E L S E V I E R Fuzzy Sets and Systems 94 (1998) 171-183

FUZZY sets and systems

On possibility analysis of fuzzy data M i i n - S h e n Y a n g * , M a n - C h u n L i u

Department of Mathematics, Chung-Yuan Christian University, Chung-Li 32023, Taiwan

Received January 1996; revised August 1996

Abstract

This paper addresses the analysis of fuzzy data from a possibilistic perspective. The concept of a double fuzzy variable is introduced and related notions are defined and investigated. The properties and fuzzy parameter estimation of the possibility distribution are also investigated. A maximum possibility likelihood principle is provided to estimate unknown fuzzy parameters. A normal possibility distribution is used as an example and estimates of its fuzzy parameters are given. The proposed possibility theory of fuzzy data constitutes a new approach to analyzing fuzzy data from a possibilistic point of view. © 1998 Elsevier Science B.V.

Keywords. Possibility space; Possibility distribution function; Fuzzy variable; Double fuzzy variable; Fuzzy parameter estimation; Possibility likelihood function

1. Introduction

Fuzzy sets introduced by Zadeh [16] give an approach to treating uncertainty which is different from that o f probability. Probability treats the uncertainty of occurrence that is described by a probability distribution function. On the other hand, a fuzzy set conveys the idea of uncertainty of belongedness described by a membership function. The use of a fuzzy set provides imprecise class membership information and is widely applied in diverse areas such as control, cluster analysis, decision making, engineering systems, etc. See, for example, [2, 3, 5, 11-13, 17].

Since the advent o f computer technology, the types of data in practical applications have become more complicated. Fuzzy data have commonly appeared in diverse systems; therefore, the analysis of fuzzy data has become increasingly important [1, 14, 15]. Although there has been much research on statistical analysis of fuzzy data [7, 8], there have been fewer discussions concerning possibility analysis of fuzzy data. The objective of this paper is to construct a possibility theory for the analysis o f fuzzy data. The concept o f a double fuzzy variable is proposed as a new approach to analyzing fuzzy data.

Zadeh [18] first proposed fuzzy sets as a basis for a theory of possibility. There are many discussions and applications of possibility theory [4]. What is possibility? What are the differences between possibility and probability? In Section 2, a simple comparison will be made and the definition of a fuzzy variable which

* Corresponding author.

0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PH S01 65 -0 11 4( 9 6 ) 0 0 2 5 9 - X

172 M.-S. Yang, M.-C Liu/Fuzzy Sets and Systems 94 (1998) 171-183

had been defined elsewhere will be presented. This fuzzy variable has been used in the context o f analyzing crisp data from a possibilistic perspective. In Section 3, the implicit features of fuzziness on a double level are interpreted and a double fuzzy variable is defined. The possibility distribution and fuzzy modal value of a double fuzzy variable are also defined and properties investigated. Consequently, a double fuzzy variable becomes a tool for handling fuzzy data in a possibility space. This constitutes a new approach to possibility analysis o f fuzzy data.

Section 4 is concerned with fuzzy parameter estimation. Similar to the maximum likelihood principle in statistics, a maximum possibility likelihood principle is provided to estimate unknown fuzzy parameters. Particular concern is given to the normal possibility distribution and to the estimation of its fuzzy parameters. Concluding remarks are given in Section 5.

2. Preliminary definitions

A "possibility" was first proposed by Zadeh [18], who used fuzzy sets as a basis for a theory of possibility. Possibility is used to treat an uncertainty which is different from that of probability. Then a possibility space and a fuzzy variable were proposed by Nahmias [9]. This fuzzy variable has been used as the analysis of crisp data from a possibilistic point o f view [4, 6, 9, 10, 18]. In this section, some preliminary definitions are given.

Definition 2.1. Let f2 be a set o f all possible outcomes of a random experiment and let F be a collection of subsets o f ~2; that is a a-field. Let P be a set function from F to [0, 1] which satisfies the following:

(a) P(12) -- 1. (b) For any event A EF, P(A)>~O. (c) For any countable collection of events {A1 . . . . . An . . . . } in F with A i f') A j = 0 for i ~ j , one has

P(Ui=~Ai) = ~, e(Ai). Then (~,F,P) is called a probability space and P is called a probability measure.

Similarly, a possibility space and a possibility measure are defined as follows [9].

Definition 2.2. Let F be a universe of discourse and let fq be the power set o f F. Let /7 be a set function from f# to [0, 1] which satisfies the following:

(a) H ( 0 ) = 0, /7 (F)= 1. (b) For any index set I with A i E ~ , iEL one has I I ( U i E I A i ) = supiEl l l ( A i ) . Then (F, f#,/7) is called a possibility space and /7 is called a possibility measure.

