In Chapter 3, we focused our attention on uncertainty affecting one vari- able. We now want to extend the definitions of random set to multi- dimensional spaces, where several uncertain variables can be described.
More precisely, let S be the Cartesian product of sets Si, i=1,…,ν where the i-th uncertain variable takes values on Si. A random relation is a random set on S= ×S1 ...Sν, i.e. it is a family of n focal elements, Ai⊆ S, and the basic probabilistic assignment, m(Ai), that satisfies the conditions:
m(∅)=0; Σi m(Ai)=1. In the following, reference will be made to the two- dimensional case, its extension to the multidimensional case being straightforward. Figure 4.1a exemplifies a case in which the random rela- tion is composed of three focal elements.
For any subset T ⊆ S, it is possible to evaluate the values of the set- valued functions Bel(T) and Pla(T) and the probability bounds on T by us- ing Eq. (3.3); therefore:
P(T) ∈ [ Bel(T), Pla(T)] (4.1) For example, inFigure 4.1a: P(T) ∈ [ m(A2), m(A1) + m(A2)].
Figure 4.1b illustrates that a focal set, Ai, projects onto the s1-axis as the set (interval if Ai is simply connected, multiple intervals if Ai is not simply connected)
( )
{ }
1i 1 1| 1, 2 i for some 2 2
A = s ∈S s s ∈A s ∈S (4.2)
and onto the s2-axis as the set
( )
{ }
2i 2 2| 1, 2 i for some 1 1
A = s ∈S s s ∈A s ∈S (4.3)
As in the case of random variables (Eqs. (2.23) and (2.24)), the marginal probability assignment, mj, on the sj-axis is defined by using the marginal (or additivity) rule (if projections of different focal sets coincide)
( ) ( )
': '
'
i
j j
i
j j
A A A
m A m A
=
= ∑ (4.4)
Random sets { (A m1i, 1i) }and { (A m2i, 2i) }are called marginal random sets.
a) b)
Fig. 4.1 a) Random relation with three focal elements; b) marginals of focal ele- ment Ai and contours of one possible probability distribution Pi∈ Ψi
From Eq. (3.11), recall that a random set can be defined by using con- vex linear combinations of all probability measures defined over the focal elements (and equal to zero elsewhere). The coefficients of the linear com- binations are fixed, and they are equal to the probability assignments (mi in Eq. (3.11)). Formally, let Pi be a probability measure in the set of prob- ability measures, Ψi, which are zero outside the focal set Ai. Figure 4.1b illustrates the contour of a distribution induced by Pi∈Ψi. A random
relation is the set, Ψ, of probability measures, PRS, obtained as convex combinations of Pi
( )
1
:
n
i i
RS RS
i
P P m A P
=
⎧ ⎫
Ψ =⎨ = ⎬
⎩ ∑ ⎭ (4.5)
Let Fk = (Ak, mk) be a marginal random set, where the focal elements are Ak
{A1k,...,Aknk}
= , and the probability assignment is mk ={m1k,...,mknk} Accord-
ing to Eq. (4.5), the set of probability measures, Ψk, associated to the k-th marginal is
( ( ) )
1
|
nk
i i i i
k k k k k k
i
m A P P
=
⎧ ⎫
⎪ ⎪
Ψ =⎨ ⋅ ∈Ψ ⎬
⎪ ⎪
⎩∑ ⎭, (4.6)
where Ψikis the set of all probability measures defined over the focal ele- ment Aki (and equal to zero elsewhere). Let us now investigate the rela- tionship between the elements in Ψ1 and those in Ψ by taking the marginal of PRS∈Ψ onto S1:
( ) ( ) 1 ( )
1 1
2 2 2
1 1 : i j
n n
i i i i
RS
i j i A A
P S m P S m P S
= = =
⋅× =∑ ⋅× =∑ ∑ ⋅× (4.7)
By equating the last expression in Eq. (4.7) to Eq. (4.6), one has:
( ) ( )
1 1
1 1 2
: i j
j j i i
i A A
m P m P S
=
⋅ = ∑ ⋅× , (4.8)
and by remembering Eq. (4.4), one obtains the final expression for an element of Ψ1i:
( )
( )
1 1
1 1
2 :
1
:
j i
j i
i i i A A j
i i A A
m P S
P m
=
=
⋅×
⋅ =
∑
∑ (4.9)
Eq. (4.9) can be interpreted as follows. Attach a mass equal to probabilistic assignment mi to each projection of a joint probability measure Pi∈Ψi over a joint focal element Ai whose projection is A1j. A marginal probabil- ity measure in Ψ1j is the centroid of this system of masses, which will
“resemble” the projection of joint probability measure(s) with the larger
probabilistic assignment(s). On the other hand, since Ψicontains all joint measures that are zero outside Ai, each probability measure in Ψ1j can be generated by means of Eq. (4.9).
