A short survey of normative properties of
possibility distributions∗
Robert Fullér
rfuller@abo.fi
Péter Majlender
peter.majlender@abo.fi
Abstract
In 2001 Carlsson and Fullér [1] introduced the possibilistic mean value,
variance and covariance of fuzzy numbers. In 2003 Fullér and Majlender [4]
introduced the notations of crisp weighted possibilistic mean value, variance
and covariance of fuzzy numbers, which are consistent with the extension
principle. In 2003 Carlsson, Fullér and Majlender [2] proved the possibilistic Cauchy-Schwartz inequality. Drawing heavily on [1, 2, 3, 4, 5] we will
summarize some normative properties of possibility distributions.
1
Probability and Possibility
In probability theory, the dependency between two random variables can be characterized through their joint probability density function. Namely, if X and Y are
two random variables with probability density functions fX (x) and fY (y), respectively, then the density function, fX,Y (x, y), of their joint random variable (X,Y ),
should satisfy the following properties
Z
R
Z
fX,Y (x,t)dt = fX (x),
R
fX,Y (t, y)dt = fY (y)
(1)
for all x, y ∈ R. Furthermore, fX (x) and fY (y) are called the the marginal probability density functions of random variable (X,Y ). X and Y are said to be independent
if
fX,Y (x, y) = fX (x) fY (y)
holds for all x, y. The expected value of random variable X is defined as
E(X) =
Z
R
∗ Appeared
x fX (x)dx,
in: B. De Baets and J. Fodor eds., Principles of Fuzzy preference Modelling and
Decision Making, Academia Press, Gent, [ISBN 90-382-0567-8], 2003, pp. 183-193.
1
and if g is a function of X then the expected value of g(X) can be computed as
E(g(X)) =
Z
R
g(x) fX (x)dx.
Furthermore, if h is a function of X and Y then the expected value of h(X,Y ) can
be computed as
Z
E(h(X,Y )) =
R2
h(x, y) fX,Y (x, y)dxdy.
Especially,
E(X +Y ) = E(X) + E(Y ),
that is, the expected value of X and Y can be determined according to their individual density functions (that are the marginal probability functions of random
variable (X,Y )).
Remark 1 The key issue here is that the joint probability distribution vanishes
(even if X and Y are not independent), because of the principle of ’falling integrals’
(1).
Let a, b ∈ R ∪ {−∞, ∞} with a ≤ b, then the probability that X takes its value from
[a, b] is computed by
P(X ∈ [a, b]) =
Z b
fX (x)dx.
a
The covariance between two random variables X and Y is defined as
Cov(X,Y ) = E (X − E(X))(Y − E(Y )) = E(XY ) − E(X)E(Y ),
and if X and Y are independent then Cov(X,Y ) = 0, since E(XY ) = E(X)E(Y ).
The variance of random variable X is defined as the covariance between X and
itself, that is
Z
2
Z
2
2
2
Var(X) = E(X ) − (E(X)) = x fX (x)dx −
x fX (x)dx .
R
R
For any random variables X,Y and real numbers λ , µ ∈ R the following relationship holds
Var(λ X + µY ) = λ 2 Var(X) + µ 2 Var(Y ) + 2λ µCov(X,Y ).
The correlation coefficient between X and Y is defined by
Cov(X,Y )
ρ(X,Y ) = p
,
Var(X)Var(Y )
2
and it is clear that −1 ≤ ρ(X,Y ) ≤ 1.
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
Rn is defined by [A]γ = {x ∈ Rn |A(x) ≥ γ} if γ > 0 and [A]γ = cl{x ∈ Rn |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 [6, 7]. Let a, b ∈
R ∪ {−∞, ∞} with a ≤ b, then the possibility that A ∈ F takes its value from [a, b]
is defined by [7]
Pos(A ∈ [a, b]) = max A(x).
x∈[a,b]
Rn
A fuzzy set B in
is said to be a joint possibility distribution of fuzzy numbers
Ai ∈ F , i = 1, . . . , n, if it satisfies the relationship
max B(x1 , . . . , xn ) = Ai (xi )
x j ∈R, j6=i
(2)
for all xi ∈ R, i = 1, . . . , n. 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, . . . , n.
