Tolerance approximation spaces
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
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TL;DR: The basic concepts of rough set theory are presented and some rough set-based research directions and applications are pointed out, indicating that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences.
2,004 citations
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..., [13,17,62,86,93,94,104,116–119,143,153,176,171,197,203,209,211–213,235,240,249,273,277,278,283–285, 289– 292,307,312,308,310,315,328,352,366,364,369,371,372]), including the various logical (see, e....
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TL;DR: Some extensions of the rough set approach are presented and a challenge for the roughSet based research is outlined and it is outlined that the current rough set based research paradigms are unsustainable.
1,161 citations
Cites background from "Tolerance approximation spaces"
...However, in the rough set approach, indiscernibility is defined relative to a given set of functionals (attributes)....
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TL;DR: A neighborhood rough set model is introduced to deal with the problem of heterogeneous feature subset selection and Experimental results show that the neighborhood model based method is more flexible to deals with heterogeneous data.
780 citations
Cites result from "Tolerance approximation spaces"
...Besides, this idea is similar to the tolerance rough sets [30]....
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TL;DR: It has been proved that the reduct of a covering is the minimal covering that generates theSame covering lower approximation or the same covering upper approximation, so this concept is also a technique to get rid of redundancy in data mining.
699 citations
Cites background from "Tolerance approximation spaces"
...To address this issue, several interesting and meaningful extensions to equivalent relation have been proposed in the past, such as tolerance relations [4,8], similarity relations [9], and others [10,13,14]....
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TL;DR: This paper reviews those techniques that preserve the underlying semantics of the data, using crisp and fuzzy rough set-based methodologies, and several approaches to feature selection based on rough set theory are experimentally compared.
Abstract: Semantics-preserving dimensionality reduction refers to the problem of selecting those input features that are most predictive of a given outcome; a problem encountered in many areas such as machine learning, pattern recognition, and signal processing. This has found successful application in tasks that involve data sets containing huge numbers of features (in the order of tens of thousands), which would be impossible to process further. Recent examples include text processing and Web content classification. One of the many successful applications of rough set theory has been to this feature selection area. This paper reviews those techniques that preserve the underlying semantics of the data, using crisp and fuzzy rough set-based methodologies. Several approaches to feature selection based on rough set theory are experimentally compared. Additionally, a new area in feature selection, feature grouping, is highlighted and a rough set-based feature grouping technique is detailed.
634 citations
Cites background from "Tolerance approximation spaces"
...This has found successful application in tasks that involve data sets containing huge numbers of features (in the order of tens of thousands), which would be impossible to process further....
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References
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Book•
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01 Aug 1996
TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
52,705 citations
Book•
11 Feb 1984
TL;DR: This invaluable reference helps readers assess and simplify problems and their essential requirements and complexities, giving them all the necessary data and methodology to master current theoretical developments and applications, as well as create new ones.
Abstract: Image Processing and Mathematical Morphology-Frank Y. Shih 2009-03-23 In the development of digital multimedia, the importance and impact of image processing and mathematical morphology are well documented in areas ranging from automated vision detection and inspection to object recognition, image analysis and pattern recognition. Those working in these ever-evolving fields require a solid grasp of basic fundamentals, theory, and related applications—and few books can provide the unique tools for learning contained in this text. Image Processing and Mathematical Morphology: Fundamentals and Applications is a comprehensive, wide-ranging overview of morphological mechanisms and techniques and their relation to image processing. More than merely a tutorial on vital technical information, the book places this knowledge into a theoretical framework. This helps readers analyze key principles and architectures and then use the author’s novel ideas on implementation of advanced algorithms to formulate a practical and detailed plan to develop and foster their own ideas. The book: Presents the history and state-of-the-art techniques related to image morphological processing, with numerous practical examples Gives readers a clear tutorial on complex technology and other tools that rely on their intuition for a clear understanding of the subject Includes an updated bibliography and useful graphs and illustrations Examines several new algorithms in great detail so that readers can adapt them to derive their own solution approaches This invaluable reference helps readers assess and simplify problems and their essential requirements and complexities, giving them all the necessary data and methodology to master current theoretical developments and applications, as well as create new ones.
9,566 citations
"Tolerance approximation spaces" refers background in this paper
...The second one is related to structural elements introduced by analogy with mathematical morphology [12]....
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...We would like to add one more condition to the de nition of set approximation which arises by analogy with mathematical morphology [12]....
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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: A generalized model of rough sets called variable precision model (VP-model), aimed at modelling classification problems involving uncertain or imprecise information, is presented and the main concepts are introduced formally and illustrated with simple examples.
1,975 citations
Book•
01 Jan 1994TL;DR: The Dempster-Shafer Theory of Evidence is applied as a guide for the management of uncertainty in knowledge-based systems.
Abstract: Partial table of contents: DEMPSTER-SHAFER THEORY OF EVIDENCE: GENERAL ISSUES. Measures of Uncertainty in the Dempster-Shafer Theory of Evidence (G. Klir). Comparative Beliefs (S. Wong, et al.). Calculus with Linguistic Probabilities and Beliefs (M. Lamata & S. Moral). FUZZIFICATION OF DEMPSTER-SHAFER THEORY OF EVIDENCE. Rough Membership Functions (Z. Pawlak & A. Skowron). DEMPSTER-SHAFER THEORY IN DECISION MAKING AND OPTIMIZATION. Decision Analysis Using Belief Functions (T. Strat). Interval Probabilities Induced by Decision Problems (T. Whalen). DEMPSTER-SHAFER THEORY FOR THE MANAGEMENT OF UNCERTAINTY IN KNOWLEDGE-BASED SYSTEMS. Using Dempster-Shafer's Belief-Function Theory in Expert Systems (P. Shenoy). Nonmonotonic Reasoning with Belief Structures (R. Yager). Index.
792 citations