Approximate Interval Method for Epistemic Uncertainty Propagation using Polynomial Chaos and Evidence Theory

A ppro ximate Interv al Method f or Epistemic Uncertainty Pr opagation using P olynomial Chaos and Evidence Theory Gabriel T erejanu, Punee t Singla, T arunraj Singh, Peter D. Scott Abstract — The paper builds upon a recent a pproach to find the app roxima te bounds of a rea l function u sing Polynomial Chaos expansions. Giv en a fu nction of random variables with compact support probability distributions, the intuition is to quantify the uncertainty in the r esponse using Polynomial Chaos expansion and discard all the inf ormation pro vided about the randomness of the out put and extract only the bound s of i ts compact support. T o solv e fo r the bou nding range of polynomials, we transfo rm th e Polynomial Chaos expansion in the Bernstein form, and use the range enclosure property of Bernstein polynomials to find the minimum and maximum value of the response. This procedure is used to propagate Dempster -Shafer structures on closed intervals through n on- linear functions and it is app lied on an algebraic challenge problem. I . I N T R O D U C T I O N In interval an alysis a fundamental problem is finding the interval bounds for the range of a real fu nction. When suc h a fu nction is m onoton e or it can be expressed in terms of arithmetic opera tions, then interval computations can be used to approx imate the b ound s of the response. Howe ver these bound s are gr oss overestimations due t o the depen dency and the wrapping ef fect [ 10]. Applications of interval methods can be f ound in estima- tion, optimiza tion techniques, robust co ntrol, robo tics and finance [15], [1 0], ju st to name a few . First, introdu ced by Moore [14], as a meth od to con trol f or nu merical erro rs in compu ters, the field of interval an alysis has ev olved with b etter app roxima tions to the range o f r eal fun ctions as presented in Ref.[19], [1 8]. In the present paper we are interested in u sing interval methods to prop agate epistemic uncertainty throug h nonlin - ear function s. In contrast to th e aleatory un certainty [26], [27], [11], d efined by variability w hich is irr educible, the epistemic uncertainty is deri ved f rom incomplete knowledge or ignor ance and can b e reduced with an increase in in- formation . D ue to their major differences, it is of great This work was supported under Contract No. HM1582-08-1-0012 from ONR. Gabriel T erejanu - Ph.D. Candidate , Department of Computer Science & Engineeri ng, Uni versit y at Buff alo, Buff alo, NY -14260, terejanu@buff alo.edu Puneet Singla - Assistant Professor , Department of Mechanica l & Aerospace Engineeri ng, Uni versi ty at Buffa lo, Buffa lo, NY -14260, psingla@buffa lo.edu T arunraj Singh - Professor , Departmen t of Mechani cal & Aerospace Engineering , Univ ersity at Buff alo, Buffa lo, NY -14260, tsingh@buffal o.edu Peter D. S cott - Associate Professor , Department of Computer Science & Engineeri ng, Uni versit y at Buff alo, Buff alo, NY -14260, peter@buffalo .edu importan ce that the two typ e o f uncertain ties to be m odeled and p ropaga ted separately [6]. Unlike the probability theory , wh ere the probability mass is assigned to singletons, in the Shafer ’ s theo ry of e viden ce [23], the prob ability mass is assigne d to sets, g iv en its power on modeling igno rance. In conjunction with the Dem pster’ s rule of combinatio ns [4] which is a generalizatio n of the of the Bayes’ ru le, the De mpster-Shafer (DS) th