Presentation on theme: "Fuzzy Measures and Integrals 1. Fuzzy Measure 2. Belief and Plausibility Measure 3. Possibility and Necessity Measure 4. Sugeno Measure 5. Fuzzy Integrals."— Presentation transcript:
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 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
Fuzzy Measure Axiomatic Definition of Fuzzy Measure Note:
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.
Belief and Plausibility Measure Properties of Belief Measure Vacuous Belief: (Total Ignorance, No Evidence)
Belief and Plausibility Measure Plausibility Measure Other Definition Properties of Plausibility Measure
How to calculate Belief Basic Probability Assignment (BPA) Note
How to calculate Belief Calculation of Bel and Pl Simple Support Function is a BPA such that Bel from such Simple Support Function
How to calculate Belief Bel from total ignorance Body of Evidence
How to calculate Belief Dempster’s rule to combine two bodies of evidence Example: Homogeneous Evidence AX A X
How to calculate Belief Example: Heterogeneous Evidence AX B X
How to calculate Belief Example: Heterogeneous Evidence
Probability Measure Theorem: The followings are equivalent.
Joint and Marginal BoE Marginal BPA Example 7.2
Possibility and Necessity Measure Consonant Bel and Pl Measure
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:
Possibility and Necessity Measure Possibility Distribution
Possibility and Necessity Measure Basic Distribution and Possibility Distribution Ex.
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