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Parameter estimations of normal fuzzy variables
Fuzzy Sets and Systems 55 (1993) 179-185 North-Holland
179
Parameter estimations of normal fuzzy variables Cai Kai-Yuan* Department of Automatic Control, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Received April 1992 Revised July 1992
Abstract Although fairly much attention has been paid to theoretical applications of Nahmias' fuzzy variables, little effort has been focused on the related parameter estimation issues. This keeps down the practical use of fuzzy variables. In this note we make a preliminary discussion on parameter estimation of normal fuzzy variables with parameters (a, b) and distinguish three situations: estimation of a with b known, estimation of b with a known, and estimation of unknown a and b. Formulae for calculating parameter estimates by use of practical collected data are presented.
Keywords: Fuzzy variable; parameter estimation.
1. Introduction
Since Nahmias introduced an elegant definition for fuzzy variables in terms of pattern space [9], theoretical applications of fuzzy variables have been drawing fairly much attention. For example, Kwakernaak defined fuzzy random variables as random variables with fuzzy variables as values [6, 7], while Cai et al. laid fuzzy variables as a basis for posbist reliability theory [4, 5]. However, little attention has been paid to parameter estimation issues of fuzzy variables except in the case of fuzzy software reliability modeling [1, 2]. This keeps down the practical use of fuzzy variables. To mention parameter estimation related work, we should note that various methods have been presented to estimate membership functions under the head of 'fuzzy statistics' [11]. Fuzzy statistics postulates that the analytic form of tXx, the membership function of fuzzy set X defined on base set U--{x}, is unknown and attempts to identify an estimate of IXx(X) for a given point x. On the other hand, Dishkant interpreted fuzzy variables in Zadeh's sense, that is, as mappings from ~ (the real number line) to the unit interval [0, 1], and discussed the corresponding parameter estimation issue [5]. He viewed a to be estimated parameter A of the membership function as a Zadeh fuzzy variable and assumed that/x(x, t), the truth value of assertion "If A has the value t, then the fuzzy variable X has the value x" is a priori known. Then a method was presented to identify an 'optimal' value for A. In fuzzy software reliability modeling [1, 2], however, Cai interpreted fuzzy variables in Nahmias' sense and viewed parameters of membership functions as real numbers. Then a method was proposed to estimate parameter a of membership function e -(x-"): through defining a scale likelihood function L(x~ . . . . , xn). In this note we interpret fuzzy variables in Nahmias' sense and confine ourselves to a discussion of parameter estimation methods for normal membership functions. In Section 2 we present preliminaries for Nahmias' fuzzy variables. In Section 3 we discuss various methods for estimating parameter a with parameter b known and present an actual example. Section 4 is devoted to estimating parameter b with parameter a known, whereas Section 5 discusses the parameter estimation problem when both Correspondence to: Dr. Cai Kai-Yuan, Department of Automatic Control, Beijing University of Aeronautics and Astronautics, Beijing 100083, China. * Supported by the National Natural Science Foundation of China.
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180
parameters a and b are unknown. An example is presented in Section 6 to illustrate the methods proposed in Sections 4 and 5. Concluding remarks are contained in Section 7.
2. Preliminaries For completeness, in this section we present preliminaries for Nahmias' fuzzy variables [9]. Definition 2.1. For a base set F, suppose that ~3 is the class of all subsets of F. Suppose a scale, o-, is defined on ~ and satisfies the following properties: (i) tr(0) = 0 and v-(F) = 1. (ii) For any arbitary collection of sets As of ~ (finite, countable or uncountable), o ' ( U Z ~ ) = sup o(A~). o~
Then o- is a scale measure and the triple (F, ~, or) is referred to as pattern space. Definition 2.2. A fuzzy variable X is a mapping from F to ~ (the real number line). Definition 2.3. The membership function of a fuzzy variable X, denoted by tZx, is a mapping from R to the unit interval [0, 1] and is given by ~ x ( x ) = o-(-/: x ( - / ) : x)
for all x E R. Note that SUPxtZx(X) = t r { Y (Y: X ( T ) = x ) } = tr(F) = 1. In general we use the briefer notation X = x to denote the subset {3': X(3,) = x} o f ~g. It has been shown that the value of tZx(X) at point x can be interpreted as the possibility that X = x holds [3], though we are not asked to adopt Zadeh's definition of possibility measure [12]. Therefore the membership function of X can be viewed as possibility distribution of X and we arrive at: Definition 2.4. The possibility distribution function of a fuzzy variable X, denoted by rex or tZx, is a mapping from R to the unit interval [0, 1] and is given by Jrx(X) = tXx(X) = t r ( X -- x) for all x E E. Now we introduce the concept and terminology of 'unrelatedness', though Rao and Rashed preferred an alternative terminology 'min-relatedness' [10]. Definition 2.5. Given a pattern space (F, ~, tr), the sets A1 . . . . , An c ~3 are said to be mutually unrelated if for any permutation of the set { 1 , . . . , n}, denoted by il . . . . . i~ for 1 ~
~r(A.~)).
