Although some information-theoretic measures of uncertainty or granularity have been proposed in rough set theory, these measures are only dependent on the underlying partition and the cardinality of the universe, independent of the lower and upper approximations. It seems somewhat unreasonable since the basic idea of rough set theory aims at describing vague concepts by the lower and upper approximations. In this paper, we thus define new information-theoretic entropy and co-entropy functions associated to the partition and the approximations to measure the uncertainty and granularity of an approximation space. After introducing the novel notions of entropy and co-entropy, we then examine their properties. In particular, we discuss the relationship of co-entropies between different universes. The theoretical development is accompanied by illustrative numerical examples.
We examine non-dual relational extensions of rough set approximations and find an extension which satisfies surprisingly many of the usual rough set properties. We then use this definition to give an explanation for an observation made by Samanta and Chakraborty in their recent paper [P. Samanta and M.K. Chakraborty. Interface of rough set systems and modal logics: A survey. Transactions on Rough Sets XIX, pages 114-137, 2015].
The notion of rough set captures indiscernibility of elements in a set. But, in many real life situations, an information system establishes the relation between different universes. This gave the extension of rough set on single universal set to rough set on two universal sets. In this paper, we introduce approximation of classifications and measures of uncertainty basing upon rough set on two universal sets employing the knowledge due to binary relations.
Northwest A & F University（西北农林科技大学）；Northwest Normal University（西北师范大学）
In this paper, we mainly construct three types of $L$-fuzzy $β$-covering-based rough set models and study the axiom sets, matrix representations and interdependency of these three pairs of $L$-fuzzy $β$-covering-based rough approximation operators. Firstly, we propose three pairs of $L$-fuzzy $β$-covering-based rough approximation operators by introducing the concepts such as $β$-degree of intersection and $β$-subsethood degree, which are generalizations of degree of intersection and subsethood degree, respectively. And then, the axiom set for each of these $L$-fuzzy $β$-covering-based rough a…
Beijing University of Posts and Telecommunications（北京邮电大学）
Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighbo…
Beijing University of Posts and Telecommunications（北京邮电大学）
In Pawlak's rough set theory, a set is approximated by a pair of lower and upper approximations. To measure numerically the roughness of an approximation, Pawlak introduced a quantitative measure of roughness by using the ratio of the cardinalities of the lower and upper approximations. Although the roughness measure is effective, it has the drawback of not being strictly monotonic with respect to the standard ordering on partitions. Recently, some improvements have been made by taking into account the granularity of partitions. In this paper, we approach the roughness measure in an axiomatic …
Dominance-based Rough Set Approach (DRSA), as the extension of Pawlak's Rough Set theory, is effective and fundamentally important in Multiple Criteria Decision Analysis (MCDA). In previous DRSA models, the definitions of the upper and lower approximations are preserving the class unions rather than the singleton class. In this paper, we propose a new Class-based Rough Approximation with respect to a series of previous DRSA models, including Classical DRSA model, VC-DRSA model and VP-DRSA model. In addition, the new class-based reducts are investigated.
