on the use of infinite random sets for bounding the probability of failure ...

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et.al. (eds.) Lisbon, Portugal, 5–8 June 2006

ON THE USE OF INFINITE RANDOM SETS FOR BOUNDING THE PROBABILITY OF FAILURE IN THE CASE OF PARAMETER UNCERTAINTY Diego A. Alvarez Institut f¨ur Technische Mathematik, Geometrie und Bauinformatik, Leopold-Franzens Universit¨at, Technikerstrasse 13 A-6020 Innsbruck, Austria, EU email: [email protected]

Keywords: probability of failure, random sets, Dempster-Shafer evidence theory, parameter uncertainty, epistemic uncertainty, aleatory uncertainty Abstract. This contribution presents a novel technique for estimating the bounds of the probability of failure of structural systems when there is aleatory and epistemic uncertainty in the representation of the basic variables. The proposed methodology allows the designer to model parameter uncertainty without making further suppositions that would be reflected in the estimated value of the probability of failure; since the method takes in consideration all possible variation due to uncertainty in the representation of the basic variables, it gives as an answer upper and lower bounds on the probability of failure. In particular, the methodology allows to represent parameter uncertainty as a possibility distribution, cumulative distribution function, probability box or family of intervals provided by experts. These four representations are special cases of the theory of random sets, which is a generalization of the theories of probability, possibility and interval analysis. Up to the best of the author’s knowledge, until now, all the papers that employ random sets in the case of uncertainty analysis have been confined to a finite random set representation, or to its analog a Dempster-Shafer body of evidence. This implies that the information provided by the experts must be discretized. In this paper, an infinite random set representation is introduced. With this new approach, the information provided does not have to be discretized. Also, a new geometrical representation of the space of basic variables is given, where the methods already existing for estimating the probability of failure and that only require the sign of the evaluations of the basic variables vector on the limit state function may work without additional overhead. Using this kind of methods, the computational cost required for the estimation of the bounds of the probability of failure could decrease notably compared with the discrete approach. Furthermore, the proposed method allows the analyst to model the lack of information about the dependence of the basic variables, and in consequence, provides a new methodology to avoid the misused assumption of independence between the basic variables and the myth that varying the correlation coefficients constitutes a sensitivity analysis for uncertainty about dependence. A benchmark example is used to demonstrate the usefulness of the method.

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Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

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Introduction

The fundamental problem in reliability analysis of structures (see e.g. [1, 2]) is the evaluation of the probability of failure: Z Pf = PX (F ) = fX˜ (x)dx (1) F

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where x ⊆ X ⊆ n is the vector of basic variables which represents the loads, material and geometric properties of the structure, F = {x : g(x) ≤ 0, x ∈ X} is the failure region, g : X → is the so-called limit state function, which determines if a condition x is safe (g(x) > 0) or unsafe (g(x) ≤ 0) and fX˜ is the joint probability density function (PDF) of the vector of basic variables1 . The evaluation of the probability of failure through (1) suffers the problem that in any given application the available information is usually incomplete and insufficient to define accurately the limit state function (model uncertainty) and the joint PDF fX˜ (parameter uncertainty) and in consequence, many of the available methods for the estimation of the probability of failure may lose applicability in real situations [3, 4]. For instance, in geotechnics, the PDFs associated to the soil parameters cannot be estimated accurately because of the limited sampling, the discrepancy between different methods of laboratory, the uncertainties in soil models, among other reasons. In addition, sometimes information cannot be expressed in a probabilistic fashion, but in terms of intervals (for example in engineering manuals) or linguistic terms (like the ones expressed by an expert). Oberguggenberger and Fellin [5; 6] showed for example, that the probability of failure of a shallow foundation may range even by orders of magnitude (they were seen to fluctuate between 10−11 and 10−3 ) when different likely PDFs related to the soil’s friction angle, and estimated using a relatively large number of tests, were considered. In this paper, we would like to show that random set (RS) theory and its close analogue, evidence theory could be a useful tool to model parameter uncertainty. These theories are generalizations of the theories of probability, possibility and interval analysis. RS theory appeared in the context of stochastic geometry theory thanks to the independent works of Kendall [7] and Matheron [8], while Dempster [9], and Shafer [10] developed what is known today as evidence theory. Those methodologies allow to represent the basic variables as possibility distributions, cumulative distribution functions (CDFs), probability boxes or family of intervals provided by experts; also, they allow to take into account the lack of knowledge about the dependence between the basic variables, whatever its nature. As a result of the analysis they allow the designer to model the most pessimistic and optimistic assumptions that can be expected. That is, RS and evidence theories are useful because they hinder the engineer to make suppositions that will be reflected later in the solution to the problem. Sentz and Ferson [11] provides an state-of-the-art review on the application of DempsterShafer evidence theory in several and diverging fields of research like cartography, classification, decision making, failure diagnosis, robotics, signal processing and risk and reliability analysis. In addition, several authors have already applied RS and evidence theories in civil and structural engineering; among their publications we have [5, 6, 12–33]. In this paper, a novel procedure for estimating bounds on the probability of failure is proposed, based on the extension of the theory of finite random sets to its infinite counterpart, carried on recently by Alvarez [34]. The main idea of this method is to represent basic variables as infinite RSs. Several interesting relations between the theories of random sets and

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Remark: along this document the tilde˜will be employed to denote random variables.

