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

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Fuzzy Measures Fuzzy Set versus Fuzzy Measure Fuzzy SetFuzzy Measure Underlying Set Vague boundaryCrisp boundary Vague boundary: Probability of fuzzy set RepresentationMembership value of an element in X Degree of evidence or belief of an element that belongs to A in X ExampleSet of large number A degree of defection of a tree Degree of Evidence or Belief of an object that is tree

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Fuzzy Measure Axiomatic Definition of Fuzzy Measure Note:

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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.

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

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Belief and Plausibility Measure Plausibility Measure Other Definition Properties of Plausibility Measure

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How to calculate Belief Basic Probability Assignment (BPA) Note

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

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How to calculate Belief Bel from total ignorance Body of Evidence

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

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How to calculate Belief Example: Heterogeneous Evidence AX B X

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How to calculate Belief Example: Heterogeneous Evidence

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Probability Measure Theorem: The followings are equivalent.

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Joint and Marginal BoE Marginal BPA Example 7.2

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Possibility and Necessity Measure Consonant Bel and Pl Measure

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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:

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Possibility and Necessity Measure Possibility Distribution

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Possibility and Necessity Measure Basic Distribution and Possibility Distribution Ex.

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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

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

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Sugeno Fuzzy Measure Sugeno’s g-lamda measure Note:

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Sugeno Fuzzy Measure Fuzzy Density Function

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Sugeno Fuzzy Measure How to construct Sugeno measure from fuzzy density

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Fuzzy Integral Sugeno Integral

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Fuzzy Integral Algorithm of Sugeno Integral

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

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