Nonspecificity for infinite random sets of indexable type Diego A. Alvarez ∗ Arbeitsbereich f¨ ur Technische Mathematik am Institut f¨ ur Grundlagen der Bauingenieurwissenschaften, Leopold-Franzens Universit¨ at, Technikerstraße 13 A-6020 Innsbruck, Austria, EU

Abstract In this document the Hartley-like-based measure of nonspecificity for finite random sets is extended to infinite random sets of indexable type. In the course of the paper, concepts defined in the realm of Dempster-Shafer evidence theory, like joint and marginal random sets and random set inclusion are also generalized. It is shown that the proposed measure is the unique one that fulfills a set of characterizing properties, and in addition corresponds to the one defined in the field of evidence theory when particularized to finite random sets. Key words: infinite random sets, nonspecificity, Dempster-Shafer evidence theory, measures of uncertainty, Hartley-like measure

1

Introduction

Recently, Alvarez [3; 4] proposed an extension of the theory of finite random sets to its infinite counterpart, in the area of uncertainty quantification. The main advantage of this generalization is that the basic variables in consideration do not require anymore to be discretized, and it serves as a unified framework for the theories of possibility, probability, probability bounds analysis, interval analysis and Dempster-Shafer evidence theory. Also this novel approach has given new insights into the theory of random sets, that may allow further theoretical development. This framework requires, in consequence, the generalization of the measures of epistemic and aleatory uncertainty, measured by the family of functions of Hartley and Shannon entropy respectively (see ∗ Corresponding author. Tel.: +43+512-5076825. Fax: +43+512-5072941. Email address: [email protected] (Diego A. Alvarez).

Preprint submitted to Fuzzy Sets and Systems

22 August 2007

e.g. [22]). For a definition of epistemic and aleatory uncertainty, the reader is referred e.g. to [19]. The Hartley family of functions measure what is called nonspecificity, which is a measure about the amount of information required to remove the epistemic uncertainty. In the discrete setting, the Hartley measure ([18, 31]) is employed, while for subsets of d , the Hartley-like measure is used. Using the Hartleylike measure ([21, 30]) it is possible to define measures of nonspecificity for finite random sets defined on d . The purpose of this paper is twofold: to extend the measures of nonspecificity to infinite random sets and to give some new generalizations of concepts that were defined in the realm of finite random sets. The plan of this document is the following. First a brief introduction to nonspecificity and Hartley-like measures (section 2), random sets (section 3), copulas (section 4) and random sets of indexable type (section 5) is done; then in section 6, concepts like joint and marginal infinite random sets are introduced. In section 7, random set inclusion is generalized to infinite random sets and then in section 8, the measure of nonspecificity for infinite random sets is proposed. In section 9, the properties of the new measure are analyzed; finally, in section 10, it is shown that the proposed specificity measure is the unique one that satisfies a defining set of properties. The article ends with conclusions and some ideas that require further development.

2

Nonspecificity

Let us consider a universal set X and its power set P(X). Suppose that we have to choose an element x∗ from a set of elements X. All that we are given is a class of possible choices F ⊆ P(X), where at least one γ ∈ F contains x∗ ; here some choices γ of F are more probable than others. Each γ ∈ F is composed by some elements, however only x∗ is the correct one and there is no way to make some distinction between the elements in γ for almost all γ ∈ F , so that one can choose x∗ . The nonspecificity is a measure of epistemic uncertainty used in cases like the this, when we have to choose a unique element x∗ from X, but we are totally indifferent about which element of the provided γ-s to choose. The nonspecificity is thus associated to the size of the elements γ ∈ F and may thus be measured by the average amount of information needed to completely remove the epistemic uncertainty from some set γ. The name nonspecificiy follows from the fact that full specificity will be only reached when at the most one element in γ remains for almost all possible choices γ in F . In contraposition, when all available information is expressed by assigning to 2

each suboption in the universal set X a relative degree of confidence (the so called probability) on the assertion that the suboption is the true one, provided that the sum of all probabilities is one, then we are dealing with aleatory uncertainty. Here several alternatives are correct, however, only one of them is the true one. It is then clear that those correct claims conflict with each other. This type of uncertainty is measured for example by Shannon’s entropy, which measures the average amount of uncertainty associated with the selection of a set γ from the set of weighted alternatives F , in the case that all γ-s are singletons and F is finite. In the following we will briefly discuss nonspecificity for finite sets and for convex subsets of d . For details the reader is referred to [22]. 2.1 Nonspecificity for finite sets Consider the following problem: given a finite set E of balls, which contains a black ball, while the rest are white, we would like to measure the amount of information H required to find the black ball. Suppose that the set of balls E has m × n elements. If this set is partitioned into n sets of m balls or into m sets of n balls, the measure of nonspecificity H characterizing all those sets should satisfy H(m × n) = H(m) + H(n) (1) Also, note that the larger the set of balls E, the less specific the predictions are, and in consequence H(n) ≤ H(n + 1) (2) where n is the cardinality of E, i.e., n := |E|. Hartley [18] proposed the formula H(n) := log2 n and R´enyi [31] showed that this is the unique expression that satisfies equations (1) and (2) up to the normalization H(2) = 1. This function is known in the literature as the Hartley measure of uncertainty, and it measures the lack of specificity of a finite set. 2.2 Nonspecificity for convex and bounded subsets of

d

In the sequel we will use the term isometric transformation in d to denote a translation followed by an orthogonal transformation (rotation, reflection). 3

The Hartley-like measure was proposed by Klir and Yuan [21] to measure nonspecificity for convex and bounded subsets of d . It is characterized by the function HL : C → [0, ∞), (

HL(A) := min log2 t∈T

"

d Y

(1 + µ(Ait )) + µ(A) −

i=1

d Y

µ(Ait )

i=1

#)

(3)

where C represents the family of all convex and bounded subsets of d , µ is the Lebesgue measure, T denotes the set of all isometric transformations and Ait denotes the i-th projection of A in the coordinate system t. Ramer and Padet [30] considered also the possibility of extending the HL measure to nonconvex subsets of d . The set function defined in (3) satisfies the following set of properties: Property HL1: Range For every A ∈ C, HL(A) ∈ [0, ∞) where HL(A) = 0 if and only if A = {x} or A = ∅ for some x ∈ d . Property HL2: Monotonicity If A ⊆ B, then HL(A) ≤ HL(B) for all A, B ∈ C. According to Ramer and Padet [30], HL is strictly monotonic on convex sets, i.e., if A & B then HL(A) < HL(B). Property HL3: Subadditivity For every A ∈ C we have that HL(A) ≤ Pd i=1 HL(Ai ) where Ai denotes the one-dimensional projection of A to dimension i in some coordinate system. Property HL4: Additivity For all A ∈ C such that A := ×di=1 Ai , then P HL(A) = di=1 HL(Ai ). Ai here has the same meaning as in Property HL3. Property HL5: Coordinate invariance HL does not depend on isometric transformations of the coordinate system. Property HL6: Continuity HL is a continuous set function. This criterion has not been explicitly defined by Klir and Yuan [21], but according to the interpretation of Ramer and Padet [30], it is intended with respect to the Hausdorff metric. n o Property HL7: Normalization When A = ×di=1 [ai , ai + 1] for any [a1 , a2 , . . . , ad ] ∈ d , then HL(A) = d, for any d ∈ .


