- Email: [email protected]
On interactive fuzzy numbers
Available online at www.sciencedirect.com
Fuzzy Sets and Systems 143 (2004) 355 – 369 www.elsevier.com/locate/fss
On interactive fuzzy numbers Robert Full%era; b;∗ , P%eter Majlendera; b; c a
Department of Operations Research, Eotvos Lorand University, Pazmany Peter set. 1C, P.O. Box 120, Budapest H-1518, Hungary b - Akademi University, Lemminkainengatan 14C, Institute for Advanced Management Systems Research, Abo - FIN-20520, Finland Abo c - Akademi University, IAMSR Datacity B 6708, Lemminkainengatan 14B, Turku Centre for Computer Science, Abo - FIN-20520, Finland Abo Received 28 March 2002; received in revised form 28 February 2003; accepted 4 April 2003 Dedicated to Professor Christer Carlsson
Abstract In this paper we will introduce a measure of interactivity between marginal distributions of a joint possibility distribution C as the expected value of the interactivity relation between the -level sets of its marginal distributions. c 2003 Elsevier B.V. All rights reserved. Keywords: Joint possibility distribution; Marginal possibility distribution; Interactivity; Covariance; Variance
1. Introduction The concept of conditional independence has been studied in depth in possibility theory, for good surveys see, e.g., Campos and Huete [1,2]. The notion of non-interactivity in possibility theory was introduced by Zadeh [9]. Hisdal [5] demonstrated the di>erence between conditional independence and non-interactivity. In this paper we will introduce a measure of interactivity between fuzzy numbers via their joint possibility distribution. A fuzzy set A in R is said to be a fuzzy number if it is normal, fuzzy convex and has an upper semi-continuous membership function of bounded support. The family of all fuzzy numbers will
Partially supported by the Hungarian Research Funds OTKA T32412 and FKFP-0157/2000. A Corresponding author. Institute for Advanced Management Systems Research, Abo Akademi University, A LemminkEainengatan 14C, Abo FIN-20520, Finland. Tel.: +358-2-2654-066; fax: +358-2-2654-809. E-mail addresses: [email protected], [email protected] (R. Full%er), [email protected], [email protected] (P. Majlender). ∗
c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/S0165-0114(03)00180-5
356
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
Fig. 1. Non-interactive possibility distributions.
be denoted by F. A -level set of a fuzzy set A in Rm is deJned by [A] = {x ∈ Rm : A(x)¿} if ¿0 and [A] = cl{x ∈ Rm : A(x)¿} (the closure of the support of A) if = 0. If A ∈ F is a fuzzy number then [A] is a convex and compact subset of R for all ∈ [0; 1]. Fuzzy numbers can be considered as possibility distributions [9]. A fuzzy set B in Rm is said to be a joint possibility distribution of fuzzy numbers Ai ∈ F; i = 1; : : : ; m, if it satisJes the relationship max
xj ∈R; j =i
B(x1 ; : : : ; xm ) = Ai (xi );
∀xi ∈ R; i = 1; : : : ; m:
Furthermore, Ai is called the ith marginal possibility distribution of B, and the projection of B on the ith axis is Ai for i = 1; : : : ; m. Fuzzy numbers Ai ∈ F, i = 1; : : : ; m, are said to be noninteractive if their joint possibility distribution is given by B(x1 ; : : : ; xm ) = min{A1 (x1 ); : : : ; Am (xm )}, for all x1 ; : : : ; xm ∈ R. Furthermore, in this case the equality [B] = [A1 ] × · · · ×[Am ] holds for any ∈ [0; 1]. If A; B ∈ F are non-interactive then their joint membership function is deJned by A × B. It is clear that in this case any change in the membership function of A does not e>ect the second marginal possibility distribution and vice versa. On the other hand, A and B are said to be interactive if they cannot take their values independently of each other [3] (Fig. 1). Denition 1.1 (Full%er and Majlender [4]). A function f : [0; 1] → R is said to be a weighting function if f is non-negative, monotone increasing and satisJes the following normalization condition
1 0
f() d = 1:
Di>erent weighting functions can give di>erent (case-dependent) importances to -levels sets of fuzzy numbers. If f is a stepwise linear function then we can have discrete weights. This deJnition is motivated in part by the desire to give less importance to the lower levels of fuzzy sets (it is why f should be monotone increasing).
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
357
2. Central values In this section, we shall introduce the notion of average value of real-valued functions on -level sets of joint possibility distributions and we shall show how these average values can be used for measuring interactions between -level sets of its marginal distributions. Let B be a joint possibility distribution in Rn , let ∈ [0; 1] and let g : Rn → R be an integrable function. It is well known from analysis that the average value of function g on [B] can be computed by 1 C[B] (g) = g(x) dx dx [B] [B] 1 = g(x1 ; : : : ; xn ) dx1 : : : dxn : dx1 : : : dxn [B] [B] We will call C as the central value operator. Note 2.1. If [B] is degenerated for some ∈ [0; 1], that is dx = 0; [B]
then we approximate [B] with non-degenerated sets and consider the limit case. Namely, let {Sk } be a sequence of sets such that [B] ⊂ Sk ⊂ Rn , 0¡ Sk dx¡∞, and lim sup{ x − s | x ∈ [B] ; s ∈ Sk } = 0:
k →∞
Then, 1 dx Sk
C[B] (g) = lim CSk (g) = lim k →∞
k →∞
Sk
g(x) dx:
If g : R → R is an integrable function and A ∈ F then the average value of function g on [A] is deJned by 1 g(x) dx: C[A] (g) = dx [A] [A] Especially, if g(x) = x, for all x ∈ R is the identity function (g = id) and A ∈ F is a fuzzy number with [A] = [a1 (); a2 ()] then the average value of the identity function on [A] is computed by a2 () 1 1 a1 () + a2 () ; C[A] (id) = x dx = x dx = a2 () − a1 () a1 () 2 dx [A] [A] which remains valid in the limit case a2 () − a1 () = 0 for some . Because C[A] (id) is nothing else, but the center of [A] we will use the shorter notation C([A] ) for C[A] (id). It is clear that C[B] is linear for any Jxed joint possibility distribution B and for any ∈ [0; 1]. Let us denote the projection functions on R2 by x and y , that is, x (u; v) = u and y (u; v) = v for u; v ∈ R.
