The fuzzy systems handbook : a practitioner's guide to building, using, and maintaining fuzzy systems / Earl Cox.

1. Introduction 1
Fuzzy System Models 1
Logic, Complexity, and Comprehension 1
The Idea of Fuzzy Sets 2
Linguistic Variables 3
Approximate Reasoning 4
Benefits of Fuzzy System Modeling 5
The Ability to Model Highly Complex Business Problems 5
Improved Cognitive Modeling of Expert Systems 6
The Ability to Model Systems Involving Multiple Experts 7
Reduced Model Complexity 7
Improved Handling of Uncertainty and Possibilities 8
Notes 8
2. Fuzziness and Certainty 9
The Different Faces of Imprecision 9
Inexactness 10
Precision and Accuracy 11
Accuracy and Imprecision 12
Measurement Imprecision and Intrinsic Imprecision 12
Ambiguity 13
Semantic Ambiguity 13
Visual Ambiguity 14
Structural Ambiguity 14
Undecidability 15
Vagueness 17
Probability, Possibility, and Fuzzy Logic 18
What Is Probability? 18
Context Terms 19
Fuzzy Logic 19
Notes 20
3. Fuzzy Sets 21
Imprecision in the Everyday World 21
Imprecise Concepts 21
The Nature of Fuzziness 23
Fuzziness and Imprecision 27
Representing Imprecision with Fuzzy Sets 30
Fuzzy Sets Versus Crisp Sets 30
Fuzzy Sets 31
Representing Fuzzy Sets in Software 33
Basic Properties and Characteristics of Fuzzy Sets 35
Fuzzy Set Height and Normalization 36
Domains, Alpha-level Sets, and Support Sets 38
The Fuzzy Set Domain 38
The Universe of Discourse 40
The Support Set 41
Fuzzy Alpha-Cut Thresholds 41
Encoding Information with Fuzzy Sets 44
Approximating a Fuzzy Concept 45
Generating Fuzzy Membership Functions 46
Linear Representations 47
S-Curve (Sigmoid/Logistic) Representations 51
S-Curves and Cumulative Distributions 53
Proportional and Frequency Representations 55
Fuzzy Numbers and "Around" Representations 61
Fuzzy Numbers 62
Fuzzy Quantities and Counts 64
Pl Curves 65
Beta Curves 69
Gaussian Curves 75
Irregularly Shaped and Arbitrary Fuzzy Sets 78
Truth Series Descriptions 83
Domain-based Coordinate Memberships 88
Triangular, Trapezoidal, and Shouldered Fuzzy Sets 94
Triangular Fuzzy Sets 95
Shouldered Fuzzy Sets 97
Notes 105
4. Fuzzy Set Operators 107
Conventional (Crisp) Set Operations 107
Basic Zadeh-Type Operations on Fuzzy Sets 110
Fuzzy Set Membership and Elements 111
The Intersection of Fuzzy Sets 111
The Union of Fuzzy Sets 118
The Complement (Negation) of Fuzzy Sets 121
Counterintuitives and the Law of Noncontradiction 125
Non-Zadeh and Compensatory Fuzzy Set Operations 130
General Algebraic Operations 133
The Mean and Weighted Mean Operators 134
The Product Operator 138
The Heap Metaphor 139
The Bounded Difference and Sum Operators 141
Functional Compensatory Classes 143
The Yager Compensatory Operators 144
The Yager AND Operator 144
The Yager OR Operator 146
The Yager NOT Operator 151
The Sugeno Class and Other Alternative NOT Operators 153
Threshold NOT Operator 154
The Cosine NOT Function 155
Notes 158
5. Fuzzy Set Hedges 161
Hedges and Fuzzy Surface Transformers 161
The Meaning and Interpretation of Hedges 162
Applying Hedges 163
Fuzzy Region Approximation 164
Restricting a Fuzzy Region 167
Intensifying and Diluting Fuzzy Regions 170
The Very Hedge 171
The Somewhat Hedge 179
Reciprocal Nature of Very and Somewhat 186
Contrast Intensification and Diffusion 186
The Positively Hedge 187
The Generally Hedge 189
Approximating a Scalar 200
Examples of Typical Hedge Operations 203
Notes 209
6. Fuzzy Reasoning 211
The Role of Linguistic Variables 213
Fuzzy Propositions 215
Conditional Fuzzy Propositions 215
Unconditional Fuzzy Propositions 216
The Order of Proposition Execution 216
Monotonic (Proportional) Reasoning 217
Monotonic Reasoning with Complex Predicates 223
The Fuzzy Compositional Rules of Inference 226
The Min-Max Rules of Implication 226
The Fuzzy Additive Rules of Implication 227
Accumulating Evidence with the Fuzzy Additive Method 227
Fuzzy Implication Example 231
Correlation Methods 235
Correlation Minimum 235
Correlation Product 236
The Minimum Law of Fuzzy Assertions 239
Methods of Decomposition and Defuzzification 245
Composite Moments (Centroid) 249
Composite Maximum (Maximum Height) 250
Hyperspace Decomposition Comparisons 251
Preponderance of Evidence Technique 252
Other Defuzzification Techniques 256
The Average of Maximum Values 257
The Average of the Support Set 257
The Far and Near Edge of the Support Set 258
The Center of Maximums 259
Singleton Geometry Representations 266
Notes 269
7. Fuzzy Models 271
The Basic Fuzzy System 271
The Fuzzy Model Overview 271
The Model Code View 273
Code Representation of Fuzzy Variables 274
Incorporating Hedges in the Fuzzy Model 277
Representing and Executing Rules in Code 278
Setting Alpha-Cut Thresholds 280
Including a Model Explanatory Facility 281
The Advanced Fuzzy Modeling Environment 285
The Policy Concept 286
Understanding Hash Tables and Dictionaries 287
Creating a Model and Associated Policies 294
Managing Policy Dictionaries 298
Loading Default Hedges 299
Fundamental Model Design Issues 301
