Fuzzy Measures and Integrals 1 2 3 4

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

Fuzzy Measures • Fuzzy Set versus Fuzzy Measure Fuzzy Set Underlying Set Representation Example Vague boundary Fuzzy Measure Crisp boundary Vague boundary: of fuzzy set Probability Membership value of an element in A Degree of evidence or belief of an element that belongs to A in X Set of large number Degree of Evidence or Belief of an object that is tree

Types of Uncertainty vagueness: fuzzy sets ambiguity: fuzzy measures Vagueness: associated with the difficulty of making sharp or precise distinctions in the world. Ambiguity: associated with one-to-many relations, i. e. difficult to make a choice between two or more alternatives.

Fuzzy Measure vs. Fuzzy Set Ex) Criminal trial: The jury members are uncertain about the guilt or innocence of the defendant. – Two crisp set: 1) the set of people who are guilty of the crime 2) the set of innocent people – The concern: - Not with the degree to which the defendant is guilty. - The degree to which the evidence proves his/her membership in either he crisp set of guilty people or in the crisp set of innocent people. - Our evidence is rarely, if ever, perfect, and some uncertainty usually prevails. – Fuzzy measure: to represent this type of uncertainty - Assign a value to each possible crisp set to which the element in question might belong, signifying the degree of evidence or belief that a particular element belongs in the set. - The degree of evidence, or certainty of the element’s membership in the set

Fuzzy Measure • Axiomatic Definition of Fuzzy Measure • Note:

Note that where P(X) is a power set of X.

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

Ex) Let the universal set X denote the set of all possible diseases P: pneumonia, B: bronchitis, E: emphysema 기관지염 m Bel P 0. 05 B 0 0 E 0. 05 PUB 0. 15 0. 2 PUE 0. 1 0. 2 BUE 0. 05 0. 1 PUBUE 0. 6 1 기종 where B: all the possible subset of A

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Given Bel(·), find m(·) where |A-B| is the size of (A-B), size: cardinality of crisp set (A-B) Ex)

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

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

How to calculate Belief • Example: Heterogeneous Evidence

Joint and Marginal Bo. E • 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 Belief Measure Necessity Measure Possibility 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

Ex) Evaluation of the desirability of houses Let , where =location and = price, = size, = facilities, = living environment, and evaluation function: where m: # of houses, and : evaluation value of attribute of the jth house, where Then, let the fuzzy measure: where g: degree of consideration (importance) of attributes in the evaluation process.

The desirability of the jth houses: General linear evaluation model: - Performs well only when the attributes of evaluation are independent and the measures of evaluation are independent.

- Practically, price ( ) and size ( ) are not independent. - Even if and are independent, the degree of consideration might not be independent, i. e. , Additivity might not be true for measures. - F. I. Models are more general than the linear models. - Problem about Fuzzy Integral Evaluation Model ① How to find out the necessary attributes for evaluation. ② How to identify the fuzzy measure.