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Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

probabilistic-based reliability analysis will be discussed, which will provide a generalization of equation (1) and solving in this way the problem of parameter uncertainty. The paper begins in section 2 with a brief introduction to the mathematical modelling of the uncertainty, and is followed in Section 3 by a succinct presentation of RS and evidence theories and their relationship with other methods for the assessment of the uncertainty like probability boxes, interval analysis and possibility distributions. Sections 5 and 6 present in detail the proposed procedure, which will be tested, which together with the analysis of results, are presented in Section 7. The document finishes in Section 8 with some conclusions, final remarks and open problems. 2

Parameter uncertainty

Helton [35] arranged uncertainty into two classes: aleatory and epistemic. On the one side, aleatory uncertainty (also called random, irreducible or inherent uncertainty) is related to the natural variability of the variables involved. Probability theory has shown to be adequate to manage this class of uncertainty, using for every random basic variable a PDF. Here every observation has to be punctual and no imprecision is permitted. This information may be difficult to obtain because small quantities of the probability of failure are associated to extreme observations of the input data which are also scarce, in consequence usually observations are not being taken in these extreme regions, hence there may be difficulties estimating the tails of the associated PDFs. On the other side, epistemic uncertainty results due to insufficient information, therefore it can be reduced when new data is available. Possibility, evidence, interval analysis and RS theories have shown to be appropriate to deal with this type of subjective uncertainty and in contraposition to aleatory uncertainty, here the information is denoted by means of intervals and linguistic terms. Given the large variety of theories to treat uncertainty, when epistemic and aleatory uncertainties are present at the same time, sometimes researchers treat them separately by trying to convert one type of uncertainty into another. RS and evidence theory are well suited to manage epistemic and aleatory uncertainties in concert without making any kind of suppositions, because it is a general framework that includes interval analysis, probability and possibility theory and probability boxes as particular cases. Therefore, they are perfect tools to deal with statistically imprecise data and to highlight links between those theories. In the following a brief introduction to these theories is presented. 3

Evidence and random set theory

Due to space constraints, this is a minimal introduction to Dempster-Shafer evidence (see e.g. [11, 36–39]) and random set (see e.g. [40, 41]) theories. For an complete overview to this topics, the reader is referred to the quoted references. 3.1

Dempster-Shafer evidence theory

Let X be an universal non-empty set, let P(X) be its power set. Let F := {Ai : Ai ∈ P(X) \ ∅, Ai 6= Aj , i, j = 1, . . . , n, i 6= j} be a collection of distinct non-empty subsets of X and P m : F → [0, 1] be a mapping called the basic mass assignment such that m(∅) = 0 and Ai ∈F m(Ai ) = 1. A pair (F , m) is called a finite body of evidence on X. Every set Ai ∈ F has an associated m(Ai ) > 0 and is called focal element. The collection F of all focal elements is referred to as focal set. Each focal set contains sets of possible values of the variable x ∈ X, and m(Ai ) expresses the probability of Ai to be the actual range of x. The following properties are true for the basic mass assignment: a) it is not required that m(X) = 1; b) B ⊆ A does not 3

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

imply that m(B) ≤ m(A); c) it is not required m(A) + m(Ac ) = 1 to hold; d) it is not required m(A) + m(B) = m(A ∪ B) + m(A ∩ B) to hold, for A, B, A ∪ B, A ∩ B ∈ F ; e) if m(X) = 1, this implies that F = {X}; f) if there is a unique focal element A ⊂ X such that m(A) = 1, then m(X) = 0; g) if n > 1 then m(X) < 1; h) if the focal set F is specific , i.e. if |Ai | = 1, for i = 1, . . . , n, then m is a probability mass function. Belief and Plausibility measures Due to our scarcity of information, the probability that an element x belongs to a set F ∈ P(X), cannot be expressed by a unique value of probability, but is contained in the interval   P (F ) ∈ Bel(F ,m) (F ), Pl(F ,m) (F ) (2) where belief and plausibility measures, Bel and Pl, are given respectively by, X X Bel(F ,m) (F ) := m(Ai ) Pl(F ,m) (F ) := m(Ai ) Ai :Ai ⊆F

(3)

Ai :Ai ∩F 6=∅

for Ai ∈ F ; The strict equality on (2) occurs when F is a specific focal set because in this case m is a probability mass function. It can be shown that belief and plausibility are dual fuzzy measures, (see e.g. [42]), that is Bel(F ,m) (F ) = 1 − Pl(F ,m) (F c ) and Pl(F ,m) (F ) = 1 − Bel(F ,m) (F c ). 3.2

Random set theory

Let (Ω, σΩ , PΩ ) be a probability space and for X ⊆ P(X), let (X , σX ) be a measurable ˜ is a (σΩ − σX )-measurable mapping Γ ˜ : Ω → X . In an analogous way to the space. A RS Γ definition of a random variable, this mapping can be used to generate a probability measure ˜ −1 on (Γ, σΓ ) given Ω ◦ Γ . This means that an event R ∈ σX has the probability n by PΓ = Po ˜ ∈ R . In other words, a RS is a set-valued random variable. PΓ (R) = PΩ ω : Γ(ω) ˜ ˜ When every element of the RS is a point, Γ(ω) becomes the random variable X(ω), and the n value of the probability of occurrence of the event F can be exactly captured by PX (F ) = o ˜ PΩ ω : X(ω) ∈ F for any F ∈ σX . In the case of random sets, it is not possible to know the exact probability of occurrence of a measurable set F but there is an upper and lower bounded like in equation (2). Dempster [9] defined those upper and lower probabilities by the formulas n o ˜ (4) Pl(F ) = PΩ ω : Γ(ω) ∩ F 6= ∅ = PΓ {γ : γ ∩ F 6= ∅} n o ˜ Bel(F ) = PΩ ω : Γ(ω) ⊆ F 6= ∅, Γ(ω) 6= ∅ = PΓ {γ : γ ⊆ F, γ 6= ∅} .

