Possibility versus probability:
falling shadows versus falling
integrals
Probability – falling integrals
In probability theory, the dependency between two
random variables X, Y with probability density functions
fX(x) and fY(y) respectively, can be characterized through
their joint probability density function fX,Y(x,y), where
∫
R
f X ,Y ( x, t )dt = f X ( x),
∫
R
f X ,Y (t , y )dt = fY ( y )
for all x, y 0 R.
fX(x) and fY(y) are called the marginal probability density
functions of joint random variable (X,Y).
X and Y are said to be independent if fX,Y(x,y) = fX(x) fY(y).
Probability – expected value
Let a, b 0 R c {–∞, ∞} with a # b, then the probability
that X takes its value from [a, b] is computed by
b
P( X ∈ [a, b]) = ∫ f X ( x)dx.
a
The expected value of random variable X is defined by
E( X ) = ∫ xf X ( x)dx.
R
If g is a function of X then the expected value of g(X) is
E( g ( X )) = ∫ g ( x) f X ( x)dx.
R
Probability – measure of dependency
If h is a function of X and Y then the expected value of
h(X,Y) can be computed by
E(h( X , Y )) = ∫ 2 h( x, y ) f X ,Y ( x, y )dxdy.
R
The covariance of two random variables X and Y is
defined by
Cov( X , Y ) = E(( X − E ( X ))(Y − E (Y )))
= ∫ 2 xyf X ,Y ( x, y )dxdy − ∫ xf X ( x)dx ∫ yfY ( y )dy.
R
R
R
Probability – measure of variability
The variance of random variable X is defined as the
covariance of X and itself, that is
Var ( X ) = E(( X − E( X )) ) = ∫ x f X ( x)dx −
2
2
R
(∫ xf
R
)
2
X
( x)dx .
For any random variables X and Y and real numbers λ and
µ the following relationship holds
Var (λX + µY ) = λ2 Var ( X ) + µ 2 Var (Y ) + 2λµCov( X , Y ).
Probability – correlation coefficient
Let X and Y be random variables with non-zero variances.
Then, the correlation coefficient of X and Y is defined by
Cov( X , Y )
ρ ( X ,Y ) =
,
Var ( X )Var (Y )
where –1 # ρ(X, Y) # 1.
If X and Y are independent random variables then
Cov( X , Y ) = ρ ( X , Y ) = 0.
Possibility – concept of fuzzy number
A fuzzy set A in R is 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 are denoted by ö.
A γ-level set of a fuzzy set A in Rm is defined by
[ A]γ = cl{x ∈ R m : A( x) > γ }.
If A 0 ö is a fuzzy number then [A]γ is a convex and
compact subset of R for all γ 0 [0,1].
Possibility – concept of joint distribution
Let a, b 0 R c {–∞, ∞} with a # b, then the possibility
that A 0 ö takes its value from [a, b] is defined by
Pos( A ∈ [a, b]) = max A( x).
x∈[ a ,b ]
A fuzzy set B in Rm is said to be a joint possibility
distribution of fuzzy numbers Ai 0 ö, i = 1, ..., m, if
max B( x1 , K, xm ) = Ai ( xi )
x j ∈R , j ≠ i
for all xi 0 R, i = 1, ..., m.
Ai is called the i-th marginal possibility distribution of B.
Possibility – falling shadows
The joint possibility distribution always uniquely defines
its marginal distributions – the shadow of B on the i-th
axis is exactly Ai –, but not vice versa.
If Ai 0 ö, i = 1, ..., m, and B is their joint possibility
distribution then
B( x1 , K , xm ) ≤ min{ A1 ( x1 ),K , Am ( xm )},
[ B]γ ⊆ [ A1 ]γ × L × [ Am ]γ
hold for all x1, ..., xm 0 R and γ 0 [0,1].
B
1
γ
A
G
B
1
γ
A
H
Possibility – concept of non-interactivity
Definition 1. Fuzzy numbers Ai 0 ö, i = 1, ..., m, are said
to be non-interactive if their joint possibility distribution B
is given by
B( x1 , K , xm ) = min{ A1 ( x1 ),K, Am ( xm )}
or, equivalently,
[ B]γ = [ A1 ]γ × L× [ Am ]γ
for all x1, ..., xm 0 R and γ 0 [0,1].
