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More About This Textbook
Overview
As a powerful approach to data reasoning, rough set theory has proven to be invaluable in knowledge acquisition, decision analysis and forecasting, and knowledge discovery. With the ability to enhance the advantages of other soft technology theories, hybrid rough set theory is quickly emerging as a method of choice for decision making under uncertain conditions.
Keeping the complicated mathematics to a minimum, Hybrid Rough Sets and Applications in Uncertain Decision-Making provides a systematic introduction to the methods and application of the hybridization for rough set theory with other related soft technology theories, including probability, grey systems, fuzzy sets, and artificial neural networks. It also:
The authors examine the scope of application of the rough set theory and discuss how the combination of variable precision rough sets and dominance relations can produce probabilistic preference rules out of preference attribute decision tables of preference actions. Complete with numerous cases that illustrate the specific application of hybrid methods, the text adopts the latest achievements in the theory, method, and application of rough sets.
Editorial Reviews
From the Publisher
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
Product Details
Meet the Author
Lirong Jian received her PhD in management science and engineering from Southeast University, Nanjing, China, in 2004. She then had two years of postdoctoral 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 books. Over the years, he has received 18 provincial and national awards for his outstanding achievements in scientific research and applications. In 2002, one of his papers was recognized by the World Organization of Systems and Cybernetics as one of the best papers of its 12th International Congress.
Dr. Liu is a member of the evaluation committee of the Natural Science Foundation of China (NSFC) and a member of the standing committee for teaching guide in management science and engineering of the Ministry of Education, China. He also serves as an expert on soft science at the Ministry of Science and Technology, China. Professor Liu currently serves as the chair of the technical committee of the IEEE SMC on Grey Systems; the president of the Grey Systems Society of China (GSSC); a vice president of the Chinese Society for Optimization, Overall Planning and Economic Mathematics (CSOOPEM); a cochair of the Beijing Chapter and the Nanjing Chapter of IEEE SMC; a vice president of the Econometrics and Management Science Society of Jiangsu Province (EMSSJS); a vice president of the Systems Engineering Society of Jiangsu Province (SESJS); and a member of the Nanjing Decision Consultancy Committee. He serves as the editor in chief of Grey Systems: Theory and Application, and as a member of the editorial boards of over 10 professional journals, including The Journal of Grey System (United Kingdom); Scientific Inquiry (United States); The Journal of Grey System (Taiwan, China); Chinese Journal of Management Science; Systems Theory and Applications; Systems Science and Comprehensive Studies in Agriculture; and the Journal of Nanjing University of Aeronautics and Astronautics.
Dr. Liu has won several accolades, such as the National Excellent Teacher in 1995, Excellent Expert of Henan Province in 1998, National Expert with Prominent Contribution in 1998, Expert Enjoying Government’s Special Allowance in 2000, xcellent Science and Technology Staff in Jiangsu Province in 2002, National Advanced Individual for Returnee and Achievement Award for Returnee in 2003, and Outstanding Managerial Personnel of China in 2005.
Yi Lin holds all his educational degrees (BS, MS, and PhD) in pure mathematics from Northwestern University, Xi’an, China and Auburn University, Alabama, and has had one year of postdoctoral experience in statistics at Carnegie Mellon University, Pittsburgh, Pennsylvania. Currently, he serves as a guest or specially appointed professor in economics, finance, systems science, and mathematics at several major universities in China, including Huazhong University of Science and Technology, Changsha National University of Defence Technology, and Nanjing University of Aeronautics and Astronautics, and as a professor of mathematics at the Pennsylvania State System of Higher Education (Slippery Rock campus). Since 1993, he has been serving as the president of the International Institute for General Systems Studies, Inc. Among his other professional endeavors, Professor Lin has had the honor of mobilizing scholars from over 80 countries representing more than 50 different scientific disciplines. Over the years, he has served on the editorial boards of 11 professional journals, including Kybernetes: The International Journal of Cybernetics, Systems and Management Science; the Journal of Systems Science and Complexity; the International Journal of General Systems, and Advances in Systems Science and Applications. He is also a coeditor of the book series entitled Systems Evaluation, Prediction and Decision-Making, published by Taylor & Francis (2008).
