A Note of the Expected Value and Variance of Fuzzy Variables

Transcription

1 ISSN (print, (online International Journal of Nonlinear Science Vol.9( No.,pp.86-9 A Note of the Expected Value and Variance of Fuzzy Variables Zhigang Wang, Fanji Tian Department of Applied Mathematics, Hainan University,Haikou,578,China Institute of Mathematics and Computer Science,Hubei University,Wuhan,36,China (Received 5 May 9, accepted 9 December 9 Abstract: The emphasis in this paper is mainly on some properties expected value operator and variance of fuzzy variables,the expceted value and variance formulas of three common types of fuzzy variables.in the end, the expceted value and variance of geometric Liu process are given. Keywords: fuzzy variables;expected value ;variance; geometric Liu process. Introduction Probability theory is a branch of mathematics for studying the behavior of random phenomena based on the normalality,nonnegativity and countable additivity axioms. however,the additivity axiom of classical measure theory has been challenged by many mathematicians.sugeno [] generalized classical measure theory to fuzzy measure theory by replacing additivity axiom with monotonicity and semicontinuty axioms.in order to deal with general uncertainty,self-duality and countable subadditivity are much more important than continuity and semicontinuity. For this reason,liu [] founded an uncertainty theory that is a branch of mathematics based on normality,monotonicity,self-duality and countable subadditivity axioms. Uncertainty provides the commonness of probability theory,credibility theory and chance theory.in fact,probability,credibility,chance,and uncertain measures meet the normality,monotonicity and self-duality axioms.the essential difference among those measures is how to determine the measure of union.since additivity and maximality are special cases of subadditivity,probability and credibility are special cases of chance measure,and three of them are in the category of uncertain measure.this fact also implies that random variable and fuzzy variable are special cases of hybrid variables,and three of them are instances of uncertain variables. The emphasis in this paper is mainly on some properties expected value operator and variance of fuzzy variables,the expceted value and variance formulas of three common types of fuzzy variables.in the end, the expceted value and variance of geometric Liu process are given. Text. Credibility measure In this section,we review some preliminiary concepts within the framework of credibility theory.([3-6] Definition (Liu[] Let Θ be a nonempty set,and P(Θ the power set of Θ.A set function Cr(.called a credibility measure if it satisfies: ((Normality Cr(. = ; ((Monotonicity Cr{A} Cr{B} whenever A B; (3(Self-Duality Cr{A} + Cr{A c } = for any event A; ((Maximality Cr{ i A i } = sup i Cr{A i }for any eventa {A i }with sup i Cr{A i } <.5. The triplet (Θ, P(Θ, Cr is called a credibility space. The law of contradiction tells us that a proposition cannot be both true and false at the same time, and the law of excluded middle tells us that a proposition is either true or false. Self-duality is in fact a generalization of the law of Corresponding author. address: wzhigang@hainu.edu.cn Copyright c World Academic Press, World Academic Union IJNS..6.3/377

2 Z. Wang, F. Tian: A Note of the Expected Value and Variance of Fuzzy Variables 87 contradiction and law of excluded middle. In other words, a mathematical system without self-duality assumption will be inconsistent with the laws. This is the main reason why self-duality axiom is assumed. Definition (Liu[] A fuzzy variable is a function from a credibility space (Θ, P,Cr to the set of real numbers. Definition 3 (Liu[] Let ξ be a fuzzy variable defined on the credibility space (Θ, P,Cr. Then its membership function is derived from the credibility measure by μ(x = (Cr{ξ = x}, x R ( In practice, a fuzzy variable may be specified by a membership function. In this case, we need a formula to calculate the credibility value of some fuzzy event. The credibility inversion theorem provides this []. For fuzzy variables, there are many ways to define an expected value operator. See, for example,dubois and Prade [7], Heilpern [8], Campos and Gonzalez [9], Gonzalez [] and Yager []. The most general definition of expected value operator of fuzzy variable was given by Liu and Liu [3].This definition is applicable to both continuous and discrete fuzzy variables.. Expected valueof fuzzy variables Definition (Liu and Liu [] Let ξ be a fuzzy variable. Then the expected value of ξ is defined by E[ξ] = + Cr{ξ r}dr Cr{ξ r}dr ( provided that at least one of the two integrals is finite. The equipossible fuzzy variable on [a, b] has an expected value (a+b. The triangular fuzzy variable (a,b, c has an expected value (a+b+c ; The trapezoidal fuzzy variable (a, b, c,d has an expected value (a+b+c+d. Let ξ be a continuous nonnegative fuzzy variable with membership function μ. If μ is decreasing on [,, then Cr{ξ x} = μ(x for any x >, and E[ξ] = μ(xdx (3 Example A fuzzy variable ξ is called exponentially distributed if it has an exponential membership function μ(x = ( + exp( x 6m, x, m > ( then the expected value is 6m ln. Theorem Let ξ be a continuous fuzzy variable with membership function μ. If its expected value exists, and there is a point x such that μ(x is increasing on (, x and decreasing on (x, +, then Proof If x,then + x E[ξ] = x + μ(xdx x μ(xdx (5 and Cr{ξ r} = μ(x, { Cr{ξ r} = [ + μ(x], if r x, μ(x. if r > x. E[ξ] = x [ μ(x]dx + + x μ(xdx = x + + x μ(xdx x μ(xdx μ(xdx If x <,a similar way may prove the equation (5. The theorem is proved. Especially,let μ(x be symmetric function on the line x = x,then E[ξ] = x. Example Let ξ be triangular fuzzy variable (a,b, c,then it has an expected value E[ξ] = b + c b x c b c dx b a x a c b a b a + b + c dx = b + + = b a IJNS homepage:

3 88 International Journal of NonlinearScience,Vol.9(,No.,pp Example 3 Let ξ be equipossible fuzzy variable on [a, b],then it has an expected value E[ξ] = a + b + b a+b dx a+b a dx = a + b Example Let ξ be trapezoidal fuzzy variable (a, b, c,d,then it has an expected value E[ξ] = b + c + c b+c dx + d c x d c d dx b x a a b a b+c b dx = a + b + c + d Especially,let ξ be triangular fuzzy variable (a,b, c and b a = c b,then it has an expected value E[ξ] = b. Let ξ be trapezoidal fuzzy variable (a, b, c,d and b a = d c,then it has an expected value E[ξ] = b+c. A fuzzy variable ξ is called normally distributed if it has a normal membership function x e μ(x = ( + exp(, x R, σ > (6 By theorem,the expected value is e. The definition of expected value operator is also applicable to discrete case. Assume that ξ be a simple fuzzy variable whose membership function is given by μ(x = μ, if x = a μ, if x = a... μ m, if x = a m where a, a,..., a m are distinct numbers.note that μ μ... μ m =. Then the expexted value of ξ is where the weights are given by m E[ξ] = ω i a i. i= ω i = ( max j m {μ j a j a i } max j m {μ j a j < a i } + max j m {μ j a j a i } max j m {μ j a j > a i } for i =,,..., m. It is easy to verity that all ω i and the sum of all weights is just..3 Some formulas variance of fuzzy variables The variance of a fuzzy variable provides a measure of the spread of the distribution around its expected value. A small value of variance indicates that the fuzzy variable is tightly concentrated around its expected value; and a large value of variance indicates that the fuzzy variable has a wide spread around its expected value. Definition 5 (Liu and Liu [] Let ξ be a fuzzy variable with finite expected value e.then the variance of ξ is defined by E[(ξ e ]. Definition 6 (Liu [] Let ξ be a fuzzy variable, and k a positive number. Then the expected value E[ξ k ] is called the kth moment. Theorem Let ξ be a normal fuzzy variable with membership function (6.Then the variance is σ. Proof E[ξ] = e, then the variance Cr{(ξ e r} = Cr({(ξ e r} {(ξ e r} = Cr{(ξ e r} Cr{(ξ e r} = Cr{(ξ e r} = ( + exp r e ( + exp r dr = σ IJNS for contribution: editor@nonlinearscience.org.uk

4 Z. Wang, F. Tian: A Note of the Expected Value and Variance of Fuzzy Variables 89 Theorem 3 Let ξ be a exponential fuzzy variable with membership function (.Then the second moment is m. Theorem Let ξ be a triangular fuzzy variable (a,b, c,then its variance , > 6, = ( , < where = b a, = c b. Proof Let m = E[ξ],when =,then m = b and then when >, m < b Cr{(ξ m r} = E[(ξ m ] = { r, r, r Cr{(ξ m }dr = 6 (i r (b m,then Cr{(ξ m r} = Cr{(ξ m r} Cr{(ξ m r} (iiwhen (b m r r s,werer r s = (+ 6,then (iii when r s r (b m,then Cr{(ξ m r} = Cr{ξ m r} = [ + r+m a ] = r+m b+ Cr{(ξ m r} = Cr{ξ m r} = [ m+ r b ] = m r+b+ Cr{(ξ m r} = Cr{ξ m r} = [ m r a ] = r+m b+ (iv when r (b m, Cr{(ξ m r} = then, E[(ξ m ] = Cr{(ξ m r}dr = when >,a similar way may prove the equation (7.The theorem is proved. Theorem 5 ([Chen 3] Let ξ be a trapezoidal fuzzy variable (a,b, c,d,then its variance (i when =, (iiwhen >, 3(b a + + ; (iii when <, { (b m3 6 [ 6 [ (a m3 { 6 [ (a m3 6 [ (a m3 (a m3 (b m3 + (b+ m3 (b m3 + (b+a m(+3 (a m3 ( ], a m < + (b+a m(+3 ( ], a m (b+a m(+3 + (b+ m3 ( ], b m > + (b+a m(+3 ( ], b m IJNS homepage:

5 9 International Journal of NonlinearScience,Vol.9(,No.,pp Theorem 6 Let ξ be a triangular fuzzy variable (a,b, c,then V [ξ] is an increasing function of and.where = b a, = c b. Proof (i when =,the equation holds trivially. (ii when >, set <,then V [ξ] is an increasing function of. On the other hand, < ( = + 3 > V [ξ] is an increasing function of. (iii A similar way may prove the result. The theorem 6 is proved. Theorem 7 Let ξ be a trapezoidal fuzzy variable (a,b, c,d.then V [ξ] is an increasing function of and.it is also an increasing function of c b. where = b a, = d c Proof ( when =,the equation holds trivially. ( where >,Let h = b a,then a m = ( h, b m = ( + h, when a m < when a m, a m = (3 + + h, b + a m( + = ( + + h( 3 [3(h (3 + + h + ( + + h + (3 + + h( + + h] 6 (6h + 66 h h 8h 3 6 h h 3 It is easy to prove that it is an increasing function of, and h,respectively. (3 when <,a slmilar way may prove the result.the theorem 7 is proved.. Expected value and variance of Geometric Liu process Definition 7 (Liu []A fuzzy process Ct is said to be a Liu process if (i C =, (ii C t has stationary and independent increments, (iii every increment C s+t C s is a normally distributed fuzzy variable with expected value et and variance σ t. The parameters e and σ are called the drift and diffusion coefficients,respectively. Liu process is said to be standard if e = and σ =. Definition 8(Liu []Let C t be a standard Liu process. Then the fuzzy process G t = exp(et + σc t is called a geometric Liu process. In 8, Li [5] proved that G t has a lognormal membership function and obtained its expeceted value and variance.in addition, alternative fuzzy stock models were introduced by Peng [6]. Theorem 8 Let G t be a geometric Liu process whose membership function ln x et μ(x = ( + exp (, x > (8 t then it has expected value E[G t ] = exp(et csc( t t, t < (. IJNS for contribution: editor@nonlinearscience.org.uk

6 Z. Wang, F. Tian: A Note of the Expected Value and Variance of Fuzzy Variables 9 Proof t < ( E[G t ] = Cr{G t r}dr = exp(et [ ( + exp( = exp(et = exp( e exp( e +x exp( e exp( e dx +x t (et ln x t t dx + = exp(et ( + x t dx = exp(et csc( t t ]dx + exp(et exp( e exp(et exp( e dx +x t (ln x et ( + exp( dx Example 5 Let G t be a geometric Liu process.li X [5] obtained the variance formula If t (, Otherwise,V [G t ] = +. t V [G t ] = E [G t] exp(et ( + exp ( + exp ( t (et ln(e[g t ] x dx + E [G t] exp(et ( + exp ( t (ln(e[g t ] + x et dx 3 Conclusions The main contribution of the present paper is to obtain the expected value and variance formulas of the triangular fuzzy variable,the trapezoidal fuzzy variable, normal fuzzy variable and geometric Liu process. On the other hand, some properities of variance are griven. Acknowledgments This work is supported by the institution of higher learning scientific research Program (No.Hjkj98 of Hainan Provincial Department of Education, the Hainan Natural Science Foundation(No.875,China. References [] Choquet G. Theory of capacities. Annals de l Institute Fourier, 5(95:3-95. [] Liu B. Uncertainty Theory,nd ed.springer-verlag. Berlin. 7. [3] Liu Y,K,and Liu B. Fuzzy random variables:a scalar expected value operator. Fuzzy Optimization and Decision Making, (3:3-6. [] Liu B, and Liu YK. Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, ((: 5-5. [5] Li X, and Liu B. A sufficient and necessary condition for credibility measures. Fuzzy Optimization and Decision Making, 5(6(: [6] Liu B. A survey of credibility theory. Fuzziness and Knowledge-Based Systems, (6(5: [7] Dubois D and Prade H. The mean value of a fuzzy number. Fuzzy Sets and Systems, (987: [8] Heilpern S. The expected value of a fuzzy number. Fuzzy Sets and Systems,7(99: [9] Campos L, and Gonzalez. A subjective approach for ranking fuzzy numbers Fuzzy Sets and Systems, 9(989:5-53. [] Gonzalez, A. A study of the ranking function approach through mean values. Fuzzy Sets and Systems, 35(99:9-. [] Yager RR. A procedure for ordering fuzzy subsets of the unit interval. Information Sciences,(98:3-6. [] Liu B. Inequalities and convergence concepts of fuzzy and rough variables. Fuzzy Optimization and Decision Making, (3(: 87-. [3] Chen Yanju. The credibility theory and its application in portfolio selection. A dissertation for the degree of M.Science,Hubei University, 5. IJNS homepage:

7 9 International Journal of NonlinearScience,Vol.9(,No.,pp [] Liu B. Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, (8(: 3-6. [5] Li xiang and Qin zhongfeng. Expected value and variance of geometric Liu process. Far East Journal of Experimental and Theoretical Artificial Intelligence, (8(: [6] J. Peng. A general stock model for fuzzy markets. Journal of Uncertain Systems, (8(: 8-5. IJNS for contribution: editor@nonlinearscience.org.uk