In comparing the definitions of probability and possibility, one finds that the additivity of a probability measure P is replaced by a supremum in a possibility measure H. This is the principal difference between them. A probability is used to represent the uncertainty of randomness, but a possibility describes the uncertainty of fuzziness. It is known that the occurrence of "head" or " ta i l" when tossing a coin is random, which is an uncertainty of probability. Natural concepts such as young, hot, good and various others of humanness, cannot always be exactly described. These are examples of uncertainties of fuzziness. I f one assigns a random restriction to a probability space, one can obtain a random variable (r.v.) with its probability distribution function. For example, if one assigns a random restriction X as the total number of occurrences of "heads" in n tosses of a coin with a probability p of occurrence of "heads" , then X is a random variable with its probability distribution function f x (x )

f x ( x ) = P ( X = x ) = (n~ p x ( 1 - p) n-x, x = 0 , 1 . . . . . n a n d 0 o t h e r w i s e . \ x /

M.-S. Yang, M.-C Liu/Fuzzy Sets and Systems 94 (1998) 171-183 173

Therefore, one simply represents the relationship as follows:

r.v.X ( f2 ,F ,P) , (~ ,B , Px ),

random restriction

where the r.v. X:(2 ~ E is a F - B measurable real function. I f one assigns a fuzzy restriction to a possibility space, one can obtain a fuzzy variable (f.v.) with its

possibility distribution function. Consider an example in Zadeh [18]: the proposition "John is young". It is known that "young" is a fuzzy term. Suppose that the membership function of "young" is

1, x~<25,

Pyoung(X) = (1 + (½(x _ 25)2)_1, 25 <x~<200.

In this case, a fuzzy restriction, "young", is associated with a fuzzy variable X defined by age. It is natural to assign a possibility distribution function 7x of X as the membership function of "young". That is, 7x =/£young. A fuzzy variable X is used to transform the membership function ]Ayoung to a possibility distribution function Vx. This relationship can be represented as follows:

f .F.X ( r , ~¢,II) , (m,B, ~x ).

fuzzy restriction

Similarly, a fuzzy variable and its possibility distribution function are defined as follows [9, 18]:

Definition 2.3. Let (F , f~ ,H) be a possibility space. A real-valued function X : F ~ ~ is called a fuzzy variable. A possibility distribution function Yx of X is defined as

7x(x) = I 1 ( { t E r l x ( t ) =x}) for any x E ~ .

It is known that 7x has two properties: (i) 0~<Tx(X)~<l for all x E ~ , and

(ii) supxER ~/x(X)= 1.

The fuzzy variable and the possibility distribution function as defined above have been used for possibility analysis of crisp data [18, 9, 6]. In the next section a double fuzzy variable which can be used for possibility analysis of fuzzy data is defined.

3. Double fuzzy variable and its properties

In Section 2, some preliminary definitions about a fuzzy variable are defined as the possibility analysis of crisp data. Since fuzzy data have commonly appeared in diverse systems, the analysis of fuzzy data has become increasingly important. In this section, the concept of a double fuzzy variable which is a fuzzy perception of a fuzzy variable is introduced and related properties are also investigated. This constitutes a new approach to the possibility analysis of fuzzy data.

Definition 3.1 (~t-cut, Z immermann [19]). The s-cut of a fuzzy set A with pA(X) as its membership function is defined by

A~ = {xlm(x)~>=} for ~E(0,1].

174 M.-S. Yang, M.-C. Liu/Fuzzy Sets and Systems 94 (1998) 171-183

Definition 3.2 (Normality, Zimmermann [19]). The fuzzy set A is normal if IXA(X) attains 1 at some x in the space over which A is defined.

Definition 3.3 (Fuzz), number, Zimmermann [ 19]). The normal fuzzy set A whose membership function, de- fined on [~, is piecewise continuous and whose e-cut A~ is a closed and finite interval in [~, is called a fuzzy number.

Throughout this paper the set of all fuzzy numbers as defined above is denoted by d~(R). Note that if

A E ,~-(~), then any ~-cut A~ can be represented by the closed interval [A~,A~] where A~ = min{A~} and

A-~ = max{A~}. Hence, the set {[A~,A-~] ] eC(0 , 1]} becomes a representation of A.

Definition 3.4 (Equality o f fuzzy numbers). Let A and B be any two fuzzy numbers in ~ ( ~ ) . I f A~ = B~

and A-~ -- B~ for all e E (0, 1], then one can say that A and B are equal. The equality of A and B is denoted b y A = B .

