulations is created. Resolution principles and resolution strategies of granulations based on the semantics of Rough logic are proposed, it includes λ-resolution strategy and lock resolution strategy. The soundness of the granular resolution is proved. Application of granulations based on the semantics of Rough logic in problem resolving, expert systems and machine theorem proving of AI could hopefully be a new research idea and new theoretical tools. Finally, further developing perspectives of granulations based on the semantics of Rough logic are also proposed in this article.
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paper_title: Granular Computing Models for Knowledge Uncertainty

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Knowledge is not only an important cornerstone that constructs the cognitive ability in human beings,but is also one of the basic issues in intelligence science.With the development of intelligence science and technology,the study of knowledge uncertainty is attracting more and more attention.Knowledge uncertainty includes the inherent uncertainty of knowledge itself and the effect of external influence on knowledge.In the view of granular computing(GrC),the knowledge uncertainties of the fuzzy set theory model,the rough set theory model,the quotient space theory model,and some other related granular computing models are studied in this paper.The state-of-the-art and key issues of knowledge uncertainty are discussed.
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paper_title: Interpreting Concept Learning in Cognitive Informatics and Granular Computing

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Cognitive informatics and granular computing are two emerging fields of study concerning information and knowledge processing. A central notion to this processing is information and knowledge granularity. Concepts, as the basic units of thought underlying human intelligence and communication, may play a fundamental role when integrating the results from the two fields in terms of information and knowledge coding, representation, communication, and processing. While cognitive informatics focuses on information processing in the abstract, in machines, and in the brain, granular computing models such processing at multiple levels of granularity. In this paper, we examine a conceptual framework for concept learning from the viewpoints of cognitive informatics and granular computing. Within the framework, we interpret concept learning based on a layered model of knowledge discovery.
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paper_title: Granular Computing: basic issues and possible solutions

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Granular computing (GrC) may be regarded as a label of theories, methodologies, techniques, and tools that make use of granules, i.e., groups, classes, or clusters of a universe, in the process of problem solving. The main objective of this paper is to discuss basic issues of GrC, with emphasis on the construction of granules and computation with granules. After a brief review of existing studies, a set-theoretic model of GrC is proposed based on the notion of power algebras.
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paper_title: Research and Development of Granularity Clustering

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The rapid development of the Internet and information systems result in massive, high-dimensional, distributed, dynamic and complex data which often have characteristics of incompleteness, unreliability, inaccuracy, inconsistency and so on. Traditional cluster methods are very difficult to satisfy these data cluster demand. The granular computing has developed the cluster analysis research area, enhanced the use value further, and made theoretical significance closer to reality. The cluster may be carried on at different levels and from different angles through the granularity transformation. This will be beneficial to the solution of the problem. This article gives an overview of the essential relationship between granular computing and clustering, the advantages and scope of granule clustering, the research results about granule clustering based on rough sets, fuzzy sets and quotient space. Then the article discusses the necessity that three granular computing models integrate into each other to be used in granule clustering through comparative analysis, and make the analysis and the forecast to establish a unified granule clustering model at last.
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paper_title: The Puzzle of Granular Computing

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Rem tene, verba sequentur (Gaius J. Victor, Rome VI century b.c.) The ultimate goal of this book is to bring the fundamental issues of information granularity, inference tools and problem solving procedures into a coherent, unified, and fully operational framework. The objective is to offer the reader a comprehensive, self-contained, and uniform exposure to the subject.The strategy is to isolate some fundamental bricks of Computational Intelligence in terms of key problems and methods, and discuss their implementation and underlying rationale within a well structured and rigorous conceptual framework as well as carefully related to various application facets. The main assumption is that a deep understanding of the key problems will allow the reader to compose into a meaningful mosaic the puzzle pieces represented by the immense varieties of approaches present in the literature and in the computational practice. All in all, the main approach advocated in the monograph consists of a sequence of steps offering solid conceptual fundamentals, presenting a carefully selected collection of design methodologies, discussing a wealth of development guidelines, and exemplifying them with a pertinent, accurately selected illustrative material.
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paper_title: Foundations of cognitive science