3.3

(5) (6) (7)

What is the relationship between Dempster-Shafer bodies of evidence and random sets?

When the cardinality of the body of evidence is finite, random sets result to be isomorphic to Dempster-Shafer bodies of evidence. That is, given a body of evidence (F , m) and a RS ˜ : Ω → X , then the following relationships are true: F ≡ X = {A1 , A2 , . . . , An } and Γ m(Ai ) ≡ PΓ (Ai ). Thus, making some abuse of notation, (F , m) ≡ (X , PΓ ) or simply (F , PΓ ). 4

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

REMARK: Along this document, we will generally refer to random sets, and will be denoted in the infinite case as (F , PΓ ) and in the finite case either as (F , m) or as (Fn , m) when emphasis on the cardinality of F is required. In some cases the superindex i will be used to represent the position of a marginal RS in a random relation. In the finite RS representation, the focal elements will be written as Ai for some i = 1, . . . , n, while in the infinite case, for consistency with the previous notation, the focal sets should be written as Aγ , however, since they are completely identified by its subindex, they will be simply symbolized as γ, except when the RS represents a possibility distribution, in which case the notation will be Aα , to agree with the established notation of α-cut. 3.4

Random relations

A random relation on X := ×ni=1 Xi is a RS (R, ρ) on the cartesian  i product X given by i i i the combination of the marginal random sets (F , m ), where F = Aji , ji = 1,
. . . , ni , for i = 1, . . . , n as, (F , m) := Aj1 ,...,jn := ×ni=1 Aiji , mj1 ,...,jn := f (m1 , . . . , mn ) . The function f contains the dependence information between the marginal RSs: for example, when the marginal RSs are independent [43]) then the basic mass assignment in F can be Qn (seei e.g. i obtained as m(Aj1 ,...,jn ) = i=1 m (Aji ), for all Aj1 ,...,jn ∈ F . 3.5

Extension principle

Given a function g : X → Y and a RS (F , m) the image RS of (F , m) through g, i.e. (R, ρ) is defined by the Papplication of the extension principle [36], that is, R := {Rj := g(Ai ) : Ai ∈ F } and ρ(Rj ) := Ai :Rj =g(Ai ) m(Ai ). When Y ⊆ , the computation of the image of the focal elements through g is usually made by one of the following techniques: the optimization method, the sampling methods (see e.g. [44]), the vertex method [45] and the function approximation method [26, 46].

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Relationship between infinite random sets and CDFs, possibility distributions, probability boxes and random intervals

Random sets can be understood as a generalization of probability, possibility, interval analysis and probability boxes theories. In this section, following Alvarez [34], these relationships will be clarified, in the frame of infinite random sets. In particular, to every focal element of an infinite RS defined on the real line (and represented by one of the four ways above listed), one can associate a unique number α ∈ (0, 1] that represents exclusively that focal element, and that induces an ordering relation in the RS. Note that [34] contains the proofs of all Lemmas and Theorems here postulated. 4.1

Relationship between random sets and possibility theory

The reader is referred to [47–49] for a complete introduction to fuzzy sets and possibility theory. A possibility distribution A of a set X is a normalized fuzzy set, that is a mapping A : X → [0, 1], where supx∈X A(x) = 1. In this case, A(x), for x ∈ X, represents the degree to which x is compatible with the concept represented by A. The α-cut of a membership function is defined as the crisp set Aα = {x ∈ X : A(x) ≥ α} for α ∈ (0, 1]. Two non-additive measures of importance in possibility theory are the possibility and the necessity. The possibility measure is a mapping PosA : X → [0, 1] given by PosA (K) = supx∈K {A(x)} while the necessity measure NecA : X → [0, 1] is defined by NecA (K) = 1 − PosA (K c ), where K ⊆ X. Let A be a possibility distribution of X, and let α ˜ : Ω → (0, 1] be a uniformly distributed ran5

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dom variable on some probability space (Ω, σΩ , P ), i.e., P {ω : α ˜ (ω) ≤ z} = z for z ∈ [0, 1]. ˜ A (ω) = {x ∈ X : A(x) ≥ α Then α ˜ induces a RS Γ ˜ (ω)}, which is simply the randomized α-cut set Aα(ω) [50]. That is, the possibility distribution A on X ⊆ can be expressed as an infinite ˜ RS (F , PΓ ) where F is the collection of all α-cuts Aα , i.e. F := {γ := Aα : α ∈ (0, 1]}. Let P be the probability measure on associated to the uniform CDF on (0, 1], Fα˜ , that is Fα˜ (α) = P (˜ α ≤ α) = α for α ∈ (0, 1], then the probability measure PΓ : σF → [0, 1] is induced as the Lebesgue integral Z PΓ ({Aα ∈ F : α ∈ G}) = dP (α) (8)

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R

G

= P (G)

(9)

where σF is a σ-algebra on F , G ⊆ (0, 1] contains the subindexes α of the focal elements which will be measured by PΓ and {Aα ∈ F : α ∈ G} is an element of σF . For every α sampled from Fα˜ , there corresponds a unique α-cut Aα and viceversa; that is, Aα and α are related one to one. This is the reason why the subindex α of Aα must be preserved inasmuch as there exist cases were different α-cuts contain the same family of elements, and therefore the subindex α distinguish them. Observe that in this case α induces an ordering in F such that, αi ≤ αj if Ai ⊇ Aj , that is, the associated RS is consonant, i.e. nested, in the sense that F is totally ordered by set inclusion and that the corresponding possibility distribution must be unimodal. Finally, it is necessary to quote the following Lemma: Lemma 4.1. Let A : X → [0, 1] be a normalized fuzzy set and (F , PΓ ) be its associated infinite RS defined on X ⊆ . The belief and plausibility of any subset F of X with regard to the RS (F , PΓ ) is equal to the necessity Nec and possibility Pos of the set F with respect to the normalized fuzzy set A, i.e., NecA (F ) = Bel(F ,PΓ ) (F ) and PosA (F ) = Pl(F ,PΓ ) (F ) for all F ⊆ X.