eory of evidence offers a powerful methodo logy for representin g and aggregating ep istemic u ncertainties. One can define Demp ster-Shafer structures on focal ele- ments that are closed interv als on the real line for example. Ferson [7] shows how this structures can be tran sformed into probability b ound s an d vice-versa b y discretization. T o propag ate these f ocal elemen ts throug h system functions, it inv olves findin g the solution to the interval p ropag ation problem . W e build upon a recent appro ach to find the approximate bound s of a real fu nction using Polynomial Chaos expansions introdu ced by Monti [13], [24] an d applied for worst-case analysis and robust stability . Given a function of random variables with c ompact suppor t p robab ility distributions, the intuition is to qu antify the uncertainty in the r esponse u sing Polynom ial Chaos expansion and discard a ll th e informa tion provided about the randomn ess o f the outpu t and e xtract only the bounds of its com pact suppor t. Introd uced b y Norbert W einer [28], Polyno mial Chaos initially co ined as the Homo geneou s Chaos was used to rep- resent a Gaussian process as a s eries of Hermite polyn omials. This metho d has been generalized to the Askey-schem e of ortho gonal polyno mials used to mod el rando m v ariables characterized by different p robab ility density fun ctions, in - cluding Beta and Unifor m which h ave compact support [29]. The Polyno mial Chaos is m athematically attr activ e due to the function al representations of the stochastic variables. It separates the determ inistic part in the polynomial coefficients and the stochastic part in the ortho gonal p olyno mial basis. This bec omes particu larly useful in char acterizing the uncer- tainty of the response of a dynam ical system rep resented by ordinar y differential equ ations with uncertain parameter s. T o so lve f or the boun ding rang e o f p olynom ials, we propo se to transfor m the Polynom ial Chaos expan sion in the Bernstein fo rm, an d use the range enclo sure pr operty of Bernstein polyno mials to find the minim um an d m aximum value of the response [2]. T he transfor mation d oes not require poly nomial e valuations and it guarantees the glob al optimality of the bounds [8] a nd it is shown to be more efficient tha n existent inter val glo bal optimizers [20]. T o dem onstrate this appr oach fo r pr opagating Dempster- Shafer structures on closed intervals throu gh nonlinear func- tions, we ap ply the propo sed m ethod o n an a lgebraic chal- lenge pr oblem used to investigate the propag ation o f ep is- temic uncertainty [1 7]. The p roblem of propa gating DS structu res on closed intervals through nonlinear fun ctions is stated in Section I I and the proposed meth od is presented in Section III. Th e numerical example is giv en in Section IV and the co nclusions and fu ture work ar e d iscussed in Section V. I I . P R O B L E M S T A T E M E N T A. Theory of Eviden ce The primiti ve fun ction in the theory of evidence is the basic pr obability assignment (b pa) , represented here by m , which is similar to the pr obability in the pro bability theo ry . The bpa for a given set can be understoo d as the weight of evidence th at the truth is in that set, evidence, wh ich cann ot be further subdivided among the members of the set. The bpa defines a map of the p ower set over the frame of d iscernment Ω to the interval [0 , 1] : m : 2 Ω → [0 , 1] . T he focal element of m is every sub set A ⊆ Ω such that m ( A ) > 0 an d the belief str ucture m verifies: X A ⊆ Ω m ( A ) = 1 where m ( A i ) = p i (1) In th is work we are co nsidering no rmalized belief struc- tures (closed -world assum ption) which satisfy the following relation: m ( φ ) = 0 , wh ere φ is the null set. As an example consider the fo llowing body of evidenc e (Ω , m ) : Ω = { M , N , P } with m ( { M , N } ) = 0 . 