Some of Lin’s research was funded by the United Nations, the State of Pennsylvania, the National Science Foundation of China, and the German National Research Center for Information Architecture and Software Technology. By the end of 2009, he had published nearly 300 research papers and over 30 monographs, and edited volumes on special topics. His works were published by such prestigious publishers as Springer, Wiley, World Scientific, Kluwer Academic (now part of Springer), Academic Press (now part of Springer), and others. Throughout his career, Lin’s scientific achievements have been recognized by various professional organizations and academic publishers. In 2001, he was inducted into the honorary fellowship of the World Organization of Systems and Cybernetics. Lin’s professional career started in 1984 when his first paper was published. His research interests are mainly in the area of systems research and applications in a wide range of disciplines of traditional science, such as mathematical modeling, foundations of mathematics, data analysis, theory and methods of predictions of disastrous natural events, economics and finance, management science, and philosophy of science.
Table of Contents
Preface
Acknowledgments
Authors
1 Introduction 1
1.1 Background and Significance of Soft Computing Technology 1
1.1.1 Analytical Method of Data Mining 2
1.1.1.1 Automatic Prediction of Trends and Behavior 2
1.1.1.2 Association Analysis 3
1.1.1.3 Cluster Analysis 3
1.1.1.4 Concept Description 4
1.1.1.5 Deviation Detection 4
1.1.2 Knowledge Discovered by Data Mining 4
1.2 Characteristics of Rough Set Theory and Current Status of Rough Set Theory Research 6
1.2.1 Characteristics of the Rough Set Theory 7
1.2.2 Current Status of Rough Set Theory Research 8
1.2.2.1 Analysis with Decision-Making 8
1.2.2.2 Non-Decision-Making Analysis 8
1.3 Hybrid of Rough Set Theory and Other Soft Technologies 9
1.3.1 Hybrid of Rough Sets and Probability Statistics 10
1.3.2 Hybrid of Rough Sets and Dominance Relation 10
1.3.3 Hybrid of Rough Sets and Fuzzy Sets 11
1.3.4 Hybrid of Rough Set and Grey System Theory 12
1.3.5 Hybrid of Rough Sets and Neural Networks 13
1.4 Summary 14
2 Rough Set Theory 17
2.1 Information Systems and Classification 17
2.1.1 Information Systems and Indiscernibility Relation 18
2.1.2 Set and Approximations of Set 18
2.1.3 Attributes Dependence and Approximation Accuracy 23
2.1.4 Quality of Approximation and Reduct 25
2.1.5 Calculation of the Reduct and Core of Information System Based on Discernable Matrix 26
2.2 Decision Table and Rule Acquisition 31
2.2.1 The Attribute Dependence, Attribute Reduct, and Core 31
2.2.2 Decision Rules 32
2.2.3 Use the Discernibility Matrix to Work Out Reducts, Core, and Decision Rules of Decision Table 33
2.3 Data Discretization 36
2.3.1 Expert Discrete Method 37
2.3.2 Equal Width Interval Method and Equal Frequency Interval Method 38
2.3.3 The Most Subdivision Entropy Method 38
2.3.4 Chimerge Method 38
2.4 Common Algorithms of Attribute Reduct 39
2.4.1 Quick Reduct Algorithm 39
2.4.2 Heuristic Algorithm of Attribute Reduct 40
2.4.3 Genetic Algorithm 41
2.5 Application Case 43
2.5.1 Data Collecting and Variable Selection 44
2.5.2 Data Discretization 44
2.5.3 Attribute Reduct 49
2.5.4 Rule Generation 52
