A Monte-Carlo-based method for the estimation of lower and upper probabilities of events using 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 Random set theory is a useful tool to quantify lower and upper bounds on the probability of the occurrence of events given uncertain information represented for example by possibility distributions, probability boxes, or Dempster-Shafer structures, among others. In this paper it is shown that the belief and plausibility estimated by Dempster-Shafer evidence theory are basically approximations by Riemann-Stieltjes sums of the integrals of the lower and upper probability employed when using infinite random sets of indexable type. In addition, it is shown that the evaluation of the lower and upper probability is more efficient if it is done by Pseudo-Monte-Carlo strategies. This discourages the use of Dempster-Shafer evidence theory and suggests the use of infinite random sets of indexable type specially in high dimensions, not only because the initial discretization step of the basic variables is not required anymore, but also because the evaluation of the lower and upper probability of events is much more efficient using the different techniques for multidimensional integration like Monte Carlo simulation. Key words: infinite random sets of indexable type, Dempster-Shafer evidence theory, Riemann-Stieltjes sums, Monte Carlo simulation

∗ Corresponding author. Tel.: +43+512-5076825. Fax: +43+512-5072941. Email address: [email protected] (Diego A. Alvarez).

Preprint submitted to Fuzzy Sets and Systems

13 August 2008

1

Introduction

When designing for example a mechanical system, engineers must cope with the problem of assessing the reliability of the structure. Usually there is uncertainty in the definitions of the parameters that describe the system; and this uncertainty might be because there is lack of information in the definition of the variable itself (epistemic uncertainty), or because of the inherent variability of it (aleatory uncertainty). Other kind of uncertainty might be because there is not an appropriate mathematical model that describes the physics of the system. In the former case, the theory of imprecise probabilities is the most general framework for dealing with parametric uncertainty. This theory is very general and in fact, it is difficult to use in practice. There exist less flexible and simpler (although expressive enough) methods, for representing uncertainty, like the theory of evidence and the theory of random sets. Random set and evidence theory constitute useful generalizations of probability theory which allow the representation of aleatory and epistemic uncertainty. Within these theories, it is possible to represent the uncertain information about a basic variable by for example cumulative distribution functions (CDFs), probability boxes, intervals, possibility distributions or DempsterShafer bodies of evidence. However, until recently, the calculation of lower and upper bounds on probability of events required the discretization of such basic variables; Tonon [36] provides an excellent example of the application of this methodology. It is the feeling within the engineering community that the quality of the calculation of the probability bounds improves as long as the granularity of the discretization increases; this however, comes at the expense of increasing the computational cost of the calculation exponentially with the dimension; we will analyze that issue in this paper. In order to alleviate this curse of dimensionality, authors like Wilson [39], have on the one hand made remarks on how to diminish the computational load in the calculation of the belief and plausibility of a set F with relation to some Dempster-Shafer structure (Fn , m), in the context of how to draw basically those focal elements Ai ∈ Fn that contribute the most to the belief and plausibility; on the other hand, some other authors (see e.g. Hall and Lawry [16], Tonon [36]) have discussed how to discretize the basic variables, such that the belief and plausibility results will contain the best possible bounds on the uncertainty of some event. Recently, Alvarez [1] proposed a method to deal with a special kind of random sets (which we call in Section 3.2 of indexable type) and which includes as particular cases CDFs, probability boxes, intervals, possibility distributions or Dempster-Shafer bodies of evidence and which does not require anymore the discretization of the basic variables. In the same reference, it is proposed 2

a method for the estimation of such probability bounds, which is based on the use of Monte Carlo integration techniques. In this paper, we will theoretically analyze the performance of such strategy. We will also characterize the different possible types of discretization of basic variables, concluding that the outer discretization is the most conservative way of discretizing the basic variables. In addition, we will analyze the classical modi operandi employed in Dempster-Shafer evidence theory, and we will conclude that belief and plausibility are just a special type of Riemann-Stieltjes approximations of the Stieltjes integrals that are used to calculate the lower and upper probabilities in the framework of infinite random sets of indexable type. Finally, we will study the numerical overhead of the discretization method and the Monte-Carlo-based method proposed in Ref. [1], concluding that, specially in high dimensions, the later is preferable not only because we will avoid the discretization step of the basic variables, but also because we can speed up the evaluation of the lower and upper probabilities by orders of magnitude. The plan of this paper is as follows: First, in Sections 2 and 3 we will make a brief introduction to copulas and random sets respectively; then, in Section 4, we will deal with discretizations of infinite random sets of indexable type. Section 5 compares the efficiency of different methods employed in the solution of the lower and upper probability integrals, including their approximation by Riemann-Stieltjes sums and the use of Monte Carlo simulation techniques. After a numerical example in Section 6, the paper ends with some conclusions and suggestions for future work.

2

Some concepts about copulas

Copulas play an important role in the specification of the dependence in random sets. The following is a succinct presentation of some key points about copulas. The reader is referred to Nelsen [27] for additional information.

2.1 The VH -volume Let H be a function H : S → , where S ⊆ d . Let B = ×di=1 [ai , bi ] be a d-box all of whose vertexes are in S. The H-volume of B, VH (B), is the d-th order difference of H on B, VH (B) =

2 X

i1 =1

···

2 X

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

id =1

3

(1)

where xj1 = aj and xj2 = bj for all j = 1, . . . , d (see e.g. Ref. [11]). When H represents a joint cumulative distribution function (CDF), VH (B) stands for the probability measure of any d-box B, and therefore VH (B) ∈ [0, 1].

2.2 Copulas A copula is a d-dimensional 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 with domain [0, 1]d which fulfills the following conditions: (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. 3

Random sets and Dempster-Shafer structures

3.1 Definition of a random set 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 say that every γ := Γ(α) ∈ F is a focal element while F is a focal set. Analogously to the definition of a random variable, this mapping can be used to define a probability measure on (F , σF ) given by PΓ := PΩ ◦ Γ−1 . That is, an event R ∈ σF has the probability PΓ (R) = PΩ { α ∈ Ω : Γ(α) ∈ R } .

(2)

The random set Γ will be called from now on also as (F , PΓ ). Depending on the cardinality of F , the random set (F , PΓ ) is named finite or infinite (in contrast to the definition employed by some other authors when the final space is finite; in fact, some authors do not mention explicitly the initial space); also, when every element of F is a singleton, then Γ becomes a random variable X, and F is said to be specific; that is, if F is specific then 4

Γ(α) = X(α) and the probability of occurrence of the event F , is PX (F ) := (PΩ ◦ X −1 )(F ) = PΩ { α : X(α) ∈ F } for any F ∈ σX . In the case of random sets, it is not feasible to calculate exactly PX (F ) but its upper and lower bounds. Dempster [7] defined those upper and lower 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= ∅ }

(3) (4) (5) (6)

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

(7)

where

Note that the equality in (7) holds for F specific. The reader is referred for instance to Refs. [5, 40] for more information about random sets. It can be shown that lower and upper probability measures are dual fuzzy measures, that is LP(F ,PΓ ) (F ) = 1 − UP(F ,PΓ ) (F c ) UP(F ,PΓ ) (F ) = 1 − LP(F ,PΓ ) (F c ) and also that lower probability is an ∞-monotone Choquet capacity and the upper probability is an ∞-alternating Choquet capacity (see e.g. Refs. [19, p.66],[28]).

