Grey Systems Analysis : Methods, Models and Applications

The book covers up-to-date theoretical and applied advances in grey systems theory from across the world and vividly presents the reader with the overall picture of this new theory and its frontier research. Many of the concepts, models and methods in the book are original by the authors, including simplified form of grey number, general grey number and the operations of grey numbers; the axiomatic system of buffer operators and a series of weakening and strengthening operators; a series of grey relational analysis models, including grey absolute, relative, synthetic, similarity, closeness, negative and three dimension degree, etc.; grey fixed weight clustering model, grey evaluation models based on center-point and end-point mixed possibility functions; original difference grey model (ODGM), even difference grey model (EDGM), discrete grey model (DGM), fractional grey models, self-memory grey models; multi-attribute intelligent grey target decision models, weight vector group with kernel and the weighted comprehensive clustering coefficient vector, and spectrum analysis of sequence operators, etc. This book will be appropriate as a reference and/or professional book for courses of grey system theory for graduate students or high-level undergraduate students, majoring in areas of science, technology, agriculture, medicine, astronomy, earth science, economics, and management. It can also be utilized by researchers and practitioners in research institutions, business entities, and government agencies. Series Preface 7 Foreword by Dr. James M. Tien 9 Foreword by Dr. Keith William Hipel 11 Foreword by Dr. Hermann Haken 14 Foreword by Dr. Robert Vallée 16 Preface 18 Acknowledgements 20 Contents 22 1 Introduction 28 1.1 The Scientific Background of the Birth of Grey System Theory 28 1.2 The Founder of Grey System Theory 29 1.3 Development of Grey Systems Theory 30 1.3.1 Building a Basic Team 30 1.3.2 Establishment of Academic Organizations 31 1.3.3 Journals and Book Series on Grey System Theory 32 1.3.4 Grey System Theory Curriculums 33 1.3.5 Researchers of Grey System Theory Are All Over the World 33 1.3.6 Papers of Grey Systems Theory Are Growing Rapidly 34 1.4 Elementary Concepts of Grey System Theory 37 1.5 Fundamental Principles of Grey System Theory 38 1.6 Main Contents of Grey System Theory 40 References 41 2 Characteristics of Grey System Theory 43 2.1 A Kind of Poor Data Analysis Method with Strong Penetration 43 2.2 Characteristics of Uncertain Systems and the Simplicity Principle in Sciences 44 2.2.1 Incomplete Information 44 2.2.2 Inaccuracies in Data 45 2.2.3 The Scientific Principle of Simplicity 46 2.2.4 Precise Models Suffer from Inaccuracies 47 2.3 Comparison of Several Uncertainty Methods 49 2.4 Deep Applications of Grey System Theory in the Fields of Social Science, Natural Science and Engineering Technology 50 2.4.1 Successful Application of Grey System Theory in the Field of Social Sciences 50 2.4.2 Deep Application of Grey System Theory in the Field of Natural Science 51 2.4.3 A Large Number of Applications of Grey System Theory in the Field of Engineering Technology 55 References 59 3 Grey Numbers and Their Operations 64 3.1 Grey Numbers 64 3.2 The Whitenization of a Grey Number and Degree of Greyness 65 3.3 Degree of Greyness Defined by Axioms 68 3.4 The Operations of Interval Grey Numbers 70 3.5 General Grey Numbers and Their Operations 72 3.5.1 Reduced Form of Interval Grey Numbers 72 3.5.2 General Grey Number and Its Reduced Form 73 3.5.3 Synthesis of Degree of Greyness and Operations of General Grey Numbers 75 References 79 4 Sequence Operators and Grey Data Mining 80 4.1 Introduction 80 4.2 Systems Under Shocking Disturbances and Buffer Operators 82 4.2.1 The Trap for Shocking Disturbed System Forecasting 82 4.2.2 Axioms of Buffer Operators 83 4.2.3 Properties of Buffer Operators 84 4.3 Construction of Practically Useful Buffer Operators 86 4.3.1 Weakening Buffer Operators 86 4.3.2 Strengthening Buffer Operators 89 4.3.3 The General Form of Buffer Operator 90 4.4 Average Operator 92 4.5 The Quasi-Smooth Sequence and Stepwise Ratio Operator 93 4.6 Accumulating and Inverse Accumulating Operators 95 4.7 Exponentiality of Accumulating Sequence 97 References 101 5 Grey Relational Analysis Models 102 5.1 Introduction 102 5.2 Grey Relational Factors and Set of Grey Relational Operators 104 5.3 Grey Relational Axioms and Deng’s Grey Relational Analysis Model 107 5.4 Grey Absolute Relational Degree 111 5.5 Grey Relative and Synthetic Relational Degree 116 5.5.1 Relative Grey Relational Degree 116 5.5.2 Grey Synthetic Relational Degree 119 5.6 Grey