Hybrid Rough Sets and Applications in Uncertain Decision-Making

Introduction
Background and Significance of Soft Computing Technology
     Analytical Method of Data Mining
          Automatic Prediction of Trends and Behavior
          Association Analysis
          Cluster Analysis
          Concept Description
          Deviation Detection
     Knowledge Discovered by Data Mining
Characteristics of Rough Set Theory and Current Status of Rough Set Theory Research 
     Characteristics of the Rough Set Theory
     Current Status of Rough Set Theory Research
          Analysis with Decision-Making
          Non-Decision-Making Analysis
Hybrid of Rough Set Theory and Other Soft Technologies
     Hybrid of Rough Sets and Probability Statistics
     Hybrid of Rough Sets and Dominance Relation
     Hybrid of Rough Sets and Fuzzy Sets
     Hybrid of Rough Set and Grey System Theory
     Hybrid of Rough Sets and Neural Networks

Rough Set Theory
Information Systems and Classification
     Information Systems and Indiscernibility Relation
     Set and Approximations of Set
     Attributes Dependence and Approximation Accuracy
     Quality of Approximation and Reduct
     Calculation of the Reduct and Core of Information System Based on Discernable Matrix
Decision Table and Rule Acquisition
     The Attribute Dependence, Attribute Reduct, and Core
     Decision Rules
      Use the Discernibility Matrix to Work Out Reducts, Core, and Decision Rules of Decision Table
Data Discretization
     Expert Discrete Method
     Equal Width Interval Method and Equal Frequency Interval Method
     The Most Subdivision Entropy Method
     Chimerge Method
Common Algorithms of Attribute Reduct
     Quick Reduct Algorithm
     Heuristic Algorithm of Attribute Reduct
     Genetic Algorithm
Application Case
     Data Collecting and Variable Selection
     Data Discretization
     Attribute Reduct
     Rule Generation
     Simulation of the Decision Rules

Hybrid of Rough Set Theory and Probability
Rough Membership Function
Variable Precision Rough Set Model
     β-Rough Approximation
     Classification Quality and β-Reduct
     Discussion about β Value
     Construction of Hierarchical Knowledge Granularity Based on VPRS 
     Knowledge Granularity
     Relationship between VPRS and Knowledge Granularity
          Approximation and Knowledge Granularity
          Classification Quality and Granularity Knowledge Granularity 
          Construction of Hierarchical Knowledge Granularity
          Methods of Construction of Hierarchical Knowledge Granularity 
          Algorithm Description
Methods of Rule Acquisition Based on the Inconsistent Information System in Rough Set 
     Bayes’ Probability
     Consistent Degree, Coverage, and Support
     Probability Rules
     Approach to Obtain Probabilistic Rules Hybrid of Rough Set and Dominance Relation

Hybrid of Rough Set and Dominance Relation
Dominance-Based Rough Set
     The Classification of the Decision Tables with Preference Attribute 
     Dominating Sets and Dominated Sets
     Rough Approximation by Means of Dominance Relations
     Classification Quality and Reduct
     Preferential Decision Rules
     Dominance-Based Variable Precision Rough Set
     Inconsistency and Indiscernibility Based on Dominance Relation 
     β-Rough Approximation Based on Dominance Relations
     Classification Quality and Approximate Reduct
     Preferential Probabilistic Decision Rules
     Algorithm Design
An Application Case
     Post Evaluation of Construction Projects Based on Dominance-Based Rough Set 
          Construction of Preferential Evaluation Decision Table
           Search of Reduct and Establishment of Preferential Rules
     Performance Evaluation of Discipline Construction in Teaching-Research Universities Based on Dominance-Based Rough Set 
          The Basic Principles of the Construction of Evaluation Index System 
          The Establishment of Index System and Determination of Weight and Equivalent
          Data Collection and Pretreatment 
          Data Discretization
          Search of Reducts and Generation of Preferential Rules
          Analysis of Evaluation Results

