Provides a unique and methodologically consistent treatment of various areas of fuzzy modeling and includes the results of mathematical fuzzy logic and linguisticsThis book is the result of almost thirty years of research on fuzzy modeling. It provides a unique view of both the theory and various types of applications. The book is divided into two parts. The first part contains an extensive presentation of the theory of fuzzy modeling. The second part presents selected applications in three important areas: control and decision-making, image processing, and time series analysis and forecasting. The authors address the consistent and appropriate treatment of the notions of fuzzy sets and fuzzy logic and their applications. They provide two complementary views of the methodology, which is based on fuzzy IF-THEN rules. The first, more traditional method involves fuzzy approximation and the theory of fuzzy relations. The second method is based on a combination of formal fuzzy logic and linguistics. A very important topic covered for the first time in book form is the fuzzy transform (F-transform). Applications of this theory are described in separate chapters and include image processing and time series analysis and forecasting. All of the mentioned components make this book of interest to students and researchers of fuzzy modeling as well as to practitioners in industry.Features:* Provides a foundation of fuzzy modeling and proposes a thorough description of fuzzy modeling methodology* Emphasizes fuzzy modeling based on results in linguistics and formal logic* Includes chapters on natural language and approximate reasoning, fuzzy control and fuzzy decision-making, and image processing using the F-transform* Discusses fuzzy IF-THEN rules for approximating functions, fuzzy cluster analysis, and time series forecastingInsight into Fuzzy Modeling is a reference for researchers in the fields of soft computing and fuzzy logic as well as undergraduate, master and Ph.D. students.Vilém Novák, D.Sc. is Full Professor and Director of the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.Irina Perfilieva, Ph.D. is Full Professor, Senior Scientist, and Head of the Department of Theoretical Research at the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.Antonín Dvorák, Ph.D. is Associate Professor, and Senior Scientist at the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.

Preface xiiiAcknowledgments xvAbout the Companion Website xviiPART I FUNDAMENTALS OF FUZZY MODELING 11 What is Fuzzy Modeling 31.1 Indeterminacy in Human Life 31.2 Fuzzy Modeling: With and Without Words 62 Overview of Basic Notions 112.1 Relations, Functions, Ordered Sets 112.1.1 Relations 112.2 Fuzzy Sets and Fuzzy Relations 142.2.1 The Concept of a Fuzzy Set 142.2.2 Operations with Fuzzy Sets 192.2.3 Fuzzy Numbers 282.2.4 Fuzzy Partition and Fuzzy Covering 312.2.5 Cartesian Product and Fuzzy Relations 322.2.6 Fuzzy Equality and Extensional Fuzzy Sets 372.3 Elements of Mathematical Fuzzy Logic 412.3.1 Structure of Truth Degrees in Mathematical Fuzzy Logic 412.3.2 Logical Inference 432.3.3 Formal Systems of MFL 452.3.4 The Concept of Fuzzy IF-THEN Rule 463 Fuzzy IF-THEN Rules in Approximation of Functions 493.1 Relational Interpretation of Fuzzy IF-THEN Rules 493.1.1 Finite Functions and Their Description 503.1.2 Relational Interpretation of Linguistic Descriptions 533.1.3 Managing More Variables 593.2 Approximation of Functions Using Fuzzy IF-THEN Rules 603.2.1 Defuzzification 603.2.2 Fuzzy Approximation 633.2.3 Choosing between DNF and CNF 693.3 Generalized Modus Ponens and Fuzzy Functions 723.4 Takagi-Sugeno Rules 743.4.1 Basic Concepts 743.4.2 Fuzzy Approximation Using TS Rules 753.4.3 Identification of TS Rules 784 Fuzzy Transform 814.1 Fuzzy Partition 814.2 The Concept of F-Transform 