On interactive fuzzy numbers ∗
Robert Fullér
rfuller@abo.fi
Péter Majlender
peter.majlender@abo.fi
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 γ-level sets of its marginal distributions.
1
Introduction
The concept of conditional independence has been studied in depth in possibility
theory, for good surveys see, e.g. Campos and Huete [5, 6]. The notion of noninteractivity in possibility theory was introduced by Zadeh [10]. Hisdal [4] demonstrated the difference 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 be denoted by F.
A γ-level set of a fuzzy set A in Rm is defined by [A]γ = {x ∈ Rm : A(x) ≥ γ}
if γ > 0 and [A]γ = cl{x ∈ Rm : A(x) > γ} (the closure of the support of A) if
γ = 0. If A ∈ F is a fuzzy number then [A]γ is a convex and compact subset of
R for all γ ∈ [0, 1]. Fuzzy numbers can be considered as possibility distributions
[10]. A fuzzy set B in Rm is said to be a joint possibility distribution of fuzzy
numbers Ai ∈ F, i = 1, . . . , m, if it satisfies the relationship
max
xj ∈R, j6=i
B(x1 , . . . , xm ) = Ai (xi ), ∀xi ∈ R, i = 1, . . . , m.
Furthermore, Ai is called the i-th marginal possibility distribution of B, and the
projection of B on the i-th axis is Ai for i = 1, . . . , m. Fuzzy numbers Ai ∈ F,
i = 1, . . . , m, are said to be non-interactive if their joint possibility distribution is
given by B(x1 , . . . , xm ) = min{A1 (x1 ), . . . , Am (xm )}, for all x1 , . . . , xm ∈ R.
∗
The final version of this paper appeared in: Fuzzy Sets and Systems, 143(2004) 355-369.
1
Furthermore, in this case the equality [B]γ = [A1 ]γ × · · · × [Am ]γ holds for any
γ ∈ [0, 1].
If A, B ∈ F are non-interactive then their joint membership function is defined
by A × B. It is clear that in this case any change in the membership function of
A does not effect the second marginal possibility distribution and vice versa. On
the other hand, A and B are said to be interactive if they can not take their values
independently of each other [1].
Figure 1: Non-interactive possibility distributions.
Definition 1.1 [2] A function f : [0, 1] → R is said to be a weighting function if
f is non-negative, monotone increasing and satisfies the following normalization
condition
Z
1
f (γ)dγ = 1.
0
Different weighting functions can give different (case-dependent) importances
to γ-levels sets of fuzzy numbers. If f is a step-wise linear function then we can
have discrete weights. This definition 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).
2
Central values
In this section we shall introduce the notion of average value of real-valued functions on γ-level sets of joint possibility distributions and we shall show how these
average values can be used for measuring interactions between γ-level sets of its
marginal distributions.
Let B be a joint possibility distribution in Rn , let γ ∈ [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]γ can be computed by
C[B]γ (g) = R
1
[B]γ
Z
dx
g(x)dx
[B]γ
1
=R
[B]γ dx1 . . . dxn
Z
g(x1 , . . . , xn )dx1 . . . dxn .
[B]γ
We will call C as the central value operator.
2
Note 2.1 If [B]γ is a degenerated set, that is
Z
dx = 0,
[B]γ
then we compute C[B]γ (g) as the limit case of a uniform approximation of [B]γ with
non-degenerated sets [3]. That is, let
S(ε) = {x ∈ Rn |∃c ∈ [B]γ kx − ck ≤ ε},
ε > 0.
Then obviously
Z
∀ε > 0,
dx > 0,
S(ε)
and we define the central value of g on [B]γ as
C[B]γ (g) = lim CS(ε) (g) = lim R
ε→0
1
Z
S(ε) dx S(ε)
ε→0
g(x)dx.
(1)
If g : R → R is an integrable function and A ∈ F then the average value of
function g on [A]γ is defined by
C[A]γ (g) = R
1
[A]γ
Z
dx
g(x)dx.
[A]γ
Especially, if g(x) = x, for all x ∈ R is the identity function (g = id) and
A ∈ F is a fuzzy number with [A]γ = [a1 (γ), a2 (γ)] then the average value of the
identity function on [A]γ is computed by
C
[A]γ
(id) = R
1
xdx =
a2 (γ) − a1 (γ)
[A]γ dx [A]γ
1
Z
Z
a2 (γ)
xdx =
a1 (γ)
a1 (γ) + a2 (γ)
,
2
which remains valid in the limit case a2 (γ) − a1 (γ) = 0 for some γ. Because
C[A]γ (id) is nothing else, but the center of [A]γ we will use the shorter notation
C([A]γ ) for C[A]γ (id).