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"Posner is to be congratulated on having brought together so many distinguished authors, each contributing a tutorial chapter on their particular branch of the science. The best of these are exemplary, and few of them fall below a high standard." -- L.H. Shaffer, "Nature" "A beautifully lucid account.... Highly recommended." -- Nick Beard, "New Scientist" What is cognitive science? Foundations of Cognitive Science answers this question in a way that gives a feeling for the excitement, ferment, and accomplishments of this new field. It is the first broad treatment of cognitive science at an advanced level. Complete and authoritative, Foundations of Cognitive Science covers the major architectures; provides background in philosophy, linguistics, cognitive psychology, and neuroscience; and deals with methods for studying both brain and mind. All of the chapters have been written especially for the book by the leading scholars in the field. " A Bradford Book"
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paper_title: Web Document Clustering Research Based on Granular Computing

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In this paper, a method of web document clustering based on granular computing (WDCGrc) is presented. The method computes the weight value of the words in documents by adopting the TF-IDF principle. Meanwhile, combinative ways defining documents threshold and average weight value are adopted to reduce dimensions and extract the keywords in each document. The paper establishes the transformation between the keywords in documents and the binary granules, and adopts the algorithm of association rules based on granular computing to obtain frequent item sets between documents. Bring in the set theory thought, numbers of the same word between documents as the document similarity and the clustering result is obtained. The experiment shows that the method is practical and feasible, with good quality of clustering.
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paper_title: Parallel Artificial Immune Clustering Algorithm Based on Dynamic Granulation

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This paper discusses measure of harmony between clustering and transcendent knowledge,and a new clustering algorithm is proposed,which is parallel artificial immune clustering algorithm based on dynamic granulation.Artificial immune system model has the characteristics,such as parallel,random search,and maintains diversity.It is unified to dynamic granulation model.In the process of granulation changing,appropriate granulation can be made by adjusting,which can ensure clustering efficiency and quality of the new algorithm.Test results show that the algorithm is reasonable when the problem is handled,which has many samples,many attributions and many classifications.
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paper_title: Spatial clustering property and its self-similarity in membrane potentials of hippocampal CA1 pyramidal neurons for a spatio-temporal input sequence

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To clarify how the information of spatiotemporal sequence of the hippocampal CA3 affects the postsynaptic membrane potentials of single pyramidal cells in the hippocampal CA1, the spatio-temporal stimuli was delivered to Schaffer collaterals of the CA3 through a pair of electrodes and the post-synaptic membrane potentials were recorded using the patch-clamp recording method. The input–output relations were sequentially analyzed by applying two measures; “spatial clustering” and its “self-similarity” index. The membrane potentials were hierarchically clustered in a self-similar manner to the input sequences. The property was significantly observed at two and three time-history steps. In addition, the properties were maintained under two different stimulus conditions, weak and strong current stimulation. The experimental results are discussed in relation to theoretical results of Cantor coding, reported by Tsuda (Behav Brain Sci 24(5):793–847, 2001) and Tsuda and Kuroda (Jpn J Indust Appl Math 18:249–258, 2001; Cortical dynamics, pp 129–139, Springer-Verlag, 2004).
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paper_title: Application of Rough Set and K-means clusteringin image segmentation

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Rough Set theory is a new mathematical tool to deal with problems on vagueness and uncertainty. An image segmentation method based on Rough Set theory and K-means clustering is presented. The original image is segmented according to the relation of equal value. By applying value reduct to the attribute values, different regions are classified based on indiscernibility. The experimental results indicate that the method can improve veracity and stability of image segmentation.
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paper_title: Fuzzy Clustering of WEB Page

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Nowdays,there are much information in Internet,but how to manage and classify it is an inevitable and meaning ful thing This paper described a method in which WEB pages are clustered or auto-classified by using FUZZY SET theory and FUZZY CLUSTERING technique. Experiment of the WEB page's FUZZY CLUSTERING indicates that FUZZY CLUSTERING is effective method for WEB page.
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paper_title: Research of text clustering based on fuzzy granular computing

