scholarly journals Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Jianguo Tang ◽  
Kun She ◽  
William Zhu
Keyword(s):  
Graph Theory ◽  
Rough Sets ◽  
Mathematical Theory ◽  
Research Field ◽  
Rank Function ◽  
Complete Graphs ◽  
Important Research ◽  
Approximation Number ◽  
Lower Approximation ◽  
The Universe

Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. Some new properties, which have not been found in rough sets, are obtained. Furthermore, by defining the concept of lower approximation number, the rank function of some subset of the universe and the approximations of the subset are connected. Finally, the relationships between the two types of matroids are discussed, and the result shows that they are just dual matroids.

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals Equivalent Characterizations of Some Graph Problems by Covering-Based Rough Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shiping Wang ◽  
Qingxin Zhu ◽  
William Zhu ◽  
Fan Min
Keyword(s):  
Graph Theory ◽  
Rough Sets ◽  
Systematic Approach ◽  
Rough Set Theory ◽  
Approximation Number ◽  
Edge Cover ◽  
Upper Approximation ◽  
Graph Problems ◽  
New Concepts ◽  
Equivalent Characterization

Covering is a widely used form of data structures. Covering-based rough set theory provides a systematic approach to this data. In this paper, graphs are connected with covering-based rough sets. Specifically, we convert some important concepts in graph theory including vertex covers, independent sets, edge covers, and matchings to ones in covering-based rough sets. At the same time, corresponding problems in graphs are also transformed into ones in covering-based rough sets. For example, finding a minimal edge cover of a graph is translated into finding a minimal general reduct of a covering. The main contributions of this paper are threefold. First, any graph is converted to a covering. Two graphs induce the same covering if and only if they are isomorphic. Second, some new concepts are defined in covering-based rough sets to correspond with ones in graph theory. The upper approximation number is essential to describe these concepts. Finally, from a new viewpoint of covering-based rough sets, the general reduct is defined, and its equivalent characterization for the edge cover is presented. These results show the potential for the connection between covering-based rough sets and graphs.

GENERALIZED FUZZY ROUGH SETS BY CONDITIONAL PROBABILITY RELATIONS

2002 ◽  
Vol 16 (07) ◽  
pp. 865-881 ◽  
Author(s):  
ROLLY INTAN ◽  
MASAO MUKAIDONO
Keyword(s):  
Conditional Probability ◽  
Rough Set ◽  
Real World ◽  
Rough Sets ◽  
Similarity Relation ◽  
Fuzzy Rough Set ◽  
Fuzzy Similarity ◽  
Fuzzy Similarity Relation ◽  
Real World Applications ◽  
The Universe

In 1982, Pawlak proposed the concept of rough sets with a practical purpose of representing indiscernibility of elements or objects in the presence of information systems. Even if it is easy to analyze, the rough set theory built on a partition induced by equivalence relation may not provide a realistic view of relationships between elements in real-world applications. Here, coverings of, or nonequivalence relations on, the universe can be considered to represent a more realistic model instead of a partition in which a generalized model of rough sets was proposed. In this paper, first a weak fuzzy similarity relation is introduced as a more realistic relation in representing the relationship between two elements of data in real-world applications. Fuzzy conditional probability relation is considered as a concrete example of the weak fuzzy similarity relation. Coverings of the universe is provided by fuzzy conditional probability relations. Generalized concepts of rough approximations and rough membership functions are proposed and defined based on coverings of the universe. Such generalization is considered as a kind of fuzzy rough set. A more generalized fuzzy rough set approximation of a given fuzzy set is proposed and discussed as an alternative to provide interval-value fuzzy sets. Their properties are examined.

scholarly journals On Some Types of Covering-Based ℐ , T -Fuzzy Rough Sets and Their Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Mohammed Atef ◽  
José Carlos R. Alcantud ◽  
Hussain AlSalman ◽  
Abdu Gumaei
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Practical Application ◽  
Fuzzy Rough Sets ◽  
Fuzzy Rough Set ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Numerical Example ◽  
Variable Precision

The notions of the fuzzy β -minimal and maximal descriptions were established by Yang et al. (Yang and Hu, 2016 and 2019). Recently, Zhang et al. (Zhang et al. 2019) presented the fuzzy covering via ℐ , T -fuzzy rough set model ( FC ℐ T FRS ), and Jiang et al. (Jiang et al., in 2019) introduced the covering through variable precision ℐ , T -fuzzy rough sets ( CVP ℐ T FRS ). To generalize these models in (Jiang et al., 2019 and Zhang et al. 2019), that is, to improve the lower approximation and reduce the upper approximation, the present paper constructs eight novel models of an FC ℐ T FRS based on fuzzy β -minimal (maximal) descriptions. Characterizations of these models are discussed. Further, eight types of CVP ℐ T FRS are introduced, and we investigate the related properties. Relationships among these models are also proposed. Finally, we illustrate the above study with a numerical example that also describes its practical application.

