Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications

Preface
	What This Book Is About
	Dedication
	Who This Book Is for
	Acknowledgements
	Short Biography of Vladik Kreinovich
Contents
Symmetries Are Important
	References
Constructive Mathematics
Constructive Continuity of Increasing Functions
	1 Introduction
	2 Continuity
	References
A Constructive Framework for Teaching Discrete Mathematics
	1 Introduction
	2 Cantorian Set Theory
	3 Skepticism About Cantorian Set Theory and ZFC
		3.1 Computational Facts and Laws
		3.2 Skepticism About Replacement
		3.3 Skepticism About Choice
		3.4 Skepticism About Power Sets
		3.5 Skepticism About Infinity
	4 The Foundational Crisis and the Rise of ZFC
		4.1 Hilbert\'s Support for ZFC
		4.2 Der Grundlagenkrise
	5 Pedagogical Issues
	6 Language K
		6.1 Universe of Discourse
		6.2 Arithmetic in K
		6.3 Tuples
		6.4 Sets and Set Operations
		6.5 Equality
		6.6 Logical Operators
		6.7 Aggregates and Set Comprehension
		6.8 Lambda Terms
		6.9 Fine Points
		6.10 Typical Exercises
	7 Language P
		7.1 Language Definition and Properties
		7.2 Finitist Semantics for P
	References
Fuzzy Techniques
Fuzzy Logic for Incidence Geometry
	1 Introduction
	2 Axiomatic Geometry and Extended Objects
		2.1 Geometric Primitives and Incidence
	3 Fuzzification of Incidence Geometry
		3.1 Proposed Fuzzy Logic
		3.2 Geometric Primitives as Fuzzy Predicates
		3.3 Formalization of Fuzzy Predicates
		3.4 Fuzzy Axiomatization of Incidence Geometry
		3.5 Equality of Extended Lines Is Graduated
		3.6 Incidence of Extended Points and Lines
		3.7 Equality of Extended Points and Lines
	4 Fuzzification of Euclid’s First Postulate
		4.1 A Euclid’s First Postulate Formalization
		4.2 Fuzzy Logical Inference for Euclid’s First Postulate
		4.3 Example
	5 Conclusion
	References
Interval Valued Intuitionistic Fuzzy Sets Past, Present and Future
	1 Introduction
	2 Past: Interval Valued Intuitionistic Fuzzy Sets—A Definition, Operations, Relations and Operators over Them
	3 Present: Interval Valued Intuitionistic Fuzzy Sets—Theory and Applications
	4 Future: Interval Valued Intuitionistic Fuzzy Sets—Open Problems and Ideas for Next Research
	5 Conclusion
	References
Strengths of Fuzzy Techniques in Data Science
	1 Introduction
	2 Knowledge Representation
		2.1 Fuzzy Sets and Possibility Degrees
		2.2 Rule Bases
		2.3 Linguistic Summaries
		2.4 Fuzzy Ontologies
		2.5 Similarity Measures
	3 Conclusion
	References
How to Enhance, Use and Understand Fuzzy Relational Compositions
	1 Introduction and Preliminaries
		1.1 Introduction
		1.2 Fuzzy Relational Compositions
		1.3 Compositions Based on Fuzzy Quantifiers
		1.4 Excluding Features in Fuzzy Relational Compositions
	2 Dragonfly Classification Problem
		2.1 Taxonomical Classification—Setting up the Problem
		2.2 Results of the Taxonomic Classification and Discussion
	3 Combinations of Fuzzy Quantifier and Excluding Features
		3.1 Direct Combination of Fuzzy Quantifier and Excluding Features
		3.2 Fuzzy Relational Compositions Using Grouping  of Features
		3.3 Experiments
	4 Conclusion and Future Work
	References
Łukasiewicz Logic and Artificial Neural Networks
	1 Introduction
	2 Preliminary
		2.1 Łukasiewicz Logic and MV-Algebras
		2.2 Multilayer Perceptrons
	3 Łukasiewicz Equivalent Neural Networks
	4 Function Approximation Problems
		4.1 Input Selection and Polynomial Completeness
		4.2 On the Number of Hidden Layers
	5 Conclusion
	References
Impact of Time Delays on Networked Control of Autonomous Systems
	1 Introduction
		1.1 Delays and Their Impact on Systems
	2 Motivation
	3 Delay Modeling
	4 Environment
		4.1 Implementation
	5 Results
	6 Conclusions
	7 Future Work
	References
Intervals and More: Aggregation Functions for Picture Fuzzy Sets
	1 Introduction
	2 Some Generalizations of Fuzzy Sets—An Overview
	3 Aggregation Functions on Intervals
	4 Aggregation Functions for Picture Fuzzy Sets
	5 Concluding Remarks
	References
The Interval Weighted Average and Its Importance to Type-2 Fuzzy Sets and Systems