It is clear that C[B]γ is linear for any fixed joint possibility distribution B and
for any γ ∈ [0, 1]. Let us denote the projection functions on R2 by πx and πy , that
is, πx (u, v) = u and πy (u, v) = v for u, v ∈ R.
The following theorems shows two important properties of the central value
operator.
3
Theorem 2.1 If A, B ∈ F are non-interactive and g = πx + πy is the addition
operator on R2 then
C[A×B]γ (πx + πy ) = C[A]γ (id) + C[B]γ (id) = C([A]γ ) + C([B]γ ),
for all γ ∈ [0, 1].
Proof 1 If A and B are non-interactive then [C]γ = [A]γ × [B]γ for all γ ∈ [0, 1],
that is,
Z
1
C[C]γ (πx + πy ) = R
(x + y)dxdy
[A]γ ×[B]γ dxdy [A]γ ×[B]γ
Z a2 (γ) Z b2 (γ)
1
=
(x + y)dxdy
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)] a1 (γ) b1 (γ)
=
1
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)]
Z
Z
× [b2 (γ) − b1 (γ)]
xdx + [a2 (γ) − a1 (γ)]
[A]γ
=
1
a2 (γ) − a1 (γ)
ydy
[B]γ
Z
xdx +
[A]γ
1
b2 (γ) − b1 (γ)
Z
ydy
[B]γ
a1 (γ) + a2 (γ) b1 (γ) + b2 (γ)
+
= C([A]γ ) + C([B]γ ),
2
2
Theorem 2.2 If A, B ∈ F are non-interactive and p = πx πy is the multiplication
operator on R2 then
=
C[A×B]γ (πx πy ) = C[A]γ (id) · C[B]γ (id) = C([A]γ ) · C([B]γ ),
for all γ ∈ [0, 1].
Proof 2 If A and B are non-interactive then [C]γ = [A]γ × [B]γ for all γ ∈ [0, 1],
that is,
Z
1
C[C]γ (πx πy ) = R
xydxdy
[A]γ ×[B]γ dxdy [A]γ ×[B]γ
Z a2 (γ) Z b2 (γ)
1
xydxdy
=
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)] a1 (γ) b1 (γ)
Z a2 (γ)
Z b2 (γ)
1
=
xdx ·
ydy
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)] a1 (γ)
b1 (γ)
=
a1 (γ) + a2 (γ) b1 (γ) + b2 (γ)
·
= C([A]γ ) · C([B]γ ).
2
2
4
The following definition is crucial for our theory.
Definition 2.1 Let C be a joint possibility distribution with marginal possibility
distributions A, B ∈ F, and let γ ∈ [0, 1]. The measure of interactivity between
the γ-level sets of A and B is defined by
R[C]γ (πx , πy ) = C[C]γ (πx − C[C]γ (πx ))(πy − C[C]γ (πy )) .
Using the definition of central value we have
1
[C]γ dxdy
Z
1
=R
[C]γ dxdy
Z
R[C]γ (πx , πy ) = R
[C]γ
(x − C[C]γ (πx )) · (y − C[C]γ (πy ))dxdy
Z
1
xydxdy − C[C]γ (πy ) · R
xdxdy
[C]γ
[C]γ dxdy [C]γ
Z
1
− C[C]γ (πx ) · R
ydxdy + C[C]γ (πx ) · C[C]γ (πy )
[C]γ dxdy [C]γ
= C[C]γ (πx πy ) − C[C]γ (πx ) · C[C]γ (πy ),
for all γ ∈ [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]γ (πx ))(y − C[C]γ (πy )),
on [C]γ . Suppose that we choose a point x ∈ R then [C]γ 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]γ (πx ), C[C]γ (πy )) are concordant (discordant). Loosely
speaking, the value of R[C]γ (πx , πy ) is positive if the ’strength’ of (x, y) and
(C[C]γ (πx ), C[C]γ (πy )) that are concordant is bigger than the strength of those ones
that are discordant.
Consider now the case depicted on Fig.2. After some calculations we find
C[C]γ (πx ) =
a1 (γ) + a2 (γ)
= C([A]γ ),
2
C[C]γ (πy ) =
b1 (γ) + b2 (γ)
= C([B]γ ).