3.2 Random sets of indexable type

Definition 1 is very general; in the next sections it is showed that particularizing this definition to Ω := (0, 1]d , σΩ := (0, 1]d ∩ B d and PΓ ≡ µC for some copula that contains the dependence information within the joint random set, and employing intervals and d-boxes as elements of F , it is sufficient to model possibility distributions, probability boxes, intervals, CDFs and Dempster-Shafer structures or their joint combinations.We will call this kind of random sets with this type of particularization of indexable type. We will denote by PΓ ≡ µC the fact that PΓ is the probability measure generated by 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 . In the rest of this article we will deal only with RSs of indexable type, inasmuch as random set of indexable type are enough to express popular engineering representations of uncertainty. 5

3.3 Relationship between random sets and probability boxes, CDFs and possibility distributions

Some particularizations of random sets of indexable type are for example probability boxes, CDFs and possibility distributions. The reader is referred to Alvarez [1] for details. In this Section, B will denote the Borel σ-algebra on and (Ω, σΩ , PΩ ) will stand for 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 CDF of a random variable α ˜ uniformly distributed on (0, 1], i.e. Fα˜ (α) := PΩ [α ˜ ≤ α] = α for α ∈ (0, 1]; that is, PΩ is a Lebesgue measure on (0, 1].

Probability boxes A probability box or p-box (see e.g. Ref. [12]) hF , F i is a set of 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 Γ : Ω → F , α 7→ Γ(α) (i.e. (F , PΓ )) defined on where F   is the class of focal elements Γ(α) := hF , F i(−1) (α) := F (−1)

(−1)

(α), F (−1) (α) for α ∈ Ω with

(α) denoting the quasi-inverses of F and F (the quasiF (−1) (α) and F inverse of the CDF F is defined by F (−1) (α) := inf { x : F (x) ≥ α }) and PΓ is specified by (2).

Cumulative distribution functions When a basic variable is expressed as a random variable on X ⊆ , the probability law of the random variable can be expressed using a CDF FX . A CDF can be represented as the random set Γ : Ω → F , α 7→ Γ(α) where F is the system of focal elements Γ(α) := (−1) FX (α) for α ∈ Ω and PΓ is defined by (2). Note that FX (x) = PΓ (X ≤ x) for x ∈ X.

Possibility distributions A possibility distribution (see e.g. Ref. [10]) with membership function A : X → (0, 1], X ⊆ can be symbolized as the random set Γ : Ω → F , α 7→ Γ(α) (i.e. (F , PΓ )) 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 (2). For a proof of this result, the reader is referred for example to Refs. [8, 15, 25, 28]. 6

3.4 Relationship between random sets and Dempster-Shafer structures RS theory is closely related to Dempster-Shafer evidence theory (see e.g. Refs. [9, 18, 20, 31, 33]). In fact, if F is a finite set, a (finite) random set results to be isomorphic to a Dempster-Shafer (DS) structure (also known as DS body of evidence). That is, given a DS structure (Fn , m) with Fn = { A1 , A2 , . . . , An } and a finite RS Γ : Ω → F , then the following relationships occur: F ≡ Fn , i.e., Aj ≡ γj for j = 1, 2, . . . , n and m(Aj ) ≡ PΓ (γj ). It is appropriate to recall that in evidence theory m is called the basic mass assignment. Observe that (4) and (6) become respectively the belief and plausibility measures of the set F with regard to the DS structure (Fn , m), i.e., Bel(Fn ,m) (F ) := LP(Fn ,m) (F ) = Pl(Fn ,m) (F ) := UP(Fn ,m) (F ) =

n X

j=1 n X

j=1

I [Aj ⊆ F ] m(Aj )

(8)

I [Aj ∩ F 6= ∅] m(Aj ).

(9)

where I stands for the indicator function. A DS structure (Fn , m) is also a special case of an infinite random set (F , PΓ ); here F is the system of focal elements Γ(α) := A∗ (α) for α ∈ (0, 1] where Pj−1 P A∗ (α) ∈ Fn is the focal element for which k=1 m(Ak ) < α ≤ jk=1 m(Ak ), P making 0k=1 m(Ak ) = 0; also PΓ is the probability measure defined on the measurable space (F , σF ) by (2), with PΩ defined as a Lebesgue measure on (0, 1] (see e.g. Ref. [1]). Note that in this case, there can be several repetitions of the focal element Ak in F ; however, here α plays here the role of an index which distinguishes the different occurrences of Ak in F ; this is the reason underlying the adjective “of indexable type”. It is appropriate to mention that the representation of Dempster-Shafer structures as random sets in the infinite case is basically a problem tackled by means of Choquet’s theorem (see for instance Refs. [26, 29]) It will be seen in the next Section that a copula C models the dependence information in joint RSs of indexable type. The specification of this copula is attached to the indexing of the elements within Fn . In order to avoid confusions in the specification of C (and also in the posterior calculation of FLP and FUP , as is discussed for example in Ref. [1]), it is recommended to induce a unique and reproducible ordering in the family of intervals of a given (Fn0 , m0 ); this is required inasmuch as in several cases families of intervals do not have a natural ordering structure. For example, if { [ai , bi ] for i = 1, . . . , s } are the enumeration of the focal elements of Fn0 , this family of intervals can be naturally ordered by the rule: [ai , bi ] ≤ [aj , bj ] if ai < aj or (ai = aj and bi ≤ bj ) forming the reordered finite random set (Fn , m). This can be performed using any sorting algorithm by using the appropriate comparison function. Other 7

sortings, like for example sorting according to the basic mass assignment are possible, but this would alter the natural representation that for example, histograms have. Also, if several focal elements have the same basic mass assignment, it would be unclear how to sort them.

3.5 Partitions and refinements The concepts of partitions and refinements will be required later in this paper when introducing discretizations on infinite random sets of indexable type. In the following, we will present these terms. A partition of an interval (a, b] is a finite sequence a = x0 < x1 < · · · < xn = b. Each (xj−1 , xj ], j = 1, . . . , n is called a subinterval of the partition. The set P = {(x0 , x1 ], (x1 , x2 ], . . . , (xn−1 , xn ]} is sometimes used to denote the partition and kPk := maxni=1 (xi − xi−1 ) denotes the norm of the partition. A refinement of the partition P 0 = {(x0 , x1 ], (x1 , x2 ], . . . , (xn−1 , xn ]} is a partition P 00 = {(y0 , y1 ], (y1 , y2 ], . . . , (ym−1 , ym ]}, x0 = y0 , xn = ym such that for every i = 0, . . . , n there is an integer r(i) such that xi = yr(i) ; this will be denoted henceforth by P 0 ≺ P 00 , since refinement induces a partial ordering. In other words, to make a refinement, one cuts the subintervals into smaller pieces and does not remove any cut. The definition of a partition and refinement of a partition can be generalized to higher dimensions. Here a box ×di=1 (ai , bi ] will be decomposed into a disjoint set of boxes, that when refined will be decomposed into boxes as well. 3.6 α-representation of Dempster-Shafer structures: partitions of (0, 1] d It has been shown in Ref. [1] that the α-representation of an infinite random set of indexable type (F , PΓ ) := ×di=1 (F i , PΓi ) (up to some dependence specification) is a representation of the points (α1 , α2 , . . . , αd ) ∈ (0, 1]d which correspond to the focal elements Γ1 (α1 ) × Γ2 (α2 ) × · · · × Γd (αd ) ∈ F for Γi (αi ) ∈ F i , i = 1, 2, . . . , d. This geometrical depiction allows to see easily which focal elements contribute to the lower or upper probabilities of an event F. induces a partition P = A DS structure (Fn , m) defined on {(0, α1 ], (α1 , α2 ], . . . , (αn−1 , αn ]}, αn = 1 of the interval (0, 1] where αk = Pk j=1 m(Aj ) for k = 1, . . . , n. We will refer to this partition as the partition of the interval (0, 1] associated to (Fn , m). This partition forms the αrepresentation of (Fn , m) which is shown in Figure 1; note that there is a 8

γ = Ai

one-to-one relationship between Aj ∈ Fn and the subinterval (αj−1 , αj ], which we will denote by Aj ↔ (αj−1 , αj ] or simply by Aj ↔ αj ; this is possible since m(Aj ) > 0. A5

0

0

α1

m(A5 )

A1

m(A2 )

A2

m(A4 )

m(A4 ) m(A3 ) m(A2 ) m(A1 ) x

A3

m(A3 )

m(A5 )

A4

m(A1 )

1

α2 α3

α4

α5 = 1

Fig. 1. Schematic representation a one-dimensional DS structure (F , m) in the α-space.