Hybrid of Rough Set Theory and Fuzzy Set Theory
The Basic Concepts of the Fuzzy Set Theory
     Fuzzy Set and Fuzzy Membership Function
     Operation of Fuzzy Subsets
     Fuzzy Relation and Operation
     Synthesis of Fuzzy Relations
     λ-Cut Set and the Decomposition Proposition
     The Fuzziness of Fuzzy Sets and Measure of Fuzziness
Rough Fuzzy Set and Fuzzy Rough Set
     Rough Fuzzy Set
     Fuzzy Rough Set
Variable Precision Rough Fuzzy Sets
     Rough Membership Function Based on λ-Cut Set
     The Rough Approximation of Variable Precision Rough Fuzzy Set 
     The Approximate Quality and Approximate Reduct of variable Precision
     The Probabilistic Decision Rules Acquisition of Rough Fuzzy Decision Table 
     Algorithm Design
Variable Precision Fuzzy Rough Set
     Fuzzy Equivalence Relation
     Precision Fuzzy Rough Model
     Acquisition of Probabilistic Decision Rules in Fuzzy Rough Decision Table 
     Measure Methods of the Fuzzy Roughness for Output Classification 
          Distance Measurement
          Entropy Measurement

Hybrid of Rough Set and Grey System
The Basic Concepts and Methods of the Grey System Theory
     Grey Number, Whitening of Grey Number, and Grey Degree
          Types of Grey Numbers
          Whitenization of Grey Numbers and Grey Degree
     Grey Sequence Generation
     GM(1, 1) Model
     Grey Correlation Analysis
     Grey Correlation Order
     Grey Clustering Evaluation
          Clusters of Grey Correlation
          Cluster with Variable Weights
          Grey Cluster with Fixed Weights
Establishment of Decision Table Based on Grey Clustering
The Grade of Grey Degree of Grey Numbers and Grey Membership Function Based on Rough Membership Function
Grey Rough Approximations
Reduced Attributes Dominance Analysis Based on Grey Correlation Analysis

A Hybrid Approach of Variable Precision Rough Set, Fuzzy Set, and Neural Network
Neural Network

Introduction
Background and Significance of Soft Computing Technology
     Analytical Method of Data Mining
          Automatic Prediction of Trends and Behavior
          Association Analysis
          Cluster Analysis
          Concept Description
          Deviation Detection
     Knowledge Discovered by Data Mining
Characteristics of Rough Set Theory and Current Status of Rough Set Theory Research 
     Characteristics of the Rough Set Theory
     Current Status of Rough Set Theory Research
          Analysis with Decision-Making
          Non-Decision-Making Analysis
Hybrid of Rough Set Theory and Other Soft Technologies
     Hybrid of Rough Sets and Probability Statistics
     Hybrid of Rough Sets and Dominance Relation
     Hybrid of Rough Sets and Fuzzy Sets
     Hybrid of Rough Set and Grey System Theory
     Hybrid of Rough Sets and Neural Networks

Rough Set Theory
Information Systems and Classification
     Information Systems and Indiscernibility Relation
     Set and Approximations of Set
     Attributes Dependence and Approximation Accuracy
     Quality of Approximation and Reduct
     Calculation of the Reduct and Core of Information System Based on Discernable Matrix
Decision Table and Rule Acquisition
     The Attribute Dependence, Attribute Reduct, and Core
     Decision Rules
      Use the Discernibility Matrix to Work Out Reducts, Core, and Decision Rules of Decision Table
Data Discretization
     Expert Discrete Method
     Equal Width Interval Method and Equal Frequency Interval Method
     The Most Subdivision Entropy Method
     Chimerge Method
Common Algorithms of Attribute Reduct
     Quick Reduct Algorithm
     Heuristic Algorithm of Attribute Reduct
     Genetic Algorithm
Application Case
     Data Collecting and Variable Selection
     Data Discretization
     Attribute Reduct
     Rule Generation
     Simulation of the Decision Rules

Hybrid of Rough Set Theory and Probability
Rough Membership Function
Variable Precision Rough Set Model
     β-Rough Approximation
     Classification Quality and β-Reduct
     Discussion about β Value
     Construction of Hierarchical Knowledge Granularity Based on VPRS 
     Knowledge Granularity
     Relationship between VPRS and Knowledge Granularity
          Approximation and Knowledge Granularity
          Classification Quality and Granularity Knowledge Granularity 
          Construction of Hierarchical Knowledge Granularity
          Methods of Construction of Hierarchical Knowledge Granularity 
          Algorithm Description
Methods of Rule Acquisition Based on the Inconsistent Information System in Rough Set 
     Bayes’ Probability
     Consistent Degree, Coverage, and Support
     Probability Rules
     Approach to Obtain Probabilistic Rules Hybrid of Rough Set and Dominance Relation

