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Fuzzy Measures and Integrals

Fuzzy Measures and Integrals. 1. Fuzzy Measure 2. Belief and Plausibility Measure 3. Possibility and Necessity Measure 4. Sugeno Measure 5. Fuzzy Integrals. Fuzzy Measures. Fuzzy Set versus Fuzzy Measure. Fuzzy Measure. Axiomatic Definition of Fuzzy Measure Note:.

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Fuzzy Measures and Integrals

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  1. Fuzzy Measures and Integrals 1. Fuzzy Measure 2. Belief and Plausibility Measure 3. Possibility and Necessity Measure 4. Sugeno Measure 5. Fuzzy Integrals

  2. Fuzzy Measures • Fuzzy Set versus Fuzzy Measure

  3. Fuzzy Measure • Axiomatic Definition of Fuzzy Measure • Note:

  4. Belief and Plausibility Measure • Belief Measure • Note: • Interpretation: Degree of evidence or certainty factor of an element in X that belongs to the crisp set A, a particular question. Some of answers are correct, but we don’t know because of the lack of evidence.

  5. Belief and Plausibility Measure • Properties of Belief Measure • Vacuous Belief: (Total Ignorance, No Evidence)

  6. Belief and Plausibility Measure • Plausibility Measure • Other Definition • Properties of Plausibility Measure

  7. How to calculate Belief • Basic Probability Assignment (BPA) • Note

  8. How to calculate Belief • Calculation of Bel and Pl • Simple Support Function is a BPA such that • Bel from such Simple Support Function

  9. How to calculate Belief • Bel from total ignorance • Body of Evidence

  10. X A A X How to calculate Belief • Dempster’s rule to combine two bodies of evidence • Example: Homogeneous Evidence

  11. How to calculate Belief • Example: Heterogeneous Evidence X B A X

  12. How to calculate Belief • Example: Heterogeneous Evidence • Example: Heterogeneous Evidence

  13. Probability Measure • Theorem: The followings are equivalent.

  14. Joint and Marginal BoE • Marginal BPA • Example 7.2

  15. Possibility and Necessity Measure • Consonant Bel and Pl Measure

  16. Possibility and Necessity Measure • Necessity and Possibility Measure • Consonant Body of Evidence • Belief Measure -> Necessity Measure • Plausibility Measure -> Possibility Measure • Extreme case of fuzzy measure • Note:

  17. Possibility and Necessity Measure • Possibility Distribution

  18. Possibility and Necessity Measure • Basic Distribution and Possibility Distribution • Ex.

  19. Fuzzy Set and Possibility • Interpretation • Degree of Compatibility of v with the concept F • Degree of Possibility when V=v of the proposition p: V is F • Possibility Measure • Example

  20. Summary Fuzzy Measure Plausibility Measure Probability Measure Possibility Measure Belief Measure Necessity Measure

  21. Sugeno Fuzzy Measure • Sugeno’s g-lamda measure • Note:

  22. Sugeno Fuzzy Measure • Fuzzy Density Function

  23. Sugeno Fuzzy Measure • How to construct Sugeno measure from fuzzy density

  24. Fuzzy Integral • Sugeno Integral

  25. Fuzzy Integral • Algorithm of Sugeno Integral

  26. Fuzzy Integral • Choquet Integral • Interpretation of Fuzzy Integrals in Multi-criteria Decision Making

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