Table of Contents
Preface vii
List of Tables xv
List of Figures xvi
Chapter 1 Introduction 1
Chapter 2 Basic Knowledge on Classical Sets 4
2.1 Classical Sets and Set Inclusion 4
2.2 Set Operations 7
2.3 Set Sequences and Set Classes 10
2.4 Set Classes Closed Under Set Operations 13
2.5 Relations, Posets, and Lattices 17
2.6 The Supremum and Infimum of Real Number Sets 20
Exercises 22
Chapter 3 Fuzzy Sets 24
3.1 The Membership Functions of Fuzzy Sets 24
3.2 Inclusion and Operations of Fuzzy Sets 27
3.3 α-Cuts 33
3.4 Convex Fuzzy Sets 36
3.5 Decomposition Theorems 37
3.6 The Extension Principle 40
3.7 Interval Numbers 42
3.8 Fuzzy Numbers and Linguistic Attribute 45
3.9 Binary Operations for Fuzzy Numbers 51
3.10 Fuzzy Integers 58
Exercises 59
Chapter 4 Set Functions 62
4.1 Weights and Classical Measures 63
4.2 Extension of Measures 66
4.3 Monotone Measures 69
4.4 λ-Measures 74
4.5 Quasi-Measures 82
4.6 Möbius and Zeta Transformations 87
4.7 Belief Measures and Plausibility Measures 91
4.8 Necessity Measures and Possibility Measures 102
4.9 κ-Interactive Measures 107
4.10 Efficiency Measures and Signed Efficiency Measures 108
Exercises 112
Chapter 5 Integrations 115
5.1 Measurable Functions 115
5.2 The Riemann Integral 123
5.3 The Lebesgue-Like Integral 128
5.4 The Choquet Integral 133
5.5 Upper and Lower Integrals 153
5.6 r-Integrals on Finite Spaces 162
Exercises 174
Chapter 6 Information Fusion 177
6.1 Information Sources and Observations 177
6.2 Integrals Used as Aggregation Tools 181
6.3 Uncertainty Associated with Set Functions 186
6.4 The Inverse Problem of Information Fusion 190
Chapter 7 Optimization and Soft Computing 193
7.1 Basic Concepts of Optimization 193
7.2 Genetic Algorithms 195
7.3 Pseudo Gradient Search 199
7.4 A Hybrid Search Method 202
Chapter 8 Identification of Set Functions 204
8.1 Identification of λ-Measures 204
8.2 Identification of Belief Measures 206
8.3 Identification of Monotone Measures 207
8.3.1 Main algorithm 210
8.3.2 Reordering algorithm 211
8.4 Identification of Signed Efficiency Measures by a Genetic Algorithm 213
8.5 Identification of Signed Efficiency Measures by the Pseudo Gradient Search 215
8.6 Identification of Signed Efficiency Measures Based on the Choquet Integral by an Algebraic Method 217
8.7 Identification of Monotone Measures Based on r-Integrals by a Genetic Algorithm 219
Chapter 9 Multiregression Based on Nonlinear Integrals 221
9.1 Linear Multiregression 221
9.2 Nonlinear Multiregression Based on the Choquet Integral 226
9.3 A Nonlinear Multiregression Model Accommodating Both Categorical and Numerical Predictive Attributes 232
9.4 Advanced Consideration on the Multiregression Involving Nonlinear Integrals 234
9.4.1 Nonlinear multiregressions based on the Choquet integral with quadratic core 234
9.4.2 Nonlinear multiregressions based on the Choquet integral involving unknown periodic variation 235
9.4.3 Nonlinear multiregressions based on upper and lower integrals 236
Chapter 10 Classifications Based on Nonlinear Integrals 238
10.1 Classification by an Integral Projection 238
10.2 Nonlinear Classification by Weighted Choquet Integrals 242
10.3 An Example of Nonlinear Classification in a Three-Dimensional Sample Space 250
10.4 The Uniqueness Problem of the Classification by the Choquet Integral with a Linear Core 263
10.5 Advanced Consideration on the Nonlinear Classification Involving the Choquet Integral 267
10.5.1 Classification by the Choquet integral with the widest gap between classes 267
10.5.2 Classification by cross-oriented projection pursuit 268
10.5.3 Classification by the Choquet integral with quadratic core 270
Chapter 11 Data Mining with Fuzzy Data 272
11.1 Defuzzified Choquet Integral with Fuzzy-Valued Integrand (DCIFI) 273
11.1.1 The α-level set of a fuzzy-valued function 274
11.1.2 The Choquet extension of μ 275
11.1.3 Calculation of DCIFI 277
11.2 Classification Model Based on the DCIFI 282
11.2.1 Fuzzy data classification by the DCIFI 283
11.2.2 GA-based adaptive classifier-learning algorithm via DCIFI projection pursuit 286
11.2.3 Examples of the classification problems solved by the DCIFI projection classifier 290
11.3 Fuzzified Choquet Integral with Fuzzy-Valued Integrand (FCIFI) 300
11.3.1 Definition of the FCIFI 300
11.3.2 The FCIFI with respect to monotone measures 303
11.3.3 The FCIFI with respect to signed efficiency measures 306
11.3.4 GA-based optimization algorithm for the FCIFI with respect to signed efficiency measures 309
11.4 Regression Model Based on the CIII 319
11.4.1 CIII regression model 319
11.4.2 Double-GA optimization algorithm 321
11.4.3 Explanatory examples 324
Bibliography 329
Index 337
