Christer Carlsson | Robert Fullér
Possibilistic correlation: illustration, explanation and computation of some important cases
TUCS Technical Report
No 872, February 2008
Possibilistic correlation: illustration, explanation and computation of some
important cases
Christer Carlsson
Institute for Advanced Management Systems Research
Department of Information Technologies, Åbo Akademi University
Joukahainengatan 3-5, 20520 Åbo, Finland
christer.carlsson@abo.fi
Robert Fullér
Institute for Advanced Management Systems Research
Department of Information Technologies, Åbo Akademi University
Joukahainengatan 3-5, 20520 Åbo, Finland
robert.fuller@abo.fi
TUCS Technical Report
No 872, February 2008
Abstract
In 1987 Dubois and Prade defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. In 2001 Carlsson and Fullér defined
an interval-valued mean value of fuzzy numbers, viewing them as possibility distributions, and they introduced the notation of crisp possibilistic mean value and
crisp possibilistic variance of continuous possibility distributions, which are consistent with the extension principle. In 2003 Fullér and Majlender introduced the
notation of weighted interval-valued possibilistic mean value of fuzzy numbers
and investigate its relationship to the interval-valued probabilistic mean. They
also introduced the notations of crisp weighted possibilistic mean value, variance
and covariance of fuzzy numbers, which are consistent with the extension principle. In 2004 Fullér and Majlender introduced the notion of covariance between
marginal distributions of a joint possibility distribution to measure the degree to
which they interact. In 2005 Carlsson, Fullér and Majlender presented the concept
of possibilistic correlation representing an average degree of interaction between
marginal distributions of a joint possibility distribution as compared to their respective dispersions. Moreover, they formulated the classical Cauchy-Schwarz
inequality in this possibilistic environment and showed that the measure of possibilistic correlation satisfies the same property as its probabilistic counterpart. In
particular, applying the idea of transforming level sets of possibility distributions
into uniform probability distributions, they identified a fundamental relationship
between our proposed possibilistic approach and the classical probabilistic approach to measuring correlation. In this work we give a geometrical interpretation
and explanation of the possibilistic correlation by some important cases.
Keywords: Possibility distribution, Possibilistic covariance, Possibilistic correlation, Uniform probability distribution
TUCS Laboratory
1 Fuzzy numbers
A fuzzy number A is a fuzzy set of the real line with a normal, (fuzzy) convex
and continuous membership function of bounded support. The family of fuzzy
numbers will be denoted by F .
Let A be a fuzzy number. Then [A]γ is a closed convex (compact) subset of R for
all γ ∈ [0, 1]. Let us introduce the notations
a1 (γ) = min[A]γ and a2 (γ) = max[A]γ .
In other words, a1 (γ) denotes the left-hand side and a2 (γ) denotes the right-hand
side of the γ-cut. It is easy to see that if α ≤ β then [A]α ⊃ [A]β . Furthermore, the
left-hand side function a1 : [0, 1] → R is monoton increasing and lower semicontinuous, and the right-hand side function a2 : [0, 1] → R is monoton decreasing
and upper semicontinuous. We shall use the notation
[A]γ = [a1 (γ), a2 (γ)].
The support of A is the open interval (a1 (0), a2 (0)).
Figure 1: A triangular fuzzy number.
Definition 1.1. A fuzzy set A is called triangular fuzzy number with peak (or
center) a, left width α > 0 and right width β > 0 if its membership function has
the following form

a−t



if a − α ≤ t ≤ a
1−


α

t−a
A(t) =
1−
if a ≤ t ≤ a + β



β



0
otherwise
and we use the notation A = (a, α, β). It can easily be verified that
[A]γ = [a − (1 − γ)α, a + (1 − γ)β], ∀γ ∈ [0, 1].
The support of A is (a − α, b + β). A triangular fuzzy number with center a may
be seen as a fuzzy quantity
”x is approximately equal to a”.