Let B denote a joint possibility distribution of A1 , A2 ∈ F . Then B should
satisfy the relationships
max B(x1 , y) = A1 (x1 ),
max B(y, x2 ) = A2 (x2 )
y∈R
y∈R
for all x1 , x2 ∈ R.
If Ai ∈ F , i = 1, . . . , n and B is their joint possibility distribution then the relationships
B(x1 , . . . , xn ) ≤ min{A1 (x1 ), . . . , An (xn )},
and
[B]γ ⊆ [A1 ]γ × · · · × [An ]γ
hold for all x1 , . . . , xn ∈ R and γ ∈ [0, 1].
In the following the biggest (in the sense of subsethood of fuzzy sets) joint
possibility distribution will play a special role among joint possibility distributions:
it defines the concept of non-interactivity of fuzzy numbers (see Fig. 1).
Definition 1 Fuzzy numbers Ai ∈ F , i = 1, . . . , n, are said to be non-interactive if
their joint possibility distribution, B, is given by
B(x1 , . . . , xn ) = min{A1 (x1 ), . . . , An (xn )},
3
Figure 1: Non-interactive possibility distributions.
or equivalently,
[B]γ = [A1 ]γ × · · · × [An ]γ
for all x1 , . . . , xn ∈ R and γ ∈ [0, 1].
Marginal probability distributions are determined from the joint one by the principle of ’falling integrals’ and marginal possibility distributions are determined from
the joint possibility distribution by the principle of ’falling shadows’ (2).
A function f : [0, 1] → R is said to be a weighting function [4] if f is nonnegative, monotone increasing and satisfies the following normalization condition
Z 1
f (γ)dγ = 1.
(3)
0
2
Possibilistic expected value, variance, covariance
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
4
function g on [B]γ can be computed by
C[B]γ (g) = R
1
[B]γ
dx
Z
[B]γ
g(x)dx
Z
1
g(x1 , . . . , xn )dx1 . . . dxn .
=R
γ
[B]γ dx1 . . . dxn [B]
We will call C as the central value operator.
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
dx
Z
[A]γ
g(x)dx.
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
[A]γ
dx
Z
[A]γ
xdx =
Z a2 (γ)
1
a1 (γ) + a2 (γ)
xdx =
,
a2 (γ) − a1 (γ) a1 (γ)
2
which remains valid in the limit case a2 (γ)−a1 (γ) = 0 for some γ ∈ [0, 1]. 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].
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
C[A]γ (g) = R
1
[A]γ
dx
Z
[A]γ
g(x)dx.
Definition 2 [5] The expected value of function g on A with respect to a weighting
function f is defined by
E f (g; A) =
Z 1
0
C[A]γ (g) f (γ)dγ =
Z 1
0
R
1
[A]γ
dx
Z
[A]γ
g(x)dx f (γ)dγ.
Especially, if g is the identity function then we get
E f (id; A) = E f (A) =
Z 1
a1 (γ) + a2 (γ)
0
5
2
f (γ)dγ,
which is the f -weighted possibilistic expected value of A introduced in [4].
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 show two important properties of the central value
operator [5].
Theorem 1 If A, B ∈ F are non-interactive fuzzy numbers 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].
Theorem 2 If A, B ∈ F are non-interactive fuzzy numbers 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].
Definition 3 [5] 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
R[C]γ (πx , πy ) = C[C]γ (πx πy ) − C[C]γ (πx ) · C[C]γ (πy )
for all γ ∈ [0, 1].
Definition 4 [5] Let C be a joint possibility distribution in R2 , and let A, B ∈ F
be 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
Cov f (A, B) =
=
Z 1
0
R[C]γ (πx , πy ) f (γ)dγ
Z 1
0
C[C]γ (πx πy ) − C[C]γ (πx ) · C[C]γ (πy ) f (γ)dγ.