Conditional Hartley-like measures The conditional Hartley-like measures express the relationship between marginal and joint Hartley-like measures. Given a set R ⊆ X × Y , whose projections on the sets X and Y are given by RX := projX (R) and RY := projY (R) where the projection operators projX : X × Y → X and projY : X × Y → Y stand for, projX (γ) := { x ∈ X : ∃y ∈ Y, (x, y) ∈ γ } projY (γ) := { y ∈ Y : ∃x ∈ X, (x, y) ∈ γ }

4

given γ ⊆ X × Y . These measures are defined by, HL(R|RY ) := HL(R) − HL(RY ) HL(R|RX ) := HL(R) − HL(RX )

3

(4) (5)

Random sets

This section is a minimal introduction to RS theory. The reader is referred to [5, 34] for information about random sets. Definition 1 Let us consider a universal set X 6= ∅ and its power set P(X). Let (Ω, σΩ , PΩ ) be a probability space and (F , σF ) be a measurable space where F ⊆ P(X). A random set Γ is a (σΩ − σF )-measurable mapping Γ : Ω → F , α 7→ Γ(α). We will call every γ ≡ Γ(α) ∈ F a focal element while F will be called a focal set. In a similar way to the definition of a random variable, the mapping Γ : Ω → F can be used to generate a probability measure on (F , σF ) given by PΓ := PΩ ◦ Γ−1 . This means that an event R ∈ σF has the probability PΓ (R) = PΩ { α : Γ(α) ∈ R } .

(6)

The random set Γ will be referred to in the following also as (F , PΓ ). A random set (F , PΓ ) can be called finite or infinite depending on the cardinality of F ; when all elements of F are singletons (points), then Γ becomes a random variable X, and F is called specific; that is, if F is specific then Γ(α) = X(α) and the value of the probability of occurrence of the event F , PX (F ), is the unique value PX (F ) := PΩ (X −1 (F )) = PΩ { α : X(α) ∈ F } for any F ∈ σX . In the case of random sets, it is not possible to compute the exact value of PX (F ) but lower and upper bounds of it. Dempster [8] defined those lower and upper probabilities by LP(F ,PΓ ) (F ) := PΩ { α : Γ(α) ⊆ F, Γ(α) 6= ∅ } = PΓ { γ : γ ⊆ F, γ 6= ∅ } UP(F ,PΓ ) (F ) := PΩ { α : Γ(α) ∩ F 6= ∅ } = PΓ { γ : γ ∩ F 6= ∅ }

(7) (8) (9) (10)

LP(F ,PΓ ) (F ) ≤ PX (F ) ≤ UP(F ,PΓ ) (F ).

(11)

where The equality in (11) happens when F is specific. 5

In the following, we will refer to LP(F ,PΓ ) (F ) and UP(F ,PΓ ) (F ) respectively as the lower and upper probability measures of the set F with respect to the random set (F , PΓ ). Note also that these measures are a generalization of the belief and plausibility measures (see equations (12) and (13)) defined by Shafer [33] in evidence theory, and which are used exclusively when dealing with finite random sets. (Infinite) random sets have already been analyzed by other authors. See for example [17, 25, 26] and references therein. 3.1 Relationship between random sets and possibility distributions, probability boxes and CDFs Possibility distributions, probability boxes and CDFs are simple particularizations of random sets of indexable type (see Section 5). The reader is referred to Alvarez [3] for details. In this Section, B will stand for the Borel σ-algebra on and (Ω, σΩ , PΩ ) will denote a probability space with Ω := (0, 1], σΩ := (0, 1] ∩ B := ∪θ∈B {(0, 1] ∩ θ} and PΩ will be a probability measure corresponding to the uniform CDF on (0, 1] of a random variable α ˜ , i.e. Fα˜ (α) := PΩ [α ˜ ≤ α] = α for α ∈ (0, 1].

Possibility distributions A possibility distribution (see e.g. [13, 23, 28]) with membership function A : X → (0, 1], X ⊆ can be symbolized as the random set (F , PΓ ) (i.e. Γ : Ω → F , α 7→ Γ(α)) defined on where F is the system of all α-cuts of A, i.e, Γ(α) ≡ Aα := {x : A(x) ≥ α, x ∈ X} for α ∈ (0, 1] and PΓ is defined by (6).

Probability boxes A probability box or p-box (see e.g. [15]) hF , F i is a set of cumulative distribution functions (CDFs) { F : F ≤ F ≤ F , F is a CDF } delimited by lower and upper CDF bounds F and F : → [0, 1]. It can be represented as the random set (F , PΓ ) (i.e. Γ : Ω → F , α 7→ Γ(α)) de(−1) fined on where F (α) :=   is the class of focal elements Γ(α) := hF , F i F

(−1)

(α), F (−1) (α) for α ∈ Ω with F (−1) (α) and F

(−1)

(α) denoting the

quasi-inverses of F and F (the quasi-inverse of the CDF F is defined by F (−1) (α) := inf { x : F (x) ≥ α }) and PΓ is specified by (6).

Cumulative distribution functions When a basic variable is expressed as a random variable on X ⊆ , the probability law of the random variable 6

can be expressed using a CDF FX . A CDF can be represented as the random set (F , PΓ ) (i.e. Γ : Ω → F , α 7→ Γ(α)) where F is the system of focal (−1) elements Γ(α) := FX (α) for α ∈ Ω and PΓ is defined by (6). Note that FX (x) = PΓ (X ≤ x) for x ∈ X.