1
Definition 1.2. A fuzzy set A is called trapezoidal fuzzy number with tolerance
interval [a, b], left width α and right width β if its membership function has the
following form

a−t


if a − α ≤ t ≤ a
1−


α



 1
if a ≤ t ≤ b
A(t) =

t−b


1
−
if a ≤ t ≤ b + β


β



0
otherwise
and we use the notation
A = (a, b, α, β).
(1)
It can easily be shown that
[A]γ = [a − (1 − γ)α, b + (1 − γ)β], ∀γ ∈ [0, 1].
The support of A is (a − α, b + β).
A trapezoidal fuzzy number may be seen as a fuzzy quantity
”x is approximately in the interval [a, b]”.
Figure 2: Trapezoidal fuzzy number.
Let A ∈ F be fuzzy number with [A]γ = [a1 (γ), a2 (γ)] and let Uγ denote a uniform probability distribution on [A]γ , γ ∈ [0, 1]. It is clear that the (probabilistic)
mean value of Uγ is equal to
a1 (γ) + a2 (γ)
,
2
and its (probabilistic) variance is
σU2 γ =
(a2 (γ) − a1 (γ))2
.
12
2
Let A and B ∈ F be fuzzy numbers with [A]γ = [a1 (γ), a2 (γ)] and [B]γ =
[b1 (γ), b2 (γ)], γ ∈ [0, 1]. In 1986 Goetschel and Voxman introduced a method for
ranking fuzzy numbers as [25]
Z
A ≤ B ⇐⇒
1
γ(a1 (γ) + a2 (γ)) dγ ≤
0
Z
1
γ(b1 (γ) + b2 (γ)) dγ
(2)
0
As was pointed out by Goetschel and Voxman this definition of ordering given in
(2) was motivated in part by the desire to give less importance to the lower levels
of fuzzy numbers.
2 Possibilistic mean value of fuzzy numbers
In 1987 Dubois and Prade [22] dened an interval-valued expectation of fuzzy
numbers, viewing them as consonant random sets. They also showed that this
expectation remains additive in the sense of addition of fuzzy numbers. In 2001
introducing the notations of lower possibilistic and upper possibilistic mean values Carlsson and Fullér defined the interval-valued possibilistic mean, crisp possibilistic mean value and crisp (possibilistic) variance of a continuous possibility
distribution, which are consistent with the extension principle and with the wellknown dentions of expectation and variance in probability theory. The theory
developed by Carlsson and Fullér in [5] was that time fully motivated by the principles introduced in [22] and by the possibilistic interpretation of the ordering
introduced in [25].
Definition 2.1. [5] Let A ∈ F be fuzzy number with [A]γ = [a1 (γ), a2 (γ)], γ ∈
[0, 1]. Then the possibilistic mean (or expected) value of fuzzy number A is defined
as
Z 1
Z 1
Z 1
a1 (γ) + a2 (γ)
E(A) =
E(Uγ )2γ dγ =
2γ dγ =
(a1 (γ) + a2 (γ))γ dγ,
2
0
0
0
where Uγ is a uniform probability distribution on [A]γ for all γ ∈ [0, 1].
We note that from the equality
E(A) =
Z
1
γ(a1 (γ) + a2 (γ))dγ =
0
Z
0
1
γ·
a1 (γ) + a2 (γ)
dγ
2
,
Z 1
γ dγ
0
it follows that E(A) is nothing else but the level-weighted average of the arithmetic means of all γ-level sets, that is, the weight of the arithmetic mean of a1 (γ)
and a2 (γ) is just γ.
3
Example 2.1. Let A = (a, α, β) be a triangular fuzzy number with center a, leftwidth α > 0 and right-width β > 0 then a γ-level of A is computed by
[A]γ = [a − (1 − γ)α, a + (1 − γ)β], ∀γ ∈ [0, 1],
Then,
E(A) =
Z
1
γ[a − (1 − γ)α + a + (1 − γ)β]dγ = a +
0
β−α
.
6
When A = (a, α) is a symmetric triangular fuzzy number we get E(A) = a.
3 Possibilistic variance of fuzzy numbers
In 2004 Fullér and Majlender [24] defined the possibilistic variance of A as the
covariance of A with itself.
Definition 3.1. [24] Let A ∈ F be fuzzy number with [A]γ = [a1 (γ), a2 (γ)],
γ ∈ [0, 1].