Now, a new variable called a double fuzzy variable is introduced.

Definition 3.5 (Double juzzy variable). Let (F,f¢,I1) be a possibility space. A fuzzy-valued function X * : F--+,~-(E) is called a double fuzzy variable (d.f.v.).

According to the definition of the d.fv. X*, it is seen that for any e c (0, 1], (X*)~ and (X*)~ are both fuzzy variables and the linear combination of d.fv.s is also a d.fv. Since any x* in Y ( N ) can be represented by {[(x*)~,(x*)~] I e E (0, 1]}, {[(X*)~,(X*)~] ] a E (0, 1]} becomes a representation of the d fv . X*. Suppose

that 7(x*) and ~"(~=5~ are possibility distribution functions of (X*)~ and (X*)~, respectively. Then a possibility

distribution function of X* is defined as follows:

Definition 3.6 (Possibility distribution Jimction of the d.fv. X* ). Let X* : F---+,~-(R) be a d.fv. X* is said to have a possibility distribution function (poss. d.f ) 7x* (x*); or equivalently X* ~ 7x* if the following holds: for any e E ( 0 , 1],

(a) ',,'~x*) is a poss. d. f of (X*)~ and

(b) ?'(~=5~ is a poss. d . f of (X*)~, with [°/(x*), 7(-2zSfl being the e-cut of 7x*(x*).

Definition 3.7 (Joint poss. d . f ) . Let Xi* : F-+o~-([~ ) be d.fv.s, i = 1 . . . . . n. (Xl*, . . . . X*) is said to have a joint poss. d.f 7(x....,x2) if the following holds: for all e E ( 0 , 1],

(a) ((XI*) , . . . . (X~*)) ~ 7((x,.),...,(x,7)), and

(b) ((X I )~, . . . . (X,;)~) ~ 7(~*) ,...,(x,*~)-

The following definition introduces min-relatedness of double fuzzy variables in a manner that is similar to that introduced by Rao and Rashed [10] for fuzzy variables.

Definition 3.8 (Min-relatedness of d.fv.s). The d.fv.s XI*, . . . . X,~ (m.m.r.) if for all e E ( 0 , 1] and (xl . . . . . x n ) E N n,

(a) 7((x,.)/...,(xT))(xl . . . . . x,,) min{7(x.) (xl) . . . . . )(x,7) ( ,,)}, and

(b) 7((x* )~,...,((a;7 )~)(xt . . . . . xn ) = min{7(~5.) (xl) . . . . . 7(x~5. ) (xn)}.

are said to be mutually min-related

M.-S. Yang, M.-C Liu/Fuzzy Sets and Systems 94 (1998) 171-183 175

Definition 3.9 (Modal value of a Juzzy variable, Rao and Rashed [10]). Let X be a fuzzy variable with a possibility distribution function 7x- A real number m is said to be a modal value of X if 7x(m) = 1. If the number m is unique, then it is said that X is unimodal and ~ ( X ) = m is written.

Lemma 3.1 (Rao and Rashed [10]). Let Xi . . . . . X~ be m.m.r, unimodal fv .s . Then ~ ( X 1 + -.- + Am) = ,¢l(x~ ) + . . . + ~,#(x, ).

Now, the fuzzy modal value of a double fuzzy variable as an extension of the modal value of a fuzzy variable is defined. It is defined by using the concept of original-perception proposed by Kwakemaak [8] and Kruse [7]. It is noted that fuzzy data usually come from the subjectivity o f human beings, the expression of natural language, or an imprecision in measurement, etc. One is interested in going back to find the original data which have produced these fuzzy perceptions. Of course, it is impossible to find the exact original. But one can find the compatibility between perception and its original. In fact, one may represent the measure of compatibility by a possibility distribution function. Assume, for example, that a double fuzzy variable X* can be considered as the perception o f a fuzzy variable X and X is considered as the original of X*. Let W be the set of all fuzzy variables on (F, f¢,/ /) , i.e.

/1" = {XIX:F-- -+R is a f .v.}.

Then one defines the compatibility Cx . (X ) of X* to its original X as

Cx* (X) = inf{px,( t)(X(t)) ] t E F},

where Px*(t) is the membership function o f the fuzzy set X*(t). A definition o f the fuzzy modal value of a d.fv. is given as follows:

Definition 3.10 (Fuzzy modal value of a d.fv. ). Let X* : F --~ Y ( N ) be a double fuzzy variable. A fuzzy number Jd(X*) in , 7 ( R ) is defined as

p //(x*)(t) = sup{Cx.(X) [X E Y',.:tl(X) = t}.

