Measures of uncertainty for a fuzzy probabilistic approximation space

2021 ◽  
pp. 1-24
Author(s):  
Lijun Chen ◽  
Damei Luo ◽  
Pei Wang ◽  
Zhaowen Li ◽  
Ningxin Xie
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fuzzy Relation ◽  
Point Of View ◽  
Approximation Space ◽  
Fuzzy Approximation ◽  
Entropy Measurement ◽  
Fuzzy Relation Matrix ◽  
Relation Matrix

 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.

Multi-Granular Computing through Rough Sets

2014 ◽  
pp. 1-34 ◽  
Author(s):  
B. K. Tripathy
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Intelligent Systems ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Point Of View ◽  
Information Granules ◽  
Computing Platform

Granular Computing has emerged as a framework in which information granules are represented and manipulated by intelligent systems. Granular Computing forms a unified conceptual and computing platform. Rough set theory put forth by Pawlak is based upon single equivalence relation taken at a time. Therefore, from a granular computing point of view, it is single granular computing. In 2006, Qiang et al. introduced a multi-granular computing using rough set, which was called optimistic multigranular rough sets after the introduction of another type of multigranular computing using rough sets called pessimistic multigranular rough sets being introduced by them in 2010. Since then, several properties of multigranulations have been studied. In addition, these basic notions on multigranular rough sets have been introduced. Some of these, called the Neighborhood-Based Multigranular Rough Sets (NMGRS) and the Covering-Based Multigranular Rough Sets (CBMGRS), have been added recently. In this chapter, the authors discuss all these topics on multigranular computing and suggest some problems for further study.

scholarly journals AN OVERVIEW OF ROUGH SET SEMANTICS FOR MODAL AND QUANTIFIER LOGICS

2000 ◽  
Vol 08 (01) ◽  
pp. 93-118 ◽  
Author(s):  
CHURN-JUNG LIAU
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Possible Worlds ◽  
Rough Set Theory ◽  
Modal Logics ◽  
The Other ◽  
Approximation Space ◽  
The Universe

In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

Representations of Vague Approximation Operators

2011 ◽  
Vol 282-283 ◽  
pp. 287-290
Author(s):  
Hai Dong Zhang ◽  
Yan Ping He
Keyword(s):  
Artificial Intelligence ◽  
Mathematical Model ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fuzzy Relation ◽  
Partial Knowledge ◽  
Data Bases ◽  
Approximation Operators ◽  
Rough Approximation

As a suitable mathematical model to handle partial knowledge in data bases, rough set theory is emerging as a powerful theory and has been found its successive applications in the fields of artificial intelligence such as pattern recognition, machine learning, etc. In the paper, a vague relation is first defined, which is the extension of fuzzy relation. Then a new pair of lower and upper generalized rough approximation operators based on the vague relation is first proposed by us. Finally, the representations of vague rough approximation operators are presented.

scholarly journals Multigranulations Rough Set Method of Attribute Reduction in Information Systems Based on Evidence Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Minlun Yan
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Decision Rules ◽  
Evidence Theory ◽  
Point Of View ◽  
Upper Approximation

Attribute reduction is one of the most important problems in rough set theory. However, from the granular computing point of view, the classical rough set theory is based on a single granulation. It is necessary to study the issue of attribute reduction based on multigranulations rough set. To acquire brief decision rules from information systems, this paper firstly investigates attribute reductions by combining the multigranulations rough set together with evidence theory. Concepts of belief and plausibility consistent set are proposed, and some important properties are addressed by the view of the optimistic and pessimistic multigranulations rough set. What is more, the multigranulations method of the belief and plausibility reductions is constructed in the paper. It is proved that a set is an optimistic (pessimistic) belief reduction if and only if it is an optimistic (pessimistic) lower approximation reduction, and a set is an optimistic (pessimistic) plausibility reduction if and only if it is an optimistic (pessimistic) upper approximation reduction.

Characteristics of Rough SetC-Means Clustering

2018 ◽  
Vol 22 (4) ◽  
pp. 551-564 ◽  
Author(s):  
Seiki Ubukata ◽  
◽  
Keisuke Umado ◽  
Akira Notsu ◽  
Katsuhiro Honda
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Malignant Disease ◽  
Rough Set Theory ◽  
Fuzzy Theory ◽  
Classification Performance ◽  
Approximation Space ◽  
Clustering Model ◽  
Positive Region ◽  
Medical Dataset

HardC-means (HCM), which is one of the most popular clustering techniques, has been extended by using soft computing approaches such as fuzzy theory and rough set theory. FuzzyC-means (FCM) and roughC-means (RCM) are respectively fuzzy and rough set extensions of HCM. RCM can detect the positive and the possible regions of clusters by using the lower and the upper areas which are respectively analogous to the lower and the upper approximations in rough set theory. RCM-type methods have the problem that the original definitions of the lower and the upper approximations are not actually used. In this paper, rough setC-means (RSCM), which is an extension of HCM based on the original rough set definition, is proposed as a rough set-based counterpart of RCM. Specifically, RSCM is proposed as a clustering model on an approximation space considering a space granulated by a binary relation and uses the lower and the upper approximations of temporal clusters. For this study, we investigated the characteristics of the proposed RSCM through basic considerations, visual demonstrations, and comparative experiments. We observed the geometric characteristics of the examined methods by using visualizations and numerical experiments conducted for the problem of classifying patients as having benign or malignant disease based on a medical dataset. We compared the classification performance by viewing the trade-off between the classification accuracy in the positive region and the fraction of objects classified as being in the positive region.