Z 1
Z 1
Z 1
(a2 (γ) − a1 (γ))2
(a2 (γ) − a1 (γ))2
2
Var(A) =
σUγ 2γ dγ =
2γ dγ =
γ dγ
12
6
0
0
0
where Uγ is a uniform probability distribution on [A]γ and σU2 γ denotes the variance of Uγ .
Note 3.1. Carlsson and Fullér originally introduced the possibilistic variance of
A ∈ F in 2001 [5] as
2 !
Z 1
a1 (γ) + a2 (γ)
− a1 (γ)
dγ
Var(A) =
Pos[A ≤ a1 (γ)]
2
0
2 !
Z 1
a1 (γ) + a2 (γ)
+
Pos[A ≥ a2 (γ)]
− a2 (γ)
dγ
2
0
Z
2
1 1
=
γ a2 (γ) − a1 (γ) dγ.
2 0
where Pos denotes possibility,i.e.
Pos[A ≤ a1 (γ)] = sup A(u) = γ,
Pos[A ≥ a2 (γ)] = sup A(u) = γ.
u≤a1 (γ)
u≥a2 (γ)
The variance of A defined in our earlier paper in 2001 can be interpreted as the
expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e. the lower possibility-weighted average of the squared
distance between the left-hand endpoint and the arithmetic mean of the endpoints
of its level sets plus the upper possibility-weighted average of the squared distance
between the right-hand endpoint and the arithmetic mean of the endpoints their
of its level sets.
4
Example 3.1. If A = (a, α, β) is a triangular fuzzy number then
Z
2
1 1
(α + β)2
Var(A) =
γ a + β(1 − γ) − (a − α(1 − γ)) dγ =
.
6 0
72
especially, if A = (a, α) is a symmetric triangular fuzzy number then
α2
Var(A) = .
18
4 Weighted possibilistic mean value
Let A ∈ F be fuzzy number with [A]γ = [a1 (γ), a2 (γ)], γ ∈ [0, 1]. 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
In 2003 Fullér and Majlender [23] defined the f -weighted possibilistic mean (or
expected) value of fuzzy number A as
Z 1
Z 1
a1 (γ) + a2 (γ)
f (γ)dγ.
Ef (A) =
E(Uγ )f (γ)dγ =
2
0
0
where Uγ is a uniform probability distribution on [A]γ for all γ ∈ [0, 1].
Example Let A = (a, b, α, β) be a fuzzy number of trapezoidal form with peak
[a, b], left-width α > 0 and right-width β > 0, and let f (γ) = (n + 1)γ n , n ≥ 0.
The γ-level of A is
[A]γ = [a − (1 − γ)α, b + (1 − γ)β], ∀γ ∈ [0, 1].
Then the weighted possibilistic mean values of A are computed by
Ef (A) =
β−α
a+b
+
.
2
2(n + 2)
So,
lim Ef (A) = lim
n→∞
n→∞
a+b
a+b
β−α
=
+
.
2
2(n + 2)
2
5 Weighted possibilistic variance
In 2004 Fullér and Majlender [24] introduced f -weighted possibilistic variance of
A by
Z 1
Z
1 1
2
Varf (A) =
σUγ f (γ)dγ =
(a2 (γ) − a1 (γ))2 f (γ)dγ,
12 0
0
5
where σU2 γ denotes the variance of Uγ . It was originally introduced by Fullér and
Majlender in 2003 [23] as
Z 1
1
Varf (A) = ×
(a2 (γ) − a1 (γ))2 f (γ)dγ.
4
0
Example 5.1. Let A = (a, b, α, β) be a trapezoidal fuzzy number and let f (γ) =
(n + 1)γ n be a weighting function. Then,
Z 1
1
Varf (A) = (n + 1) ×
×
(a2 (γ) − a1 (γ))2 γ n dγ
12
0
Z 1
n+1
[(b − a) + (α + β)(1 − γ)]2 γ n dγ
=
12 0
1
b−a
α + β 2 1 (n + 1)(α + β)2
= ×
+
.
