On possibilistic mean value and variance of fuzzy numbers

Transcription

1 On possibilistic mean value and variance of fuzzy numbers Supported by the E-MS Bullwhip project TEKES 4965/98 Christer Carlsson Institute for Advanced Management Systems Research, Robert Fullér Institute for Advanced Management Systems Research, TUCS Turku Centre for Computer Science TUCS Technical Report No 99 August 1999 ISBN ISSN

2 Abstract Dubois and Prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions. In this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval-valued possibilistic mean and investigate its relationship to the interval-valued probabilistic mean. We also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions, which are consistent with the extension principle. We also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory. Keywords: Fuzzy number, possibility distribution, mean value, variance TUCS Research Group Institute for Advanced Management Systems Research

3 Keywords: Fuzzy numbers; Possibility theory 1 Introduction In 1987 Dubois and Prade [] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers. In this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval-valued possibilistic mean, crisp possibilistic mean value and crisp (possibilistic) variance of a continuous possibility distribution, which are consistent with the extension principle and with the well-known defintions of expectation and variance in probability theory. The theory developed in this paper is fully motivated by the principles introduced in [] and by the possibilistic interpretation of the ordering introduced in [3]. A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by F. Aγ-level set of a fuzzy number A is defined by [A] γ = {t R A(t) γ} if γ> and [A] γ = cl{t R A(t) > } (the closure of the support of A) if γ =. It is well-known that if A is a fuzzy number then [A] γ is a compact subset of R for all γ [, 1]. Let A and B F be fuzzy numbers with [A] γ = [a 1 (γ),a (γ)] and [B] γ = [b 1 (γ),b (γ)], γ [, 1]. In 1986 Goetschel and Voxman introduced a method for ranking fuzzy numbers as [3] A B γ(a 1 (γ)+a (γ)) dγ γ(b 1 (γ)+b (γ)) dγ (1) As was pointed out by Goetschel and Voxman this definition of ordering given in (1) was motivated in part by the desire to give less importance to the lower levels of fuzzy numbers. Possibilistic mean value of fuzzy numbers We explain now the way of thinking that has led us to the introduction of notations of lower and upper possibilitistic mean values. First, we note that from the equality M(A) := γ(a 1 (γ)+a (γ))dγ = γ a1(γ)+a (γ) dγ, () γ dγ it follows that M(A) is nothing else but the level-weighted average of the arithmetic means of all γ-level sets, that is, the weight of the arithmetic mean of a 1 (γ) and a (γ) is just γ. 1

4 Second, we can rewrite M(A) as M(A) = γ(a 1 (γ)+a (γ))dγ = = 1 = 1 γa 1 (γ)dγ + γa 1 (γ)dγ + 1 γa 1 (γ)dγ + γdγ γa (γ)dγ γa (γ)dγ 1 γa (γ)dγ. γdγ Third, let us take a closer look at the right-hand side of the equation for M(A). The first quantity, denoted by M (A) can be reformulated as M (A) = = = where Pos denotes possibility, i.e. γa 1 (γ)dγ = γa 1 (γ)dγ γdγ Pos[A a 1 (γ)]a 1 (γ)dγ Pos[A a 1 (γ)]dγ Pos[A a 1 (γ)] min[a] γ dγ, Pos[A a 1 (γ)]dγ Pos[A a 1 (γ)] = Π((,a 1 (γ]) = sup A(u) =γ. u a 1 (γ) (since A is continuous!) So M (A) is nothing else but the lower possibilityweighted average of the minima of the γ-sets, and it is why we call it the lower possibilistic mean value of A. In a similar manner we introduce M (A), the upper possibilistic mean value of A,

5 as M (A) = = = where we have used the equality γa (γ)dγ = γa (γ)dγ γdγ Pos[A a (γ)]a (γ)dγ Pos[A a (γ)]dγ Pos[A a (γ)] max[a] γ dγ, Pos[A a (γ)]dγ Pos[A a (γ)] = Π([a (γ, )) = Let us introduce the notation M(A) =[M (A),M (A)]. sup A(u) =γ. u a (γ) that is, M(A) is a closed interval bounded by the lower and upper possibilistic mean values of A. Definition.1. We call M(A) the interval-valued possibilistic mean of A. If A is the characteristic function of the crisp interval [a, b] then M((a, b,, )) = [a, b], that is, an interval is the possibilistic mean value of itself. We will now show that M is a linear function on F in the sense of the extension principle. Theorem.1. Let A and B be two non-interactive fuzzy numbers and let λ R be a real number. Then M(A + B) =M(A)+M(B), M(λA) =λm(a), i.e. M (A + B) =M (A)+M (B), M (A + B) =M (A)+M (B), and [M (λa),m (λa)] = { [λm (A), λm (A)] if γ [λm (A), λm (A)] if γ< where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension principle [4]. 3