Then ,¢/(X*) is called a fuzzy modal value o f X* and the symbol ,//g is called a modal operator.

Next, some results which relate to the modal operator ~ are given.

Theorem 3.1. Let X* be a d.fv. with its juzzy modal value .~(X*). Then, for any :~E(0, 1], one has (a) J l ( (X* )~) = ,~¢Z(min(X* )~) = min(~#(X* ))~ = (.A'(X*)~), and

(b) ,d//((X* )~ ) = ~tl(max(X* )~) = max( J l (X* ))~ = (J/{(X*)~).

Proofi It is known that (X*)~ eY" with Cx.((X*)~) = e. Assume that ,~P/((X*)~)=t. That is, 7 ( x * ) ( t ) = 1.

Then I~//(x*)(t) = sup{Cx.(X)I X E f , . J / / ( X ) = t} = c~. But ~/z(x.)(t) = sup{~'Ii,#(x.l)~,(t)lcd E (0, 1]} where l(i/(x*)); is the indicator function, and (Jd(X*))~ is a closed interval. It is concluded that t = min(otd(X*))~ = (Jl(X*))~; therefore the assertion (a) holds. The proof o f assertion (b) is similar to (a).

[]

Note that Theorem 3.1 presents the exchangeability of operators vii and "min" (or "max").

Theorem 3.2. Let X* be a d.fv. with its fuzzy modal value ,rid(X*). Then, Jor any ~ E (0, 1], one has (~,sl(x*)) ~ = [ ~ ( ( x * ) ~ ), J/~((x* )~)].

176 M.-S. Yano, M.-C Liu/Fuzzy Sets and Systems 94 (1998) 171-183

Proof. Since J / (X* ) is in ~(g~), then for any ~ c ( 0 , 1], the a-cut (J//(X*))~ of J t (X*) can be expressed as a closed interval. Then,

( J t (X* ))~ = [min(~'(X* ))~, max(J l (X* ))~].

--- [J t (min(X*)~),J / / (max(X*)~)] (By Theorem 3.1)

= [~#((x*)~), ~ ' ( (x*)~) ] . []

Theorem 3.3. Let X~ . . . . . X; be m.m.r, unimodal d.fv.s and 2i >O for i = 1 . . . . . n. Then J//(21XI* + . . . + 2 n X ; ) = ~ l ~ / ~ ( Y l * ) --~- • • • ~- 2 n ~ / [ ( X ; ) .

Proof. First, it is claimed that for any aE(0, 1],

(x~* + . . . + x,*)~ = (x~)~ + . . . + (x;)~. (1)

As n = 2, for any ~ c F and a E ( 0 , 1], it is shown that

(X~*(:,) + X2"(7))~ = (X~*(7))~ + (X:*(~,))~.

(a) Let xj C(Xl*(7))~ and x2 E(X:*(7))~. Then,

#x*(-,.)+~*(;,)(x, +x2) = sup min{px*(:,)(t),p~*(,;)(s)} x, +x2 = t + s

>~ min{px,*().)(xl ),/~*(~,)(x2)} >/min{~, a} = a.

That is, xl +x2 E(XI*(7)+X2"(7))~; therefore,

(x,*(~))~ + (x:*(~))~ c (x~*(,/) + x2"(7))~.

(b) Let xE(XI*(7 ) +X2"(7))~. That is,

#x,*(;,)+~*(;.)(x) = sup min{Itx,*(.~,)(t),pxi(.,,)(s)} >_.o~. X = l + s

t oo • Then, there exists a sequence { k}k=t so that for all k, tk 6 (X, (7))~(1-~) and x - tk 6(X2"(7))~O_~). Since Xl*(7) E ~(g¢), (X,*(7))~(,_½) is a closed interval. Therefore, there is a subsequence {vk}~l of {tk}~l SO

V oe that {Vk}~-l and {m,*(;,)( k)}k=l are convergent. One has

I~x'*(')(t)=l~x'*(')(lim vk) >~ li~m t~x'*(';)(vk)>>" lim : t ( 1 - 1 ) k oc k--~

One can choose another suhsequence {cok}~c_l of {tk}~_-i SO that {~ok}~=l and {/tx*(7)(x - ~ok)}~_-i are con- vergent. Then,

px:(~,)(x t)=llx:C,) (x l ima)t~ lim px.(~.)(x- o~)>_-lim e (1 ~ k ) - ~ k ~ /I ~---.*~¢~ k - - - - ~

That is, t E (Xt*(7))~ and x - t ~ (X2"(7))~. Then x = t + (x - t)~(X~*(7))~ + (X2"(7))~, i.e. (X~*(7) +X2"(7))~ c(x,*(~))~ + (x~*(~/))~.

