EE 788 Robot Cognition and Planning Prof J

EE 788 Robot Cognition and Planning, Prof. J. -H. Kim Lecture 18. Fuzzy Measures n n n Belief Measure, Plausibility Measure, Probability Measure, Possibility Measure, Necessity Measure, -Fuzzy Measure, Fuzzy Integral Robot Intelligence Technology Lab.

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. Robot Intelligence Technology Lab. 2

Fuzzy Measure vs. Fuzzy Set Ex) Criminal trial: The jury members are uncertain about the guilt or innocence of the defendant. l Two crisp set: 1) the set of people who are guilty of the crime 2) the set of innocent people l The concern: - Not with the degree to which the defendant is guilty. - The degree to which the evidence proves his/her membership in either the 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. l 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 Robot Intelligence Technology Lab. 3

Fuzzy Measure vs. Fuzzy Set Note that where P(X) is a power set of X. Robot Intelligence Technology Lab. 4

Fuzzy Measure Def: A F. M. g(. ) is defined by a set ftn. that satisfies the following axioms: g 1: Boundary condition g(f) = 0, g(X) = 1 The element in question belongs to the universal set X. g 2: Monotonicity For every A, B then Robot Intelligence Technology Lab. , if , 5

Fuzzy Measure g 3: Continuity For every sequence if either or ( i. e. sequence is monotonic), then Robot Intelligence Technology Lab. of subset of X, 6

Fuzzy Measure A fuzzy measure is often defined more generally as a function: Def: Borel filed 1. 2. 3. is a family of subset of X: and If , then is closed under the operation of set union. (i. e. and , then ) Robot Intelligence Technology Lab. 7

Fuzzy Measure n Remarks: (a) Since and (by monotonic property, g 2), (b) Since and also (g 2) (c) The F. M. was introduced by Sugeno (’ 77) to exclude the “additivity” requirement of the standard measure, if , then Robot Intelligence Technology Lab. , that is, . 8

Fuzzy Measure Ex) Consider 3 poles/sticks of length 1, 2, and 3 (inch or meter). Let “length” measure be g(. ). Then, or depending on how to measure because the intersection of g(length=1) and g(length=2) is f. Robot Intelligence Technology Lab. 9

Belief Measure is a ftn. such that Bel: P(X) [0, 1] that satisfies axioms g 1 to g 3 and an additional g 4 (subadditivity axiom): g 4: for every and for every collection of subsets of X. Robot Intelligence Technology Lab. 10

Belief Measure Ex) n=2, Let Lower probability: Robot Intelligence Technology Lab. 11

Plausibility Measure Associated with belief measure is a plausibility measure (Schafer, ’ 76) defined as Similarly, Def: Plausibility Measure is a ftn. such that satisfies axioms g 1 to g 3 and an additional g 5: for every and every collection of subsets of X. Robot Intelligence Technology Lab. 12

Belief Measure For Let , , Upper Probability: Robot Intelligence Technology Lab. 13

Basic Probability Assignment Express both Belief M. and its dual Plausibility M. in terms of another set ftn. : such that Remarks: (a) m(A) is interpreted as the degree of evidence supporting the claim that a specific element belongs to the set A only and not to any special subset of A. is called basic probability assignment (or basic assignment). Robot Intelligence Technology Lab. 14

Basic Probability Assignment (b) It is not required that m(X) = 1 (c) It is not required that : g 1 (d) No relationship between (e) (f) is not a F. M. can be determined from Robot Intelligence Technology Lab. uniquely: 15

Basic Probability Assignment Distinguish among m(A), Bel(A), and Pl(A). – m(A) characterizes the DOE (Degree of Evidence) or belief that the element x belongs to the set A alone. – Bel(A) represents the total evidence or belief that the element x belongs to the set A as well as to the various special subsets of A. – Pl(A) represents not only the total evidence that x belongs to the set A and its special subsets of set A and also the additional evidence or belief associated with sets that overlap with set A. Robot Intelligence Technology Lab. 16

Basic Probability Assignment Ex) Let the universal set X denote the set of all possible diseases P: pneumonia, B: bronchitis, E: emphysema 기관지염 기종 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 Robot Intelligence Technology Lab. where B: all the possible subset of A 17

Basic Probability Assignment Robot Intelligence Technology Lab. 18

Basic Probability Assignment Given Bel(·), find m(·) where |A-B| is the size of (A-B), size: cardinality of crisp set (A-B) Ex) Robot Intelligence Technology Lab. 19

Basic Probability Assignment Def: Every for which m(A) > 0 is called a focal element (↔ support in F. S. ) n Focal elements are subset of X on which the available evidence focuses. n When X is finite, m can be characterized by a list of focal elements A and its corresponding values of m(A) Form an order pair (F, m) which denotes a “body of evidence, ” where F denotes a set of focal elements and m its associated values of basic assignment. n § Total ignorance is expressed in terms of m(·) by m(X) = 1, m(f) = 0, and m(A) = 0 for all A ≠X Robot Intelligence Technology Lab. 20

