Nonlinear Integrals and Their Applications in Data Mining

The following sections are included:

• Preface

• Contents

• List of Tables

• List of Figures

The traditional aggregation tool in information treatment is the weighted average, or more general, the weighted sum. That is, if the numerical information received from diverse information sources x₁, x₂, ⋯, x_n are f(x₁), f(x₂), ⋯, f(x_n) respectively, then the synthetic amount, weighted sum y, of the information is calculated by

The following sections are included:

• Classical Sets and Set Inclusion

• Set Operations

• Set Sequences and Set Classes

• Set Classes Closed Under Set Operations

• Relations, Posets, and Lattices

• The Supremum and Infimum of Real Number Sets

• Exercises

The following sections are included:

• The Membership Functions of Fuzzy Sets

• Inclusion and Operations of Fuzzy Sets

• α-Cuts

• Convex Fuzzy Sets

• Decomposition Theorems

• The Extension Principle

• Interval Numbers

• Fuzzy Numbers and Linguistic Attribute

• Binary Operations for Fuzzy Numbers

• Fuzzy Integers

• Exercises

The following sections are included:

• Weights and Classical Measures

• Extension of Measures

• Monotone Measures

• λ-Measures

• Quasi-Measures

• Möbius and Zeta Transformations

• Belief Measures and Plausibility Measures

• Necessity Measures and Possibility Measures

• k-Interactive Measures

• Efficiency Measures and Signed Efficiency Measures

• Exercises

The following sections are included:

• Measurable Functions

• The Riemann Integral

• The Lebesgue-Like Integral

• The Choquet Integral

• Upper and Lower Integrals

• r-Integrals on Finite Spaces

• Exercises

The following sections are included:

• Information Sources and Observations

• Integrals Used as Aggregation Tools

• Uncertainty Associated with Set Functions

• The Inverse Problem of Information Fusion

The following sections are included:

• Basic Concepts of Optimization

• Genetic Algorithms

• Pseudo Gradient Search

• A Hybrid Search Method

The following sections are included:

• Identification of λ-Measures

• Identification of Belief Measures

• Identification of Monotone Measures
  • Main algorithm
  • Reordering algorithm

• Identification of Signed Efficiency Measures by a Genetic Algorithm

• Identification of Signed Efficiency Measures by the Pseudo Gradient Search

• Identification of Signed Efficiency Measures Based on the Choquet Integral by an Algebraic Method

• Identification of Monotone Measures Based on r-Integrals by a Genetic Algorithm

The following sections are included:

• Linear Multiregression

• Nonlinear Multiregression Based on the Choquet Integral

• A Nonlinear Multiregression Model Accommodating Both Categorical and Numerical Predictive Attributes

• Advanced Consideration on the Multiregression Involving Nonlinear Integrals
  • Nonlinear multiregressions based on the Choquet integral with quadratic core
  • Nonlinear multiregressions based on the Choquet integral involving unknown periodic variation
  • Nonlinear multiregressions based on upper and lower integrals

The following sections are included:

• Classification by an Integral Projection

• Nonlinear Classification by Weighted Choquet Integrals

• An Example of Nonlinear Classification in a Three-Dimensional Sample Space

• The Uniqueness Problem of the Classification by the Choquet Integral with a Linear Core

• Advanced Consideration on the Nonlinear Classification Involving the Choquet Integral
  • Classification by the Choquet integral with the widest gap between classes
  • Classification by cross-oriented projection pursuit
  • Classification by the Choquet integral with quadratic core

The following sections are included:

• Defuzzified Choquet Integral with Fuzzy-Valued Integrand (DCIFI)
  • The α-level set of a fuzzy-valued function
  • The Choquet extension of μ
  • Calculation of DCIFI

• Classification Model Based on the DCIFI
  • Fuzzy data classification by the DCIFI
  • GA-based adaptive classifier-learning algorithm via DCIFI projection pursuit
  • Examples of the classification problems solved by the DCIFI projection classifier

• Fuzzified Choquet Integral with Fuzzy-Valued Integrand (FCIFI)
  • Definition of the FCIFI
  • The FCIFI with respect to monotone measures
  • The FCIFI with respect to signed efficiency measures
  • GA-based optimization algorithm for the FCIFI with respect to signed efficiency measures

• Regression Model Based on the CIII
  • CIII regression model
  • Double-GA optimization algorithm
  • Explanatory examples

The following sections are included:

• Bibliography

• Index