Membership Function of a Special Conditional Uncertain Set

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1 Membership Function of a Special Conditional Uncertain Set Kai Yao School of Management, University of Chinese Academy of Sciences, Beijing , China yaokai@ucas.ac.cn Abstract Uncertain set is a set-valued function on an uncertainty space, and attempts to model unsharp concepts. Conditional uncertain set, derived from an uncertain set restricted to a conditional uncertainty space given an uncertain event, plays a crucial role in uncertain inference systems. This paper studies the membership function of a conditional uncertain set, and discovers its relationship with the membership function of the original uncertain set. Keywords: uncertain set; membership function; conditional uncertain set; conditional uncertainty 1 Introduction Probability theory has been used to model indeterminacy phenomena for a long time when a lot of samples are available to estimate the probability distribution. In our daily life, we are sometimes in a situation where only few or even no samples are available due to the economical or technological difficulties. In this case, we have to rely on the domain experts belief degree that each possible event may occur. In order to deal with the belief degree, Liu [4] founded an uncertainty theory as a branch of axiomatic mathematics, which was refined by Liu [6]. In the framework of uncertainty theory, a concept of uncertain measure was first defined to indicate the belief degree that an uncertain event occurs, and an uncertainty space was thus clearly defined. As a measurable function from an uncertainty space to the set of real numbers, an uncertain variable was proposed by Liu [6] to model a quantity with human uncertainty. After that, concepts of uncertainty distribution, expected value, variance, entropy were employed to describe an uncertain variable. Peng and Iwamura [13] gave a sufficient condition for a function being an uncertainty distribution. Liu and Ha [12] gave a formula to calculate the expected value of a function of uncertain variables. Yao [19] gave a formula to calculate the variance of an uncertain variable via inverse uncertainty distribution. Dai and Chen [1] verified the linearity of entropy operator of uncertain variables. As a generalization of uncertain variable, a concept of uncertain set was defined by Liu [7] to model the 1

2 unsharp concepts such as tall, fast and old. Essentially, it is a set-valued measurable function on an uncertainty space. Meanwhile, Liu [7] proposed a concept of expected value for a nonempty uncertain set. After that, Liu [8] proposed concepts of variance and entropy for uncertain sets. In 2012, Liu [9] recast the uncertain set theory by revising the definition of membership function of uncertain set. Then Yao and Ke [17] proved the linearity of entropy operator for uncertain set in the new framework of uncertain set theory, and Wang and Ha [15] and Yao [18] proposed concepts of entropy in the forms of quadratic function and sine function, respectively. In order to determine the membership function of an uncertain set, Liu [8] designed a questionnaire survey to collect the experiential data, and introduced linear interpolation method. After that, Wang and Wang [16] introduced the Delphi method. So far, uncertain set has been applied to uncertain logic and uncertain inference. In order to deal with human language, Liu [8] proposed uncertain logic and designed a linguistic summarizer to extract linguistic summary from a collection of raw data. In order to derive consequences from human knowledge, Liu [7] proposed a concept of uncertain inference system, and designed some inference rules, which were generalized by Gao et al. [2] later. Recently, Peng and Chen [14] proved that the uncertain inference system is a universal approximator. By means of uncertain inference rules, Gao [3] balanced the inverted pendulum. Conditional uncertain set plays a crucial role in uncertain inference rules. In this paper, we will study conditional uncertain set and its membership function in the new framework of uncertain set theory. The rest of this paper is organized as follows. Section 2 will introduce some basic concepts about uncertainty space and uncertain set. Section 3 will give the membership function of a conditional uncertain set. Finally, some remarks are made in Section 4. 2 Preliminary In order to provide a quantitative measurement that an uncertain event occurs, an uncertain measure was defined as below. Definition 1. (Liu [4]) Let L be a σ-algebra on a nonempty set Γ. A set function M : L [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1: (Normality Axiom) M{Γ = 1 for the universal set Γ. Axiom 2: (Duality Axiom) + M{Λ c = 1 for any event Λ. Axiom 3: (Subadditivity Axiom) For every countable sequence of events Λ 1, Λ 2,, we have { M Λ i M {Λ i. The triple (Γ, L, M) is called an uncertainty space. The product uncertain measure on the product σ- algebra L was defined by Liu [5] as follows, 2