3.2 Relationship between random sets and Dempster-Shafer structures

Random set theory is tightly related to Dempster-Shafer evidence theory (see e.g. [12, 20, 24, 29, 32]). In fact, when F has a finite number of elements, random sets result to be mathematically isomorphic to Dempster-Shafer (DS) structures, although with somewhat different notation. That is, given a DS structure (Fn , m) with Fn = { A1 , A2 , . . . , An } we can view it as a finite RS Γ : Ω → F , where the following relationships hold: F ≡ Fn , i.e., Aj ≡ γj for j = 1, 2, . . . , n and m(Aj ) ≡ PΓ (γj ). Remember that in evidence theory m is called the basic mass assignment. Also note that in this case the lower and upper probability measures (8) and (10) are now called the belief and plausibility measures, that is, Bel(Fn ,m) (F ) = Pl(Fn ,m) (F ) =

n X

j=1 n X

I [Aj ⊆ F ] m(Aj )

(12)

I [Aj ∩ F 6= ∅] m(Aj );

(13)

j=1

here I represents the indicator function. A finite random set (Fn0 , m0 ) can be typified as an infinite random set (F , PΓ ) (i.e. Γ : Ω → F , α 7→ Γ(α)). For this purpose, it is necessary to induce in (Fn0 , m0 ) an ordering. If { [ai , bi ] for i = 1, . . . , s } are the enumeration of the focal elements of Fn0 , this family of intervals can be naturally sorted by the criteria: [ai , bi ] ≤ [aj , bj ] if ai < aj or (ai = aj and bi ≤ bj ) forming the rearranged finite random set (Fn , m). This can be done using any sorting algorithm by using the appropriate comparison function. The purpose of this ordering is to generate a unique and reproducible sorting in the family of intervals; this is mandatory because in many cases families of intervals do not have a natural ordered structure. It will be clear after reading Section 4 that an ordering strategy is necessary because different orderings require different copulas to specify adequately the dependence information (see other motivations of why to perform this ordering in [3]). On the one side F is the class of focal elements Γ(α) := A∗ (α) for α ∈ Ω where A∗ (α) ∈ Fn is the focal P P Pj−1 m(Ak ) < α ≤ jk=1 m(Ak ), making 0k=1 m(Ak ) = 0; element for which k=1 on the other side, PΓ is defined by (6). 7

4

Some concepts about copulas

The following is a brief presentation of some key points on the theory of copulas. The reader is referred to Nelsen [27] for more information. 4.1 VH -volume and Lebesgue-Stieltjes measure Let H be a mapping H : S → , where S ⊆ d . Let B = ×di=1 [ai , bi ] be a d-box all of whose vertices are in S. The H-volume of B, VH (B), is the d-th order difference of H on B, VH (B) = ∆badd ∆bad−1 · · · ∆ba11 H d−1 where the d-first order differences of H are defined as, ∆bakk H = H(t1 , . . . , tk−1 , bk , tk+1 , . . . , td ) − H(t1 , . . . , tk−1 , ak , tk+1 , . . . , td ) for k = 1, . . . , d. More explicitly VH (B) =

2 X

i1 =1

···

2 X

(−1)i1 +...+id H(x1i1 , . . . , xdid )

id =1

where xj1 = aj and xj2 = bj for all j = 1, . . . , d (see e.g. [14]). For example, when d = 2, then B = [a1 , b1 ] × [a2 , b2 ] and the H-volume of the rectangle B is given by the second order difference of H on B, VH (B) = H(a2 , b2 ) − H(a1 , b2 ) − H(a2 , b1 ) + H(a1 , b1 ). Note that if H is a joint CDF, VH (B) denotes the probability of any d-box B, and in consequence VH (B) ∈ [0, 1]. All d-boxes in d form a semiring S that can be extended to a ring R of all elementary sets that result from the finite union of disjoint elements of S. This measure can be extended by Carath´eodory’s extension theorem, to the σ-algebra generated by R. In particular that σ-algebra contains all Borel subsets of d . The extension of VH is unique and is called the Lebesgue-Stieltjes measure µH corresponding to the function H, and H is called the generating function of µH . Note that if H is a joint CDF, then µH is a probability measure. 4.2 Support of a cumulative distribution function It was said before that each joint CDF H : d → [0, 1] induces a probability measure in d given by µH . The support of a CDF F (x), is the complement 8

of the union of all open subsets of

d

with null µH -measure.

4.3 Copulas A copula is a multivariate CDF C : [0, 1]d → [0, 1] such that each of its marginal CDFs are uniform on the interval [0, 1]. Alternatively, a copula is any function C : [0, 1]d → [0, 1] which has the following three properties: (1) C is grounded, i.e., for every α ∈ [0, 1]d , C(α) = 0 if there exists an i ∈ {1, 2, . . . , d} with αi = 0. (2) If all coordinates of α ∈ [0, 1]d are 1 except αi then C(α) = αi . (3) C is d-increasing, i.e., for all [a1 , . . . , ai , . . . , ad ], [b1 , . . . , bi , . . . , bd ] ∈ [0, 1]d such that ai ≤ bi for i = 1, . . . , d we have that VC (×di=1 [ai , bi ]) ≥ 0. 4.4 Relation between copulas and finite random sets Alvarez [3] showed that copulas may specify the dependence information within infinite random sets. When particularized to finite random sets, the basic mass assignment associated to a joint focal element Aj1 ,j2 ,...,jd is given by the µC measure of its associated box Bj1 ,j2 ,...,jd in the α-representation (see [4, Chapter 5] for details), where µC is the Lebesgue-Stieltjes measure generated by the copula C, i.e., m(Aj1 ,j2 ,...,jd ) = µC (Bj1 ,j2 ,...,jd ).

5

Random sets of indexable type

The formulation of random sets sketched above is very general. Alvarez [4, Chapter 5] showed that choosing in Definition 1, Ω := (0, 1]d , σΩ := (0, 1]d ∩ n o B d := ∪θ∈B d (0, 1]d ∩ θ and PΩ := µC for some copula that contains the dependence information within the joint random set, is sufficient to model possibility distributions, probability boxes, intervals, CDFs and DempsterShafer structures or their joint combinations. Note that in the unidimensional case PΩ is a probability measure on corresponding to the uniform CDF on (0, 1], i.e. Fα (α) = PΩ (α ≤ α) = α, for α ∈ (0, 1], that is a Lebesgue measure on (0, 1]. We will say that this kind of random sets with Ω = (0, 1]d , σΩ = (0, 1]d ∩ B d and PΩ = µC is of indexable type. Sometimes we will denote by PΓ ≡ µC the fact that PΓ is the probability measure generated by the 9

probability measure PΩ which is defined by the Lebesgue-Stieltjes measure corresponding to the copula C, i.e. µC . In other words, PΓ (Γ(G)) = µC (G) for G ∈ σΩ . Alvarez [3] also showed that using the α-representation of random sets of indexable type, the d-box (0, 1]d contains the regions FLP and FUP (denoted in that reference as FBel and FPl correspondingly) which are respectively composed of all those points whose corresponding focal elements are completely contained in the set F or have in common at least one point with F ; in that paper is also proved that FLP ⊆ FUP and that both sets are independent of the copula C that relates the basic variables α1 , . . . , αd ; in this case, the lower (equation (7)) and upper probabilities (equation (9)) of a set F can be calculated by LP(F ,PΓ ) (F ) = UP(F ,PΓ ) (F ) =

Z

Z

(0,1]d (0,1]d

I [α ∈ FLP ] dC (α) = µC (FLP ) I [α ∈ FUP ] dC (α) = µC (FUP ).

provided FLP and FUP are µC -measurable sets. Unidimensional infinite random sets of indexable type have already been analyzed for example in Miranda et al. [25, Section 4.1].

