{"product_id":"uncertainty-and-information-9780471748670","title":"Uncertainty and Information","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThe principle aim of this book is to provide readers with a comprehensive and in-depth overview of GIT research to create a unified framework of the technology's diverse uncertainty theories.   Through the use of examples and exercises, the author discusses probability theory, fuzzy set theory, evidence theory and possibility theory.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"..will establish a better understanding of the complex concepts…will make significant contributions toward stimulating research in the area of generalized information theory.\" (\u003ci\u003eComputing Reviews.com\u003c\/i\u003e, October 17, 2006)  \u003cp\u003e\"…contains comprehensive and up-to-date coverage…can serve as a graduate-level text and a reference for researchers and practitioners…\" (\u003ci\u003eIEEE Computer Magazine\u003c\/i\u003e, February 2006)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003eAcknowledgments xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1. Uncertainty and Its Significance 1\u003c\/p\u003e \u003cp\u003e1.2. Uncertainty-Based Information 6\u003c\/p\u003e \u003cp\u003e1.3. Generalized Information Theory 7\u003c\/p\u003e \u003cp\u003e1.4. Relevant Terminology and Notation 10\u003c\/p\u003e \u003cp\u003e1.5. An Outline of the Book 20\u003c\/p\u003e \u003cp\u003eNotes 22\u003c\/p\u003e \u003cp\u003eExercises 23\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Classical Possibility-Based Uncertainty Theory 26\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1. Possibility and Necessity Functions 26\u003c\/p\u003e \u003cp\u003e2.2. Hartley Measure of Uncertainty for Finite Sets 27\u003c\/p\u003e \u003cp\u003e2.2.1. Simple Derivation of the Hartley Measure 28\u003c\/p\u003e \u003cp\u003e2.2.2. Uniqueness of the Hartley Measure 29\u003c\/p\u003e \u003cp\u003e2.2.3. Basic Properties of the Hartley Measure 31\u003c\/p\u003e \u003cp\u003e2.2.4. Examples 35\u003c\/p\u003e \u003cp\u003e2.3. Hartley-Like Measure of Uncertainty for Infinite Sets 45\u003c\/p\u003e \u003cp\u003e2.3.1. Definition 45\u003c\/p\u003e \u003cp\u003e2.3.2. Required Properties 46\u003c\/p\u003e \u003cp\u003e2.3.3. Examples 52\u003c\/p\u003e \u003cp\u003eNotes 56\u003c\/p\u003e \u003cp\u003eExercises 57\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Classical Probability-Based Uncertainty Theory 61\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1. Probability Functions 61\u003c\/p\u003e \u003cp\u003e3.1.1. Functions on Finite Sets 62\u003c\/p\u003e \u003cp\u003e3.1.2. Functions on Infinite Sets 64\u003c\/p\u003e \u003cp\u003e3.1.3. Bayes’ Theorem 66\u003c\/p\u003e \u003cp\u003e3.2. Shannon Measure of Uncertainty for Finite Sets 67\u003c\/p\u003e \u003cp\u003e3.2.1. Simple Derivation of the Shannon Entropy 69\u003c\/p\u003e \u003cp\u003e3.2.2. Uniqueness of the Shannon Entropy 71\u003c\/p\u003e \u003cp\u003e3.2.3. Basic Properties of the Shannon Entropy 77\u003c\/p\u003e \u003cp\u003e3.2.4. Examples 83\u003c\/p\u003e \u003cp\u003e3.3. Shannon-Like Measure of