scholarly journals Geometric Lattice Structure of Covering-Based Rough Sets through Matroids

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Aiping Huang ◽  
William Zhu
Keyword(s):  
Rough Sets ◽  
Rough Set Theory ◽  
Lattice Structure ◽  
Search Algorithm ◽  
Algorithm Design ◽  
Geometric Lattice ◽  
Lattice Structures ◽  
Upper Approximation ◽  
The Relationship ◽  
Geometric Lattices

Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets.

scholarly journals Geometric Lattice Structure of Covering and Its Application to Attribute Reduction through Matroids

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Aiping Huang ◽  
William Zhu
Keyword(s):  
Rough Sets ◽  
Optimization Problems ◽  
Lattice Structure ◽  
Attribute Reduction ◽  
Algorithm Design ◽  
Geometric Lattice ◽  
Closed Sets ◽  
Decision Systems ◽  
Dependence Space ◽  
Geometric Lattices

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice, to study the attribute reduction issues of information systems.

scholarly journals Matroidal Structure of Generalized Rough Sets Based on Tolerance Relations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Hui Li ◽  
Yanfang Liu ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Matroid Theory ◽  
Upper Approximation ◽  
Tolerance Relation ◽  
The Family ◽  
Generalized Rough Set ◽  
The Relationship ◽  
Generalized Rough Sets

Rough set theory provides an effective tool to deal with uncertain, granular, and incomplete knowledge in information systems. Matroid theory generalizes the linear independence in vector spaces and has many applications in diverse fields, such as combinatorial optimization and rough sets. In this paper, we construct a matroidal structure of the generalized rough set based on a tolerance relation. First, a family of sets are constructed through the lower approximation of a tolerance relation and they are proved to satisfy the circuit axioms of matroids. Thus we establish a matroid with the family of sets as its circuits. Second, we study the properties of the matroid including the base and the rank function. Moreover, we investigate the relationship between the upper approximation operator based on a tolerance relation and the closure operator of the matroid induced by the tolerance relation. Finally, from a tolerance relation, we can get a matroid of the generalized rough set based on the tolerance relation. The matroid can also induce a new relation. We investigate the connection between the original tolerance relation and the induced relation.

scholarly journals Covering-Based Rough Sets on Eulerian Matroids

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bin Yang ◽  
Ziqiong Lin ◽  
William Zhu
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Closure Operator ◽  
Matroid Theory ◽  
Upper Approximation ◽  
Significant Generalization ◽  
Eulerian Matroid ◽  
Vital Structure

Rough set theory is an efficient and essential tool for dealing with vagueness and granularity in information systems. Covering-based rough set theory is proposed as a significant generalization of classical rough sets. Matroid theory is a vital structure with high applicability and borrows extensively from linear algebra and graph theory. In this paper, one type of covering-based approximations is studied from the viewpoint of Eulerian matroids. First, we explore the circuits of an Eulerian matroid from the perspective of coverings. Second, this type of covering-based approximations is represented by the circuits of Eulerian matroids. Moreover, the conditions under which the covering-based upper approximation operator is the closure operator of a matroid are presented. Finally, a matroidal structure of covering-based rough sets is constructed. These results show many potential connections between covering-based rough sets and matroids.

Model of Covering Rough Sets in the Machine Intelligence

2012 ◽  
Vol 548 ◽  
pp. 735-739
Author(s):  
Hong Mei Nie ◽  
Jia Qing Zhou
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Machine Intelligence ◽  
Upper Approximation ◽  
Generalized Rough Sets

Rough set theory has been proposed by Pawlak as a useful tool for dealing with the vagueness and granularity in information systems. Classical rough set theory is based on equivalence relation. The covering rough sets are an improvement of Pawlak rough set to deal with complex practical problems which the latter one can not handle. This paper studies covering-based generalized rough sets. In this setting, we investigate common properties of classical lower and upper approximation operations hold for the covering-based lower and upper approximation operations and relationships among some type of covering rough sets.

ON ATTRIBUTE REDUCTION WITH INTUITIONISTIC FUZZY ROUGH SETS

2012 ◽  
Vol 20 (01) ◽  
pp. 59-76 ◽  
Author(s):  
ZHIMING ZHANG ◽  
JINGFENG TIAN
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Theoretical Foundation ◽  
Attribute Reduction ◽  
Upper Approximation ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Rough Sets ◽  
Axiomatic Approaches

Intuitionistic fuzzy (IF) rough sets are the generalization of traditional rough sets obtained by combining the IF set theory and the rough set theory. The existing research on IF rough sets mainly concentrates on the establishment of lower and upper approximation operators using constructive and axiomatic approaches. Less effort has been put on the attribute reduction of databases based on IF rough sets. This paper systematically studies attribute reduction based on IF rough sets. Firstly, attribute reduction with traditional rough sets and some concepts of IF rough sets are reviewed. Then, we introduce some concepts and theorems of attribute reduction with IF rough sets, and completely investigate the structure of attribute reduction. Employing the discernibility matrix approach, an algorithm to find all attribute reductions is also presented. Finally, an example is proposed to illustrate our idea and method. Altogether, these findings lay a solid theoretical foundation for attribute reduction based on IF rough sets.

Topological Approaches to the Second Type of Covering-Based Rough Sets

2011 ◽  
Vol 204-210 ◽  
pp. 1781-1784
Author(s):  
Bin Chen
Keyword(s):  
Data Mining ◽  
Information Systems ◽  
Rough Sets ◽  
Relative Interior ◽  
Upper Approximation ◽  
Definition Of ◽  
The Relationship

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper studies covering-based rough sets from the topological view. We explore the relationship between the relative closure and the second type of covering upper approximation. The major contributions of this paper are that we use the definition of the relative closure and the relative interior to discuss the conditions under which the relative operators satisfy certain classical properties. The theorems we get generalize some of the results in Zhu’s paper.