Basic Probability Assignment Def: A basic assignment m is said to be simple support ftn. focused at A if only one subset 1. 2. 3. m(A) = s m(X) = 1 – s m(B) = 0 for all other sets, where Robot Intelligence Technology Lab. 21

Basic Probability Assignment Ex) A m(A) Bel(A) Pl(A) P 0. 05 0. 9 0. 1 0. 95 B 0 0 0. 8 0. 2 1 E 0. 05 0. 8 0. 2 0. 95 PUB 0. 15 0. 2 0. 95 0. 05 0. 8 PUE 0. 1 0. 2 1 0 0. 8 BUE 0. 05 0. 1 0. 95 0. 05 0. 9 PUBUE 0. 6 1 1 0 0 Robot Intelligence Technology Lab. Pl(A)+Pl(A)≥ 1 22

Basic Probability Assignment Ex) SET A m(A) Bel(A) Pl(A) {0} {a} 0 0. 1 0 0. 5 1 0. 9 {b} 0 0 0. 6 0. 4 1 {c} 0 0 0. 5 1 {d} 0. 2 0. 4 0. 6 0. 8 {a, b} 0 0. 1 0. 8 0. 2 0. 9 {a, c} 0. 1 0. 2 0. 8 {a, d} 0 0. 3 0. 7 {b, c} 0. 3 0. 7 {b, d} 0 0. 2 0. 8 {c, d} 0 0. 2 0. 9 0. 1 0. 8 {a, b, c} 0. 1 0. 6 0. 8 0. 2 0. 4 {a, b, d} 0. 2 0. 5 1 0 0. 5 {a, c, d} 0 0. 4 1 0 0. 6 {b, c, d} 0 0. 5 0. 9 0. 1 0. 5 {a, b, c, d} 0 1 1 0 0 Robot Intelligence Technology Lab. 23

Joint Basic Assignment n Given two basic assignments and on which are evidences from two independent sources. Find joint basic assignment. All the possible subsets Focus on set A Normalization factor for , and where Expert 1 (Observe A) Expert 2 and => Dempster’s rule of combination ( or Dempster-Shafter Theory, Shafer ’ 76, Pearl ’ 88) Robot Intelligence Technology Lab. 24

Joint Basic Assignment Ex) Focal elements A B Expert 1 0. 3 0. 6 i) Calculate 1 -K Expert 2 0. 3 0. 4 0. 3 Combined evidence ? conflict ii) Robot Intelligence Technology Lab. 25

Joint Basic Assignment Ex) Assume someone discovers an old painting by Raphael. Question 1) Painting done by Raphael (R) 2) Painting done by his disciples (D) 3) Painting is a counterfeit (C) A R D C 0. 05 0. 15 0 0. 05 0. 21 0. 01 0. 09 0. 84 0. 5 0. 66 0. 15 0. 1 0. 05 0. 2 0. 4 0. 12 0. 06 0. 34 0. 5 0. 16 0. 91 0. 99 0. 79 0. 6 1 0. 5 1 0. 31 1 1 Robot Intelligence Technology Lab. 26

Joint Basic Assignment All the possible subsets such that intersection = 0: Robot Intelligence Technology Lab. 27

Probability Measure - Add the additivity requirement to the belief measures - Recall that Defn. : A Probability Measure (Bayesian belief measure) is a ftn. that satisfies g 1~g 3 of F. M. and the following axiom (g 6): Robot Intelligence Technology Lab. 28

Probability Measure Thm 1. A belief measure Bel(. ) on a finite power set P(X) is a probability measure iff its basic assignment m(. ) is given by m({x})=Bel({x}) and m(A) = 0 for all subsets of X that are not singletons. Explanation : A is a singleton. If Basic assignment is a p. d. f. Robot Intelligence Technology Lab. 29

Probability Measure l As a result, we can define a function p in the elements of X instead of the subsets of X: such that p(x) = m({x}), where this function p(x) is called the probability distribution function (pdf). l If m(·) focuses on singletons, then we have Robot Intelligence Technology Lab. 30

Probability Measure l If we denote Probability Measure as P, then l Total ignorance - For m(·), - For p(x), Remark: F. M. → (Belief Measure) & (Plausibility Measure) Special subclass of B. M. → Necessity Measure Special subclass of Pl. M. → Possibility Measure B. M. N. M. Robot Intelligence Technology Lab. Pl. M. P. M. Poss. M. 31

Possibility and Necessity Measures n Concept of nested sets: l A family of subsets of a universal set X is nested if these subsets can be ordered in such a way that each is contained within the next. Thus, are nested sets. Robot Intelligence Technology Lab. 32

Possibility and Necessity Measures n n When the focal elements of a body of evidence (F, m) are nested, the associated belief and plausibility measures are called consonant. There are not conflicts in a consonant body of evidence that is, degrees of evidence do not conflict with each other. Thm 2: Given a consonant body of evidence the associated consonant belief and plausibility measures possess the following properties: ① ② Robot Intelligence Technology Lab. 33