+ ×
3
2
2(n + 2)
3 4(n + 2)2 (n + 3)
So,
1
lim Varf (A) = lim ×
n→∞
n→∞ 3
b−a
α+β 2
(n + 1)(α + β)2
b−a
=
+
+
.
2
2
2(n + 2)
4(n + 2) (n + 3)
6
6 Possibilistic covariance
Let C be a joint possibility distribution with marginal possibility distributions A
and B, and let f be a weighting function.
Definition 6.1. [24] The measure of covariance between A and B (with respect to
their joint distribution C and weighting function f ) is defined by
Z 1
Covf (A, B) =
cov(Xγ , Yγ )f (γ)dγ,
(3)
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 C = A × B then Covf (A, B) = 0 for any weighting function f . Really,
in this case [C]γ = [A]γ × [B]γ and Xγ and Yγ are independent, and therefore
cov(Xγ , Yγ ) = 0, for any γ ∈ [0, 1].
Equation (3) improves our earlier concepts of covariance
Z
1 1
Cov(A, B) =
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ
2 0
originally introduced by Carlsson and Fullér in 2001 in [5]; and
Z 1
a2 (γ) − a1 (γ) b2 (γ) − b1 (γ)
Covf (A, B) =
·
f (γ)dγ.
2
2
0
originally introduced by Fullér and Majlender in 2003 [23].
6
7 Possibilistic correlation
The f -weighted possibilistic correlation of A, B ∈ F , (with respect to their joint
distribution C) is defined in 2005 by Carlsson, Fullér and Majlender [19] by
R1
Covf (A, B)
cov(Xγ , Yγ )f (γ)dγ
0
ρf (A, B) = p
= R
1/2 R
1/2 ,
1 2
1 2
Varf (A) Varf (B)
σ f (γ)dγ
σ f (γ)dγ
0 Uγ
0 Vγ
where Uγ is a uniform probability distribution on [A]γ and Vγ is a uniform probability distribution on [B]γ , and Xγ and Yγ are random variables whose joint probability distribution is uniform on [C]γ for all γ ∈ [0, 1].
Theorem 7.1. [19] Let C be a joint possibility distribution in R2 with marginal
possibility distributions A, B ∈ F , and let f be a weighting function. Let us
assume that Varf (A) 6= 0 and Varf (B) 6= 0, and that [C]γ is a convex set for all
γ ∈ [0, 1]. Then,
−1 ≤ ρf (A, B) = p
Covf (A, B)
Varf (A) Varf (B)
≤ 1.
Moreover, ρf (A, B) = −1 if and only if A and B are completely negatively
correlated, and ρf (A, B) = 1 if and only if A and B are completely positively
correlated. Furthermore, if C = A × B then ρf (A, B) = 0 for any weighting
function f .
8 Illustrations of possibilistic correlation
Let C be a joint possibility distribution in R2 with marginal possibility distributions A = πx (C), B = πy (C) ∈ F , and let [A]γ = [a1 (γ), a2 (γ)] and [B]γ =
[b1 (γ), b2 (γ)], γ ∈ [0, 1]. First, let us assume that A and B are non-interactive, i.e.
C = A × B. This situation is depicted in Fig. 3. Then [C]γ = [A]γ × [B]γ ([C]γ
is rectangular subset of R2 ) for any γ ∈ [0, 1], and from
cov(Xγ , Yγ ) = 0
for all γ ∈ [0, 1] we have
Covf (A, B) =
Z
1
cov(Xγ , Yγ )f (γ)dγ = 0
0
and
ρf (A, B) = 0
for any weighting function f .
7
Figure 3: The case of ρf (A, B) = 0.
In the case, depicted in Fig. 4, 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
ρf (A, B) = 1,
for any weighted function f . Really, the variances are computed by
Z
1 1
Varf (A) =
[a2 (γ) − a1 (γ)]2 f (γ)dγ,
12 0
Z
1 1
[b2 (γ) − b1 (γ)]2 f (γ)dγ
Varf (B) =
12 0
for any weighting function f .