Now, consider the DS structure (Fn , m) := ×di=1 (Fni i , mi ) defined as the random relation of d unidimensional DS structures, up to some dependence specification. Every DS structure (Fni i , mi ) has an associated partition of the interval (0, 1], which forms a system of sets P i formed by the subintervals of the partition, i.e., P i = {(α0i , α1i ], (α1i , α2i ], . . . , (αni i −1 , αni i ]}. The direct product P = P 1 × P 2 × · · ·× P d will form a rectangular grid of (0, 1]d , composed by the disjoint boxes Bj1 ,j2 ,...,jd := ×di=1 (αji i −1 , αji i ] where ji = 1, . . . , ni ; this partition of (0, 1]d forms the α-representation of (Fn , m) which is shown in Figure 2 in the two dimensional case; note that there is a one-to-one association between PSfrag replacements α2 1 x m2

(αi1 , αj2 )

s

m2j

Bi,j

m21 0 m1i

m11 0

α1

m1r 1

Fig. 2. Schematic representation a two-dimensional DS structure (F , m) in the α-space.

Aj1 ,j2 ,...,jd ∈ Fn , (here Aj1 ,j2 ,...,jd := ×di=1 Aiji and Aiji ∈ Fni i ) and Bj1 ,j2 ,...,jd which we will denote by Aj1 ,j2 ,...,jd ↔ Bj1 ,j2 ,...,jd or just by the lattice point 9









αj11 , αj22 , . . . , αjdd , i.e. Aj1 ,j2 ,...,jd ↔ αj11 , αj22 , . . . , αjdd .

Finally, we will define the image of ∅ in the X-space to be ∅ in the α-space and viceversa, i.e., ∅ ↔ ∅. 3.7 Copulas and Dempster-Shafer structures According to Sklar’s theorem (see Refs. [34, 35]), copulas are functions that relate a joint distributions function with its marginals, carring in this way the dependence information in the joint CDF. Ferson et al. [13, p. 68] showed that the dependence information between basic variables modelled by Dempster-Shafer structures can also be specified by a copula C. In fact, they showed that the basic mass assignment associated to the joint focal element Aj1 ,j2 ,...,jd can be constructed by the VC volume of its associated box Bj1 ,j2 ,...,jd , that is, m(Aj1 ,j2 ,...,jd ) = VC (Bj1 ,j2 ,...,jd ) where the VC volume of the d-box Bj1 ,j2 ,...,jd := ×di=1 (αji i −1 , αji i ] is defined by the d-th order difference of C on Bj1 ,j2 ,...,jd , αdj

αd−1 j

d−1 VC (Bj1 ,j2 ,...,jd ) = ∆αdd ∆αd−1 jd −1

jd−1 −1

α1

· · · ∆αj11 C j1 −1

where αij

∆αi i C = C(α1 , . . . , αi−1 , αji i , αi+1 , . . . , αd )−C(α1 , . . . , αi−1 , αji i −1 , αi+1 , . . . , αd ) ji −1

for i = 1, . . . , d, which reduces to the application of equation (1) for H := C. Inasmuch as C is a joint CDF, it follows that VC denotes the probability of any d-box B, and therefore VC (B) ∈ [0, 1]. All d-boxes in [0, 1]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. The measure generated by VC 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 [0, 1]d , that is [0, 1]d ∩ B d . The extension of VC is unique and is called the (Lebesgue-) Stieltjes measure µC corresponding to C and C is referred to either as the generating function of µC (in the terminology of Kolmogorov and Fomin [21]) or as the distribution function of µC (in the terminology of Lo`eve [23]), in consequence m(Aj1 ,j2 ,...,jd ) = µC (Bj1 ,j2 ,...,jd ). (10) Note that since C is a joint CDF, then µC is a probability measure. Inasmuch as m is equivalent to VC and PΓ is equivalent to µC (sometimes denoted in this document as PΓ ≡ µC ), then PΓ can also be regarded as the 10

extension of m defined on the σ-algebra σF .

3.8 Lower and upper probabilities for infinite random sets of indexable type In Ref. [1] it was shown that using the α-representation of a RS (F , PΓ ), it can be seen that the d-box Ω := (0, 1]d contains the regions FLP := {α ∈ Ω : Γ(α) ⊆ F, Γ(α) 6= ∅} and FUP := {α ∈ Ω : Γ(α) ∩ F 6= ∅} 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 ; some authors like Nguyen [28] call these sets the lower and upper inverses of F respectively; note that the set FLP is contained in FUP and both sets do not depend on the copula C that relates the basic variables α1 , . . . , αd ; in this case, the lower (3) and upper (5) probability measures of F can be calculated by LP(F ,PΓ ) (F ) = UP(F ,PΓ ) (F ) =

Z

Z

(0,1]d

I [α ∈ FLP ] dC (α) = µC (FLP )

(11)

(0,1]d

I [α ∈ FUP ] dC (α) = µC (FUP ).

(12)

provided that FLP and FUP are µC -measurable sets. This is a good point to mention that the fact that the lower and upper probability measures can be expressed as Riemann-Stieltjes integrals can also be deduced from the results in De Cooman et al. [6]; and that some definitions of random sets have been proposed via some measurability conditions on the upper and lower inverses (see for instance Himmelberg [17]).

4

Discretization of infinite random sets of indexable type

The use of Dempster-Shafer structures, or equivalently finite random sets requires the discretization of every basic variable in the cases when they represent possibility distributions, probability boxes or continuous CDFs. A basic variable can be discretized in several ways, which can be gathered into three main groups, outer, inner and intermediate discretization. Suppose that the basic variable X defined on d , and represented by the infinite RS of indexable type (F , PΓ ), is to be discretized into a DS structure (Fn , m). Given a partition P of (0, 1]d , and a box Bi ∈ P, the focal element Ai ∈ Fn such that Ai ↔ Bi , may be defined by one of the three methods of focal approximation that are proposed in the following: 11

• Outer focal approximation: in this case Ai := A∗i =

[

Γ(α);

(13)

Γ(α)

(14)

α∈Bi

• Inner focal approximation: in this case Ai := Ai∗ =

\

α∈Bi

whenever Ai∗ 6= ∅. • Intermediate focal approximation: here Ai∗ ⊆ Ai ⊂ A∗i ;

(15)

and Ai 6= ∅. In all three cases Ai is m-measurable with m(Ai ) = VC (Bi ) > 0.