Hybrid of Rough Set and Dominance Relation
Dominance-Based Rough Set
     The Classification of the Decision Tables with Preference Attribute 
     Dominating Sets and Dominated Sets
     Rough Approximation by Means of Dominance Relations
     Classification Quality and Reduct
     Preferential Decision Rules
     Dominance-Based Variable Precision Rough Set
     Inconsistency and Indiscernibility Based on Dominance Relation 
     β-Rough Approximation Based on Dominance Relations
     Classification Quality and Approximate Reduct
     Preferential Probabilistic Decision Rules
     Algorithm Design
An Application Case
     Post Evaluation of Construction Projects Based on Dominance-Based Rough Set 
          Construction of Preferential Evaluation Decision Table
           Search of Reduct and Establishment of Preferential Rules
     Performance Evaluation of Discipline Construction in Teaching-Research Universities Based on Dominance-Based Rough Set 
          The Basic Principles of the Construction of Evaluation Index System 
          The Establishment of Index System and Determination of Weight and Equivalent
          Data Collection and Pretreatment 
          Data Discretization
          Search of Reducts and Generation of Preferential Rules
          Analysis of Evaluation Results

Hybrid of Rough Set Theory and Fuzzy Set Theory
The Basic Concepts of the Fuzzy Set Theory
     Fuzzy Set and Fuzzy Membership Function
     Operation of Fuzzy Subsets
     Fuzzy Relation and Operation
     Synthesis of Fuzzy Relations
     λ-Cut Set and the Decomposition Proposition
     The Fuzziness of Fuzzy Sets and Measure of Fuzziness
Rough Fuzzy Set and Fuzzy Rough Set
     Rough Fuzzy Set
     Fuzzy Rough Set
Variable Precision Rough Fuzzy Sets
     Rough Membership Function Based on λ-Cut Set
     The Rough Approximation of Variable Precision Rough Fuzzy Set 
     The Approximate Quality and Approximate Reduct of variable Precision
     The Probabilistic Decision Rules Acquisition of Rough Fuzzy Decision Table 
     Algorithm Design
Variable Precision Fuzzy Rough Set
     Fuzzy Equivalence Relation
     Precision Fuzzy Rough Model
     Acquisition of Probabilistic Decision Rules in Fuzzy Rough Decision Table 
     Measure Methods of the Fuzzy Roughness for Output Classification 
          Distance Measurement
          Entropy Measurement

Hybrid of Rough Set and Grey System
The Basic Concepts and Methods of the Grey System Theory
     Grey Number, Whitening of Grey Number, and Grey Degree
          Types of Grey Numbers
          Whitenization of Grey Numbers and Grey Degree
     Grey Sequence Generation
     GM(1, 1) Model
     Grey Correlation Analysis
     Grey Correlation Order
     Grey Clustering Evaluation
          Clusters of Grey Correlation
          Cluster with Variable Weights
          Grey Cluster with Fixed Weights
Establishment of Decision Table Based on Grey Clustering
The Grade of Grey Degree of Grey Numbers and Grey Membership Function Based on Rough Membership Function
Grey Rough Approximations
Reduced Attributes Dominance Analysis Based on Grey Correlation Analysis

A Hybrid Approach of Variable Precision Rough Set, Fuzzy Set, and Neural Network

Introduction
Background and Significance of Soft Computing Technology
     Analytical Method of Data Mining
          Automatic Prediction of Trends and Behavior
          Association Analysis
          Cluster Analysis
          Concept Description
          Deviation Detection
     Knowledge Discovered by Data Mining
Characteristics of Rough Set Theory and Current Status of Rough Set Theory Research 
     Characteristics of the Rough Set Theory
     Current Status of Rough Set Theory Research
          Analysis with Decision-Making
          Non-Decision-Making Analysis
Hybrid of Rough Set Theory and Other Soft Technologies
     Hybrid of Rough Sets and Probability Statistics
     Hybrid of Rough Sets and Dominance Relation
     Hybrid of Rough Sets and Fuzzy Sets
     Hybrid of Rough Set and Grey System Theory
     Hybrid of Rough Sets and Neural Networks

