A Rough Set-Based Clustering Method with Modification of Equivalence Relations

This paper presents a rough set-based fuzzy clustering algorithm in which the objects of fuzzy clustering are initial clusters obtained in terms of equivalence relations. Initial clustering is performed directly by judging whether equivalence relations are equal, not computing the intersection of equivalence classes as usual, and the correctness of the theory is proved using rough set theory. Excessive generation of some small classes is suppressed by secondary clustering on the basis of defining fuzzy similarity between two initial clusters. Consequently the dimension of fuzzy similarity matrix is reduced. The definition of integrated approximation precision is given as evaluation of clustering validity. The algorithm can dynamically adjust parameter to get the optimal result. Some experiments were performed to validate this method. The results showed that the algorithm could handle preferably the clustering problems of both numerical data and nominal data.

Rough neighborhood graph: A method for proximity modeling and data clustering

The paper presents a novel, rough set-based clustering method that enables the evaluation of classification knowledge simplicity during the clustering procedure. The method iteratively refines equivalence relations so that they become a more simple set of relations that give adequate coarse classification to the objects. At each step of the iteration, the importance of the equivalence relation is evaluated on the basis of the newly introduced measure, indiscernibility degree. An indiscernibility degree is defined as a ratio of equivalence relations that classify the two objects into the same equivalence class. If an equivalence relation has the ability to discern two objects that have a high indiscernibility degree, a very fine classification is performed and then modified to regard them as indiscernible objects. The refinement is repeated, decreasing the threshold level of indiscernibility degree, and finally simple clusters can be obtained. Experimental results on the artificial data shows that iterative refinement of equivalence relation leads to successful generation of coarse clusters that can be represented by simple knowledge.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

The definition of basic rough sets [3] depends upon either a single equivalence relation defined on a universe or several equivalence relations defined over the universe, taken one each taken at a time. In the view of granular computing, classical rough set theory is based upon single granulation. Extending this notion, a rough set model based on multi-granulations (MGRS) was introduced in [5]. In this approach, approximations of sets were defined through multiple equivalence relations on the universe and their properties were investigated. Using hybridization of fuzzy set [13] with rough set the concept of rough fuzzy set was introduced by Dubois and Prade [1]. Recently, a Rough Fuzzy Set Model was introduced and studied by Wu and Kou [12], which is based on Multiple Granulation. Topological properties of rough sets introduced by Pawlak in terms of their types were recently studied by Tripathy and Mitra [10]. These were extended to the context of incomplete multi granulation by Tripathy and Raghavan [11]. In this paper we introduce incomplete multigranulation on rough fuzzy sets, study their basic properties and extend the topological properties in [11] to this context. Our findings are true for both complete and incomplete fuzzy rough set models based upon multi granulation.KeywordsRough SetsFuzzy rough setsequivalence relationstolerance relationstype of rough setsmulti granular fuzzy rough sets

In this paper we present a hybrid text categorization method based on Rough Sets theory. A central problem in good text classification for information filtering and retrieval (IF/IR) is the high dimensionality of the data. It may contain many unnecessary and irrelevant features. To cope with this problem, we propose a hybrid technique using Latent Semantic Indexing (LSI) and Rough Sets theory (RS) to alleviate this situation. Given corpora of documents and a training set of examples of classified documents, the technique locates a minimal set of co-ordinate keywords to distinguish between classes of documents, reducing the dimensionality of the keyword vectors. This simplifies the creation of knowledge-based IF/IR systems, speeds up their operation, and allows easy editing of the rule bases employed. Besides, we generate several knowledge base instead of one knowledge base for the classification of new object, hoping that the combination of answers of the multiple knowledge bases result in better performance. Multiple knowledge bases can be formulated precisely and in a unified way within the framework of RS. This paper describes the proposed technique, discusses the integration of a keyword acquisition algorithm, Latent Semantic indexing (LSI) with Rough Set-based rule generate algorithm, and provides experimental results. The test results show the hybrid method is better than the previous rough set-based approach.