By (a) and (b), one has

(x~(~) + xi(,/))~ = (x~*(~))~ + (x:*(~))~.

M.-S. Yan9, M.-C. Liu/Fuzzy Sets and Systems 94 (1998) 171-183

Then,

(x? (~) + . . . + x;(~,))~

= ((x,*(7) + . . . + x,*_~(7)) + x,*(~))~ = (x~*(7) + . . - + x,*_,(~,))~ + (x,*(~))~

. . . . . (x?(~))~ + . . . + (x; (~))~.

That is, for any ~E(0, 1], one has

(x? + . . . + x ; ) ~ = (x~*)~ + . . . + (x;)~.

(Jz(;~lx? + . . . + ~ ,x; ) )~

= [Jg((21Xl* + . . . + 2,X,*)~),~/g((21Xl* + . . . + 2,X,*)~)] (By Theorem 3.2)

= [Jg(()qX1*)~ + . . . + (2,X*)~),J/t'((2,Xl*)~ + - . . + (2,X*)~)] (By Eq. (1))

= [ J ( ( , ~ l X l * ) : ( ~ - ' - " -I- (/~nX*)),J//[((21X~)g -~"" ~- ( 2 . X * ) a ) ]

= [ ~ ( ( ~ x ~ * ) ) + . . . + Jz(O~,X,*)~), :z((~lx~*)~) + . . . + Jz((;~,xyL)]

: [/~1 J~[ ( (Xl*)a ) "~- • " • ~- /~ .J~/ / ( (Xn*)~) , )q ~ / / ( ( X l * ) a ) -I- - • - -}- )~ .~{ ( (Xn*)~ ) ]

= hl [ J # ( ( X l * )~ ), J / / /((Xl* )x )] -~- " " ' -[- ")~ n [ ~ { ( ( X ; )x ), ~ { ( (Xn*)x )]

= / ~ l ( ~ ( X l * ) ) x A w . . . -[- ~ n ( ~ / ' ( X ] * ) ~ ) .

Therefore, J//()qXl* + . . . + 2nX~, ) = 21./#(Xl*) + " " + 2,,J//(X~, ). []

(by Lemma 3.1 )

(Since •i > O)

177

4. Fuzzy parameter estimation

The concept of a fuzzy variable has been used to analyze crisp data from a possibilistic point of view. A double fuzzy variable as introduced in Section 3 is proposed as a tool for possibility analysis of fuzzy data. Let X be a fuzzy variable with possibility distribution function 7x(x; 0), where 0 is an unknown parameter. Problems which deal with the estimation of 0 were addressed in [9, 10]. Let X* be a double fuzzy variable with possibility distribution function 7x*(x*; 0"). In this section, the estimation of the fuzzy parameter 0* is investigated. Particularly, the focus is directed toward the normal possibility distribution function.

Definition 4.1 ( N o r m a l f u z z y variable). Let X be a fuzzy variable with a possibility distribution function

It is said that X has a normal possibility distribution function, and X is called a normal fuzzy variable, denoted by X ~ N(a , b).

Note that a normal fuzzy variable represents fuzzy observations which are "approximate a" with a dispersion Ibl in a bell shape.

178 M.-S. Yang, M.-C~ Liu/Fuzzy Sets and Systems 94 (1998) 171 183

Definition 4.2. A double fuzzy variable X* is said to have a poss. d.f yx*(X*; 0") if for any ~ E (0, 1],

(Y*)~ ~ y(x.)(x;(O*)~) and (X*)~ ~ y(-2-z5 (x;(0*)~).

The correspondence between X* and its poss. d.f is denoted by X* ~ 7x*(x*; 0").

Definition 4.3 (Normal d.fv. ). A double fuzzy variable X* is called a normal d.fv. and denoted by X* N(a*,b*) i f for any ~ C (0, 1],

(X*) ~N((a*)~,(b*)~) and (X*)~N((a*)~ , (b*)~) .

L e m m a 4.1 (Nahmias [9]). Let X1 . . . . . X~ be mutually rain-related (m.m.r.) normal fuzzy variables with poss. d.fs. N(al,bl ) . . . . . N(a, ,b , ) , respectively. Let 21 . . . . . 2n be positive scalars. Then, ~i~1 2iXi is a normal

N n 2 n fuzz)' variable with (}-~'~i=l iai, }-~i=1 2ibi) as its poss. d.f

Theorem 4.1. Let X* and Y* be mutually min-related (m.m.r.) normal d.fv.s with X* ~ N(aT,b~) and Y* ~ N(a~, b~). Then, X* + Y* ~ N(a~ + a~, b~ + b~).