358
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
The following theorems shows two important properties of the central value operator. Theorem 2.1. If A; B ∈ F are non-interactive and g = x + y is the addition operator on R2 then C[A×B] (x + y ) = C[A] (id) + C[B] (id) = C([A] ) + C([B] ) for all ∈ [0; 1]. Proof. If A and B are non-interactive then [C] = [A] × [B] for all ∈ [0; 1], that is, 1 (x + y) dx dy C[C] (x + y ) = dx dy [A] ×[B] [A] ×[B] a2 () b2 () 1 (x + y) dx dy = [a2 () − a1 ()][b2 () − b1 ()] a1 () b1 () 1 [a2 () − a1 ()][b2 () − b1 ()] × [b2 () − b1 ()] x dx + [a2 () − a1 ()] y dy [A] [B] 1 1 = x dx + y dy a2 () − a1 () [A] b2 () − b1 () [B]
=
=
a1 () + a2 () b1 () + b2 () + = C([A] ) + C([B] ): 2 2
Theorem 2.2. If A; B ∈ F are non-interactive and p = x y is the multiplication operator on R2 then C[A×B] (x y ) = C[A] (id)C[B] (id) = C([A] )C([B] ) for all ∈ [0; 1]. Proof. If A and B are non-interactive then [C] = [A] × [B] for all ∈ [0; 1], that is, 1 xy dx dy C[C] (x y ) = dx dy [A] ×[B] [A] ×[B] a2 () b2 () 1 xy dx dy = [a2 () − a1 ()][b2 () − b1 ()] a1 () b1 () a2 () b2 () 1 x dx y dy = [a2 () − a1 ()][b2 () − b1 ()] a1 () b1 () =
a1 () + a2 () b1 () + b2 () = C([A] )C([B] ): 2 2
The following deJnition is crucial for our theory.
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
359
Denition 2.1. Let C be a joint possibility distribution with marginal possibility distributions A; B ∈ F, and let ∈ [0; 1]. The measure of interactivity between the -level sets of A and B is deJned by R[C] (x ; y ) = C[C] ((x − C[C] (x ))(y − C[C] (y ))): Using the deJnition of central value we have R[C] (x ; y ) =
1 dx dy
=
1 dx dy
[C]
[C]
[C]
[C]
− C[C] (x )
[C]
(x − C[C] (x ))(y − C[C] (y )) dx dy xy dx dy − C 1 dx dy
[C]
[C]
(y )
[C]
1 dx dy
[C]
x dx dy
y dx dy + C[C] (x )C[C] (y )
= C[C] (x y ) − C[C] (x )C[C] (y ) for all ∈ [0; 1]. Recall that two ordered pairs (x1 ; y1 ) and (x2 ; y2 ) of real numbers are concordant if (x1 − x2 ) (y1 − y2 )¿0; and discordant if (x1 − x2 )(y1 − y2 )¡0. Note 2.2. The interactivity relation computes the average value of the interactivity function g(x; y) = (x − C[C] (x ))(y − C[C] (y )) on [C] . Suppose that we choose a point x ∈ R then [C] will impose a constraint on possible values of y. The value of g(x; y) is positive (negative) if and only if (x; y) and (C[C] (x ); C[C] (y )) are concordant (discordant). Loosely speaking, the value of R[C] (x ; y ) is positive if the ‘strength’ of (x; y) and (C[C] (x ); C[C] (y )) that are concordant is bigger than the strength of those ones that are discordant. Consider now the case depicted in Fig. 2. After some calculations we Bnd C[C] (x ) =
a1 () + a2 () = C([A] ) 2
C[C] (y ) =
b1 () + b2 () = C([B] ): 2
and
If A takes u then B can take only a single value v. We can easily see that in this case g(x; y) = (x − C([A] ))(y − C([B] )) will always be positive.
360
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
Let A; B ∈ F with [A] = [a1 (); a2 ()] and [B] = [b1 (); b2 ()]. In [4] we introduced the notion of f-weighted covariance of possibility distributions A and B as 1 a2 () − a1 () b2 () − b1 () Covf (A; B) = f() d: 2 2 0 The main drawback of that deJnition is that Covf (A; B) is always non-negative for any A and B. Based on the notion of central values we shall introduce a novel deJnition of covariance, that agrees with the principle of ‘falling shadows.’ Furthermore, we shall use this new deJnition of covariance to measure interactions between two fuzzy numbers. Denition 2.2. Let C be a joint possibility distribution in R2 . Let A; B ∈ F denote its marginal possibility distributions. The covariance of A and B with respect to a weighting function f (and with respect to their joint possibility distribution C) is deJned by 1 R[C] (x ; y )f() d Covf (A; B) = =
0
1 0
[C[C] (x y ) − C[C] (x )C[C] (y )]f() d:
(1)
If A and B are non-interactive then from Theorem 2.2 it follows that R[C] (x ; y ) = C[A×B] (x y ) − C([A] )C([B] ) = 0 for all ∈ [0; 1], which directly implies the following theorem: Theorem 2.3. If A; B ∈ F are non-interactive then Covf (A; B) = 0 for any weighting function f. Note 2.3. If we consider (1) with the constant weighting function f() = 1 for all ∈ [0; 1] then we get 1 R[C] (x ; y ) d: Covf (A; B) = 0
That is, Covf (A; B) simple sums up the interactivity measures levelwisely. In (1) f() weights the importance of the interactivity between the -level sets of A and B. It can happen that we care more about interactivites on higher levels than on lower levels. Now we illustrate our approach to interactions by several simple examples. Let A; B ∈ F and suppose that their joint possibility distribution C is deJned by (see Fig. 2) [C] = {t(a1 (); b1 ()) + (1 − t)(a2 (); b2 ()) | t ∈ [0; 1]} for all ∈ [0; 1]. It is easy to check that A and B are really the marginal distributions of C. In this case A and B are interactive. We will compute the measure of interactivity between A and B.
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
361
B 1
γ
b1
C
A
b2
y
v
a1 a2
u
x
Fig. 2. Joint possibility distribution C.