6
Figure 2: The case of ρ f (A, B) = 0 for interactive fuzzy numbers.
In [5] we proved that if A, B ∈ F are non-interactive then Cov f (A, B) = 0.
However, 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 (hence, line defined by {(a,t)|t ∈ R} is the axis of symmetry of
[C]γ ). It can be shown [3] that in this case the interactivity relation of [A]γ and [B]γ
vanishes, i.e. R[C]γ (πx , πy ) = 0 (see Fig 2).
In many papers authors consider 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,
(4)
max T (A(x), B(y)) = B(y), ∀y ∈ R,
(5)
y∈R
and
x∈R
for example a triangular norm.
7
Remark 2 In this case the joint distribution depends barely on the membership
values of its marginal distributions.
Whatever is the definition of the join
C, we always have the following rela
A +C B ⊆ A +
that is, (A +C B)(y) ≤ (A + B)(y) f
In this Section we have shown a fa
distributions, for which the equality
A +C B = A +
holds. Namely, we have proved that
are completely positively correlated t
non-interactive sums have the same m
IV. S UBSTRACTION OF COMPLETEL
Fig. 5.
Completely positively correlated fuzzy numbers.
Figure 3: The case of ρ f (A, B) = 1.
NUMBERS
Let us consider now the subtraction
correlated fuzzy numbers. Let A, B
Let A and B be fuzzy numbers, where the membership correlated fuzzy numbers, let their join
function
B is defined
by case the covariance between its marginal
In [3] we haveofshown
that in this
be defined by (4), and let
! one of"its marginal distributions is symdistributions will be zero whenever at least
x−r
g(x1 , x2 ) = f (x1 , −x2 )
B(x) = A
,
metrical.
q
be the subtraction operator in R2 . Th
anyLet
x A,
∈ R,
for let
anytheir
q >joint
0 wepossibility
find
Theorem for
3 [3]
B ∈then
F and
distribution C be defined
gC (A, B)(y) = (A −C B)(y) =
by
[A + B]γ = [A]γ + [B]γ
C(x, y) = T (A(x),
B(y)),
= [A]γ + q[A]γ + r
That is,
γ
for x, y ∈ R, where T is a function=satisfying
(4) and (5). If A is a
(q + 1)[A]conditions
+r
(A −C B)(y) = sup A(x1 ) · χ
symmetrical fuzzy number then
y=x1 −x2
= [A +C B]γ .
for all γ ∈ [0, 1]. So, Cov f (A, B) = 0
Then for a γ-level set of A −C B we
we get that their interactive and non-interactive sums are
8 for q = −2 we get,
usually not equal. For example,
that is, the fuzziness of A −C B vani
Remark 4.1: We have just proved
positively correlated fuzzy numbers
ship function, that is,
[A −C B]γ = cl{x1 − x2 ∈ R|A(x1
A + B = A + B.
for any fuzzy number B, aggregator CT , and weighting function f .
= (1 − q)[A]γ − r
that is, the membership function of the interactive sum of two γ 2
Let us denote
R[A]γ (id,positively
id) the average
valuefuzzy
of function
g(x)
= (x − by
C ([A]for)) allonγ ∈ [0, 1].
completely
correlated
numbers
(defined
the γ-level(1)
setand
of an
individual
fuzzy
A. That
is, of their non(4))
is equal to
the number
membership
function
In particular, if A and B are comple
interactive sum (defined by
their
sup-min
convolution).
with q = 1, i.e.
2
Z
Z
1 completely
1 correlated,
2
However,
if
they
are
negatively
that
[B]γ = [A]γ +
R
R
R[A]γ (id, id) =
x dx −
xdx .