6 Proof. Really, from the equation [A + B] γ =[a 1 (γ)+b 1 (γ),a (γ)+b (γ)], we have and M (A + B) = = M (A + B) = γ(a 1 (γ)+b 1 (γ))dγ γa 1 (γ)dγ + = M (A)+M (B), = γ(a (γ)+b (γ))dγ γa (γ)dγ + = M (A)+M (B), γb 1 (γ)dγ γb (γ)dγ furthermore, from [λa] γ = λ[a] γ = λ[a 1 (γ),a (γ)] = for λ we get and for γ< M (λa) = M (λa) = M (λa) = M (λa) = Which ends the proof. γ(λa 1 (γ))dγ =λ γ(λa (γ))dγ =λ γ(λa (γ))dγ =λ γ(λa 1 (γ))dγ =λ { [λa1 (γ), λa (γ)] if γ [λa (γ), λa 1 (γ)] if γ< γa 1 (γ)dγ = λm (A). γa (γ)dγ = λm (A). γa (γ)dγ = λm (A). γa 1 (γ)dγ = λm (A). We introduce the crisp possibilistic mean value of A by () as the arithemtic mean of its lower possibilistic and upper possibilistic mean values, i.e. M(A) = M (A)+M (A). The following theorem shows two very important properties of M : F R. 4

7 Theorem.. Let [A] γ =[a 1 (γ),a (γ)] and [B] γ =[b 1 (γ),b (γ)] be fuzzy numbers and let λ R be a real number. Then M(A + B) = M(A)+ M(B), and M(λA) =λ M(A). Proof. First we find M(A + B) = γ(a 1 (γ)+a (γ))dγ + and for λ we get M(λA) = γ(a 1 (γ)+b 1 (γ)+a (γ)+b (γ))dγ = γ(λa 1 (γ)+λa (γ))dγ = λ and, finally, for γ< we have M(λA) = Which ends the proof. γ(λa (γ)+λa 1 (γ))dγ = λ γ(b 1 (γ)+b (γ))dγ = M(A)+ M(B), γ(a 1 (γ)+a (γ))dγ = λ M(A). γ(a 1 (γ)+a (γ))dγ = λ M(A). Example.1. Let A =(a, α, β) be a triangular fuzzy number with center a, leftwidth α> and right-width β> then a γ-level of A is computed by that is, and therefore, and, finally, M(A) = [A] γ =[a (1 γ)α, a + (1 γ)β], γ [, 1], M (A) = M (A) = M(A) = γ[a (1 γ)αdγ = a α 3, γ[a + (1 γ)β]dγ = a + β 3, [ a α 3,a+ β ], 3 γ[a (1 γ)α + a + (1 γ)β]dγ = a + β α 6. Specially, when A =(a, α) is a symmetric triangular fuzzy number we get M(A) = a. If A is a symmetric fuzzy number with peak [q,q + ] then the equation always holds. M(A) = q + q +. 5

8 3 Relation to upper and lower probability mean values We show now an important relationship between the interval-valued expectation E(A) = [E (A),E (A)] introduced in [] and the interval-valued possibilistic mean M(A) =[M (A),M (A)] for LR-fuzzy numbers with strictly decreasing shape functions. An LR-type fuzzy number A F can be described with the following membership function [1] ( ) q u L if q α u q α 1 if u [q,q + ] A(u) = ( ) u q+ R if q + u q + + β β otherwise where [q,q + ] is the peak of A; q and q + are the lower and upper modal values; L, R: [, 1] [, 1], with L() = R() = 1 and L(1) = R(1) = are nonincreasing, continuous mappings. We shall use the notation A =(q,q +, α, β) LR. The closure of the support of A is exactly [q α, q + + β]. If L and R are strictly decreasing functions then we can easily compute the γ-level sets of A. That is, [A] γ =[q αl 1 (γ),q + + βr 1 (γ)], γ [, 1]. Following [] (page 93) the lower and upper probability mean values of A F are computed by E (A) =q α L(u)du, E (A) =q + + β R(u)du, (note that the support of A is bounded) and the lower and upper possibilistic mean values are obtained as M (A) = M (A) = γ(q αl 1 (γ))dγ = q α γ(q + + βr 1 (γ))dγ = q + + β Therefore, we can state the following lemma. γl 1 (γ)dγ γr 1 (γ)dγ Lemma 3.1. If A F is a fuzzy number of LR-type with strictly decreasing (and continuous) shape functions then its interval-valued possibilistic mean is a proper subset of its interval-valued probabilistic mean, E(A) M(A). 6