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The traditional K-means is very sensitive to initial clustering centers and the clustering result will wave with the different initial input.To remove this sensitivity,a new method was proposed to get initial clustering centers.This method is as follows:provide a normalized distance function in the fuzzy granularity space of data objects,then use the function to do a initial clustering work to these data objects who has a less distance than granularity d_λ,then get the initial clustering centers.The test shows this method has such advantages on increasing the rate of accuracy and reducing the program times.
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paper_title: Boosting of granular models

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In this study, we are concerned with the design of granular modeling being originally proposed by Pedrycz and Vasilakos. The enhancement of the development process comes in the form of the boosting mechanism applied to the generic model. In comparison with the original topology of the model studied so far, we augment it by a bias term and investigate its role in the overall architecture. Second, treating the granular model as a weak learner, we discuss the underlying mechanisms of boosting which in this setting has to be refined so that it handles the continuous case. From the design standpoint, we are interested in studying the following issues: (a) effectiveness of boosting when applied in this modeling framework along with its numeric quantification, and (b) impact of information granularity at which the granular models are developed on the improvements offered by boosting procedure itself. Numeric experiments help quantify the performance of the boosted granular models and gain a detailed view at the efficiency of the boosting strategy vis-a-vis different design scenarios.
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paper_title: Survey of Clustering Algorithms in Data Mining

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Clustering is an important technique in Data Mining(DM) for the discovery of data distribution and latent data pattern.This paper provides a detailed survey of current clustering algorithms in DM at first,then it makes a comparison(among) them,illustrates the merits existing in them,and identifies the problems to be solved and the new directions in the future according to the application requirements in multimedia domain.
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paper_title: Principle of Granularity in Clustering and Classification

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From the view of information granularity, it is clear that there exists a tight link between clustering and classification. Clustering is a computation under the same granularity, while classification is under different ones. This paper proposes a new classification algorithm based on theory of information granularity, and it shows better generalization ability in experiments.
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paper_title: A new fuzzy clustering algorithm for optimally finding granular prototypes

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Prototype Reasoning using granular objects is an important technology for knowledge discovery. Fuzzy clustering can be used to generate prototypes with different granularities. In order to find optimal granular prototypes through fuzzy clustering, for given data, two conditions are necessary: a good cluster validity function, which can be applied to evaluate the goodness of cluster schemes for varying number of clusters (different granularity); a good cluster algorithm that can produce an optimal solution for a fixed number of clusters. To satisfy the first condition, a new validity measure called granularity-dissimilarity (GD) measure is proposed, which is stable in evaluating granularities and works well even when the number of clusters is very large. For the second condition, we propose a new algorithm called multi-step maxmin and merging algorithm (3M algorithm). Experiments show that, when used in conjunction with the new cluster validity measure, 3M algorithm produces better results on the experimental data sets than several alternatives.
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paper_title: Knowledge acquisition through information granulation for imbalanced data

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When learning from imbalanced/skewed data, which almost all the instances are labeled as one class while far few instances are labeled as the other class, traditional machine learning algorithms tend to produce high accuracy over the majority class but poor predictive accuracy over the minority class. This paper proposes a novel method called ‘knowledge acquisition via information granulation’ (KAIG) model which not only can remove some unnecessary details and provide a better insight into the essence of data but also effectively solve ‘class imbalance’ problems. In this model, the homogeneity index (H-index) and the undistinguishable ratio (U-ratio) are successfully introduced to determine a suitable level of granularity. We also developed the concept of sub-attributes to describe granules and tackle the overlapping among granules. Seven data sets from UCI data bank, including one imbalanced diagnosis data (pima-Indians-diabetes), are provided to evaluate the effectiveness of KAIG model. By using different performance indexes, overall accuracy, G-mean and Receiver Operation Characteristic (ROC) curve, the experimental results comparing with C4.5 and Support Vector Machine (SVM) demonstrate the superiority of our method.
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paper_title: Soft-Link Spectral Clustering for Information Extraction