And in this case [C]γ is a line segment in R2 , which can be represented by
[C]γ = {t(a1 (γ), b1 (γ)) + (1 − t)(a2 (γ), b2 (γ))|t ∈ [0, 1]}
(4)
where [A]γ = [a1 (γ), a2 (γ)] and [B]γ = [b1 (γ), b2 (γ)].
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
c(y) =
y − b1
a2 − a1
a1 b2 − a2 b1
b2 − y
a1 +
a2 =
y+
.
b2 − b1
b2 − b1
b2 − b1
b2 − b1
8
Figure 4: The case of ρf (A, B) = 1.
Since all γ-level sets of C are degenerated, i.e. their integrals vanish, the following
formal calculations can be done
Z
dxdy =
[C]γ
Z
b2
c(y)
[x]c(y) dy,
b1
Z
xydxdy =
[C]γ
Z
b2
b1
"
x2
y
2
#c(y)
dy.
b2
c(y)+ǫ
c(y)
Now we compute the expected value of the product Xγ Yγ ,
1
E(Xγ Yγ ) = R
dxdy
[C]γ
1
=
b2 − b1
Z
Z
b2
[C]γ
xydxdy = lim Z
ǫ→0
1
b2 Z
b1
c(y)+ǫ
dxdy
Z
b1
Z
c(y)−ǫ
c(y)−ǫ
yc(y)dy
b1
"
#
1
1
1
=
(a2 − a1 )(b32 − b31 ) + (a1 b2 − a2 b1 )(b22 − b21 )
2
(b2 − b1 ) 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
=
9
xydxdy
And the expected value of Xγ is computed by
Z b2 Z c(y)+ǫ
Z
1
1
xdxdy
E(Xγ ) = Z
xdxdy = lim Z b2 Z c(y)+ǫ
ǫ→0
b1
c(y)−ǫ
[C]γ
dxdy
dxdy
[C]γ
= lim Z
b1
1
b2
ǫ→0
2ǫdy
Z
b2
b1
c(y)−ǫ
Z b2 1
2c(y)ǫdy =
b2 − b1
b1
a2 − a1
a1 b2 − a2 b1
dy
y+
b2 − b1
b2 − b1
b1
1
a2 − a1 b22 − b21 a1 b2 − a2 b1
=
×
+
(b2 − b1 )
b2 − b1 b2 − b1
2
b2 − b1
that is,
E(Xγ ) =
a1 + a2
.
2
E(Yγ ) =
b1 + b2
.
2
In a similar way we get,
Hence, we have
Cov(Xγ , Yγ ) = E(Xγ Yγ ) − E(Xγ )E(Yγ )
(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
In particular, from the definition (4) of joint possibility distribution C we find that
there exists a constant ϑ ∈ R, ϑ ≥ 0 such that
b2 (γ) − b1 (γ) = ϑ(a2 (γ) − a1 (γ)),
∀γ ∈ [0, 1].
Thus, we obtain
Covf (A, B) = ϑ Varf (A),
Varf (B) = ϑ2 Varf (A),
which implies
Covf (A, B)
ϑ Varf (A)
ρf (A, B) = p
=p
=1
Varf (A) Varf (B)
Varf (A) ϑ2 Varf (A)
for any weighting function f .
10
Figure 5: The case of ρf (A, B) = −1.
Note 8.1. If u ∈ [A]γ for some u ∈ R then there exists a unique v ∈ R that B can
take. Furthermore, if u is moved to the left (right) then the corresponding value
(that B can take) will also move to the left (right). This property can serve as a
justification of the principle of (complete positive) correlation of A and B.
In the case, depicted in Fig. 3, the covariance of A and B with respect to their
joint possibility distribution D is
Z
1 1
Covf (A, B) = −
[a2 (γ) − a1 (γ)][b2 (γ) − b1 (γ)]f (γ)dγ,
12 0
ρf (A, B) = −1,
for any weighted function f .
Really, in this case [C]γ is a line segment in R2 , which can be represented by
[C]γ = {t(a1 (γ), b2 (γ)) + (1 − t)(a2 (γ), b1 (γ))|t ∈ [0, 1]},
(5)
where [A]γ = [a1 (γ), a2 (γ)] and [B]γ = [b1 (γ), b2 (γ)].