B
1
γ
A
AHB
Non-interactive possibility distributions
Possibility – central value
Definition 2. Let A 0 ö be a fuzzy number with [A]γ =
[a1(γ), a2(γ)], then the central value of [A]γ is defined by
γ
C ([ A] ) =
1
∫
[ A ]γ
∫
dx
[ A ]γ
a1 (γ ) + a2 (γ )
.
xdx =
2
Definition 6. Let A1, ..., An 0 ö be fuzzy numbers, let B
be their joint possibility distribution, and let g: Rn 6 R be
a continuous function. The central value of g on [B]γ is
1
C[ B ]γ ( g ) =
g ( x)dx.
γ
∫
[ B]
dx
∫ γ
[ B]
Possibility – central value
The central value operator is linear for any fixed joint
possibility distribution B and for any γ 0 [0,1].
Let the projection functions on R2 be denoted by πx and
πy, that is, πx (u,v) = u and πy (u,v) = v for u,v 0 R. Let A,
B 0 ö be non-interactive fuzzy numbers, then
C[ A× B ]γ (π x + π y ) = C[ A×B ]γ (π x ) + C[ A× B ]γ (π y )
= C[ A]γ (id ) + C[ B ]γ (id ) = C ([ A]γ ) + C ([ B ]γ ),
C[ A×B ]γ (π xπ y ) = C[ A]γ (id )C[ B ]γ (id ) = C ([ A]γ )C ([ B ]γ ).
Possibility – mean value
Definition 3. A function f : [0,1] 6 R is said to be a
weighting function if f is non-negative, monotone
increasing, and satisfies
1
∫
0
f (γ )dγ = 1.
Definition 4. The expected (or mean) value of A with
respect to a weighting function f is defined by
a1 (γ ) + a2 (γ )
E f ( A) = ∫ C ([ A] ) f (γ )dγ = ∫
f (γ )dγ .
0
2
0
1
γ
1
Possibility – mean value
Definition 6. Let A1, ..., An 0 ö be fuzzy numbers, let B
be their joint possibility distribution, and let g: Rn 6 R be
a continuous function. The expected (or mean) value of g
on B with respect to a weighting function f is defined by
1
E f ( g ; B) = ∫ C[ B ]γ ( g ) f (γ )dγ
0
1
=∫
0
1
∫
[ B ]γ
∫
dx
γ
[B]
g ( x)dx f (γ )dγ .
Possibility – measure of interactivity
Definition 8. Let A, B 0 ö be fuzzy numbers, and let C
be their joint possibility distribution. The interactivity
relation of [A]γ and [B]γ is defined by
R[C ]γ (π x , π y ) = C[C ]γ ((π x − C[C ]γ (π x ))(π y − C[C ]γ (π y )))
= C[C ]γ (π xπ y ) − C[C ]γ (π x )C[C ]γ (π y ).
The covariance of A and B with respect to a weighting
function f is defined by
1
Cov f ( A, B) = ∫ R[C ]γ (π x , π y ) f (γ )dγ .
0
Possibility – measure of variability
The variance of A 0 ö is defined by
1
Var f ( A) = Cov f ( A, A) = ∫ R[ A]γ (id, id ) f (γ )dγ .
0
Theorem 3. Let A, B 0 ö be fuzzy numbers, let C be their
joint possibility distribution, and let λ, µ 0 R. Then,
R[C ]γ (λπ x + µπ y , λπ x + µπ y )
= λ2 R[C ]γ (π x , π x ) + µ 2 R[C ]γ (π y , π y ) + 2λµR[C ]γ (π x , π y ),
Var f (λA + µB) = λ2 Var f ( A) + µ 2 Var f ( B) + 2λµCov f ( A, B).
Possibility – correlation coefficient
Theorem 4. Let A, B 0 ö be fuzzy numbers with nonzero variances, and let C be their joint possibility
distribution. Then, the correlation coefficient of A and B
ρ f ( A, B) =
Cov f ( A, B)
Var f ( A)Var f ( B)
satisfies the property –1 # ρ f (A, B) # 1 for any weighting
function f.
If A, B 0 ö are non-interactive fuzzy numbers then
Cov f ( A, B ) = ρ f ( A, B ) = 0.
B
1
γ
A
C
The case of ρf(A, B) = 1
B
1
γ
A
D
The case of ρf(A, B) = –1
B
1
A
γ
E
The case of ρf(A, B) = 1/3
B
1
A
γ
F
The case of ρf(A, B) = –1/3
B
1
γ
A
I
The case of interactivity when ρf(A, B) = 0