Geometry optimization for tunable band gap and wave guiding in periodic grid structures

2021 ◽  
Author(s):  
Shuai Yang ◽  
Xiao-Liang Zhou ◽  
Chang-Qing Li ◽  
Shi-Ke Zhang
Keyword(s):  
Band Gap ◽  
Lattice Structure ◽  
Lower Boundary ◽  
Geometry Optimization ◽  
Geometric Lattice ◽  
Homogeneous Material ◽  
Lattice Structures ◽  
Perfect Lattice ◽  
Tunable Property ◽  
Geometric Defect

A proper lattice structure consisting of homogeneous material is designed in this paper to investigate the maximum bandwidth of perfect lattice structures and tunable property of waveguide with linear geometric defect by means of selecting optimal geometric lattice cell. A simulation model based on finite element method is used to calculate dispersion curves and transmission spectrums of lattice structures with different geometric parameters. Meanwhile, a simplified theoretical model of unit cell, which considers the mass of grid bar and stiffness of node area, is applied to validate the accuracy of simulation result and may provide an effective approach for prediction of band gap lower boundary. Then, the validated numerical results show different orders of widest band gap that can be realized by different optimal geometric structures. Moreover, waveguide property can be effectively controlled and manipulated by changing defect parameters. The present study may establish theoretical and simulation foundation to control and manipulate band structures and other acoustic propagation characteristics of waveguide devices.

scholarly journals A Novel Method of the Generalized Interval-Valued Fuzzy Rough Approximation Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Tianyu Xue ◽  
Zhan’ao Xue ◽  
Huiru Cheng ◽  
Jie Liu ◽  
Tailong Zhu
Keyword(s):  
Rough Sets ◽  
Rough Set Theory ◽  
Binary Relations ◽  
Approximation Space ◽  
Upper Approximation ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Fuzzy Binary Relation ◽  
Novel Method ◽  
Interval Valued

Rough set theory is a suitable tool for dealing with the imprecision, uncertainty, incompleteness, and vagueness of knowledge. In this paper, new lower and upper approximation operators for generalized fuzzy rough sets are constructed, and their definitions are expanded to the interval-valued environment. Furthermore, the properties of this type of rough sets are analyzed. These operators are shown to be equivalent to the generalized interval fuzzy rough approximation operators introduced by Dubois, which are determined by any interval-valued fuzzy binary relation expressed in a generalized approximation space. Main properties of these operators are discussed under different interval-valued fuzzy binary relations, and the illustrative examples are given to demonstrate the main features of the proposed operators.

scholarly journals Rough Sets Data Analysis in Knowledge Discovery: A Case of Kuwaiti Diabetic Children Patients

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Aboul ella Hassanien ◽  
Mohamed E. Abdelhafez ◽  
Hala S. Own
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Confusion Matrix ◽  
Decision Rules ◽  
Minimal Subset ◽  
Discretization Algorithm ◽  
Diabetic Children ◽  
The Relationship ◽  
Accuracy Rates

The main goal of this study is to investigate the relationship between psychosocial variables and diabetic children patients and to obtain a classifier function with which it was possible to classify the patients on the basis of assessed adherence level. The rough set theory is used to identify the most important attributes and to induce decision rules from 302 samples of Kuwaiti diabetic children patients aged 7–13 years old. To increase the efficiency of the classification process, rough sets with Boolean reasoning discretization algorithm is introduced to discretize the data, then the rough set reduction technique is applied to find all reducts of the data which contains the minimal subset of attributes that are associated with a class label for classification. Finally, the rough sets dependency rules are generated directly from all generated reducts. Rough confusion matrix is used to evaluate the performance of the predicted reducts and classes. A comparison between the obtained results using rough sets with decision tree, neural networks, and statistical discriminate analysis classifier algorithms has been made. Rough sets show a higher overall accuracy rates and generate more compact rules.

scholarly journals Design of Customized TPU Lattice Structures for Additive Manufacturing: Influence on the Functional Properties in Elastic Products

Polymers ◽  
2021 ◽  
Vol 13 (24) ◽  
pp. 4341
Author(s):  
Sergio de la Rosa ◽  
Pedro F. Mayuet ◽  
José Ramón Méndez Salgueiro ◽  
Lucía Rodríguez-Parada
Keyword(s):  
Lattice Structure ◽  
Reaction Force ◽  
Statistical Treatment ◽  
Lattice Structures ◽  
Compression Process ◽  
Element Simulation ◽  
Relationship Of ◽  
The Relationship ◽  
Manufacturing Parameters ◽  
Elastic Lattice

This work focuses on evaluating and establishing the relationship of the influence of geometrical and manufacturing parameters in stiffness of additively manufactured TPU lattice structures. The contribution of this work resides in the creation of a methodology that focuses on characterizing the behavior of elastic lattice structures. Likewise, resides in the possibility of using the statistical treatment of results as a guide to find favorable possibilities within the range of parameters studied and to predict the behavior of the structures. In order to characterize their behavior, different types of specimens were designed and tested by finite element simulation of a compression process using Computer Aided Engineering (CAE) tools. The tests showed that the stiffness depends on the topology of the cells of the lattice structure. For structures with different cell topologies, it has been possible to obtain an increase in the reaction force against compression from 24.7 N to 397 N for the same manufacturing conditions. It was shown that other parameters with a defined influence on the stiffness of the structure were the temperature and the unit size of the cells, all due to the development of fusion mechanisms and the variation in the volume of material used, respectively.

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