Proof . Since X* ~ N(a'(,b~) and Y* ~ N(a~,b~), then for any c~ E (0, 1]

* * b* (X*) ~ N ( ( a l ) , ( b l ) ~ ) , ( X * ) ~ N ( ( a ~ ) ~ , ( 1 ) ~ ) ,

(Y*)~ ~ N((a~) ,(b~)~), (Y*)~ ~ N((a~)~,(b~)~).

By Lemma 4.1, one has

and

(X*)~+(Y*)~ ~ N((a~)~+(a~) ,(b~)~+(b~) )

ta*~ tb*~ + ( b 2 ) ~ ) . ( X * ) ~ + ( Y * ) ~ N ( ( a ~ ) ~ + ~ 2,~,~ 1,~ *

By Eq. (1),

( X * + Y * ) ~ N ( ( a ~ + a ~ _ ) , ( b ~ + b ~ ) ), ( X * + Y * ) ~ N ( ( a ~ + a ~ ) ~ , ( b ~ + b ~ ) x ) .

By Definition 4.3 it is concluded that

X* + Y* ~ N(a~ + a~,b~ + b~). []

Theorem 4.2. Let X* be a normal d.fv. with X* ~ N(a*, b*). Let 2 be a positive scalar. Then, 2X* N(2a*, 2b*).

Proof . X* ~ N(a*,b*) means that for any ~ E (0, 1], (X*)~ ~ N((a*)~,(b*)~) and (X*)~ ~ N((a*)~,(b*)~).

By Lemma 4.1 one has )~(X*) ~N(2(a*) ,2(b*)~) and 2(X*)~N(2(a*)~,2(b*)~) . That is, 2 X * ~

N(2a*,2b*). []

Combining Theorems 4.1 and 4.2, one has the following theorem.

M.-S. Yang, M.-C. Liu/Fuzzy Sets and Systems 94 (1998) 171-183 179

Theorem 4.3. L e t Xl*, . . . . X ; be m.m.r, normal d . f v . s with poss. d . fs . denoted by N ( a ~ , b ~ ) , . . . , N ( a ~ , b ~ ) , respectively. Le t 21,...,)~n be posi t ive scalars. Then,

£ ~ i X i * ~ N ( ~ i a T ' ~ i b ; ) i=1 i=1

Theorem 4.4. Le t X * be a normal d . fv . with X * * * = N ( a ,b ). Then, / g ( X * ) a*.

Proof . By Theorem 3.2, for any ~ d (0, 1],

(~¢~(X*))~ = [ ~ ( ( X * )~), ~ ( ( X * )~)].

X* ~ N(a* , b*) means that

( X * ) ~ N ( ( a * ) , (b*)~) and (X*)~ ~ N((a*)~, (b*)~) .

But, (X*)~ and (X*)~ are fuzzy variables. Hence, by Definition 3.9

~#((X*)~) = (a*)~ and ~ ( ( X * ) ~ ) = (a*)~.

a* Thus, (~#(X*))~ = [( ) , ( a* )~ ] = (a*)~. i.e. ~/ ' (X*) = a*. []

According to the concepts of original and perception, a normal double fuzzy variable with its fuzzy modal value shall be a perception of a normal fuzzy variable with its modal value. That is, if X ~ N(a , b) and X * ~ N ( a * , b * ) then ~ g ( X ) = a and J 4 ' ( X * ) = a* where a* is a fuzzy perception of a.

Now the estimation problem of a fuzzy parameter 0* is studied. The possibility likelihood function (PLF) is defined, and the maximum possibility likelihood (MPL) principle is proposed.

Definition 4.4. Let XI* . . . . . X* be m.m.r, d . f v . s all with the same poss. d . f yx . (x*;O*) , O* c ~ (~) . The quantity T , ( X I * , . . . , X * ) of XI* . . . . . X* which does not involve any unknown fuzzy parameter 0* is called a fuzzy estimator of 0*.

* * *. * Definition 4.5. Let X1* . . . . ,X~ be d . f v . s with a joint poss. d . f 7x....,x,7(x I . . . . ,Xn, 0 ). A possibility likelihood function (PLF) is defined as

L(O ;x 1 . . . . . x . ) = yx,*,...,x,7(x 1 . . . . . x . , ).