2
and
5
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]γ ))(y − C([B]γ ),
will always be positive.
Let A, B ∈ F with [A]γ = [a1 (γ), a2 (γ)] and [B]γ = [b1 (γ), b2 (γ)]. In [2] we
introduced the notion of f -weighted covariance of possibility distributions A and
B as
Z 1
a2 (γ) − a1 (γ) b2 (γ) − b1 (γ)
Covf (A, B) =
·
f (γ)dγ.
2
2
0
The main drawback of that definition 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
definition of covariance, that agrees with the principle of ’falling shadows’. Furthermore, we shall use this new definition of covariance to measure interactions
between two fuzzy numbers.
Definition 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 distributioin C) is
defined by
Z 1
R[C]γ (πx , πy )f (γ)dγ
Covf (A, B) =
0
(2)
Z 1
=
C[C]γ (πx πy ) − C[C]γ (πx ) · C[C]γ (πy ) f (γ)dγ.
0
If A and B are non-interactive then from Theorem 2.2 it follows that
R[C]γ (πx , πy ) = C[A×B]γ (πx πy ) − C([A]γ ) · C([B]γ ) = 0
for all γ ∈ [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 (2) with the constant weighting function f (γ) = 1 for all
γ ∈ [0, 1] then we get
1
Z
R[C]γ (πx , πy )dγ.
Covf (A, B) =
0
6
That is, Covf (A, B) simple sums up the interactivity measures level-wisely. In (2)
f (γ) weights the importance of the interactivity between the γ-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 defined
by (see Fig. 2)
[C]γ = {t(a1 (γ), b1 (γ)) + (1 − t)(a2 (γ), b2 (γ)) | t ∈ [0, 1]},
for all γ ∈ [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.
Figure 2: Joint possibility distribution C.
Let γ be arbitrarily fixed, and let a1 = a1 (γ), a2 = a2 (γ), b1 = b1 (γ), b2 =
b2 (γ). Then the γ-level set of C can be calculated as {(c(y), y)|b1 ≤ y ≤ b2 }
where
b2 − y
y − b1
a2 − a1
a1 b2 − a2 b1
c(y) =
a1 +
a2 =
y+
.
b2 − b1
b2 − b1
b2 − b1
b2 − b1
Since all γ-level sets of C are degenerated, i.e. their integrals vanish, the following
formal calculations can be done
Z
Z b2
Z
Z b2 " 2 #c(y)
x
c(y)
dxdy =
dy,
[x]c(y) dy,
xydxdy =
y
2
[C]γ
b1
[C]γ
b1
c(y)
which turns into
C[C]γ (πx πy ) =
=
=
=
=
Z
1
R
xydxdy
[C]γ dxdy [C]γ
Z b2
1
yc(y)dy
b2 − b1 b1
"
#
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
7
Hence, we have
R[C]γ (πx , πy ) = C[C]γ (πx πy ) − C[C]γ (πx )C[C]γ (πy )
=
=
(a2 − a1 )(b2 − b1 ) a1 b2 + a2 b1 (a1 + a2 )(b1 + b2 )
+
−
3
2
4
(a2 − a1 )(b2 − b1 )
,
12
and, finally, the covariance of A and B with respect to their joint possibility distribution C is
Z
1 1
Covf (A, B) =
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)]f (γ)dγ.
12 0
Note 2.4 In this case Covf (A, B) > 0 for any weighting function f ; and if
A(u) ≥ γ 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 defined
by (see Fig. 3)
[D]γ = {t(a1 (γ), b2 (γ)) + (1 − t)(a2 (γ), b1 (γ)) | t ∈ [0, 1]}.
for all γ ∈ [0, 1]. In this case A and B are interactive. We will compute now the
measure of interactivity between A and B.
Figure 3: Joint possibility distribution D.
After similar calculations we get
C[D]γ (πx πy ) = −
(a2 − a1 )(b2 − b1 ) a1 b1 + a2 b2
+
,
3
2
which implies that
R[D]γ (πx , πy ) = C[D]γ (πx πy ) − C[D]γ (πx )C[D]γ (πy ) = −
(a2 − a1 )(b2 − b1 )
,
12
and we find
Covf (A, B) = −
1
12
Z
1
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)]f (γ)dγ.