Rough Set Theory
Information Systems and Classification
     Information Systems and Indiscernibility Relation
     Set and Approximations of Set
     Attributes Dependence and Approximation Accuracy
     Quality of Approximation and Reduct
     Calculation of the Reduct and Core of Information System Based on Discernable Matrix
Decision Table and Rule Acquisition
     The Attribute Dependence, Attribute Reduct, and Core
     Decision Rules
      Use the Discernibility Matrix to Work Out Reducts, Core, and Decision Rules of Decision Table
Data Discretization
     Expert Discrete Method
     Equal Width Interval Method and Equal Frequency Interval Method
     The Most Subdivision Entropy Method
     Chimerge Method
Common Algorithms of Attribute Reduct
     Quick Reduct Algorithm
     Heuristic Algorithm of Attribute Reduct
     Genetic Algorithm
Application Case
     Data Collecting and Variable Selection
     Data Discretization
     Attribute Reduct
     Rule Generation
     Simulation of the Decision Rules

Hybrid of Rough Set Theory and Probability
Rough Membership Function
Variable Precision Rough Set Model
     β-Rough Approximation
     Classification Quality and β-Reduct
     Discussion about β Value
     Construction of Hierarchical Knowledge Granularity Based on VPRS 
     Knowledge Granularity
     Relationship between VPRS and Knowledge Granularity
          Approximation and Knowledge Granularity
          Classification Quality and Granularity Knowledge Granularity 
          Construction of Hierarchical Knowledge Granularity
          Methods of Construction of Hierarchical Knowledge Granularity 
          Algorithm Description
Methods of Rule Acquisition Based on the Inconsistent Information System in Rough Set 
     Bayes’ Probability
     Consistent Degree, Coverage, and Support
     Probability Rules
     Approach to Obtain Probabilistic Rules Hybrid of Rough Set and Dominance Relation

Hybrid of Rough Set and Dominance Relation
Dominance-Based Rough Set
     The Classification of the Decision Tables with Preference Attribute 
     Dominating Sets and Dominated Sets
     Rough Approximation by Means of Dominance Relations
     Classification Quality and Reduct
     Preferential Decision Rules
     Dominance-Based Variable Precision Rough Set
     Inconsistency and Indiscernibility Based on Dominance Relation 
     β-Rough Approximation Based on Dominance Relations
     Classification Quality and Approximate Reduct
     Preferential Probabilistic Decision Rules
     Algorithm Design
An Application Case
     Post Evaluation of Construction Projects Based on Dominance-Based Rough Set 
          Construction of Preferential Evaluation Decision Table
           Search of Reduct and Establishment of Preferential Rules
     Performance Evaluation of Discipline Construction in Teaching-Research Universities Based on Dominance-Based Rough Set 
          The Basic Principles of the Construction of Evaluation Index System 
          The Establishment of Index System and Determination of Weight and Equivalent
          Data Collection and Pretreatment 
          Data Discretization
          Search of Reducts and Generation of Preferential Rules
          Analysis of Evaluation Results

Hybrid of Rough Set Theory and Fuzzy Set Theory
The Basic Concepts of the Fuzzy Set Theory
     Fuzzy Set and Fuzzy Membership Function
     Operation of Fuzzy Subsets
     Fuzzy Relation and Operation
     Synthesis of Fuzzy Relations
     λ-Cut Set and the Decomposition Proposition
     The Fuzziness of Fuzzy Sets and Measure of Fuzziness
Rough Fuzzy Set and Fuzzy Rough Set
     Rough Fuzzy Set
     Fuzzy Rough Set
Variable Precision Rough Fuzzy Sets
     Rough Membership Function Based on λ-Cut Set
     The Rough Approximation of Variable Precision Rough Fuzzy Set 
     The Approximate Quality and Approximate Reduct of variable Precision
     The Probabilistic Decision Rules Acquisition of Rough Fuzzy Decision Table 
     Algorithm Design
Variable Precision Fuzzy Rough Set
     Fuzzy Equivalence Relation
     Precision Fuzzy Rough Model
     Acquisition of Probabilistic Decision Rules in Fuzzy Rough Decision Table 
     Measure Methods of the Fuzzy Roughness for Output Classification 
          Distance Measurement
          Entropy Measurement

