Presentation on theme: "Fuzzy Measures and Integrals"— Presentation transcript:
1 Fuzzy Measures and Integrals 2. Belief and Plausibility Measure3. Possibility and Necessity Measure4. Sugeno Measure5. Fuzzy Integrals
2 Fuzzy Measures Fuzzy Set versus Fuzzy Measure Fuzzy Set Fuzzy Measure UnderlyingSetVague boundaryCrisp boundaryVague boundary: Probability of fuzzy setRepresentationMembership value of an element in XDegree of evidence orbelief of an element that belongs to A in XExampleSet of large numberA degree of defection of a treeDegree of Evidence orBelief of an object that is tree
3 Fuzzy MeasureAxiomatic Definition of Fuzzy MeasureNote:
4 Belief and Plausibility Measure Belief MeasureNote: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 MeasureVacuous Belief: (Total Ignorance, No Evidence)
6 Belief and Plausibility Measure Other DefinitionProperties of Plausibility Measure
7 How to calculate Belief Basic Probability Assignment (BPA)Note
8 How to calculate Belief Calculation of Bel and PlSimple Support Function is a BPA such thatBel from such Simple Support Function
9 How to calculate Belief Bel from total ignoranceBody of Evidence
10 How to calculate Belief Dempster’s rule to combine two bodies of evidenceExample: Homogeneous EvidenceAX
11 How to calculate Belief Example: Heterogeneous EvidenceXBAX
12 How to calculate Belief Example: Heterogeneous Evidence
13 Probability MeasureTheorem: The followings are equivalent.
15 Possibility and Necessity Measure Consonant Bel and Pl Measure
16 Possibility and Necessity Measure Necessity and Possibility MeasureConsonant Body of EvidenceBelief Measure -> Necessity MeasurePlausibility Measure -> Possibility MeasureExtreme case of fuzzy measureNote:
17 Possibility and Necessity Measure Possibility Distribution
18 Possibility and Necessity Measure Basic Distribution and Possibility DistributionEx.
19 Fuzzy Set and Possibility InterpretationDegree of Compatibility of v with the concept FDegree of Possibility when V=v of the proposition p: V is FPossibility MeasureExample