Some Criteria for Equality of Possibilistic Variables
Robert Fullér
Abstract— Two random variables are said to be equal if they
are equal as functions on their probability space. In this paper
we will show that similar conditons hold for the equality of
possibilistic variables.
I. I NTRODUCTION
and
There are several different senses in which random variables can be considered to be equivalent. Two random
variables can be equal, equal almost surely, equal in mean,
or equal in distribution [8]. Let X and Y be two random
variables with cumulative distribution functions FX and FY ,
respectively. We say that X and Y are equal in distribution
if
P (Y ≤ y) = P (X ≤ y),
(1)
for all y ∈ R. If, furthermore, there exist some constants
a > 0 and b ∈ R such that Y = aX + b, (when their
correlation coefficient, corr(X, Y ), is equal to one) then we
have
FY (y) = P (Y ≤ y) = P (aX + b ≤ y)
y−b
y−b
= FX
=P X≤
a
a
Let X and Y be possibilistic variables in R with possibility
distributions πX and πY , respectively. We shall use the
notations
Pos(X = x) = πX (x),
(2)
for all x, y ∈ R. So, we can easily state the following
theorem,
Pos(Y = y) = πY (y),
where Pos denotes possibility.
Definition II.1. Let X and Y be possibilistic variables in
R with possibility distributions πX and πY , respectively. A
two-place possibility distribution πX,Y is said to be a joint
possibility distribution of possibilistic variables X and Y , if
it satisfies the relationships
max{x | πX,Y (x, y)} = πY (y),
and
max{y | πX,Y (x, y) = πX (x),
for all x, y ∈ R. Furthermore, πX and πY are called the
marginal possibility distributions of πX,Y .
Definition II.2. We say that two possibilistic variables X
and Y are equal in distribution if they have the same
distribution function
πX (z) = πY (z)
Theorem I.1. If two random variables X and Y are equal
in distribution and corr(X, Y ) = 1 then according to (1-2)
we get that a = 1 and b = 0, and therefore, X and Y are
equal with probability one.
for all z ∈ R. Furthermore, we say that X and Y are equal
(and write X = Y ) if they equal in distribution and if one
takes a value z ∈ R then the other can only take the same
value z.
In this paper we will show that similar conditons hold for
the equality of possibilistic variables.
We recall that a function f : [0, 1] → R is said to be a
weighting function if f is non-negative, monoton increasing
and satisfies the following normalization condition
Z 1
f (γ)dγ = 1.
II. P OSSIBILITY DISTRIBUTIONS
A fuzzy number is a fuzzy set of the real line R with a
normal, fuzzy convex and upper-semicontinuous membership
function of bounded support. Fuzzy numbers can be considered as possibility distributions. Let X be a possibilistic
variable in R with possibility distributions πX . The degree
of possibility that X takes a value x is defined by πX (x).
A γ-level set of a possibility distribution πX is denoted by
[πX ]γ .
Robert Fullér is with the Institue for Advanced Management Systems
Research, Åbo Akademi University, Joukahaisenkatu 3-5, FIN-20520 Åbo,
Finland (email: robert.fuller@abo.fi) and with the Department of Operations
Research, Eötvös Loránd University, Pázmány Péter sétány 1C, H-1117
Budapest, Hungary (email: rfuller@cs.elte.hu).
The final version of this paper appeared in: Proceedings of the Seventh
International Symposium on Intelligent Systems and Informatics, September
25-26, 2009, Subotica, Serbia
0
Different weighting functions can give different (casedependent) importances to γ-levels sets of possibility distributions. It is motivated in part by the desire to give less
importance to the lower levels of fuzzy sets [7] (it is why f
should be monotone increasing). Let X be a possibilistic
variable with possibility distribution πX . The f -weighted
possibilistic variance of X, defined in [5], can be written
as
Z 1
2
Varf (X) =
σU
f (γ)dγ
γ
0
where Uγ is a uniform probability distribution on the γlevel sets of πX for all γ ∈ [0, 1]. The f -weighted measure
of possibilistic covariance between possibilistic variables X
Fig. 1.