Let be arbitrarily Jxed, and let a1 = a1 (); a2 = a2 (); b1 = b1 (); b2 = b2 (). Then the -level set of C can be calculated as {(c(y); y) | b1 6y6b2 } where c(y) =
b2 − y y − b1 a2 − a1 a 1 b2 − a 2 b1 a1 + a2 = y+ : b2 − b 1 b2 − b 1 b2 − b 1 b2 − b 1
Since all -level sets of C are degenerated, i.e. their integrals vanish, the following formal calculations can be done:
[C]
dx dy =
b2 b1
c(y) [x]c(y)
dy;
[C]
xy dx dy =
b2 b1
x2 y 2
c(y) c(y)
dy;
which turns into C[C] (x y ) =
[C]
1 dx dy
1 = b2 − b 1
b2 b1
[C]
xy dx dy
yc(y) dy
1 1 1 3 3 2 2 (a2 − a1 )(b2 − b1 ) + (a1 b2 − a2 b1 )(b2 − b1 ) = (b2 − b1 )2 3 2 =
2(a2 − a1 )(b21 + b1 b2 + b22 ) + 3(a1 b2 − a2 b1 )(b1 + b2 ) 6(b2 − b1 )
=
(a2 − a1 )(b2 − b1 ) a1 b2 + a2 b1 + : 3 2
362
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
B 1
γ A
D
b1
a1
b2
v
y
a2 u
x
Fig. 3. Joint possibility distribution D.
Hence, we have R[C] (x ; y ) = C[C] (x y ) − C[C] (x )C[C] (y ) =
(a2 − a1 )(b2 − b1 ) a1 b2 + a2 b1 (a1 + a2 )(b1 + b2 ) + − 3 2 4
=
(a2 − a1 )(b2 − b1 ) ; 12
and Jnally, the covariance of A and B with respect to their joint possibility distribution C is 1 1 [a2 () − a1 ()][b2 () − b1 ()]f() d: Covf (A; B) = 12 0 Note 2.4. In this case Covf (A; B)¿0 for any weighting function f; and if A(u)¿ for some u ∈ R then there exists a unique v ∈ R that B can take (see Fig. 2), furthermore, if u is moved to the left (right) then the corresponding value (that B can take) will also move to the left (right). Let A; B ∈ F and suppose that their joint possibility distribution D is deJned by (see Fig. 3) [D] = {t(a1 (); b2 ()) + (1 − t)(a2 (); b1 ()) | t ∈ [0; 1]} for all ∈ [0; 1]. In this case A and B are interactive. We will compute now the measure of interactivity between A and B. After similar calculations we get C[D] (x y ) = −
(a2 − a1 )(b2 − b1 ) a1 b1 + a2 b2 + ; 3 2
which implies that R[D] (x ; y ) = C[D] (x y ) − C[D] (x )C[D] (y ) = −
(a2 − a1 )(b2 − b1 ) 12
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
363
Fig. 4. Joint possibility distribution F.
and we Jnd Covf (A; B) = −
1 12
1 0
[a2 () − a1 ()][b2 () − b1 ()]f() d:
Note 2.5. In this case Covf (A; B)¡0 for any weighting function f; and if A(u)¿ for some u ∈ R then there exists a unique v ∈ R that B can take (see Fig. 3), furthermore, if u is moved to the left (right) then the corresponding value (that B can take) will move to the right (left). Now consider the case when A(x) = B(x) = (1 − x) [0;1] (x) for x ∈ R, that is, [A] = [B] = [0; 1 − ] for ∈ [0; 1]. Suppose that their joint possibility distribution is given by F(x; y) = (1 − x − y) T (x; y); where T = {(x; y) ∈ R2 | x¿0; y¿0; x + y61}. This situation is depicted in Fig. 4, where we have shifted the fuzzy sets to get a better view of the situation. It is easy to check that A and B are really the marginal distributions of F. A -level set of F is computed by [F] = {(x; y) ∈ R2 | x¿0; y¿0; x + y61 − }. Then R
[F]
(x ; y ) =
[F]
1 dx dy
−
[F]
where, [F]
dx dy =
(1 − )2 2
[F]
1 dx dy
xy dx dy
[F]
x dx dy
[F]
1 dx dy
[F]
y dx dy ;
364
and
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
1 1− xy dx dy = xy dx dy = x(1 − − x)2 dx 2 0 0 0 [F] 1 1− 3 (1 − )4 1 − 1− 2 x dx − x dx = ; = 2 2 0 24 0
furthermore, x dx dy = [F]
=
1 2
1−
1− − x
[F]
y dx dy =
1− 0
1− 0
1− − x 0
(1 − − x)2 dx =
y dx dy
(1 − )3 6
for any ∈ [0; 1]. Hence, the covariance between A and B is, 1 1 1 Covf (A; B) = R[F] (x ; y )f() d = − (1 − )2 f() d 36 0 0 for any weighting function f. Let us take a closer look at joint possibility distribution F. It is clear that, C[F] (x ) = C[F] (y ) = (1 − )=3 and if A takes a value u such that A(u)¿, then B can take its values from the interval [0; 1 − − u]. The average value of the interactivity function 1− 1− g(x; y) = x − y− 3 3 on [F] can be computed as the sum of its average values on H1 , H2 , H3 and H4 (see Fig. 5), where g is positive on H1 and H4 (and negative on H2 and H3 ). Now consider the case when A(1 − x) = B(x) = x [0;1] (x) for x ∈ R, that is, [A] = [0; 1 − ] and [B] = [; 1], for ∈ [0; 1]. Let E(x; y) = (y − x) S (x; y); where S = {(x; y) ∈ R2 | x¿0; y61; y − x¿0}. This situation is depicted in Fig. 6, where we have shifted the fuzzy sets to get a better view of the situation. We can easily see that max E(x; y) = y [0;1] (y) = B(y); x∈R
max E(x; y) = (1 − x) [0;1] (x) = A(x): y∈R
A -level set of E is computed by [E] = {(x; y) ∈ R2 | x¿0; y61; y − x¿}.
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
365
Fig. 5. Partition of [F] .
Fig. 6. Joint possibility distribution E.