3 and m ( { N , P } ) = 0 . 7 . Based on the mass function two new fun ctions can be induced . The belief function o r the lo wer bo und , B e l , which quantifies the total amoun t of support gi ven to the set of interest A : B e l ( A ) = X B ⊆ A m ( B ) (2) The plausibility functio n or the upp er b ound , P l , which quantifies the maxim um amo unt o f p otential given to the set of inter est A (here ¯ A is the comp lement o f A ): P l ( A ) = X B ∩ A 6 = φ m ( B ) = 1 − B el ( ¯ A ) (3) The precise pr obability is bo unded by the two q uantities defined above, and when the eq uality is satisfied then the belief m easure is just a pro bability mea sure and all the focal elements are singletons. B e l ( A ) ≤ prob ( A ) ≤ P l ( A ) (4) Giv en two bpa’ s m 1 and m 2 based on indepen dent argu- ments on the same frame of discernment, the De mpster’ s rule of com bination provides the means to calcu late the aggregation o f th e two belief stru ctures: m 12 ( A 6 = φ ) = 1 1 − K X B ∩ C = A m 1 ( B ) m 2 ( C ) (5) m 12 ( φ ) = 0 where K = P B ∩ C = φ m 1 ( B ) m 2 ( C ) r epresents the amoun t of pr obability mass due to c onflict. While a numb er of com bination rules have been derived to aggr egate inform ation [2 2], we present also the mixing rule which is used in the nu merical example. Given n belief structures to be aggregated, the formula for the mixing rule is given by: m 1 ...n ( A ) = 1 n n X i =1 w i m i ( A ) (6) where w i are the correspon ding weights propo rtional with the reliability of the sour ces. B. DS structur es on closed intervals Giv en two non decreasing fun ctions F an d F , where F , F : R → [0 , 1] and F ( x ) ≤ F ( x ) for all x ∈ R , we can represent the imprecision in the cumu lativ e distribution function (CDF), F ( x ) = P rob ( X ≤ x ) , by the pro bability box ( p-bo x) [ F , F ] as fo llows: F ( x ) ≤ F ( x ) ≤ F ( x ) [7]. A Dempster-Shafer structure on closed intervals can in- duce a uniq ue p -box, wh ile the inv erse is not uniqu ely determined . Many Demp ster-Shafer structures exist for the same p-bo x. Given the following body of evidence,  ([ x 1 , x 1 ] , p 1 ) , ( [ x 2 , x 2 ] , p 2 ) , . . . ([ x n , x n ] , p n )  , the cumulative belief func tion ( CBF) and th e cumulative plausi- bility fun ction (CPF) are define d by: C B F ( x ) = F ( x ) = X x i ≤ x p i (7) C P F ( x ) = F ( x ) = X x i ≤ x p i (8) Similarly one can obtain the com plementary cu mulative belief function (CCBF) and the comp lementary cum ulativ e plausibility f unction (CCPF). C C B F ( x ) = 1 − C P F ( x ) = X x i >x p i (9) C C P F ( x ) = 1 − C B F ( x ) = X x i >x p i (10) Thus the comp lementary cu mulative distribution fun ction (CCDF), F c ( x ) = P r ob ( X > x ) , is boun ded a s fo llows: C C B F ( x ) ≤ F c ( x ) ≤ C C P F ( x ) (11) Example : Consider th e following bod y of evidence  ([1 , 4] , 2 / 3) , ([3 , 6] , 1 / 3)  , the lower and the upper cumulative func tions are plo tted in Fig.1. C. Map ping of DS structur es Consider the f ollowing functio n: y = f ( a, b ) , where f : R 2 → R , and a and b are g iv en by th e following bodies of evidence  ([ a 1 , a 1 ] , p a 1 ) , . . . ([ a n , a n ] , p a