The Genetic Algorithm

2019 ◽  
pp. 154-178
Author(s):  
Burcu Adıguzel Mercangöz ◽  
Ergun Eroglu
Keyword(s):  
Genetic Algorithm ◽  
Mathematical Models ◽  
Portfolio Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Biological Evolution ◽  
Research Field ◽  
Risk Level ◽  
Important Research ◽  
Portfolio Optimization Problem

The portfolio optimization is an important research field of the financial sciences. In portfolio optimization problems, it is aimed to create portfolios by giving the best return at a certain risk level from the asset pool or by selecting assets that give the lowest risk at a certain level of return. The diversity of the portfolio gives opportunity to increase the return by minimizing the risk. As a powerful alternative to the mathematical models, heuristics is used widely to solve the portfolio optimization problems. The genetic algorithm (GA) is a technique that is inspired by the biological evolution. While this book considers the heuristics methods for the portfolio optimization problems, this chapter will give the implementing steps of the GA clearly and apply this method to a portfolio optimization problem in a basic example.

scholarly journals On the Lattice of Intervals and Rough Sets

2009 ◽  
Vol 17 (4) ◽  
pp. 237-244 ◽  
Author(s):  
Adam Grabowski ◽  
Magdalena Jastrzębska
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Algebraic Theory ◽  
Algebraic Models ◽  
Lower Approximation ◽  
Definable Sets

On the Lattice of Intervals and Rough Sets Rough sets, developed by Pawlak [6], are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].

Improving the Performance of an Associative Classifier by Gamma Rough Sets Based Instance Selection

2017 ◽  
Vol 32 (01) ◽  
pp. 1860009 ◽  
Author(s):  
Jarvin A. Antón-Vargas ◽  
Yenny Villuendas-Rey ◽  
Cornelio Yáñez-Márquez ◽  
Itzamá López-Yáñez ◽  
Oscar Camacho-Nieto
Keyword(s):  
Information Systems ◽  
Rough Sets ◽  
Nearest Neighbor ◽  
Management Information Systems ◽  
Computational Cost ◽  
Instance Selection ◽  
Management Information ◽  
Similarity Relations ◽  
The Universe ◽  
Selection Of

This paper introduces the Gamma Rough Sets for management information systems where the universe objects are represented by continuous attributes and are connected by similarity relations. Some properties of such sets are demonstrated in this paper. In addition, Gamma Rough Sets are used to improve the Gamma associative classifier, by selecting instances. The results indicate that the selection of instances significantly reduces the computational cost of the Gamma classifier without affecting its effectiveness. The results also suggest that the selection of instances using Gamma Rough Sets favors other lazy learners, such as Nearest Neighbor and ALVOT.

scholarly journals Application of MD Simulations to Predict Membrane Properties of MOFs

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Elda Adatoz ◽  
Seda Keskin
Keyword(s):  
Metal Organic Frameworks ◽  
Experimental Studies ◽  
Molecular Simulations ◽  
Md Simulations ◽  
Research Field ◽  
Computational Approach ◽  
Membrane Properties ◽  
Important Research ◽  
Zeolite Membranes ◽  
Metal Organic

Metal organic frameworks (MOFs) are a new group of nanomaterials that have been widely examined for various chemical applications. Gas separation using MOF membranes has become an increasingly important research field in the last years. Several experimental studies have shown that thin-film MOF membranes can outperform well known polymer and zeolite membranes due to their higher gas permeances and selectivities. Given the very large number of available MOF materials, it is impractical to fabricate and test the performance of every single MOF membrane using purely experimental techniques. In this study, we used molecular simulations, Monte Carlo and Molecular Dynamics, to estimate both single-gas and mixture permeances of MOF membranes. Predictions of molecular simulations were compared with the experimental gas permeance data of MOF membranes in order to validate the accuracy of our computational approach. Results show that computational methodology that we described in this work can be used to accurately estimate membrane properties of MOFs prior to extensive experimental efforts.

Making the Black Box Transparent

1999 ◽  
Vol 92 (9) ◽  
pp. 780-784
Author(s):  
Lawrence M. Lesser
Keyword(s):  
Mathematical Theory ◽  
Black Box ◽  
Mathematical Understanding ◽  
Complete Graphs ◽  
Interpolating Polynomials ◽  
Mathematics Curricula ◽  
Best Fit ◽  
Shed Light

Although introducing technology into our mathematics curricula allows us to tackle problems of size and complexity as never before, we face a danger of introducing tools to students before they have a sufficient understanding of how mathematics content within their reach can be used to shed light on the algorithms within the tools or on the use of the tools themselves. Fortunately, we can view mathematical theory and technology not as opponents but rather as partners that make the whole of mathematical understanding richer than the sum of its parts. Indeed, bringing technology into our classrooms can encourage new questions that technology-free mathematics must answer. This article focuses mainly on a common example in technology-rich mathematics curricula, namely, the line of best fit, followed by a discussion of two additional examples—interpolating polynomials and complete graphs. In each case, connections between theory and technology do not appear to be as widely known and used as they could be.

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