Measures of uncertainty for a fuzzy probabilistic approximation space

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

Different kinds of generalized rough sets based on neighborhoods with a medical application

Approximation space can be said to play a critical role in the accuracy of the set’s approximations. The idea of “approximation space” was introduced by Pawlak in 1982 as a core to describe information or knowledge induced from the relationships between objects of the universe. The main objective of this paper is to create new types of rough set models through the use of different neighborhoods generated by a binary relation. New approximations are proposed representing an extension of Pawlak’s rough sets and some of their generalizations, where the precision of these approximations is substantially improved. To elucidate the effectiveness of our approaches, we provide some comparisons between the proposed methods and the previous ones. Finally, we give a medical application of lung cancer disease as well as provide an algorithm which is tested on the basis of hypothetical data in order to compare it with current methods.

On Single-Valued Neutrosophic Closure Spaces

This paper aims to mark out new terms of single-valued neutrosophic notions in a Šostak sense called single-valued neutrosophic semi-closure spaces. To achieve this, notions such as β£-closure operators and β£-interior operators are first defined. More precisely, these proposed contributions involve different terms of single-valued neutrosophic continuous mappings called single-valued neutrosophic (almost β£, faintly β£, weakly β£) and β£-continuous. Finally, for the purpose of symmetry, we define the single-valued neutrosophic upper, single-valued neutrosophic lower and single-valued neutrosophic boundary sets of a rough single-valued neutrosophic set αn in a single-valued neutrosophic approximation space (F˜,δ). Based on αn and δ, we also introduce the single-valued neutrosophic approximation interior operator intαnδ and the single-valued neutrosophic approximation closure operator Clαnδ.

A New Single-Valued Neutrosophic Rough Sets and Related Topology

(Fuzzy) rough sets are closely related to (fuzzy) topologies. Neutrosophic rough sets and neutrosophic topologies are extensions of (fuzzy) rough sets and (fuzzy) topologies, respectively. In this paper, a new type of neutrosophic rough sets is presented, and the basic properties and the relationships to neutrosophic topology are discussed. The main results include the following: (1) For a single-valued neutrosophic approximation space U , R , a pair of approximation operators called the upper and lower ordinary single-valued neutrosophic approximation operators are defined and their properties are discussed. Then the further properties of the proposed approximation operators corresponding to reflexive (transitive) single-valued neutrosophic approximation space are explored. (2) It is verified that the single-valued neutrosophic approximation spaces and the ordinary single-valued neutrosophic topological spaces can be interrelated to each other through our defined lower approximation operator. Particularly, there is a one-to-one correspondence between reflexive, transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces.

On local multigranulation covering decision-theoretic rough sets

Multi-granulation decision-theoretic rough sets uses the granular structures induced by multiple binary relations to approximate the target concept, which can get a more accurate description of the approximate space. However, Multi-granulation decision-theoretic rough sets is very time-consuming to calculate the approximate value of the target set. Local rough sets not only inherits the advantages of classical rough set in dealing with imprecise, fuzzy and uncertain data, but also breaks through the limitation that classical rough set needs a lot of labeled data. In this paper, in order to make full use of the advantage of computational efficiency of local rough sets and the ability of more accurate approximation space description of multi-granulation decision-theoretic rough sets, we propose to combine the local rough sets and the multigranulation decision-theoretic rough sets in the covering approximation space to obtain the local multigranulation covering decision-theoretic rough sets model. This provides an effective tool for discovering knowledge and making decisions in relation to large data sets. We first propose four types of local multigranulation covering decision-theoretic rough sets models in covering approximation space, where a target concept is approximated by employing the maximal or minimal descriptors of objects. Moreover, some important properties and decision rules are studied. Meanwhile, we explore the reduction among the four types of models. Furthermore, we discuss the relationships of the proposed models and other representative models. Finally, illustrative case of medical diagnosis is given to explain and evaluate the advantage of local multigranulation covering decision-theoretic rough sets model.

N-soft rough sets and its applications

In this paper, we propose a new hybrid model called N-soft rough sets, which can be seen as a combination of rough sets and N-soft sets. Moreover, approximation operators and some useful properties with respect to N-soft rough approximation space are introduced. Furthermore, we propose decision making procedures for N-soft rough sets, the approximation sets are utilized to handle problems involving multi-criteria decision-making(MCDM), aiming at electing the optional objects and the possible optional objects based on their attribute set. The algorithm addresses some limitations of the extended rough sets models in dealing with inconsistent decision problems. Finally, an application of N-soft rough sets in multi-criteria decision making is illustrated with a real life example.

Minimal structure approximation space and some of its application

Topological concepts play an important role in applications and solving real-life problems. Among of these concepts are neighbourhood and minimal structure. In this paper, we introduce a new space-based on a generalized system with a binary relation on a nonempty set by using the concept of a minimal structure, which is called a minimal structure approximation space (briefly, MSAS), and study some of its properties. Also, we compare the advantages of MSAS with neighbourhood approximation space which are based on the same starting point, and apply the concept of MSAS in some examples of chemistry to extraction and reduct the information. Finally, we investigate the concepts of the separation axioms on MSAS and study some of its properties in the information system as the process of approximation of information.

INCREMENTALLY UPDATING APPROXIMATION IN INCOMPLETE INFORMATION SYSTEMS UNDER THE VARIATION OF OBJECTS

Rough membership functions in covering approximation space not only give numerical characterizations of covering-based rough set approximations, but also establish the relationship between covering-based rough sets and fuzzy covering-based rough sets. In this paper, we introduce a new method to update the approximation sets with rough membership functions in covering approximation space. Firstly, we present the third types of rough membership functions and study their properties. And then, we consider the change of them while simultaneously adding and removing objects in the information system. Based on that change, we propose a method for updating the approximation sets when the objects vary over time.