n )  and  ([ b 1 , b 1 ] , p b 1 ) , . . . ([ b n , b n ] , p b n )  respectively . W e are interested in finding the induced Dempster-Shafer structu re 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 x CDF Cumulative belief function Cumulative plausibility function Fig. 1. P-box induced by the Dempster-Sha fer structure in the y v ariable. The basic probability assi gnm ent describing y is gi ven b y [3 0]: m f ( Y ) = X f ( A i ,B j )= Y m 1 ( A i ) | {z } p a i m 2 ( B j ) | {z } p b j (12) where Y = [ y , y ] , A i = [ a i , a i ] an d B j = [ b j , b j ] Thus the p roblem of finding the mapping of a bod y of evidence on closed intervals is redu ced to interval pro p- agation [12]. T his problem can be solved using the ad- vanced techn iques de veloped in the in terval analy sis field [10]. Howev er, due to the depend ency pr oblem the ob tained bound s are conservativ e which is detrim ental to th e belief structure, since th e evidence is assigned automatically to other elements wh ich are n ot in the body of e vidence. This problem beco mes more ac ute when th e unc ertainty has to be propag ated over a period of time. I I I . P R O P O S E D A P P ROA C H W e prop ose a new approa ch in approxima ting the pr op- agation of intervals u sing a non -intrusive polyn omial chaos method [3] in comb ination with Bernstein p olyno mials [25]. The ap proach o f usin g po lynomial chaos in propag ating epistemic u ncertainty has been consid ered previously in Ref.[13] f or a limited num ber of arithm etic op erations and relies on sampling or glo bal optimiza tion in the general case which is inaccurate for small n umber of samp les and computatio nally expe nsiv e. A. Non-in trusive polyn omial chao s The pro blem in solving the mapping of DS structur es as giv en in Eq.(12) is to find Y = [ y , y ] such that: Y = f ( A, B ) (13) where A = [ a , a ] an d B = [ b, b ] . While in this paper we present only the biv ariate case, the method ca n be scaled up to th e desired number of v ariables. The problem can be transformed into finding the stochastic response y by defin ing a ∼ U ( a , a ) and b ∼ U ( b, b ) : y = f ( a, b ) (14) and write the polynom ial ch aos e xpan sion for the un certain arguments an d the response: a = p − 1 X i =0 a i ψ i ( ξ 1 ) where ξ 1 ∼ U ( − 1 , 1) (15) b = p − 1 X j =0 b j ψ j ( ξ 2 ) where ξ 2 ∼ U ( − 1 , 1) (16) y = P − 1 X k =0 y k ψ k ( ξ ) where P = ( n + p )! n ! p ! (17) For this pap er we are on ly concern ed with the Uniform distribution, howe ver due to the spe cificity of th is application any o ther probability distribution with c ompact support c an be used (eq. Beta). The sensiti vity of the me thod with respect to the shape of the p robab ility density func tion over the compact supp ort remain s to be stud ied. Here n is the numbe r of uncertain input variables and p the order of the po lynomia l chaos expansion. The basis function ψ n is the n -th degree Legendre poly nomial and the polyno mial coe fficients of the inpu t variables ar e g iv en by: a 0 = a + a 2 , a 1 = a − a 2 , a 2 = . . . = a p − 1 = 0 (18) b 0 = b + b 2 , b 1 = b − b 2 , b 2 = . . . = b p − 1 = 0 (19) The first six multidimensiona l Legendre polyn omials for the bi variate case are g iv en by: ψ 0 ( ξ ) = ψ 0 ( ξ 1 ) ψ 0 ( ξ 2 ) = 1 (20) ψ 1 ( ξ ) = ψ 1 ( ξ 1 ) ψ 0 ( ξ 2 ) = ξ 1 (21) ψ 2 ( ξ ) = ψ 2 ( ξ 1 ) ψ 0 ( ξ 2 ) = 1 2 (3 ξ 2 1 − 1) (22) ψ 3 ( ξ ) = ψ 0 ( ξ 1 ) ψ 1 ( ξ 2 ) = ξ 2 (23) ψ 4 ( ξ ) = ψ 1 ( ξ 1 ) ψ 1 ( ξ 2 ) = ξ 1 ξ 2 (24) ψ 5 ( ξ ) = ψ 2 ( ξ 1 ) ψ 1 ( ξ 