Fuzzy random variables revisited - Semantic Scholar

Fuzzy Random Variables Revisited Dan A. R&lescu*t Department of Systems Science Tokyo Institute of Technology 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan and T h e Institute of Statistical Mathematics 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

1 Introduction

Abstract We review the concept of fuzzy random variable, its expected value, and limit theorems for sequences of fuzzy random variables. We point out shortcomings in some concepts and results that have been defined in the literature. Finally, we study two inequalities involving the expected value of a fuzzy random variable: the Brunn-Minkowski inequality and the Jensen inequality. We explore different extensions of the former, and give an analog for the latter. Potential applications of our results are to the analysis of fuzzy random variables and to statistics with inexact data.

*On leave from the Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025,U.S.A. ‘Research supported by the National Science Foundation Grant INT-9303202.

Fuzzy-valued functions, as an extension of set-valued functions, were considered in connection with a categorical theory of fuzzy sets, since the mid 1970’s (see Negoita and Ralescu [16].Indeed, even earlier, fuzzy-valued functions with some special properties were studied: fuzzy relations (Zadeh [28]). However, fuzzy-valued functions defined on a probability space, came a t a later date. Different concepts: fuzzy variables, linguistic variables, fuzzy random sets, fuzzy random variables, probabilistic sets, were introduced in the late 1970’s. F6ron [5] defined a fuzzy random set as a random element with values L-fuzzy sets in a topological space. However, he only studied some algebraic properties, and he did not define any concept of expected value. Nahmias [15]defined fuzzy variables as ordinary random variables on a space endowed with both a probability and a pos-

0-7803-2461-7/95/$4.00 0 1995 IEEE

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sibility measure. His is not a concept of fuzzy-valued function. Hirota [6], one of the first to study relationships between probability and fuzziness defined probabilistic sets, but he did not discuss any concept of expected value, and he concentrated on properties which are very similar to those of fuzzy relations. Kwakernaak [13] introduced the concept of f i z z y random variable whose values are fuzzy sets on the real line. His concept of measurability is such, however, that it cannot be extended beyond fuzzy numbers in W. As a matter of fact, that concept of measurability is not the proper concept needed in this framework (see later). Also his extensive discussion is often obscuring in what is the simplest example of fuzzy random variable (i.e. a family of random intervals in W). Since 1980 (see the early papers [20], (211) we have defined and studied the concept of fuzzy random variable (FRV) whose values are fuzzy subsets of (or, more generally, of a Banach space). We have investigated the proper notions of measurability, integrability, metric concepts. We have defined the expected value and studied limit properties (Puri and Ralescu [19]). Our concept is much more general, and it includes [13]. An important subject in this framework is that of limit theorems for fuzzy random variables (e.g. strong law of large numbers and central limit theorem). In [21]we have announced a strong law of large numbers (SLLN) for fuzzy random variables in W".Independently, Kruse [ll]proved an SLLN for fuzzy random variables whose

w"

w).

values are convex fuzzy numbers (in His methods, however, cannot be used in more general cases and his SLLN does not include the corresponding result for random sets, thus it does not include the classical SLLN. T h e result announced in [21] was proved by Klement, Puri, and Ralescu [9], [lo]. Note that the latter result is valid for fuzzy not necessarily convex. It subsets of completely subsumes the result of Kruse [ll] (see below) and it extends the SLLN for random sets of Artstein and Vitale [l]. Miyakoshi and Shimbo [14] gave a result similar to that of [ll]. Let us recall that Klement, Puri and Ralescu [9], [lo] were the first to prove a central limit theorem (CLT) for fuzzy random variables in A few years later, Boswell and Taylor [3] gave a CLT by using the concept of FRV of Kwakernaak [13] and a new concept of independence that they have proposed. We show below that their concept of independence is so strong that it collapses and so their result is nothing else but the classical CLT for random variables! A law of large numbers for fuzzy variables (Nahmias, [15]) was given by Stein and Talati [23]. However, their result is just the well known Bernoulli law of large numbers. Niculescu and Viertl [17] have proved an extension of the Bernoulli law of large numbers for a very particular (essentially translates of a single membership function) kind of fuzzy random variables in W. We show that their result is a simple consequence of the Klement, Puri, Ralescu

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w",

w".