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Unsupervised spectral clustering methods can yield good performance when identifying crisp clusters with low complexity since the learning algorithm does not rely on finding the local minima of an objective function and rather use spectral properties of the graph. Nonetheless, the performance of such approaches are usually affected by their uncertain parameters. Using the underlying structure of a general spectral clustering method, in this paper a new soft-link spectral clustering algorithm is introduced to identify clusters based on fuzzy k-nearest neighbor approach. We construct a soft weight matrix of a graph by identifying the upper and lower boundaries of learning parameters of the similarity function, specifically the fuzzifier parameter (fuzziness) of the Fuzzy k-Nearest Neighbor algorithm. The algorithm allows perturbations on the graph Laplace during the learning stage by the changes on these learning parameters. With the empirical analysis using an artificial and a real textual entailment dataset, we demonstrate that our initial hypothesis of implementing soft links can improve the classification performance of final outcome.
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paper_title: Efficient uncertainty minimization for fuzzy spectral clustering.

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Spectral clustering uses the global information embedded in eigenvectors of an inter-item similarity matrix to correctly identify clusters of irregular shape, an ability lacking in commonly used approaches such as k -means and agglomerative clustering. However, traditional spectral clustering partitions items into hard clusters, and the ability to instead generate fuzzy item assignments would be advantageous for the growing class of domains in which cluster overlap and uncertainty are important. Korenblum and Shalloway [Phys. Rev. E 67, 056704 (2003)] extended spectral clustering to fuzzy clustering by introducing the principle of uncertainty minimization. However, this posed a challenging nonconvex global optimization problem that they solved by a brute-force technique unlikely to scale to data sets having more than O(10;{2}) items. Here we develop a method for solving the minimization problem, which can handle data sets at least two orders of magnitude larger. In doing so, we elucidate the underlying structure of uncertainty minimization using multiple geometric representations. This enables us to show how fuzzy spectral clustering using uncertainty minimization is related to and generalizes clustering motivated by perturbative analysis of almost-block-diagonal matrices. Uncertainty minimization can be applied to a wide variety of existing hard spectral clustering approaches, thus transforming them to fuzzy methods.
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paper_title: Fuzzy Clustering With Viewpoints

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In this study, we introduce a certain knowledge-guided scheme of fuzzy clustering in which domain knowledge is represented in the form of so-called viewpoints. Viewpoints capture a way in which the user introduces his/her point of view at the data by identifying some representatives, which, being treated as externally introduced prototypes, have to be included in the clustering process. More formally, the viewpoints (views) augment the original, data-based objective function by including the term that expresses distances between data and the viewpoints. Depending upon the nature of domain knowledge, the viewpoints are represented either in a plain numeric format (considering that there is a high level of specificity with regard to how one establishes perspective from which the data need to be analyzed) or through some information granules (which reflect a more relaxed way in which the views at the data are being expressed). The detailed optimization schemes are presented, and the performance of the method is illustrated through some numeric examples. We also elaborate on a way in which the clustering with viewpoints enhances fuzzy models and mechanisms of decision making in the sense that the resulting constructs reflect the preferences and requirement that are present in the modeling environment.
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paper_title: Spectral clustering with fuzzy similarity measure

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Spectral clustering algorithms have been successfully used in the field of pattern recognition and computer vision. The widely used similarity measure for spectral clustering is Gaussian kernel function which measures the similarity between data points. However, it is difficult for spectral clustering to choose the suitable scaling parameter in Gaussian kernel similarity measure. In this paper, utilizing the prototypes and partition matrix obtained by fuzzy c-means clustering algorithm, we develop a fuzzy similarity measure for spectral clustering (FSSC). Furthermore, we introduce the K-nearest neighbor sparse strategy into FSSC and apply the sparse FSSC to texture image segmentation. In our experiments, we firstly perform some experiments on artificial data to verify the efficiency of the proposed fuzzy similarity measure. Then we analyze the parameters sensitivity of our method. Finally, we take self-tuning spectral clustering and Nystrom methods for baseline comparisons, and apply these three methods to the synthetic texture and remote sensing image segmentation. The experimental results show that the proposed method is significantly effective and stable.
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paper_title: A Data Reduction Algorithm Using Clustering Based on Rough Set Theory