2 ) = 1 2 (3 ξ 2 1 − 1) ξ 2 (25) W e are interested in finding the poly nomial coefficients y k which char acterize the stochastic behavior of the ou tput variable. Using th e Galer kin pr ojection and the ortho gonality proper ty of the polynom ials one can isolate th e coefficients y k as shown in Ref. [5]: y k = < f , ψ k > < ψ 2 k > = 1 < ψ 2 k > Z Ω f ψ k ς ( ξ )d ξ (26) where ς ( ξ ) = Q n i =1 ς i ( ξ i ) is the join t probability de nsity function . Th e integral can be evaluated usin g sampling or quadra ture tech niques. W e sh ow that by br inging the p olynom ial chaos expan sion to a Bernstein form u sing the Garloff ’ s meth od [ 8], we can efficiently find the min imum and the max imum value of the compact su pport than ks to the properties o f the Bernstein polyno mials: the smallest and the largest coefficient bo und the output o f the function modeled . T o transform our expansion f rom Legendre polynomial ba- sis to Bernstein p olyno mial b asis we expand our Po lynomia l Chaos expansion, Eq.(17) into a simple po wer series and identify the new coefficients: y = X I ≤ N α I ξ I (27) where the multi-in dex I = ( i 1 , . . . , i n ) ∈ N n and N = ( n 1 , . . . , n n ) ∈ N n is the multi-index of maximum degrees; thus th e maximu m degree of ξ k is g iv en by n k . Here , we denote ξ I = ξ i 1 1 · . . . · ξ i n n and the inequality I ≤ J implies i 1 ≤ j 1 , . . . , i n ≤ j n . B. Garloff ’s method to ca lculate th e Bernstein coefficients W e a re in terested in the transfo rmation of the power series in E q.(27) into its Bernstein form: y = X I ≤ N β I B N I ( ξ ) (28) where B N I ( ξ ) is the I th Bernstein poly nomial of d egree N on th e general box G = [ ξ , ξ ] . In ou r b i-variate case G = [ − 1 , 1] × [ − 1 , 1 ] since ξ 1 , ξ 2 ∼ U ( − 1 , 1) . B N I ( ξ ) = B n 1 i 1 ( ξ 1 ) · . . . · B n n i n ( ξ n ) (29) The univ ariate Bernstein p olyno mial B n k ( ξ ) on the gener al interval [ ξ , ξ ] is giv en by : B n k ( ξ ) =  n k  ( ξ − ξ ) k ( ξ − ξ ) n − k ( ξ − ξ ) n (30) The Bernstein coefficients β I are gi ven b y: β I = X J ≤ I ≤ N  I J   N J  ˆ α J (31) where we write  I J  =  i 1 j 1  · . . . ·  i n j n  . The scaled coefficients ˆ α I are obtained as describ ed in Ref.[1] fro m the α I coefficients in Eq.(27) and the box G : ˆ α I = ˜ α I ( ξ − ξ ) I (32) ˜ α I = X I ≤ J ≤ N  J I  α J ξ J − I (33) Example : Consider the following Polyno mial Chaos ex- pansion given by n = 2 , p = 3 and P = 10 : y = 5 ψ 0 ( ξ ) + ψ 1 ( ξ ) + ψ 3 ( ξ ) + ψ 4 ( ξ ) T ransform ing this expansion into a simple power series we o btain the folowing poly nomial: y = 5 + ξ 1 + ξ 2 + ξ 1 ξ 2 where a 00 = 5 , a 10 = 1 , a 01 = 1 , an d a 11 = 1 . Her e the multi-index of max imum degree is N = (1 , 1) . The following interm ediate c oefficients are o btain from Eq.(33) in order to perform the scaling operation : ˜ a 00 = 4 , ˜ a 01 = 0 , ˜ a 10 = 0 , and ˜ a 11 = 1 . Th e final set of power- coefficients is given by Eq.(3 2) : ˆ a 00 = 4 , ˆ a 01 = 0 , ˆ a 10 = 0 , and ˆ a 11 = 4 . Finally , the Bernstein coefficients ar e obtain using Eq.(31): β 00 = 4 , β 01 = 4 , β 10 = 4 , an d β 11 = 8 , and the Bernstein basis is given by: B 11 00 = 1 16 (1 − ξ 1 )(1 − ξ 2 ) B 11 01 = 1 16 (1 − ξ 1 )( ξ 2 + 1) B 11 10 = 1 16 ( ξ 1 + 1)(1 − ξ 2 ) B 11 11 = 1 16 ( ξ 1 + 1)( ξ 2 + 1) C. Bo unding the r ange of polynomials Giv en the Bernstein expansion in Eq.