Note that l f X 1 . . . . . X,~ are m . m . r . d . f v . s all with the same poss. d . f c x . ( x ;0 ) then for any ~ c (0, 1],

* 0 * , = min ~'(x*) (xi;( )~), L((O*)~;Xl . . . . x~) y(x?),..,(x,7) (xt . . . . . x . ; (O )~) = ]<.i<.,,

and

, rain (0")~). L( ( O* )~; Xl . . . ,xn ) = yix?)~,...,(x;L(xl . . . . . xn; (0")~) = l ~i <,n Y(-YzS~ (xi;

Let fuzzy data x]' . . . . . xT, be m.m.r, from the poss. d . f 7x*(x*;O*). Then, a fuzzy estimate Tn(x'( . . . . . x~) of 0* which maximizes the PLF L(O*;x T . . . . . x~) would seemingly be a good and reasonable estimate of 0* because it would provide the largest possibility of these particular fuzzy observations x~ . . . . . x~. This maximum possibility likelihood (MPL) principle is defined as follows:

180 M.-S. Yang, M.-C. L i u / F u z z y Sets and Systems 94 (1998) 171 183

Defin i t ion 4.6 ( M P L principle). The op t imiza t ion procedure max0* L(O*;x~ . . . . . x~) which leads to the fuzzy es t imate Tn(x~ . . . . . x~) o f 0* is cal led the M P L principle , where max0. L(O*;x~ . . . . . x~) means that for any

C (0, 1],

and

max L((O )~, (x I )~ . . . . . (x , ) ) (0*___2)

max L((O* )~; (x~')~ . . . . . (xT,)~). (0")~

Now, N(a*,b*) is taken as an example , and a me thod o f f inding fuzzy es t imators o f a* and b* is given.

E x a m p l e 4.1. Let XI*, . . . . X,* be mutua l ly min- re la ted (m.m.r.) normal d.fv.s all with the normal poss. d . f N(a*,b*). Then, for any ~ C (0, 1], (Xl*) , . . . . (An*) are m.m.r, normal f v . s all wi th the normal poss. d . f

L,(b )O. N((a*) , (b*)~) and ~tX*~I J~, • • •, ~IX*~ j~ are m.m.r, normal f v . s all wi th the normal poss. d . f N((a* *

and

The fo l lowing three cases wil l be o f concern: ( a ) F ind a fuzzy es t imator o f a* when b* is known: For any c¢ C (0, 1], one has

* • * * min exp - L((a ) ~ , ( x l ) , . . . . (x,)~) = l<~i<-%n ~ (b*)a

. . . . min (exp[ a* ll } L((a )~,(x 1 )~ . . . . . (x,)~) = l<~i~n = ~ , r - . (b )~

Then max(a.__2) ~ L((a*) ; (x T)~ . . . . . ( x ~ ) ) is equivalent to

min m a x ] (x~')~ - (a*)~l. (a*) l<~i<~n

A That is, the op t imizer (a*)~ is

, l max * + min ( x * ) (a )~ = 5 \ l <~i<~n (xi )~ l<~i<~n

A X* Similar ly , one has an op t imizer (a*)~ o f L((O*)~; ( 1 )~ . . . . . (x~)~) as

- / ) • l m a x ( x ] ' ) ~ + m i n ( x * ) ~ .

A A a* a* .. •, That is, for any ~ E (0, 1], one has (a*)~ = [( ) , ( )~]. A s s u m e that the member sh ip functions/~x~, #x; o f

fuzzy data x~ . . . . . x~, are un imoda l wi th the same shape. Let xl . . . . . xn be the moda l values o f x T . . . . . x~, respect ively . Then, one defines maxl<~i<nx~ as the fuzzy number with the same shape whose

M.-S. Yan 9, M.-C. L i u / F u z z y Sets and Sys tems 94 (1998) 171 183 181

modal value is a maximum of {xl . . . . . Xn} and minl~i~<nx* as that whose modal value is a minimum of {xl . . . . ,xn}. Therefore, one has the fuzzy estimator 4" of a* with

4 " = ½ ( m a x * + m i n x * ) . \ l ~ i < ~ n x i l<~i<~n

(b) Find a fuzzy estimator of b* when a* is known: Let fuzzy variables )(1 . . . . . Xn be m.m.r, all with the normal poss. d . f N (a ,b ) where a is known. Then,