3 Axiom 4: (Product Axiom) Let (Γ k, L k, M k ) be uncertainty spaces for k = 1, 2, Then the product uncertain measure M on the product σ-algebra satisfies { M Λ k = M k {Λ k where Λ k are arbitrarily chosen events from L k for k = 1, 2,, respectively. k=1 Definition 2. (Liu [6]) The uncertain events Λ 1, Λ 2,, Λ n are said to be independent if { n n M = M i {Λ i where Λ i are arbitrarily chosen from {Λ i, Λ c i, Γ for i = 1, 2,, n, respectively, and Γ denote the universal set. Λ i k=1 Definition 3. (Liu [4]) Let (Γ, L, M) be an uncertainty space, and Λ 1, Λ 2 be two uncertain events. Then the conditional uncertain measure of Λ 1 given Λ 2 is M{Λ 1 Λ 2 = M{Λ 1 Λ 2, if M{Λ 1 Λ 2 < 0.5 M{Λ 2 M{Λ 2 1 M{Λc 1 Λ 2, if M{Λc 1 Λ 2 < 0.5 M{Λ 2 M{Λ 2 0.5, otherwise (1) provided M{Λ 2 > 0. In this case, the triple (Γ, L, M{ Λ 2 ) is also an uncertainty space. Definition 4. (Liu [9]) An uncertain set is a measurable function ξ from an uncertainty space (Γ, L, M) to a collection of sets of real numbers, i.e., for any Borel set B of real numbers, the following two sets {ξ B = {γ Γ ξ(γ) B, {B ξ = {γ Γ B ξ(γ) are events. Definition 5. (Liu [9]) An uncertain set ξ is said to have a membership function µ if the equations hold for any Borel set B of real numbers. M{B ξ = inf x B µ(x), M{ξ B = 1 sup Liu [7] proved that a real-valued function µ is a membership function if and only if 0 µ(x) 1. A membership function is said to be regular if there exists a point x 0 such that µ(x 0 ) = 1 and µ(x) is unimodal with respect to the point x 0. 3

4 Definition 6. (Liu [9]) Let ξ be an uncertain set with a membership function µ. Then the set-valued function is called the inverse membership function of ξ. µ 1 (α) = {x R µ(x) α, α [0, 1] If µ is a regular membership function, then the function µ 1 l (α) = inf µ 1 (α) is called the left inverse membership function, and the function µ 1 r (α) = sup µ 1 (α) is called the right inverse membership function. Note that µ 1 l (α) is increasing and µ 1 r (α) is decreasing with respect to α. Definition 7. (Liu [10]) The uncertain sets ξ 1, ξ 2,, ξ n are said to be independent if for any Borel sets B 1, B 2,, B n, we have { n n M (ξi B i ) = M{ξi B i, { n n M (ξi B i ) = M{ξi B i, where ξ i are arbitrarily chosen from {ξ i, ξi c, i = 1, 2,, n, respectively. For regular independent uncertain sets ξ and η with membership functions µ(x) and ν(x), respectively, Liu [9] proved that the union ξ η has a membership function µ(x) ν(x), the intersection ξ η has a membership function µ(x) ν(x), and the complement ξ c has a membership function 1 µ(x). Theorem 1. (Liu [9]) Let ξ 1, ξ 2,, ξ n be independent uncertain sets with regular membership functions µ 1, µ 2,, µ n, respectively. If the function f is increasing with respect to x 1, x 2,, x m and decreasing with respect to x m+1, x m+2,, x n, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain set with a membership function µ where µ 1 l (α) = f ( µ 1 1l (α),, µ 1 ml (α), µ 1 ( µ 1 1r (α),, µ 1 mr(α), µ 1 µ 1 r (α) = f m+1,r m+1,l (α),, µ 1 nr (α) ), ) (α),, µ 1 nl (α). Definition 8. (Liu [11]) Let ξ be an uncertain set on an uncertainty space (Γ, L, M) and Λ be an uncertain event with > 0. Then the conditional uncertain set ξ Λ on the uncertainty space (Γ, L, M{ Λ) is said to have a membership function ν(x) if we have M{B ξ Λ = inf x B ν(x), for any Borel set B. M{ξ B Λ = 1 sup x B c ν(x) 4

5 3 Membership Function of Conditional Uncertain Set Definition 9. An uncertain set ξ and an uncertain event Λ are said to be independent if for any Borel set B, the uncertain events {B ξ and Λ are independent, and the uncertain events {ξ B and Λ are also independent. Theorem 2. Let ξ be an uncertain set with a membership function µ(x) on an uncertainty space (Γ, L, M), and Λ be an uncertain event independent of ξ. Then the conditional uncertain set ξ Λ the uncertainty space (Γ, L, M{ Λ) has a membership function µ(x), if µ(x) < 1 2 ν(x) = µ(x) + 1, if µ(x) > Proof. According to Definition 8, we just need to prove that M{B ξ Λ = inf ν(x), (3) x B M{ξ B Λ = 1 sup x B c ν(x) (4) hold for any Borel set B. We first prove the Equation (3). For any Borel Set B, according to the Definition 3 of conditional uncertain measure, M{(B ξ) Λ, if M{B ξ Λ = M{B ξ Λ = M{(B ξ) Λ < M{(B ξ)c Λ, if M{(B ξ)c Λ < 1 2 Since the uncertain events {B ξ and Λ are independent, we have M{B ξ, if M{B ξ < M{(B ξ)c, if M{(B ξ)c < 1 2 Noting that the uncertain set ξ has a membership function µ(x) on the uncertainty space (Γ, L, M), we on (2) have M{B ξ Λ = inf x B inf µ(x) x B, µ(x) + 1 if inf µ(x) < 1 x B 2, if inf x B µ(x) >

