Rough sets and topological spaces based on similarity
01 Oct 2013-International Journal of Machine Learning and Cybernetics (Springer Berlin Heidelberg)-Vol. 4, Iss: 5, pp 451-458
TL;DR: If the function between two topological spaces is bijective, open and continuous, then the image and the inverse image of a rough set are also rough.
Abstract: Ordinary topology now has been used in many sub-fields of artificial intelligence, such as knowledge representation, spatial reasoning etc. In this paper, we discuss the relationships between the theory of rough sets and topological spaces. We obtain the concepts of rough sets from topology, also we obtain the concepts of topological space from rough sets. Furthermore, if the function between two topological spaces is bijective, open and continuous, then the image and the inverse image of a rough set are also rough.
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TL;DR: The basic concepts, operations and characteristics on the rough set theory are introduced, and then the extensions of rough set model, the situation of their applications, some application software and the key problems in applied research for the roughSet theory are presented.
185 citations
TL;DR: New types of neighborhoods called containment neighborhoods are introduced depending on the inclusion relations between j-neighborhoods under arbitrary binary relation and it is proved that a C j -accuracy measure is the highest in cases of j = i, 〈 i 〉 .
65 citations
TL;DR: A new similarity measure is proposed for Atanassov’s intuitionistic fuzzy sets by the relationship between entropy and similarity measure and an approach to derive the relative importance weights of experts is given.
Abstract: In this paper, we propose a new similarity measure for Atanassov’s intuitionistic fuzzy sets by the relationship between entropy and similarity measure. With respect to multi-attribute group decision making problem, we then give an approach to derive the relative importance weights of experts. This approach takes into account decision information from three aspects: the uncertainty degrees of individual expert’s assessing information for alternatives, the similarity degree of the assessing information for alternatives provided by individual expert, and the similarity degree of the individual expert’s assessing information to all the others’. Finally, we establish a method for handling multi-attribute group decision making problem with Atanassov’s intuitionistic fuzzy information, and adopt an illustrative example to demonstrate its rationality and effectiveness.
40 citations
TL;DR: In this article , a subset neighborhood is defined under an arbitrary binary relation using the inclusion relations between Nρ-neighborhoods, and Sρ-accuracy and roughness measures are derived.
Abstract: We present a novel kind of neighborhood, named subset neighborhood and denoted as Sρ-neighborhood. It is defined under an arbitrary binary relation using the inclusion relations between Nρ-neighborhoods. We study its relationships with some kinds of neighborhood systems given in the literature. Then, we formulate the concepts of Sρ-lower and Sρ-upper approximations, and Sρ-accuracy and roughness measures based on Sρ-neighborhoods. We show in which cases the Sρ-accuracy measure is the highest among related approximations and investigate under which conditions the Sρ-accuracy and Sρ-roughness measures are monotonic. Moreover, we compare our approach with two existing ones and elucidate the advantages of our technique to obtain accuracy measures under some specific relations. To support the obtained results, we provide two medical examples.
28 citations
15 Oct 2021
TL;DR: In this paper, the authors apply a topological concept called "somewhere dense sets" to improve the accuracy of rough set theory, which is a non-statistical approach to handle uncertainty and uncertain knowledge.
Abstract: Rough set theory is a non-statistical approach to handle uncertainty and uncertain knowledge. It is characterized by two methods called classification (lower and upper approximations) and accuracy measure. The closeness of notions and results in topology and rough set theory motivates researchers to explore the topological aspects and their applications in rough set theory. To contribute to this area, this paper applies a topological concept called “somewhere dense sets” to improve the approximations and accuracy measure in rough set theory. We firstly discuss further topological properties of somewhere dense and cs-dense sets and give explicitly formulations to calculate S-interior and S-closure operators. Then, we utilize these two sets to define new concepts in rough set context such as SD-lower and SD-upper approximations, SD-boundary region, and SD-accuracy measure of a subset. We establish the fundamental properties of these concepts as well as show their relationships with the previous ones. In the end, we compare the current method of approximations with the previous ones and provide two examples to elucidate that the current method is more accurate.
28 citations
References
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Book•
31 Oct 1991
TL;DR: Theoretical Foundations.