2.5.5 Simulation of the Decision Rules 52
2.6 Summary 55
3 Hybrid of Rough Set Theory and Probability 57
3.1 Rough Membership Function 57
3.2 Variable Precision Rough Set Model 60
3.2.1 β-Rough Approximation 61
3.2.2 Classification Quality and β-Reduct 62
3.2.3 Discussion about β-Value 64
3.3 Construction of Hierarchical Knowledge Granularity Based on VPRS 67
3.3.1 Knowledge Granularity 67
3.3.2 Relationship between VPRS and Knowledge Granularity 68
3.3.2.1 Approximation and Knowledge Granularity 68
3.3.2.2 Classification Quality and Granularity Knowledge Granularity 68
3.3.3 Construction of Hierarchical Knowledge Granularity 69
3.3.3.1 Methods of Construction of Hierarchical Knowledge Granularity 69
3.3.3.2 Algorithm Description 70
3.4 Methods of Rule Acquisition Based on the Inconsistent Information System in Rough Set 73
3.4.1 Bayes' Probability 73
3.4.2 Consistent Degree, Coverage, and Support 74
3.4.3 Probability Rules 75
3.4.4 Approach to Obtain Probabilistic Rules 76
3.5 Summary 78
4 Hybrid of Rough Set and Dominance Relation 79
4.1 Dominance-Based Rough Set 79
4.1.1 The Classification of the Decision Tables with Preference Attribute 80
4.1.2 Dominating Sets and Dominated Sets 81
4.1.3 Rough Approximation by Means of Dominance Relations 82
4.1.4 Classification Quality and Reduct 83
4.1.5 Preferential Decision Rules 84
4.2 Dominance-Based Variable Precision Rough Set 86
4.2.1 Inconsistency and Indiscernibility Based on Dominance Relation 86
4.2.2 β-Rough Approximation Based on Dominance Relations 87
4.2.3 Classification Quality and Approximate Reduct 89
4.2.4 Preferential Probabilistic Decision Rules 89
4.2.5 Algorithm Design 89
4.3 An Application Case 93
4.3.1 Post-Evaluation of Construction Projects Based on Dominance-Based Rough Set 93
4.3.1.1 Construction of Preferential Evaluation Decision Table 94
4.3.1.2 Search of Reduct and Establishment of Preferential Rules 102
4.3.2 Performance Evaluation of Discipline Construction in Teaching-Research Universities Based on Dominance-Based Rough Set 103
4.3.2.1 The Basic Principles of the Construction of Evaluation Index System 104
4.3.2.2 The Establishment of Index System and Determination of Weight and Equivalent 105
4.3.2.3 Data Collection and Pretreatment 115
4.3.2.4 Data Discretization 115
4.3.2.5 Search of Reducts and Generation of Preferential Rules 115
4.3.2.6 Analysis of Evaluation Results 119
4.4 Summary 122
5 Hybrid of Rough Set Theory and Fuzzy Set Theory 125
5.1 The Basic Concepts of the Fuzzy Set Theory 126
5.1.1 Fuzzy Set and Fuzzy Membership Function 126
5.1.2 Operation of Fuzzy Subsets 128
5.1.3 Fuzzy Relation and Operation 131
5.1.4 Synthesis of Fuzzy Relations 132
5.1.5 λ-Cut Set and the Decomposition Proposition 133
5.1.6 The Fuzziness of Fuzzy Sets and Measure of Fuzziness 134
5.2 Rough Fuzzy Set and Fuzzy Rough Set 137
5.2.1 Rough Fuzzy Set 137
5.2.2 Fuzzy Rough Set 138
5.3 Variable Precision Rough Fuzzy Sets 138
5.3.1 Rough Membership Function Based on λ-Cut Set 139
5.3.2 The Rough Approximation of Variable Precision Rough Fuzzy Set 140
5.3.3 The Approximate Quality and Approximate Reduct of Variable Precision Rough Fuzzy Set 141
5.3.4 The Probabilistic Decision Rules Acquisition of Rough Fuzzy Decision Table 141
5.3.5 Algorithm Design 142
5.4 Variable Precision Fuzzy Rough Set 145
