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Witold Pedrycz - One of the best experts on this subject based on the ideXlab platform.
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augmentation of the reconstruction performance of fuzzy c means with an optimized Fuzzification factor vector
Knowledge Based Systems, 2021Co-Authors: Witold PedryczAbstract:Abstract Information granules have been considered as the fundamental constructs of Granular Computing. As a useful unsupervised learning technique, Fuzzy C-Means (FCM) is one of the most frequently used methods to construct information granules. The FCM-based granulation–degranulation mechanism plays a pivotal role in Granular Computing. In this paper, to enhance the quality of the degranulation (reconstruction) process, we augment the FCM-based degranulation mechanism by introducing a vector of Fuzzification factors (Fuzzification factor vector) and setting up an adjustment mechanism to modify the prototypes and the partition matrix. The design is regarded as an optimization problem, which is guided by a reconstruction criterion. In the proposed scheme, the initial partition matrix and prototypes are generated by the FCM. Then a Fuzzification factor vector is introduced to form an appropriate Fuzzification factor for each cluster to build up an adjustment scheme of modifying the prototypes and the partition matrix. With the supervised learning mode of the granulation–degranulation process, we construct a composite objective function of the Fuzzification factor vector, the prototypes and the partition matrix. Subsequently, the particle swarm optimization is employed to optimize the Fuzzification factor vector to refine the prototypes and develop the optimal partition matrix. Finally, the reconstruction performance of the FCM algorithm is enhanced. Overall, we show that the enhanced version of the degranulation process is beneficial to reduce the deterioration of the reconstruction results and improve the performance of the mechanism of granulation–degranulation, which is also meaningful for transforming data between numeric form and granular format. We offer a thorough analysis of the developed scheme. In particular, we show that the classical FCM algorithm forms a special case of the proposed scheme. Experiments completed for both synthetic and publicly available datasets demonstrate that the proposed approach outperforms the generic data reconstruction approach.
H. De Meyer - One of the best experts on this subject based on the ideXlab platform.
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On the transitivity of a parametric family of cardinality-based similarity measures
International Journal of Approximate Reasoning, 2009Co-Authors: B. De Baets, Saskia Janssens, H. De MeyerAbstract:We introduce a parametric family of cardinality-based similarity measures for ordinary sets (on a finite universe) harbouring numerous well-known similarity measures. We characterize the Lukasiewicz-transitive and product-transitive members of this family. Their importance derives from their one-to-one correspondence with pseudo-metrics. Fuzzification schemes based on a commutative quasi-copula are then used to transform these similarity measures for ordinary sets into similarity measures for fuzzy sets, rendering them applicable on graded feature set representations of objects. The main result of this paper is that transitivity, and hence also the corresponding dual metric interpretation, is preserved along this Fuzzification process.
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Transitivity-preserving Fuzzification schemes for cardinality-based similarity measures
European Journal of Operational Research, 2005Co-Authors: B. De Baets, H. De MeyerAbstract:Abstract A family of Fuzzification schemes is proposed that can be used to transform cardinality-based similarity measures for ordinary sets into similarity measures for fuzzy sets in a finite universe. The family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets. In particular, the fuzzy set intersections are pointwisely generated by Frank t -norms. The Fuzzification schemes are applied to a variety of previously studied rational cardinality-based similarity measures for ordinary sets and it is demonstrated that transitivity is preserved in the Fuzzification process.
B. De Baets - One of the best experts on this subject based on the ideXlab platform.
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On the transitivity of a parametric family of cardinality-based similarity measures
International Journal of Approximate Reasoning, 2009Co-Authors: B. De Baets, Saskia Janssens, H. De MeyerAbstract:We introduce a parametric family of cardinality-based similarity measures for ordinary sets (on a finite universe) harbouring numerous well-known similarity measures. We characterize the Lukasiewicz-transitive and product-transitive members of this family. Their importance derives from their one-to-one correspondence with pseudo-metrics. Fuzzification schemes based on a commutative quasi-copula are then used to transform these similarity measures for ordinary sets into similarity measures for fuzzy sets, rendering them applicable on graded feature set representations of objects. The main result of this paper is that transitivity, and hence also the corresponding dual metric interpretation, is preserved along this Fuzzification process.