SLLN. That also gives us the extension of the Bernoulli law to more general fuzzy sets in R". Interesting extensions of the SLLN were obtained by Inoue [7]. An in-depth study of the expected value was done in [27] and various measurability concepts were investigated by Butnariu [4]. An ergodic the* rem for fuzzy random variables in Banach space appears in B i n [2]. Recently, a number of papers have " r e discovered" some of the old results (a Sample of these papers: [24], (291, [30]). Essentially, they do not contain anything new. The remainder of this paper contains: Section 2, in which we very briefly describe the mathematical framework we need and we state the Klement, Puri, and -1escu SLLN; Section 3, in which we critically discuss various other concepts and results as suggested above. Finally, in Section 4 we explore the Brunn-Minkowski and the Jensen inequalities for fuzzy random variables.

2

The law of large numbers

where d H is the Hausdorff distance. Let d l be the distance [9]:

1

1

d l ( u ,),, =

d H ( ~ , u , ~ , ~ ) d(2)~

Definition 1. ((211, 1191). Any measurable map : R -+ F(R") is called a f u z z y r a n d o m variable (FRV).

x

Definition 2. ([21], [19]). The e w c t e d value of the FRV X is the unique fuzzy set E X such that L , ( E X ) = E ( L d ) for (2 E [O,11, where E ( L , X ) is the Kudo [12] integral (see [I]).

Theorem (SLLN Of 1912 [lo]). Let { X k l k 2 1) be i n d e p e n d e n t a n d identically distributed FRV'S Such that EIIsuPpX1II < O0. Then

(x1+ x2+ . . + x,,, E ( c o X d ) ,

71

-+

0

(3) a l m o s t surely.

Here d stands for d l or d, (in the latter case we have to assume that { x k } are separably valued).

Let (R, A, P ) be a probability space, and 3 Critical discussion of let F(Rn)be the space of all fuzzy sets u : Other W" -+ [0,1] whose levels L,u = { z I u ( z ) 2 CY} are compact, and nonempty for a > 0, (a) Kwakernaak [13] coined the term fuzzy and whose support is compact. Let dm be random variable. He defined his concept the distance in F(Rn)first introduced in for F ( W ) only, i.e. X : R -+ F ( W ) . His Puri and Ralescu [NI): measurability concept c a n n o t be extended actually that measurability conto F(Rn); doo(ur') = sup U>O d H ( L a u ' (') cept is unrelated to any of the (equivalent)

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measurability concepts for spaces of sets of independence. Let us briefly show that this independence concept is so strong (cf. [81). For example, consider the random set that it can only be satisfied if the FRV’s are ordinary random variables. So their X = [ A ,B ] U [C,D ] result collapses into the classical, well where A , B , C , D : 0 -+ and A , D are known CLT.

w,

We don’t even have to consider FRV’s; measurable, while B = -C are nonmeasurable, and A I I D. Then random sets will suffice. Their concept of Kwakernaak, is: X , Y =e “i~dependent”if every sex is an FRV in the

but the distance dist(0,X) = c is not measurable. Actually, in the of Definition 1,in the particular case n = 1 and convex Values, an FRV X has levels Lax = [U,, V,], where U,, V, are ordinary random variables.

’

‘

e lector f x and are independent. They do this, to obtain E ( X Y ) = E X E Y , for “independent” x i y .It is not to show

x,

ProPosition 1 (i) if { I } are %dependent”, then X = {q) (i.e. x is a single random variable. (b) The Kmse LLN [ll]is proved for (ii) If X , Y are “independent”, then FRV’s which are convez and valued in a X = {q}, Y = { I } . subset F1 C F(R).Also convergence is pointwise convergence of the membership (d) The Niculescu- v i e d l [iq] extension functions. The first drawback is that space of the Bernoulli LLN can be deduced as F1 does not include the ordinary intervals a particular case of the Klement, puri, - thus this SLLN does not extend the Clas- Ralescu SLLN (Theorem 1). Their idea k sical SLLN of probability theory. At any rate, it is possible to show that is t o extend ~ I A ( X = ~ )‘‘total “km of i=l in F1 convergence in metric d , implies x i ’ s which belong to A” (where A c I[$, pointwise convergence, thus the SLLN of x i , . . . , zk E R) to fuzzy sets, by using the [9]contains the result of [ll]. extension principle. It can be shown that, Another shortcoming is that we don’t at least for the class FA of membership really know if class F1 is closed with refunctions U such that supAU and sup-U spect to operations of fuzzy arithmetic, are both attained, their concept can %e and if X E F1 whether E X E F1 or not? k = ‘‘number of The proper framework is to use a distance expressed as between fuzzy sets, such as (1) or (2). i=l which belong to A ” . (c) The Boswell-Taylor CLT [3]. These Then, a simple application of the Kleauthors also use the Kwakernaak FRV ment, Puri, Ralescu SLLN (Theorem 1) concept, but they define their own notion gives: If { X $ 2 1) are i.i.d. FRV’s val-

