일반화된 삼각퍼지집합에 대한 정규퍼지확률
Normal fuzzy probability for generalized triangular fuzzy sets
강철*․윤용식**†
Chul Kang and Yong Sik Yun†
*Department of Applied Mathematics, Hankyong National University
**Department of Mathematics, Jeju National University
요약
확률공간 ℱ 위에 정의된 퍼지집합을 퍼지이벤트라 한다. Zadeh는 확률 를 이용하여 퍼지이벤트 에 대한 확률 을 정의하였다. 우리는 일반화된 삼각퍼지집합을 정의하고 거기에 확장된 대수적 작용소를 적용하였다. 일반화된 삼각퍼지 집합은 대칭적이지만 함숫값으로 1을 갖지 않을 수 있다. 두 개의 일반화된 삼각퍼지집합 와 에 대하여 와
는 일반화된 사다리꼴퍼지집합이 되었지만, ⋅와 는 일반화된 삼각퍼지집합도 되지 않았고 일반화된 사다리꼴퍼지집합도 되지 않았다. 그리고 정규분포를 이용하여 ℝ위에서 정규퍼지확률을 정의하였다. 그리고 일반화된 삼 각퍼지집합에 대한 정규퍼지확률을 계산하였다.
Abstract
A fuzzy set defined on a probability space ℱ is called a fuzzy event. Zadeh defines the probability of the fuzzy event using the probability . We define the generalized triangular fuzzy set and apply the extended algebraic operations to these fuzzy sets. A generalized triangular fuzzy set is symmetric and may not have value . For two generalized triangular fuzzy sets and , and become generalized trapezoidal fuzzy sets, but ⋅ and need not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set. We define the normal fuzzy probability on ℝ using the normal distribution. And we calculate the normal fuzzy probability for generalized triangular fuzzy sets.
Key Words : fuzzy event, generalized triangular fuzzy set, normal fuzzy probability
접수일자: 2011년 12월 26일 심사(수정)일자: 2012년 3월 23일 게재확정일자 : 2012년 3월 28일
†교신저자
This work was supported by the research grant of Jeju National University in 2011.
1. Introduction
We define the generalized triangular fuzzy set and calculate four operations of two generalized triangular fuzzy sets([4]). Four operations are based on the Zadeh's extension principle([6], [7], [8]). And Zadeh defines the probability of fuzzy event as follows.
Let ℱ be a probability space, where denotes the sample space, ℱ the -algebra on , and a probability measure. A fuzzy set on is called a fuzzy event. Let μ⋅ be the membership function of the fuzzy event .
Then the probability of the fuzzy event is defined by Zadeh([9]) as
μ μ⋅ →We defined the normal fuzzy probability using the normal distribution and calculated the normal fuzzy probability for quadratic fuzzy number([2]). And then we had the explicit formula for the normal fuzzy probability for trigonometric fuzzy number.
Furthermore we calculated the normal fuzzy probability for trigonometric fuzzy numbers driven by the above four operations([3]).
In this paper, we calculate the normal fuzzy probability for generalized triangular fuzzy sets.
2. Preliminaries
Let ℱ be a probability space, and be a random variable defined on it. Let be a real-valued Borel-measurable function on ℝ. Then is also a random variable.
Definition 2.1. We say that the mathematical
expectation of exists if
is finite.
We note that a random variable defined on
ℱ induces a measure on a Borel set Β∈B defined by the relation Then
becomes a probability measure on B and is called the probability distribution of . It is known that if
exists, then is also integrable over ℝ with respect to . Moreover, the relation
ℝ (2.1)holds. We note that the integral on the right-hand side of (2.1) is the Lebesgue- Stieltjes integral of with respect to . In particular, if is continuous on ℝ and exists, we can rewrite (2.1) as follows
ℝ
∞∞ (2.2)where is the distribution function corres- ponding to , and the last integral is a Riemann-Stieltjes integral.
Let be absolutely continuous on ℝ with probability density function ′ . Then
exists if and only if the integral
∞∞ is finite and in that case we have
∞
∞
Example 2.2. Let the random variable (denoted
∼ ) have the normal distribution with the probability density function
∈ℝ
where and ∈ℝ. Then ∞ for every , and we have
and
The induced measure is called the normal distribution.
A fuzzy set on is called a fuzzy event. Let
μ․ be the membership function of the fuzzy event . Then the probability of the fuzzy event is defined by Zadeh([9]) as
μ μ⋅ →Definition 2.3. The normal fuzzy probability of a fuzzy set on ℝ is defined by
ℝμ (2.3)where is the normal distribution.
Definition 2.4. A triangular fuzzy number is a fuzzy number having membership function
μ
≤
≤
≤
The above triangular fuzzy number is denoted by
.