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To the question of the optimization of the enterprise resources, the study presented in this paper reduces the original data with clustering algorithm firstly, and eliminates the suspicious information, which makes the data consistent. Then the data are made a qualitative analysis and reduced based on rough set theory. The data are reduced in both horizontal and vertical directions by using hierarchical clustering and rough set methods. The favorable economic profit can be obtained by using this algorithm in the process of the optimization of the enterprise resources configuration.
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paper_title: Interval set clustering

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Rough k-means clustering describes uncertainty by assigning some objects to more than one cluster. Rough cluster quality index based on decision theory is applicable to the evaluation of rough clustering. In this paper we analyze rough k-means clustering with respect to the selection of the threshold, the value of risk for assigning an object and uncertainty of objects. According to the analysis, clusters presented as interval sets with lower and upper approximations in rough k-means clustering are not adequate to describe clusters. This paper proposes an interval set clustering based on decision theory. Lower and upper approximations in the proposed algorithm are hierarchical and constructed as outer-level approximations and inner-level ones. Uncertainty of objects in out-level upper approximation is described by the assignment of objects among different clusters. Accordingly, ambiguity of objects in inner-level upper approximation is represented by local uniform factors of objects. In addition, interval set clustering can be improved to obtain a satisfactory clustering result with the optimal number of clusters, as well as optimal values of parameters, by taking advantage of the usefulness of rough cluster quality index in the evaluation of clustering. The experimental results on synthetic and standard data demonstrate how to construct clusters with satisfactory lower and upper approximations in the proposed algorithm. The experiments with a promotional campaign for the retail data illustrates the usefulness of interval set clustering for improving rough k-means clustering results.
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paper_title: Rough entropy hierarchical agglomerative clustering in image segmentation

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In data clustering there is a constant demand on development of new algorithmic schemes capable of robust and correct data handling. This demand has been additionally highly fueled and increased by emerging new technologies in data imagery area. Hierarchical clustering represents established data grouping technique with a wide spectrum of application, especially in image analysis branch. In the paper, a new algorithmic rough entropy framework has been applied in the hierarchical clustering setting. During cluster merges the quality of the resultant merges has been assessed on the base of the rough entropy. Incorporating rough entropy measure as the evaluation of cluster quality takes into account inherent uncertainty, vagueness and impreciseness. The experimental results suggest that hierarchies created during rough entropy based merging process are robust and of high quality, giving possible area for future research applications in real implementations.
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paper_title: A rough set approach for selecting clustering attribute

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A few of clustering techniques for categorical data exist to group objects having similar characteristics. Some are able to handle uncertainty in the clustering process while others have stability issues. However, the performance of these techniques is an issue due to low accuracy and high computational complexity. This paper proposes a new technique called maximum dependency attributes (MDA) for selecting clustering attribute. The proposed approach is based on rough set theory by taking into account the dependency of attributes of the database. We analyze and compare the performance of MDA technique with the bi-clustering, total roughness (TR) and min-min roughness (MMR) techniques based on four test cases. The results establish the better performance of the proposed approach.
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paper_title: A Rough Set-Based Hierarchical Clustering Algorithm

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Rough set theory is a new mathematical tool to deal with vagueness and uncertainty It has received considerable attention and has been applied in a variety of areas in recent years In this paper, rough set theory is applied to clustering analysis in knowledge discovery A lot of definitions such as the local indiscernibility relation, the local and total indiscernibility degree between two objects, the indiscernibility degree between two clusters and the integrated approximation rate of the clustering result are given Based on these definitions, a rough set based hierarchical clustering algorithm is proposed It can automatically adjust the parameter in order to get the more optimum result Some experiments are made on the data sets in UCI (University of California, Irvine) machine learning repository The experimental results show that the algorithm is feasible and has good clustering performance especially for symbolic attributes
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paper_title: Research of Individual Neural Network Generation and Ensemble Algorithm Based on Quotient Space Granularity Clustering

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The aim of this paper is to develop an individual neural network generation and ensemble algorithm based on quotient space granularity clustering. Firstly, we give the characteristics of the quotient space granularity and affinity propagation(AP) clustering. Secondly, we introduce the quotient space concept to the AP clustering analysis, which can find an optimal granularity from all possible granularities. Then using improved AP clustering algorithm to seek optimal results of sample clustering and using different individual neural network to learn different categories of samples so that the degree of difference between networks and the generalization ability of neural network ensemble(NNE) can be improved. Further, according to the degree of correlation between the input data and the sample category to adaptively adjust ensemble weights. The algorithm proposed here is not only a method of generating the individual neural networks, but also can adaptively adjust ensemble weights of individual neural network. Experiments show that our proposed method is validity and correctness.
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paper_title: Covering clustering algorithm based on quotient space granularity

