Explainable Uncertain Rule-Based Fuzzy Systems

• Autore: Mendel Jerry M
• Pubblicazione: Cham : , : Springer International Publishing AG, , 2024; ©2024
• Edizione: 3rd ed.
• Descrizione fisica: 1 online resource (598 pages)
• Disciplina: 511.313
• Nota di contenuto: Intro -- Preface -- References -- Contents -- About the Author -- 1: Introduction -- 1.1 What This Book Is About -- 1.1.1 Rules -- 1.1.2 Partitions and Sets -- 1.1.2.1 Crisp Partitions -- 1.1.2.2 First-Order Uncertainty Partitions -- 1.1.2.3 Second-Order Uncertainty Partitions: Uniformly Weighted -- 1.1.2.4 Second-Order Uncertainty Partitions: Nonuniformly Weighted -- 1.1.2.5 Footprint of Uncertainty (FOU) -- 1.1.2.6 Comments -- 1.2 The Structure of a Rule-Based Fuzzy System -- 1.3 A New Direction for Fuzzy Systems -- 1.4 Fundamental Design Requirement -- 1.5 Advisable Design Approaches -- 1.6 Understanding the Potential for Improved Performance -- 1.7 Explainable Fuzzy Systems -- 1.8 An Impressionistic Brief History of Type-1 Fuzzy Sets and Fuzzy Logic -- 1.9 Literature on Type-2 Fuzzy Sets and Fuzzy Systems -- 1.9.1 Early Literature: 1975-1992 -- 1.9.2 Publications that Heavily Influenced the First Edition of This Book -- 1.9.3 Most Cited Articles -- 1.10 Coverage -- 1.11 Applicability Outside of Rule-Based Fuzzy Systems -- 1.12 Computation -- References -- 2: Type-1 Fuzzy Sets and Fuzzy Logic -- 2.1 Crisp Sets -- 2.2 Type-1 Fuzzy Sets and Associated Concepts -- 2.2.1 Lotfi A. Zadeh -- 2.2.2 Type-1 Fuzzy Set Defined -- 2.2.3 Type-1 Fuzzy Numbers -- 2.2.4 Linguistic Variables -- 2.2.5 Returning to Linguistic Labels from Numerical Values of MFs -- 2.3 Set Theoretic Operations for Crisp Sets -- 2.4 Set Theoretic Operations for Type-1 Fuzzy Sets -- 2.5 Crisp Relations and Compositions on the Same Product Space -- 2.6 Fuzzy Relations and Compositions on the Same Product Space -- 2.7 Crisp Relations and Compositions on Different Product Spaces -- 2.8 Fuzzy Relations and Compositions on Different Product Spaces -- 2.9 Hedges -- 2.10 Extension Principle -- 2.11 α-Cuts -- 2.12 Representing Type-1 Fuzzy Sets Using α-Cuts. ; 2.13 Functions of Type-1 Fuzzy Sets Computed by Using α-Cuts -- 2.14 Multivariable MFs and Cartesian Products -- 2.15 Crisp Logic -- 2.16 From Crisp Logic to Fuzzy Logic -- 2.17 Mamdani (Engineering) Implications -- 2.18 Remarks -- 1.1 Laws That Are Satisfied -- 1.2 Laws That Are Not Satisfied -- Appendix 2: Cardinality and Similarity -- 2.1 Cardinality of Type-1 Fuzzy Sets -- 2.2 Similarity of Type-1 Fuzzy Sets -- References -- 3: Type-1 Fuzzy Systems -- 3.1 Type-1 Fuzzy Systems -- 3.2 Rules -- 3.3 Fuzzifier -- 3.4 Fuzzy Inference Engine -- 3.4.1 General Results -- 3.4.2 Fuzzification and Its Effects on Inference -- 3.4.2.1 Singleton Fuzzifier -- 3.4.2.2 Non-Singleton Fuzzifier -- 3.5 Combining Fired-Rule Output Sets on the Way to Defuzzification -- 3.5.1 Mamdani Fuzzy System: Combining Using Set Theoretic Operations -- 3.5.2 Mamdani Fuzzy System: Combining Using a Weighted Combination -- 3.5.3 Mamdani and TSK Fuzzy Systems: Combining During Defuzzification -- 3.6 Defuzzifier -- 3.6.1 Mamdani Fuzzy System: Centroid Defuzzifier -- 3.6.2 Mamdani Fuzzy System: Height Defuzzifier -- 3.6.3 Mamdani Fuzzy System: COS Defuzzifier -- 3.6.4 TSK Fuzzy System