Possibility and Necessity Measures Ex) m(·) Bel(·) Pl(·) {p} 0 0 0. 6 {q} 0. 15 1 {r} 0 0 0. 7 {s} 0 0 0. 85 {p, q} 0 0. 15 1 {p, r} 0 0 0. 7 {p, s} 0 0 0. 85 {q, r} 0 0. 15 1 {q, s} 0. 15 0. 3 1 {r, s} 0 0 0. 85 {p, q, r} 0 0. 15 1 {p, q, s} 0 0. 3 1 {q, r, s} 0. 1 0. 4 1 {p, r, s} 0 0 0. 85 {p, q, r, s} 0. 6 1 1 Robot Intelligence Technology Lab. Focal elements: {q}, {q, s}, {q, r, s}, {p, q, r, s} Consonant body of evidence 34

Possibility and Necessity Measures Robot Intelligence Technology Lab. 35

Possibility and Necessity Measures - Consonant Belief Measure Necessity M. , N(A) Consonant Plausibility Measure Possibility M. , - From the previous theorem, - Recall that Robot Intelligence Technology Lab. 36

Possibility and Necessity Measures Properties: x is necessarily not in - Like probability, we can define a ftn. on the elements of X. Thm. 3: Every Possibility Measure on can be uniquely determined by a possibility distribution ftn. using the formula Robot Intelligence Technology Lab. 37

Possibility and Necessity Measures Q. E. D. Robot Intelligence Technology Lab. 38

Possibility and Necessity Measures Ex. ) X={0, 1, 2, ……, 10} x 0 1 2 3 4 0 0 0 5 6 7 0. 1 0. 5 0. 8 8 1 9 10 0. 8 0. 5 is the possibility distribution ftn. that x is close to 8. and N(A) are Possi. M. and N. M. that a set A contains an integer close to 8. When A={2, 5, 9}, Robot Intelligence Technology Lab. 39

Clarification on Terminology n Some clarification on terminology l pdf (probability distribution function): l df (possibility distribution function): (A) Relationship between and (B) Possibility theory (Zadeh); Relationship between and Robot Intelligence Technology Lab. 40

(A) Relationship between and - We can order the elements in X Robot Intelligence Technology Lab. 41

- Let denote the set of all ordered possibility distribution of length n (# of elements=n), and let - Given two possibility distribution of length n then we define Robot Intelligence Technology Lab. 42

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Ex) Robot Intelligence Technology Lab. 44

- Define a basic distribution: - Remarks ① ② ③ Robot Intelligence Technology Lab. 45

Q: To find the relationship between R and M There exists a one-to-one correspondence between R & M. Robot Intelligence Technology Lab. 46

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Ex) 0 0. 3 0. 4 0 0 0. 1 0. 2 1 1 0. 7 0. 3 0. 2 A Robot Intelligence Technology Lab. 48

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Then there exists a 1 -1 correspondence (mapping), pm Robot Intelligence Technology Lab. 51

-Fuzzy Measure (Sugeno Measure) n n Axioms g 1 to g 3 of Fuzzy Measures Another axiom: - By this axiom, we can calculate Robot Intelligence Technology Lab. 52

Sugeno Measure, - For for ( -fuzzy measure) , Sugeno’s measures are Brief Measures. , Sugeno’s measures are Plausibility Measures. , Sugeno’s measures are Probability Measures. - n Similar to an ordinary integral, a Lebesque integral, defined based on “measures, ” is used to define a F. I. using F. M. Robot Intelligence Technology Lab. 53

Fuzzy Integrals Def: Let h be a mapping from X to [0, 1]. The fuzzy integral of h over a subset A X with respect to a fuzzy measure g is defined: Where , and A: the domain of integration Robot Intelligence Technology Lab. 54

Fuzzy Integrals Ex. ) If h = a [0, 1] is a constant, Properties (Monotonicity): 1. If A 2. If B, then (by axiom g 2 of F. M. ) Robot Intelligence Technology Lab. 55

Fuzzy Integrals Lebesque Integral (u: Lebesque measure) 1. If , then 2. (by the additivity of Lebesque measures) n Fuzzy Integrals are the generalization of Lebesque integrals. Robot Intelligence Technology Lab. 56

Fuzzy Integrals n Let X be a finite set, w. l. o. g. , assume , then where Robot Intelligence Technology Lab. 57

Fuzzy Integrals 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. Robot Intelligence Technology Lab. 58

Fuzzy Integrals 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. Robot Intelligence Technology Lab. 59

Fuzzy Integrals - 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. Robot Intelligence Technology Lab. 60

Fuzzy Measures of Fuzzy Sets Def: Fuzzy measure of the fuzzy set where - : -cut of A, and is : power set of fuzzy sets of X g(A): i) Grade of certainty of the fuzzy event “x belongs to the fuzzy set A” ii) Grade of membership of x in the fuzzy set of elements which “more or less surely belong” to A (by Dubois and Prade ’ 78) iii) Recall : grade of membership of x in the fuzzy set A Robot Intelligence Technology Lab. 61