A law of large numbers for fuzzy numbers

Fuzzy Sets and Systems 45 (1992) 299-303 North-Holland

299

A law of large numbers for fuzzy numbers* R. F u l l 6 r

dual property with respect to (1):

Department of Operations Research, Computer Center, L. E6tvOs University, H-1502 Budapest 112, P.O. Box 157, Hungary

Nes(~[D) = 1 -

Received August 1989 Revised May 1990

Abstract: We study the problem: if ~1, ~2. . . . are fuzzy numbers with modal values M1, M2. . . . . then what is the strongest t-norm for which l i m N e s ( m , , - c -~< ~ l + " 'n+ ~ n < ~ m , , + e ) = l ,

for any e > 0

where m, = (M~ + • • • + M,,)/n, the arithmetic mean ( ~ + • .. + ~,,)/n is defined via sup-t-norm convolution and Nes denotes necessity.

Keywords: Possibility; probability; necessity; fuzzy number; triangular norm; law of large numbers; sequence of fuzzy numbers; convergence theorem.

1. Definitions A fuzzy number ~ is a fuzzy set of the real line R with a unimodal, normalized (i.e. there exists unique a • R such that ~ ( a ) = 1) and uppersemicontinuous membership function [4]. Given a subset D c I~, the grade of possibility of the statement 'D contains the value of ~' is defined by [11] Pos(~ID ) = sup ~(x).

(1)

x~.D

The quantity 1-Pos(~tD), where /5 is the complement of D, is denoted by Nes(~lD ) and is interpreted as the grade of necessity of the statement 'D contains the value of ~'. It satisfies * This work has been supported by the Hungarian Young Scholars' Fund under No. 400-0113 and the Hungarian Scholars' Foundation at project OTKA-606-1986.

Pos(~[/)).

If D = [a, b] c ~, then instead of Nes(~[[a, b]) we shall write Nes(a ~< ~ < b ) and if D = {x}, x • ~ we write Nes(~ = x). Let ~1, ~2 . . . . be a sequence of fuzzy numbers. We say that {~,} converges pointwise to a fuzzy set ~ (and write limn~o~ ~, = ~) if lim,__,~ ~n(x) = ~(x), for all x • I~. Given two fuzzy numbers, ~ and r/, their T-sum ~ + r1 is defined by [3, 12] (~ + r/)(z) = sup T(~(x), tl(y)),

x, y, z • ~,

x+y=z

where T is a triangular norm [10] (t-norm for short), i.e. T is a two-place function from [0, 1] x [0, 1] to [0, 1] such that T is symmetric, associative, non-decreasing and T(x, 1) = x, x • [ 0 , 11. The function H~ :[0, 1] x [0, 1]-+ [0, 1], where y > 0, defined by H,(u, v ) =

UU

y + (1 - 7)(u + v - uv)

is called the Hamacher norm with parameter (Hr-norm for short) [6]. A symmetric triangular fuzzy number denoted by (a, or) is defined as

~(t)={~-Ja-t[/°l

if l a - t l <-o~, otherwise,

where a • II~ is the modal value and 2o~ > 0 is the spread of ~. Let T~, T2 be t-norms. We say that T, is weaker than T2 (and write T1 <~ Tz) if Tl(x, y) ~< T2(x, y) for each x, y • [0, 1]. If ~ is a fuzzy number then its support denoted by supp ~ is defined as supp ~ = {t • IR : ~(t) > 0}.

0165-0114/92/$05.00 I~) 1992--Elsevier Science Publishers B.V. All rights reserved

3OO

R. Fuller / L a w o f large numbers for f u z z y numbers

2. Chebyshev's form o f the law o f large numbers

Proof. From the definition of necessity we have

Nes(~ = x) = 1 - Pos(~lR \{x}) = 1 - sup ~(t) t-~ x

We shall provide a fuzzy analogue of the following theorem [1].