Uncertainty for Infinite Sets  91\u003c\/p\u003e \u003cp\u003eNotes  95\u003c\/p\u003e \u003cp\u003eExercises  97\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Generalized Measures and Imprecise Probabilities 101\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1. Monotone Measures 101\u003c\/p\u003e \u003cp\u003e4.2. Choquet Capacities 106\u003c\/p\u003e \u003cp\u003e4.2.1. Möbius Representation 107\u003c\/p\u003e \u003cp\u003e4.3. Imprecise Probabilities: General Principles 110\u003c\/p\u003e \u003cp\u003e4.3.1. Lower and Upper Probabilities 112\u003c\/p\u003e \u003cp\u003e4.3.2. Alternating Choquet Capacities 115\u003c\/p\u003e \u003cp\u003e4.3.3. Interaction Representation 116\u003c\/p\u003e \u003cp\u003e4.3.4. Möbius Representation 119\u003c\/p\u003e \u003cp\u003e4.3.5. Joint and Marginal Imprecise Probabilities 121\u003c\/p\u003e \u003cp\u003e4.3.6. Conditional Imprecise Probabilities 122\u003c\/p\u003e \u003cp\u003e4.3.7. Noninteraction of Imprecise Probabilities 123\u003c\/p\u003e \u003cp\u003e4.4. Arguments for Imprecise Probabilities 129\u003c\/p\u003e \u003cp\u003e4.5. Choquet Integral 133\u003c\/p\u003e \u003cp\u003e4.6. Unifying Features of Imprecise Probabilities 135\u003c\/p\u003e \u003cp\u003eNotes 137\u003c\/p\u003e \u003cp\u003eExercises 139\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Special Theories of Imprecise Probabilities 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1. An Overview 143\u003c\/p\u003e \u003cp\u003e5.2. Graded Possibilities 144\u003c\/p\u003e \u003cp\u003e5.2.1. Möbius Representation 149\u003c\/p\u003e \u003cp\u003e5.2.2. Ordering of Possibility Profiles 151\u003c\/p\u003e \u003cp\u003e5.2.3. Joint and Marginal Possibilities 153\u003c\/p\u003e \u003cp\u003e5.2.4. Conditional Possibilities 155\u003c\/p\u003e \u003cp\u003e5.2.5. Possibilities on Infinite Sets 158\u003c\/p\u003e \u003cp\u003e5.2.6. Some Interpretations of Graded Possibilities 160\u003c\/p\u003e \u003cp\u003e5.3. Sugeno l-Measures 160\u003c\/p\u003e \u003cp\u003e5.3.1. Möbius Representation 165\u003c\/p\u003e \u003cp\u003e5.4. Belief and Plausibility Measures 166\u003c\/p\u003e \u003cp\u003e5.4.1. Joint and Marginal Bodies of Evidence 169\u003c\/p\u003e \u003cp\u003e5.4.2. Rules of Combination 170\u003c\/p\u003e \u003cp\u003e5.4.3. Special Classes of Bodies of Evidence 174\u003c\/p\u003e \u003cp\u003e5.5. Reachable Interval-Valued Probability Distributions 178\u003c\/p\u003e \u003cp\u003e5.5.1. Joint and Marginal Interval-Valued Probability Distributions 183\u003c\/p\u003e \u003cp\u003e5.6. Other Types of Monotone Measures 185\u003c\/p\u003e \u003cp\u003eNotes 186\u003c\/p\u003e \u003cp\u003eExercises 190\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Measures of Uncertainty and Information 196\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1. General Discussion 196\u003c\/p\u003e \u003cp\u003e6.2. Generalized Hartley Measure for Graded Possibilities 198\u003c\/p\u003e \u003cp\u003e6.2.1. Joint and Marginal U-Uncertainties 201\u003c\/p\u003e \u003cp\u003e6.2.2. Conditional U-Uncertainty 203\u003c\/p\u003e \u003cp\u003e6.2.3. Axiomatic Requirements for the U-Uncertainty 205\u003c\/p\u003e \u003cp\u003e6.2.4. U-Uncertainty for Infinite Sets 206\u003c\/p\u003e \u003cp\u003e6.3. Generalized Hartley Measure in Dempster–Shafer Theory 209\u003c\/p\u003e \u003cp\u003e6.3.1. Joint and Marginal Generalized Hartley Measures 209\u003c\/p\u003e \u003cp\u003e6.3.2. Monotonicity of the Generalized Hartley Measure 211\u003c\/p\u003e \u003cp\u003e6.3.3. Conditional Generalized Hartley Measures 213\u003c\/p\u003e \u003cp\u003e6.4. Generalized Hartley Measure for Convex Sets of Probability Distributions 214\u003c\/p\u003e \u003cp\u003e6.5. Generalized Shannon Measure in Dempster-Shafer Theory 216\u003c\/p\u003e \u003cp\u003e6.6. Aggregate Uncertainty in Dempster–Shafer Theory 226\u003c\/p\u003e \u003cp\u003e6.6.1. General Algorithm for Computing the Aggregate Uncertainty 230\u003c\/p\u003e \u003cp\u003e6.6.2. Computing the Aggregated Uncertainty in Possibility Theory 232\u003c\/p\u003e \u003cp\u003e6.7. Aggregate Uncertainty for Convex Sets of Probability Distributions 234\u003c\/p\u003e \u003cp\u003e6.8. Disaggregated Total Uncertainty 238\u003c\/p\u003e \u003cp\u003e6.9. Generalized Shannon Entropy 241\u003c\/p\u003e \u003cp\u003e6.10. Alternative View of Disaggregated Total Uncertainty 248\u003c\/p\u003e \u003cp\u003e6.11. Unifying Features of Uncertainty Measures 253\u003c\/p\u003e \u003cp\u003eNotes 253\u003c\/p\u003e \u003cp\u003eExercises 255\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Fuzzy Set Theory 260\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1. An Overview 260\u003c\/p\u003e \u003cp\u003e7.2. Basic Concepts of Standard Fuzzy Sets 262\u003c\/p\u003e \u003cp\u003e7.3. Operations on Standard Fuzzy Sets 266\u003c\/p\u003e \u003cp\u003e7.3.1. Complementation Operations 266\u003c\/p\u003e \u003cp\u003e7.3.2. Intersection and Union Operations 267\u003c\/p\u003e \u003cp\u003e7.3.3. Combinations of Basic Operations 268\u003c\/p\u003e \u003cp\u003e7.3.4. Other Operations 269\u003c\/p\u003e \u003cp\u003e7.4. Fuzzy Numbers and Intervals 270\u003c\/p\u003e \u003cp\u003e7.4.1. Standard Fuzzy Arithmetic 273\u003c\/p\u003e \u003cp\u003e7.4.2. Constrained Fuzzy Arithmetic 274\u003c\/p\u003e \u003cp\u003e7.5. Fuzzy Relations 280\u003c\/p\u003e \u003cp\u003e7.5.1. Projections and Cylindric Extensions 281\u003c\/p\u003e \u003cp\u003e7.5.2. Compositions, Joins, and Inverses 284\u003c\/p\u003e \u003cp\u003e7.6. Fuzzy Logic 286\u003c\/p\u003e \u003cp\u003e7.6.1. Fuzzy Propositions 287\u003c\/p\u003e \u003cp\u003e7.6.2. Approximate Reasoning 293\u003c\/p\u003e \u003cp\u003e7.7. Fuzzy Systems 294\u003c\/p\u003e \u003cp\u003e7.7.1. Granulation 295\u003c\/p\u003e \u003cp\u003e7.7.2. Types of Fuzzy Systems 297\u003c\/p\u003e \u003cp\u003e7.7.3. Defuzzification 298\u003c\/p\u003e \u003cp\u003e7.8. Nonstandard Fuzzy Sets 299\u003c\/p\u003e \u003cp\u003e7.9. Constructing Fuzzy Sets and Operations 303\u003c\/p\u003e \u003cp\u003eNotes 305\u003c\/p\u003e \u003cp\u003eExercises 308\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Fuzzification of Uncertainty Theories 315\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1. Aspects of Fuzzification 315\u003c\/p\u003e \u003cp\u003e8.2. Measures of Fuzziness 