Definition 2.5. The addition, subtraction, multi-plication, and division of two fuzzy sets are defined as
1. Addition :
μ sup min
μ μ
2. Subtraction :
μ sup min
μ μ
3. Multiplication ⋅ :
μ⋅ sup ⋅min
μ μ
4. Division :
μ sup min
μ μ
Example 2.6. ([2]) For two triangular fuzzy number
and , we have 1. Addition : . 2. Subtraction : . 3. Multiplication :
μ⋅
≤
≤
≤
Note that ⋅ is not a triangular fuzzy number.
4. Division :
μ
≤
≤
≤
Note that is not a triangular fuzzy number.
3. Generalized triangular fuzzy sets
In this section, we define the generalized triangular fuzzy set and apply the extended algebraic operations to these fuzzy sets. A generalized triangular fuzzy set is symmetric and may not have value .
Definition 3.1. A generalized triangular fuzzy set is a symmetric fuzzy set having membership function
μ
≤
≤
≤
(3.1)
where ∈ℝ and
The above generalized triangular fuzzy set is denoted by
Theorem 3.2. ([4]) For two generalized triangular fuzzy sets and , if
≤ and μ ≥ in , we have the followings.
1.
, i.e., is a generalized trapezoidal fuzzy set.
2.
, i.e., is a generalized trapezoidal fuzzy set.
3. ⋅ is a fuzzy set on , but need not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set.
4. is a fuzzy set on
, but need not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set.
Example 3.3. ([4]) Let
and
be generalized triangular fuzzy sets, i.e.,
μ
≤
≤
≤
and
μ
≤
≤
≤
Let and be the -cuts of and , respectively. Let and . Since
and
, we have
Since
and
, we have
Then we have the followings.
1. Addition :
μ
≤
≤
≤
≤
i.e.,
2. Subtraction :
μ
≤
≤
≤
≤
i.e.,
3. Multiplication : μ⋅
≤
≤
≤
≤
Thus ⋅ is a fuzzy set on , but not a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set.
4. Division :
μ
≤
≤
≤
≤
Thus is a fuzzy set on
, but not a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set.
4. Normal fuzzy probability
In this section, we derive the explicit formula for the normal fuzzy probability for generalized triangular fuzzy sets and give some examples.
Theorem 4.1. Let ∼ and
be generalized triangular fuzzy set. Then the normal fuzzy probability of a generalized triangular fuzzy set
is
(4.1)
,
where is the standard normal distribution, that is,
∞ exp .Proof. Since
μ
≤
≤
≤
we have
∞∞μ
where
and
exp
. Putting
, then
. Thus the proof is complete.
Example 4.2. 1. Let
be a generalized triangular fuzzy number. Then the normal fuzzy probability of with respect to ∼ is
. In fact, putting
, we have
2. Let
be a generalized triangular fuzzy number. Then the normal fuzzy probability of
with respect to ∼ is . In fact, putting
, we have
References
[1] A. Kaufmann and M. M. Gupta, Introduction To Fuzzy Arithmetic : Theory and Applications, Van Nostrand Reinhold Co., New York, 1985.
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Journal of fuzzy logic and intelligent systems, vol. 15, no. 3, pp. 277-281, 2005.
[3] Y. S. Yun, J. C. Song and S. U. Ryu, “Normal fuzzy probability for trigonometric fuzzy number,”
Journal of applied mathematics and computing, vol. 19, no. 1-2, pp. 513-520, 2005.
[4] Y. S. Yun, S. U. Ryu and J. W. Park, “The generalized triangular fuzzy sets,” Journal of the Chungcheong Mathematical Society, vol. 22, no. 2, pp. 161-170, 2009.
[5] H. J. Zimmermann, Fuzzy set Theory - and Its Applications, Kluwer-Nijhoff Publishing, Boston- Dordrecht-Lancaster, 1985.
[6] L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning - I,”
Information Sciences, vol. 8, pp. 199-249, 1975.
[7] L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning - II,”
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and its application to approximate reasoning - III,” Information Sciences, vol. 9, pp. 43-80, 1975.
[9] L. A. Zadeh, “Probability measures of fuzzy events,”
J. Math. Anal. Appl. vol. 23, pp. 421-427, 1968.
저 자 소 개
강철(Chul Kang)
2008년～현재 : 한경대학교 응용수학과 교수
관심분야 : Statistics, Fuzzy Probability Phone : 031-670-5342
E-mail : [email protected]
윤용식(Yong Sik Yun)
2006년～현재 : 제주대학교 수학과 교수
관심분야 : Fuzzy Probability, Probability Theory Phone : 064-754-3565
E-mail : [email protected]