Defuzzifiers -- 3.7 Comprehensive Example -- 3.8 Fuzzy Basis Functions -- 3.9 Sculpting the State Space and the Potential for Improved Performance over a Non-Fuzzy System -- 3.9.1 Course Sculpting of the State Space -- 3.9.2 Fine Sculpting of the State Space -- 3.9.3 Observations -- 3.10 Remarks and Insights -- 3.10.1 Unique Features of Type-1 Fuzzy Systems -- 3.10.2 Layered Architecture Interpretations of a Fuzzy System -- 3.10.3 Functional Equivalence to Other Machine Learning Methods -- 3.10.4 Universal Approximation by Fuzzy Systems -- 3.10.5 Continuity of Fuzzy Systems -- 3.10.6 Rule Explosion and Some Ways to Control It -- 3.10.7 Interpretable and Explainable T1 Fuzzy Systems -- 3.10.7.1 Introduction. ; 3.10.7.2 On Interpretable -- 3.10.7.3 On Explainable -- 3.10.8 A Top-Down Approach to T1 Fuzzy Systems -- 1.1 Evaluation of Sup-Star Composition for Minimum t-Norm -- 1.2 Evaluation of Sup-Star Composition for Product t-Norm -- 1.3 A Novel Suggestion -- Appendix 2: Constructing Type-1 Rule Partitions -- 2.1 Singleton Fuzzification: T1 First-Order Rule Partitions -- 2.2 Singleton Fuzzification: T1 Second-Order Rule Partitions -- 2.3 Non-Singleton Fuzzification: T1 First-Order Rule Partitions -- 2.4 Non-Singleton Fuzzification: T1 Second-Order Rule Partitions -- 2.5 Rule Crossover Phenomenon -- Appendix 3: Procedure for Determining the Active Rules in a First-Order Rule Partition -- 3.1 First-Order Rule Partition Information Table -- 3.2 Indexing Rules -- 3.3 Determining Rules Associated with x = x′ -- References -- 4: Type-1 Fuzzy Systems: Design Methods and Case Studies -- 4.1 Designing Type-1 Fuzzy Systems -- 4.1.1 Design Choices and Complexity -- 4.1.2 An Interpretation for the Design of a Type-1 Fuzzy System -- 4.1.3 Recapitulation of Mamdani and TSK Fuzzy Systems -- 4.1.4 Number of Design Degrees of Freedom and a Design Principle -- 4.1.5 High-Level Design Statements and Design Approaches -- 4.2 Some Design Methods -- 4.2.1 One-Pass Methods -- 4.2.1.1 Data Assignment Method -- 4.2.1.2 WM Method -- 4.2.2 Clustering Using Fuzzy c-Means (FCM) -- 4.2.3 Least Squares (LS) Method -- 4.2.4 Derivative-Based Methods (Back-Propagation) -- 4.2.5 Derivative-Free Methods -- 4.2.6 Hybrid Design Methods -- 4.2.6.1 Adaptive Network Fuzzy Inference System (ANFIS) -- 4.2.6.2 Structure Identification and Feature Extraction (SIFE) for TSK Systems -- 4.2.7 Remarks -- 4.3 Case Study: Forecasting of Time-Series -- 4.3.1 Mackey-Glass Chaotic Time Series -- 4.3.2 One-Pass Design: Singleton Fuzzification -- 4.3.3 Derivative-Based (BP) Design: Singleton Fuzzification. ; 4.3.4 A Change in the Measurements -- 4.3.5 One-Pass Design: Non-singleton Fuzzification -- 4.3.6 Derivative-Based (BP) Design: Non-singleton Fuzzification -- 4.3.7 Final Remark -- 4.4 Case Study: Knowledge Mining Using Surveys -- 4.4.1 Methodology for Knowledge Mining -- 4.4.2 Survey Results -- 4.4.3 Determining Type-1 Fuzzy Sets from Survey Results -- 4.4.4 What Does One Do with a Histogram of Responses? -- 4.4.5 Averaging the Responses: Consensus FLAs -- 4.4.6 Preserving All of the Responses -- 4.4.7 On Multiple Indicators -- 4.4.8 How to Use an FLA -- 4.4.9 Connections to the Perceptual Computer -- 4.5 Case Study: Rule-Based Classification of Video Traffic -- 4.5.1 