321\u003c\/p\u003e \u003cp\u003e8.3. Fuzzy-Set Interpretation of Possibility Theory 326\u003c\/p\u003e \u003cp\u003e8.4. Probabilities of Fuzzy Events 334\u003c\/p\u003e \u003cp\u003e8.5. Fuzzification of Reachable Interval-Valued Probability Distributions 338\u003c\/p\u003e \u003cp\u003e8.6. Other Fuzzification Efforts 348\u003c\/p\u003e \u003cp\u003eNotes 350\u003c\/p\u003e \u003cp\u003eExercises 351\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Methodological Issues 355\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1. An Overview 355\u003c\/p\u003e \u003cp\u003e9.2. Principle of Minimum Uncertainty 357\u003c\/p\u003e \u003cp\u003e9.2.1. Simplification Problems 358\u003c\/p\u003e \u003cp\u003e9.2.2. Conflict-Resolution Problems 364\u003c\/p\u003e \u003cp\u003e9.3. Principle of Maximum Uncertainty 369\u003c\/p\u003e \u003cp\u003e9.3.1. Principle of Maximum Entropy 369\u003c\/p\u003e \u003cp\u003e9.3.2. Principle of Maximum Nonspecificity 373\u003c\/p\u003e \u003cp\u003e9.3.3. Principle of Maximum Uncertainty in GIT 375\u003c\/p\u003e \u003cp\u003e9.4. Principle of Requisite Generalization 383\u003c\/p\u003e \u003cp\u003e9.5. Principle of Uncertainty Invariance 387\u003c\/p\u003e \u003cp\u003e9.5.1. Computationally Simple Approximations 388\u003c\/p\u003e \u003cp\u003e9.5.2. Probability–Possibility Transformations 390\u003c\/p\u003e \u003cp\u003e9.5.3. Approximations of Belief Functions by Necessity Functions 399\u003c\/p\u003e \u003cp\u003e9.5.4. Transformations Between l-Measures and Possibility Measures 402\u003c\/p\u003e \u003cp\u003e9.5.5. Approximations of Graded Possibilities by Crisp Possibilities 403\u003c\/p\u003e \u003cp\u003eNotes 408\u003c\/p\u003e \u003cp\u003eExercises 411\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Conclusions 415\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1. Summary and Assessment of Results in Generalized Information Theory 415\u003c\/p\u003e \u003cp\u003e10.2. Main Issues of Current Interest 417\u003c\/p\u003e \u003cp\u003e10.3. Long-Term Research Areas 418\u003c\/p\u003e \u003cp\u003e10.4. Significance of GIT 419\u003c\/p\u003e \u003cp\u003eNotes 421\u003c\/p\u003e \u003cp\u003eAppendix A Uniqueness of the U-Uncertainty 425\u003c\/p\u003e \u003cp\u003eAppendix B Uniqueness of Generalized Hartley Measure in the Dempster–Shafer Theory 430\u003c\/p\u003e \u003cp\u003eAppendix C Correctness of Algorithm 6.1 437\u003c\/p\u003e \u003cp\u003eAppendix D Proper Range of Generalized Shannon Entropy 442\u003c\/p\u003e \u003cp\u003eAppendix E Maximum of GSa in Section 6.9 447\u003c\/p\u003e \u003cp\u003eAppendix F Glossary of Key Concepts 449\u003c\/p\u003e \u003cp\u003eAppendix G Glossary of Symbols 455\u003c\/p\u003e \u003cp\u003eBibliography 458\u003c\/p\u003e \u003cp\u003eSubject Index 487\u003c\/p\u003e \u003cp\u003eName Index 494\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49402668417367,"sku":"9780471748670","price":121.46,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780471748670.jpg?v=1730481172","url":"https:\/\/bookcurl.com\/products\/uncertainty-and-information-9780471748670","provider":"Book Curl","version":"1.0","type":"link"}