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This paper introduced the basic idea of the covering algorithm and some theories of the quotient space,put forward a new covering clustering algorithm based on quotient space granularity.And the results show the feasibility and effectiveness of the covering clustering algorithm.By this means,it is efficient for the large number of observations.
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paper_title: An Alternative Covering Design Algorithm of Multi-layer Neural Networks

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In this paper, the authors present a new principle for designing a sort of multi layer weight sum and threshold neural networks. The design process consists of two steps. First, each class of the given training samples is covered by using neighbor coverings as less as possible. Then a neural network is specifically designed by an approach called alternative covering algorithm. The simulation results of two representative hard classification problems are shown. One is so called “two spirals separation” problem, the other is “the learning problem of infinitetraining samples”. These simulation results are given to illustrate the effectiveness of the new method.
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paper_title: An entropy weighting mixture model for subspace clustering of high-dimensional data

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In high-dimensional data, clusters of objects usually exist in subspaces; besides, different clusters probably have different shape volumes. Most existing methods for high-dimensional data clustering, however, only consider the former factor. They ignore the latter factor by assuming the same shape volume value for different clusters. In this paper we propose a new Gaussian mixture model (GMM) type algorithm for discovering clusters with various shape volumes in subspaces. We extend the GMM clustering method to calculate a local weight vector as well as a local variance within each cluster, and use the weight and variance values to capture main properties that discriminate different clusters, including subsets of relevant dimensions and shape volumes. This is achieved by introducing negative entropy of weight vectors, along with adaptively-chosen coefficients, into the objective function of the extended GMM. Experimental results on both synthetic and real datasets show that the proposed algorithm outperforms its competitors, especially when applying to high-dimensional datasets.
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paper_title: Enhanced soft subspace clustering integrating within-cluster and between-cluster information

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While within-cluster information is commonly utilized in most soft subspace clustering approaches in order to develop the algorithms, other important information such as between-cluster information is seldom considered for soft subspace clustering. In this study, a novel clustering technique called enhanced soft subspace clustering (ESSC) is proposed by employing both within-cluster and between-class information. First, a new optimization objective function is developed by integrating the within-class compactness and the between-cluster separation in the subspace. Based on this objective function, the corresponding update rules for clustering are then derived, followed by the development of the novel ESSC algorithm. The properties of this algorithm are investigated and the performance is evaluated experimentally using real and synthetic datasets, including synthetic high dimensional datasets, UCI benchmarking datasets, high dimensional cancer gene expression datasets and texture image datasets. The experimental studies demonstrate that the accuracy of the proposed ESSC algorithm outperforms most existing state-of-the-art soft subspace clustering algorithms.
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Title: Survey on Granularity Clustering

Section 1: Introduction
Description 1: Write about the background, challenges, and opportunities in processing high-dimensional massive data using granularity clustering.

Section 2: Granular Computing and Clustering Analysis
Description 2: Discuss the essence of granular computing and clustering analysis, their relationship, and the extension of clustering to the field of soft computing.

Section 3: Principle of Granularity in Clustering
Description 3: Explain the concept and principles of granularity in clustering, its philosophical background, and how it affects clustering results.

Section 4: Granularity Clustering Theories
Description 4: Introduce the various theories and methods in granularity clustering including fuzzy clustering analysis, rough clustering, clustering based on quotient space, and hybrid approaches like rough-fuzzy sets and fuzzy quotient space.

Section 5: Granular Computing for Subspace Clustering
Description 5: Explore the application of granular computing in subspace clustering, discussing techniques for finding subspaces and classifying samples within those subspaces.

Section 6: Conclusions and Prospect
Description 6: Summarize the importance of clustering in handling massive and complex data, the role of granularity clustering, and future research directions in this field.