Compressed Video Traffic -- 4.5.2 High-Level Video Classification Problem -- 4.5.3 Selected Features -- 4.5.4 MFs for the Features -- 4.5.5 Rules and Their Parameters -- 4.5.6 Computational Formulas for the RBC -- 4.5.7 Optimization of Rule Design Parameters -- 4.5.8 Testing the FL RBC -- 4.5.9 Results and Conclusions -- 4.6 Case Study: Fuzzy Logic Control -- 4.6.1 Early History of Fuzzy Control -- 4.6.2 What Is a Type-1 Fuzzy Logic Controller (FLC)? -- 4.6.3 Fuzzy PID Control -- 4.6.3.1 Background -- 4.6.3.2 General Structure of Fuzzy PID Controller -- 4.6.3.3 Conventional and Fuzzy PID Controller Design Methods -- 4.6.3.4 Simulation Results (T1-FPID Versus PID) -- 4.7 Case Study: Explainable Type-1 Fuzzy System -- 4.7.1 Computations Common to Both Fuzzy Systems -- 4.7.1.1 Firing Levels for the Active Rules -- 4.7.1.2 Similarities -- 4.7.2 Mamdani with Centroid Defuzzification -- 4.7.2.1 Computation of yc (2.4, 5.4, 9) -- 4.7.2.2 Explaining yc (2.4,5.4,9) -- 4.7.2.3 Quality of Explanation -- 4.7.3 Mamdani with COS Defuzzification -- 4.7.3.1 Computation of yCOS(2.4,5.4,9) -- 4.7.3.2 Explaining yCOS(2.4,5.4,9) -- 4.7.3.3 Observations -- 1.1 Count of MF Parameters. ; 1.2 T1 MF Constraints -- 1.3 Determine If Satisfying All of the Constraints Is Possible -- 1.4 Constraints Almost Always-Satisfied Parameters (CAASPs) -- 1.5 Comments -- 1.6 Optimizing T1 MF Parameters -- References -- 5: Sources of Uncertainty and Membership Functions -- 5.1 Uncertainties in a Fuzzy System -- 5.1.1 Uncertainty: General Discussions -- 5.1.2 Uncertainties and Sets -- 5.1.3 Uncertainties in a Fuzzy System -- 5.2 Words Mean Different Things to Different People -- 5.2.1 Collecting Word Data by Means of a Survey -- 5.2.2 Making Use of Word Uncertainties -- 5.2.3 Conclusion -- 5.3 Words Must Also Mean Similar Things to Different People -- 5.3.1 Probability-Based Solution of (5.1) -- 5.3.2 Iterative Solution of (5.1)s -- 5.3.3 Example -- 5.4 From Interval Data to a T1 FS -- 5.4.1 Mean and Standard Deviation for Each Data Interval -- 5.4.2 T1 FS Models and Their Mean and Standard Deviation -- 5.4.3 Computation of MF Parameters -- 5.4.4 Choice of T1 MF -- 5.4.5 Ensemble of T1 MFs -- References -- 6: Type-2 Fuzzy Sets Including Word Models -- 6.1 The Concept of a Type-2 Fuzzy Set -- 6.2 Definitions of a General Type-2 Fuzzy Set and Associated Concepts -- 6.3 Definitions of an IT2 FS and Associated Concepts -- 6.4 Examples of Two Popular FOUs -- 6.5 Interval Type-2 Fuzzy Numbers -- 6.6 Different Kinds of T2 FSs: Hierarchy -- 6.7 Mathematical Representations for T2 FSs -- 6.7.1 Vertical Slice Representation -- 6.7.2 Wavy Slice Representations -- 6.7.2.1 General Case -- 6.7.2.2 Covering the FOU -- 6.7.2.3 Minimal Coverings -- 6.7.2.4 Comments -- 6.7.3 Horizontal Slice Representation -- 6.7.4 Modeling Secondary MFs -- 6.8 Representing Non-T2 FSs as T2 FSs -- 6.9 Returning to Linguistic Labels for General T2 FSs -- 6.10 Multivariable Membership Functions -- 6.11 IT2 FS Word Models -- References -- 7: Working with Type-2 Fuzzy Sets. ; 7.1 Introduction and Guide for the Reader.
• Titolo autorizzato: Explainable Uncertain Rule-Based Fuzzy Systems
• ISBN: 3-031-35378-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910831020703321
• Lo trovi qui: Univ. Federico II