From the representation (5) of joint possibility distribution C we can see that there
exists a constant ϑ ∈ R, ϑ ≥ 0 such that (8) holds. Hence, we have
Covf (A, B) = −ϑ Varf (A),
therefore, we find that
11
Varf (B) = ϑ2 Varf (A),
ρf (A, B) = p
Covf (A, B)
Varf (A) Varf (B)
= −1
for any weighting function f .
Note 8.2. If u ∈ [A]γ for some u ∈ R then there exists a unique v ∈ R that B
can take. Furthermore, if u is moved to the left (right) then the corresponding
value (that B can take) will move to the right (left). This property can serve as a
justification of the principle of (complete negative) correlation of A and B.
Zero correlation does not always imply non-interactivity. Let A, B ∈ F be fuzzy
numbers, let C be their joint possibility distribution, and let γ ∈ [0, 1]. Suppose
that [C]γ is symmetrical, i.e. there exists a ∈ R such that
C(x, y) = C(2a − x, y),
for all x, y ∈ [C]γ (the line defined by {(a, t)|t ∈ R} is the axis of symmetry of
[C]γ ). In this case cov(Xγ , Yγ ) = 0. Indeed, let
H = {(x, y) ∈ [C]γ |x ≤ a},
then
Z
xydxdy =
[C]γ
Z
xdxdy =
[C]γ
Z
Z
xy + (2a − x)y dxdy = 2a
H
Z
x + (2a − x) dxdy = 2a
H
ydxdy = 2
[C]γ
Z
Z
ydxdy,
H
Z
ydxdy,
H
Z
dxdy,
H
dxdy = 2
[C]γ
Z
dxdy,
H
therefore, we obtain
1
cov(Xγ , Yγ ) = Z
dxdy
Z
xydxdy
[C]γ
[C]γ
−Z
1
dxdy
[C]γ
Z
[C]γ
xdxdy Z
1
dxdy
Z
ydxdy = 0.
[C]γ
[C]γ
For example, let G be a joint possibility distribution with a symmetrical γ-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),
12
Figure 6: A case of ρf (A, B) = 0 for interactive fuzzy numbers.
for all x, y ∈ [G]γ , where (a, b) is the center of the set [G]γ , In Fig. 6, the joint
possibility distribution is defined from marginal distributions as
G(x, y) = TW (A(x), B(y)),
where TW denotes the weak t-norm.
Consider now joint possibility distributions that are derived from given marginal
distributions by aggregating their membership values.
Namely, let A, B ∈ F . We will say that their joint possibility distribution C is
directly defined from its marginal distributions if
C(x, y) = T (A(x), B(y)), x, y ∈ R,
where T : [0, 1] × [0, 1] → [0, 1] is a function satisfying the properties
max T (A(x), B(y)) = A(x), ∀x ∈ R,
(6)
max T (A(x), B(y)) = B(y), ∀y ∈ R,
(7)
y
and
x
for example a triangular norm.
In this case the joint distribution depends barely on the membership values of
its marginal distributions. Furthermore, the covariance (and, consequently, the
correlation) between its marginal distributions will be zero whenever at least one
of its marginal distributions is symmetrical.
13
Theorem 8.1. [17] Let A, B ∈ F and let their joint possibility distribution C be
defined by
C(x, y) = T (A(x), B(y)),
for x, y ∈ R, where T is a function satisfying conditions (6) and (7). If A is a
symmetrical fuzzy number then
Covf (A, B) = 0,
for any fuzzy number B, aggregator T , and weighting function f .
Really, if A is a symmetrical fuzzy number with center a such that A(x) = A(2a−
x) for all x ∈ R then,
C(x, y) = T (A(x), B(y)) = T (A(2a − x), B(y)) = C(2a − x, y),
that is, C is symmetrical. Hence, considering the results obtained above we have
cov(Xγ , Yγ ) = 0,
and, therefore,
Covf (A, B) = 0,
for any weighting function f .
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. 7, 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 γ-level set of F is computed by
[F ]γ = {(x, y) ∈ R2 |x ≥ 0, y ≥ 0, x + y ≤ 1 − γ}.
