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Random Sets and Random Fuzzy Sets As Ill-Perceived Random Variables : An Introduction for Ph. D. Students and Practitioners.

By: Contributor(s): Material type: TextTextSeries: SpringerBriefs in Applied Sciences and Technology SeriesPublisher: Cham : Springer International Publishing AG, 2014Copyright date: ©2014Edition: 1st edDescription: 1 online resource (104 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9783319086118
Subject(s): Genre/Form: Additional physical formats: Print version:: Random Sets and Random Fuzzy Sets As Ill-Perceived Random VariablesDDC classification:
  • 519.22
LOC classification:
  • Q342
Online resources:
Contents:
Intro -- Preface -- Contents -- 1 Introduction -- [DELETE] -- References -- 2 Random Sets as Ill-Perceived Random Variables -- 2.1 Introduction: Different Applications of Random Sets -- 2.1.1 Two Uses of Sets -- 2.1.2 Canonical Examples -- 2.1.3 Encoding Random Sets by Random Vectors -- 2.2 Basic Formal Definitions -- 2.2.1 Upper and Lower Inverses -- 2.2.2 Random Sets -- 2.2.3 Upper and Lower Probabilities -- 2.3 Three Sets of Probabilities Associated with a Random Set -- 2.4 Expectation and Variance of Random Sets -- 2.4.1 The Conjunctive Versus the Disjunctive Approach -- 2.4.2 Examples -- 2.4.3 Relating Aumann Expectation and Choquet Integral -- 2.5 Conditioning -- 2.5.1 Conditioning a Conjunctive Random Set -- 2.5.2 Conditioning an Ill-Known Random Variable: Prediction -- 2.5.3 Conditioning an Ill-Known Random Variable: Revision -- 2.6 Independence -- 2.6.1 Random Set Independence -- 2.6.2 Independence Between Ill-Known Variables -- 2.7 Exercises -- References -- 3 Random Fuzzy Sets as Ill-Perceived Random Variables -- 3.1 Introduction: Two Views of Fuzzy Random Variables -- 3.1.1 The Different Semantics of Fuzzy Sets -- 3.1.2 Different Interpretations of Fuzzy Random Variables -- 3.2 Prerequisites -- 3.2.1 Possibility and Necessity Measures -- 3.2.2 Fuzzy-Valued Probabilities of Events -- 3.3 The Acceptability Function of a Fuzzy Random Variable -- 3.4 Second-Order Possibility Measures Associated to a Fuzzy Random Variable -- 3.4.1 The Fuzzy Probability Envelope -- 3.4.2 The Fuzzy Probability of Events Induced by a Fuzzy Random Variable -- 3.4.3 The Fuzzy Credal Set Induced by an FRV -- 3.4.4 Statistical Parameters Under the Epistemic Approach -- 3.5 Discrepancies Between Fuzzy Random Variables and Their Associated Fuzzy Credal Sets -- 3.5.1 Probability Envelopes Versus Credal Sets: The Case of Random Sets -- 3.5.2 Ill-Known Entropy.
3.5.3 Probability Envelopes Versus Credal Sets: Fuzzy Random Variables -- 3.5.4 Probability Envelopes Versus Credal Sets: Extreme Discrepancies -- 3.5.5 Joint Probability Envelopes and Credal Sets of FRV's -- 3.5.6 Fuzzy Random Variables Versus Random Fuzzy Sets -- 3.5.7 A Summary Example -- 3.6 Fuzzy Random Variables as Crisp Credal Sets -- 3.7 Conclusion -- 3.8 Exercises -- References -- Solutions to Exercises.
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Intro -- Preface -- Contents -- 1 Introduction -- [DELETE] -- References -- 2 Random Sets as Ill-Perceived Random Variables -- 2.1 Introduction: Different Applications of Random Sets -- 2.1.1 Two Uses of Sets -- 2.1.2 Canonical Examples -- 2.1.3 Encoding Random Sets by Random Vectors -- 2.2 Basic Formal Definitions -- 2.2.1 Upper and Lower Inverses -- 2.2.2 Random Sets -- 2.2.3 Upper and Lower Probabilities -- 2.3 Three Sets of Probabilities Associated with a Random Set -- 2.4 Expectation and Variance of Random Sets -- 2.4.1 The Conjunctive Versus the Disjunctive Approach -- 2.4.2 Examples -- 2.4.3 Relating Aumann Expectation and Choquet Integral -- 2.5 Conditioning -- 2.5.1 Conditioning a Conjunctive Random Set -- 2.5.2 Conditioning an Ill-Known Random Variable: Prediction -- 2.5.3 Conditioning an Ill-Known Random Variable: Revision -- 2.6 Independence -- 2.6.1 Random Set Independence -- 2.6.2 Independence Between Ill-Known Variables -- 2.7 Exercises -- References -- 3 Random Fuzzy Sets as Ill-Perceived Random Variables -- 3.1 Introduction: Two Views of Fuzzy Random Variables -- 3.1.1 The Different Semantics of Fuzzy Sets -- 3.1.2 Different Interpretations of Fuzzy Random Variables -- 3.2 Prerequisites -- 3.2.1 Possibility and Necessity Measures -- 3.2.2 Fuzzy-Valued Probabilities of Events -- 3.3 The Acceptability Function of a Fuzzy Random Variable -- 3.4 Second-Order Possibility Measures Associated to a Fuzzy Random Variable -- 3.4.1 The Fuzzy Probability Envelope -- 3.4.2 The Fuzzy Probability of Events Induced by a Fuzzy Random Variable -- 3.4.3 The Fuzzy Credal Set Induced by an FRV -- 3.4.4 Statistical Parameters Under the Epistemic Approach -- 3.5 Discrepancies Between Fuzzy Random Variables and Their Associated Fuzzy Credal Sets -- 3.5.1 Probability Envelopes Versus Credal Sets: The Case of Random Sets -- 3.5.2 Ill-Known Entropy.

3.5.3 Probability Envelopes Versus Credal Sets: Fuzzy Random Variables -- 3.5.4 Probability Envelopes Versus Credal Sets: Extreme Discrepancies -- 3.5.5 Joint Probability Envelopes and Credal Sets of FRV's -- 3.5.6 Fuzzy Random Variables Versus Random Fuzzy Sets -- 3.5.7 A Summary Example -- 3.6 Fuzzy Random Variables as Crisp Credal Sets -- 3.7 Conclusion -- 3.8 Exercises -- References -- Solutions to Exercises.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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