	1 Introduction
	2 Formulation of the IWA
	3 Computing the IWA
	4 Properties of the IWA
	5 Algorithms for Finding the Switch Points
	6 Centroid Type-Reduction of an IT2 Fuzzy Set
	7 Type-Reduction in IT2 Fuzzy Systems
	8 Remarks
	9 Centroid Type-Reduction for GT2 FSs
	10 Type-Reduction in a GT2 Fuzzy System
	11 Conclusions
	References
Fuzzy Answer Set Programming: From Theory to Practice
	1 Introduction
	2 Modeling Problems as Logic Programs
	3 Syntax and Semantics of FASP
	4 Solving FASP Programs
		4.1 Non-disjunctive Programs
		4.2 Disjunctive Programs
	5 An Application of FASP in Biological Network Modeling
	6 Conclusion
	References
Impact and Applications of Fuzzy Cognitive Map Methodologies
	1 Introduction
	2 Generic Structure of FCM and Its Development
	3 Design FCMs Based on Experts
	4 Generalization of FCM Topology and Design
		4.1 Enhancement, Generalization of Individual Units and New Topologies (Architectures)
		4.2 Timed Fuzzy Cognitive Maps
	5 Synergies of FCMs with Other Methods for Improved Efficiency
		5.1 Competitive Fuzzy Cognitive Maps with Case Based Reasoning
		5.2 Timed Fuzzy Cognitive Maps with Hidden Markov Models
	6 Applications Areas
	7 Main Future Directions
	References
Interval Computations
Rigorous Global Filtering Methods with Interval Unions
	1 Introduction
		1.1 Context
		1.2 Interval Unions and Related Work
		1.3 Contribution
		1.4 Notation
	2 Interval Unions
	3 Interval Union and CSPs
		3.1 The Forward-Backward Constraint Propagation
		3.2 The Interval Union Newton Operator
		3.3 Gap Filling
	4 GloptLab
	5 Numerical Experiments
		5.1 The COCONUT Test Set
		5.2 Forward-Backward Constraint Propagation
		5.3 Interval Union Newton Method
	References
On the Computational Complexity  of the Range Computation Problem
	1 Introduction
	2 How Can One Measure the Complexity of Computation Problems?
		2.1 Some Notation
		2.2 Encoding the Input and Output of Computation Problems by Strings
		2.3 Some Discrete Complexity Classes
		2.4 Polynomial Time Computable Real Numbers  and NP-Real Numbers
	3 Some of the Known Results Concerning the Complexity of the Range Computation Problem for Polynomials
		3.1 The General Range Computation Problem  for Polynomials
		3.2 Linear Functions
		3.3 Polynomials with a Fixed Number of Variables
		3.4 The Lower Complexity Bound of Gaganov
	4 An Upper Bound for the Complexity of the Approximate Range Computation Problem
	5 On the Complexity of the Range Computation Problem for a Fixed Sequence of Polynomials
		5.1 Noncomputable Sequences Versus Polynomial Time Computable Sequences
		5.2 The Problem for a Fixed Sequence of Linear Functions
		5.3 The Problem for a Fixed Sequence of Polynomials  of Degree at Least 2
		5.4 The Problem for a Fixed Sequence of Polynomials  and a Fixed Sequence of Interval Boxes
		5.5 Summary
	6 Proofs
	References
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
	1 Introduction
	2 Tridiagonal Matrices