0
8
Note 2.5 In this case Covf (A, B) < 0 for any weighting function f ; and if
A(u) ≥ γ 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) · χ[0,1] (x) for x ∈ R,
that is, [A]γ = [B]γ = [0, 1 − γ] for γ ∈ [0, 1]. Suppose that their joint possibility
distribution is given by
F (x, y) = (1 − x − y) · χT (x, y),
where T = {(x, y) ∈ R2 |x ≥ 0, y ≥ 0, x + y ≤ 1}. This situation is depicted on
Fig. 4, where we have shifted the fuzzy sets to get a better view of the situation.
Figure 4: Joint possibility distribution F .
It is easy to check that A and B are really the marginal distributions of F . A γlevel set of F is computed by [F ]γ = {(x, y) ∈ R2 |x ≥ 0, y ≥ 0, x + y ≤ 1 − γ}.
Then
Z
1
R[F ]γ (πx , πy ) = R
xydxdy
[F ]γ dxdy [F ]γ
!
!
Z
Z
1
1
R
xdxdy
ydxdy ,
− R
[F ]γ dxdy [F ]γ
[F ]γ dxdy [F ]γ
where,
Z
dxdy =
[F ]γ
(1 − γ)2
,
2
and,
1−γ
1−γ−x
Z
1 1−γ
xydxdy =
xydxdy =
x(1 − γ − x)2 dx
2 0
[F ]γ
0
0
Z
Z
1 1−γ 3
(1 − γ)4
1 − γ 1−γ 2
=
x dx −
x dx =
,
2
2 0
24
0
Z
Z
Z
furthermore,
Z
Z
Z
xdxdy =
[F ]γ
Z
1−γ−x
ydxdy =
[F ]γ
=
1−γ
1
2
Z
ydxdy
0
0
1−γ
(1 − γ − x)2 dx =
0
9
(1 − γ)3
,
6
Figure 5: Partition of [F ]γ .
for any γ ∈ [0, 1]. Hence, the covariance between A and B is,
1
Z
Covf (A, B) =
0
1
R[F ]γ (πx , πy )f (γ)dγ = −
36
1
Z
(1 − γ)2 f (γ)dγ,
0
for any weighting function f .
Let us take a closer look at joint possibility distribution F . It is clear that,
C[F ]γ (πx ) = C[F ]γ (πy ) = (1 − γ)/3,
and if A takes a value u such that A(u) ≥ γ, then B can take its values from the
interval [0, 1 − γ − u]. The average value of the interactivity function
1−γ
1−γ
g(x, y) = x −
y−
,
3
3
on [F ]γ 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 · χ[0,1] (x) for x ∈ R, that
is, [A]γ = [0, 1 − γ] and [B]γ = [γ, 1], for γ ∈ [0, 1]. Let
E(x, y) = (y − x) · χS (x, y),
where S = {(x, y) ∈ R2 |x ≥ 0, y ≤ 1, y − x ≥ 0}. This situation is depicted on
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 · χ[0,1] (y) = B(y),
x∈R
max E(x, y) = (1 − x) · χ[0,1] (x) = A(x).
y∈R
A γ-level set of E is computed by [E]γ = {(x, y) ∈ R2 |x ≥ 0, y ≤ 1, y − x ≥ γ}.
Figure 6: Joint possibility distribution E.
After some calculations we get
Z
Covf (A, B) =
0
1
1
R[E]γ (πx , πy )f (γ)dγ =
36
10
Z
0
1
(1 − γ)2 f (γ)dγ,
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]γ
Z
1
C[A]γ (g) = R
g(x)dx.
[A]γ dx [A]γ
Definition 2.3 The expected value of function g on A with respect to a weighting
function f is defined by
Z 1
Z 1
Z
1
R
Ef (g; A) =
C[A]γ (g)f (γ)dγ =
g(x)dxf (γ)dγ.
0
0
[A]γ dx [A]γ
Especially, if g is the identity function then we get
Z 1
a1 (γ) + a2 (γ)
Ef (id; A) = Ef (A) =
f (γ)dγ,
2
0
which is thef -weighted possibilistic expected value value of A introduced in [2].
Let us denote R[A]γ (id, id) the average value of function g(x) = (x−C([A]γ ))2
on the γ-level set of an individual fuzzy number A. That is,
2
Z
Z
1
1
2
R[A]γ (id, id) = R
x dx − R
xdx .