CzA(u,)

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depends on the dimension n. We can now prove

ued in FA,then -

1

n

n .

IA(Xi)

+

E[IA(Xl)]

Theorem 2 (The Bmnn-Minkowski inequality for fuzzy sets): Let U , v E F ( P ) . Then

1=1

= Probability (XI is A ) . k

V,[Au

If we take ~ I A ( u , )as the definition for "the number i=lof ui's which belong to A"

2

+ (1- X)v] (5)

[x (Vn(U))ll" + (1 - A) ( v n ( v ) ) 1 q n

(which with the Other definition if where Vn denotes the volume of a &zzy set, the ui's are numbers or ordinary sets), then the restriction of values to .FA is unProof: We have successively necessary, and we obtain the general rev& (1- X)v] sult. Incidentally, the restriction A C R is = [ h O , l ] (Vn(LCr(Au also unnecessary and we can assume A C R". +(1- A ) v ) ) ) 1 / n d a l n

+

Actually an application of Theorem 1 gives us the more general result

I n

-C v ( x i ) n .

-+

~ ( v ( ~ 1 )almost )

=

[J[O,l,vn ( A L d

+(1 - A)L,v))l'n

surely,

[x ( V n ( L a 4 ) l l n

2 [Jp,,,

where 2=1 v is a fuzzy set. Here the sum is interpreted as "number of Xi's which are v" (e.g. "number of short people who are smart").

da]

+(1- A) (v,(L,v)l~n] d a I n =

[x (Vn(U))l/"

+(1- A) (Vn(v))1/nln.

4

The

Brunn-Minkowski

We have used the classical BrunnMinkowski inequality as well as L,(Xu (1- A)v) = XLau (1 - A)L,v. It now follows that for a fuzzy random We first define the volume of a fuzzy set variable x we have U E F(R") as

and the Jensen inequalities

vn(u)=

[lo,ll

+

Vn(EX)

'I

2 (E(VJ)l/n)n

+

(6)

(4) which extends the result for random sets in [25]. where V,, inside the integral is ordinary Our next goal is to extend the Jensen volume. Note that this concept of volume inequality to FRV's. The main result is (vn (Lau))l'"da

997

interesting discussions with Anca Ralescu, as well as from the kind hospitality of Michi0 Sugeno.

(7)

References

An important example of such a convex function p is the Steiner point map; in that case (o is actually additive and positively homogeneous (i.e. "linear"). For this choice of p (7) becomes an equality. Our extension of the Jensen inequality seems to be unknown even if X is a random set. Potential applications of these inequalities are in the field of statistics with inexact data and, more specifically, at the evaluation of the power function of statistical tests based on fuzzy data (see the recent papers [22] and [26]).

5

Conclusions

We showed that the proper concept of fuzzy random variable [19] (in R" or in a Banach space) involves distances on spaces of fuzzy sets and measurability of random elements valued in a metric space. The most general limit theorems for fuzzy random variables are those in [lo]; all other results in this direction can be derived from them. Finally, we extended the BrunnMinkowski and the Jensen inequalities to fuzzy random variables. Acknowledgements. While working on this paper I greatly benefitted from many

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[7] Inoue, H. A strong law of large numbers for fuzzy random sets, Fuzzy Sets Syst. 41(1991) 285-291. [8] Klein, E. and Thompson, A.C. Theory of Correspondences, Wiley, New York, 1984.

(91 Klement, E.P., Puri, M.L., and Ralescu, D.A. Law of large numbers

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SBrie I,

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[18] Puri, M.L. and Ralescu, D. A. [27] Xue, X., Ha, M., and Ma, M. RanDiffkrentielle d'une fonction floue,

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