(2 8), the r ange en- closing property [ 9] gi ves a bound on the polyno mial in terms of the Bernstein coefficients: min I ≤ N β I ≤ y ( ξ ) ≤ max J ≤ N β J ∀ ξ ∈ G = [ ξ , ξ ] (34) Provided that the initial box is small enough , the range provided by the Bern stein form is exact. Com pared with other forms in e stimating the ran ge, it is exper imentally shown in Ref. [25] that the Bernstein form provid es the smallest average overestimation erro r in the univariate case. For the previous example th e ran ge of y is bo unded by [4 , 8] . T ighter boun ds can be obtaine d by subdivision of th e initial box and choosing the minim um and the max imum of all the Bernstein co efficients correspondin g to each sub- box. An efficient algor ithm fo r r ange co mputation tha t in- corpor ates a nu mber of featur es such a s subdivision, cut- off test, simplified vertex test, mono tonicity test an d others is provided in Ref.[20]. Therefo re, getting b ack to our pro blem in mapping DS structures on closed intervals, E q.(12) and Eq.(13) , the output interval o r the focal element Y = [ y , y ] is gi ven by : y = min I ≤ N β I and y = max J ≤ N β J (35) This meth odolog y is ap plied to ma p all the fo cal elements in the initial body of e vidence through the nonlinear function . Their correspo nding masses are o btained u sing Eq.(12). T his way a b ody of evidence fo r the response is c onstructed . I V . N U M E R I C A L R E S U L T S T o p rove the concep t, we h av e selected an algebr aic problem f rom a set of challeng es used to investigate the propag ation of epistemic uncertain ty [17]. The presen ted problem has been inv estigated pre viou sly in the literature by Oberka mpf and Helto n [16]. In the present pap er we are using the exact paramete rs for the simu lation as in Ref.[16]. Consider th e following mappin g: y = f ( a, b ) = ( a + b ) a (36) where the inform ation co ncerning a an d b is provided by th e following sou rces and their corre spondin g bpa: A 1 :   [0 . 6 , 0 . 9] , 1 . 0   A 2 :   [0 . 1 , 0 . 5] , 0 . 2  ,  [0 . 5 , 1 . 0] , 0 . 8   B 1 :   [0 . 3 , 0 . 5] , 0 . 1  ,  [0 . 6 , 0 . 8] , 0 . 9   B 2 :   [0 . 2 , 0 . 4] , 0 . 1  ,  [0 . 4 , 0 . 6] , 0 . 7  ,  [0 . 6 , 1 . 0] , 0 . 2   B 3 :   [0 . 0 , 0 . 2] , 1 3  ,  [0 . 2 , 0 . 4] , 1 3  ,  [0 . 3 , 0 . 5] , 1 3   Giv en the above infor mation, we are looking to bound the pro bability of the resp onse in th e unsafe region when y > 1 . 7 . The function and the desired unsafe region are shown in Fig.2. Fig. 2. The functi on and the unsafe region The bpa for a and b is obtained by agg regating th e informa tion from th e first two so urces and th e last three sources respe ctiv ely using the mix ing ru le in Eq.(6) u nder the equal reliability assumptio n. Thus the n ew DS stru ctures obtained ar e given b y: A :   [0 . 1 , 0 . 5] , 0 . 1  ,  [0 . 5 , 1 . 0] , 0 . 4  ,  [0 . 6 , 0 . 9] , 0 . 5   B :   [0 . 0 , 0 . 2] , 0 . 11 1  ,  [0 . 2 , 0 . 4] , 0 . 14 4  ,  [0 . 3 , 0 . 5] , 0 . 14 4  ,  [0 . 4 , 0 . 6] , 0 . 23 3  ,  [0 . 6 , 0 . 8] , 0 . 3  ,  [0 . 6 , 1 . 0] , 0 . 06 7   The fo cal elements o f the DS stru cture Y are obtained b y propag ating th e pro duct space A × B throu gh the n onlinear function in Eq.(36 ), and the basic prob ability assign ment is obtained using Eq.