A and B are non-interactive possibility distributions.
Fig. 2. Completely negatively correlated possibility distributions with q =
−1.
and Y with possibility distributions πX and πY , respectively
(with respect to their joint distribution πX,Y ), defined by [6],
can be written as
Z 1
Covf (X, Y ) =
cov(Xγ , Yγ )f (γ)dγ,
0
where Xγ and Yγ are random variables whose joint distribution is uniform on the γ-level set of πX,Y for all γ ∈ [0, 1],
and cov(Xγ , Yγ ) denotes their covariance.
The f -weighted possibilistic correlation between possibilistic variables X and Y with possibility distributions πX
and πY , respectively (with respect to their joint distribution
πX,Y ), defined in [2], can be written as
Covf (X, Y )
p
.
ρf (X, Y ) = p
Varf (X) Varf (Y )
Fig. 3. Completely negatively correlated possibility distributions with q 6=
−1.
Two possibilistic variables X and Y with possibility distributions πX and πY , respectively, are said to be noninteractive if their joint possibility distribution πX,Y satisfies
the relationship
πX,Y (x, y) = min{πX (x), πY (y)},
stands for the characteristic function of the line
{(x, y) ∈ R2 |qx + r = y}.
In this case we have,
or, equivalently,
[πX,Y ] = [πX ] × [πY ] ,
γ
γ
γ
πY (x) = πX
for all x, y ∈ R2 and γ ∈ [0, 1]. It is clear that if
two possibilistic variables X and Y are non-interactive (or
independent) then their correlation coefficient is equal to zero
(see [2] for details).
Definition II.3 ([1]). We say that two possibilistic variables
X and Y are completely correlated if there exist q, r ∈ R,
q 6= 0 such that their joint possibility distribution is defined
by
πX,Y (x, y) = πX (x) · χ{qx+r=y} (x, y) =
πY (y) · χ{qx+r=y} (x, y),
where
χ{qx+r=y} ,
(3)
x−r
,
q
for all x ∈ R. Furthermore, for q > 0 we have
ρf (X, Y ) = 1
and for q < 0 we have
ρf (X, Y ) = −1.
for any weighting function f . The weighting function f does
not matter in the case of completely correlated possibility
variables. It is why we will simple write ρ(X, Y ) for completely correlated possibilistic variables.
If X and Y are equal in distribution, (πY (x) = πX (x)
for all x ∈ R) and ρ(X, Y ) = 1 then q = 1 and r = 0 and
IV. S ECOND
QUALITY of
C ONDITION
FOR P OSSIBILISTIC
Whatever
is theEdefinition
the joint possibility
distribution
V
ARIABLES
C, we always have the following relationship,
We say that a possibilistic variable X is equal to zero, if it
A +C B ⊆ A + B,
can only take zero (and with
possibility one), that is X = 0
if and
that
is, only
(A +if
B)(y)
≤
(A
+
B)(y)
for all y ∈ R.
C
Pos(X
= 0)a=family
1, of joint possibility
In this Section we have
shown
distributions, for which the equality
and
0,
APos(X
+C B =
=x)
A=
+ B,
holds.
for all Namely,
x 6= 0. we have proved that if two fuzzy numbers
are completely positively correlated then their interactive and
Definition IV.1 ([1]). Let X and Y be possibilistic variables
non-interactive sums have the same membership function.
with joint possibility distribution πX,Y and with marginal
IV.
S UBSTRACTION
OF πCOMPLETELY
CORRELATED
FUZZY
possibility
distributions
let
X and πY , and
Fig. 5. Completely positively correlated fuzzy numbers.
Fig. 4. Completely positively correlated possibility distributions.
Let A and B be fuzzy numbers, where the membership
their joint
possibility
πX,Y , is uniquely defined
function
of B
is defineddistribution,
by
by (see [1])
!