(16)

We will say that the DS structure (Fn , m) is an outer discretization of (F , PΓ ) if all focal elements were obtained by outer focal approximations; if at least one focal element Ai ∈ Fn was defined by an intermediate focal approximation, then we will say that the discretization is intermediate. One special case of the intermediate discretization is the inner discretization, which happens when all focal elements were defined by inner focal approximations. Figure 3 illustrates these types of discretization for the one dimensional case. Observe that according to equations (14) and (15) sometimes Ai could be an empty set. In this case the discretization in consideration does not exist inasmuch as Ai = ∅ is represented in the α-space also by an empty set, namely Bi = ∅, and according to (16) then µC (∅) > 0, which contradicts the Kolmogorov’s second axiom of probability. Also note that Ai does not necessarily belong to F . It is a good point to comment that the outer discretization of probability boxes coincides with the outer discretization method proposed by Tonon [36]; and his averaging discretization method is a particularization of the intermediate discretization of a probability box; also, the upper and lower approximation of possibility distributions described for example by Baudrit et al. [3], correspond respectively to the outer and inner discretizations of possibility distributions, as shown in Figure 3. We want to analyze what is the relation between the focal elements of Fn and the regions FLP and FUP for the outer and intermediate types of focal approximation. Then four cases must be analyzed, as shown in Table 1; this table shows that when a focal element is created by the outer focal approximation, 12

Inner discretization

Intermediate discretization 1

1

1

0

x

0

x

0

x

1

1

1

This discretization does not always exist

0

x

0

x

0

x

1

1

1

This discretization CDF

PSfrag replacements

Probability box

Possibility distribution

Outer discretization

does not always exist

0

x

0

x

0

x

Fig. 3. Outer, inner and intermediate discretization of CDFs, probability boxes and possibility distributions. Note that the inner discretization of probability boxes sometimes does not exist. The inner discretization of CDFs only exists when the “random” variable is a constant. In this Figure we employ boxes to define the discretization of the basic variable. The side of the boxes parallel to the x axis, and shown here with a thick line, denotes the focal element A i obtained in the discretization. The other side of the box depicts the basic mass assignment m corresponding to the element, i.e., m(Ai ).

then there are certain relations between its associated box in the α-space and the sets FLP and FUP . Note that the outer discretization is the most conservative way to discretize a basic variable, inasmuch as the following result holds:

Theorem 2 Let (F , PΓ ) be a random set of indexable type and let the DS structure (Fn , m) be any of its outer discretizations. Then h

i

h

i

LP(F ,PΓ ) (F ), UP(F ,PΓ ) (F ) ⊆ Bel(Fn ,m) (F ), Pl(Fn ,m) (F ) .

PROOF. Given some partition P of (0, 1]d , let (Fn , m) be the outer disUP cretization of (F , PΓ ) associated to P; here n := |P|. Let ϕLP ∗ and ϕ∗ be two simple functions defined on (0, 1]d with range {0, 1} defined correspondingly 13

FLP

FUP

outer focal approx.

intermediate focal approx.

If Ai ⊆ F then for all α ∈ Bi , Γ(α) ⊆ F otherwise Γ(α) may or may not be contained in F

In this case, independently of the fact that Ai ⊆ F or Ai * F we have that Γ(α) may or may not be contained in F , for all α ∈ Bi

If Ai ∩ F 6= ∅ then Γ(α) may or may not be intersected with F otherwise Γ(α) ∩ F = ∅, for all α ∈ Bi

In this case, independently of the fact that Ai ∩F 6= ∅ or Ai ∩F = ∅ we have that Γ(α) may or may not be intersected with F , for all α ∈ Bi

Table 1 Four possible cases of focal approximation of a focal element. The Table depicts the relation between the focal elements of the discretization (F n , m) and the regions FLP and FUP generated by (F , PΓ ) and the set F . Here α ∈ Bi and Ai ↔ Bi .

by ϕLP ∗ (α) :=

and ϕUP ∗ (α) :=

 1

if Ai ⊆ F and Bi 3 α 0 otherwise.

 1

if Ai ∩ F 6= ∅ and Bi 3 α 0 otherwise.

(17)

(18)

Here Ai ∈ Fn , Bi ∈ P and Ai ↔ Bi . Note that the images of Bi , through the above-defined functions, fulfill the relations ϕLP ∗ (Bi ) = I[Ai ⊆ F ]

(19)

and ϕUP ∗ (Bi ) = I[Ai ∩ F 6= ∅]. UP The fact that ϕLP ∗ (α) ≤ I[α ∈ FLP ] and ϕ∗ (α) ≥ I[α ∈ FUP ] for all α ∈ d (0, 1] (see Figure 4), follows easily taking into account that ϕLP ∗ (α) = 1 if α ∈ Bi and Ai ⊆ F ; however since Γ(α) ⊆ Ai , by means of equation (13), then Γ(α) ⊆ F , and therefore I [α ∈ FLP ] = 1. Also I [α ∈ FUP ] = 1 if Γ(α) ∩ F 6= ∅; but then this implies that Ai ∩ F 6= ∅, for the Ai corresponding to the Bi 3 α, and therefore ϕUP ∗ (α) = 1.

14

α2

α2

1

1

supp(ϕLP ∗ ) FLP

supp(ϕUP ∗ ) FUP

PSfrag replacements

0

0

α1 0

α1 0

1

1

Fig. 4. Outer discretizations in the α-space. The
left picture shows the support of
LP , and F ; note that supp ϕLP ⊆ F . The right one shows the ϕLP , supp ϕ LP
LP ∗ ∗ ∗
UP , and F UP . support of ϕUP UP ; note that FUP ⊆ supp ϕ∗ ∗ , supp ϕ∗

The result that LP(F ,PΓ ) (F ) ≥ Bel(Fn ,m) (F ) follows from the fact that Z

(0,1]d

I [α ∈ FLP ] dC(α) = ≥ =

Z

(0,1]d

Z

(0,1]d

=

ϕLP ∗ (α) dµC (α)

X Z

Bi ∈P

=

I [α ∈ FLP ] dµC (α)

ϕLP ∗ (α) dµC (α)

(20)

ϕLP ∗ (Bi )µC (Bi )

(21)

I[Ai ⊆ F ]m(Ai )

(22)

X

Bi ∈P n X i=1

Bi

The third equality follows from the fact that ϕLP ∗ is, according to (17), constant on Bi , and the last one from (10) and (19) since Ai ↔ Bi . Similar considerations can be done to show that UP(F ,PΓ ) (F ) ≤ Pl(Fn ,m) (F ). 2

NOTE: The same result has been derived in Tonon [37], using the fact that a random set can be interpreted as a credal set, and thereafter as a probability box hF , F i. However, the demonstrations of Tonon [37] require in addition F and F to be continuous and strictly monotonically increasing, assumption which is not required inhere, making the proposed solution of more general nature. Finally, as a hint, the above result might also be deduced as well using the notion of natural extensions of Walley [38]. 15

5

Approximation of the lower and upper probability integrals

In most of the cases integrals (11) and (12) are impossible to solve analytically. Therefore, we require numerical methods to approximate those integrals, like Riemann-Stieltjes sums, or Monte Carlo simulation techniques. Both methods are analyzed in the following.