The aim of this paper is to introduce nine types of rough Pythagorean fuzzy sets in UP-algebras and nine types of rough sets in UP-algebras. Then we study relation of these rough Pythagorean fuzzy sets and new types of Pythagorean fuzzy UP-filter under equivalence (congruence) relation. Moreover, we will also discuss \(t\)-level subsets of rough Pythagorean fuzzy sets in UP-algebras to study the relationships between rough Pythagorean fuzzy sets and rough sets in UP-algebras which we defined them above. Finally, we discuss the concept of homomorphisms between IUP-algebras and also study the direct and inverse images of four special subsets.

The further investigation of covering-based rough sets: Uncertainty characterization, similarity measure and generalized models

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

Rough sets were introduced by Pawlak as a model to capture impreciseness in data and several techniques in different directions have been developed to perform operations on such data through the model. But the basic model of rough set introduced is unigranular from the granular computing point of view. So, attempts have been made to introduce multigranulation using rough sets. Two models called optimistic multigranular rough sets and pessimistic multigranular rough sets were introduce by Qian et al in 2006 and 2010 respectively. However, like the basic case, the information granules generated were introduced by equivalence relations and as a result the granules are members of partitions induced on the universe. However, such concepts have limited applications due to the rarity of equivalence relations or equivalently partitions of universes. The notion of cover is more general than partition and covers are available in abundance in real life situations. So, even in the case of basic rough sets the definition has been extended to develop covering based rough sets. Recently, the concept of multigranularity has been extended by Liu et al to introduce four types of covering based optimistic multigranular rough sets (CBOMGRS). In this paper we introduce the notion of Covering Based Pessimistic Multigranular Rough Sets (CBPMGRS) and proved some properties. The important observation is that some of the properties of basic rough sets, which were not true for CBOMGRS, are true for CBPMGRS. So, CBPMGRS seems to be more natural extensions of the basic concepts than CBOMGRS.

Rough set model introduced by Pawlak in 1982 was dependent upon equivalence relations (equivalently on partitions), which had restricted its application due to its stringent requirements. The notion of covering is an extension of that of a partition and is generated by relations much less restricted than equivalence relations. Several covering based rough sets are found in literature. As the notion of rough sets, basic or otherwise are unigranular from the granular computing point of view, in order to handle more than one granular structures on a universe simultaneously, optimistic and pessimistic multigranular computing were introduced by Qian et al in 2006 and 2010 respectively. Combining the two concepts of covering and multigranulation, covering based multigranular models were introduced by Liu et al in 2012. The notion of mathematical equality of concepts is too stringent and less applicable in real life situations. In order to incorporate human knowledge into it, four types of approximate equalities basing upon rough sets were introduced by Novotny and Pawlak in 1985 and by Tripathy in 2011. In this paper, we study the covering based pessimistic multigranular approximate rough equalities and establish several of their properties and provide suitable examples for illustration and in constructing counter examples in the proofs. This is an attempt to generalize the notion of approximate equalities by one more level in order to extend their applicability.

Rough sets and matroids from a lattice-theoretic viewpoint

Several tasks under human activities need to be performed in a sequence of navigation and manipulation of objects. In several applications of human activities like robotics monitoring plays an important role. So, in these applications, processing of sequential data is of utmost importance. Because of the presence of imprecision intelligent clustering approaches using fuzzy or rough set techniques play a major role. The basic rough sets which are defined by using equivalence relations is less useful because of their scarcity in real life scenarios. As a result, covering based rough sets have been introduced which are more general and applicable to real world problems. In this paper, covering rough set based clustering approach is introduced and studied using refined first type of covering based rough sets. Through experimental analysis,illustrated the efficiency of proposed algorithm and provided a comparative analysis of this algorithm with other existing algorithms.

Rough sets, a technique of granular computing, deal with the vagueness and granularity in information systems. They are based on equivalence relations on a set, or equivalently, on a partition on the set. Covering is an extension of a partition and a more feasible concept for coping with incompleteness in information, thus the classical rough sets based on partition are extended to covering based rough sets. When a covering is introduced, there are more than one possibility to define the upper approximation. It is necessary to study the properties of these different types of upper approximations and the relation- ships among them. This paper presents three kinds of covering generalized rough sets and explores the relationships among them. The main results are conditions under which two different types of upper approximation operations are identical.