After some calculations we get Covf (A; B) =
1 0
R
[E]
1 (x ; y )f() d = 36
1 0
(1 − )2 f() d
for any weighting function f. We can also use the principle of central values to introduce the notion of expected value of functions on fuzzy sets. Let g : R → R be an integrable function and let A ∈ F. Let us consider again the average value of function g on [A] 1 C[A] (g) = g(x) dx: dx [A] [A]
366
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
Denition 2.3. The expected value of function g on A with respect to a weighting function f is deJned by Ef (g; A) =
1 0
C[A] (g)f() d =
1 0
1
[A]
dx
[A]
g(x) dxf() d:
Especially, if g is the identity function then we get Ef (id; A) = Ef (A) =
1 0
a1 () + a2 () f() d; 2
which is the f-weighted possibilistic expected value of A introduced in [4]. Let us denote R[A] (id; id) the average value of function g(x) = (x − C([A] ))2 on the -level set of an individual fuzzy number A. That is, R[A] (id; id) =
[A]
1 dx
[A]
x2 dx −
[A]
2
1 dx
[A]
x dx
:
Denition 2.4. The variance of A is deJned as the expected value of function g(x) = (x − C([A] ))2 on A. That is, Var f (A) = Ef (g; A) =
1 0
R[A] (id; id)f() d:
From the equality, 2 a2 () 1 x dx − x dx a2 () − a1 () a1 () a1 () a21 () + a1 ()a2 () + a22 () a1 () + a2 () 2 = − 3 2
1 R[A] (id; id) = a2 () − a1 ()
a2 ()
2
(a2 () − a1 ())2 a21 () − 2a1 ()a2 () + a22 () = ; 12 12
= we get, Var f (A) =
1 0
(a2 () − a1 ())2 f() d: 12
Note 2.6. The expected value of the identity function on an individual fuzzy number (with respect to a weighting function f ) coincides with the deBnition of f-weighted possibilistic mean value of fuzzy numbers introduced in [4]. The f-weighted variance is almost the same as in [4] (the same up to a scalar multiplier).
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
367
Fig. 7. Joint possibility distribution G.
Note 2.7. The covariance between marginal distributions A and B of a joint possibility distribution C is nothing else but the expected value of their interactivity function on C (with respect to a weighting function f ). The next theorem about the bilinearity of the interactivity relation operator can easily be proved. Theorem 2.4. Let C be a joint possibility distribution in R2 , and let $; % ∈ R. Then R[C] ($x + %y ; $x + %y ) = $2 R[C] (x ; x ) + %2 R[C] (y ; y ) + 2$%R[C] (x ; y ): Zero covariance does not always imply non-interactivity. Really, let G be a joint possibility distribution with a symmetrical -level set, i.e., there exist a; b ∈ R such that G(x; y) = G(2a − x; y) = G(x; 2b − y) = G(2a − x; 2b − y) for all x; y ∈ [G] , where (a; b) is the center of the set [G] (see Fig. 7). In this case R[G] (x ; y ) = 0. In fact, introducing the notation H = {(x; y) ∈ [G] | x 6 a; y 6 b}; we Jnd,
[G]
xy dx dy =
H
(xy + (2a − x)y + x(2b − y) + (2a − x)(2b − y)) dx dy
= 4ab
H
dx dy;
[G]
x dx dy = 2
H
(x + (2a − x)) dx dy = 4a
H
dx dy;
368
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
[G]
[G]
y dx dy = 2
H
dx dy = 4
(y + (2b − y)) dx dy = 4b
That is, [G]
1 dx dy
−
dx dy;
dx dy:
H
R[G] (x ; y ) =
H
[G]
[C]
1 dx dy
xy dx dy
[C]
x dx dy
[G]
1 dx dy
[C]
y dx dy
= 0:
If all -level sets of G are symmetrical then the covariance between its marginal distributions A and B becomes zero for any weighting function f, that is, Covf (A; B) = 0; even though A and B may be interactive. Note 2.8. Consider a symmetrical joint possibility distribution with center (0; 0), that is, G(x; y) = G(−x; y) = G(x; −y) = G(−x; −y), for all x; y. So, if (u; v) ∈ [G] for some u; v ∈ R then (−u; v), (u; −v) and (−u; −v) will also belong to [G] , therefore, simple algebra shows that the equation xy dx dy = 0 [G]
holds for any ∈ [0; 1]. In a similar manner we can prove the relationships x dx dy = y dx dy = 0 [G]
[G]
for any ∈ [0; 1]. So, the average value of function g(x; y) = (x − 0)(y − 0) = xy on [G] is zero (independently of ). Note 2.9. Let X and Y be random variables with joint probability distribution function H and marginal distribution functions F and G, respectively. Then there exists a copula C (which is uniquely determined on Range F × Range G) such that H (x; y) = C(F(x); G(y)) for all x; y [8]. Based on this fact, several indices of dependencies were introduced in [6,7]. Furthermore, any copula C satisBes the relationship [7] W (x; y) = max{x + y − 1; 0} 6 C(x; y) 6 M (x; y) = min{x; y}: For uniform distributions on [a1 (); a2 ()] and [b1 (); b2 ()] we look for a joint distribution deBned on the Cartesian product of this level intervals (with uniform marginal distributions on [a1 (); a2 ()]
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
369
and [b1 (); b2 ()]): this joint distribution can always be characterized by a unique copula C, which may depend also on (see [6]). For non-interactive situation, C is just the product (with zero covariance, this is Theorem 2.3); Fig. 2 is linked to copula M (with correlation one); and Fig. 3 is linked to copula W (with correlation minus one). 3. Summary We have introduced the notion of covariance between marginal distributions of a joint possibility distribution C as the expected value of their interactivity function on C. We have interpreted this covariance as a measure of interactivity between marginal distributions. We have shown that non-interactivity entails zero covariance; however, zero covariance does not always imply non-interactivity. The measure of interactivity is positive (negative) if the expected value of the interactivity relation on C is positive (negative). The concept introduced in this paper can be used in capital budgeting decisions where (estimated) future costs and revenues are modelled by interactive fuzzy numbers. Acknowledgements The authors thank the anonymous referee for his=her helpful note (Note 2.9) and suggestions on the earlier version of this paper. References [1] L.M. de Campos, J.F. Huete, Independence concepts in possibility theory: Part I, Fuzzy Sets and Systems 103 (1999) 127–152. [2] L.M. de Campos, J.F. Huete, Independence concepts in possibility theory: Part II, Fuzzy Sets and Systems 103 (1999) 487–505. [3] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988. [4] R. Full%er, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374. [5] E. Hisdal, Conditional possibilities independence and noninteraction, Fuzzy Sets and Systems 1 (1978) 283–297. [6] R.B. Nelsen, An introduction to copulas, in: Lecture Notes in Statistics, Vol. 139, Springer, New York, 1999. % [7] R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodr%Sguez-Lallena, M. Ubeda-Flores, Distribution functions of copulas: a class of bivariate probability integral transforms, Statist. Probab. Lett. 54 (2001) 277–282. [8] A. Sklar, Fonctions de r%epartition aT n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231. [9] L.A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning, I, II, III, Inform. Sci. 8 (1975) 199–249, 301–357; L.A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning, I, II, III, Inform. Sci. 9 (1975) 43–80.