[A]γ dx [A]γ
[A]γ dx [A]γ
Definition 2.4 The variance of A is defined as the expected value of function
g(x) = (x − C([A]γ ))2 on A. That is,
Z 1
Varf (A) = Ef (g; A) =
R[A]γ (id, id)f (γ)dγ.
0
From the equality,
2
Z a2 (γ)
1
x dx −
xdx
a2 (γ) − a1 (γ) a1 (γ)
a1 (γ)
a2 (γ) + a1 (γ)a2 (γ) + a22 (γ)
a1 (γ) + a2 (γ) 2
= 1
−
3
2
2
2
a (γ) − 2a1 (γ)a2 (γ) + a2 (γ) (a2 (γ) − a1 (γ))2
=
,
= 1
12
12
1
R[A]γ (id, id) =
a2 (γ) − a1 (γ)
Z
a2 (γ)
2
we get,
Z
Varf (A) =
0
1
(a2 (γ) − a1 (γ))2
f (γ)dγ.
12
11
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 definition of f -weighted
possibilistic mean value of fuzzy numbers introduced in [2]. The f -weighted variance is almost the same as in [2] (the same up to a scalar multiplier).
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 λ, µ ∈ R.
Then
R[C]γ (λπx + µπy , λπx + µπy ) =
λ2 R[C]γ (πx , πx ) + µ2 R[C]γ (πy , πy ) + 2λµR[C]γ (πx , πy ).
Zero covariance does not always imply non-interactivity. Really, let G be a
joint possibility distribution with a symmetrical γ-level set, i.e., there exist a, b ∈
Rt such that
G(x, y) = G(2a − x, y) = G(x, 2b − y) = G(2a − x, 2b − y)
for all x, y ∈ [G]γ , where (a, b) is the center of the set [G]γ (see Fig. 7).
In this case R[G]γ (πx , πy ) = 0. In fact, introducing the notation
H = {(x, y) ∈ [G]γ |x ≤ a, y ≤ b},
we find,
Z
Z
xydxdy =
xy + (2a − x)y + x(2b − y) + (2a − x)(2b − y) dxdy
[G]γ
H
Z
= 4ab
dxdy,
H
Z
Z
Z
xdxdy = 2
x + (2a − x) dxdy = 4a
dxdy,
[G]γ
H
H
Z
Z
Z
ydxdy = 2
y + (2b − y) dxdy = 4b
dxdy,
[G]γ
H
H
Z
Z
dxdy = 4
dxdy.
[G]γ
H
12
Figure 7: Joint possibility distribution G.
That is,
Z
1
xydxdy
[G]γ dxdy [C]γ
Z
Z
1
1
R
− R
xdxdy
ydxdy = 0.
[G]γ dxdy [C]γ
[G]γ dxdy [C]γ
R[G]γ (πx , πy ) = R
If all γ-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]γ for some u, v ∈ R then (−u, v), (u, −v) and (−u, −v) will also
belong to [G]γ , therefore, simple algebra shows that the equation
Z
xydxdy = 0,
[G]γ
holds for any γ ∈ [0, 1]. In a similar manner we can prove the relationships
Z
Z
xdxdy =
ydxdy = 0
[G]γ
[G]γ
for any γ ∈ [0, 1]. So, the average value of function
g(x, y) = (x − 0)(y − 0) = xy,
on [G]γ is zero (independently of γ).
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 [9]. Based on this fact, several indeces of
dependencies were introduced in [7, 8]. Furthermore, any copula C satisfies the
relationship [8]
W (x, y) = max{x + y − 1, 0} ≤ C(x, y) ≤ M (x, y) = min{x, y},
13
For uniform distributions on [a1 (γ), a2 (γ)] and [b1 (γ), b2 (γ)] we look for a joint
distribution defined on the Cartesian product of this level intervals (with uniform
marginal distributions on [a1 (γ), a2 (γ)] and [b1 (γ), b2 (γ)]): this joint distribution
can always be characterized by a unique copula C, which may depend also on γ
(see [7]). 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 1); and Fig. 3
is linked to copula W (with correlation -1).
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.
4
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] D. Dubois and H. Prade, Possibility Theory: An Approach to Computerized
Processing of Uncertainty, Plenum Press, New York, 1988.
[2] R. Fullér and P. Majlender, On weighted possibilistic mean and variance of
fuzzy numbers, Fuzzy Sets and Systems. (to appear)
[3] R. Fullér and P. Majlender, Correction to: “On interactive fuzzy numbers” [Fuzzy Sets and Systems, 143(2004) 355-369] Fuzzy Sets and Systems
152(2005) 159-159.