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4.2

Relationship between random sets and probability boxes 

A probability box or p-box (see e.g. [51, 52]) F , F is a class of cumulative distribution  functions (CDFs) F : F ≤ F ≤ F , F is a CDF bounded by upper and lower CDFs F and F : → [0, 1]. This class of CDFs jointly

represents the epistemic uncertainty about the CDF of a random variable. Given a p-box F , F such that F and F are piecewise continuous from the left, the quasi-inverses of F and F are given respectively by, F (−1) (α) := inf {x : F (x) ≥ α}  (−1) (α) := inf x : F (x) ≥ α for α ∈ (0, 1]. and F There is a close relationship between random sets and probability boxes. Every RS generates a unique probability box whose component CDFs are all those in agreement with the evidence. In turn, every probability box generates an equivalence class of random intervals consistent with it [53]. Alvarez [34] proposed the following representation of a probability box as an infinite RS: a probability box hF , F i on a subset of can be represented as an infinite RS (F , PΓ ) where i o nh (−1) (α), F (−1) (α) : α ∈ (0, 1] , (10) F = F

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R

α

and PΓ is defined in an similar way to the case of possibility distributions, by equation (8), i.e., Z PΓ ({γ ∈ F : α ∈ G}) = dP (α) = P (G) (11) G

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It must be observed that for every α sampled at random from Fα˜ , there corresponds a unique focal element [·, ·]α . This relationship is one to one if the subindex [·, ·]α holds. Observe that in this particular case, α induces a partial ordering in F such that if [a1 , b1 ]α1 and [a2 , b2 ]α2 are elements of F , then it follows that if α1 < α2 then a1 ≤ a2 and b1 ≤ b2 . Finally, it is necessary to quote the following Lemma: Lemma 4.2. Let hF , F i be a probability box and (F , PΓ ) be its associated infinite RS both defined on X ⊆ . The belief and plausibility of (−∞, x] with respect to (F , PΓ ) is equal to F (x) and F (x) respectively, that is, F (x) = Bel(F ,PΓ ) ((−∞, x]) and F (x) = Pl(F ,PΓ ) ((−∞, x]) for all x ∈ X.

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4.3

Relationship between finite random sets and families of intervals

In practice, sometimes experts provide finite families of intervals with a corresponding a priori information about the confidence of their opinions on those intervals; in addition, a histogram belongs to this category. This information is contained in a finite RS (Fs , m′ ). A single interval estimate can be regarded as a RS with a unique element A with m′ (A) = 1. When a set of s intervals is available, every interval is considered to be a focal element Ai with a corresponding m′ (Ai ) = 1/s. In the case there is evidence that supports the fact that the occurrence of an interval is more probable than another, then the corresponding basic assignment might be modified accordingly. In order to define an indexation by α, we have to induce in (Fs , m′ ) an ordering. If {[ai , bi ] for i = 1, . . . , s} are the enumeration of the focal elements of Fs , this family of intervals can be sorted by the criteria: [ai , bi ] ≤ [aj , bj ] if ai < aj or (ai = aj and bi ≤ bj ). This can be performed using a standard sorting algorithm by using the appropriate comparison function. 4.4

Relationship between finite random sets and cumulative distribution functions

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A basic variable can be expressed as a random variable on X ⊆ , with CDF FX˜ (x) = ˜ ≤ x) such that x ∈ X for some probability measure PΓ , and with quasi-inverse given PΓ (X (−1) by FX˜ . It is well known that if α ˜ is a uniformly distributed random variable on (0, 1], then (−1) ˜ := F ˜ is a random variable X (˜ α) is distributed according to FX˜ , or equivalently, if X ˜ X with CDF FX˜ then FX˜ (x) is a realization of a random variable uniformly distributed on (0, 1]. In consequence, this random variable can be represented by an infinite RS (F , PΓ ) with the n o (−1) specific focal set F = x := FX˜ (α) : α ∈ (0, 1] . Note that if αi has an associated xi for i = 1, 2 and if α1 < α2 , then x1 ≤ x2 . Note in addition that this is a particular case of the probability box when F = F . It is important to quote the following Lemma:

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Lemma 4.3. Let FX˜ be a CDF and (F , PΓ ) be its associated infinite RS defined of X ⊆ . Then both belief and plausibility of (−∞, x] with respect to (F , PΓ ) are equal to FX˜ (x), i.e., FX˜ (x) = Bel(F ,PΓ ) ((−∞, x]) = Pl(F ,PΓ ) ((−∞, x]), for all x ∈ X. 5

Sampling from an infinite random set

Given an infinite RS (F , PΓ ), one could be interested in sampling from it. This process is analogous to drawing a sample from a given CDF. Let (Fn , m) be a sample of (F , PΓ ) which contains n elements. In particular, Fn = {Ai , i = 1, . . . , n}. The belief and plausibility

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functions of the sample are given by the equations Bel(Fn ,m) (F ) =

n X

I [Ai ⊆ F ] m(γi )

Pl(Fn ,m) (F ) =

n X

I [Ai ∩ F 6= ∅] m(γi )

(12)

i=1

i=1

Since (Fn , m) is a random sample from (F , PΓ ), then m(γi ) = 1/|Fn | = 1/n for all i = 1, . . . , n, that is, m gives the same weight to every one of the samples. Now, rewriting (12), it follows, n