6

Joint and marginal infinite random sets

The definitions of joint and marginal infinite random sets follow directly from the analogous definitions for Dempster-Shafer bodies of evidence (see e.g. [22]). In this section we will consider two marginal infinite random sets. A generalization to more dimensions follows the same considerations. Given the joint infinite random set (F XY , PΓXY ) on X × Y , the associated marginal random sets (F X , PΓX ) and (F Y , PΓY ), on X and Y respectively, are defined by the focal sets, F X := { projX (γ) : γ ∈ F XY } F Y := { projY (γ) : γ ∈ F XY }

10

and the probability measures PΓX : σF X → [0, 1], and PΓY : σF Y → [0, 1], PΓX (G X )

:= =

PΓY (G Y ) := =

Z

h

i

h

i

I projX (γ) ∈ G X dPΓXY (γ)

F XYn PΓXY γ

Z

: projX (γ) ∈ G X , γ ∈ F XY

I projY (γ) ∈ G Y dPΓXY (γ)

F XYn PΓXY γ

: projY (γ) ∈ G Y , γ ∈ F XY

o

(14)

o

(15)

for all G X ∈ σF X and all G Y ∈ σF Y and where σF X , σF Y are σ-algebras on F X and F Y respectively. Note that in the finite case |F X | ≤ |F XY | and |F Y | ≤ |F XY |, and that (14) and (15) are a generalization of the definitions of marginal random sets for finite RSs, namely mX (A) =

I[projX (A0 ) = A]mXY (A0 )

X

A0 ∈F XY Y

m (B) =

I[projY (A0 ) = B]mXY (A0 )

X

A0 ∈F XY

for all A ∈ F X and all B ∈ F Y . When (F XY , PΓXY ), (F X , PΓX ) and (F Y , PΓY ) are of indexable type, we can associate correspondingly the unique copulas C XY , C X and C Y , to the probability measures PΓXY , PΓX and PΓY . We will define the conditional copulas C X|Y and C Y |X , respectively, by:

C X|Y (α|β) =

 XY  C (α,β)

C Y |X (β|α) =

 XY  C (α,β)

and

C Y (β)

0

C X (α)

0

for C Y (β) 6= 0 otherwise for C X (α) 6= 0 otherwise

It is important to observe the following relationships between the lower and upper probabilities of (F X , PΓX ), (F Y , PΓY ) and (F XY , PΓXY ), LP(F X ,PΓX ) (A) = LP(F XY ,PΓXY ) (A × Y )

(16)

LP(F Y ,PΓY ) (B) = LP(F XY ,PΓXY ) (X × B)

(17)

UP(F X ,PΓX ) (A) = UP(F XY ,PΓXY ) (A × Y )

(18)

UP(F Y ,PΓY ) (B) = UP(F XY ,PΓXY ) (X × B)

(19)

11

For example, (16) can be shown as follows: LP(F X ,PΓX ) (A) = n

= PΓX γ X : γ X ⊆ A, γ X ∈ F X , γ X 6= ∅ n

n

o

o

= PΓXY γ : projX (γ) ∈ γ X : γ X ⊆ A, γ X ∈ F X , γ X 6= ∅ , γ ∈ F XY n

= PΓXY γ : γ ⊆ A × Y, γ ∈ F XY , γ 6= ∅ = LP(F XY ,PΓXY ) (A × Y )

o

o

The deduction of equations (17), (18) and (19) follow the same considerations.

Non-interactivity There is a special case of relationship between marginal random sets called non-interactivity. The probability measures corresponding to the marginal infinite random sets (F X , PΓX ) and (F Y , PΓY ) are called non-interactive if and only if for all G X ∈ σF X , G Y ∈ σF Y and ΓXY ∈ F XY we have that PΓXY (G XY ) = PΓX (G X )PΓY (G Y ) if G XY = G X × G Y , otherwise PΓXY (G XY ) = 0. Alvarez [4, Chapter 6] has shown that this concept is equivalent to the extension of the concept of random set independence (see e.g. Couso et al. [6], Fetz and Oberguggenberger [16]) to infinite random sets 1 . In the case of random sets of indexable type, PΓXY , PΓX and PΓY will have the associated copulas C XY , C X and C Y correspondingly; in this case C XY = C X C Y .

7

Random set inclusion for infinite random sets of indexable type

In this section, the concept of finite random set inclusion proposed by Delgado and Moral [7], Dubois and Prade [10; 12], Yager [35] (see Definition 3) is generalized to infinite random sets of indexable type. Definition 2 Let (F A , PΓA ) and (F B , PΓB ) be two infinite RSs of indexable type defined on X ⊆ d . Then (F A , PΓA ) is said to be included in (F B , PΓB ), 1

There are two other types of independence for infinite random sets, namely strong independence and fuzzy set independence; we will not deal inhere with those cases since we will allow all possible probability measures residing in the marginal focal elements and also we will allow arbitrary dependence for the combination of those marginal probability measures. When these matters are taken into consideration, strong independence plays a main role. Also, we will not deal with fuzzy set independence since we will not restrict our analysis to consonant focal sets.

12

denoted as (F A , PΓA ) ⊆ (F B , PΓB ) if and only if the following three conditions hold: (1) For all γ A ∈ F A there exists a set γ B ∈ F B such that γ A ⊆ γ B . (2) For all γ B ∈ F B there exists a set γ A ∈ F A such that γ A ⊆ γ B . (3) In the space (0, 1]d × (0, 1]d corresponding to the α-representation of (F A , PΓA ) × (F B , PΓB ) respectively, there exists a copula F : [0, 1]d × [0, 1]d → [0, 1], (α, β) 7→ F (α, β) whose support is the set {(α, β) : ΓA (α) ⊆ ΓB (β), α, β ∈ [0, 1]d , ΓA (α) ∈ F A , ΓB (α) ∈ F B } such that: PΓA



A

A



Γ (G ) =

Z

GA

Z

(0,1]d

dF (α, β)

(20)

for each GA ∈ [0, 1]d ∩ B d , ΓA (GA ) ∈ σF A and PΓB



B

B



Γ (G ) =

Z

(0,1]d

Z

GB

dF (α, β)

(21)



(22)

for each GB ∈ [0, 1]d ∩ B d , ΓB (GB ) ∈ σF B . Observe that the copula F has the form, 

F (α, β) := C AB C A (α), C B (β)

where C A : [0, 1]d → [0, 1] and C B : [0, 1]d → [0, 1] are respectively the copulas associated to the RSs (F A , PΓA ) and (F B , PΓB ) by PΓA ≡ µC A and PΓB ≡ µC B , and C AB is a two-dimensional copula.  A B B A B A|B AB C (α)|C (β) C (β) = C (α), C (β) = C Since F (α, β) = C 