In this case, the probabilistic expected values of marginal distributions of Xγ and
Yγ are equal to (1 − γ)/3 see (Fig. 8). And the covariance between Xγ and Yγ is
positive on H1 and H4 and negative on H2 and H3 .
In this case we get,
14
Figure 7: ρf (A, B) = −1/3.
Figure 8: Partition of [F ]γ .
15
cov(Xγ , Yγ ) = Z
−Z
1
dxdy
Z
[F ]γ
[F ]γ
1
dxdy
Z
xydxdy
[F ]γ
[F ]γ
1
xdxdy × Z
dxdy
Z
ydxdy.
[F ]γ
[F ]γ
That is,
2
×
cov(Xγ , Yγ ) =
(1 − γ)2
Z
1−γ
2
cov(Xγ , Yγ ) =
×
(1 − γ)2
Z
1−γ
0
Z
0
1−γ−x
xydxdy −
0
(1 − γ)2
.
9
(1 − γ)2
.
x(1 − γ − x)dx −
9
(1 − γ)2
(1 − γ)2 (1 − γ)2
−
=−
12
9
36
After some calculations (see Fig. 8) we get
cov(Xγ , Yγ ) =
1
Covf (A, B) = −
36
Z
1
(1 − γ)2 f (γ)dγ,
0
1
Varf (A) = Varf (B) =
12
and, therefore,
Z
1
(1 − γ)2 f (γ)dγ
0
R1
− 1/36 0 (1 − γ)2 f (γ)dγ
ρf (A, B) = p
=
= −1/3,
R1
Varf (A) Varf (B)
1/12 0 (1 − γ)2 f (γ)dγ
Covf (A, B)
for any weighting function f .
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. 9, where we have shifted the fuzzy sets to get a
better view of the situation.
16
Figure 9: ρf (A, B) = 1/3.
Figure 10: Partition of [E]γ .
17
A γ-level set of E is computed by
[E]γ = {(x, y) ∈ R2 |x ≥ 0, y ≤ 1, y − x ≥ γ}.
In this case, the probabilistic expected value of marginal distribution Xγ is equal
to (1 − γ)/3 and the probabilistic expected value of marginal distribution of Yγ is
equal to 2(1 − γ)/3 see (Fig. 10).
And the covariance between Xγ and Yγ is positive on H1 and H4 and negative on
H2 and H3 . After some calculations (see Fig. 10) we get
1
Covf (A, B) =
36
Z
1
(1 − γ)2 f (γ)dγ,
0
1
Varf (A) = Varf (B) =
12
and, therefore,
Covf (A, B)
=
ρf (A, B) = p
Varf (A) Varf (B)
for any weighting function f .
Z
1
(1 − γ)2 f (γ)dγ
0
1/36
R1
(1 − γ)2 f (γ)dγ
R01
1/12 0 (1
− γ)2 f (γ)dγ
= 1/3,
Note 8.3. Between 2002 and 2006 Carlsson, Fullér and Majlender [13, 14, 15,
16, 17, 18, 20, 21] showed some other important aspects of possibilistic expected
value, variance, covariance and correlation.