5.1 Approximation of the lower and upper probability integrals using Riemann-Stieltjes sums

Suppose we are given a partition P of (0, 1]d and a d-dimensional CDF H; let f be a function defined on (0, 1]d . For each Bi ∈ P, let ξi be an arbitrary point of Bi . Then any sum of the form S(f, P) :=

X

f (ξi )µH (Bi )

(23)

Bi ∈P

is called a Riemann-Stieltjes sum of f relative to P with respect to the CDF H (here we are considering just a special case of the Riemann-Stieltjes sum. More general conditions on H are possible). For any kind of discretization, given the partition P of (0, 1]d , let f := ϕLP ∗ , defined by equation (17), and H be the copula C. Then Bel(Fn ,m) (F ) =

X

ϕLP ∗ (ξ i )µC (Bi )

Bi ∈P

is a Riemann-Stieltjes sum that approximates integral (11), where ξ i ∈ Bi . This follows from equations (20), (21) and (22) and the fact that the selection of ξi ∈ Bi is irrelevant inasmuch as ϕLP is constant on Bi . Analogously, if ∗ UP f := ϕ∗ , as defined by equation (18), then Pl(Fn ,m) (F ) =

X

ϕUP ∗ (ξ i )µC (Bi )

Bi ∈P

approximates integral (12). In other words, the belief and plausibility of a set F with respect to the outer discretization (Fn , m) of an infinite random set (F , PΓ ) are simply Riemann-Stieltjes sums that approximate integrals (11) and (12) respectively. I [α ∈ FLP ] is a discontinuous function on the boundary ∂FLP of FLP . Therefore, the size of the partition P together with the regularity of the boundary ∂FLP may have a substantial impact on the discretization error of Riemann16

Stieltjes sums. In fact Z (0,1]d

I [α ∈

LP FLP ] dC(α) − ϕ∗ (ξ i ) µC (Bi ) Bi ∈P X X

≤

Bi ∈P

I [Bi ∩ ∂FLP 6= ∅] µC (Bi )

for any kind of approximation ϕLP ∗ and for any ξ i ∈ Bi . Similar considerations can be done with regard to the calculation of the plausibility. With regard to equation (23), suppose that C = Π (here Π is the product copula which is useful to model random set independence (see Ref. [2, Chapter 6])), then µC reduces to the Lebesgue measure and then the Riemann-Stieltjes sums become Riemann sums. According to Krommer and Ueberhuber [22], the convergence of multivariate integration rules based on simple d-dimensional Riemann sums is unacceptably slow for most practical problems. Suppose now that in addition to C = Π, f is a continuous function whose second derivatives ∂ 2 f /∂αi2 for i = 1, . . . , d are also continuous on (0, 1]d and P is a regular grid, i.e. it is composed of hypercubes of the same size. The trapezoidal rule can be used to integrate f and in this case, if N is the number of discretizations in every variable, then the accuracy of the approximation is of the order O(n−2/d ) (see e.g. Niederreiter [30]), where n := N d is the number of evaluations of f required. Thus, this algorithm is efficient only in very low dimensions. For example, according to Niederreiter [30], to guarantee an absolute error of less than 0.01, we must roughly use 10d nodes, which clearly shows the curse of dimensionality in the method. If ∂ 4 f /∂αi4 is continuous for i = 1, . . . , d, changing the method to Simpson’s rule of integration, does not improve much things, inasmuch as in this case the error bound will be of the order O(n−4/d ). Another trouble with Riemann-Stieltjes sums strategy is that one has to decide beforehand how fine the discretization P that defines (Fn , m) is; then one is compelled to evaluating all of the focal elements Ai in Fn . In other words, with a grid it is not possible to sample until some convergence or termination criterion is met. Monte Carlo simulation techniques overcome this problem.

5.2 Approximation of the lower and upper probability integrals using Monte Carlo simulation techniques For decades, Monte Carlo simulation (MCS) techniques have been reliable and robust methods to estimate integrals when their analytical solution is too complicated (see e.g. Refs. [14, 32]). Their performance depends on how the points 17

αi ∈ (0, 1]d , i = 1, . . . , n are sampled. These points can be chosen for example by simple Monte Carlo sampling, Latin hypercube sampling, importance sampling, quasi Monte Carlo sampling, among others.

Simple Monte Carlo sampling In Ref. [1] it is explained how to approximate integrals (11) and (12) by means of simple Monte Carlo sampling. Basically, the method consists in sampling n points from the copula C, namely α1 , α2 , . . . , αn (Nelsen [27] provides methods to do it), and then retrieve the corresponding focal elements Ai := Γ(αi ), i = 1, . . . , n from F , to form the focal set Fn . Thereafter to each of those focal elements in Fn , a basic mass assignment of m(Ai ) := 1/n is given. Then integrals (11) and (12) are estimated by computing the belief and plausibility of F with respect to (Fn , m) using (8) and (9), that is, n n X 1X ˆ (F ,P ) (F ) = Bel(Fn ,m) (F ) = 1 I [A ⊆ F ] = I [αj ∈ FLP ] LP j Γ n j=1 n j=1

n n 1X 1X ˆ UP(F ,PΓ ) (F ) = Pl(Fn ,m) (F ) = I [Aj ∩ F 6= ∅] = I [αj ∈ FUP ] n j=1 n j=1

In Ref. [1], is also shown that Bel(Fn ,m) (F ) and Pl(Fn ,m) (F ) are unbiased estimators of LP(F ,PΓ ) (F ) and UP(F ,PΓ ) (F ) respectively. Let C be a copula; let f : Ω → be any random variable defined in the probability space (Ω, σΩ , µC ), where Ω := (0, 1]d , σΩ := (0, 1]d ∩ B d and µC is the Lebesgue-Stieltjes measure corresponding to the copula C. The expected value of the random variable f is defined by E(f ) =

Z

Ω

f (α) dµC (α).

provided f is µC -integrable. Using the Monte Carlo method, this value can be estimated by n 1X f (αi ) E(f, Wn ) = n i=1 where Wn := {α1 , . . . , αn } ∈ Ω is a set of samples from Ω which are independent and identically distributed according to the copula C. By the strong law of large numbers, if n tends to infinity, then E(f, Wn ) will tend to E(f ) µ∞ C -almost everywhere, i.e., µ∞ C where α

:=



α∈Ω

∞



: lim En (f, α) = E(f ) = 1 n→∞

(α1 , α2 , . . . , αn , . . .)

∈ 18

Ω∞

=:

×∞i=1Ω,

En (f, α)

:=

E(f, {α1 , α2 , . . . , αn }) = E(f, Wn ) and µ∞ is the product measure of countN∞ C ∞ able many copies of µC , i.e., µC = i=1 µC .

We would like to estimate the error in the approximation of E(f ) by E(f, Wn ) for some sample Wn . The variance of the random variable f , σ 2 (f ) :=

Z

Ω

(f (α) − E(f ))2 dµC (α),

is finite provided that Ω |f (α)|2 dµC (α) < ∞; therefore, for any n ≥ 1, it follows that (Niederreiter [30, p. 4]), R

Z

Ω

···

Z

Ω

n 1X f (αi ) − E(f ) n i=1

!2

dµC (α1 ) . . . dµC (αn ) =

σ 2 (f ) . n

This equation says that the q error in the estimation of E(f ) is on average √ σ(f )/ n where σ(f ) := σ 2 (f ). If σ(f ) < ∞, the central limit theorem states that the probability that of E(f ), E(f, Wn ), √ √ the Monte Carlo estimate lies between E(f ) − aσ(f )/ n and E(f ) + bσ(f )/ n satisfies lim µ∞ n→∞ C

α∈Ω

∞

n σ(f ) σ(f ) 1X : −a √ ≤ f (αi ) − E(f ) ≤ b √ n n i=1 n ! Z b 1 t2 =√ dt (24) exp − 2 2π −a

!

