An attempt of reconstruction of object-oriented rough set models

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..

Quantitative rough sets based on subsethood measures

The paper aims to establish topological links between perception of objects (as it is defined in the framework of near sets) and classification of these objects (as it is defined in the framework of rough sets). In the near set approach, the discovery of near sets (i.e. sets containing objects with similar descriptions) starts with the selection of probe functions which provide a basis for describing and discerning objects. On the other hand, in the rough set approach, the classification of objects is based on object attributes which are collected into information systems (or data tables). As is well-known, an information system can be represented as a topological space (U, τ E ). If we pass froman approximation space (U,E) to the quotient space U/E, where points represent indiscernible objects of U, then U/E will be endowed with the discrete topology induced (via the canonical projection) by τ E . The main objective of this paper is to show how probe functions can provide new topologies on the quotient set U/E and, in consequence, new (perceptual) topologies on U.

Covering based intuitionistic fuzzy (IF) rough set is a generalization of granular computing and covering based rough sets. By combining covering based rough sets, IF sets and fuzzy rough sets, we introduce three classes of coverings based IF rough set models via IF$$\beta $$-neighborhoods and IF complementary $$\beta $$-neighborhood (IFC$$\beta $$-neighborhood). The corresponding axiomatic systems are investigated, respectively. In particular, the rough and precision degrees of covering based IF rough set models are discussed. The relationships among these types of coverings based IF rough set models and covering based IF rough set models proposed by Huang et al. (Knowl Based Syst 107:155–178, 2016). Based on the theoretical analysis for coverings based IF rough set models, we put forward intuitionistic fuzzy TOPSIS (IF-TOPSIS) methodology to multi-attribute decision-making (MADM) problem with the evaluation of IF information problem. An effective example is to illustrate the proposed methodology. Finally, we deal with MADM problem with the evaluation of fuzzy information based on CFRS models. By comparative analysis, we find that it is more effective to deal with MADM problem with the evaluation of IF information based on CIFRS models than the one with the evaluation of fuzzy information based on CFRS models.

The basic notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation. This paper proposes an intuitionistic fuzzy rough set (IFRS) model which is a combination of intuitionistic fuzzy set (IFS) and rough set. We approximate an IFS by using soft binary relations instead of fixed binary relations. By using this technique, we get two pairs of intuitionistic fuzzy (IF) soft sets, called the upper approximation and lower approximation with respect to foresets and aftersets. Properties of newly defined rough set model (IFRS) are studied. Similarity relations between IFS with respect to this rough set model (IFRS) are also studied. Finally, an algorithm is constructed depending on these approximations of IFSs and score function for decision-making problems, although a method of decision-making algorithm has been introduced for fuzzy sets already. But, this new IFRS model is more accurate to solve the problem because IFS has degree of nonmembership and degree of hesitant.

Six-Set Approximation Theorem of neighborhood related rough sets

The uncertainty of probabilistic rough sets in multi-granulation spaces

Rough set theory for the interval-valued fuzzy information systems

Optimal approximations with Rough Sets and similarities in measure spaces

It is an important subject to mine valuable knowledge from complex and massive data in the era of big data. Rough set theory is a new mathematical tool for dealing with uncertain and inaccurate data, decision‐theoretic rough set model (DTRS), as an extension of classical rough set model, is used to analyze decision information systems and multi‐granulation decision‐theoretic rough set model (MG‐DTRS) can analyze and process target concepts from different angles and levels. However, the classical DTRS model exists some limitations in dealing numerical or hybrid‐valued data. Considering the different influence of numerical features and symbolic features on decision‐making, the paper proposes an incomplete neighborhood multi‐granulation decision‐theoretic rough set model in hybrid‐valued decision system through integrating MG‐DTRS with neighbourhood rough sets, and two types of neighborhood multi‐granulation decision‐theoretic set models are emphatically analysed. Furthermore, taking pessimistic and optimistic neighborhood multi‐granulation decision‐theoretic rough sets as examples, the implementation algorithms and related properties of the two type of models are studied. Finally, the relationship between the proposed model and other models is analyzed through formula derivation. The model proposed in this paper can effectively solve the decision‐making problem of hybrid‐valued incomplete information system through multi‐angle and multi‐level analysis.

Most previous studies on rough sets focused on attribute reduction and decision rule mining on a single concept level. Data with attribute value taxonomies (AVTs) are, however, commonly seen in real-world applications. In this paper, we extend Pawlak's rough set model, and propose a novel multi-level rough set model (MLRS) based on AVTs and a full-subtree generalization scheme. Paralleling with Pawlak's rough set model, some conclusions related to the MLRS are given. Meanwhile, a novel concept of cut reduction based on MLRS is presented. A cut reduction can induce the most abstract multi-level decision table with the same classification ability on the raw decision table, and no other multi-level decision table exists that is more abstract. Furthermore, the relationships between attribute reduction in Pawlak's rough set model and cut reduction in MLRS are discussed. We also prove that the problem of cut reduction generation is NP-hard, and develop a heuristic algorithm named CRTDR for computing the cut reduction. Finally, an approach named RMTDR for mining multi-level decision rule is provided. It can mine decision rules from different concept levels. Example analysis and comparative experiments show that the proposed methods are efficient and effective in handling the problems where data is associated with AVTs.

Based on the analysis of the rough set model on a tolerance relation and the fuzzy rough set, two types of fuzzy rough sets models on tolerance relations are constructed and researched. Then we propose the optimistic and pessimistic multi-granulation fuzzy rough sets models in a fuzzy tolerance approximation space with the point view of granular computing. In these models, the fuzzy lower and upper approximations of a fuzzy set are defined on multiple fuzzy tolerance relations. It follows the research on the properties of the fuzzy lower and upper approximations of the new models. The fuzzy rough set model, rough set model on a tolerance relation and multi-granulation rough sets models are special cases of the new ones from the perspective of the considered concepts, related relations and granular computing. The new models are meaningful generalizations of the classical rough sets.

A neighborhood rough set model with nominal metric embedding

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.

In order to effectively handle the real-valued data sets in practice, it is valuable from theoretical and practical aspects to combine fuzzy rough set and variable precision rough set so that a powerful tool can be developed. That is, the model of fuzzy variable precision rough set, which not only can handle numerical data but also is less sensitive to misclassification and perturbation,In this paper, we propose a new variable precision rough fuzzy set by introducing the variable precision parameter to generalized rough fuzzy set, i.e., the variable precision rough fuzzy set based on general relation. We, respectively, define the variable precision rough lower and upper approximations of any fuzzy set and it level set with variable precision parameter by constructive approach. Also, we present the properties of the proposed model in detail. Meanwhile, we establish the relationship between the variable precision rough approximation of a fuzzy set and the rough approximation of the level set for a fuzzy set. Furthermore, we give a new approach to uncertainty measure for variable precision rough fuzzy set established in this paper in order to overcome the limitations of the traditional methods. Finally, some numerical example are used to illuminate the validity of the conclusions given in this paper.