~> 1 - sup r/(t) = Nes(T/= x), t--#x

t h e o r e m . If ~ , ~2 . . . .

is a sequence of pairwise independent random variables having finite variances bounded by the same constant, Chebyshev's

D~I<~C,

D~2<~C ....

,

The proof of the next two lemmas follows from the definition of T-sum of fuzzy numbers. L e m m a 2. Let TI, T2 be t-norms and let ~1, ~2 be f u z z y numbers. If T1 <~ T2, then

D~n<~C .....

and

(~1 ÷ ~2)1 ~ (~1 ÷ ~2)2

MI+'" "+Mn M = lira n---*oo n

exists, then for any

which ends the proof.

where (~1+ ~2)i denotes the Ti-sum of f u z z y numbers ~, ~l, i = 1, 2.

positive constant e,

3. Let ~i=(ai,~r), i = 1 . . . . . n, be f u z z y numbers of symmetric triangular f o r m and let T be a t-norm. Then Lemma

limProb( ~1+"'+~ n--,~

MI+'"+Mn

n


n

=1

supp(~ + . . .

where Mn = M~n and Prob denotes probability.

+ ~n)

~_ supp ~ + . • • + supp ~n + [a,, - a¢, a,~ + ol]

= [al - a~, al + a:] + . . .

= [a~ + . • • + a n - n o ~ , al + • • • a,, +not] 3. A law o f large numbers for fuzzy numbers

In this section we shall prove that if ~1 = (MI, o0, ~2--(M2, o 0 " " is a sequence of symmetric triangular fuzzy numbers and T is a t-norm (by which the sequence of arithmetic means ((~l + ' " • + ~,,)/n} is defined) then (a) the relation

limNes(m,, ~ l + e <~ '''+~n
--1,

\

where the sum ~1 + " " " + ~ is defined via sup-T convolution. Lemma

1, 2 . . . . .

'l. = ~1 + "" "+~n,

A.

= a 1 + • • • ÷ an,

we have

(i)

+e)

4. Let T = H o and ~i=(ai, o¢), i = n. Then with the notations

,Tn(z)=

n

for a n y e > 0

f

(2)

holds for any T ~
1+

IAnn o-~ zl IAn - zl (n - 1 ) n~

0

otherwise,

(ii)

1 - IA~/n - zl Ol

L e m m a 1. Let ~, rI be f u z z y sets of ~. If ~ ~ r1

(i.e. ~(x) <~rl(x), for each x e R) then Nes(~ = x) t> Nes(r/= x),

for each x e •.

if IAn - zl < nol,

1 + (n - 1)

IAn/n - zl

if IA./n - z[ < ol,

O~

otherwise.

R. Full& / L a w o f large numbers f o r f u z z y numbers

Proof. We prove (i) by making an induction argument on n. Let n = 2. Then according to Lemma 3 we need to determine the value of r/2(z) from the following relationship:

1

~l(X) "+ ~2(Y) - ~l(x)~2(y)

x+y=z

= sup x+e=~

IA. - x l

1

1 --4 ~l(x)

1 + (n - 1) - no:

1

1

~2(Y)

1

if z e (al + a2 - 2o~, al + a 2 + 2o0 and r/2(z) = 0 otherwise. According to the decomposition rule of fuzzy numbers into two separate parts [3], r/z(z), g E ( a 1 + a 2 - - 2o6 a~ + a 2 ] , is equal to the value of the following mathematical programming problem: 1 1

max

1

a~-x 1 - - -

1+

a 2 -

2o: a,

+ a 2 -

z

1

14--2tr

2tr

z

1

x -- a 1

1---

1

4 1

z - x - a2

,* max

(P2)

z -A: 2tr

z -A2 l + - 2a~

z - A,,+I (n + 1)a

if A,,+1 < z < A , + l + (n + 1)a,

Z-An+ 1

l + n - (n + 1)0c 0

otherwise,

1

IA, - nzl nt~

(rln/n)(z) -

1 + ( n - l) 1

]A, - nz[

noc

I A , / n - zl O~

1

subject to al < x < a l + tx, az < z - x < a 2 + o:. In a similar manner we get that the value of (P2) is 1


z e ~ [2], we have

and the solution of (P1) is x = ½ ( a l - a z + z ) (where the first derivative vanishes). If al + a 2 < - z < a l + a 2 + 2 t r , then we need to solve the following problem:

1

if An+ l -- (n + 1)tr

and the solution of (P3) is x = ( A , - - a n + 1 + z ) / ( n + 1) (where the first derivative vanishes). (ii) From the relationship ( r l , / n ) ( z ) = ri,(nz),

2a~

A 2 -

(n + 1)tr (n + 1)a

,1.+,(z) ='

A 2 -- Z

1---

An+ 1 - z

l +nA.+~-z

o{

z

1

1

(P3)

1

is

a~ +

la,+l-z +xl tx

subject to IA, - x l
subject t o a I - f f < x ~
1

Im,-xl na~

1

4

-----> m a x

(P1)

1

a2-z+x

1-

o/

and the solution of (P2) is x = ½ ( a l - a 2 + z ) (where the first derivative vanishes). Let us assume that (i) holds for some n ~ [~. Then

~n+l(Z) = (?~n "1- ~n+l)(Z)

~l(x)~z(Y)

r/2(z) = sup

301

l+(n-1)

[ A , / n - z[

if [A, - nz[ < n a , and O % / n ) ( z ) = 0 otherwise. This ends the proof. The following lemma shows that if instead of Necessity we used Possibility in (2), then every sequence of fuzzy numbers would obey the law of large numbers.

302

R. Fulldr / Law of large numbers for fuzzy numbers

5. Let T be a t-norm and let ~1, ~2. . . . be a sequence of fuzzy numbers with modal values M1, M2 . . . . . Then with the notations

e

Lemma

1---O(

=1 l+(n-1)

-e

~/n=~+ "''+~n,

O~

mn= (M, + " " + M,)/n

and, consequently,

we have

lim N e s ( m n - e<~rln <-mn + e)

Pos(r/n/n = mn) = 1,

n ---~

n e N.

n

e

From

1----

lemmas it follows that O1Jn)(m,) = 1, n e ~. This ends the proof.

Proof.

the

o~

= 1 - lim = 1. "--'= 1 + (n _ 1 ) _e O~

The theorem in question can be stated as follows: Theorem

1 (Law of large numbers for fuzzy

numbers). Let T <~Ho and let ~i = (Mi, o0, i • be fuzzy numbers. If M = lim n--,~

(ii) Since

IN

1

lim rln/n)(z)= lim . . . . . . 1 =

n

+ e / = 1,

x{M~(z),

Nes(!im ~ =

M ) = I - s u p r a ( J i m = 71n/n)(z )

/

= 1 -

(ii) Nes(lim r/n= M ) = 1, \,--,= n

sup XM(Z) = 1, z~M

where and

+ (n - 1 Im. ) - -zl

it follows that

(i) lim N e s ( m , - e ~ < r / ' ~ < m , n--,~ , n

MI+'" "+M~

oc o~

M,+.. "+M n

exists, then for any e > O,

mn =

Imn -- Z]

r/, = ~1 + . • • + ~,.

where Z{M} is the characteristic function of {M} (see Figure 1). This ends the proof.

n

1. Theorem 1 can be interpreted as a law of large numbers for mutually T-related fuzzy variables [9]. Strong laws of large numbers for fuzzy random variables were proved in [7, 8]. Remark

(i) If e I> cr then we get (i) trivially. Let e < oc. Then from Lemmas 1 and 2 it follows that we need to prove (i) only for T = Ho. Using Lemma 4 we get Proof.