Fuzzy Sets and Systems 143 (2004) 355 – 369 www.elsevier.com/locate/fss
On interactive fuzzy numbers Robert Full%era; b;∗ , P%eter Majlendera; b; c a
Department of Operations Research, Eotvos Lorand University, Pazmany Peter set. 1C, P.O. Box 120, Budapest H-1518, Hungary b - Akademi University, Lemminkainengatan 14C, Institute for Advanced Management Systems Research, Abo - FIN-20520, Finland Abo c - Akademi University, IAMSR Datacity B 6708, Lemminkainengatan 14B, Turku Centre for Computer Science, Abo - FIN-20520, Finland Abo Received 28 March 2002; received in revised form 28 February 2003; accepted 4 April 2003 Dedicated to Professor Christer Carlsson
Abstract In this paper we will introduce a measure of interactivity between marginal distributions of a joint possibility distribution C as the expected value of the interactivity relation between the -level sets of its marginal distributions. c 2003 Elsevier B.V. All rights reserved. Keywords: Joint possibility distribution; Marginal possibility distribution; Interactivity; Covariance; Variance
1. Introduction The concept of conditional independence has been studied in depth in possibility theory, for good surveys see, e.g., Campos and Huete [1,2]. The notion of non-interactivity in possibility theory was introduced by Zadeh [9]. Hisdal [5] demonstrated the di>erence between conditional independence and non-interactivity. In this paper we will introduce a measure of interactivity between fuzzy numbers via their joint possibility distribution. A fuzzy set A in R is said to be a fuzzy number if it is normal, fuzzy convex and has an upper semi-continuous membership function of bounded support. The family of all fuzzy numbers will
Partially supported by the Hungarian Research Funds OTKA T32412 and FKFP-0157/2000. A Corresponding author. Institute for Advanced Management Systems Research, Abo Akademi University, A LemminkEainengatan 14C, Abo FIN-20520, Finland. Tel.: +358-2-2654-066; fax: +358-2-2654-809. E-mail addresses: [email protected], [email protected] (R. Full%er), [email protected], [email protected] (P. Majlender). ∗
c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/S0165-0114(03)00180-5
356
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
Fig. 1. Non-interactive possibility distributions.
be denoted by F. A -level set of a fuzzy set A in Rm is deJned by [A] = {x ∈ Rm : A(x)¿} if ¿0 and [A] = cl{x ∈ Rm : A(x)¿} (the closure of the support of A) if = 0. If A ∈ F is a fuzzy number then [A] is a convex and compact subset of R for all ∈ [0; 1]. Fuzzy numbers can be considered as possibility distributions [9]. A fuzzy set B in Rm is said to be a joint possibility distribution of fuzzy numbers Ai ∈ F; i = 1; : : : ; m, if it satisJes the relationship max
xj ∈R; j =i
B(x1 ; : : : ; xm ) = Ai (xi );
∀xi ∈ R; i = 1; : : : ; m:
Furthermore, Ai is called the ith marginal possibility distribution of B, and the projection of B on the ith axis is Ai for i = 1; : : : ; m. Fuzzy numbers Ai ∈ F, i = 1; : : : ; m, are said to be noninteractive if their joint possibility distribution is given by B(x1 ; : : : ; xm ) = min{A1 (x1 ); : : : ; Am (xm )}, for all x1 ; : : : ; xm ∈ R. Furthermore, in this case the equality [B] = [A1 ] × · · · ×[Am ] holds for any ∈ [0; 1]. If A; B ∈ F are non-interactive then their joint membership function is deJned by A × B. It is clear that in this case any change in the membership function of A does not e>ect the second marginal possibility distribution and vice versa. On the other hand, A and B are said to be interactive if they cannot take their values independently of each other [3] (Fig. 1). Denition 1.1 (Full%er and Majlender [4]). A function f : [0; 1] → R is said to be a weighting function if f is non-negative, monotone increasing and satisJes the following normalization condition
1 0
f() d = 1:
Di>erent weighting functions can give di>erent (case-dependent) importances to -levels sets of fuzzy numbers. If f is a stepwise linear function then we can have discrete weights. This deJnition is motivated in part by the desire to give less importance to the lower levels of fuzzy sets (it is why f should be monotone increasing).
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
357
2. Central values In this section, we shall introduce the notion of average value of real-valued functions on -level sets of joint possibility distributions and we shall show how these average values can be used for measuring interactions between -level sets of its marginal distributions. Let B be a joint possibility distribution in Rn , let ∈ [0; 1] and let g : Rn → R be an integrable function. It is well known from analysis that the average value of function g on [B] can be computed by 1 C[B] (g) = g(x) dx dx [B] [B] 1 = g(x1 ; : : : ; xn ) dx1 : : : dxn : dx1 : : : dxn [B] [B] We will call C as the central value operator. Note 2.1. If [B] is degenerated for some ∈ [0; 1], that is dx = 0; [B]
then we approximate [B] with non-degenerated sets and consider the limit case. Namely, let {Sk } be a sequence of sets such that [B] ⊂ Sk ⊂ Rn , 0¡ Sk dx¡∞, and lim sup{ x − s | x ∈ [B] ; s ∈ Sk } = 0:
k →∞
Then, 1 dx Sk
C[B] (g) = lim CSk (g) = lim k →∞
k →∞
Sk
g(x) dx:
If g : R → R is an integrable function and A ∈ F then the average value of function g on [A] is deJned by 1 g(x) dx: C[A] (g) = dx [A] [A] Especially, if g(x) = x, for all x ∈ R is the identity function (g = id) and A ∈ F is a fuzzy number with [A] = [a1 (); a2 ()] then the average value of the identity function on [A] is computed by a2 () 1 1 a1 () + a2 () ; C[A] (id) = x dx = x dx = a2 () − a1 () a1 () 2 dx [A] [A] which remains valid in the limit case a2 () − a1 () = 0 for some . Because C[A] (id) is nothing else, but the center of [A] we will use the shorter notation C([A] ) for C[A] (id). It is clear that C[B] is linear for any Jxed joint possibility distribution B and for any ∈ [0; 1]. Let us denote the projection functions on R2 by x and y , that is, x (u; v) = u and y (u; v) = v for u; v ∈ R.