Definition 2.6. Given a pattern space (F, cg, o-), the fuzzy variables X1 . . . . . Xn are said to be (mutually) unrelated if for any permutation of the set {1. . . . . n}, denoted by il, • • . , ik for 1 ~ k ~
Cai Kai-Yaan / Parameter estimations of normal fuzzy variables
181
Definition 2.7. Let X be a fuzzy variable with m e m b e r s h i p function/Xx. A real n u m b e r m is said to be a modal value of X, i f / x x ( m ) = 1. Definition 2.8. A fuzzy variable X is said to be unimodal, if there exists a unique a c ~ such that txx(a) = 1. At this time we signify •X = a. Finally we introduce the notion of normal fuzzy variables.
Definition 2.9. A fuzzy variable X is said to be normal if the membership function is of form
where a e N and b > 0. At this time we briefly say that X is a normal fuzzy variable with p a r a m e t e r s (s, b) or denote X as N(a, b). Then we have the following result [9]:
Lemma 2.1. Let Xx . . . . . Xn be unrelated f u z z y variables N(al, bl) . . . . . N(a,,, bn), and al . . . . . a,, non-zero scalars. Then Z = ~,i'- i ~iXi becomes N(~,'i'_l aiai, ~ ' 1 ~ibi).
3, Estimation of parameter a with b known Let X be a normal fuzzy variable defined on pattern space (F, 5, o-) and gx(X)=~(X=x)=exp
-
T
'
X can be imagined as some quantitative representation of an object. To estimate a under the condition that p a r a m e t e r b is assigned a known value, we take experiments on the object for n times with Xi being the corresponding quantitative representation for the i-th experiment. We assume that all the experiments are taken in an unrelated way, that is, X~ . . . . . X,, are unrelated. Furthermore we assume that X1 . . . . . Xn are N(a, b) and the i-th experiment stops with Xi -- xi. Now we can proceed to discuss how to estimate p a r a m e t e r a with the above information.
3.1. Point estimation m e t h o d ( P E M ) Let Z = Z ( X j . . . . . X,,) denote an estimate of some p a r a m e t e r Z by use of X1 . . . . . X,,. Then X,,) is called a point estimate of Z. To identify a point estimate of p a r a m e t e r a in our regime, we put
Z(X~ . . . . .
Z=
X l +...
+X n
Since X~ . . . . . X,, are unrelated N(a, b), then from L e m m a 2.1 we know that so is Z. This implies IEZ = ~X = a. This is a desirable property. It just means that the expected value of Z is consistent with that of X and is equal to a. At this time Z is said to be an unbiased estimate of a. So with {xl . . . . . x,,}, a realization of
182
Cai K a i - Y u a n / Parameter estimations o f normal f u z z y variables
{X~ . . . . , An}, a point estimate of parameter a is given by X I + • . . @Xn H
3.2. Maximum scale likelihood (MSL) method Let L(xl . . . . . X n ) = O ' ( X 1 - - X l . . . . , X n ~ - X n ) where o ' ( X l = x l , . . . , X , = x n ) is the scale that X1 = x l . . . . , X , = x , . We call L ( x l , . . . , x,) scale likelihood function. Evidently L ( X l , . . . , x , ) represents the possibility of {X1 = Xl . . . . , X,, = x,} and the estimate of parameter a should make the possibility achieve its maximum. Since X1 . . . . . Xn are unrelated N(a, b), we have L(Xl,
. . . , x,)
=
i (xi a)2]
l~i~,mino'(Xi = xi) = l~i~,mine x p , -
-7
.