Integrating Application Code with the Modeling System 302
Tasks at the Module Main Program Level 302
Connecting the Model to the System Control Blocks 303
Allocating and Installing the Policy Structure 304
Defining Solution (Output) Variables 304
Creating and Storing Fuzzy Sets in Application Code 304
Creating and Storing Fuzzy Sets in a Policy's Dictionary 306
Loading and Creating Hedges 307
Segmenting Application Code into Modules 310
Maintaining Addressability to the Model 310
Establish the Policy Environment 311
Initialize the Fuzzy Logic Work Area for the Policy 311
Locate the Necessary Fuzzy Sets and Hedges 312
Exploring a Simple Fuzzy System Model 313
Exploring a More Extensive Pricing Policy 325
Fuzzy Set and Data Representational Issues 335
Fuzzy Sets and Model Variables 336
Semantic Decomposition of PROFIT 337
Fuzzy Set Naming Conventions 340
The Meaning and Degree of Fuzzy Set Overlap 341
Control Engineering Perspectives on Overlap and Composition 346
Highly Overlapping Fuzzy Regions 349
Boolean and Semi-Fuzzy Variables 350
Using Boolean Filters 350
Applying Explicit Degrees of Membership 351
Uncertain and Noisy Data 353
Fuzzy and Uncertain Numbers 353
Handling Uncertain and Noisy Data 357
Inferencing with Fuzzy Data 358
The Interpretation of Model Results 360
Undecidable Models 361
Compatibility Index Metrics 365
The Idea of a Compatibility Index 365
The Unit Compatibility Index 366
Scaling Expected Values by the Compatibility Index 375
The Statistical Compatibility Index 376
Selecting Height Measurements 377
Notes 377
8. Fuzzy Systems: Case Studies 379
A Fuzzy Steam Turbine Controller 379
The Fuzzy Control Model 379
The Fuzzy Logic Controller 380
The Conventional PID Controller 381
The Stream Turbine Plant Process 382
Designing the Fuzzy Logic Controller 382
Running the Steam Turbine FLC Logic 386
The New Product Pricing Model (Version 1) 389
Model Design and Objectives 389
The Model Execution Logic 390
Create the Basic Price Fuzzy Sets 391
Create the Run-Time Model Fuzzy Sets 392
Execute the Price Estimation Rules 392
Defuzziły to Find Expected Value for Price 398
Evaluating Defuzzification Strategies 399
The New Product Pricing Model (Version 2) 413
Model Design Strategies 413
The Model Execution Logic 414
Create the Basic Fuzzy Sets 414
Create the Run-Time Model Fuzzy Sets 415
Execute the Price Estimation Rules 417
The New Product Pricing Model (Version 3) 423
The Model Execution Logic 423
Execute the Price Estimation Rules423
Defuzzify to Find Expected Value for Price 429
The New Product Pricing Model (P&L Version) 430
Design for the P&L Model 430
Model Execution and Logic 432
Using Policies to Calculate Price and Sales Volume 434
A Project Risk Assessment Model 436
The Model Design 437
Model Application Issues 438
Model Execution Logic 440
Executing the Risk Assessment Rules 443
Notes 448
9. Building Fuzzy Systems 449
Evaluating Fuzzy System Projects 449
The Ideal Fuzzy System Problem 450
Fuzzy Model Characteristics 450
Fuzzy Control Parameters 450
Multiple Experts 453
Elastic Relationships Among Continuous Variables 454
Complex, Poorly Understood, or Nonlinear Problems 455
Uncertainties, probabilities, and possibilities in data 455
Building Fuzzy System Models 457
The Fuzzy Design Methodology 459
Define the Model Functional and Operational Characteristics 459
Define the System in Terms of an Input-Process-Output Model 459
Localize the Model in the Production System 460
Segment Model into Functional and Operational Components 461
Isolate the Critical Performance Variables 461
Choose the Mode of Solution Variables 461
Resolve Basic Performance Criteria 462
Decide on a Level of Granularity 462
Determine Domain of the Model Variables 462
Determine the Degree of Uncertainty in the Data 463
Define the Limits of Operability 463
Establish Metrics for Model Performance Requirements 464
Define the Fuzzy Sets 464
Determine the Type of Fuzzy Measurement 464
Choose the Shape of the Fuzzy Set (Its Surface Morphology) 465
Elicit a Fuzzy Set Shape 466
Select an Appropriate Degree of Overlap 467
Decide on the Space Correlation Metrics 467
Ensure That the Sets Are Conformally Mapped 467
Write the Rules 468
Write the Ordinary Conditional Rules 469
Enter Any Unconditional Rules 469
Select Compensatory Operators for Special Rules 469
Review the Rule Set and Add Any Hedges 469
Add Any Alpha Cuts to Individual Rules 470
Enter the Rule Execution Weights 470
Define the Defuzzification Method for Each Solution Variable 470
Notes 471
10. Using the Fuzzy Code Libraries 473
Linking Code to your Application 473
Formal and Warning Messages 474
System and Client Error Diagnostics 474
Software Status Codes 476
Modeling and Utility Software 477
Symbolic Constants, Global Data, and Prototypes 477
Data Structures 477
Fuzzy Logic Functions 478
The Fuzzy System Modeling Functions 481
Miscellaneous Tools and Utilities 482
Demonstration and Fuzzy Model Programs 484
Description of Fuzzy Logic Functions 486