C B|A C B (β)|C A (α) C A (α), where C A|B and C B|A are the conditional copulas associated to C AB , using the relations between joint, conditional and marginal CDFs, we retrieve the next basic relations from integral (20) (and similarly for (21)), PΓA (ΓA (GA )) = = =

Z

Z

Z

GA GA GA

Z

Z

(0,1]d (0,1]d



dC AB C A (α), C B (β) 





dC B|A C B (β)|C A (α) dC A (α)

dC A (α)

= µC A (GA ) and PΓB (ΓB (GB )) = µC B (GB ) for GA , GB ∈ [0, 1]d ∩ B d . 13

Examples of random set inclusion Possibility distributions Let A and B be two possibility distributions such that A(x) ≤ B(x) for all x ∈ X. In this case, A is said to be a normalized fuzzy subset of B (see e.g. [22, p. 262]). If A and B are associated to the random sets (F A , PΓA ) and (F B , PΓB ), then (F A , PΓA ) ⊆ (F B , PΓB ), or shortly A ⊆ B, because the three requisites for random set inclusion are satisfied, namely, 1. 2. if αA ∈ (0, 1] is the index associated to the α-cuts AαA ∈ F A and αB ∈ (0, 1] is the index associated to BαB ∈ F B then for all αA = αB = α, such that α ∈ (0, 1], we have that Aα ⊆ Bα . 3. Since d = 1, then the copulas C A and C B in (22) will correspond to uniform CDFs on [0, 1] and therefore F will be any copula with support supp(F ) := {(αA , αB ) : AαA ⊆ BαB , αA , αB ∈ [0, 1]}. Consider two possibility distributions that have a trapezoidal shape, A := trap(3, 4, 6, 9) and B := trap(2, 3, 7, 9). Clearly A ⊆ B. Figure 1 shows those possibility distributions and its associated set supp(F ).

Probability boxes Let hA, Ai and hB, Bi two probability boxes with infinite random set representation (F A , PΓA ) and (F B , PΓB ), such that B ≤ A ≤ A ≤ B, that is, the probability box hA, Ai is contained in the probability box hB, Bi. We will say in this case that the probability box hA, Ai is a subset of the probability box hB, Bi, i.e., hA, Ai ⊆ hB, Bi, based on the fact that (F A , PΓA ) ⊆ (F B , PΓB ); the demonstration is analogous to the one ofpossibil ity distributions, with Aα replaced by ΓA (α) := A B



replaced by Γ (α) := B

(−1)

(α), B

(−1)

(−1)

(α), A(−1) (α) and Bα



(α) .

:= Figure 1 shows two probability boxes hA, Ai hT (90, 150, 200), T (100, 160, 210)i and hB, Bi := hT (80, 140, 200), T (100, 170, 220)i; here T (a, b, c) stands for the formulation of a triangular CDF corresponding to the triangular PDF t(a, b, c) with lower limit a, mode b and upper limit c. It can be shown that hA, Ai ⊆ hB, Bi. Figure 1 shows those probability boxes along with its associated set supp(F ).

Dempster-Shafer structures In this case, when the RSs are finite, the concept of random set inclusion reduces to the one proposed by Delgado and Moral [7], Dubois and Prade [12], Yager [35]:

14

A⊆B

1

A B

0.8

0.6

0.6

αB

αA , αB

0.8

0.4

Sfrag replacements 0.2 2

4

X

6

8

0

10

hA, Ai ⊆ hB, Bi

1

0.2

0.4

αA

0.6

0.8

1

0.8

1

supp(F )

0.8

0.6

0.6

A A B B

0.4 0.2

80

100

120

140

160

X

180

200

αB

αA , αB

0

1

0.8

0

0.4 0.2

0

Ai,j

supp(F )

1

0.4 0.2 0

220

0

0.2

0.4

αA

0.6

Fig. 1. Upper row: Left: possibility distributions A := trap(3, 4, 6, 9) and B := trap(2, 3, 7, 9), here A ⊆ B. Right: support of the copula F . Lower row: Left: probability boxes hA, Ai := hT (90, 150, 200), T (100, 160, 210)i and hB, Bi := hT (80, 140, 200), T (100, 170, 220)i, here hA, Ai ⊆ hB, Bi. Right: support of the copula F .

Definition 3 ([12]) The finite random set (F A , mA ) is said to be included in the finite random set (F B , mB ), denoted as (F A , mA ) ⊆ (F B , mB ) if and only if the following three conditions hold: (1) For all A ∈ F A there exists a set B ∈ F B such that A ⊆ B. (2) For all B ∈ F B there exists a set A ∈ F A such that A ⊆ B. (3) There exists an assignment matrix W : F A × F B → [0, 1], (Ai , Bj ) 7→ W (Ai , Bj ) such that: mA (Ai ) = mB (Bj ) =

s X

j=1 r X

W (Ai , Bj ) for all Ai ∈ F A

(23)

for all Bj ∈ F B

(24)

W (Ai , Bj )

i=1

and W (Ai , Bj ) = 0 if Ai 6⊂ Bj , i.e., if Ai is not strictly contained in Bj .

15

Equations (23) and (24) follow directly from (20) and (21) by observing that a finite RS induces a partition of the α-space. Every entry W (Ai , Bj ) will correspond to the Lebesgue-Stieltjes measure of the box in the α-space associated to the focal element Ai × Bj of the RS (F A , mA ) × (F B , mB ) with respect to the related copula F (see [4, Chapter 5]).

8

Measures of nonspecificity for infinite random sets

In this section a measure of nonspecificity for infinite random sets is proposed, which coincides with the definitions of nonspecificity when particularized to finite random sets. Let (F , PΓ ) be an infinite random set defined on X ⊆ d , the nonspecificity for (F , PΓ ) will be defined by NL((F , PΓ )) := E(F ,PΓ ) [HL(γ)] =

Z

HL(γ) dPΓ (γ)

F

(25)

where HL(γ) stands for the Hartley-like measure of the set γ ∈ F . That is, the NL measure is the expectation of the HL measure of the focal elements within the focal set F . In the following we will assume that all focal elements are convex and compact; if they are not, we will employ their closures. If (F , PΓ ) is of indexable type, then using the α-representation of infinite random sets, integral (25) is equivalent to NL((F , PΓ )) :=

Z

(0,1]d

HL(Γ(α)) dC(α)

(26)

If it happens that d = 1, then NL((F , PΓ )) =

Z

1 0

HL(Γ(α)) dα

(27)

Examples

Equation (26) can be particularized to the special cases when (F , PΓ ) represents possibility distributions, probability boxes or Dempster-Shafer structures, as outlined below. 16

Possibility distributions If (F , PΓ ) represents the unidimensional possibility distribution A, equation (27) turns into NL(A) = =

Z

Z

1 0 1 0

HL(Aα ) dα log2 (1 + µ(Aα )) dα

In the generalized information theory literature this measure is known as the UL-uncertainty (see for example [22, p. 206]).