Abstract: I. Theoretical Foundations.- 1. Knowledge.- 1.1. Introduction.- 1.2. Knowledge and Classification.- 1.3. Knowledge Base.- 1.4. Equivalence, Generalization and Specialization of Knowledge.- Summary.- Exercises.- References.- 2. Imprecise Categories, Approximations and Rough Sets.- 2.1. Introduction.- 2.2. Rough Sets.- 2.3. Approximations of Set.- 2.4. Properties of Approximations.- 2.5. Approximations and Membership Relation.- 2.6. Numerical Characterization of Imprecision.- 2.7. Topological Characterization of Imprecision.- 2.8. Approximation of Classifications.- 2.9. Rough Equality of Sets.- 2.10. Rough Inclusion of Sets.- Summary.- Exercises.- References.- 3. Reduction of Knowledge.- 3.1. Introduction.- 3.2. Reduct and Core of Knowledge.- 3.3. Relative Reduct and Relative Core of Knowledge.- 3.4. Reduction of Categories.- 3.5. Relative Reduct and Core of Categories.- Summary.- Exercises.- References.- 4. Dependencies in Knowledge Base.- 4.1. Introduction.- 4.2. Dependency of Knowledge.- 4.3. Partial Dependency of Knowledge.- Summary.- Exercises.- References.- 5. Knowledge Representation.- 5.1. Introduction.- 5.2. Examples.- 5.3. Formal Definition.- 5.4. Significance of Attributes.- 5.5. Discernibility Matrix.- Summary.- Exercises.- References.- 6. Decision Tables.- 6.1. Introduction.- 6.2. Formal Definition and Some Properties.- 6.3. Simplification of Decision Tables.- Summary.- Exercises.- References.- 7. Reasoning about Knowledge.- 7.1. Introduction.- 7.2. Language of Decision Logic.- 7.3. Semantics of Decision Logic Language.- 7.4. Deduction in Decision Logic.- 7.5. Normal Forms.- 7.6. Decision Rules and Decision Algorithms.- 7.7. Truth and Indiscernibility.- 7.8. Dependency of Attributes.- 7.9. Reduction of Consistent Algorithms.- 7.10. Reduction of Inconsistent Algorithms.- 7.11. Reduction of Decision Rules.- 7.12. Minimization of Decision Algorithms.- Summary.- Exercises.- References.- II. Applications.- 8. Decision Making.- 8.1. Introduction.- 8.2. Optician's Decisions Table.- 8.3. Simplification of Decision Table.- 8.4. Decision Algorithm.- 8.5. The Case of Incomplete Information.- Summary.- Exercises.- References.- 9. Data Analysis.- 9.1. Introduction.- 9.2. Decision Table as Protocol of Observations.- 9.3. Derivation of Control Algorithms from Observation.- 9.4. Another Approach.- 9.5. The Case of Inconsistent Data.- Summary.- Exercises.- References.- 10. Dissimilarity Analysis.- 10.1. Introduction.- 10.2. The Middle East Situation.- 10.3. Beauty Contest.- 10.4. Pattern Recognition.- 10.5. Buying a Car.- Summary.- Exercises.- References.- 11. Switching Circuits.- 11.1. Introduction.- 11.2. Minimization of Partially Defined Switching Functions.- 11.3. Multiple-Output Switching Functions.- Summary.- Exercises.- References.- 12. Machine Learning.- 12.1. Introduction.- 12.2. Learning From Examples.- 12.3. The Case of an Imperfect Teacher.- 12.4. Inductive Learning.- Summary.- Exercises.- References.
7,826 citations
TL;DR: The metric and dimensional assumptions that underlie the geometric representation of similarity are questioned on both theoretical and empirical grounds and a set of qualitative assumptions are shown to imply the contrast model, which expresses the similarity between objects as a linear combination of the measures of their common and distinctive features.
Abstract: The metric and dimensional assumptions that underlie the geometric representation of similarity are questioned on both theoretical and empirical grounds. A new set-theoretical approach to similarity is developed in which objects are represented as collections of features, and similarity is described as a feature-matching process. Specifically, a set of qualitative assumptions is shown to imply the contrast model, which expresses the similarity between objects as a linear combination of the measures of their common and distinctive features. Several predictions of the contrast model are tested in studies of similarity with both semantic and perceptual stimuli. The model is used to uncover, analyze, and explain a variety of empirical phenomena such as the role of common and distinctive features, the relations between judgments of similarity and difference, the presence of asymmetric similarities, and the effects of context on judgments of similarity. The contrast model generalizes standard representations of similarity data in terms of clusters and trees. It is also used to analyze the relations of prototypicality and family resemblance
7,251 citations
TL;DR: This paper presents a framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations, and introduces a special class of neighborhood system, called 1-neighborhood systems.
967 citations
Additional excerpts
...Thus the classical rough approximations has been extended to the similarity relation based rough sets [2, 30, 31], the tolerance relation based rough sets [28], the arbitrary binary relation based rough sets [3, 19, 38, 41, 42] and the covering-based rough sets [5, 6, 8, 18, 40, 43]....
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TL;DR: New definitions of lower and upper approximations are proposed, which are basic concepts of the rough set theory and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.
Abstract: This paper proposes new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the concept of ambiguity introduced in this paper. The new definitions are compared to the classical definitions and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.
963 citations
"Rough sets and topological spaces b..." refers background in this paper
...The pair (U, s) is called a space, the elements of U are called points of the space, the subsets of U belonging to s are called open sets in the space, and their complement are called closed sets in the space; the family s of open subsets of U is also called a topology for U [30]....
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...3 [30] Let (U, s) be a topological space, a closure (resp....
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...Thus the classical rough approximations has been extended to the similarity relation based rough sets [2, 30, 31], the tolerance relation based rough sets [28], the arbitrary binary relation based rough sets [3, 19, 38, 41, 42] and the covering-based rough sets [5, 6, 8, 18, 40, 43]....
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TL;DR: In tolerance approximation spaces the lower and upper set approximations are defined and the tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process.
Abstract: We generalize the notion of an approximation space introduced in [8] In tolerance approximation spaces we define the lower and upper set approximations We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems The tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process We obtain the solutions of the reduction problems by applying boolean reasoning [1] The solutions are represented by tolerance reducts and relative tolerance reducts
955 citations
"Rough sets and topological spaces b..." refers background in this paper
...[28, 29] generalized the classical approximation spaces to tolerance E....
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...Thus the classical rough approximations has been extended to the similarity relation based rough sets [2, 30, 31], the tolerance relation based rough sets [28], the arbitrary binary relation based rough sets [3, 19, 38, 41, 42] and the covering-based rough sets [5, 6, 8, 18, 40, 43]....
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