(12). Th us the induced bo dy o f e vidence for th e respon se y is gi ven in T able I. The following para meters have been u sed in obtainin g the p olyno mial chao s expansion of the resp onse and the afferent boun ds: n = 2 , p = 5 , and P = 19 such that the to tal degree of the polyno mial is n o greater than 5 . The integrals in Eq. (26) have b een nume rically e valuated using Gauss-Legendr e q uadratur e rule with 20 po ints in each dire ction, a nd the Bernstein coefficients have been obtain using 1 1 subdivisions in each directio n. Th e obtained intervals a re comp ared with the intervals given by in terval analysis (INT LAB [21]), an d with the r eference intervals provided by a genetic alg orithm (GA) . In T able I, given the i th focal elemen t of A an d the j th focal elem ent of B , th e Bo x# is given by 3( j − 1) + i . The number s in b old indicate that a smaller lo wer boun d o r a larger upper bound has been foun d b y the genetic a lgorithm. In this p articular exam ple we have overestimated most o f the lower b ounds and we have provide d no underestimation for the upper b ound s. The focal elements from th e DS structure in T able I are also graph ically pr esented in Fig.3. T he wider bo xes represent the bo unds fou nd b y the prop osed appr oach while the narrow boxes d epict the bou nds retu rned by the genetic algorithm , an d the lines show the rang e comp uted u sing interval a rithmetics. T ABLE I I N D U C E D D S S T R U C T U R E F O R T H E R E S P O N S E y Box# y y m f Box# y y m f 1 0.687 0.909 0.011 10 0.890 1.061 0.023 2 0.721 1.222 0.044 11 0.967 1.630 0.093 3 0.741 1.097 0.056 12 1.007 1.450 0.117 4 0.804 0.961 0.014 13 0.953 1.152 0.030 5 0.853 1.426 0.058 14 1.069 1.834 0.120 6 0.880 1.275 0.072 15 1.123 1.623 0.150 7 0.850 1.012 0.014 16 0.952 1.236 0.007 8 0.912 1.528 0.058 17 1.068 2.039 0.027 9 0.945 1.363 0.072 18 1.122 1.794 0.033 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0.5 1 1.5 2 Box# y Unsafe Region Reference Interval GA Interval IA Interval PC−Bernstein Pl(y>1.7) Fig. 3. Focal elements from the induced DS s tructure T o com pute the lowest and th e high est prob ability o f y > 1 . 7 , we use the Fig.3 to sum over all the bp a’ s of the intervals that are pr operly includ ed in the unsafe region for th e lower bound an d for the upper bo und, sum over all the bpa’ s of the intervals that intersect the unsafe region. No intervals are proper ly includ ed in th e unsafe region, thu s the lo wer bound is 0 . 0 . Three intervals are found to intersect the u nsafe region, boxes: 14 , 17 and 1 8 . Sum ming over their bp a’ s we found the up per bound to be 0 . 18 . All three methods g iv e 0 . 0 ≤ P r ob ( y > 1 . 7) ≤ 0 . 1 8 , w hich is in agre ement with the result p ublished by Oberkampf in Ref.[16]. Both th e CCBF and the CCPF giv en b y Eq.( 9)-(10) are plotted in Fig.4 along with the marking for the u nsafe region. A reason of concern is th e ov erestimation of th e lo wer bound, due to th e finite polynomial chaos expansion, wh ich in this example may p rovide a larger lo wer bound for the probab ility of failure. Observe th e gross bou nds pr ovided by the in terval arithmetics d ue to dependen cy effect. 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 y BEL/PL Bel : PC−Bernstein Pl : PC−Bernstein Bel : GA Pl : GA Bel : Interval Analysis Pl : Interval Analysis Unsafe Region Fig. 4. CCBF and CCPF from the induced DS structure V . C O N C L U S I O N S A new approach in app roximatin g the interval prop agation for epistemic uncer tainty quantification has been presented. The input variables ar e represente d as a p olyno mial expan- sion of rand om variables on comp act supp ort, and b y apply - ing the Galer kin p rojection in a non-intru si ve way we find the response of the system also as a polyn omial expan sion, here in the Legendre basis. 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