9 Proof. From the relationships L(u)du = L 1 (γ)dγ and R(u)du = R 1 (γ)dγ. we get γl 1 (γ)dγ < L(u)du and γr 1 (γ)dγ < R(u)du. Which ends the proof. Lemma 3.1 reflects on the fact that points with small membership degrees are considered to be less important in the definition of lower and upper possibilistic mean values than in the definition of probabilistic ones. In the limit case, when A =(q,q +,, ), the possibilistic and probablistic mean values are equal, and the equality E(A) =M(A) =[q,q + ] holds. Example 3.1. Let A =(a, α, β) be a triangular fuzzy number with center a, leftwidth α> and right-width β> then M(A) = [ a α 3,a+ β ] [ E(A) = a α 3,a+ β ] and M(A) =a + β α 6 Ē(A) =E (A)+E (A) However, when A is a symmetric fuzzy number then the equation = a + β α 4. M(A) =Ē(A). always holds. 7

10 4 Variance of fuzzy numbers We introduce the (possibilistic) variance of A F as ( 1 [a1 (γ)+a (γ) Var(A) = Pos[A a 1 (γ)] ( 1 [a1 (γ)+a (γ) + Pos[A a (γ)] ([ ] a1 (γ)+a (γ) = γ a 1 (γ) [ ] a1 (γ)+a (γ) ) + a (γ) dγ = 1 γ ( a (γ) a 1 (γ) ) dγ. ] ) a 1 (γ) dγ ] ) a (γ) dγ The variance of A is defined as the expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e. the lower possibility-weighted average of the squared distance between the left-hand endpoint and the arithmetic mean of the endpoints of its level sets plus the upper possibilityweighted average of the squared distance between the right-hand endpoint and the arithmetic mean of the endpoints their of its level sets. The standard deviation of A is defined by σ A = Var(A). For example, if A =(a, α, β) is a triangular fuzzy number then Var(A) = 1 γ ( a + β(1 γ) (a α(1 γ)) ) (α + β) dγ =. 4 especially, if A =(a, α) is a symmetric triangular fuzzy number then Var(A) = α 6. If A is the characteristic function of the crisp interval [a, b] then Var(A) = 1 ( ) b a γ(b a) dγ = that is, σ A = b a, a + b M(A) =. In probability theory, the corresponding result is: if the two possible outcomes of a probabilistic variable have equal probabilities then the expected value is their arithmetic mean and the standard deviation is the half of their distance. 8

11 We show now that the variance of a fuzzy number is invariant to shifting. Let A F and let θ be a real number. If A is shifted by value θ then we get a fuzzy number, denoted by B, satisfying the property B(x) =A(x θ) for all x R. Then from the relationship we find Var(B) = 1 [B] γ =[a 1 (γ)+θ, a (γ)+θ] = 1 γ ( (a (γ)+θ) (a 1 (γ)+θ) ) dγ γ ( a (γ) a 1 (γ) ) dγ = Var(A). The covariance between fuzzy numbers A and B is defined as Cov(A, B) = 1 γ(a (γ) a 1 (γ))(b (γ) b 1 (γ))dγ The covariance measures how much the endpoints of the γ-level sets of two fuzzy numbers move in tandem. Let A =(a, α) and B =(b, β) be symmetric trianglar fuzzy numbers. Then Cov(A, B) = αβ 6. The following theorem shows that the variance of linear combinations of fuzzy numbers can easily be computed (in the same manner as in probability theory). Theorem 4.1. Let λ, µ R and let A and B be fuzzy numbers. Then Var(λA + µb) =λ Var(A)+µ Var(B)+ λµ Cov(A, B) where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension principle. Proof. Suppose λ< and µ<. Then we find Var(λA + µb) = 1 λ 1 1 γ ( λa 1 (γ)+µb 1 (γ) λa (γ) µb (γ) ) dγ = γ ( λ(a 1 (γ) a (γ)) + µ(b 1 (γ) b (γ)) ) dγ = γ(a 1 (γ) a (γ)) dγ + µ 1 λµ 1 γ(b 1 (γ) b (γ)) dγ+ γ(a 1 (γ) a (γ))(b 1 (γ) b (γ))dγ = 9