In consequence, choosing n points independently and identically distributed according to the copula C, leads to an error term in Monte Carlo integration of the order O(n−1/2 ). Note that this error estimate is not a rigorous bound, but a probabilistic bound; thus the error term should be taken only as a rough indicator of the probable error. Also, this error estimate is independent of the dimension, and therefore it does not have the curse of dimensionality. This error term should be compared with the O(n−2/d ) error bound for the Riemann sum, discussed in the previous section, when f is a continuous function whose second derivatives ∂ 2 f /∂αi2 for i = 1, . . . , d are also continuous on (0, 1]d and C = Π. Note that the Monte Carlo method only requires the function |f |2 to be µC -integrable. Note that in our particular case f (α) := I[α ∈ FLP ]

(25)

or f (α) := I[α ∈ FUP ]. (26) This shows that use of simple Monte Carlo methods for the approximation of integrals (11) and (12) has as advantage over the discretization method its lack of sensibility to the dimensionality of the problem. Note that when f is 19

Basic

Representation

Variable i

of (F i , PΓi )

1, 2, 3

hUnifCDF(−1.0, 0.5), UnifCDF(−0.5, 1.0)i

4, 5, 6

TriangCDF(0.7, 1.0, 1.3)

7, 8, 9

TrapzPD(0.0, 0.4, 0.8, 1.2)

10, 11, 12

TriangPD(−0.25, 0.5, 1.25)

Table 2 Marginal basic variables considered in the numerical example of Section 6. Basic variables 1, 2 and 3 are described by a probability box defined by uniform CDFs, variables 4, 5 and 6 are represented by a triangular CDFs, variables 7, 8 and 9 are defined by trapezoidal possibility distributions and variables 10, 11 and 12 are defined by triangular possibility distributions.

a continuous function whose second derivatives ∂ 2 f /∂αi2 for i = 1, . . . , d are also continuous, C = Π and the discretization is formed by Riemann sums then, the Monte Carlo method is more efficient for dimensions d ≥ 5. In practice, one does not have the exact value of σ(f ), and for some sample Wn one uses the estimate σ ˆ (f ) =

n 1 X (f (αi ) − E(f, Wn ))2 n − 1 i=1

!1/2

(27)

instead. In the particular case when f is random variable that represents a Bernoulli trial (i.e., the range of f is {0, 1}), like in the case of equations (25) and (26), then equation(27) simplifies to n E(f, Wn ) (1 − E(f, Wn )) σ ˆ (f ) = n−1 

6

1/2

.

(28)

Numerical example

In order to compare the proposed approach with the classical approach, namely to discretize the basic variables and thereafter form Dempster-Shafer structures, we will set up the following problem. Consider the marginal basic variables expressed by the random sets (F i , PΓi ), i = 1, 2, . . . , 12 described in Table 2 and depicted in Figure 5. Our problem consists in assessing the lower and upper probabilities of the set F := [0, 1]12 i i with respect to the joint random set (F , PΓ ) := ×12 i=1 (F , PΓ ) using the dependence relations given by the copulas: 20

Sfrag replacements

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−2

−1

0

1

2

0.5

1

1, 2, 3 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

0.5

1.5

4, 5, 6

1

−0.5

0

7, 8, 9

0.5

1

1.5

10, 11, 12

Fig. 5. Marginal basic variables described in Table 2.

• Product copula: C1 (u) := • Clayton copula: C2 (u) :=

Qd

i=1 P d

ui ;

−δ i=1 ui − d + 1



• Gumbel copula: C3 (u) := exp −

P

−1/δ

d δ i=1 (− ln ui )

, using δ = 1.0;

1/δ 

, using δ = 2.0

In all three cases d = 12. For comparative purposes, the problem will be solved analytically, and with the discretization and Monte Carlo sampling methods.

6.1 Analytical solution The simplicity of the problem allows us to calculate LP(F ,PΓ ) (F ) and UP(F ,PΓ ) (F ) exactly, by LP(F ,PΓ ) (F ) = VC (BLP ) UP(F ,PΓ ) (F ) = VC (BUP )

(29) (30)

where BLP = [2/3, 1]3 × (0, 1/2]3 × [1/2, 1]3 × [1/3, 1]3 and BUP = [1/3, 1]3 × [0, 1/2]3 × (0, 1]3 × (0, 1]3 . Table 3 shows the values of LP(F ,PΓ ) (F ) and UP(F ,PΓ ) (F ) for the different copulas. 21

Copula

LP(F ,PΓ ) (F )

UP(F ,PΓ ) (F )

C1

0.00017147

0.037037

C2

0.00020998

0.025

C3

0.00004918

0.027383

Table 3 Exact values of LP(F ,PΓ ) (F ) and UP(F ,PΓ ) (F ) for the different copulas, in the numerical example considered in Section 6. These values were calculated using equations (29) and (30)

6.2 The discretization method The lower and upper probabilities (29) and (30) were estimated by discretizing the basic variables using the outer and intermediate discretization of (F , PΓ ), forming the corresponding Dempster-Shafer structure (Fn , m) as described in Section 4 and then estimating the corresponding belief Bel(Fn ,m) (F ) and plausibility Pl(Fn ,m) (F ) by equations (8) and (9) respectively. Remember that in this case the basic mass assignment of the focal element Ai1,...,i12 is calculated using equation (10). In order to test the influence of the size and type of the discretization, each basic variable was discretized using N = 2, 3, 4 and 5 focal elements, as explained in Table 4. The results of the evaluation using Dempster-Shafer structures are shown in Table 5. The estimations of the lower and upper probabilities using discretization of the basic variables requires the calculation of the VC -volume of the corresponding B-box of every focal element. This induces in the computation additional overhead, since the copula must be evaluated (2N )d -times (N d times for each focal element and 2d times for each evaluation of VC , using equation (1), in every focal element). This operation becomes specially slow when either the size of the discretization N or the dimension d or both increase and is extremely slow when the copula does not have an explicit formula like in the case of the Gaussian or the Student’s t copulas (however in this particular case, it is easy to simulate samples from those copulas). For example, in the case of the evaluation of Table 5 using the Clayton copula, we employed 29 seconds, 21 minutes and 10.3 hours for discretization sizes N of 2, 3 and 4 respectively using a personal computer with an AMD Processor of 2.4 GHz of velocity and 512 Mb of RAM running Fedora Core 4 Linux, C++ and the GNU Scientific Library. The estimated time for a discretization of size N = 5 was approximately 6 days. However, we calculated this number employing parallel computing using the LAM/MPI library and a cluster of 8 machines). Note that five is a relatively low number of discretizations to represent adequately a basic variable, compared for example with the one hundred discretizations proposed for example by Ferson et al. [12, p. 18] in the definition of the canonical Dempster-Shafer structure employed in the description of probability boxes. 22

Finally, analyzing the results in Table 5 one can observe that the outer discretization approach gives more conservative results than the intermediate discretization method; however, in this particular example, the results obtained with the intermediate discretization strategy were a little bit more accurate. 6.3 The Monte Carlo sampling method The probabilities (29) and (30) were also estimated using simple Monte Carlo sampling techniques, as shown in Table 6. Simple methods to simulate from the Clayton and the Gumbel copulas are provided in McNeil et al. [24]. In contrast to the discretization method, the evaluation of Table 6 using 5000000 Monte Carlo samples and the personal computer last just 20 seconds, several orders of magnitude faster than the discretization approach. As already commented, these estimator are unbiased; note for example that even with 50000 Monte Carlo samples, the evaluation is more accurate using copulas C2 and C3 than the evaluation using approximately 244.1 million of focal elements (N = 5) with the discretization method. In addition, Table 6 shows that the accuracy of the Monte Carlo simulation approach depends on the number of simulations and the exact value of the lower and upper probabilities. In a further step this value can be employed in equation (24) to estimate probabilistic bounds (in comparison to the hard bounds of the outer discretization method, which can be estimated at the expense of an increasingly high computational cost) on the value of the lower and upper probabilities. These results discourage the use of the discretization of the basic variables specially in high dimensional problems and suggest instead to use the Monte Carlo approximation methods, not only because the latter is much faster but also more accurate than the former technique. In higher dimensions even more striking differences are to be expected. In low dimensions (d ≤ 4), the discretization method can be efficient as well.