"
x−r
B(x) = A
,
πX,Y (x, y) = πX (x)χ
q {x=y} (x, y)
(4)
Y (y)χ
for any x ∈ R, then for any=qπ>
0 we{x=y}
find (x, y)
[A +
B]γ χ
={x=y}
[A]γ +
[B]γ for the characteristic
for all x, y ∈ R,
where
stands
function of the line x − =
y=
[A]0.γ + q[A]γ + r
= (q + 1)[A]γ + r
III. F IRST E QUALITY C ONDITIONγ FOR P OSSIBILISTIC
= [A +C B] .
VARIABLES
for all γ ∈ [0, 1]. So,
We will show some equivalent conditions for equality of
A +We
=A
+ B.
possibilistic variables.
now
in the postion to prove
C Bare
the next theorem.
that is, the membership function of the interactive sum of two
completely
numbers variables
(defined by
Theorem positively
III.1. Let correlated
X and Y fuzzy
be possibilistic
in
(1)R.and
(4))Xis=equal
theonly
membership
their nonThen
Y if, to
and
if, they arefunction
equal inofdistribution
interactive
by their sup-min convolution).
and ρ(X,sum
Y ) =(defined
1.
However, if they are completely negatively correlated, that
If Xfrom
andthe
Y are
equal then ρ(X, Y ) = 1 and their
isProof
q < 0,1.then
inequality
joint possibility distribution, πX,Y , is uniquely defined by (4).
γ
γ
γ
[A](x,
+y)
q[A]
(q6=+y1)[A]
In this case πX,Y
= 0 #=
if x
(that ,is, X and Y can
takethat
different
values), andand
if z non-interactive
= x = y then sums are
wenotget
their interactive
usually not equal. For example, for q = −2 we get,
πX,Y (z, z) = πX (z) = πY (z)
[A]γ +q[A]γ = [A]γ −2[A]γ = [a1 (γ)−2a2 (γ), a2 (γ)−2a1 (γ)]
for all z ∈ R. We have just shown that in this case they
and
are equal in distribution.
But if X and Y are are equal in
(q + 1)[A]γ = −[A]γ = [−a (γ), −a (γ)].
distribution then ρ(X, Y ) = 1 if, 2and only1 if, their joint
is given by (4). Which completes the
It possibility
is easy to distribution
see that,
proof.
[a1 (γ) − 2a2 (γ), a2 (γ) − 2a1 (γ)] #= [−a2 (γ), −a1 (γ)].
Note III.1. Now let X and Y be possibilistic variables in
Remark 3.3: It is clear that, in the general case, the interR. In possibility theory, equality in distribution can not be
active sum,
defined in the same way as in probability theory, since the
equality (A +C B)(y) = sup C(x1 , x2 ),
y=x1 +x2
Pos(X ≤ x) = Pos(Y ≤ x), ∀x,
can be very different from the non-interactive sum,
can be (A
valid
even if=the sup
right-hand
sides1 ),ofB(x
πX2 and
+ B)(y)
min{A(x
)}. πY are
y=x1 +x2
very different.
NUMBERS
f (x, y) = x − y,
Let us consider now the subtraction operator on completely
the substraction
operator in
. Then
discorrelated
fuzzy numbers.
LetR2A,
B ∈ the
F possibility
be completely
tribution fuzzy
of X numbers,
− Y , denoted
by joint
πX−Y
, is defined
by the
correlated
let their
possibility
distribution
generalized
as [3]
be
defined byextension
(4), and principle
let
πX−Y (z)g(x
= Pos(X
z) =
max
πX,Y
(x=
x1 −
x2 , (x, y).
1 , x2 ) =−fY
1 , −x
2) =
x−y=z
(5)
be the
operator
in R2 . Then,
Let subtraction
us consider now
the subtraction
operator on completely
correlated
possibilistic
variables
X
and
Y with
possigC (A, B)(y) = (A −C B)(y) = sup
C(xjoint
1 , x2 ).
y=x1 −x2(4). Then,
bility distribution πX,Y defined by equation
That is,
Pos(X − Y = z) = max π
(x, y).
x−y=z
X,Y
(A −C B)(y) = sup A(x1 ) · χ{qx1 +r=x2 } (x1 , x2 ).