Then we an see that to make L(Xl . . . . , xn) achieve its maximum is just equivalent to making max Jxi- a l
e~ =
l<~i<_n
achieve its minimum. This implies that the estimate of a is given by
1
/
~=~
xi+ m i n x i • l<~i<~n
/
3.3. Interval estimation method (1EM) As in the case of probabilistic statistics, sometimes point estimates are not desired. In this situation we prefer interval estimates. Let
Z=
Xl+...+X
n
n
Since Z is a N(a, b), we can show that given o~, a unique e . can be determined to satisfy
This is just to say that a takes value beyond the interval [Z - bed, Z + be~] with possibility a. Then ~ can be interpreted as the risk that a does not lie in the interval [ Z - b e ~ , Z + b e ~ ] . We call [Z - bed, Z + be,] confidence interval with risk a. It just indicates that we can 'almost surely' assert that Z - be~ <~a <~Z + be~ is valid. Given risk c~, the confidence interval [Z - bed, Z + be~] is accordingly identified. And given { X l , . . . , x n } , a realization of { X 1 , . . . , X,}, the corresponding realization can be determined.
3.4. Example 1 It has been shown that fuzzy variables provide a powerful tool for characteizing software reliability behavior [1, 2]. In the software validation phase, the time to the next software failure, denoted by T, can be viewed as a normal fuzzy variable with parameters (a, 1), that is
tZr(t) = e -(t-a)2 where a can be interpreted as the expected time to the next software failure (ETSF) and is a key metric to represent the software reliability level. An important task in the software validation phase is to estimate ETSF. Let us consider an actual example [8]. It is decided that the objective value of ETSF of
Cai Kai-Yuan / Parameter estimations of normal fuzzy variables
183
a software package is 10 CPU hours. In the software validation phase testing activities are conducted such that three software failures are experienced with 1] -- 8, T2 -- 16, T3 = 62, in units of CPU hours. Now the problem is whether this software has met its ETSF objective. By use of the PE method, we can obtain 8 + 16 + 62
86
3
3
Then the answer is positive as a result of 86/3 > 10. According to the MSL method, we arrive at a = 1(max{8, 16, 62} + rain{8, 16, 62}) = 35. This indicates the answer is also positive. To apply the IEM method, we choose a - - e -4. Then we can say that a lies beyond [ 8 6 / 3 - 2 , 8 6 / 3 +2] with possibility e 4. Since 8 0 / 3 > 10, we are convinced that the software has surpassed its objective. In sum, conclusions drawn through PEM, MSL and IEM methods are consistent. The ETSF objective of the software has been met.
4. Estimation o f parameter b with a k n o w n
As formulated in the last section, here we also assume that X I , . . . , Xn are unrelated N(a, b) and x~ . . . . , x~ are their realizations. Then given possibility ~, we can identify e, such that e=,i
1,...
=
=a.
That is, with risk a, we assert that ~-~
~
This implies maxl~i~
Then how to determine an estimate of b? We note that tr(L(xi - a ) / b l > e~) just represents the possibility that xi lies beyond the confidence interval [ a - b e , , a + be~] with e~ predetermined. Obviously an estimate of b should make the length of the confidence interval achieve its minimum. This implies that the estimate of b is determined by 6 = maxl~i~. [xi - a[
5. Estimation o f u n k n o w n parameters a and b Let X1 . . . . . X,, be unrelated N(a, b) and x~, . . . , xn are their realizations. Then given possibility c~, we can identify e~ such that
-I
i--
"I
''I
°
184
Cai K a i - Y u a n / Parameter estimations o f normal f u z z y variables
Then with risk a, we can assert I(xi - a)/br <~ e~ or a - bed <~xi <<-a + bed. That is, maxl~i~n x i - minl~i~
Similar as in the last section, an estimate of b should make the length of the confidence interval [a - bed, a + bed] achieve its minimum. That is, the estimate of b is given by /3 = maxl~
max xi + m i n
2 \l~i~n
Xi •
1~'~
In this way, there hold a -/3e~ = minl_<~ xi and a +/3e, = m a x ~ < ~ x~.
6. Example 2 Consider a hardware reliability experiment. Suppose six hardware components are exposed to the experiment and their times to failure {X1,.. •, X6} are just fuzzy variables with membership function [_(x tzx,(x)=exp
- a~ 2]
\
b
/ J'
i = l . . . . ,6.