7

Conclusions

In this paper we characterized the different possible types of discretizations of basic variables, concluding that the outer discretization is the most conservative of the types of discretizations for a basic variable, in view of the fact that it does not entail loss of information. Also it was shown that the classical discretization of infinite random sets is 23

equivalent to an approximation by Riemann-Stieltjes sums of the integrals (11) and (12). In addition, it was shown that more efficient algorithms like Monte Carlo methods exist for the evaluation of the lower and upper probability measures. This is a strong result, that would discourage in the future the use of the classical discretization of the basic variables employed for example in Demspter-Shafer evidence theory, and that suggest the use of infinite random sets of indexable type as a method for a faster evaluation of the lower and upper probability of events, specially in high dimensional problems. In low dimensions (d ≤ 4), the discretization method can be efficient as well. Monte Carlo simulation is the most basic algorithm for integration of functions in high dimensions. This is a current topic of research. We may benefit of any additional development in the field of high dimensional integration that outperforms Monte Carlo simulation. For example, it is known that variance reduction techniques, like stratified or importance sampling, or even adaptive Monte Carlo methods like the VEGAS algorithm, outperform simple Monte Carlo integration of Riemann integrals. It is left as an open problem to include, in the introduced methodology, fuzzy random variables. The reader is referred to Baudrit et al. [4] for some ideas in this direction.

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. I would also like to thank Mr. Marcos Valdebenito for allowing me to run the simulations in the cluster of his Institute.

References [1] D. A. Alvarez. On the calculation of the bounds of probability of events using infinite random sets. International Journal of Approximate Reasoning, 43(3):241–267, December 2006. [2] D. A. Alvarez. Infinite random sets and applications in uncertainty analysis. PhD thesis, Leopold-Franzens Universit¨at Innsbruck, Innsbruck, Austria, April 2007. [3] C. Baudrit, D. Dubois, and D. Guyonnet. Joint propagation and exploitation of probabilistic and possibilistic information in risk assessment. IEEE Transactions on Fuzzy Systems, 14(5):593–608, October 2006. 24

[4] C. Baudrit, D. Dubois, and N. Perrot. Representing parametric probabilistic models tainted with imprecision. Fuzzy Sets and Systems, 159 (15):1913–1928, August 2008. [5] C. Bertoluzza, M. A. Gil, and D. A. Ralescu, editors. Statistical modeling, analysis and management of fuzzy data, volume 87 of Studies in fuzziness and soft computing. Physica Verlag, Heidelberg; New York, 2002. [6] G. De Cooman, M. Troffaes, and E. Miranda. n-monotone lower previsions and lower integrals. In Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, ISIPTA’05, Pittsburgh, Pennsylvania, 2005. Available at http://www.sipta.org/ isipta05/proceedings/017.html. [7] A. P. Dempster. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics, 38:325–339, 1967. [8] D. Dubois and H. Prade. The mean value of a fuzzy number. Fuzzy Sets and Systems, 24(3):279–300, 1987. [9] D. Dubois and H. Prade. Random sets and fuzzy interval analysis. Fuzzy Sets and Systems, 42(1):87–101, July 1991. [10] D. Dubois and H. Prade. Possibility Theory. Plenum Press, New York, 1988. [11] P. Embrechts, A. McNeil, and D. Straumann. Correlation and dependence in risk management: properties and pitfalls. In M. Dempster, editor, Risk Management: Value at Risk and Beyond, pages 176–223. Cambridge University Press, Cambridge, UK, 2002. [12] S. Ferson, V. Kreinovich, L. Ginzburg, D. S. Myers, and K. Sentz. Constructing probability boxes and Dempster-Shafer structures. Report SAND2002-4015, Sandia National Laboratories, Albuquerque, NM, January 2003. Available at http://www.ramas.com/unabridged.zip. [13] S. Ferson, R. B. Nelsen, J. Hajagos, D. J. Berleant, J. Zhang, W. T. Tucker, L. R. Ginzburg, and W. L. Oberkampf. Dependence in probabilistic modelling, Dempster-Shafer theory and probability bounds analysis. Report SAND2004-3072, Sandia National Laboratories, Albuquerque, NM, October 2004. Available at http://www.ramas.com/depend.zip. [14] G. S. Fishman. Monte Carlo - Concepts, Algorithms and Applications. Springer-Verlag, New York, 1996. [15] I. Goodman. Fuzzy sets as equivalent classes of random sets. In R. Yager, editor, Fuzzy Sets and Possibility Theory: recent developments, pages 327– 343. Pergamon, New York, 1982. [16] J. W. Hall and J. Lawry. Generation, combination and extension of random set approximations to coherent lower and upper probabilities. Reliability Engineering and System Safety, 85(1–3):89–101, 2004. [17] C. J. Himmelberg. Measurable relations. Fundamenta Mathematicae, 87: 53–72, 1975. [18] C. Joslyn and J. M. Booker. Generalized information theory for engineering modeling and simulation. In E. Nikolaidis and D. Ghiocel, editors, Engineering Design Reliability Handbook, pages 9:1–40. CRC Press, 2004. 25

[19] G. J. Klir. Uncertainty and Information : Foundations of Generalized Information Theory. John Wiley and Sons, New Jersey, 2006. [20] G. J. Klir and M. J. Wierman. Uncertainty-Based Information: Elements of Generalized Information Theory, volume 15 of Studies in Fuzziness and Soft Computing). Physica-Verlag, Heidelberg, Germany, 1998. [21] A. N. Kolmogorov and S. V. Fomin. Introductory Real Analysis. Dover Publications, New York, 1970. ISBN O-486-61226-0. [22] A. R. Krommer and C. W. Ueberhuber. Computational integration. SIAM, Philadelphia, 1998. [23] M. Lo`eve. Probability theory I. Springer Verlag, Berlin, 4th edition, 1977. [24] A. J. McNeil, R. Frey, and P. Embrechts. Quantitative risk management: concepts, techniques and tools. Princeton Series in Finance. Princeton University Press, Princeton, NJ, 2005. [25] E. Miranda, I. Couso, and P. Gil. A random set characterisation of possibility measures. Information Sciences, 168(1–4):51–75, 2004. [26] I. S. Molchanov. Theory of Random Sets. Springer, 2005. [27] R. B. Nelsen. An Introduction to Copulas, volume 139 of Lectures Notes in Statistics. Springer Verlag, New York, 1999. [28] H. Nguyen. On random sets and belief functions. Journal of Mathematical Analysis and Applications, 1978. [29] H. T. Nguyen. An introduction to random sets. Chapman and Hall CRC, Boca Raton, FL, 2006. [30] H. Niederreiter. Random number generation and quasi-Monte Carlo methods. Number 63 in CBMS-NSF regional conference series in applied mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pensilvania, 1992. [31] W. L. Oberkampf, J. C. Helton, and K. Sentz. Mathematical representation of uncertainty. In Non-deterministic approached Forum 2001, AIAA-2001-1645, Seattle, WA, 2001. American Institute of Aeronautics and Astronautics. April 16-19. [32] R. Rubinstein and D. Kroese. Simulation and the Monte Carlo Method. John Wiley & Sons, New York, 2 edition, 2007. [33] K. Sentz and S. Ferson. Combination of evidence in Dempster-Shafer theory. Report SAND2002-0835, Sandia National Laboratories, Albuquerque, NM, 2002. [34] A. Sklar. Fonctions de r´epartition a` n dimensions et leurs marges. Publ. Inst. Statis. Univ. Paris, 8:229–231, 1959. [35] A. Sklar. Random variables, distribution functions, and copulas - a personal look backward and forward. In L. R¨ uschendorf, B. Schweizer, and M. Taylor, editors, Distributions with fixed marginals and related topics, pages 1–14. Institute of Mathematical Statistics, Hayward, CA, 1996. [36] F. Tonon. Using random set theory to propagate epistemic uncertainty through a mechanical system. Reliability Engineering and System Safety, 85(1–3):169–181, 2004. [37] F. Tonon. Some properties of a random set approximation to upper 26

and lower distribution functions. International Journal of Approximate Reasoning, 48(1):174–184, April 2007. [38] P. Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991. [39] N. Wilson. Algorithms for Dempster-Shafer theory. In J. Kohlas and S. Moral, editors, Handbook of Defeasible Reasoning and Uncertainty Management Systems, volume 5: Algorithms for Uncertainty and Defeasible Reasoning, pages 421–476. Kluwer, Dordrecht, 2000. Available at http://4c.ucc.ie/web/upload/publications/inBook/ Handbook00wilson.pdf. [40] O. Wolkenhauer. Data Engineering. John Wiley and Sons, New York, 2001.