Bel(Fn ,m) (F ) =

1X I [Ai ⊆ F ] n i=1

n

Pl(Fn ,m) (F ) =

1X I [Ai ∩ F 6= ∅] n i=1

(13)

Alvarez [34] showed that Bel(Fn ,m) (F ) and Pl(Fn ,m) (F ) are unbiased estimators of Bel(F ,PΓ ) (F ) and Pl(F ,PΓ ) (F ) respectively, and also the following Theorem 5.1. The belief (plausibility) of the RS (Fn , m), defined as a sample from the infinite RS (F , PΓ ) on X, converges as n → ∞ almost surely to the belief (plausibility) of (F , PΓ ), i.e. Bel(F ,PΓ ) (F ) = lim Bel(Fn ,m) (F )

Pl(F ,PΓ ) (F ) = lim Pl(Fn ,m) (F )

n→∞

n→∞

(14)

for all F ∈ P(X). 5.1

Sampling from a basic variable

In this subsection, for the sake of simplicity in the notation, the subindex i corresponding to the i-th basic variable will be omitted (see Section 6.1). Sampling a focal element from a possibility distribution: sampling of a focal element from a basic variable represented by a possibility distribution consists in drawing a realization of α ˜ from Fα˜ and choosing the corresponding α-cut Aα (see Figure 1b). This is confirmed by the following Lemma 5.2. Let (F , PΓ ) be an infinite RS representing a possibility distribution A defined on X ⊆ and (Fn , m) be a finite sample of this RS with n elements. The belief and plausibility F ⊆ X with respect to (Fn , m) will converge almost surely to the necessity Nec and possibility Pos of the set F with respect to possibility distribution A, that is, NecA (F ) = limn→∞ Bel(Fn ,m) (F ) and PosA (F ) = limn→∞ Pl(Fn ,m) (F ) almost surely for all F ⊆ X.

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Sampling a focal element from a probability box: Alvarez [34] proposed the following method to sample from a probability box: followingh the inversion method i (see e.g. [54]), (−1) (−1) (α), F (α) is retrieved. This a sample α from Fα˜ is obtained and then the interval F α interval will be considered as the drawnhfocal element inasmuch as it contains the samples for all i  (−1) (α), F (−1) (α) = x : F (x) = α, F ∈ hF , F i the CDFs in the probability box, i.e., F α (see Figure 1a). This is confirmed by the following Lemma: Lemma 5.3. Let hF , F i be a probability box, (F , PΓ ) be its corresponding infinite RS defined on X ⊆ and (Fn , m) be a finite sample of this RS with n elements. In the limit, when

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1

A(x)

1 h (−1) i (−1) γ = F X (α), F X (α)

α

α

γ = Aα

α

F 0

1

F γ

A5 γ = Ai

α

A4 A3 A1

γ

X

γ

1

m(A5 )

A2 0

0

X

(−1)

α

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

γ = FX

0

X

γ

(α)

X

Figure 1: Sampling of focal elements. a) from a probability box. b) from a possibility distribution. c) from a family of intervals. d) from a CDF.

an infinite number of focal sets is sampled, the belief and plausibility of (−∞, x] with respect to the sampled RS will converge almost surely to F (α) and F (α) respectively, that is, F (x) = limn→∞ Bel(Fn ,m) ((−∞, x]) and F (x) = limn→∞ Pl(Fn ,m) ((−∞, x]) almost surely for all x ∈ X at which F and F are continuous, respectively.

Sampling a focal element from a finite family of intervals: After ordering of the family of focal sets, as explained in Section 4.3, one obtains the new subindexes i1 , i2 , . . . is . Similarly to the cases described before, a RS (Fs , m′ ) can be sampled (to form the RS (Fn , m)) by drawing an P α from Fα˜ , a uniformly P distributed CDF on (0, 1] and then selecting the j-th focal j j−1 ′ ′ m (A ) < α ≤ element if k k=1 m (Ak ) where j = i1 , i2 , . . . is and by convention k=1 P0 ′ k=1 m (Ak ) = 0. This kind of sampling is illustrated in Figure 1d. Also, the following Lemma shows the convergence of the method,

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Lemma 5.4. Let (Fs , m′ ) be an finite RS defined on X ⊆ and (Fn , m) be a finite sample of this RS with n elements. Then, the belief and plausibility with respect to the sampled RS (Fn , m) converges almost surely to the belief and plausibility with regard to the RS (Fs , m′ ) i.e., Bel(Fs ,m′ ) (F ) = limn→∞ Bel(Fn ,m) (F ) and Pl(Fs ,m′ ) (F ) = limn→∞ Pl(Fn ,m) (F ) for all F ⊆ X. Sampling a focal element from a CDF: As in the above cases, a focal element of a RS can be sampled by drawing an α from a uniform CDF on (0, 1], Fα˜ and selecting the associated focal element from F (see Figure 1c). The convergence of the method follows directly from Lemma 5.3.

