Fuzzy Approaches for Soft Computing and Approximate Reasoning: Theories and Applications

Foreword
Preface
Contents
Contributors
The Fuzzy Theoretic Turn
Membership Functions
	1 Introduction
	2 Mathematical Aspects of Membership Functions
		2.1 Classical Versus Gradual Properties
		2.2 Graduality and Partial Pre-orderings
		2.3 Membership Scale: [0, 1] or Not
		2.4 Bipolar Scale or Not?
		2.5 Beyond Scalar Membership Grades
	3 Finding Membership Grades
		3.1 Based on Similarity
		3.2 Fuzzy Sets as Possibility Distributions
		3.3 Fuzzy Sets in Preference Evaluation
	4 Concluding Remarks
	References
The Evolution of the Notion of Overlap Functions
	1 The Origin of the Notion of Overlap Function
	2 The Mathematical Definition of Overlap Functions
	3 Overlap Functions Versus Triangular Norms
	4 n-Dimensional Overlap Functions
	5 Applications of Overlap Functions
	6 Conclusions
	References
Interpolative Reasoning: Valid, Specificity-Gradual and Similarity-Based
	1 Introduction
	2 Interpolative Reasoning
		2.1 Similarity-Based Interpolative Reasoning
	3 Similarity Based Axioms for Interpolative Reasoning
		3.1 Behaviour When Observation Equals a Premise  of a Known Rule
		3.2 Homogeneity of Observation-Premise Specificity  with Respect to Conclusions
		3.3 Monotonicity with Respect to Observation-Conclusion Specificities
		3.4 Valid Convex Conclusions
		3.5 Graduality with Respect to Rule\'s Proximity
	4 Success and Limitations of Similarity-Based Interpolative Reasoning
		4.1 Non-convex Solutions
		4.2 Valid and Specificity-Gradual Solution
	5 Conclusions
	References
A Similarity-Based Three-Valued Modal Logic Approach to Reason with Prototypes and Counterexamples