Similarity, Closeness and Three-Dimensional Relational Degree 120 5.6.1 Grey Relational Analysis Models Based on Similarity and Closeness 120 5.6.2 Grey Three-Dimension Degree of Relational Degree 124 5.7 Negative Grey Relational Analysis Models 126 5.8 Superiority Analysis 134 5.9 Practical Application 140 References 148 6 Grey Clustering Evaluation Models 150 6.1 Introduction 150 6.2 Grey Relational Clustering Model 151 6.3 Common Possibility Functions 155 6.4 Variable Weight Grey Clustering Model 157 6.5 Fixed Weight Grey Clustering Model 161 6.6 Grey Clustering Evaluation Models Based on Mixed Possibility Functions 165 6.6.1 Grey Clustering Evaluation Model Based on End-Point Mixed Possibility Functions 165 6.6.2 Grey Clustering Evaluation Model Based on Center-Point Mixed Possibility Functions 167 6.7 Practical Applications 169 References 175 7 Series of GM Models 177 7.1 Introduction 177 7.2 The Four Basic Forms of GM(1,1) 178 7.2.1 The Basic Forms of Model GM(1,1) 178 7.2.2 Properties and Characteristics of the Basic Model 180 7.3 Suitable Ranges of Different GM(1,1) 185 7.3.1 Suitable Sequences of Different GM(1,1) 185 7.3.2 Applicable Ranges of EGM 193 7.4 Remnant GM(1,1) Model 196 7.5 Group of GM(1,1) Models 201 7.6 The Fractional Grey Model 204 7.7 The Models of GM(r,h) 207 7.7.1 The Model of GM(0,N) 207 7.7.2 The Model of GM(1, N) 209 7.7.3 The Grey Verhulst Model 211 7.7.4 The Self-memory Grey Model 214 7.7.5 The Models of GM(r,h) 216 7.8 Practical Applications 218 References 225 8 Combined Grey Models 227 8.1 Grey Econometrics Models 227 8.1.1 Determination of Variables Using the Grey Relational Principles 227 8.1.2 Grey Econometrics Models 228 8.2 Combined Grey Linear Regression Models 231 8.3 Grey Cobb–Douglas Model 234 8.4 Grey Artificial Neural Network Models 235 8.4.1 BP Artificial Neural Model and Computational Schemes 235 8.4.2 Steps in Grey BP Neural Network Modeling 236 8.5 Grey Markov Model 238 8.5.1 Grey Moving Probability Markov Model 238 8.5.2 Grey State Markov Model 239 8.6 Combined Grey-Rough Model 241 8.6.1 Rough Membership, Grey Membership and Grey Numbers 241 8.6.2 Grey Rough Approximation 243 8.6.3 Combined Grey Clustering and Rough Set Model 246 8.7 Practical Applications 247 References 252 9 Techniques for Grey Systems Forecasting 253 9.1 Introduction 253 9.2 Interval Forecasting 255 9.3 Grey Distortion Forecasting 259 9.4 Wave Form Forecasting 262 9.5 System Forecasting 264 9.5.1 The Five-Step Modeling Process 264 9.5.2 System Models for Prediction 266 9.6 Practical Applications 267 References 269 10 Grey Models for Decision-Making 271 10.1 Introduction 271 10.2 Grey Target Decisions 273 10.3 Other Approaches to Grey Decision 278 10.3.1 Grey Relational Decision 278 10.3.2 Grey Development Decision 282 10.3.3 Grey Clustering Decision 284 10.4 Multi-attribute Intelligent Grey Target Decision Model 285 10.4.1 The Uniform Effect Measure 286 10.4.2 The Weighted Synthetic Effect Measure 288 10.5 On Paradox of Rule of Maximum Value and Its Solution 291 10.5.1 The Weight Vector Group with Kernel 292 10.5.2 The Weighted Comprehensive Clustering Coefficient Vector 294 10.5.3 Several Functional Weight Vector Groups with Kernel 295 10.6 Practical Applications 296 References 298 11 Grey Control Systems 300 11.1 Introduction 300 11.2 Controllability and Observability of Grey System 301 11.3 Transfer Functions of Grey System 303 11.3.1 Grey Transfer Function 303 11.3.2 Transfer Functions of Typical Links 304 11.3.3 Matrices of Grey Transfer Functions 308 11.4 Robust Stability of Grey System 309 11.4.1 Robust Stability of Grey Linear Systems 309 11.4.2 Robust Stability of Grey Linear Time-Delay Systems 311 11.4.3 Robust Stability of Grey Stochastic Linear Time-Delay System 314 11.5 Several Typical Grey Control Models 318 11.5.1 Control of Redundancy Removal 319 11.5.2 Grey Relational Control 319 11.5.3 Control of Grey Prediction 320 References 325 12 Spectrum Analysis of Sequence Operators 327 12.1 Introduction 327 12.2 Spectrum Analysis of Time Series Data 328 12.3 Filtering Effect of Mean Operator and Accumulation Operator 330 12.3.1 Filtering Effect of Mean Operator 330 12.3.2 Filtering Effect of Accumulation Operator 331 12.3.3 Filtering Effect of Series Operator 333 12.4 Spectrum Analysis of Buffer Operator 334 References 336 Appendix Introduction to Grey Systems Modeling Software 337 A.1 Introduction 337 A.2 Software Features and Functions 338 A.3 Main Components 340 A.4 Operation Guide 341 A.4.1 The Confirmation System 342 A.4.2 Using the Software Package 344 Memorabilia of the Establishment and Development of Grey System Theory (1982–2021) 353 Farewell to Our Tutor 356 Bibliography 359 Index 378