	3 M-matrices and H-matrices
	4 Inverse Nonnegative Matrices
	5 Totally Positive Matrices
	6 P-Matrices
	7 Diagonally Interval Matrices
	8 Nonnegative Matrices
	9 Inverse M-matrices
	10 Parametric Matrices
	11 Conclusion
	References
Interval Methods for Solving Various Kinds of Quantified Nonlinear Problems
	1 Introduction
	2 Generic Algorithm
	3 The Second Phase—Quantifier Elimination
		3.1 Herbrand Expansion
		3.2 Shared Quantities
		3.3 Existentially Quantified Formulae
		3.4 When Is the Second Phase Not Necessary?
	4 Necessary Conditions
	5 Seeking Local Optima of a Function
	6 Example Heuristics
	7 Conclusions
	8 Further Studies
	References
High Speed Exception-Free Interval Arithmetic, from Closed and Bounded Real Intervals to Connected Sets of Real Numbers
	1 Remarks on the History of Interval Arithmetic
	2 High Speed Interval Arithmetic by Exact Evaluation of Dot Products
	3 From Closed Real Intervals to Connected Sets of Real Numbers
	4 Computing Dot Products Exactly
	5 Early Super Computers
	6 Conclusion
	References
Guaranteed Nonlinear Parameter Estimation with Additive Gaussian Noise
	1 Introduction
	2 Set Inversion for Nonlinear Gaussian Estimation
	3 Linearization Method
	4 Test-Cases
		4.1 Test-Case 1
		4.2 Test-Case 2
		4.3 Test-Case 3
	5 Conclusion
	References
Influence of the Condition Number  on Interval Computations: Illustration  on Some Examples
	1 Introduction
	2 Condition Number and Interval Computations
		2.1 Condition Number of a Problem
		2.2 Amplification Factor for Interval Computations
	3 Summation
	4 Solving Linear Systems
	5 Univariate Nonlinear Equations
	6 Conclusion and Future Work
	References
Interval Regularization for Inaccurate Linear Algebraic Equations
	1 Problem Statement
	2 Idea of the Solution
	3 Implementation of the Idea
	4 Tolerable Solution Set for Interval Linear Systems  of Equations
	5 Recognizing Functional and Its Application
	6 Formal (algebraic) Approach
	References
Uncertainty in General and its Applications
Probabilistic Solution of Yao\'s Millionaires\' Problem
	1 Introduction
	2 Protocol
	3 Probabilities
		3.1 What if Alice Does Not Walk?
		3.2 The Case of n43 Steps in Random Walks
		3.3 The Case of n53 Steps in Random Walks
		3.4 The Probability to Guess the Other Party\'s Number
	4 Suggested Parameters for Practical Use and Experimental Results
	5 Conclusions
	References
Measurable Process Selection Theorem and Non-autonomous Inclusions
	1 Introduction
	2 Global Processes
	3 Local Processes
	4 Example
	References
Handling Uncertainty When Getting Contradictory Advice from Experts
	1 Introduction
	2 Definition of Selectors
	3 Deterministic Selectors and Downward Self-Reducibility
	4 Probabilistic Selectors for Languages Beyond PSPACE
	References
Characterizing Uncertainties in the Geophysical Properties of Soils in the El Paso, Texas Region
	1 Introduction
	2 Background
	3 Analysis of Magnetic Data
	4 Conclusions
	References
Why Sparse?
