Generalization of Rough Sets Using Weak Fuzzy Similarity Relations

Author(s):  
Rolly Intan ◽  
Y. Y. Yao ◽  
Masao Mukaidono
Author(s):  
ROLLY INTAN ◽  
MASAO MUKAIDONO

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.


Author(s):  
Angelo Ciaramella ◽  
Angelo Riccio ◽  
Stefano Galmarini ◽  
Giulio Giunta ◽  
Slawomir Potempski

Author(s):  
Mohamed El Alaoui ◽  
Khalid El Yassini

Similarity is an ambiguous term that can be interpreted differently depending on the context of use. In this chapter, the authors review some of its uses before focusing on decision making. Ignoring the uncertainty of human knowledge would be denying a major attribute. Thus, they linked it to the fuzzy context. However, even taking this aspect into consideration, the opinion itself must be relevant. Therefore, the more a decision is similar to other opinions, the more it is coherent. Hence, there is a need to measure the similarity between each couple of expressed opinions.


Author(s):  
Jarvin A. Antón-Vargas ◽  
Yenny Villuendas-Rey ◽  
Cornelio Yáñez-Márquez ◽  
Itzamá López-Yáñez ◽  
Oscar Camacho-Nieto

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.


2022 ◽  
Vol 7 (4) ◽  
pp. 5790-5807
Author(s):  
Imran Javaid ◽  
◽  
Shahroz Ali ◽  
Shahid Ur Rehman ◽  
Aqsa Shah

<abstract><p>In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.</p></abstract>


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