Probability boxes If (F , PΓ ) is an infinite RS defined on sents the probability box hF , F i, equation (27) turns into NL(hF , F i) = =

Z

Z

1 0

0

1

HL



F

(−1)



log2 1 + F

(α), F (−1)

(−1)

(α)

(α) − F



(−1)

which repre-

dα 

(α) dα

(28)

Note that using (28), the nonspecificity of a CDF is 0. This confirms the fact that probability theory cannot measure nonspecificity, inasmuch as it requires perfect knowledge in the specification of the CDFs.

Dempster-Shafer structures If (F , PΓ ) represents the Dempster-Shafer structure (Fn , m) integral (25) turns into NL((Fn , m)) =

n X

HL (Ai ) m(Ai )

i=1

which is already recognized within generalized information theory (see e.g. [24, p. 67]). See also Dubois and Prade [9; 11] who deal with specificity for belief functions.

9

Properties fulfilled by the NL measure

According to [22, p. 196] every measure of uncertainty is required to fulfill a set of defining properties. In the rest of this article it will be shown that (26) satisfies the following ones: Property NL1: Range For each random set (F , PΓ ) of indexable type, whose focal elements are bounded and convex subsets of d , we have that NL((F , PΓ )) is nonnegative and finite; in addition NL((F , PΓ )) = 0 if and only if almost all elements of F are singletons. 17

Property NL2: Monotonicity For any pair of infinite random sets of indexable type (F A , PΓA ) and (F B , PΓB ) such that (F A , PΓA ) ⊆ (F B , PΓB ), it holds that NL((F A , PΓA )) ≤ NL((F B , PΓB )). Property NL3: Subadditivity For any joint infinite RS (F XY , PΓXY ) of indexable type defined on X × Y and its associated marginals (F X , PΓX ) and (F Y , PΓY ) defined on X and Y respectively, we have that, NL((F XY , PΓXY )) ≤ NL((F X , PΓX )) + NL((F Y , PΓY )) Property NL4: Additivity For any non-interactive (independent in the sense of RS independence) infinite random sets (F X , PΓX ) and (F Y , PΓY ) defined on X and Y respectively, and the associated joint infinite RS (F XY , PΓXY ) defined on X ×Y , where F XY = F X ×F Y and PΓXY (A×B) = PΓX (A)PΓY (B) for all A ∈ σF X and all B ∈ σF Y , we have that, NL((F XY , PΓXY )) = NL((F X , PΓX )) + NL((F Y , PΓY )) Property NL5: Coordinate invariance The functional NL does not change under isometric transformations of the focal elements. o n Property NL7: Normalization When F = ×di=1 [ai , ai + 1] for any [a1 , a2 , . . . , ad ] ∈ d , then NL((F , PΓ )) = d, for any d ∈ . Property NL8: Branching Suppose we have the infinite random set of indexable type (F , PΓ ), and any partition of F into n sets F i ; then to each of those sets F i we will assign a probability measure PΓ (G|F i ) := PΓ (G ∩ F i )/PΓ (F i ), for G ∈ σF i , so that (F , PΓ ) is decomposed into the random sets (F i , PΓ (·|F i )). The NL-uncertainty of (F , PΓ ) can be computed by individually calculating the uncertainty of each partition (F i , PΓ (·|F i )) and then making a weighted addition of those uncertainties according to the importance that F i has in the representation of F , i.e.,


NL((F , PΓ )) =

n X i=1





NL (F i , PΓ (·|F i )) PΓ (F i )

Note: If (F , PΓ ) contains a unique bounded and convex focal element, i.e., F = {A} and PΓ (A) = 1 then Properties NL i reduce to Properties HL i, for i = 1, . . . , 7. In this and in the following section, we will show that the NL measure in fact satisfies those properties. Finally, there is an additional property namely Property NL6: Continuity NL is a continuous function. which is expected to be fulfilled. However, its proof is left as an open problem. 18

9.1 Monotonicity Random set inclusion implies monotonicity in the Hartley-like measure, i.e., if (F 1 , PΓ1 ) ⊆ (F 2 , PΓ2 ) then NL((F 1 , PΓ1 )) ≤ NL((F 2 , PΓ2 )). The following theorem is an extension of a theorem proposed by Klir [22, p. 213] for the case of finite random sets. Theorem 4 For any pair of infinite random sets of indexable type (F A , PΓA ) and (F B , PΓB ) such that (F A , PΓA ) ⊆ (F B , PΓB ), it holds that NL((F A , PΓA )) ≤ NL((F B , PΓB )).

PROOF. Let C A and C B be the copulas associated to PΓA and PΓB respectively. Then, Z

FA

HL(γ A ) dPΓA (γ A ) = = = = ≤ =

Z

Z

Z

(0,1]d (0,1]d (0,1]d

Z

Z

(0,1]d

(0,1]d

HL(ΓA (α)) dC A (α) Z

Z

(0,1]d (0,1]d

Z

(0,1]d









HL(ΓA (α)) dC B|A C B (β)|C A (α) dC A (α) HL(ΓA (α)) dF (α, β)

HL(ΓB (β)) dC A|B C A (α)|C B (β) dC B (β)

HL(ΓB (β)) dC B (β)

(29)

The inequality (29) follows from Property HL2 and the fact that the point (α, β) belongs to the support of the copula F and in consequence, ΓA (α) ⊆ ΓB (β). 2

9.2 Subadditivity and additivity: relationship between the joint and the marginal generalized Hartley-like measures The following theorem shows that the NL measure of a joint infinite random set of indexable type (F XY , PΓXY ) is subadditive with regard to the NL measure of its corresponding marginal infinite random sets. Theorem 5 Let (F XY , PΓXY ) be a joint infinite random set of indexable type defined on X × Y , X ⊆ a , Y ⊆ b and its associated marginal infinite 19

random sets (F X , PΓX ) and (F Y , PΓY ) defined on X and Y respectively. Then: NL((F XY , PΓXY )) ≤ NL((F X , PΓX )) + NL((F Y , PΓY ))

PROOF. NOTE: sometimes we will write C XY (α, β) as C XY ((α, β)) to give a representation of C XY as a multivariate CDF. Let C XY , C X and C Y be the copulas associated to the probability measures PΓXY , PΓX and PΓY respectively. Then NL((F

X

, PΓX ))

= = =

Z

Z

Z

(0,1]a (0,1]a

HL(ΓX (α)) dC X (α) Z

(0,1]b

(0,1]a+b

HL(ΓX (α)) dC Y |X (β|α) dC X (α) 



HL projX Γ((α, β)) dC XY ((α, β))

using the fact that ΓX (α) = projX Γ((α, β)). Similarly, NL((F

Y

, PΓY ))