That is,
y=x1 −x2
Then for
a γ-level
setz)of=A max
−C BπX
we· χ
get,
Pos(X
−Y =
{qx+r=y} (x, y).
x−y=z
[A −C B]γ = cl{x1 − x2 ∈ R|A(x1 ) > γ, qx1 + r = x2 }
Then for a γ-level set of πγX−Y we get,
= (1 − q)[A] − r
[πX−Y ]γ = cl{x − y ∈ R|πX > γ, qx + r = y}
for all γ ∈ [0, 1].
−B
q)[π
]γ − r
In particular, if=A(1
and
areX completely
positively correlated
with
q
=
1,
i.e.
for all γ ∈ [0, 1].
γ
[B]γY=are
[A]completely
+ r,
In particular, if X and
positively correlated
with
q
=
1,
i.e.
∀γ ∈ [0, 1] then
γ C B]γ =γ −r,
[AY ]−
[π
= [πX ] + r,
that
is,[0,
the1],fuzziness
of A −C B vanishes.
∀γ ∈
then
Remark 4.1: We have just proved that if two completely
[πX−Y
]γ =numbers
−[r]γ = have
−{r, the
r} same memberpositively correlated
fuzzy
ship function, that is,
that is,
Pos(X
−=
Y B(x),
= z) = 1
A(x)
if zall=xr ∈and
for
R, then their (interactive) difference will be (crisp)
Pos(X − Y = z) = 0
zero.
On the other hand, for q < 0 we get
if z 6= r.
γ
γ
We are now [A
in the
position
prove
− B]
= [A]to
−
[B]γthe second condition
γ
of equality of possibilistic=variables.
[A] − q[A]γ − r
Theorem IV.1. X = Y if=and
if γX−−r Y = 0.
(1 −only
q)[A]
Proof 2. Really, from
So,
= [A −C B]γ .
πX,Y (x, y) = πX (x)χ{x=y} (x, y)
A −C=Bπ =(y)χ
A − B, (x, y)
Y
{x=y}
and (5) we get
Pos(X − Y = z) = max πX,Y (x, y)
x−y=z
1 if z = 0
=
0 otherwise.
(6)
So if X = Y then X −Y = 0. On the other hand, if X −Y =
0 then πX,Y is uniquely defined, since its support is a subset
of the line x − y = 0, which can only happen if πX,Y is
defined by (4).
We note that equation (6) can be written in the form
Pos(X − Y 6= 0) = 0 [4]. That is, two possibilistic variables
X and Y are equal if, and only if, the possibility that they are
different is zero. On the other hand, two random variables
are equal almost surely if, and only if, the probability that
they are different is zero.
R EFERENCES
[1] C. Carlsson, R. Fullér and P. Majlender, Additions of Completely
Correlated Fuzzy Numbers, in: FUZZY IEEE 2004 CD-ROM Conference Proceedings Budapest, July 26-29, 2004, IEEE Catalog Number:
04CH37542C.
[2] C. Carlsson, R. Fullér and P. Majlender, On possibilistic correlation,
Fuzzy Sets and Systems, 155(2005) 425-445.
[3] C. Carlsson, R. Fullér, On additions of interactive fuzzy numbers, Acta
Polytechnica Hungarica, 2(2005) 59-73.
[4] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized
Processing of Uncertainty, Plenum Press, New York, 1988.
[5] R. Fullér and P. Majlender, On weighted possibilistic mean and variance
of fuzzy numbers, Fuzzy Sets and Systems, 136(2003) 363-374.
[6] R. Fullér and P. Majlender, On interactive fuzzy numbers, Fuzzy Sets
and Systems, 143(2004) 355-369.
[7] R. Goetschel and W. Voxman, Elementary Fuzzy Calculus, Fuzzy Sets
and Systems, 18(1986) 31-43.
[8] O. Kallenberg, Random Measures, Academic Press, New York, London,
1986.