358
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
The following theorems shows two important properties of the central value operator. Theorem 2.1. If A; B ∈ F are non-interactive and g = x + y is the addition operator on R2 then C[A×B] (x + y ) = C[A] (id) + C[B] (id) = C([A] ) + C([B] ) for all ∈ [0; 1]. Proof. If A and B are non-interactive then [C] = [A] × [B] for all ∈ [0; 1], that is, 1 (x + y) dx dy C[C] (x + y ) = dx dy [A] ×[B] [A] ×[B] a2 () b2 () 1 (x + y) dx dy = [a2 () − a1 ()][b2 () − b1 ()] a1 () b1 () 1 [a2 () − a1 ()][b2 () − b1 ()] × [b2 () − b1 ()] x dx + [a2 () − a1 ()] y dy [A] [B] 1 1 = x dx + y dy a2 () − a1 () [A] b2 () − b1 () [B]
=
=
a1 () + a2 () b1 () + b2 () + = C([A] ) + C([B] ): 2 2
Theorem 2.2. If A; B ∈ F are non-interactive and p = x y is the multiplication operator on R2 then C[A×B] (x y ) = C[A] (id)C[B] (id) = C([A] )C([B] ) for all ∈ [0; 1]. Proof. If A and B are non-interactive then [C] = [A] × [B] for all ∈ [0; 1], that is, 1 xy dx dy C[C] (x y ) = dx dy [A] ×[B] [A] ×[B] a2 () b2 () 1 xy dx dy = [a2 () − a1 ()][b2 () − b1 ()] a1 () b1 () a2 () b2 () 1 x dx y dy = [a2 () − a1 ()][b2 () − b1 ()] a1 () b1 () =
a1 () + a2 () b1 () + b2 () = C([A] )C([B] ): 2 2
The following deJnition is crucial for our theory.
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
359
Denition 2.1. Let C be a joint possibility distribution with marginal possibility distributions A; B ∈ F, and let ∈ [0; 1]. The measure of interactivity between the -level sets of A and B is deJned by R[C] (x ; y ) = C[C] ((x − C[C] (x ))(y − C[C] (y ))): Using the deJnition of central value we have R[C] (x ; y ) =
1 dx dy
=
1 dx dy
[C]
[C]
[C]
[C]
− C[C] (x )
[C]
(x − C[C] (x ))(y − C[C] (y )) dx dy xy dx dy − C 1 dx dy
[C]
[C]
(y )
[C]
1 dx dy
[C]
x dx dy
y dx dy + C[C] (x )C[C] (y )
= C[C] (x y ) − C[C] (x )C[C] (y ) for all ∈ [0; 1]. Recall that two ordered pairs (x1 ; y1 ) and (x2 ; y2 ) of real numbers are concordant if (x1 − x2 ) (y1 − y2 )¿0; and discordant if (x1 − x2 )(y1 − y2 )¡0. Note 2.2. The interactivity relation computes the average value of the interactivity function g(x; y) = (x − C[C] (x ))(y − C[C] (y )) on [C] . Suppose that we choose a point x ∈ R then [C] will impose a constraint on possible values of y. The value of g(x; y) is positive (negative) if and only if (x; y) and (C[C] (x ); C[C] (y )) are concordant (discordant). Loosely speaking, the value of R[C] (x ; y ) is positive if the ‘strength’ of (x; y) and (C[C] (x ); C[C] (y )) that are concordant is bigger than the strength of those ones that are discordant. Consider now the case depicted in Fig. 2. After some calculations we Bnd C[C] (x ) =
a1 () + a2 () = C([A] ) 2
C[C] (y ) =
b1 () + b2 () = C([B] ): 2
and
If A takes u then B can take only a single value v. We can easily see that in this case g(x; y) = (x − C([A] ))(y − C([B] )) will always be positive.
360
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
Let A; B ∈ F with [A] = [a1 (); a2 ()] and [B] = [b1 (); b2 ()]. In [4] we introduced the notion of f-weighted covariance of possibility distributions A and B as 1 a2 () − a1 () b2 () − b1 () Covf (A; B) = f() d: 2 2 0 The main drawback of that deJnition is that Covf (A; B) is always non-negative for any A and B. Based on the notion of central values we shall introduce a novel deJnition of covariance, that agrees with the principle of ‘falling shadows.’ Furthermore, we shall use this new deJnition of covariance to measure interactions between two fuzzy numbers. Denition 2.2. Let C be a joint possibility distribution in R2 . Let A; B ∈ F denote its marginal possibility distributions. The covariance of A and B with respect to a weighting function f (and with respect to their joint possibility distribution C) is deJned by 1 R[C] (x ; y )f() d Covf (A; B) = =
0
1 0
[C[C] (x y ) − C[C] (x )C[C] (y )]f() d:
(1)
If A and B are non-interactive then from Theorem 2.2 it follows that R[C] (x ; y ) = C[A×B] (x y ) − C([A] )C([B] ) = 0 for all ∈ [0; 1], which directly implies the following theorem: Theorem 2.3. If A; B ∈ F are non-interactive then Covf (A; B) = 0 for any weighting function f. Note 2.3. If we consider (1) with the constant weighting function f() = 1 for all ∈ [0; 1] then we get 1 R[C] (x ; y ) d: Covf (A; B) = 0
That is, Covf (A; B) simple sums up the interactivity measures levelwisely. In (1) f() weights the importance of the interactivity between the -level sets of A and B. It can happen that we care more about interactivites on higher levels than on lower levels. Now we illustrate our approach to interactions by several simple examples. Let A; B ∈ F and suppose that their joint possibility distribution C is deJned by (see Fig. 2) [C] = {t(a1 (); b1 ()) + (1 − t)(a2 (); b2 ()) | t ∈ [0; 1]} for all ∈ [0; 1]. It is easy to check that A and B are really the marginal distributions of C. In this case A and B are interactive. We will compute the measure of interactivity between A and B.
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
361
B 1
γ
b1
C
A
b2
y
v
a1 a2
u
x
Fig. 2. Joint possibility distribution C.