At the beginning of the experiment all the components are perfectly functioning and put into operation. The experiment goes on until all the components fail to function and their times to failure are recorded. In this experiment the times to failure for the six components are, respectively, 8, 23, 38, 80, 115, 200 hours. Since both parameters a and b are known, we employ the method proposed in Section 5 to determine their estimates 6 and/3. Let a = e-4; then e~ = 2. In this way we have d = ~(max{8, 23, 38, 80, 115,200} + min{8, 23, 38, 80, 115,200}) = 104, /3 = J(max{8, 23, 38, 80, 115,200} - rain{8, 23, 38, 80, 115,200}) = 48. Now we assume that parameter a is known with a = 104. Then we can use the method proposed in Section 4 to estimate parameter b. That is /3 = maxl~i~<6 IXi 2
--
1041 ---=96 48. 2
7. Concluding remarks Although fairly much attention has been paid to theoretical applications of Nahmias' fuzzy variables, little effort has has been focused on the related parameter estimation issues. This limits the practical use of fuzzy variables. In the sections presented above, we have discussed parameter estimation issues for normal fuzzy variables and considered three distinct situations: estimation of a with b known, estimation of b with a known, and estimation of unknown a and b. This may construct a bridge linking theoretical studies and engineering practice. However, we should also note that parameter estimations of non-normal fuzzy variables and stability behavior of parameter estimates arc still falling within the scope of open problems.
Cai Kai-Yuan / Parameter estimations of normal fuzzy variables
185
References [1] K.Y. Cai, A fuzzy model for software reliability validation, Acta Aeronautica et Astronautica Sinica (1993, in press) (in Chinese). [2] K.Y. Cai, C.Y. Wen and M.L. Zhang, A novel approach to software reliability modeling, Microelectronics and Reliability (1993, in press). [3] K.Y. Cai, C.Y. Wen and M.L. Zhang, Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets and Systems 42 (1991) 145-172. [4] K.Y. Cai, C.Y. Wen and M.L. Zhang, Posbist reliability behavior of typical systems with two types of failures, Fuzzy Sets and Systems 43 (1991) 17-32. [5] H. Dishkant, About membership functions estimation, Fuzzy Sets and Systems 5 (1981) 141-147. [6] H. Kwakernaak, Fuzzy random variables- I, lnform. Sci. 15 (1978) 1-29. [7] H. Kwakernaak, Fuzzy random variables - II, Inform. Sci. 17 (1979) 253-278. [8] J.D. Musa and A.F. Ackerman, Quantifying software validation: When to stop testing?, IEEE Software 6 (3) (1989) 19-27. [9] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems 1 (1978) 97-110. [10] M.B. Rao and A. Rashed, Some comments on fuzzy variables, Fuzzy Sets and Systems 6 (1981) 285-292. [1l] P.Z. Wang, Fazzy Set Theory and Its Applications (Shanghai Publishing House of Science and Technology, 1983) (in Chinese). [12] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.
179
Parameter estimations of normal fuzzy variables Cai Kai-Yuan* Department of Automatic Control, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Received April 1992 Revised July 1992
Abstract Although fairly much attention has been paid to theoretical applications of Nahmias' fuzzy variables, little effort has been focused on the related parameter estimation issues. This keeps down the practical use of fuzzy variables. In this note we make a preliminary discussion on parameter estimation of normal fuzzy variables with parameters (a, b) and distinguish three situations: estimation of a with b known, estimation of b with a known, and estimation of unknown a and b. Formulae for calculating parameter estimates by use of practical collected data are presented.
Keywords: Fuzzy variable; parameter estimation.