1.2

Nes( mn-e<~rln <'mn + = 1- Pos(~ ---

(-oo, m , , - e ) U ( m , , + e , oo))

("%) x¢[mn-e,mn+e]\ n /

0.8

0,6

sup

1

Ira,, - (m,, + e)l

0.4

0,2

of

=1

i

1 + (n - 1)

Im, - (m, + e)l 0¢

i

i

M Fig. 1. The limit distribution of

~./n if T ~Ho.

R. Fuller / Law of large numbers for fuzzy numbers

303

4. Q u e s t i o n lr

Let T be a t-norm such that H0< T < ' m i n ' and let ~ l = ( M l , or), ~2 = (M2, o:) . . . . be a sequence of symmetric triangular fuzzy numbers. Does this sequence obey the law of large numbers, i.e. does l i m , ~ r / , / n =%{Mr follow from l i m , ~ rn, = M?

0.8 0.6 0.4

0.2

5. G u e s s

0 M--o(

M

M-~-(x

Fig. 2. The limit distribution of rl, ln if T = 'min'.

2. Especially, if T(u, v ) = Hi(u, v ) = uv, then we get [5] Remark

~imNes(mn-e<~<~m.+e) = 1 - (rl./n)(m. - e) = 1 - l i m (1 - t i e r ) n = 1.

The following theorem shows that if T = 'min', then the sequence ~1= (Mi, re), ~2= (ME, tr) . . . . does not obey the law of large numbers for fuzzy numbers. 2. Let T(u, v ) = min{u, v} and ~i = (Mi, ol), i e ~. Then for any positive e such that e < o: we have

Theorem

im~ Nes m~

-

n

Nes(lim r/n = M) = 0. \n~ n Proof. The proof of this theorem follows from the equalities ~ln/n =(mn, re), n e ~ and l i m , ~ On/n = (M, ol) (see Figure 2). R e m a r k 3. From the addition rule of LR-type fuzzy numbers via sup-min convolution [2], it follows that Theorem 2 remains valid for any sequence ~1 = (MI, tr)LL, ~2 = (M2, tr)LL. . . . of LL-type fuzzy numbers with continuous shape function L.

If T is an Archimedean t-norm [10], then every sequence of fuzzy numbers ff~, ~2. . . . . such that the diameter (diam) of their supports are bounded by the same constant:

diam(supp ~1) ~ C, diam(supp ~2) ~< C . . . . . obeys the law of large numbers for fuzzy numbers. References

[1] P.L. Chebyshev, On mean quantities, Mat. Sb. 2 (1867); Complete works, 2 (1948). [2] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). [3] D. Dubois and H. Prade, Additions of interactive fuzzy numbers, IEEE Trans. Autom. Control 26 (1981) 926-936. [4] D. Dubois and H. Prade, Linear programming with fuzzy data, in: J.C. Bezdek, Ed., Analysis of Fuzzy Information, Vol. 3: Applications in Engineering and Science (CRC Press, Boca Raton, FL, 1987) 241-261. [5] R. Full6r, On product-sum of triangular fuzzy numbers, Fuzzy Sets and Systems 41 (1991) 83-87. [6] H. Hamacher, (lber logische Aggregationen nicht biniir explizierter Entscheidung-kriterien (Rita G. Fischer Verlag, Frankfurt, 1978). [7] R. Kruse, The strong law of large numbers for fuzzy random variables, Inform. Sci. 28 (1982) 233-241. [8] M. Miyakoshi and M. Shimbo, A strong law of large numbers for fuzzy random variables, Fuzzy Sets and Systems 12 (1984) 133-142. [9] M.B. Rao and A. Rashed, Some comments on Fuzzy variables, Fuzzy Sets and Systems 6 (1981) 285-292. [10] B. Schweizer and A. Sklar, Associative functions and abstracts semigroups, Publ. Math. Debrecen 10 (1963) 69-81. [11] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [12] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, I, Inform. Sci. 8 (1975) 199-249; II, Inform. Sci. g (1975) 301-357; III, Inform. Sci. 9 (1975) 43-80.