Let be arbitrarily Jxed, and let a1 = a1 (); a2 = a2 (); b1 = b1 (); b2 = b2 (). Then the -level set of C can be calculated as {(c(y); y) | b1 6y6b2 } where c(y) =
b2 − y y − b1 a2 − a1 a 1 b2 − a 2 b1 a1 + a2 = y+ : b2 − b 1 b2 − b 1 b2 − b 1 b2 − b 1
Since all -level sets of C are degenerated, i.e. their integrals vanish, the following formal calculations can be done:
[C]
dx dy =
b2 b1
c(y) [x]c(y)
dy;
[C]
xy dx dy =
b2 b1
x2 y 2
c(y) c(y)
dy;
which turns into C[C] (x y ) =
[C]
1 dx dy
1 = b2 − b 1
b2 b1
[C]
xy dx dy
yc(y) dy
1 1 1 3 3 2 2 (a2 − a1 )(b2 − b1 ) + (a1 b2 − a2 b1 )(b2 − b1 ) = (b2 − b1 )2 3 2 =
2(a2 − a1 )(b21 + b1 b2 + b22 ) + 3(a1 b2 − a2 b1 )(b1 + b2 ) 6(b2 − b1 )
=
(a2 − a1 )(b2 − b1 ) a1 b2 + a2 b1 + : 3 2
362
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
B 1
γ A
D
b1
a1
b2
v
y
a2 u
x
Fig. 3. Joint possibility distribution D.
Hence, we have R[C] (x ; y ) = C[C] (x y ) − C[C] (x )C[C] (y ) =
(a2 − a1 )(b2 − b1 ) a1 b2 + a2 b1 (a1 + a2 )(b1 + b2 ) + − 3 2 4
=
(a2 − a1 )(b2 − b1 ) ; 12
and Jnally, the covariance of A and B with respect to their joint possibility distribution C is 1 1 [a2 () − a1 ()][b2 () − b1 ()]f() d: Covf (A; B) = 12 0 Note 2.4. In this case Covf (A; B)¿0 for any weighting function f; and if A(u)¿ for some u ∈ R then there exists a unique v ∈ R that B can take (see Fig. 2), furthermore, if u is moved to the left (right) then the corresponding value (that B can take) will also move to the left (right). Let A; B ∈ F and suppose that their joint possibility distribution D is deJned by (see Fig. 3) [D] = {t(a1 (); b2 ()) + (1 − t)(a2 (); b1 ()) | t ∈ [0; 1]} for all ∈ [0; 1]. In this case A and B are interactive. We will compute now the measure of interactivity between A and B. After similar calculations we get C[D] (x y ) = −
(a2 − a1 )(b2 − b1 ) a1 b1 + a2 b2 + ; 3 2
which implies that R[D] (x ; y ) = C[D] (x y ) − C[D] (x )C[D] (y ) = −
(a2 − a1 )(b2 − b1 ) 12
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
363
Fig. 4. Joint possibility distribution F.
and we Jnd Covf (A; B) = −
1 12
1 0
[a2 () − a1 ()][b2 () − b1 ()]f() d:
Note 2.5. In this case Covf (A; B)¡0 for any weighting function f; and if A(u)¿ for some u ∈ R then there exists a unique v ∈ R that B can take (see Fig. 3), furthermore, if u is moved to the left (right) then the corresponding value (that B can take) will move to the right (left). Now consider the case when A(x) = B(x) = (1 − x) [0;1] (x) for x ∈ R, that is, [A] = [B] = [0; 1 − ] for ∈ [0; 1]. Suppose that their joint possibility distribution is given by F(x; y) = (1 − x − y) T (x; y); where T = {(x; y) ∈ R2 | x¿0; y¿0; x + y61}. This situation is depicted in Fig. 4, where we have shifted the fuzzy sets to get a better view of the situation. It is easy to check that A and B are really the marginal distributions of F. A -level set of F is computed by [F] = {(x; y) ∈ R2 | x¿0; y¿0; x + y61 − }. Then R
[F]
(x ; y ) =
[F]
1 dx dy
−
[F]
where, [F]
dx dy =
(1 − )2 2
[F]
1 dx dy
xy dx dy
[F]
x dx dy
[F]
1 dx dy
[F]
y dx dy ;
364
and
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
1 1− xy dx dy = xy dx dy = x(1 − − x)2 dx 2 0 0 0 [F] 1 1− 3 (1 − )4 1 − 1− 2 x dx − x dx = ; = 2 2 0 24 0
furthermore, x dx dy = [F]
=
1 2
1−
1− − x
[F]
y dx dy =
1− 0
1− 0
1− − x 0
(1 − − x)2 dx =
y dx dy
(1 − )3 6
for any ∈ [0; 1]. Hence, the covariance between A and B is, 1 1 1 Covf (A; B) = R[F] (x ; y )f() d = − (1 − )2 f() d 36 0 0 for any weighting function f. Let us take a closer look at joint possibility distribution F. It is clear that, C[F] (x ) = C[F] (y ) = (1 − )=3 and if A takes a value u such that A(u)¿, then B can take its values from the interval [0; 1 − − u]. The average value of the interactivity function 1− 1− g(x; y) = x − y− 3 3 on [F] can be computed as the sum of its average values on H1 , H2 , H3 and H4 (see Fig. 5), where g is positive on H1 and H4 (and negative on H2 and H3 ). Now consider the case when A(1 − x) = B(x) = x [0;1] (x) for x ∈ R, that is, [A] = [0; 1 − ] and [B] = [; 1], for ∈ [0; 1]. Let E(x; y) = (y − x) S (x; y); where S = {(x; y) ∈ R2 | x¿0; y61; y − x¿0}. This situation is depicted in Fig. 6, where we have shifted the fuzzy sets to get a better view of the situation. We can easily see that max E(x; y) = y [0;1] (y) = B(y); x∈R
max E(x; y) = (1 − x) [0;1] (x) = A(x): y∈R
A -level set of E is computed by [E] = {(x; y) ∈ R2 | x¿0; y61; y − x¿}.
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
365
Fig. 5. Partition of [F] .
Fig. 6. Joint possibility distribution E.