γ
γ
γ dx
is q < 0, then from [A]
the
inequality
[A]
[A]γ dx [A]
∀γ ∈ [0, 1] then
γ
γ
[A] + q[A] #= (q + 1)[A]γ ,
[A −C B]γ = −
[A]γ +q[A]γ = [A]γ −2[A]γ = [a1 (γ)−2a2 (γ), a2 (γ)−2a1 (γ)]
and
(q + 1)[A]γ = −[A]γ = [−a2 (γ), −a1 (γ)].
It is easy to see that,
A(x) = B(x)
for all x ∈ R, then their (interactive) d
zero.
Definition 5 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
Var f (A) = E f (g; A) =
0
R[A]γ (id, id) f (γ)dγ.
Figure 4: The case of ρ f (A, B) = −1.
From the equality
R[A]γ (id, id) =
we get
Var f (A) =
(a2 (γ) − a1 (γ))2
12
Z 1
(a2 (γ) − a1 (γ))2
f (γ)dγ.
12
In [5] we proved that the ’principle of central values’ leads us to the same relationships in possibilistic environment as in probabilitic one. It is why we can claim
that the principle of ’central values’ should play an important role in defining possibilistic interactivities.
0
Theorem 4 [5] Let C be a joint possibility distribution in R2 , and let λ , µ ∈ R.
Then
R[C]γ (λ πx + µπy , λ πx + µπy )
= λ 2 R[C]γ (πx , πx ) + µ 2 R[C]γ (πy , πy ) + 2λ µR[C]γ (πx , πy ).
9
Figure 5: The case of ρ f (A, B) = 1/3.
Furthermore, in [2] we have proven the following theorem.
Theorem 5 Let A, B ∈ F be fuzzy numbers (with Var f (A) 6= 0, Var f (B) 6= 0) with
joint possibility distribution C. Then, the correlation coefficient between A and B,
defined by
Cov f (A, B)
ρ f (A, B) = p
.
Var f (A)Var f (B)
satisfies the property
−1 ≤ ρ f (A, B) ≤ 1.
for any weighting function f .
Let us consider three interesting cases. In [4] we proved that if A and B are independent, that is, their joint possibility distribution is A × B then ρ f (A, B) = 0.
Consider now the case depicted in Fig. 2. It can be shown [2] that in this case
ρ f (A, B) = 1. Consider now the case depicted in Fig. 2. It can be shown [2] that in
this case ρ f (A, B) = −1. Consider now the case depicted in Fig. 2. It can be shown
that in this case ρ f (A, B) = 1/3.
3
Summary
We have illustrated some important feautures of possibilistic mean value, covariance, variance and correlation by several examples. We have shown that zero cor10
relation does not always imply non-interactivity. We have also shown the limitations of direct definitions of joint possibility distributions from individual fuzzy
numbers, for example, when one simply aggregates the membership values of two
fuzzy numbers by a triangular norm.
References
[1] C. Carlsson, R. Fullér, On possibilistic mean value and variance of fuzzy
numbers, Fuzzy Sets and Systems, 122(2001) 315-326.
[2] C. Carlsson, R. Fullér and P. Majlender, On possibilistic correlation, Fuzzy
Sets and Systems, 155(2005) 425-445.
[3] C. Carlsson, R. Fullér and P. Majlender, A normative view on possibility distributions, in: Masoud Nikravesh, Lotfi A. Zadeh and Victor Korotkikh eds.,
Fuzzy Partial Differential Equations and Relational Equations: Reservoir
Characterization and Modeling, Studies in Fuzziness and Soft Computing ,
Vol. 142, Springer Verlag, [ISBN 3-540-20322-2], 2004 186-205.
[4] R. Fullér and P. Majlender, On weighted possibilistic mean and variance of
fuzzy numbers, Fuzzy Sets and Systems, 136(2003) 363-374.
[5] R. Fullér and P. Majlender, On interactive fuzzy numbers, Fuzzy Sets and
Systems (to appear).
[6] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.
[7] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and
Systems, 1(1978) 3-28.
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