Nonlinear Integrals And Their Applications In Data Mining

by Wang, Zhenyuan; Yang, Rong; Leung, Kwong-Sak

• ISBN13: 9789812814678
• ISBN10: 9812814671
• Format: Hardcover
• Copyright: 2010-06-30
• Publisher: World Scientific Pub Co Inc

Summary

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.

Table of Contents

Preface	p. vii
List of Tables	p. xv
List of Figures	p. xvi
Introduction	p. 1
Basic Knowledge on Classical Sets	p. 4
Classical Sets and Set Inclusion	p. 4
Set Operations	p. 7
Set Sequences and Set Classes	p. 10
Set Classes Closed Under Set Operations	p. 13
Relations, Posets, and Lattices	p. 17
The Supremum and Infimum of Real Number Sets	p. 20
Exercises	p. 22
Fuzzy Sets	p. 24
The Membership Functions of Fuzzy Sets	p. 24
Inclusion and Operations of Fuzzy Sets	p. 27
¿-Cuts	p. 33
Convex Fuzzy Sets	p. 36
Decomposition Theorems	p. 37
The Extension Principle	p. 40
Interval Numbers	p. 42
Fuzzy Numbers and Linguistic Attribute	p. 45
Binary Operations for Fuzzy Numbers	p. 51
Fuzzy Integers	p. 58
Exercises	p. 59
Set Functions	p. 62
Weights and Classical Measures	p. 63
Extension of Measures	p. 66
Monotone Measures	p. 69
¿-Measures	p. 74
Quasi-Measures	p. 82
Möbius and Zeta Transformations	p. 87
Belief Measures and Plausibility Measures	p. 91
Necessity Measures and Possibility Measures	p. 102
¿-Interactive Measures	p. 107
Efficiency Measures and Signed Efficiency Measures	p. 108
Exercises	p. 112
Integrations	p. 115
Measurable Functions	p. 115
The Riemann Integral	p. 123
The Lebesgue-Like Integral	p. 128
The Choquet Integral	p. 133
Upper and Lower Integrals	p. 153
r-Integrals on Finite Spaces	p. 162
Exercises	p. 174
Information Fusion	p. 177
Information Sources and Observations	p. 177
Integrals Used as Aggregation Tools	p. 181
Uncertainty Associated with Set Functions	p. 186
The Inverse Problem of Information Fusion	p. 190
Optimization and Soft Computing	p. 193
Basic Concepts of Optimization	p. 193
Genetic Algorithms	p. 195
Pseudo Gradient Search	p. 199
A Hybrid Search Method	p. 202
Identification of Set Functions	p. 204
Identification of ¿-Measures	p. 204
Identification of Belief Measures	p. 206
Identification of Monotone Measures	p. 207
Main algorithm	p. 210
Reordering algorithm	p. 211
Identification of Signed Efficiency Measures by a Genetic Algorithm	p. 213
Identification of Signed Efficiency Measures by the Pseudo Gradient Search	p. 215
Identification of Signed Efficiency Measures Based on the Choquet Integral by an Algebraic Method	p. 217
Identification of Monotone Measures Based on r-Integrals by a Genetic Algorithm	p. 219
Multiregression Based on Nonlinear Integrals	p. 221
Linear Multiregression	p. 221
Nonlinear Multiregression Based on the Choquet Integral	p. 226
A Nonlinear Multiregression Model Accommodating Both Categorical and Numerical Predictive Attributes	p. 232
Advanced Consideration on the Multiregression Involving Nonlinear Integrals	p. 234
Nonlinear multiregressions based on the Choquet integral with quadratic core	p. 234
Nonlinear multiregressions based on the Choquet integral involving unknown periodic variation	p. 235
Nonlinear multiregressions based on upper and lower integrals	p. 236
Classifications Based on Nonlinear Integrals	p. 238
Classification by an Integral Projection	p. 238
Nonlinear Classification by Weighted Choquet Integrals	p. 242
An Example of Nonlinear Classification in a Three-Dimensional Sample Space	p. 250
The Uniqueness Problem of the Classification by the Choquet Integral with a Linear Core	p. 263
Advanced Consideration on the Nonlinear Classification Involving the Choquet Integral	p. 267
Classification by the Choquet integral with the widest gap between classes	p. 267
Classification by cross-oriented projection pursuit	p. 268
Classification by the Choquet integral with quadratic core	p. 270
Data Mining with Fuzzy Data	p. 272
Defuzzified Choquet Integral with Fuzzy-Valued Integrand (DCIFI)	p. 273
The ¿-level set of a fuzzy-valued function	p. 274
The Choquet extension of ¿	p. 275
Calculation of DCIFI	p. 277
Classification Model Based on the DCIFI	p. 282
Fuzzy data classification by the DCIFI	p. 283
GA-based adaptive classifier-learning algorithm via DCIFI projection pursuit	p. 286
Examples of the classification problems solved by the DCIFI projection classifier	p. 290
Fuzzified Choquet Integral with Fuzzy-Valued Integrand (FCIFI)	p. 300
Definition of the FCIFI	p. 300
The FCIFI with respect to monotone measures	p. 303
The FCIFI with respect to signed efficiency measures	p. 306
GA-based optimization algorithm for the FCIFI with respect to signed efficiency measures	p. 309
Regression Model Based on the CIII	p. 319
CIII regression model	p. 319
Double-GA optimization algorithm	p. 321
Explanatory examples	p. 324
Bibliography	p. 329
Index	p. 337
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