Reduction in a fuzzy probabilistic information system

2021 ◽  
pp. 1-17
Author(s):  
Damei Luo ◽  
Zhaowen Li ◽  
Liangdong Qu
Keyword(s):  
Information System ◽  
Probability Distribution ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Mathematical Tool ◽  
Fuzzy Relations ◽  
Probabilistic Information ◽  
Entropy Measurement ◽  
The Universe

An information system (IS) is an important mathematical tool for artificial intelligence. A fuzzy probabilistic information system (FPIS), the combination of some fuzzy relations in the same universe which satisfies the probability distribution, can be seen as an IS with fuzzy relations. A FPIS overcomes the shortcoming that rough set theory assumes elements in the universe with equal probability and leads to lose some useful information. This paper integrates the probability distribution into the fuzzy relations in a FPIS and studies its reduction. Firstly, the concept of a FPIS is introduced and its reduction is proposed. Then, the fuzzy relations in a FPIS are divided into three categories (P-necessary, P-relatively necessary and P-unnecessary fuzzy relations) according to their importance. Next, entropy measurement for a FPIS is investigated. Moreover, some reduction algorithms are constructed. Finally, an example is given to verify the effectiveness of these proposed algorithms.

A Glance to the Goldman’s Testors from the Point of View of Rough Set Theory

2016 ◽  
pp. 189-197 ◽  
Author(s):  
Manuel S. Lazo-Cortés ◽  
José Francisco Martínez-Trinidad ◽  
Jesús Ariel Carrasco-Ochoa
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Point Of View

Uncertainty Measures for Multigranulation Approximation Space

2015 ◽  
Vol 23 (03) ◽  
pp. 443-457 ◽  
Author(s):  
Guoping Lin ◽  
Jiye Liang ◽  
Yuhua Qian
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Information Entropy ◽  
Rough Set Theory ◽  
Mathematical Tool ◽  
Approximation Space ◽  
Uncertainty Measures ◽  
Multigranulation Rough Set ◽  
Essential Properties ◽  
The Relationship

Multigranulation rough set theory is a relatively new mathematical tool for solving complex problems in the multigranulation or distributed circumstances which are characterized by vagueness and uncertainty. In this paper, we first introduce the multigranulation approximation space. According to the idea of fusing uncertain, imprecise information, we then present three uncertainty measures: fusing information entropy, fusing rough entropy, and fusing knowledge granulation in the multigranulation approximation space. Furthermore, several essential properties (equivalence, maximum, minimum) are examined and the relationship between the fusion information entropy and the fusion rough entropy is also established. Finally, we prove these three measures are monotonously increasing as the partitions become finer. These results will be helpful for understanding the essence of uncertainty measures in multigranulation rough space and enriching multigranulation rough set theory.

scholarly journals Multiset concepts in two-universe approximation spaces

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
O. A. Embaby ◽  
Nadya A. Toumi
Keyword(s):  
Decision Making ◽  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Approximation Space ◽  
Approximation Spaces ◽  
Lower And Upper Approximations ◽  
Two Universes ◽  
Basic Sets

Abstract Rough set theory over two universes is a generalization of rough set model to find accurate approximations for uncertain concepts in information systems in which uncertainty arises from existence of interrelations between the three basic sets: objects, attributes, and decisions. In this work, multisets are approximated in a crisp two-universe approximation space using binary ordinary relation and multi relation. The concept of two universe approximation is applied for defining lower and upper approximations of multisets. Properties of these approximations are investigated, and the deviations between them and corresponding notions are obtained; some counter examples are given. The suggested notions can help in the modification of the decision-making for events in which objects have repetitions such as patients visiting a doctor more than one time; an example for this case is given.

scholarly journals On Fuzzy Rough Sets and Their Topological Structures

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Weidong Tang ◽  
Jinzhao Wu ◽  
Dingwei Zheng
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Fuzzy Relations ◽  
Fuzzy Rough Sets ◽  
Approximation Spaces ◽  
Topological Structures ◽  
Fuzzy Approximation ◽  
Approximation Operators

The core concepts of rough set theory are information systems and approximation operators of approximation spaces. Approximation operators draw close links between rough set theory and topology. This paper is devoted to the discussion of fuzzy rough sets and their topological structures. Fuzzy rough approximations are further investigated. Fuzzy relations are researched by means of topology or lower and upper sets. Topological structures of fuzzy approximation spaces are given by means of pseudoconstant fuzzy relations. Fuzzy topology satisfying (CC) axiom is investigated. The fact that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies satisfying (CC) axiom is proved, the concept of fuzzy approximating spaces is introduced, and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained, which illustrates that we can research fuzzy relations or fuzzy approximation spaces by means of topology and vice versa. Moreover, fuzzy pseudoclosure operators are examined.

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