Nonlinear Integrals and Their Applications in Data Mining

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More About This Textbook

Overview

Regarding the set of all feature attributes in a given database as the universal set, this monograph discusses various nonadditive set functions that describe the interaction among the contributions from feature attributes towards a considered target attribute. Then, the relevant nonlinear integrals are investigated. These integrals can be applied as aggregation tools in information fusion and data mining, such as synthetic evaluation, nonlinear multiregressions, and nonlinear classifications. Some methods of fuzzification are also introduced for nonlinear integrals such that fuzzy data can be treated and fuzzy information is retrievable.

The book is suitable as a text for graduate courses in mathematics, computer science, and information science. It is also useful to researchers in the relevant area.

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Product Details

  • ISBN-13: 9789812814678
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 6/9/2010
  • Pages: 360
  • Product dimensions: 6.10 (w) x 9.00 (h) x 0.90 (d)

Table of Contents

Preface

List of Tables

List of Figures

Chapter 1 Introduction 1

Chapter 2 Basic Knowledge on Classical Sets 4

2.1 Classical Sets and Set Inclusion 4

2.2 Set Operations 4

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 Mobius and Zeta Transformations 87

4.7 Belief Measures and Plausibility Measures 91

4.8 Necessity Measures and Possibility Measures 102

4.9 k-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

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