1. Introduction
Since Nahmias introduced an elegant definition for fuzzy variables in terms of pattern space [9], theoretical applications of fuzzy variables have been drawing fairly much attention. For example, Kwakernaak defined fuzzy random variables as random variables with fuzzy variables as values [6, 7], while Cai et al. laid fuzzy variables as a basis for posbist reliability theory [4, 5]. However, little attention has been paid to parameter estimation issues of fuzzy variables except in the case of fuzzy software reliability modeling [1, 2]. This keeps down the practical use of fuzzy variables. To mention parameter estimation related work, we should note that various methods have been presented to estimate membership functions under the head of 'fuzzy statistics' [11]. Fuzzy statistics postulates that the analytic form of tXx, the membership function of fuzzy set X defined on base set U--{x}, is unknown and attempts to identify an estimate of IXx(X) for a given point x. On the other hand, Dishkant interpreted fuzzy variables in Zadeh's sense, that is, as mappings from ~ (the real number line) to the unit interval [0, 1], and discussed the corresponding parameter estimation issue [5]. He viewed a to be estimated parameter A of the membership function as a Zadeh fuzzy variable and assumed that/x(x, t), the truth value of assertion "If A has the value t, then the fuzzy variable X has the value x" is a priori known. Then a method was presented to identify an 'optimal' value for A. In fuzzy software reliability modeling [1, 2], however, Cai interpreted fuzzy variables in Nahmias' sense and viewed parameters of membership functions as real numbers. Then a method was proposed to estimate parameter a of membership function e -(x-"): through defining a scale likelihood function L(x~ . . . . , xn). In this note we interpret fuzzy variables in Nahmias' sense and confine ourselves to a discussion of parameter estimation methods for normal membership functions. In Section 2 we present preliminaries for Nahmias' fuzzy variables. In Section 3 we discuss various methods for estimating parameter a with parameter b known and present an actual example. Section 4 is devoted to estimating parameter b with parameter a known, whereas Section 5 discusses the parameter estimation problem when both Correspondence to: Dr. Cai Kai-Yuan, Department of Automatic Control, Beijing University of Aeronautics and Astronautics, Beijing 100083, China. * Supported by the National Natural Science Foundation of China.
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180
parameters a and b are unknown. An example is presented in Section 6 to illustrate the methods proposed in Sections 4 and 5. Concluding remarks are contained in Section 7.
2. Preliminaries For completeness, in this section we present preliminaries for Nahmias' fuzzy variables [9]. Definition 2.1. For a base set F, suppose that ~3 is the class of all subsets of F. Suppose a scale, o-, is defined on ~ and satisfies the following properties: (i) tr(0) = 0 and v-(F) = 1. (ii) For any arbitary collection of sets As of ~ (finite, countable or uncountable), o ' ( U Z ~ ) = sup o(A~). o~
Then o- is a scale measure and the triple (F, ~, or) is referred to as pattern space. Definition 2.2. A fuzzy variable X is a mapping from F to ~ (the real number line). Definition 2.3. The membership function of a fuzzy variable X, denoted by tZx, is a mapping from R to the unit interval [0, 1] and is given by ~ x ( x ) = o-(-/: x ( - / ) : x)
for all x E R. Note that SUPxtZx(X) = t r { Y (Y: X ( T ) = x ) } = tr(F) = 1. In general we use the briefer notation X = x to denote the subset {3': X(3,) = x} o f ~g. It has been shown that the value of tZx(X) at point x can be interpreted as the possibility that X = x holds [3], though we are not asked to adopt Zadeh's definition of possibility measure [12]. Therefore the membership function of X can be viewed as possibility distribution of X and we arrive at: Definition 2.4. The possibility distribution function of a fuzzy variable X, denoted by rex or tZx, is a mapping from R to the unit interval [0, 1] and is given by Jrx(X) = tXx(X) = t r ( X -- x) for all x E E. Now we introduce the concept and terminology of 'unrelatedness', though Rao and Rashed preferred an alternative terminology 'min-relatedness' [10]. Definition 2.5. Given a pattern space (F, ~, tr), the sets A1 . . . . , An c ~3 are said to be mutually unrelated if for any permutation of the set { 1 , . . . , n}, denoted by il . . . . . i~ for 1 ~
~r(A.~)).
Definition 2.6. Given a pattern space (F, cg, o-), the fuzzy variables X1 . . . . . Xn are said to be (mutually) unrelated if for any permutation of the set {1. . . . . n}, denoted by il, • • . , ik for 1 ~ k ~
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181
Definition 2.7. Let X be a fuzzy variable with m e m b e r s h i p function/Xx. A real n u m b e r m is said to be a modal value of X, i f / x x ( m ) = 1. Definition 2.8. A fuzzy variable X is said to be unimodal, if there exists a unique a c ~ such that txx(a) = 1. At this time we signify •X = a. Finally we introduce the notion of normal fuzzy variables.