	1 Formulation of the Problem
	2 Main Idea
	3 Main Result: Formulation and Discussion
	4 Proof
	References
The Kreinovich Temporal Universe
	1 Vladik Kreinovich and His Wife. Autumn, 1986
	2 The Kreinovich Articles on Chronogeometry
		2.1 Observable Causality Implies Lorentz Group
		2.2 Approximately Measured Causality Implies the Lorentz Group
		2.3 Stochastic Causality is Inconsistent with the Lorentz Group
	3 Past Does Not Restore
	4 Time Machine
		4.1 A Natural Time Machine in Simply-Connected Space-Time can Exist only in Extremal Conditions
		4.2 Time Machine Construction Using Resilient Leaf in 5-Dimensional Hyperspace M5
	5 Antigravitation
	References
Bilevel Optimal Tolls Problems with Nonlinear Costs: A Heuristic Solution Method
	1 Introduction
	2 The Toll Optimization Problem
	3 Linear-Quadratic Bilevel Program Reformulation
	4 The Heuristic Algorithms
	5 Description of the Algorithms
		5.1 Algorithm 1
		5.2 Algorithm 2
		5.3 The Algorithm for Calculating the ARSBs
		5.4 The Procedure for Finding the Jacobian Matrices
		5.5 Filled Function Algorithm
	6 Numerical Results
	7 Conclusions
	References
Enhancement of Cross Validation Using Hybrid Visual and Analytical Means with Shannon Function
	1 Introduction
		1.1 Preliminaries
		1.2 Challenges of k-Fold Cross Validation
		1.3 k-Fold Cross Validation Process
	2 Method
		2.1 Shannon Function
		2.2 Alternative Algorithms
		2.3 Interactive Hybrid Algorithm
	3 Case Studies
		3.1 Case Study 1: Linear SVM and LDA in 2-D on Modeled Data
		3.2 Case Study 2: LDA and Visual Classification in 4-D on Iris Data
		3.3 Case Study 3: GLC-AL and LDA in 9-D on Wisconsin Breast Cancer Diagnostic Data
	4 Discussion and Conclusion
	Appendix
		Appendix 1: Base GLC-L Algorithm
		Appendix 2: Algorithm GLC-AL for Automatic Discovery of Relation Combined with Interactions
	References
Conditional Event Algebras: The State-of-the-Art
	1 Introduction
	2 What is a Conditional Event and Why?
	3 Two Main Approaches to Defining Conditional Events
		3.1 A Non Boolean Structure for Conditional Events
		3.2 The Product Space Approach to Conditional Events
	4 Implications of Conditional Event Algebras
	5 Related Research Issues
	References
Beyond Integration: A Symmetry-Based Approach to Reaching Stationarity in Economic Time Series
	1 Formulation of the Problem
	2 Analysis of the Problem
	3 A General Approach to Reaching Stationarity
	References
Risk Analysis of Portfolio Selection Based on Kernel Density Estimation
	1 Introduction
		1.1 Research Process
		1.2 The Chapter Structure
	2 Literature Review and Basic Concepts
		2.1 Kernel Smoothing
		2.2 Copula Method
		2.3 Value at Risk (VaR)
		2.4 Comparison
		2.5 Kernel Density Estimation
	3 Computation of Value at Risk
		3.1 Definition
		3.2 Delta Normal Model
		3.3 Monte Carlo Simulation
	4 Solving Portfolio with VaR
		4.1 Analytical Method
		4.2 VaR Computation Based on Copula Model
		4.3 Back Testing
	5 Numerical Experiment
		5.1 Normality Test and Correlation Analysis
		5.2 Augmented Dickey-Fuller (ADF) Unit Root Test
		5.3 Normality Test
		5.4 Evaluation of Marginal Distribution and Copula Parameter
		5.5 The Selection of Optimal Copula
		5.6 Tail Dependence Research
		5.7 Value at Risk Calculation
	6 Conclusion
	References
Minimax Context Principle
	1 Contexts and Their Supports in Classical Realm
	2 Device Resolution Order
	3 Redundancy Order
	4 Minimax Context Principle for Classical Systems
	5 TFTP and Daseinisation
	6 Minimax Principle in General TFTP Operationalistic Models
	7 A Toy Model of `Topologimeter\'
	References
Neural Networks
Why Rectified Linear Neurons Are Efficient: A Possible Theoretical Explanation
	1 Rectified Linear Neurons: Formulation of the Problem
	2 Our Explanation
	3 Auxiliary Arguments in Favor of Rectified Linear Neurons
		3.1 Symmetry-Based Argument
		3.2 Complexity-Based Argument
		3.3 Fuzzy-Based Argument
	References
Quasiorthogonal Dimension
	1 Introduction
	2 Orthogonal and Quasiorthogonal Geometry
	3 Graph Theoretic Aspects of Quasiorthogonality
	4 Quasiorthogonal Sets in Hamming Cubes
	5 Construction of Sparse Ternary Quasiorthognal Sets
	6 Vector Space Models of Word Semantics
	7 Some Variants of Orthogonality
	References
Integral Transforms Induced  by Heaviside Perceptrons
	1 Introduction
	2 Preliminaries
	3 Infinite Heaviside Perceptron Networks
	4 Generalizing the Integral Formula
	5 Network Complexity
	6 Discussion
	References