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6

Simulation techniques applied to the evaluation of belief and plausibility measures of the failure region

In Alvarez [34], the extension of the theory of finite random sets to infinite random sets in risk analysis was proposed. The use of this technique gives new insights to the theory of probabilistic-based structural reliability analysis, allowing a new formulation of integral (1), and solving in this way the problem of parameter uncertainty; as a result, the method provides bounds for the probability of failure. In the following, the principal results developed in [34] will be briefly presented, and the application of this generalization will be carried out. Integral (1), defines the probability of failure of an structural system. Note that this integral can also be written as Z Z Pf = I[x ∈ F ]dFX˜ (x) = I[x ∈ F ]dPX˜ (x) (15) X

X

 ˜ ≤ x and I stands for the indicator function. It has been argued in the where FX˜ (x) = PX˜ X present document that the RS theory will allow to estimate, subject to the limitations of knowledge due to parameter uncertainty, the bounds on Pf , which are provided by the plausibility and belief of the failure region F ∈ P(X), that is using (2), 

Bel(F ,PΓ ) (F ) ≤ PX ˜ (F ) ≤ Pl(F ,PΓ ) (F )

(16)

where Bel(F ,PΓ ) (F ) = PΓ (γ ⊆ F : γ ∈ F ) PX ˜ (F )

= PX ˜ (x ∈ F : x ∈ X)

= =

Pl(F ,PΓ ) (F ) = PΓ (γ ∩ F 6= ∅ : γ ∈ F ) =

Z

ZF X

Z

I [γ ⊆ F ] dPΓ (γ),

(17)

I [x ∈ F ] dPX ˜ (x),

(18)

I [γ ∩ F 6= ∅] dPΓ (γ)

(19)

F

Here γ ∈ F and (F , PΓ ) contains the information about the basic variables. The evaluation of integrals (17) and (19) is not direct, so it is better to look for a different formulation for those integrals. If every RS could be represented as a single point, then the evaluation of (17) and (19) could be done by the standard methods used to solve (18). 6.1

Combination of focal elements

The samples γi drawn from each basic variable are combined to form the joint focal elements; they are given by ×ni=1 γi where each γi is either an interval or a point. Note that every γi has an associated αi , that is the realization of the uniform random variable on (0, 1] used to sample γi . Inasmuch as every sample of a basic variable can be symbolized by γi or by the associated αi , the joint focal element can be represented either by the (usually) hypercube ×ni=1 γi in the space X of basic variables or by the point [α1 , α2 , . . . , αn ] in the space α ≡ (0, 1]n . Note that the representation in the space α makes (F , PΓ ) a consistent RS, and therefore, the rules of the classical theory of probability are applicable. These two alternative representations of a joint focal element are sketched in Figure 2. In the space α there exists a joint CDF which contains the dependence information about the basic variables, that is, Fα˜ 1 ,...,α˜ n (α1 , . . . , αn ) = C(Fα˜ 1 (α1 ), . . . , Fα˜ n (αn )) 10

(20)

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

α2

X2

1 FBel g(X) = 0 (failure surface)

FPl

failure region

safe region

0 X1

Space X

0 Space α

1

α1

Figure 2: Alternative and equivalent representations of a focal element a) in space X b) in space α

where n is the number of basic variables in consideration, Fα˜ i (αi ) = αi is the uniform CDF defined on (0, 1] associated to the RS representation of the i-th basic variable for i = 1, . . . , n, and C is a function that joins the uniform marginal CDFs Fα˜ 1 , . . . Fα˜ n . Since they are uniform CDFs on the interval (0, 1], C is a copula [55]. Recall that a copula is a CDF on a unit cube (0, 1]n all whose marginal distribution are uniform on the interval (0, 1]. Observe that Fα˜ 1 ,...,α˜ n = C. For example, when Qnindependent, the product copula is used, Qn it is assumed that all basic variables are Fα˜ 1 ,...,α˜ n = i=1 Fα˜ i that is Fα˜ 1 ,...,α˜ n (α1 , . . . , αn ) = i=1 αi for αi ∈ (0, 1] and i = 1, . . . , n. To define the associated PΓ of the joint focal set F in the space X, we make use of the copula C on the space α. The associated PΓ : σF → [0, 1] is calculated as: Z PΓ ({γ ∈ F : γ := [α1 , . . . , αn ] ∈ G}) = dC (α1 , . . . , αn ) (21) G

n

where σF is a σ-algebra on F , G ⊆ (0, 1] contains the points [α1 , . . . , αn ] corresponding to the focal elements which will be evaluated in the integral and {γ ∈ F : γ := [α1 , . . . , αn ] ∈ G} ∈ σF . 6.2

An alternative representation of the belief and plausibility integrals

One must bear in mind that the failure surface {x ∈ X : g(x) = 0} divides X into two regions: the safe and the failure region. The evaluation of the probability of failure by means of equation (18) takes advantage of this fact, since the integral does not take into account the magnitude g(x), but only its sign. Remember also that our objective is to develop an efficient method for the evaluation of Bel(F ,PΓ ) (F ) and Pl(F ,PΓ ) (F ) according to equations (17) and (19) respectively. The solution of those integrals is not straightforward, however, using the proposed representation of the RS in space α, those integrals can be recast as the Stieltjes integrals, Z 1 Z 1 I [[α1 , . . . , αn ] ∈ FBel ] dC (α1 , . . . , αn ) (22) Bel(F ,PΓ ) (F ) = ··· 0+ 0+ | {z } Pl(F ,PΓ ) (F ) =

Z

|

n−times Z 1 1

0+

··· {z

n−times

I [[α1 , . . . , αn ] ∈ FPl ] dC (α1 , . . . , αn )

0+

} 11

(23)