=

Z

(0,1]a+b





HL projY Γ((α, β)) dC XY ((α, β))

(30)

Hence, NL((F X , PΓX )) + NL((F Y , PΓY )) = = ≥

Z

Z(0,1]

a+b

(0,1]a+b









HL projX Γ((α, β)) + HL projY Γ((α, β)) dC XY ((α, β)) 



HL Γ((α, β)) dC XY ((α, β))

= NL((F XY , PΓXY ));

here, the last inequality follows after using the subadditivity property HL3 of the HL measure (see Section 2.2). 2 Theorem 6 Given the marginal infinite random sets of indexable type (F X , PΓX ) and (F Y , PΓY ) which are combined under the condition of random set independence to form the joint infinite random set (F XY , PΓXY ) on X × Y , then NL((F XY , PΓXY )) = NL((F X , PΓX )) + NL((F Y , PΓY ))

PROOF. Since γ = γ X × γ Y and since PΓX and PΓY are non-interactive, then C XY ((α, β)) = C X (α)C Y (β) and using the additivity property HL4 it 20

follows: NL((F XY , PΓXY )) = = = = =

Z

a+b

Z(0,1] Z (0,1]a

Z

Z

(0,1]a

Z





HL Γ((α, β)) dC XY ((α, β))

(0,1]b (0,1]b

HL(ΓX (α) × ΓY (β)) dC X (α) dC Y (β) HL(ΓX (α)) + HL(ΓY (β)) dC X (α) dC Y (β)

X

(0,1]a

X

HL(Γ (α)) dC (α) +

= NL((F X , PΓX )) +

Z

(0,1]b Y Y NL((F , PΓ ))

HL(ΓY (β)) dC Y (β)

2 9.3 Coordinate invariance The NL measure does not change under isometric transformations of the focal elements. To see this, let T be the set of all isometries on X. Notice that according to property HL5, if A ⊆ X, then HL(T (A)) = HL(A) for all T ∈ T . Now, let (F , PΓ ) be an infinite RS defined also on X. Suppose that for every focal element γ ∈ F , we choose any isometry Tγ ∈ T (can be one different for every element; this is the reason of the subindex γ). In consequence, we can form a new random set (F 0 , PΓ0 ) with F 0 := {γ 0 := TΓ (γ) : γ ∈ F } and PΓ0 := PΓ ; in consequence, NL((F , PΓ )) = = =

Z

ZF

ZF

HL(γ) dPΓ (γ) HL(TΓ (γ)) dPΓ (γ)

F0

HL(γ 0 ) dPΓ0 (γ 0 )

= NL((F 0 , PΓ0 )) This shows property NL5. 9.4 Range If all focal elements are bounded, then the NL measure will be nonnegative and finite. In fact, using Property HL1, the HL measure will be nonnegative and finite for all bounded focal elements; therefore, 0 ≤ HL(γ) ≤ UR for all γ and for some U < ∞. Thus, according to (26), 0 ≤ NL((F , PΓ )) ≤ (0,1]d U dC(α) = U. 21

Suppose that almost all (with regard to PΓ ) focal elements of (F , PΓ ) are singletons; these singletons can be translated using Property NL5 of coordinate invariance to some {x}. This element will have probability one, i.e., PΓ ({x}) = 1 while all others will have a null PΓ -measure. In consequence NL((F , PΓ )) = HL({x}) = 0. Conversely, suppose that NL((F , PΓ )) = 0, then this implies that HL(γ) = 0 for almost all γ ∈ F , since NL((F , PΓ )) ≥ 0 and since HL(γ) = 0 if and only if γ is a singleton or the empty set. The justification for Property NL1 follows.

10

Uniqueness of the generalized NL measure for infinite random sets of indexable type

Our motivation for this section comes from the following quote from Klir [22, p. 198]: “... the strongest justification of a functional as a meaningful measure of the amount of uncertainty of a considered type in a given uncertainty theory is obtained when we can prove that it is the only functional that satisfies the relevant axiomatic requirements and measures the amount of uncertainty in some specific measurement unit”. The goal of this section is to show that the generalized nonspecificity measure NL defined by equation (26), is the unique measure of nonspecificity for infinite random sets of indexable type that satisfies the Properties NL1 to NL5, NL7 and NL8. The following result, shown in [4, Chapter 5], will be required in the proof of uniqueness: Theorem 7 Let (F 1 , m1 ) ≺ (F 2 , m2 ) ≺ · · · ≺ (F i , mi ) ≺ · · · be a sequence of every-time-refining finite RS of indexable type defined on X ⊆ d which converges to (F , PΓ ). Here m1 , m2 , . . . , mi , . . . and PΓ are probability measures generated by the copula C, i.e., mi ≡ µC for all i = 1, 2, . . . and PΓ ≡ µC . Let f : P(X) → [0, ∞) be a bounded and continuous function with regard to the P Hausdorff metric.R Then the sequence { Ai ∈F i f (Ai ) mi (Ai )}i∈ converges to the unique limit (0,1]d f (Γ(α)) dC(α) as i → ∞. 

The main theorem of this section follows:

Theorem 8 Given some function HL that fulfills the set of Properties HL1 to HL7, there is a unique function NL : M → [0, ∞) which satisfies the Prop22

erties to NL1 to NL5, NL7 and NL8, where M is the system of all infinite random sets of indexable type with convex and bounded focal elements. Furthermore, for any (F , PΓ ) ∈ M, with PΓ ≡ µC , we have Z

NL((F , PΓ )) =

(0,1]d

HL(Γ(α)) dC(α)

(31)

PROOF. In the following proof, ({A}, 1) will represent a random set whose unique focal element is the set A. (i) NL for sets. Consider a RS of the form ({A}, 1). In this special case NL1 to NL5 reduces to HL1 to HL5, and therefore, NL (({A}, 1)) := HL (A)

(32)

where HL is any function that fulfills HL1 to HL5 and HL7 2 (ii) NL for finite random sets. Let (F 2 , PΓ2 ) be a RS such that F 2 = {A21 , A22 }, σF 2 = {∅, F 2 , A21 , A22 } and PΓ2 (∅) = 0, PΓ2 (F 2 ) = 1, PΓ2 (A21 ) = m1 and PΓ2 (A22 ) = m2 , with m1 + m2 = 1. Using the branching Property NL8, it follows that, NL



F 2 , PΓ2







= PΓ2 A21 NL



{A21 }, 1







+ PΓ2 A22 NL



{A22 }, 1



(33)

Using (32) in (33) it yields NL



F 2 , PΓ2

















= PΓ2 A21 HL A21 + PΓ2 A22 HL A22



Suppose that for every finite random set with n focal elements, F n = {An1 , An2 , . . . , Ani , . . . , Ann }, NL ((F n , PΓn )) =

n X

PΓn (Ani ) HL (Ani )

i=1

n+1 = Ani for i = holds. Also, let us consider F = F n ∪ An+1 n+1 where Ai 1, . . . , n and PΓn+1 (G|F n ) = PΓn (G) for all G ∈ σF n . Using again the branching n+1

2

The Hartley-like measure (3) is one function that satisfies HL1 to HL7, if in addition A is convex and bounded (there is not guarantee that the Hartley-like measure is the unique function that fulfills that set of properties).