Changsha University of Science and Technology（长沙理工大学）；Hunan University（湖南大学）
In practical situations, interval-valued fuzzy sets are frequently encountered. In this paper, firstly, we present shadowed sets for interpreting and understanding interval fuzzy sets. We also provide an analytic solution to computing the pair of thresholds by searching for a balance of uncertainty in the framework of shadowed sets. Secondly, we construct errors-based three-way approximations of interval-valued fuzzy sets. We also provide an alternative decision-theoretic formulation for calculating the pair of thresholds by transforming interval-valued loss functions into single-valued loss f…
Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough sets. In this paper, we propose two family of sets and study the conditions that these two sets become some lattice structures. These two sets are consisted by the fixed point of the lower approximations of the first type and the sixth type of covering-based rough sets, respectively. These two sets are called the fixed point set of neighborhoods and the fixed p…
At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to mat…
The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery and data mining. The theory of matroids is a generalization of linear independence in vector spaces. It has been used in combinatorial optimization and algorithm design. In order to take advantages of both rough sets and matroids, in this paper we propose a matroidal structure of rough sets based on a serial and transitive relation on a universe. We define the family of all minimal neighborhoods of a relat…
Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Topology, one of the most important subjects in mathematics, provides mathematical tools and interesting topics in studying information systems and rough sets. In this paper, we present the topological characterizations to three types of covering approximation operators. First, we study the properties of topology induced by the sixth type of covering lower approximation operator. Second, some topological characterizations to the covering lower approximation operator to be…
Recently, in order to broad the application and theoretical areas of rough sets and matroids, some authors have combined them from many different viewpoints, such as circuits, rank function, spanning sets and so on. In this paper, we connect the second type of covering-based rough sets and matroids from the view of closure operators. On one hand, we establish a closure system through the fixed point family of the second type of covering lower approximation operator, and then construct a closure operator. For a covering of a universe, the closure operator is a closure one of a matroid if and on…
Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a combinatorial generalization of linear independence in vector spaces. In this paper, we define a parametric set family, with any subset of a universe as its parameter, to connect rough sets and matroids. On the one hand, for a universe and an equivalence relation on the universe, a parametric set family is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore it can generate a ma…
Indian Statistical Institute, Kolkata（印度统计研究所加尔各答分所）
Rational approximations are introduced and studied in granular graded rough sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between granularity, mereology and approximations in the context. In addition, a framework for rational approximations is introduced by her in the mentioned paper(s). Granular approximations constructed as per the procedures of variable precision rough sets (VPRS) are likely to be more rational than those constructed from a classical perspective under certain condi…
Lattice-theoretic ideals have been used to define and generate non granular rough approximations over general approximation spaces over the last few years by few authors. The goal of these studies, in relation based rough sets, have been to obtain nice properties comparable to those of classical rough approximations. In this research paper, these ideas are generalized in a severe way by the present author and associated semantic features are investigated by her. Granules are used in the construction of approximations in implicit ways and so a concept of co-granularity is introduced. Knowledge …
Rough sets over generalized transitive relations like proto-transitive ones had been initiated by the present author in the year 2012. Subsequently, approximation of proto-transitive relations by other relations was investigated and the relation with rough approximations was developed towards constructing semantics that can handle fragments of structure. It was also proved that difference of approximations induced by some approximate relations need not induce rough structures. In this research we develop different semantics of proto transitive rough sets (PRAX) after characterizing the structu…
Extreme learning machine (ELM) is a new single hidden layer feedback neural network. The weights of the input layer and the biases of neurons in hidden layer are randomly generated, the weights of the output layer can be analytically determined. ELM has been achieved good results for a large number of classification tasks. In this paper, a new extreme learning machine called rough extreme learning machine (RELM) was proposed. RELM uses rough set to divide data into upper approximation set and lower approximation set, and the two approximation sets are utilized to train upper approximation neur…
Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the \emph{inverse problem}. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical perspective, a few (as opposed to one) theoretical frameworks may be suitable for formulating the problem itself. \emph{Granular operator spaces} have been recently introduced and investigated by t…
Performance evaluation of various organizations especially educational institutions is a very important area of research and needs to be cultivated more. In this paper, we propose a performance evaluation for educational institutions using rough set on fuzzy approximation spaces with ordering rules and information entropy. In order to measure the performance of educational institutions, we construct an evaluation index system. Rough set on fuzzy approximation spaces with ordering is applied to explore the evaluation index data of each level. Furthermore, the concept of information entropy is u…
HBCSE, Tata Institute of Fundamental Research（塔塔基础研究院 HBCSE）；University of Miskolc（米什科尔茨大学）
In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better behaved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approximations, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the…
The study of mereology (parts and wholes) in the context of formal approaches to vagueness can be approached in a number of ways. In the context of rough sets, mereological concepts with a set-theoretic or valuation based ontology acquire complex and diverse behavior. In this research a general rough set framework called granular operator spaces is extended and the nature of parthood in it is explored from a minimally intrusive point of view. This is used to develop counting strategies that help in classifying the framework. The developed methodologies would be useful for drawing involved conc…
Up-directed rough sets are introduced and studied by the present author in earlier papers. This is extended by her in two different granular directions in this research, with a surprising algebraic semantics. The granules are based on ideas of generalized closure under up-directedness that may be read as a form of weak consequence. This yields approximation operators that satisfy cautious monotony, while pi-groupoidal approximations (that additionally involve strategic choice and algebraic operators) have nicer properties. The study is primarily motivated by possible structure of concepts in d…