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Transitivity-preserving Fuzzification schemes for cardinality-based similarity measures
European Journal of Operational Research, 2005Co-Authors: B. De Baets, H. De MeyerAbstract:Abstract A family of Fuzzification schemes is proposed that can be used to transform cardinality-based similarity measures for ordinary sets into similarity measures for fuzzy sets in a finite universe. The family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets. In particular, the fuzzy set intersections are pointwisely generated by Frank t -norms. The Fuzzification schemes are applied to a variety of previously studied rational cardinality-based similarity measures for ordinary sets and it is demonstrated that transitivity is preserved in the Fuzzification process.
Jerry M Mendel - One of the best experts on this subject based on the ideXlab platform.
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non singleton Fuzzification made simpler
Information Sciences, 2021Co-Authors: Jerry M MendelAbstract:Abstract Non-singleton Fuzzification is used in rule-based fuzzy systems when the measurements that activate them are imperfect or uncertain or when their inputs are words. It models such measurements or words as fuzzy numbers or more general fuzzy sets so that, regardless of the cause of a measurement’s or word’s imperfections or uncertainties, they are treated within the framework of fuzzy sets and systems. Since 2011, there has been a resurgence of interest in both type-1 and interval type-2 non-singleton fuzzy systems. This paper removes a computational bottleneck associated with computing the firing level or firing interval for such fuzzy systems, by providing closed-form formulas for them when the involved fuzzy sets are trapezoidal or triangular, which are widely used fuzzy sets. This is done for both the minimum and product t-norms. It is also demonstrated that a non-singleton fuzzy system that uses the product t-norm has the potential to outperform a non-singleton fuzzy system that uses the minimum t-norm. The results in this paper greatly simplify non-singleton fuzzy systems, which should make them much more popular.
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Comparing the Performance Potentials of Singleton and Non-singleton Type-1 and Interval Type-2 Fuzzy Systems in Terms of Sculpting the State Space
IEEE Transactions on Fuzzy Systems, 2020Co-Authors: Jerry M Mendel, Ravikiran Chimatapu, Hani HagrasAbstract:This paper provides a novel and better understanding of the performance potential of a nonsingleton (NS) fuzzy system over a singleton (S) fuzzy system. It is done by extending sculpting the state space works from S to NS Fuzzification and demonstrating uncertainties about measurements, modeled by NS Fuzzification: first, fire more rules more often, manifested by a reduction (increase) in the sizes of first-order rule partitions for those partitions associated with the firing of a smaller (larger) number of rules-the coarse sculpting of the state space; second, this may lead to an increase or decrease in the number of type-1 (T1) and interval type-2 (IT2) first-order rule partitions, which now contain rule pairs that can never occur for S Fuzzification-a new rule crossover phenomenon-discovered using partition theory; and third, it may lead to a decrease, the same number, or an increase in the number of second-order rule partitions, all of which are system dependent-the fine sculpting of the state space. The authors' conjecture is that it is the additional control of the coarse sculpting of the state space, accomplished by prefiltering and the max-min (or max-product) composition, which provides an NS T1 or IT2 fuzzy system with the potential to outperform an S T1 or IT2 system when measurements are uncertain.
