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Proceedings ArticleDOI

Generalised approximate equalities based on rough fuzzy sets & rough measures of fuzzy sets

TL;DR: This paper introduces and discusses the rough measures of basic sets, fuzzy sets and interpret four types of approximate equalities in terms of the accuracy measure as well as rough measures.
Abstract: In an attempt to incorporate user knowledge in order to decide about the equality of sets, the concepts of approximate equalities using rough sets were introduced. These notions have been generalised in several ways and very recently [1] extended four types of approximate equalities using rough fuzzy sets instead of only rough sets. To be precise, a concept of leveled approximate equality was introduced and properties were studied. In this paper we extend this work with case studies to illustrate the applications of the concepts and compare them respectively. We also introduce and discuss the rough measures of basic sets, fuzzy sets and interpret four types of approximate equalities in terms of the accuracy measure as well as rough measures. The analysis had provided a clear distinguish notion in terms of the measures.
Citations
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Book ChapterDOI
01 Jan 2016
TL;DR: An application of fuzzy soft sets in decision making is provided which substantially improve and is more realistic than the algorithm proposed earlier by Maji et al.
Abstract: Soft set theory is a new mathematical approach to vagueness introduced by Molodtsov. This is a parameterized family of subsets defined over a universal set associated with a set of parameters. In this paper, we define membership function for fuzzy soft sets. Like the soft sets, fuzzy soft set is a notion which allows fuzziness over a soft set model. So far, more than one attempt has been made to define this concept. Maji et al. defined fuzzy soft sets and several operations on them. In this paper we followed the definition of soft sets provided by Tripathy et al. through characteristic functions in 2015. Many related concepts like complement of a fuzzy soft set, null fuzzy soft set, absolute fuzzy soft set, intersection of fuzzy soft sets and union of fuzzy soft sets are redefined. We provide an application of fuzzy soft sets in decision making which substantially improve and is more realistic than the algorithm proposed earlier by Maji et al.

24 citations

Book ChapterDOI
01 Jan 2018
TL;DR: Covering-based pessimistic multigranular (CBPMG) approximate rough equivalence is introduced and several of their properties are established.
Abstract: The multigranular rough set (MGRS) models of Qian et al. were extended to put forth covering-based multigranular rough sets (CBMGRS) by Liu et al. in 2012. The equality of sets, which is restrictive and redundant, was extended first in Pawlak (Rough sets: theoretical aspects of reasoning about data. Kluwer, London, 1991) and subsequently in Tripathy (Int J Adv Sci Technol 31:23–36, 2011) to propose four types of rough set-based approximate equalities. These basic concepts of rough equalities have been extended to several generalized rough set models. In this paper, covering-based pessimistic multigranular (CBPMG) approximate rough equivalence is introduced and several of their properties are established. Real life examples are taken for constructing counter examples and also for illustration. We have also discussed how these equalities can be applied in approximate reasoning and our latest proposal is no exception.

1 citations

References
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Book
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
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

Journal ArticleDOI
TL;DR: This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.
Abstract: Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

7,185 citations


"Generalised approximate equalities ..." refers background in this paper

  • ...Properties (Both direct and interchanged) for these are established by [3]–[6] and were extended and analysed in [7]–[9]....

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  • ...This notion has two distinct advantages over the earlier such notions defined in [3]–[9] In this paper we extend the study in [1]....

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  • ...2) General Analysis: The essence in the extension of the approximate equality of Novotny and Pawlak (NP) is to introduce the other three types of approximate equalities of sets....

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  • ...The notion of rough sets introduced by Pawlak [3] depends upon the concept of equivalence relations defined over a universe U ....

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  • ...Definition 3 is a direct extension of the corresponding notion on sets introduced by Novotny and Pawlak [3], [13]....

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Journal ArticleDOI
TL;DR: An integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set, and several properties of this new measure are established.

209 citations


"Generalised approximate equalities ..." refers background or methods or result in this paper

  • ...A study later by Huynh and Nakamori [11] found out that the approach [10] is not in the same lines as for the basic rough sets of Pawlak....

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  • ...In this connection it is worth noting that the approximate equalities of fuzzy sets were first studied by Banerjee and Pal [10]....

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  • ...Banerjee and Pal [10] introduced the concept of rough measures of fuzzy sets....

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  • ...equalities of fuzzy sets were first studied by Banerjee and Pal [10]....

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  • ...Also, the notion of graded approximate equalities introduced there extends the definition of Banerjee et al [10], which is a special case....

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Journal ArticleDOI
TL;DR: A fuzzy qualitative approach to vision-based human motion analysis with an emphasis on human motion recognition is proposed by combining fuzzy qualitative robot kinematics with human motion tracking and recognition algorithms and consistently outperforms conventional hidden Markov model and fuzzy HMM.
Abstract: This paper proposes a fuzzy qualitative approach to vision-based human motion analysis with an emphasis on human motion recognition. It achieves feasible computational cost for human motion recognition by combining fuzzy qualitative robot kinematics with human motion tracking and recognition algorithms. First, a data-quantization process is proposed to relax the computational complexity suffered from visual tracking algorithms. Second, a novel human motion representation, i.e., qualitative normalized template, is developed in terms of the fuzzy qualitative robot kinematics framework to effectively represent human motion. The human skeleton is modeled as a complex kinematic chain, and its motion is represented by a series of such models in terms of time. Finally, experiment results are provided to demonstrate the effectiveness of the proposed method. An empirical comparison with conventional hidden Markov model (HMM) and fuzzy HMM (FHMM) shows that the proposed approach consistently outperforms both HMMs in human motion recognition.

69 citations


"Generalised approximate equalities ..." refers background in this paper

  • ...Our future work is looking forward to apply this into human motion analysis [15]–[17] and complex scene understanding context [18]....

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