	1 Introduction
	2 Prototypes, Counter-Examples and Borderline Cases: A Simple 3-Valued Model for Gradual Properties
		2.1 A 3-Valued Approach
		2.2 A Refresher on 3-Valued Łukasiewicz Logic Ł3
		2.3 Dealing with both Prototypes and Counter-Examples: The Logic Łleq3
	3 A Similarity-Based Refined Framework
	4 A Multi-modal Approach to Reason About  the Similarity-Based Graded Entailments
	5 Conclusions and Dedication
	References
Analogy
	1 Introducing Analogy as One of Two Main Ways of Thinking
	2 The Necessity of Fuzzy Logic Computation of Analogical Reasoning and Schemes
	3 What Analogy is and What Analogy is Not
	4 Models of Solving Analogies and Metaphors for Fuzzy Inference Making
	5 Discussion
	References
The Role of the Context in Decision and Optimization Problems
	1 Introduction
	2 The General Decision Problem
	3 An Illustrative Example
		3.1 Induced Framework
		3.2 Ethical Framework
		3.3 Sustainability Framework
		3.4 Corporate Social Responsibility Framework
	4 Conclusions
	References
Decision Rules Under Vague and Uncertain Information
	1 Introduction
	2 Coherent Conditional Decomposable Measures
		2.1 Definitions and Main Results
		2.2 Coherent Extensions
	3 An Interpretation of Fuzzy Sets in Terms of Likelihood Function
		3.1 Conditional Probability and Possibility of Fuzzy Events
	4 Probabilistic and Possibilistic Fuzzy IF-THEN Rules
	5 Conclusions
	References
Abstract Models for Systems Identification
	1 Introduction
	2 Background of the Minimal Realization Problem
	3 Different Types of Identification Models
		3.1 Relational Models
		3.2 Order Models
		3.3 Categorical Models
	4 Concluding Remarks
	References
Fuzzy Systems Interpretability: What, Why and How
	1 Introduction
	2 Some Basic Concepts on Fuzzy Systems
	3 What Is Interpretability?
	4 Why Do We Need Interpretable Systems?
	5 How to Evaluate Interpretability?
	6 How to Build Interpretable Fuzzy Systems?
	7 Conclusions
	References
Fuzzy Clustering Models and Their Related Concepts
	1 Introduction
	2 Additive Clustering Model
	3 Additive Fuzzy Clustering Model
	4 Kernel Fuzzy Clustering Model
	5 Generalized Fuzzy Clustering Model
	6 Generalized Aggregation Operator Based Nonlinear Fuzzy Clustering Model
	7 Conclusion
	References
Fast Cluster Tendency Assessment  for Big, High-Dimensional Data
	1 Introduction
	2 Preliminaries
		2.1 Random Projection
		2.2 Visual Assessment of Tendency (VAT) and Its Relative Methods
	3 siVAT Algorithms
		3.1 siVAT1
		3.2 siVAT2
	4 Experiments
		4.1 Datasets
		4.2 Distance Matrix in Downspace Versus Distance Matrix  in Upspace
		4.3 Cluster Assessment Using siVAT, siVAT1 and siVAT2
	5 Conclusions
	References
An Introduction to Linguistic Summaries
	1 Introduction
	2 Basic Ideas of Linguistic Summaries
	3 Information Content of Linguistic Summaries
	4 Concepts and Hierarchies
	5 Conclusion
	References
Graduality in Data Sciences: Gradual Patterns
	1 Introduction
	2 Definitions
	3 Quality Criterion
	4 Advanced Protoforms for Gradual Patterns
	5 Conclusion
	References
Evolving Systems
	1 Introduction
	2 Rule-Based Evolving Systems
	3 Neural Evolving Fuzzy Systems
	4 Evolving Fuzzy Linear Regression Trees
	5 Conclusion
	References
Control: Advances on Fuzzy Model-Based Observers
	1 Motivation
	2 Introduction
		2.1 Notations and Preliminaries
	3 Measured Premise Variables
		3.1 The Continuous Case
		3.2 The Discrete Case
	4 Unmeasured Premise Variables
	5 Example
	6 Concluding Remarks
	References
Fuzzy Extensions of Databases
	1 Introduction
	2 Fuzzy Querying of Classical Databases
		2.1 Fuzzy Extension of SQL
		2.2 Fuzzy Query Optimization
	3 Possibilistic Databases
		3.1 About Uncertain Databases and Possible Worlds
		3.2 Possibilistic Database Models
	4 Cooperative Query-Answering
	5 Conclusion
	References
On Maxitive Image Processing
	1 Introduction
	2 Preliminary Considerations, Definitions and Notations
		2.1 Notations
		2.2 Digital Images
		2.3 Capacities and Expectations
		2.4 Summative and Maxitive Kernels
		2.5 Crisp and Fuzzy Partitions
	3 From Continuous to Digital Image Processing
		3.1 Continuous Image/Digital Image
		3.2 Kernel-Based Image Processing
		3.3 Example: Rigid Transformations
	4 Conclusion and Discussion
	References
F-Transform Representation of Nonlocal Operators with Applications to Image Restoration
	1 Introduction
	2 Preliminaries
		2.1 Fm-transform
		2.2 F0- and F1-transforms for Multivariate Functions
	3 F-transform-Based Representation of Nonlocal Operations
	4 Image Restoration by the Patch-Based-Inpainting
	5 Examples
	6 Conclusion
	References
Forensic Identification by Craniofacial Superimposition Using Fuzzy Set Theory
	1 Introduction
	2 Sources of Uncertainty
		2.1 Stage 1: Acquisition and Processing of AM-PM Materials
		2.2 Stage 2: Face and Skull Examination
		2.3 Stage 3: Skull-Face Overlay
		2.4 Stage 4: Decision Making
	3 State of the Art Approaches Modeling Craniofacial Superimposition Uncertainties
		3.1 Modeling the Uncertainty Related to the Location of Facial Landmarks
		3.2 Modeling the Landmark Matching Uncertainty
		3.3 Fuzzy Hierarchical Decision Support Framework
	4 Discussion and Conclusions
	References
On the Applicability of Fuzzy Rule Interpolation and Wavelet Analysis in Colorectal Image Segment Classification
	1 Introduction
	2 Fuzzy Sets and Reasoning
		2.1 Fuzzy Rule Interpolation
		2.2 Antecedents
	3 Fuzzy Rules
	4 Results
	5 Conclusion
	References
Association Rule Mining for Unknown Video Games
	1 Introduction
	2 General Video Game AI
	3 Frequent Pattern Mining (FPM) and Association Rule Analysis
	4 Extracting Play-Traces Patterns and Game Rules
	5 Properties of the Termination Set
	6 Applications of Association Rule Analysis for General Game AI
	7 Analysis of Fuzzy Methods
	8 Conclusions and Outlook
	References
Semantic Web: Graphs, Imprecision and Knowledge Generation
	1 Introduction
	2 Knowledge Generation Framework: Introduction
	3 Knowledge Graphs: Semantic Web Representation of Data
		3.1 Representation of Actual Data
		3.2 Representation of Concept Definitions
		3.3 Integration of Data and Conceptual Graphs
	4 Logic Structure: Topos and Fuzziness
		4.1 Logic of Topos
		4.2 Concept Signatures
		4.3 Fuzzy Term Monads
	5 Knowledge Generation Processes
	6 Conclusion
	References
Z-Numbers: How They Describe Student Confidence and How They Can Explain (and Improve) Laplacian and Schroedinger Eigenmap Dimension Reduction in Data Analysis
	1 Introduction
	2 Need to Take into Account Accuracy and Reliability When Processing Data
		2.1 Need for Data Processing
		2.2 How to Take Accuracy into Account: Probabilistic Case
		2.3 How to Take Accuracy into Account: Fuzzy Case
		2.4 Need to Take Reliability into Account
	3 Z-Numbers
		3.1 What Do We Know About the Possible Outliers
		3.2 Probabilistic Case
		3.3 Fuzzy Case
		3.4 General Case
		3.5 Problem
		3.6 Idea
		3.7 Resulting Algorithm
		3.8 How Good Is This Algorithm?
	4 Examples and Case Study
		4.1 Z-Numbers and Teaching
		4.2 Case Study: Dimension Reduction
		4.3 Reformulating the Problem in Terms of Z-Numbers
		4.4 Our Algorithm Leads to a Known Successful Heuristic
		4.5 Taking into Account that Some Objects May Be Not Relevant
		4.6 Can We Go Beyond Justification of Existing Approaches?
	5 Conclusion
	References