844.2.1 Direct F-Transform 844.2.2 Inverse F-Transform 854.3 Discrete F-Transform 884.4 F-Transform of Functions of Two Variables 894.5 F1-Transform 914.6 Methodological Remarks to Applications of the F-Transform 945 Fuzzy Natural Logic and Approximate Reasoning 975.1 Linguistic Semantics and Linguistic Variable 975.1.1 Linguistic Variable 985.1.2 Intension, Context, Extension 985.1.3 Refined Definition of Linguistic Variable 1005.2 Theory of Evaluative Linguistic Expressions 1015.2.1 The Concept and Structure of Evaluative Expressions 1015.2.2 Evaluative Linguistic Predications 1055.2.3 Mathematical Model of the Semantics of Evaluative Linguistic Expressions 1065.3 Interpretation of Fuzzy/Linguistic IF-THEN Rules 1175.3.1 Linguistic Description 1175.3.2 Intension of Fuzzy/Linguistic IF-THEN Rules 1185.4 Approximate Reasoning with Linguistic Information 1195.4.1 Basic Principle of Approximate Reasoning 1195.4.2 Perception-Based Logical Deduction 1205.4.3 Formalization of the Perception-Based Logical Deduction 1245.4.4 Comparison of Two Interpretations of Fuzzy IF-THEN Rules 1286 Fuzzy Cluster Analysis 1376.1 Basic Notions 1376.2 Fuzzy Clustering Algorithms 1396.3 The Algorithm of Fuzzy c-Means 1406.4 The Gustafson-Kessel Algorithm 1426.5 How the Number of Clusters Can Be Determined 1446.6 Construction of Fuzzy Rules Based on Found Clusters 144PART II SELECTED APPLICATIONS 1497 Fuzzy/Linguistic Control and Decision-Making 1517.1 The Principle of Fuzzy Control 1517.1.1 Control in a Closed Feedback Loop 1537.1.2 A General Scheme of Fuzzy Controller 1547.2 Fuzzy Controllers 1577.2.1 Variables 1577.2.2 Basic Types of Classical Controllers 1587.2.3 Basic Types of Fuzzy Controllers 1597.3 Design of Fuzzy/Linguistic Controller 1617.3.1 Determination of Variables and Linguistic Context 1617.3.2 Choosing Fuzzy Action Unit 1627.3.3 Formation of Knowledge Base 1637.3.4 Tuning Linguistic Description 1667.4 Learning 1717.4.1 Modification and Learning of Linguistic Context 1717.4.2 Learning Linguistic Description 1747.4.3 Practical Experiences with Control Using Linguistic Fuzzy Action Unit 1777.5 Decision-Making Using Linguistic Descriptions 1807.5.1 Introduction 1807.5.2 Hierarchy of Linguistic Descriptions in Decision-Making 1817.5.3 Demonstration of the Decision-Making Methodology Using Linguistic Descriptions 1828 F-Transform in Image Processing 1898.1 Image and Its Basic Processing Using F-Transform 1898.2 F-Transform-Based Image Compression and Reconstruction 1908.2.1 Basic Principles of Image Compression 1908.2.2 Simple F-Transform Compression 1918.2.3 Advanced Image Compression 1918.3 F1-Transform Edge Detector 1938.4 F-Transform-Based Image Fusion 1958.4.1 Basic Idea of Image Fusion 1958.4.2 Simple F-Transform-Based Fusion Algorithm 1978.4.3 Complete F-Transform-Based Fusion Algorithm 1998.4.4 Enhanced Simple Fusion Algorithm 2018.5 F-Transform-Based Corrupted Image Reconstruction 2038.5.1 The Reconstruction Problem 2038.5.2 F-Transform-Based Reconstruction 2048.5.3 Demonstration Examples 2069 Analysis and Forecasting of Time Series 2099.1 Classical Versus Fuzzy Models of Time Series 2109.1.1 Definition of Time Series 2109.1.2 Classical Models of Time Series 2109.1.3 Fuzzy Models of Time Series 2119.2 Analysis of Time Series Using F-Transform 2129.2.1 Decomposition of Time Series 2129.2.2 Extraction of Trend-Cycle and Trend Using F-Transform 2149.3 Time Series Forecasting 2199.3.1 Decomposition of Time Domain 2199.3.2 Forecast of Trend-Cycle 2209.3.3 Forecast of Seasonal Component 2239.3.4 Forecast of the Whole Time Series 2259.4 Characterization of Time Series in Natural Language 2259.4.1 Sentences Characterizing Trend 2269.4.2 Automatic Generation of Sentences Characterizing Trend 2289.4.3 Mining Information from Time Series 230References 235Index 243