Machine generated contents note:

Preface xv Acknowledgments xvii PART I FUNDAMENTALS OF FUZZY MODELING 1 What is fuzzy modeling 3 1.1 Indeterminacy in human life 3 1.2 Fuzzy modeling: with and without words 6 2 Overview of basic notions 11 2.1 Relations, functions, ordered sets 11 2.2 Fuzzy sets and fuzzy relations 14 2.2.1 The concept of a fuzzy set 15 2.2.2 Operations with fuzzy sets 20 2.2.3 Fuzzy numbers 29 2.2.4 Fuzzy partition and fuzzy covering 33 2.2.5 Cartesian product and fuzzy relations 34 2.2.6 Fuzzy equality and extensional fuzzy sets 41 2.3 Elements of mathematical fuzzy logic 44 2.3.1 Structure of truth degrees in mathematical fuzzy logic 44 2.3.2 Logical inference 47 2.3.3 Formal systems of MFL 48 2.3.4 The concept of fuzzy IF-THEN rule 50 3 Fuzzy IF-THEN rules in approximation of functions 53 3.1 Relational interpretation of fuzzy IF-THEN rules 53 3.1.1 Finite functions and their description 54 3.1.2 Relational interpretation of linguistic descriptions 57 3.1.3 Managing more variables 64 3.2 Approximation of functions using fuzzy IF-THEN rules 64 3.2.1 Defuzzification 64 3.2.2 Fuzzy approximation 67 3.2.3 Construction of approximating function 67 3.2.4 Choosing between DNF and CNF. 74 3.3 Generalized modus ponens and fuzzy functions 77 3.4 TakagiSugeno rules 80 3.4.1 Basic concepts 80 3.4.2 Fuzzy approximation using TSrules 80 3.4.3 Identification of TSrules 84 4 Fuzzy transform 87 4.1 Fuzzy partition 88 4.2 The concept of F-transform 90 4.2.1 Direct F-transform 90 4.2.2 Inverse F-transform 92 4.3 Discrete F-transform 95 4.4 F-transform of functions of two variables 96 4.5 F1transform 98 4.6 Methodological remarks to applications of the F-transform 101 5 Fuzzy natural logic and approximate reasoning 103 5.1 Linguistic semantics and linguistic variable 103 5.1.1 Linguistic variable 104 5.1.2 Intension, context, extension 105 5.1.3 Refined definition of linguistic variable 106 5.2 Theory of evaluative linguistic expressions 108 5.2.1 The concept and structure of evaluative expressions 108 5.2.2 Evaluative linguistic predications 111 5.2.3 Mathematical model of the semantics of evaluative linguistic expressions 113 5.3 Interpretation of fuzzy/linguistic IF-THEN rules 124 5.3.1 Linguistic description 124 5.3.2 Intension of fuzzy/linguistic IF-THEN rules 125 5.4 Approximate reasoning with linguistic information 126 5.4.1 Basic principle of approximate reasoning 126 5.4.2 Perceptionbased logical deduction 127 5.4.3 Formalization of the perceptionbased logical deduction 131 5.4.4 Comparison of two interpretations of fuzzy IF-THEN rules 136 6 Fuzzy cluster analysis 145 6.1 Basic notions 145 6.2 Fuzzy clustering algorithms 147 6.3 The algorithm of fuzzy cmeans 148 6.4 The GustafsonKessel algorithm 151 6.5 How the number of clusters can be determined 152 6.6 Construction of fuzzy rules based on found clusters 153 PART II SELECTED APPLICATIONS 7 Fuzzy/linguistic control and decisionmaking 159 7.1 The principle of fuzzy control 159 7.1.1 Control in a closed feedback loop 161 7.1.2 A general scheme of fuzzy controller 162 7.2 Fuzzy controllers 165 7.2.1 Variables 166 7.2.2 Basic types of classical controllers 167 7.2.3 Basic types of fuzzy controllers 167 7.3 Design of fuzzy/linguistic controller 169 7.3.1 Determination of variables and linguistic context 169 7.3.2 Choosing fuzzy action unit 171 7.3.3 Formation of knowledge base 172 7.3.4 Tuning linguistic description 177 7.4 Learning 180 7.4.1 Modification and learning of linguistic context 180 7.4.2 Learning linguistic description 183 7.4.3 Practical experiences with control using linguistic fuzzy action unit 188 7.5 Decisionmaking using linguistic descriptions 190 7.5.1 Introduction 190 7.5.2 Hierarchy of linguistic descriptions in decisionmaking 191 7.5.3 Demonstration of the decisionmaking methodology using linguistic descriptions 193 8 F-transform in image processing 197 8.1 Image and its basic processing using F-transform 197 8.2 F-transform based image compression and reconstruction 198 8.2.1 Basic principles of image compression 198 8.2.2 Simple F-transform compression 199 8.2.3 Advanced Image Compression 200 8.3 F1transform edge detector 201 8.4 F-transform based image fusion 204 8.4.1 Basic idea of image fusion 204 8.4.2 Simple F-transform based fusion algorithm 205 8.4.3 Complete F-transform based fusion algorithm 207 8.4.4 Enhanced simple fusion algorithm 209 8.5 F-transform based corrupted image reconstruction 211 8.5.1 The reconstruction problem 212 8.5.2 F-transform based reconstruction 212 8.5.3 Demonstration examples 214 9 Analysis and forecasting of time series 219 9.1 Classical vs. fuzzy models of time series 220 9.1.1 Definition of time series 220 9.1.2 Classical models of time series 220 9.1.3 Fuzzy models of time series 221 9.2 Analysis of time series using F-transform 222 9.2.1 Decomposition of time series 222 9.2.2 Extraction of trendcycle and trend using F-transform 224 9.3 Time series forecasting 229 9.3.1 Decomposition of time domain 229 9.3.2 Forecast of trendcycle 230 9.3.3 Forecast of seasonal component 234 9.3.4 Forecast of the whole time series 235 9.4 Characterization of time series in natural language 236 9.4.1 Sentences characterizing trend 236 9.4.2 Automatic generation of sentences characterizing trend 238 9.4.3 Mining information from time series 241 References 245 Index 255 .