After some calculations we get Covf (A; B) =
1 0
R
[E]
1 (x ; y )f() d = 36
1 0
(1 − )2 f() d
for any weighting function f. We can also use the principle of central values to introduce the notion of expected value of functions on fuzzy sets. Let g : R → R be an integrable function and let A ∈ F. Let us consider again the average value of function g on [A] 1 C[A] (g) = g(x) dx: dx [A] [A]
366
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
Denition 2.3. The expected value of function g on A with respect to a weighting function f is deJned by Ef (g; A) =
1 0
C[A] (g)f() d =
1 0
1
[A]
dx
[A]
g(x) dxf() d:
Especially, if g is the identity function then we get Ef (id; A) = Ef (A) =
1 0
a1 () + a2 () f() d; 2
which is the f-weighted possibilistic expected value of A introduced in [4]. Let us denote R[A] (id; id) the average value of function g(x) = (x − C([A] ))2 on the -level set of an individual fuzzy number A. That is, R[A] (id; id) =
[A]
1 dx
[A]
x2 dx −
[A]
2
1 dx
[A]
x dx
:
Denition 2.4. The variance of A is deJned as the expected value of function g(x) = (x − C([A] ))2 on A. That is, Var f (A) = Ef (g; A) =
1 0
R[A] (id; id)f() d:
From the equality, 2 a2 () 1 x dx − x dx a2 () − a1 () a1 () a1 () a21 () + a1 ()a2 () + a22 () a1 () + a2 () 2 = − 3 2
1 R[A] (id; id) = a2 () − a1 ()
a2 ()
2
(a2 () − a1 ())2 a21 () − 2a1 ()a2 () + a22 () = ; 12 12
= we get, Var f (A) =
1 0
(a2 () − a1 ())2 f() d: 12
Note 2.6. The expected value of the identity function on an individual fuzzy number (with respect to a weighting function f ) coincides with the deBnition of f-weighted possibilistic mean value of fuzzy numbers introduced in [4]. The f-weighted variance is almost the same as in [4] (the same up to a scalar multiplier).
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
367
Fig. 7. Joint possibility distribution G.
Note 2.7. The covariance between marginal distributions A and B of a joint possibility distribution C is nothing else but the expected value of their interactivity function on C (with respect to a weighting function f ). The next theorem about the bilinearity of the interactivity relation operator can easily be proved. Theorem 2.4. Let C be a joint possibility distribution in R2 , and let $; % ∈ R. Then R[C] ($x + %y ; $x + %y ) = $2 R[C] (x ; x ) + %2 R[C] (y ; y ) + 2$%R[C] (x ; y ): Zero covariance does not always imply non-interactivity. Really, let G be a joint possibility distribution with a symmetrical -level set, i.e., there exist a; b ∈ R such that G(x; y) = G(2a − x; y) = G(x; 2b − y) = G(2a − x; 2b − y) for all x; y ∈ [G] , where (a; b) is the center of the set [G] (see Fig. 7). In this case R[G] (x ; y ) = 0. In fact, introducing the notation H = {(x; y) ∈ [G] | x 6 a; y 6 b}; we Jnd,
[G]
xy dx dy =
H
(xy + (2a − x)y + x(2b − y) + (2a − x)(2b − y)) dx dy
= 4ab
H
dx dy;
[G]
x dx dy = 2
H
(x + (2a − x)) dx dy = 4a
H
dx dy;
368
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
[G]
[G]
y dx dy = 2
H
dx dy = 4
(y + (2b − y)) dx dy = 4b
That is, [G]
1 dx dy
−
dx dy;
dx dy:
H
R[G] (x ; y ) =
H
[G]
[C]
1 dx dy
xy dx dy
[C]
x dx dy
[G]
1 dx dy
[C]
y dx dy
= 0:
If all -level sets of G are symmetrical then the covariance between its marginal distributions A and B becomes zero for any weighting function f, that is, Covf (A; B) = 0; even though A and B may be interactive. Note 2.8. Consider a symmetrical joint possibility distribution with center (0; 0), that is, G(x; y) = G(−x; y) = G(x; −y) = G(−x; −y), for all x; y. So, if (u; v) ∈ [G] for some u; v ∈ R then (−u; v), (u; −v) and (−u; −v) will also belong to [G] , therefore, simple algebra shows that the equation xy dx dy = 0 [G]
holds for any ∈ [0; 1]. In a similar manner we can prove the relationships x dx dy = y dx dy = 0 [G]
[G]
for any ∈ [0; 1]. So, the average value of function g(x; y) = (x − 0)(y − 0) = xy on [G] is zero (independently of ). Note 2.9. Let X and Y be random variables with joint probability distribution function H and marginal distribution functions F and G, respectively. Then there exists a copula C (which is uniquely determined on Range F × Range G) such that H (x; y) = C(F(x); G(y)) for all x; y [8]. Based on this fact, several indices of dependencies were introduced in [6,7]. Furthermore, any copula C satisBes the relationship [7] W (x; y) = max{x + y − 1; 0} 6 C(x; y) 6 M (x; y) = min{x; y}: For uniform distributions on [a1 (); a2 ()] and [b1 (); b2 ()] we look for a joint distribution deBned on the Cartesian product of this level intervals (with uniform marginal distributions on [a1 (); a2 ()]
R. Fuller, P. Majlender / Fuzzy Sets and Systems 143 (2004) 355 – 369
369
and [b1 (); b2 ()]): this joint distribution can always be characterized by a unique copula C, which may depend also on (see [6]). For non-interactive situation, C is just the product (with zero covariance, this is Theorem 2.3); Fig. 2 is linked to copula M (with correlation one); and Fig. 3 is linked to copula W (with correlation minus one). 3. Summary We have introduced the notion of covariance between marginal distributions of a joint possibility distribution C as the expected value of their interactivity function on C. We have interpreted this covariance as a measure of interactivity between marginal distributions. We have shown that non-interactivity entails zero covariance; however, zero covariance does not always imply non-interactivity. The measure of interactivity is positive (negative) if the expected value of the interactivity relation on C is positive (negative). The concept introduced in this paper can be used in capital budgeting decisions where (estimated) future costs and revenues are modelled by interactive fuzzy numbers. Acknowledgements The authors thank the anonymous referee for his=her helpful note (Note 2.9) and suggestions on the earlier version of this paper. References [1] L.M. de Campos, J.F. Huete, Independence concepts in possibility theory: Part I, Fuzzy Sets and Systems 103 (1999) 127–152. [2] L.M. de Campos, J.F. Huete, Independence concepts in possibility theory: Part II, Fuzzy Sets and Systems 103 (1999) 487–505. [3] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988. [4] R. Full%er, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374. [5] E. Hisdal, Conditional possibilities independence and noninteraction, Fuzzy Sets and Systems 1 (1978) 283–297. [6] R.B. Nelsen, An introduction to copulas, in: Lecture Notes in Statistics, Vol. 139, Springer, New York, 1999. % [7] R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodr%Sguez-Lallena, M. Ubeda-Flores, Distribution functions of copulas: a class of bivariate probability integral transforms, Statist. Probab. Lett. 54 (2001) 277–282. [8] A. Sklar, Fonctions de r%epartition aT n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231. [9] L.A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning, I, II, III, Inform. Sci. 8 (1975) 199–249, 301–357; L.A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning, I, II, III, Inform. Sci. 9 (1975) 43–80.