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

The meaning of FBel and FPl is described in the following. Note that every element of F is a point in the space α, i.e. in (0, 1]n . In equation (17), I [γ ⊆ F ] takes the value 1 when the focal set is totally contained in the failure region F ; otherwise it takes 0. This is equivalent to say that there is a region of the space α called FBel which contains all points whose corresponding focal elements are completely contained in the failure region, that is, I [[α1 , . . . , αn ] ∈ FBel ] is equivalent to I [γ ⊆ F ] of equation (17). Analogous considerations apply to the evaluation of the plausibility by means of integral (19), and in consequence, I [[α1 , . . . , αn ] ∈ FPl ] ≡ I [γ ∩ F 6= ∅]. Since the set {γ ∈ F : γ ⊆ F } is contained in the set {γ ∈ F : γ ∩ F 6= ∅} it follows that FBel ⊆ FPl . Now, with regard to the evaluation of the belief, since in the space α all focal sets of F are symbolized by a point, and since there is a region FBel which can be understood as a failure region, many of the algorithms developed to evaluate (18), and which only consider the sign of g(x) (like for example importance sampling) may be used in the evaluation of Bel(F ,PΓ ) (F ) by means of (22). The same considerations apply to the evaluation of the plausibility Pl(F ,PΓ ) (F ) according to (23). This algorithm will select some key points in (0, 1]n which must be checked whether they belong to FBel , FPl or not. This is verified using one of the traditional methods like the vertex, sampling, optimization or response surface method, already mentioned in section 3.5. Observe that when all basic variables are random, the representation in both spaces X and α is equivalent up to a transformation given by the copula C and also FBel is equal to FPl . In this sense, the proposed method is a generalization of the traditional methodology employed to evaluate the probability of failure of a structural system by means of integral (1). 6.3

On the dependence between the basic variables

Note that in equations (22) and (23) both FBel and ⊆ FPl are not functions of the copula C that which constraints the dependence information between α1 , . . . , αn . With regard to C, we have three possible cases: a) we perfectly know it, b) we don’t have any information at all about the copula, or c) we have some information about it. In the former case, the calculation of (22) and (23) is straightforward. In the second case, the solution follows directly from the theory of imprecise probabilities and the theory of sets of probability measures (see e.g. [56]). The central idea of the theory of the sets of probability measures is to relax the precision inherent in a probabilistic model, and to replace a single probability measure by a set of probabilities measures P (also known as credal set), which is closed and convex. This credal set P defines the coherent upper and lower probabilities of the set F according to the formulas P (F ) = supP ∈P P (F ) and P (F ) = inf P ∈P P (F ) respectively. That is, this theory specifies an interval of probability [P (F ), P (F )] where P (F ) could lay. One important property of coherent upper and lower probabilities is that they are dual measures, i.e., P (F ) = 1 − P (F c ). Observe that since we are considering that there is no information available about C, then it is required to consider all the possible copulas and look for the ones that minimize and maximize the integrals (22) and (23) respectively, on the set of all copulas C. In consequence, following the same steps of the theory of sets of

12

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

probability measures, we obtain the following upper and lower probabilities of failure: Z 1 Z 1 Pf = sup ··· I [[α1 , . . . , αn ] ∈ FBel ] dC (α1 , . . . , αn ) C∈C 0+ 0+ | {z } Pf = inf

C∈C

Z

n−times Z 1 1

··· + | 0 {z

I [[α1 , . . . , αn ] ∈ FPl ] dC (α1 , . . . , αn )

(24)

(25)

0+

n−times

}

  In consequence, interval Pf , Pf is assured to contain the probability of failure no matter which type of dependence exists between the basic variables, without any assumption whatever about dependence and in addition, it will produce the tightest possible bounds in agreement with the available information. Notice that (24) and (25) are linear programming problems in infinite dimensions with infinite constraint. The analysis of those CDFs may give us intrinsic information about which is the most critical dependence condition between the implied basic variables. In the third case, any additional information about dependence can be taken into account including them as constraints to the optimization problems (24) and (25). 7

Example

To test the proposed approach, the “Challenge Problem B” of the benchmark proposed by Oberkampf et al. [57] was solved. For the sake of completeness, the formulation of this problem will be repeated here. Consider the linear mass-spring-damper system subjected to a forcing function Y cos(ωt) and depicted in Figure 3. The system has a mass m, a stiffness constant k, a damping constant c and the load has an oscillation frequency ω. The task for this problem is to estimate the uncertainty in the steady-state magnification factor DS , which is defined as the ratio of the amplitude of the steady-state response of the system to the static displacement of the system, i.e. k (26) DS = q (k − mω 2 )2 + (cω)2 To solve the problem we have to use exclusively the information provided, and we have to avoid any extra supposition on the data given. If it is so, they must be clearly specified. The objective of the benchmark is to develop techniques to solve this parameter-uncertainty-problem, using

x, x˙ c m

Y cos(ωt)

k

Figure 3: Mass-spring-damper system acted on by an excitation function.