the PLF L(b;Xl . . . . . xn) is

, min exp - . L(b;xl . ,x , ) = l<i<n

Thus, maxbL(b, xl . . . . . x , ) is equivalent to minb maxl <~i<<., ] (x i - a)/bl. Note that h ( b ) = maxl .<i<, I ( x i - a)/bl is a decreasing function of b over b > 0, i.e. h(b) --+ 0 which is minimized when b --* oc. But this unbounded minimizer (estimator) o f b ---, oc is not meaningful. The problem is how large b is chosen. It would seemly be suitable to restrict the bound of b with maxl <i<, ]xi-al so that the estimator/5 of b with D = maxl <i<,]xi-al is chosen. The chosen estimator D becomes the distance between the modal value a and the point o f data set which is farthest from a. It has been mentioned that a normal d.fv. can be viewed as a perception of a normal fuzzy variable. A new operator " m a x d " is defined as follows: Let Xl . . . . . x, and a be the modal values of x• . . . . . x* and a*, respectively. Let Xk be such that [xk - a[ = maxl <~i<~, [xi - al. One defines maxl<~i<~nd(xT,a* ) = x k*-a* if lxk--a[ = x k - - a and maxl~i<~nd(x[,a*) = a* - x k* if lxk--a[ = a - - x k where " - " is an "extended subtraction" fuzzy operator. Thus, one has a fuzzy estimate b* of b* corresponding to /~ = maxl~i~<, [xi - a[ as

= max d(x i , a ). I <~i<~n

That is, the fuzzy estimate b* is defined as the maximum fuzzy distance between fuzzy data {x T . . . . . x~} and the known modal value a*.

(c) Find fuzzy estimates 6* and/~* of a* and b* when a* and b* are both unknown: Since )(1" . . . . ,An* are m.m.r, normal d. fv .s all with the normal poss. d . f N(a*,b*), by Theorem 4.3 one has

~-~ * ( ~ - ~ a* ~ ) Xin ~ N n ' = N(a*, b*).

i=1 i=1 i=1

j g , n n Therefore, ( ~ i = l Xi*/n) = a*, and one has a fuzzy estimate ~i* of a* as 4" = Y'~i=l Xi*/n. By case (b) one b* has the fuzzy estimate b* of b* as = maxl <~i~, d(x[,d*). That is,

d* ~ - [ x * and b* ( * ~ 2 - 2 ~ ) = - - = max d xi , . gl I <~ i <~ n i=1 \ j=l

Example 4.2. A "comfortable" feeling is studied. Assume that the fuzzy term "comfortable" has a normal possibility distribution function and the collected data are fuzzy data. That is, one has a normal double fuzzy variable X* with the normal poss. d . f N(a*, b*). One would like to know what a* - "the most comfortable temperature" and b* - " the dispersion of approximate a" are? Assume that the fuzzy data set is given as follows:

23" ,26" ,25" , 27",29", . . . . . . 26"(°C) 24 ,25 ,30 ,28 ,24 ,28 ,

182 M.-S. Yang, M.-C. Liu/Fuzzy Sets and Systems 94 (1998) 171-183

where fuzzy data xT, i = 1, . . . , 12 are all assumed triangular fuzzy numbers T(x,c~,/3) with 2 = / 3 = 3. Note that a fuzzy number T(x, c~,/3) is called a triangular fuzzy number if its membership function is o f the form

{ 1 - ( ~ - ~ ) for y~<x(~ > 0),

r e ( y ) =

1 - ( X ~ - f ) f o r x < ~ y ( / 3 > O ) .

Since a* and b* are both unknown, by Example 4.1

10 d . = 1 ~-'~ •

- ~ x i = 26.25" = T(26.25, 3, 3), i=1

~ * . ^ , = max d(x i ,a ) = 30* - 26.25* = T(30,3 ,3) - T(26.25,3,3) = T(3.75,6,6).

i<~i<~12

5. Conclusions

Fuzzy set theory has been used to handle uncertainty which is different from that o f randomness. Similar to a random variable, the fuzzy variable proposed by Nahmias [9] is used to carry out the possibility analysis o f crisp data. Possibility theory has become an approach to computerized processing of uncertainty which is suitable for the analysis of complex systems [4]. But in real complex systems fuzzy data have commonly appeared more than crisp data. Fuzzy data are easily found in natural language, social science, psychomet- rics, environmetrics, econometrics, etc. In this paper a double fuzzy variable has been proposed as a fuzzy perception of a fuzzy variable. This double fuzzy variable can be used to carry out the possibility analysis of fuzzy data. Fuzzy parameters appear in a possibility distribution function when the observed data are fuzzy. Fuzzy estimations concerning these fuzzy parameters have also been investigated. We expect that this initial mathematical construction on possibility analysis of fuzzy data can be successfully used in solving problems of real complex systems. This approach to fuzzy data analysis will be applied to build a Chinese medicine diagnostic system in a future research project.

Acknowledgements

The authors are grateful to the referees for suggesting improvements and critical reading of the manuscript.

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