Hybrid of Rough Set and Grey System
The Basic Concepts and Methods of the Grey System Theory
     Grey Number, Whitening of Grey Number, and Grey Degree
          Types of Grey Numbers
          Whitenization of Grey Numbers and Grey Degree
     Grey Sequence Generation
     GM(1, 1) Model
     Grey Correlation Analysis
     Grey Correlation Order
     Grey Clustering Evaluation
          Clusters of Grey Correlation
          Cluster with Variable Weights
          Grey Cluster with Fixed Weights
Establishment of Decision Table Based on Grey Clustering
The Grade of Grey Degree of Grey Numbers and Grey Membership Function Based on Rough Membership Function
Grey Rough Approximations
Reduced Attributes Dominance Analysis Based on Grey Correlation Analysis

A Hybrid Approach of Variable Precision Rough Set, Fuzzy Set, and Neural Network
Neural Network
     An Overview of the Development of Neural Network
     Structure and Types of Neural Network
     Perceptron
          Perceptron Neuron Model
          Network Structure of Perceptron Neutral Network
          Learning Rules of Perceptron Neutral Network
     Back Propagation Network
          BP Neuron Model
          Network Structure of BP Neutral Network
          BP Algorithm
     Radial Basis Networks
          Radial Basis Neurons Model
          The Network Structure of the RBF
     Realization of the Algorithm of RBF Neural Network
     Probabilistic Neural Network
          PNN Structure
          Realization of PNN Algorithm
Knowledge Discovery in Databases Based on the Hybrid of VPRS and Neural Network 
     Collection, Selection, and Pretreatment of the Data 
     Construction of Decision Table
     Searching of β-Reduct and Generation of Probability Decision Rules 
     Searching of β-Reduct
          Learning and Simulation of the Neural Network
System Design Methods of the Hybrid of Variable Precision Rough Fuzzy and Neutral Network 
     Construction of Variable Precision Rough Fuzzy Neutral Network
     Training Algorithm of the Variable Precision Rough Fuzzy Neutral Network

Application Analysis of Hybrid Rough Set
A Survey of Transport Scheme Choice
Transport Scheme Choice Decision Undertaking No Consideration into Preference Information
     Choice Decision Based on Rough Set
     Probability Choice Decision Based on VPRS
     Choice Decision Based on Grey Rough Set
     Probability Choice Decision Based on the Hybrid of VPRS and Probabilistic Neural Network
Transport Scheme Choice Decision Undertaking Consideration into Preference Information
     Choice Decision Based on the Dominance Rough Set
     Choice Decision Based on the Dominance-Based VPRS

Bibliography
Index

Biography

Lirong Jian received her PhD in management science and engineering from Southeast University, Nanjing, China, in 2004. She then had two years ofpostdoctoral experience specializing in management science and engineering at Nanjing University of Aeronautics and Astronautics, China. At present, she is serving as a professor at the College of Economics and Management of Nanjing University of Aeronautics and Astronautics; she is also working as a guide for doctoral students in management science and systems engineering.Dr. Jian is principally engaged in forecasting and decision-making methods, soft computing, and project management and system modeling. She has also directed and/or participated in nearly 20 projects at the national, provincial, and ministerial levels, for which she received four provincial awards in scientific research and applications. Over the years, she has published over 40 research papers and 6 books.Sifeng Liu received his bachelor's degree in mathematics from Henan University, Kaifeng, China in 1981, and his MS in economics and his PhD in systems engineering from Huazhong University of Science and Technology, Wuhan, China, in 1986 and 1998, respectively. He has been to Slippery Rock University, Pennsylvania, and to Sydney University, Australia, as a visiting professor. At present, Professor Liu is the director of the Institute for Grey Systems Studies and the dean of the College of Economics and Management of Nanjing University of Aeronautics and Astronautics. He is also a distinguished professor and guide for doctoral students in management science and systems engineering.Dr. Liu's main research activities are in grey systems theory and in regional technical innovation management. He has directed more than 50 projects at the national, provincial, and ministerial levels, has participated in international collaboration projects, and has published over 200 research papers and 16 bo

The book presents the mathematical theory of rough sets: its interpretation, properties and applications for data and reasoning, especially for decision analysis and forecasting. Also, the relation between rough set theory (RST) and other soft computing theories, such as fuzzy set theory, grey systems, neural networks and probability and statistics, is considered as a tool to manage uncertainty and incomplete information. … This book especially targets postgraduates interested in activities such as economic management, information sciences, social sciences or applied mathematics, and aims to draw their attention to the soft computing approach.
—Maria-Teresa Lamata, in Mathematical Reviews, Issue 2012D