Definition 2.9. A fuzzy variable X is said to be normal if the membership function is of form
where a e N and b > 0. At this time we briefly say that X is a normal fuzzy variable with p a r a m e t e r s (s, b) or denote X as N(a, b). Then we have the following result [9]:
Lemma 2.1. Let Xx . . . . . Xn be unrelated f u z z y variables N(al, bl) . . . . . N(a,,, bn), and al . . . . . a,, non-zero scalars. Then Z = ~,i'- i ~iXi becomes N(~,'i'_l aiai, ~ ' 1 ~ibi).
3, Estimation of parameter a with b known Let X be a normal fuzzy variable defined on pattern space (F, 5, o-) and gx(X)=~(X=x)=exp
-
T
'
X can be imagined as some quantitative representation of an object. To estimate a under the condition that p a r a m e t e r b is assigned a known value, we take experiments on the object for n times with Xi being the corresponding quantitative representation for the i-th experiment. We assume that all the experiments are taken in an unrelated way, that is, X~ . . . . . X,, are unrelated. Furthermore we assume that X1 . . . . . Xn are N(a, b) and the i-th experiment stops with Xi -- xi. Now we can proceed to discuss how to estimate p a r a m e t e r a with the above information.
3.1. Point estimation m e t h o d ( P E M ) Let Z = Z ( X j . . . . . X,,) denote an estimate of some p a r a m e t e r Z by use of X1 . . . . . X,,. Then X,,) is called a point estimate of Z. To identify a point estimate of p a r a m e t e r a in our regime, we put
Z(X~ . . . . .
Z=
X l +...
+X n
Since X~ . . . . . X,, are unrelated N(a, b), then from L e m m a 2.1 we know that so is Z. This implies IEZ = ~X = a. This is a desirable property. It just means that the expected value of Z is consistent with that of X and is equal to a. At this time Z is said to be an unbiased estimate of a. So with {xl . . . . . x,,}, a realization of
182
Cai K a i - Y u a n / Parameter estimations o f normal f u z z y variables
{X~ . . . . , An}, a point estimate of parameter a is given by X I + • . . @Xn H
3.2. Maximum scale likelihood (MSL) method Let L(xl . . . . . X n ) = O ' ( X 1 - - X l . . . . , X n ~ - X n ) where o ' ( X l = x l , . . . , X , = x n ) is the scale that X1 = x l . . . . , X , = x , . We call L ( x l , . . . , x,) scale likelihood function. Evidently L ( X l , . . . , x , ) represents the possibility of {X1 = Xl . . . . , X,, = x,} and the estimate of parameter a should make the possibility achieve its maximum. Since X1 . . . . . Xn are unrelated N(a, b), we have L(Xl,
. . . , x,)
=
i (xi a)2]
l~i~,mino'(Xi = xi) = l~i~,mine x p , -
-7
.
Then we an see that to make L(Xl . . . . , xn) achieve its maximum is just equivalent to making max Jxi- a l
e~ =
l<~i<_n
achieve its minimum. This implies that the estimate of a is given by
1
/
~=~
xi+ m i n x i • l<~i<~n
/
3.3. Interval estimation method (1EM) As in the case of probabilistic statistics, sometimes point estimates are not desired. In this situation we prefer interval estimates. Let
Z=
Xl+...+X
n
n
Since Z is a N(a, b), we can show that given o~, a unique e . can be determined to satisfy
This is just to say that a takes value beyond the interval [Z - bed, Z + be~] with possibility a. Then ~ can be interpreted as the risk that a does not lie in the interval [ Z - b e ~ , Z + b e ~ ] . We call [Z - bed, Z + be,] confidence interval with risk a. It just indicates that we can 'almost surely' assert that Z - be~ <~a <~Z + be~ is valid. Given risk c~, the confidence interval [Z - bed, Z + be~] is accordingly identified. And given { X l , . . . , x n } , a realization of { X 1 , . . . , X,}, the corresponding realization can be determined.