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uncertain rule based fuzzy logic systems introduction and new directions
2001Co-Authors: Jerry M MendelAbstract:(NOTE: Each chapter concludes with Exercises.) I: PRELIMINARIES. 1. Introduction. Rule-Based FLSs. A New Direction for FLSs. New Concepts and Their Historical Background. Fundamental Design Requirement. The Flow of Uncertainties. Existing Literature on Type-2 Fuzzy Sets. Coverage. Applicability Outside of Rule-Based FLSs. Computation. Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic. Primer on Fuzzy Sets. Primer on FL. Remarks. 2. Sources of Uncertainty. Uncertainties in a FLS. Words Mean Different Things to Different People. 3. Membership Functions and Uncertainty. Introduction. Type-1 Membership Functions. Type-2 Membership Functions. Returning to Linguistic Labels. Multivariable Membership Functions. Computation. 4. Case Studies. Introduction. Forecasting of Time-Series. Knowledge Mining Using Surveys. II: TYPE-1 FUZZY LOGIC SYSTEMS. 5. Singleton Type-1 Fuzzy Logic Systems: No Uncertainties. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. DeFuzzification. Possibilities. Fuzzy Basis Functions. FLSs Are Universal Approximators. Designing FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. A Final Remark. Computation. 6. Non-Singleton Type-1 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Possibilities. FBFs. Non-Singleton FLSs Are Universal Approximators. Designing Non-Singleton FLSs. Case Study: Forecasting of Time-Series. A Final Remark. Computation. III: TYPE-2 FUZZY SETS. 7. Operations on and Properties of Type-2 Fuzzy Sets. Introduction. Extension Principle. Operations on General Type-2 Fuzzy Sets. Operations on Interval Type-2 Fuzzy Sets. Summary of Operations. Properties of Type-2 Fuzzy Sets. Computation. 8. Type-2 Relations and Compositions. Introduction. Relations in General. Relations and Compositions on the Same Product Space. Relations and Compositions on Different Product Spaces. Composition of a Set with a Relation. Cartesian Product of Fuzzy Sets. Implications. 9. Centroid of a Type-2 Fuzzy Set: Type-Reduction. Introduction. General Results for the Centroid. Generalized Centroid for Interval Type-2 Fuzzy Sets. Centroid of an Interval Type-2 Fuzzy Set. Type-Reduction: General Results. Type-Reduction: Interval Sets. Concluding Remark. Computation. IV: TYPE-2 FUZZY LOGIC SYSTEMS. 10. Singleton Type-2 Fuzzy Logic Systems. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Type-Reduction. DeFuzzification. Possibilities. FBFs: The Lack Thereof. Interval Type-2 FLSs. Designing Interval Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. Computation. 11. Type-1 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-1 Non-Singleton Type-2 FLSs. Designing Interval Type-1 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Final Remark. Computation. 12. Type-2 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-2 Non-Singleton Type-2 FLSs. Designing Interval Type-2 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Computation. 13. TSK Fuzzy Logic Systems. Introduction. Type-1 TSK FLSs. Type-2 TSK FLSs. Example: Forecasting of Compressed Video Traffic. Final Remark. Computation. 14. Epilogue. Introduction. Type-2 Versus Type-1 FLSs. Appropriate Applications for a Type-2 FLS. Rule-Based Classification of Video Traffic. Equalization of Time-Varying Non-linear Digital Communication Channels. Overcoming CCI and ISI for Digital Communication Channels. Connection Admission Control for ATM Networks. Potential Application Areas for a Type-2 FLS. A. Join, Meet, and Negation Operations For Non-Interval Type-2 Fuzzy Sets. Introduction. Join Under Minimum or Product t-Norms. Meet Under Minimum t-Norm. Meet Under Product t-Norm. Negation. Computation. B. Properties of Type-1 and Type-2 Fuzzy Sets. Introduction. Type-1 Fuzzy Sets. Type-2 Fuzzy Sets. C. Computation. Type-1 FLSs. General Type-2 FLSs. Interval Type-2 FLSs. References. Index.
Alejandro Ramossoto - One of the best experts on this subject based on the ideXlab platform.
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characterizing quantifier Fuzzification mechanisms a behavioral guide for applications
Fuzzy Sets and Systems, 2018Co-Authors: Felix Diazhermida, Martin Pereirafarina, Juan C Vidal, Alejandro RamossotoAbstract:Important advances have been made in the fuzzy quantification field. Nevertheless, some problems remain when we face the decision of selecting the most convenient model for a specific application. In the literature, several desirable adequacy properties have been proposed, but theoretical limits impede quantification models from simultaneously fulfilling every adequacy property that has been defined. Besides, the complexity of model definitions and adequacy properties makes very difficult for real users to understand the particularities of the different models that have been presented. In this work we will present several criteria conceived to help in the process of selecting the most adequate Quantifier Fuzzification Mechanisms for specific practical applications. In addition, some of the best known well-behaved models will be compared against this list of criteria. Based on this analysis, some guidance to choose fuzzy quantification models for practical applications will be provided.