9 Applications
The possibilistic expected value, variance, covariance and correlation have been
extensively used by the authors for real option valuation [2, 6, 11], project selection [9, 10], capital budgeting [1], taming the bullwhip effect [3, 4, 7, 8]. The concept of possibilistic mean value and variance has been utilized in many different
areas and by many different authors [26-73] (see [12]). For example, these notions
are applied by Zhang, et al [27] when they investigate possibilistic mean-variance
models in portfolio selection problems; Dutta et al [28] when they investigate
a continuous review inventory model in a mixed fuzzy and stochastic environment; Thiagarajah et al [31] when they introduce an option valuation model with
adaptive fuzzy numbers; Dubois [34] when he discusses possibility theory and
statistical reasoning; Beynon and Munday [36] when they elucidate of multipliers
and their moments in fuzzy closed Leontief input-output systems; Zarandi et al
[51] when they present an intelligent agent-based system for reduction of the bullwhip effect in supply chains; Lazo et al [61] when they determine of real options
value by Monte Carlo simulation and fuzzy numbers. Furthermore, when Fang
18
et al [32] presented a portfolio rebalancing model with transaction costs based on
fuzzy decision theory; Yoshida et al [33] showed a new evaluation of mean value
for fuzzy numbers and its application to American put option under uncertainty;
Hashemi et al [37] investigated a fully fuzzied linear programming, solution and
duality; Ayala et al [42] considered different averages of a fuzzy set with an application to vessel segmentation; Jahanshahloo, Soleimani-damaneh and Nasrabadi
[44] investigated measure of efficiency in DEA with fuzzy input-output levels;
Collan [64] applied fuzzy real investment valuation model for very large industrial real investments; Fang, Lai and Wang [68] offered a fuzzy approach to portfolio rebalancing with transaction costs; Peschland and Schweiger [71] performed
a reliability analysis in geotechnics with finite elements comparing probabilistic,
stochastic and fuzzy set methods. The notion of possibilistic correlation is used
in [74, 75, 76, 77] and the f -weighted possibilistic mean value and variance are
used in [79] - [90]. For example, Liu [80] computes the maximum entropy parameterized interval approximation of fuzzy numbers; Ayala, Leon and Zapater [81]
consider different averages of a fuzzy set with an application to vessel segmentation; Cheng [82] constructs control charts with fuzzy numbers in fuzzy process
controls. Zhang and Li [86] use these notions to portfolio selections problems with
quadratic utility function in a fuzzy environment; Garcia [89] uses these notions
to fuzzy real option valuation in a power station reengineering project; Wang, Xu
and Zhang [90] analysis a class of weighted possibilistic mean-variance portfolio
selection problems.
References
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cash ows, Mathware and Soft Computing, 6(1999) 81-89.
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November 2, 2000 69-77.
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Meeting on Cybernetics and Systems Research, Vienna, April 25 - 28, 2000,
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19
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2006
[81] Ayala G, Leon T, Zapater V, Different averages of a fuzzy set with an application to vessel segmentation, IEEE TRANSACTIONS ON FUZZY SYSTEMS, 13(3): 384-393 JUN 2005
[82] Cheng CB, Fuzzy process control: construction of control charts with fuzzy
numbers, FUZZY SETS AND SYSTEMS 154 (2): 287-303 SEP 1 2005
[83] Bodjanova S, Median value and median interval of a fuzzy number, INFORMATION SCIENCES 172 (1-2): 73-89 JUN 9 2005
[84] Hong DH, Kim KT, A note on weighted possibilistic mean, FUZZY SETS
AND SYSTEMS 148 (2): 333-335 DEC 1 2004
[85] Wang X, Xu WJ, Zhang WG, et al., Weighted possibilistic variance of fuzzy
number and its application in portfolio theory, LECTURE NOTES IN ARTIFICIAL INTELLIGENCE 3613: 148-155 2005
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model and possibilistic efficient frontier LECTURE NOTES IN COMPUTER SCIENCE 3521: 203-213 2005
[87] Zhang, J.-P., Li, S.-M. Portfolio selection with quadratic utility function under fuzzy enviornment 2005 International Conference on Machine Learning
and Cybernetics, ICMLC 2005, pp. 2529-2533. 2005
[88] Blankenburg, B., Klusch, M. BSCA-F: Efficient fuzzy valued stable coalition forming among agents Proceedings - 2005 IEEE/WIC/ACM International Conference on Intelligent Agent Technology, IAT’05, 2005, art. no.
1565632, pp. 732-738. 2005
[89] Garcia, F.A.A. Fuzzy real option valuation in a power station reengineering
project Soft Computing with Industrial Applications - Proceedings of the
Sixth Biannual World Automation Congress, pp. 281-287. 2004
[90] Wang, X., Xu, W.-J., Zhang, W.-G. A class of weighted possibilistic meanvariance portfolio selection problems Proceedings of 2004 International
Conference on Machine Learning and Cybernetics, 4, pp. 2036-2040. 2004
27
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• Department of Computer Science
• Institute for Advanced Management Systems Research
Turku School of Economics and Business Administration
• Institute of Information Systems Sciences
ISBN 978-952-12-2042-5
ISSN 1239-1891