27

Outer discretization N 2

3

4

5

1, 2, 3

4, 5, 6

7, 8, 9

10, 11, 12

[-0.2500, 1.0000]

[ 1.0000, 1.3000]

[ 0.2000, 1.0000]

[ 0.1250, 0.8750]

[-1.0000, 0.2500]

[ 0.7000, 1.0000]

[ 0.0000, 1.2000]

[-0.2500, 1.2500]

[ 0.0000, 1.0000]

[ 1.0551, 1.3000]

[ 0.2667, 0.9333]

[ 0.2500, 0.7500]

[-0.5000, 0.5000]

[ 0.9449, 1.0551]

[ 0.1333, 1.0667]

[ 0.0000, 1.0000]

[-1.0000, 0.0000]

[ 0.7000, 0.9449]

[ 0.0000, 1.2000]

[-0.2500, 1.2500]

[ 0.1250, 1.0000]

[ 1.0879, 1.3000]

[ 0.3000, 0.9000]

[ 0.3125, 0.6875]

[-0.2500, 0.6250]

[ 1.0000, 1.0879]

[ 0.2000, 1.0000]

[ 0.1250, 0.8750]

[-0.6250, 0.2500]

[ 0.9121, 1.0000]

[ 0.1000, 1.1000]

[-0.0625, 1.0625]

[-1.0000, -0.1250]

[ 0.7000, 0.9121]

[ 0.0000, 1.2000]

[-0.2500, 1.2500]

[ 0.2000, 1.0000]

[ 1.1103, 1.3000]

[ 0.3200, 0.8800]

[ 0.3500, 0.6500]

[-0.1000, 0.7000]

[ 1.0317, 1.1103]

[ 0.2400, 0.9600]

[ 0.2000, 0.8000]

[-0.4000, 0.4000]

[ 0.9683, 1.0317]

[ 0.1600, 1.0400]

[ 0.0500, 0.9500]

[-0.7000, 0.1000]

[ 0.8897, 0.9683]

[ 0.0800, 1.1200]

[-0.1000, 1.1000]

[-1.0000, -0.2000]

[ 0.7000, 0.8897]

[ 0.0000, 1.2000]

[-0.2500, 1.2500]

Intermediate discretization N 2

3

4

5

1, 2, 3

4, 5, 6

7, 8, 9

10, 11, 12

[ 0.1250, 0.6250]

[ 1.0000, 1.3000]

[ 0.3000, 0.9000]

[ 0.3125, 0.6875]

[-0.6250, -0.1250]

[ 0.7000, 1.0000]

[ 0.1000, 1.1000]

[-0.0625, 1.0625]

[ 0.2500, 0.7500]

[ 1.0551, 1.3000]

[ 0.3333, 0.8667]

[ 0.3750, 0.6250]

[-0.2500, 0.2500]

[ 0.9449, 1.0551]

[ 0.2000, 1.0000]

[ 0.1250, 0.8750]

[-0.7500, -0.2500]

[ 0.7000, 0.9449]

[ 0.0667, 1.1333]

[-0.1250, 1.1250]

[ 0.3125, 0.8125]

[ 1.0879, 1.3000]

[ 0.3500, 0.8500]

[ 0.4062, 0.5938]

[-0.0625, 0.4375]

[ 1.0000, 1.0879]

[ 0.2500, 0.9500]

[ 0.2188, 0.7812]

[-0.4375, 0.0625]

[ 0.9121, 1.0000]

[ 0.1500, 1.0500]

[ 0.0312, 0.9688]

[-0.8125, -0.3125]

[ 0.7000, 0.9121]

[ 0.0500, 1.1500]

[-0.1562, 1.1562]

[ 0.3500, 0.8500]

[ 1.1103, 1.3000]

[ 0.3600, 0.8400]

[ 0.4250, 0.5750]

[ 0.0500, 0.5500]

[ 1.0317, 1.1103]

[ 0.2800, 0.9200]

[ 0.2750, 0.7250]

[-0.2500, 0.2500]

[ 0.9683, 1.0317]

[ 0.2000, 1.0000]

[ 0.1250, 0.8750]

[-0.5500, -0.0500]

[ 0.8897, 0.9683]

[ 0.1200, 1.0800]

[-0.0250, 1.0250]

[-0.8500, -0.3500]

[ 0.7000, 0.8897]

[ 0.0400, 1.1600]

[-0.1750, 1.1750]

28

Table 4 Outer and intermediate discretization of the basic variables considered in Table 2. Here N represents the number of discretizations of every basic variable. Note that in the case of the intermediate discretization of variables 4, 5 and 6, described by a triangular CDF, an outer discretization was employed.

Copula

C1

C2

C3

n

Intermediate discr.

Outer discretization

ˆ (F ,m) (F ) LP n

ˆ (F ,m) (F ) UP n

ˆ (F ,m) (F ) LP n

ˆ (F ,m) (F ) UP n

212

2.4414e-3

0.125

0

1

312

1.2042e-3

0.0877

1.5053e-5

0.2962

412

1.0299e-3

0.1779

3.0517e-5

0.1779

512

1.9110e-3

0.0466

7.0778e-6

0.1105

212

3.4965e-4

0.25

0

1

312

2.3480e-5

0.0772

1.8482e-6

0.4

412

1.0326e-4

0.1841

4.2080e-5

0.1841

512

8.8183e-5

0.0333

2.3079e-6

0.110823

212

1.6692e-4

0.3246

0

1

312

2.9413e-6

0.0948

4.9725e-8

0.4955

412

1.0069e-5

0.2307

3.9306e-6

0.2307

512

2.1120e-5

0.0357

3.3363e-8

0.1420

Table 5 Estimations of the lower and upper probabilities using discretization of the basic variables, in the numerical example of Section 6. Take into consideration that 2 12 = 4096, 312 = 531441, 412 = 16777216, 512 = 244140625.

29

Copula

C1

C2

C3

n

ˆ (F ,P ) (F ) LP Γ

ˆ (F ,P ) (F ) UP Γ

5000

0

0.0414000

50000

0.0001200

0.0385600

500000

0.0001920

0.0373080

5000000

0.0001766

0.0370564

5000

0.0002000

0.0260000

50000

0.0001400

0.0247400

500000

0.0001680

0.0250780

5000000

0.0002024

0.0249836

5000

0

0.0278000

50000

0.0000400

0.0267000

500000

0.0000460

0.0273320

5000000

0.0000462

0.0274142

Table 6 Estimations of the lower and upper probabilities using simple Monte Carlo simulations, in the numerical example of Section 6. Here n represents the number of simulations employed. Probabilistic intervals on the error of these estimators can be calculated by means of equations (28) and (24).

30