5.4.1 Fuzzy Equivalence Relation 145
5.4.2 Variable Precision Fuzzy Rough Model 146
5.4.3 Acquisition of Probabilistic Decision Rules in Fuzzy Rough Decision Table 148
5.4.4 Measure Methods of the Fuzzy Roughness for Output Classification 148
5.4.4.1 Distance Measurement 149
5.4.4.2 Entropy Measurement 149
5.5 Summary 152
6 Hybrid of Rough Set and Grey System 155
6.1 The Basic Concepts and Methods of the Grey System Theory 155
6.1.1 Grey Number, Whitening of Grey Number, and Grey Degree 156
6.1.1.1 Types of Grey Numbers 156
6.1.1.2 Whitenization of Grey Numbers and Grey Degree 157
6.1.2 Grey Sequence Generation 160
6.1.3 GM(1, 1) Model 162
6.1.4 Grey Correlation Analysis 165
6.1.5 Grey Correlation Order 172
6.1.6 Grey Clustering Evaluation 175
6.1.6.1 Clusters of Grey Correlation 175
6.1.6.2 Cluster with Variable Weights 176
6.1.6.3 Grey Cluster with Fixed Weights 182
6.2 Establishment of Decision Table Based on Grey Clustering 183
6.3 The Grade of Grey Degree of Grey Numbers and Grey Membership Function Based on Rough Membership Function 185
6.4 Grey Rough Approximations 188
6.5 Reduced Attributes Dominance Analysis Based on Grey Correlation Analysis 193
6.6 Summary 196
7 A Hybrid Approach of Variable Precision Rough Sets, Fuzzy Sets, and Neural Networks 199
7.1 Neural Network 200
7.1.1 An Overview of the Development of Neural Network 201
7.1.2 Structure and Types of Neural Network 202
7.1.3 Perceptron 204
7.1.3.1 Perceptron Neuron Model 205
7.1.3.2 Network Structure of Perceptron Neutral Network 206
7.1.3.3 Learning Rules of Perceptron Neutral Network 207
7.1.4 Back Propagation Network 208
7.1.4.1 BP Neuron Model 208
7.1.4.2 Network Structure of BP Neutral Network 209
7.1.4.3 BP Algorithm 210
7.1.5 Radial Basis Networks 212
7.1.5.1 Radial Basis Neurons Model 212
7.1.5.2 The Network Structure of the RBF 214
7.1.5.3 Realization of the Algorithm of RBF Neural Network 215
7.1.6 Probabilistic Neural Network 217
7.1.6.1 PNN Structure 217
7.1.6.2 Realization of PNN Algorithm 219
7.2 Knowledge Discovery in Databases Based on the Hybrid of VPRS and Neural Network 220
7.2.1 Collection, Selection, and Pretreatment of the Data 222
7.2.2 Construction of Decision Table 222
7.2.3 Searching of β-Reduct and Generation of Probability Decision Rules 223
7.2.3.1 Searching of β-Reduct 227
7.2.3.2 Learning and Simulation of the Neural Network 227
7.3 System Design Methods of the Hybrid of Variable Precision Rough Fuzzy Set and Neutral Network 230
7.3.1 Construction of a Variable Precision Rough Fuzzy Neutral Network 233
7.3.2 Training Algorithm of the Variable Precision Rough Fuzzy Neutral Network 237
7.4 Summary 238
8 Application Analysis of Hybrid Rough Set 239
8.1 A Survey of Transport Scheme Choice 239
8.2 Transport Scheme Choice Decision Undertaking No Consideration into Preference Information 241
8.2.1 Choice Decision Based on Rough Set 241
8.2.2 Probability Choice Decision Based on VPRS 241
8.2.3 Choice Decision Based on Grey Rough Set 243
8.2.4 Probability Choice Decision Based on the Hybrid of VPRS and Probabilistic Neural Network 243
8.3 Transport Scheme Choice Decision Undertaking Consideration into Preference Information 245
8.3.1 Choice Decision Based on the Dominance Rough Set 245
8.3.2 Choice Decision Based on the Dominance-Based VPRS 246
Bibliography 249
Index 261