Similar to the one-dimensional case dealt with in Section 3.2.3.2 (page 35), a selector of a random relation {(Ai, mi)} is a random vector, V = {(vi, mi)} (Section 2.4 page 23), whose values vi are included in the focal elements Ai. Call SCT the class of selectors; marginal selectors are marginals of V∈ SCT.
If the focal elements of a random relation {(Ai, mi)} can be ordered in a nested sequence, such that Ai⊆ Ai+1, i = 1, 2, ...n-1, the random relation is termed consonant, and properties similar to the one-dimensional case hold (see Section 3.2.4). In particular, information given by the random relation is equivalent to the point-valued contour function, i.e. the possibility val- ues, π(s1, s2), of the singletons {(s1, s2)}, which is the membership function μF(s1, s2) = π(s1, s2) of a fuzzy relation F with (see Eq. (3.24)):
( ) ( { } ) ( )
(1 2)
1 2 1 2
: ,
, ,
i i
i F
A s s A
s s Pla s s m A
μ
∈
= = ∑ (4.10)
Fig. 4.2 Consonant random relation with three focal elements and its marginal consonant random sets
The focal elements are the α-cuts of the fuzzy relation for the finite se- quence α1= 1, α2<α1 , ..., αn+1 < αn , αn+1= 0, with probabilistic as- signment m(αiA) = αi - αi+1 (Eq. (3.27)). Figure 4.2 illustrates a case in which n =3.
For a consonant random relation, the marginals are consonant random sets that are fuzzy sets, F1 and F2, whose membership functions are simply defined by the following equations (S1 and S2 are finite sets; for infinite sets, the “sup” operator should be substituted for “max”):
( ) ( )
2
1 1 max 1, 2
S
s s s
μ = μ ; ( ) ( )
1
2 2 max 1, 2
S
s s s
μ = μ (4.11)
i.e. they are the projections (Klir and Yuan 1995) of the fuzzy relation onto the space of the single variables, see Figure 4.2.
When all focal elements Ai are nested Cartesian products, the random re- lation F = {(Ai, mi)} is termed consonant random Cartesian product or fuzzy Cartesian product (Figure 4.3). Section 4.3.5 (page 174) deals with the case in which the marginals are given and the fuzzy Cartesian product is derived.
Fig. 4.3 Consonant random Cartesian product with three focal elements and its marginal consonant random sets
Once a random relation is assigned, its marginal random sets are always uniquely determined by Eqs. (4.2) through (4.4). However, if only mar- ginal random sets are given on S1 and S2, the available information does not uniquely define the information on the joint space S = S1×S2 for two distinct reasons:
a) Unless a rule is known or assumed a priori, the marginal focal elements do not uniquely determine the focal elements for the random relation.
The Cartesian product is just an example of such a rule: Aij =A1i×A2j. b) Per the additivity rule (Eq. (4.4)), a marginal focal element, say A1i,
could be the projection of more than one focal element, say A1i and A1j, among which it is thus necessary to apportion the marginal basic prob- abilistic assignment m1(A1i) .
In the theory of precise probability, it is the second reason that brings about the indeterminateness of the joint probability distribution when only the marginal distributions are given. On the contrary, the first reason does not apply because the marginal focal elements are singletons (say
{ }
1 1
i i
A = s ; A2j ={ }s2j ), and thus the focal element is always uniquely de- termined (Ai j, ={s1i,s2j}).
In order to understand the implications of combining two marginal ran- dom sets on a joint space, it is necessary to make an excursion into the wider context of imprecise probabilities, similar to the approach taken in Section 3.3. Section 4.2 explains that in the theory of imprecise probabili- ties the concept of independence is not unique. In Section 4.3, the first is- sue is solved by using the Cartesian product of marginal focal elements, and the concepts of independence in the theory of imprecise probabilities are used to overcome the second source of indeterminateness. In Section 4.4, the hypothesis of Cartesian product will be relaxed in the investigation of correlation.