12 λ Var(A)+µ Var(B)+λµ 1 γ(a (γ) a 1 (γ))(b (γ) b 1 (γ))dγ = λ Var(A)+µ Var(B)+λµCov(A, B) = λ Var(A)+µ Var(B)+ λµ Cov(A, B). Similar reasoning holds for the case λ> and µ>. Suppose now that λ< and γ>. Then we get Var(λA + µb) = 1 λ 1 1 γ ( λa 1 (γ)+µb (γ) λa (γ) µb 1 (γ) ) dγ = γ ( λ(a 1 (γ) a (γ)) + µ(b (γ) b 1 (γ) ) dγ = γ(a 1 (γ) a (γ)) dγ + µ 1 λµ 1 λ Var(A)+µ Var(B) λµ 1 Which ends the proof. γ(b (γ) b 1 (γ)) dγ+ γ(a 1 (γ) a (γ))(b (γ) b 1 (γ))dγ = γ(a (γ) a 1 (γ))(b (γ) b 1 (γ))dγ = λ Var(A)+µ Var(B)+ λµ Cov(A, B). As a special case of Theorem 4.1 we get Var(λA) =λ Var(A) for any λ R and Var(A + B) = Var(A) + Var(B) + Cov(A, B). Let A =(a, α) and B =(b, β) be symmetric triangular fuzzy numbers and λµ be real numbers. Then Var(λA + µb) =λ α 6 β + µ 6 + λµ αβ 6 ( λ α + µ β) =, 6 which coincides with the variance of the symmetric triangular fuzzy number λa + µb =(λa + µb, λ α + µ β). Another important question is the relationship between the subsethood and the variance of fuzzy numbers. One might expect that A B (that is A(x) B(x) for all x R) should imply the relationship Var(A) Var(B) because A is considered a stronger restriction than B. The following theorem shows that subsethood does entail smaller variance. Theorem 4.. Let A, B F with A B. Then Var(A) Var(B). 1

13 Proof. From A B it follows that b 1 (γ) a 1 (γ) a (γ) b (γ), for all γ [, 1]. That is, for all γ [, 1], and therefore, a (γ) a 1 (γ) b (γ) b 1 (γ) Var(A) = 1 1 Which ends the proof. γ ( a (γ) a 1 (γ) ) dγ γ ( b (γ) b 1 (γ) ) dγ = Var(B). Remark 4.1. Alternatively, we could also introduce the variance of A F as Var (A) = = γ ( [ M(A) a 1 (γ)] +[ M(A) a (γ)] ) dγ γ(a 1(γ)+a (γ))dγ E (A), i.e. the possibility-weighted average of the squared distance between the expected value and the left hand and right hand endpoints of its level sets; and the covariance as Cov (A, B) = γ ( [ M(A) a 1 (γ)][ M(B) b 1 (γ)] + [ M(A) a (γ)][ M(B) b (γ)] ) dγ Then the following theorem would hold Theorem 4.3. Let λ, µ R such that λµ > and let A and B be fuzzy numbers. Then Var (λa + µb) =λ Var (A)+µ Var (B)+λµCov (A, B) where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension principle. Proof. Suppose λ< and µ<. Using the linearity of the expected value we 11

14 find Var (λa + µb) = λµ γ ( (E(λA+µB) λa (γ) µb (γ)) +(E(λA+µB) λa 1 (γ) µb 1 (γ)) ) dγ = γ ( λ M(A)+µ M(B) λa (γ) µb (γ) ) dγ + γ ( λ M(A)+µ M(B) λa 1 (γ) µb 1 (γ) ) dγ = γ ( λ( M(A) a (γ)) + µ( M(B) b (γ)) ) dγ + γ ( λ( M(A) a 1 (γ)) + µ( M(B) b 1 (γ)) ) dγ = λ γ ( [ M(A) a 1 (γ)] +[ M(A) a (γ)] ) dγ + µ γ ( [ M(B) b 1 (γ)] +[ M(B) b (γ)] ) dγ + γ ( [ M(A) a 1 (γ)][ M(B) b 1 (γ)]+[ M(A) a (γ)][ M(B) b (γ)] ) dγ = λ Var (A)+µ Var (B)+λµCov (A, B). Similar reasoning holds for the case λ> and µ>. Which ends the proof. However, nothing can be said about Var (λa + µb) if λµ <, and it is not clear if subsethood entails smaller variance. However, if A =(a, α, β) is a triangular fuzzy number then Var (A) = α + β + αβ, 18 and, therefore, any triangular subset of A will have smaller variance. 5 Summary We have introduced the notation of interval-valued possibilistic mean of fuzzy numbers and investigated its relationship to interval-valued probabilistic mean. We have proved that the proposed concepts behave properly (in a similar way as their probabilistic counterparts). 6 Acknowledgment The authors are thankful to Didier Dubois for his comments and suggestions on the earlier versions of this paper. 1

15 References [1] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 198). [] D. Dubois and H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems 4(1987) [3] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1986) [4] L.A. Zadeh, Fuzzy Sets, Information and Control, 8(1965)

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18 Turku Centre for Computer Science Lemminkäisenkatu 14 FIN-5 Turku Finland University of Turku Department of Mathematical Sciences Åbo Akademi University Department of Computer Science Institute for Advanced Management Systems Research Turku School of Economics and Business Administration Institute of Information Systems Science