[4] E. Hisdal, Conditional possibilities independence and noninteraction, Fuzzy
Sets and Systems, 1(1978) 283-297.
14
[5] L. M. de Campos and J. F. Huete, Independence concepts in possibility theory: Part I, Fuzzy Sets and Systems, 103(1999) 127-152.
[6] L. M. de Campos, J. F. Huete, Independence concepts in possibility theory:
Part II, Fuzzy Sets and Systems, 103(1999) 487-505.
[7] R.B. Nelsen, An introduction to copulas, Lecture Notes in Statistics, 139,
Springer-Verlag, New York, 1999.
[8] R.B. Nelsen, J. J. Quesada-Molina, J. A. Rodrı́guez-Lallena and M. ÚbedaFlores, Distribution functions of copulas: a class of bivariate probability integral transforms, Statist. Probab. Lett., 54 (2001) 277-282.
[9] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst.
Statist. Univ. Paris, 8(1959) 229-231.
[10] L. A. Zadeh, Concept of a linguistic variable and its application to approximate reasoning, I, II, III, Information Sciences, 8(1975) 199-249, 301-357;
9(1975) 43-80.
Main results
The main results of this paper are,
• The f -weighted possibilistic variance of fuzzy number A is defined by
Z
1
Varf (A) =
0
(a2 (γ) − a1 (γ))2
f (γ)dγ.
12
• The measure of covariance between A and B (with respect to their joint
distribution C and weighting function f ) by
Z 1
Z 1
Covf (A, B) =
R[C]γ (πx , πy )f (γ)dγ =
cov(Xγ , Yγ )f (γ)dγ,
0
0
where Xγ and Yγ are random variables whose joint distribution is uniform
on [C]γ for all γ ∈ [0, 1]; furthermore cov(Xγ , Yγ ) denotes the probabilistic
covariance between marginal random variables Xγ and Yγ . If f (γ) = 2γ
then we write simple Cov(A, B), that is
Z
Cov(A, B) =
1
2γcov(Xγ , Yγ )dγ,
0
15
Citations
[A7] Robert Fullér and Péter Majlender, On interactive fuzzy numbers, FUZZY
SETS AND SYSTEMS 143(2004) 355-369. [MR2052672]
in journals
A7-c6 Hong, D.H., Kim, K.T., A maximal variance problem, APPLIED MATHEMATICS LETTERS, 20 (10), pp. 1088-1093. 2007
http://dx.doi.org/10.1016/j.aml.2006.12.008
Fullér and Majlender [A7] presented the idea of interaction between a marginal distribution of a joint possibility distribution.
They introduced the notion of covariance between fuzzy numbers via their joint possibility distribution to measure the degree
to which the fuzzy numbers interact. (page 1088)
A7-c5 Jian-li ZHAO, Zhi-gang JIA, Ying LI, Weighted Ranking and Simple Operation Approaches of Fuzzy-number Based on Four-dimensional Representations, FUZZY SYSTEMS AND MATHEMATICS, 2007, Vol.21 No.2 pp.
97-101.
A7-c4 B. Bede and J. Fodor, Product Type Operations between Fuzzy Numbers
and their Applications in Geology, ACTA POLYTECHNICA HUNGARICA, Vol. 3, No. 1, pp. 123-139. 2006
http://www.bmf.hu/journal/Bede_Fodor_5.pdf
A7-c3 Przemysław Grzegorzewski and Edyta Mrówka, Trapezoidal approximations of fuzzy numbers, FUZZY SETS AND SYSTEMS 153(1): 115-135
JUL 1 2005
http://dx.doi.org/10.1016/j.fss.2004.02.015
in proceedings
A7-c2 Pierpaolo D’Urso, Fuzzy Clustering of Fuzzy Data, in: J. Valente de
Oliveira and W. Pedrycz eds., Advances in Fuzzy Clustering and its Applications, John Wiley & Sons, [ISBN 978-0-470-02760-8], pp. 155-192.
2007
16
A7-c1 J. Fodor, B. Bede, Arithmetics with Fuzzy Numbers: a Comparative Overview,
SAMI 2006 conference, Herlany, Slovakia, [ISBN 963 7154 44 2], pp.54-68.
2006
http://www.bmf.hu/conferences/sami2006/Fodor.pdf
17