23

theorem NL8, NL



F n+1 , PΓn+1







n n = PΓn+1 F n+1 \ An+1 n+1 NL ((F , PΓ ))









+ PΓn+1 An+1 n+1 NL = PΓn+1 (F n ) NL



{An+1 n+1 }, 1











F n , PΓn+1 (·|F n )

n+1 + PΓn+1 An+1 n+1 HL An+1

n+1 n n PΓn+1 satisfies PΓn+1 (G) = PΓn+1 (G|F  ) PΓ (F ) where G ∈ σF n , σF n ⊆ n+1 n+1 n+1 n+1 σF n+1 and PΓ An+1 = 1 − PΓ F n+1 \ An+1 = 1 − PΓn+1 (F n ) then

NL



F n+1 , PΓn+1



= PΓn+1 (F n )

n X



i=1

+

PΓn+1

=

n+1 X



An+1 n+1 





HL An+1 n+1 





PΓn+1 An+1 HL An+1 i i

i=1

This shows that NL ((Fn , m)) =

n X





PΓn+1 An+1 |F n HL An+1 i i





HL (Ai ) m (Ai )

i=1

is the unique measure of nonspecificity for finite random sets up to the specification of the function HL. (iii) NL for infinite random sets. Suppose now that (F 1 , m1 ) ≺ (F 2 , m2 ) ≺ · · · ≺ (F i , mi ) ≺ · · · is a sequence of every-time refining finite RSs of indexable type 3 which converges to (F , PΓ ). Here m1 , . . . , mi , . . . and PΓ are all probability measures generated by the copula C, i.e., mi ≡ µC for all i = 1, 2, . . . and PΓ ≡ µC . Such a sequence can always be created. Since HL is a bounded function and continuous in the sense of Hausdorff (by Property HL6), according to Theorem 7 we have that the limit of NL ((F i , PΓi )) as i → ∞ exists, is unique, and is given by the integral Z

(0,1]d

HL (Γ (α)) dC (α) = lim

i→∞

X

Ai ∈F i

= lim NL i→∞







HL Ai PΓi Ai



F i , PΓi





In consequence, (31) appears as the result of this limiting process and fulfills HL1 to HL6 (which are just a particularization of NL1 to NL6), NL7 and NL8. The other properties are shown to be satisfied in Section 9. 3

The symbol ≺ and the term sequence of every-time refining finite random sets are defined in [4, Chapter 5]

24

11

Conditional generalized Hartley-like measures

Based on the concept of conditional HL measure, according to the formulas (4) and (5), we can analogously define the conditional generalized measures NL((F XY , PΓXY )|(F Y , PΓY )) and NL((F XY , PΓXY )|(F Y , PΓX )) by the formulas, NL((F XY , PΓXY )|(F Y , PΓY )) := := E(F XY ,PΓXY ) [HL(γ|γ Y )] = E(F XY ,PΓXY ) [HL(γ)] − E(F XY ,PΓXY ) [HL(γ Y )] = NL((F XY , PΓXY )) − E(F XY ,PΓXY ) [HL(projY γ)] = NL((F XY , PΓXY )) − NL((F Y , PΓY )) where the last equality follows from (30); similarly, NL((F XY , PΓXY )|(F Y , PΓX )) := NL((F XY , PΓXY )) − NL((F X , PΓX )) NL((F XY , PΓXY )|(F Y , PΓY )) is used to measure the uncertainty after pinching (F Y , PΓY ) towards a random set with full specify (i.e, with null nonspecificity), like a constant in the case of intervals or a CDF in the case when (F , PΓ ) represents a probability box.

12

Final comments

In this article, the Hartley-like based measure of nonspecificity for infinite random sets was proposed. This is a measure of lack of information, that is, a measure of epistemic uncertainty. During the development of the paper, some concepts related to random set theory like joint and marginal infinite random sets and random set inclusion were extended to infinite random sets of indexable type. A mathematical derivation of the formulas for nonspecificity of possibility distributions and probability boxes was performed. It was also shown that the proposed measure corresponds to the one defined in the realm of Dempster-Shafer evidence theory when dealing with finite random sets. It was shown that the proposed measure is the only one that fulfills a set of characterizing properties. Note that the notion of nonspecificity is used for example in [1, 22] as one of the two measures required for the definition of a total uncertainty measure (the other one was conflict, which is a type of aleatory uncertainty). It is left as open problem to show if the Hartley-like measure is a continuous function; this property requires the definition of an appropriate metric between 25

random sets of indexable type. Also, further work is required on extending the measures of aleatory uncertainty for infinite random sets. Note also that the treatment of random sets of indexable type is restricted to convex and compact focal elements. It should be studied whether it can be extended to more general types of focal elements. Additional work is required to define a nonspecificity measure for sets of probability measures which are defined on some universal set X ⊆ d . Since random sets of indexable are a particularization of sets of probability measures (see e.g. [16], who showed for the one dimensional case that Dempster-Shafer structures, probability boxes and possibility distributions can be represented as sets of probability measures) that measure must coincide with the one inhere defined when the particularization is done. Note that the definition of a nonspecificity measure when X is finite has already been analyzed by [2] and further commented in [22]. Finally, the reader is referred to [4, Chapter 9] for an application of the NL measure in the area of sensibility analysis.

Acknowledgements This research was supported by the Programme Alßan, European Union Programme of High Level Scholarships for Latin America, identification number E03X17491CO. The helpful advice of Professors Michael Oberguggenberger and Thomas Fetz and the anonymous reviewers is gratefully acknowledged.

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Nomenclature employed in the paper µ

Lebesgue measure

µH

Lebesgue-Stieltjes measure with regard to the generating function H.

C

copula

σF

σ-algebra with regard to the set F

F

focal set

γ := Γ(α) (Fn , m)

Γ (F , PΓ ) (F X , PΓX ), (F XY , PΓXY ) ⊆

focal element of an infinite random set Finite random set (equivalent to Dempster-Shafer structure). Here the focal set Fn has cardinality n, and m denotes the basic mass assignment random set when seen as a function Ω → F random set when seen as a focal set with an associated measure P Γ marginal and joint random set (see Section 6) subset, also denotes random set inclusion

hF , F i

probability box

HL, NL

Hartley-like measure for sets and for random sets

(0, 1]d ∩ B d

Borel set defined on (0, 1]d

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