Covering Based Pessimistic Multigranular Approximate Rough Equalities and Their Properties

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

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.

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

The variable precision rough set (VPRS) model generalizes the Pawlak rough set model with variable parameters. The Bayesian rough set (BRS) model improves the VPRS model with non-parametric modification by using the prior probability as a reference. This paper presents two research results related to rough set model and rough vague sets. One is a modification to the Bayesian rough set model and the other is a modification to rough vague sets. First, the Bayesian rough set model is analyzed and discussed. Second, a modification to this model is proposed and verified. Finally, a modification to rough vague sets is presented and its related properties are discussed..

Binary information table, multi-valued information table and set-valued information table are three kinds of information systems often encountered in information processing. For any information system, we can often induce different information granular structures, and then construct the corresponding rough set models. Generally speaking, for the same information system, three models of Pawlak rough set, covering rough set and multi-granulation rough set can be induced according to different rules. These three kinds of rough set models are effective tools for data mining and information processing. This paper studies the relationship among Pawlak rough set, covering rough set and multi-granularity rough set induced in binary information table, multi-valued information table and set-valued information table, and obtains many important conclusions. The research content of this paper effectively connects the theories, methods and applications of Pawlak rough set, covering rough set and multi-granularity rough set, which not only enriches the rough set theory, but also expands the application prospect of rough set.

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.

Rough set theory for the interval-valued fuzzy information systems

Reasoning about knowledge and knowledge representation has been an issue of concern in Artificial Intelligence for over two decades. More recently, researchers have realized that these issues also play a crucial role in other subfields of computer science, including cryptography, distributed computation, data base theory, and expert systems.Any knowledge representation system provides information, usually incomplete, about some parts of perceivable reality, and different authors provide different formal models for the knowledge analysis.Recently, the notion of rough sets [Pawlak 1982] was introduced, which provides a systematic framework for the study of the problems arising from inprecise and insufficient knowledge.A rough set, like fuzzy set, is a matematical model used to deal with approximate classification. These concepts have been proved to be independent of one another [Pawlak 1985(2)].A number of experimental systems has been implemented based on the deterministic rough-set theory. These applications include analysis of medical data of patients with dudendal ulcer [Fibak 1986], control algorithm aquisition in the process of kiln production [Mrózek 1985],linguistic pattern recognition [Wojcik 1986], and approximate reasoning [Rasiowa 1987].A natural, probabilistic extention of the rough-set model has been proven to be useful mathematical tool for dealing with some problems occuring in machine learning like generation of decision rules from inconsistent training examples [Wong 1986(1)] or database design [Yasdi 1987].The problem of teaching a student by several imperfect teachers ([Pettorossi,Raś, Zemankova 1987], [Raś, Zemankova 1986]) is handled also within rough-set framework. The authors deal with the case when the teacher and the student understand each other only partially and define a formal system which provides the basis for writing procedures which generate rules of a knowledge base.The model used in all these investigations assume that in the process of perception one distinguishes entities (objects) and their properties. Properties of objects are perceived through assignment of some characteristics (attributes) and their values of the objects. In this way a universe of discourse (a problem domain) consisting of objects and elementary information items providing characterization of these objects in terms of attributes and attributes values is established. In general, information about objects obtained in this way is not sufficient to characterize objects uniquely; that is, it is not possible to distinguish all the objects by means of the admitted attributes and their values. This means that objects are recognized up to indiscernibility relation determined by elementary information items. Any two objects are indiscernible whehever they assume the same values for all the attributes under consideration.Next we form concepts; that is, we agregate some objects into sets. Information about a concept is composed from information about objects which are instances of concepts. Since objects are not necessarily distinguishable, information characterizing a concept may be ambigous to some extent. In this case we want to have at least some approximation of our information and we express it in terms of rough sets.All cited above approaches use purely semantical methods in their investigations, exept [Pettorossi,Raś, Zemankova 1987], [Raś, Zemankova 1986] where mixed semantical and syntactical (formal system) methods are used.In our paper we focus on purely syntactical methods and problems which can be solved by them. One of the problems is a problem of static learning as defined in [Pawlak 1985(1)].Its main idea is as follows. Suppose now that we are given a finite subset U of the set of objects O B. Elements of U will be called training examples and O B is called a training set.Assume futher that O B is classified into non-empty, disjoint subsets O 1, O 2, ..., O n , (n>2) by a teacher (or expert). The classification represents teacher's knowledge of objects from O B.Let's now assume that there is another person, a student, who is able to characterize each object from O B in terms of attributes from a set A. Description of objects in terms of attributes from A represents student's knowledge of objects from O B.Now we want to know if it is possible to describe the classification O 1, O 2, ..., O n provided by the teacher in terms of attributes from A, or more exactly, to find a classification algorithm which provides teacher's classification on the basis of properties of objects expressed in terms of attributes defined in the student's system S.We give here a purely syntactical, easy programable method to do so. The method consists of two procedures. First procedure, lets call it Procedure one generates, for a given term of a standard term language for a given system, its equivalent normal form.The next procedure, called Procedure Two will verify the correctness of proper decision algorithm.

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.

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.

Soft rough fuzzy sets and soft fuzzy rough sets

The notion of Rough sets introduced by Pawlak has been extended in many directions to enhance its modelling power. One such approach is to reduce the restriction of the base relation being an equivalence relation. Adding the flavour of fuzzy sets to it a fuzzy proximity relation was used to generate a fuzzy approximation space by De et al. in 1999 and hence the rough sets on fuzzy approximation spaces could be generated. These are much more general than the basic rough sets and also the rough sets defined on proximity relations. However, some of the results established in this direction by De et al. have been found to be faulty. In this paper we show through examples that the results are actually faulty and provide their correct versions. Also, we establish some more properties of these rough sets. A real life application is provided to show the application of the results.

In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

Rough sets and matroids from a lattice-theoretic viewpoint