# Fuzzy Measures and Integrals

## Presentation on theme: "Fuzzy Measures and Integrals"— Presentation transcript:

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

Fuzzy Measures Fuzzy Set versus Fuzzy Measure Fuzzy Set Fuzzy Measure
Underlying Set Vague boundary Crisp boundary Vague boundary: Probability of fuzzy set Representation Membership value of an element in X Degree of evidence or belief of an element that belongs to A in X Example Set 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
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 A X

How to calculate Belief
Example: Heterogeneous Evidence X B A 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

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

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

Sugeno Fuzzy Measure Fuzzy Density Function

Sugeno Fuzzy Measure How to construct Sugeno measure from fuzzy density

Fuzzy Integral Sugeno Integral

Fuzzy Integral Algorithm of Sugeno Integral

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