13

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

the minimum number of suppositions, therefore there is not a unique answer to it, because different methods could give different results, depending on the assumptions used. The parameters m, k, c and ω are independent, that is, the knowledge about the value of one parameter implies nothing about the value of the other. The information for each parameter is as follows: • Parameter m. It is given by a triangular PDF defined on the interval [mmin , mmax ] = [10, 12] and with mode mmod = 11. • Parameter k. It is stated by three equally credible and independent sources of information. Sources agree on that k is given by a triangular PDF, however each of them gives a closed interval for the different parameters mmin , mmod and mmax , i.e.: – Source 1: mmin = [90, 100], mmod = [150, 160] and mmax = [200, 210]. – Source 2: mmin = [80, 110], mmod = [140, 170] and mmax = [200, 220]. – Source 3: mmin = [60, 120], mmod = [120, 180] and mmax = [190, 230]. • Parameter c. Three equally credible and independent sources of information are available. Each source provided an interval for c, as follows: – Source 1: m = [5, 10]. – Source 2: m = [15, 20]. – Source 3: m = [25, 25]. • Parameter ω. It is modelled by a triangular PDF defined on the interval [mmin , mmax ] and with mode mmod . The values of mmin , mmod and mmax are given respectively by the intervals [2, 2.3], [2.5, 2.7] and [3.0, 3.5]. Note that the external amplitude Y does not appear in equation (26). Some remarks are necessary on the implementation of the proposed approach. The parameter m was modelled simply as a CDF. For modelling k, the information provided by every source was represented by a probability box and in a further step, they were combined using the intersection rule for aggregation of probability boxes [52]. The parameter C was modelled as a finite RS with focal elements [5, 10], [15, 20] and [25, 25], every one of them with a probability mass assignment of 1/3. Finally, the parameter ω was modelled as a probability box. The image of all focal elements was calculated using the optimization method (see Section 3.5). In this case, only the belief and plausibility bounds for a given region F = [Ds , ∞) were computed. The values Ds = 2.0, 2.5 and 3.0 were chosen. In Table 1, the belief and plausibility bounds obtained by the proposed methodology are shown. In relation to the proposed algorithm, according to Section 6.2, the region FBel is contained in the region FPl . This is graphically shown in Figure 4. This confirms the relation shown in Figure 2. The result shows wide intervals containing the probability of failure. This should not be taken as an argument against random set theory though. What it is does show is the danger, even in simple problems, of assuming precise parameters in order to obtain a unique value of the probability of failure at the end. The width of those intervals can be reduced if additional information about the basic variables is obtained.

14

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

FPl 1

0.8

0.8

0.6

0.6

αω

αω

FBel 1

0.4

0.2

0

0.4

0.2

0

0.2

0.4

0.6

0.8

0

1

0

αk

0.2

0.4

0.6

0.8

1

αk

Figure 4: Regions FBel and FPl . These graphics, in the space α, were calculated from the example by means of 20000 Monte Carlo simulations, setting αm = 0.95 and αc = 0.20. In this case the failure region was defined by Ds = 2.8, and in consequence, Bel(F ) = 0.02015 and Pl(F ) = 0.45045.

8

Conclusions and final remarks

In this document a novel method was proposed for the calculation of the bounds of the probability of failure of structural systems when there is either epistemic or aleatory uncertainty in the definition of the basic variables. The proposed method allows the designer to model parameter uncertainty without making further suppositions that would be reflected in the final value of the probability of failure. The method allows to model the available information about the basic variables using probability boxes, possibility and probability distribution functions and families of intervals provided by experts. Since the method takes in consideration all possible variation due to the uncertainty in the representation of the basic variables employed in the calculation of the probability of failure, it gives an interval as an answer, not a unique value of Pf . In addition techniques for sampling from a probability box, random set and possibility distributions were proposed. In comparison with the discrete approach employed by other authors, the proposed method introduces a new geometrical interpretation of the space of basic variables, where the methods designed for calculating the probability of failure and that only require the sign of an evaluation of the limit state function may work without additional overhead. Using that kind of methods, Table 1: Belief and plausibility bounds of the region F = [Ds , ∞) obtained by applying the methodology of infinite random sets. The values shown were calculated using 100000 simulations (focal element evaluations) of a direct Monte Carlo simulation.

Ds 2.0 2.5 3.0

Direct MCS Bel Pl 0.0652 0.4252 0.0111 0.1809 0.0015 0.1001

15

Alvarez. On the use of infinite random sets for bounding the probability of failure . . .

the computational cost required for estimating the belief and plausibility of the failure region could decrease notably. In addition, the estimated bound does not depend anymore on the granularity of the discretization in the representation of the basic variables, in comparison to the finite RS approach. Further research is required in developing efficient methods for the calculations of equations (22), (23), (24) and (25). Also, further investigation about efficient methods for the application of the extension principle of random sets are required. 9

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 and the comments on the manuscript of Professors Michael Oberguggenberger and Thomas Fetz is gratefully acknowledged. References [1] O. Ditlevsen and H. O. Madsen. Structural Reliability Methods. John Wiley and Sons, New York, 1996. ISBN 0-471-96086-1. 384 p. [2] Robert E. Melchers. Structural Reliability Analysis and Prediction. John Wiley and Sons, New York, 2nd edition, April 1999. ISBN 0-471-98771-9. 456 p. [3] D. I. Blockley. The nature of structural design and safety. Ellis Horwood, Chichester, 1980. [4] D. I. Blockley. Risk based structural reliability methods in context. Structural safety, 21: 335–348, 1999. [5] M. Oberguggenberger and W. Fellin. From probability to fuzzy sets: the struggle for meaning in geotechnical risk assessment. In R. P¨otter, H. Klapperich, and H. F. Schweiger, editors, Probabilistics in geotechnics: technical and economic risk estimation, pages 29– 38, Essen, 2002. Verlag Gl¨uckauf GmbH. [6] M. Oberguggenberger and W. Fellin. The fuzziness and sensitivity of failure probabilities. In W. Fellin, H. Lessmann, M. Oberguggenberger, and R. Vieider, editors, Analyzing Uncertainty in Civil Engineering, pages 33–48, Berlin, 2004. Springer-Verlag. [7] D. G. Kendall. Foundations of a theory of random sets. In E. F. Harding and D. G. Kendall, editors, Stochastic geometry, pages 322–376, London, 1974. Wiley. [8] G. Matheron. Random sets and integral geometry. Wiley, New York, 1975. [9] Arthur P. Dempster. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics, 38:325–339, 1967. [10] Glenn Shafer. A mathematical theory of evidence. Princeton University Press, Princeton, NJ, 1976. [11] K. Sentz and S. Ferson. Combination of evidence in Dempster-Shafer theory. Report SAND2002-0835, Sandia National Laboratories, Albuquerque, NM, 2002.

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