6 which just equals to inf ν(x). So the Equation (3) is proved. Now, we turn to Equation (4). For any x B Borel Set B, according to the Definition 3 of conditional uncertain measure, M{ξ B Λ = M{(ξ B) Λ, if M{(ξ B) Λ < M{(ξ B)c Λ, if M{(ξ B)c Λ < 1 2 Since the uncertain events {ξ B and Λ are independent, we have M{ξ B Λ = M{ξ B, if M{ξ B < M{(ξ B)c, if M{(ξ B)c < 1 2 Noting that the uncertain set ξ has a membership function µ(x) on the uncertainty space (Γ, L, M), we have M{ξ B Λ = 1 sup sup, if sup > 1 1 2, if sup < 1 2 Besides, we have sup ν(x) = x B c sup µ(x) x B c, if sup µ(x) < 1 x B c 2 sup µ(x) + 1 x B c, if sup µ(x) > 1 1 x B c 2 0.5, otherwise, and 1 sup x B c ν(x) = sup 1 sup, if sup < 1 2, if sup > So the Equation (4) is proved. The proof is thus completed. 6

7 Theorem 3. Let ξ and η be two independent uncertain sets with membership functions µ(x) and ν(x), respectively. Then for a Borel set B, the conditional uncertain set ξ = ξ B η has a membership function µ (x) = µ(x) if µ(x) < ν(y), inf y B µ(x) + inf y B ν(y) 1 inf ν(y), if µ(x) > 1 y B 1 2 inf y B ν(y) 1 2 inf y B ν(y) Proof. Take Λ = {B η in Theorem 2. Since η has a membership function ν(x), we have (5) M{B η = inf y B ν(y). Then this theorem follows immediately from Theorem 2. Example 1. Let ξ and η be two independent uncertain sets with membership functions µ(x) and ν(x), respectively. Then for a given real number y, the conditional uncertain set ξ = ξ y η has a membership function µ (x) = µ(x) ν(y), if µ(x) < ν(y)/2 µ(x) + ν(y) 1, if µ(x) > 1 ν(y)/2 ν(y) 1/2, otherwise. Theorem 4. Let ξ and η be two independent uncertain sets with membership functions µ(x) and ν(x), respectively. Then for a Borel set B, the conditional uncertain set ξ = ξ η B has a membership function µ(x) 1, if µ(x) < 1 sup ν(y) sup ν(y) y B c y B c µ µ(x) sup ν(y) (x) = y B c (7) 1 sup y B c ν(y), if µ(x) > sup y B c ν(y) Proof. Take Λ = {η B in Theorem 2. Since η has a membership function ν(x), we have (6) M{η B = 1 sup y B c ν(y). Then this theorem follows immediately from Theorem 2. 4 Conclusions This paper showed that if an uncertain set has a membership function, then its conditional uncertain set given an independent event also has a membership function which could be derived from the original 7

8 membership function. The results obtained in this paper provide a theoretic foundation for uncertain inference control. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No ). References [1] Dai W., and Chen X.W., Entropy of function of uncertain variables, Mathematical and Computer Modelling, Vol.55, Nos.3-4, , [2] Gao X., Gao Y., and Ralescu D.A., On Liu s inference rule for uncertain systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.18, No.1, 1-11, [3] Gao Y., Uncertain inference control for balancing an inverted pendulum, Fuzzy Optimization and Decision Making, Vol.11, No.4, , [4] Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, [5] Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No.1, 3-10, [6] Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, [7] Liu B., Uncertain set theory and uncertain inference rule with application to uncertain control, Journal of Uncertain Systems, Vol.4, No.2, 83-98, [8] Liu B., Uncertain logic for modeling human language, Journal of Uncertain Systems, Vol.5, No.1, 3-20, [9] Liu B., Membership functions and operational law of uncertain sets, Fuzzy Optimization and Decision Making, Vol.11, No.4, , [10] Liu B., A new definition of independence of uncertain sets Fuzzy Optimization and Decision Making, Vol.12, No.4, , [11] Liu B., Uncertainty Theory, 4th ed., Springer-Verlag, Berlin, [12] Liu Y.H., and Ha M.H., Expected value of function of uncertain variables, Journal of Uncertain Systems, Vol.4, No.3, , [13] Peng Z.X., and Iwamura K., A sufficient and necessary condition of uncertainty distribution, Journal of Interdisciplinary Mathematics, Vol.13, No.3, , [14] Peng Z.X., and Chen X.W., Uncertain systems are universal approximiators, Journal of Uncertainty Analysis and Applications, Vol.2, Article 13, [15] Wang X.S., and Ha M.H., Quadratic entropy of uncertain sets, Fuzzy Optimization and Decision Making, Vol.12, No.1, ,

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