3.4. Example 1 It has been shown that fuzzy variables provide a powerful tool for characteizing software reliability behavior [1, 2]. In the software validation phase, the time to the next software failure, denoted by T, can be viewed as a normal fuzzy variable with parameters (a, 1), that is
tZr(t) = e -(t-a)2 where a can be interpreted as the expected time to the next software failure (ETSF) and is a key metric to represent the software reliability level. An important task in the software validation phase is to estimate ETSF. Let us consider an actual example [8]. It is decided that the objective value of ETSF of
Cai Kai-Yuan / Parameter estimations of normal fuzzy variables
183
a software package is 10 CPU hours. In the software validation phase testing activities are conducted such that three software failures are experienced with 1] -- 8, T2 -- 16, T3 = 62, in units of CPU hours. Now the problem is whether this software has met its ETSF objective. By use of the PE method, we can obtain 8 + 16 + 62
86
3
3
Then the answer is positive as a result of 86/3 > 10. According to the MSL method, we arrive at a = 1(max{8, 16, 62} + rain{8, 16, 62}) = 35. This indicates the answer is also positive. To apply the IEM method, we choose a - - e -4. Then we can say that a lies beyond [ 8 6 / 3 - 2 , 8 6 / 3 +2] with possibility e 4. Since 8 0 / 3 > 10, we are convinced that the software has surpassed its objective. In sum, conclusions drawn through PEM, MSL and IEM methods are consistent. The ETSF objective of the software has been met.
4. Estimation o f parameter b with a k n o w n
As formulated in the last section, here we also assume that X I , . . . , Xn are unrelated N(a, b) and x~ . . . . , x~ are their realizations. Then given possibility ~, we can identify e, such that e=,i
1,...
=
=a.
That is, with risk a, we assert that ~-~
~
This implies maxl~i~
Then how to determine an estimate of b? We note that tr(L(xi - a ) / b l > e~) just represents the possibility that xi lies beyond the confidence interval [ a - b e , , a + be~] with e~ predetermined. Obviously an estimate of b should make the length of the confidence interval achieve its minimum. This implies that the estimate of b is determined by 6 = maxl~i~. [xi - a[
5. Estimation o f u n k n o w n parameters a and b Let X1 . . . . . X,, be unrelated N(a, b) and x~, . . . , xn are their realizations. Then given possibility c~, we can identify e~ such that
-I
i--
"I
''I
°
184
Cai K a i - Y u a n / Parameter estimations o f normal f u z z y variables
Then with risk a, we can assert I(xi - a)/br <~ e~ or a - bed <~xi <<-a + bed. That is, maxl~i~n x i - minl~i~
Similar as in the last section, an estimate of b should make the length of the confidence interval [a - bed, a + bed] achieve its minimum. That is, the estimate of b is given by /3 = maxl~
max xi + m i n
2 \l~i~n
Xi •
1~'~
In this way, there hold a -/3e~ = minl_<~ xi and a +/3e, = m a x ~ < ~ x~.
6. Example 2 Consider a hardware reliability experiment. Suppose six hardware components are exposed to the experiment and their times to failure {X1,.. •, X6} are just fuzzy variables with membership function [_(x tzx,(x)=exp
- a~ 2]
\
b
/ J'
i = l . . . . ,6.
At the beginning of the experiment all the components are perfectly functioning and put into operation. The experiment goes on until all the components fail to function and their times to failure are recorded. In this experiment the times to failure for the six components are, respectively, 8, 23, 38, 80, 115, 200 hours. Since both parameters a and b are known, we employ the method proposed in Section 5 to determine their estimates 6 and/3. Let a = e-4; then e~ = 2. In this way we have d = ~(max{8, 23, 38, 80, 115,200} + min{8, 23, 38, 80, 115,200}) = 104, /3 = J(max{8, 23, 38, 80, 115,200} - rain{8, 23, 38, 80, 115,200}) = 48. Now we assume that parameter a is known with a = 104. Then we can use the method proposed in Section 4 to estimate parameter b. That is /3 = maxl~i~<6 IXi 2
--
1041 ---=96 48. 2
7. Concluding remarks Although fairly much attention has been paid to theoretical applications of Nahmias' fuzzy variables, little effort has has been focused on the related parameter estimation issues. This limits the practical use of fuzzy variables. In the sections presented above, we have discussed parameter estimation issues for normal fuzzy variables and considered three distinct situations: estimation of a with b known, estimation of b with a known, and estimation of unknown a and b. This may construct a bridge linking theoretical studies and engineering practice. However, we should also note that parameter estimations of non-normal fuzzy variables and stability behavior of parameter estimates arc still falling within the scope of open problems.
Cai Kai-Yuan / Parameter estimations of normal fuzzy variables
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