Sets of Joint Probability Measures Generated by Weighted Marginal Focal Sets

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1 Sets of Joint Probability Measures Generated by Weighted Marginal Focal Sets Thomas Fetz Institut für Technische Mathemati, Geometrie und Bauinformati Universität Innsbruc, Austria Abstract This paper is devoted to the construction of sets of joint probability measures for the case that the marginal sets of probability measures are generated by weighted focal sets. Different conditions on the choice of the weights of the joint focal sets on the probability measures on these sets lead to different types of independence such as strong independence, rom set independence, fuzzy set independence unnown interaction. As an application the upper probabilities of failure of a beam are computed. Keywords. Weighted focal sets, possibility measures, plausibility measures, lower upper probabilities, sets of probability measures. Introduction Precise probability theory alone often does not suffice for modeling the uncertainties arising in civil engineering problems such as the reliability analysis of structures much more in soil mechanics. One of the most difficult problems here is to analyze the behavior of the soil or roc during the construction of a tunnel where the soil properties are only very imprecisely nown. The goal is to have practicable measures for the ris of failure in the case where the material properties are not or only partly given by precise values or probability measures. One should also have the possibility to assess subjective nowledge expert estimates. It is usually easy for the planning engineer to provide such information by using weighted focal sets to model the fluctuations of the parameters involved. In most cases intervals are used for the focal sets which has the advantage that the computations can be performed by interval analysis []. This leads, if the intervals are nested, to fuzzy numbers possibility measures or, if not nested, to plausibility measures evidence theory. Fuzzy numbers or possibility measures [7, 8] are used in [3, 4, 5, 9, 0, ]. Plausibility measures [7] are used in [6, 4, 5]. Using the more general plausibility measure has the advantage that we can mix e.g. fuzzy numbers with histograms or probability measures directly without transforming the probability measures into fuzzy numbers neglecting information. It is often more practicable to interpret these measures as upper probabilities as done in [6, 4, 5]. It is easy to do that if one can start the computations with given weighted joint focal sets, c.f. [6]. But in many cases the weighted focal sets are given only for each uncertain parameter separately. If the marginal focal sets are nested, we have a joint possibility measure a joint plausibility measure. Unfortunately these two measures are not the same in general, which leads to ambiguities in interpreting both measures as upper probabilities. The plan of this paper is as follows: Section is devoted to weighted focals sets to the notation we will use. In Section 3 we present a civil engineering example with two uncertain parameter separately given by weighted focal sets. In Section 4 we construct sets of joint probability measures by means of weighted joint focal sets. We list different conditions on choosing the weights of the joint focal sets the probability measures on these sets. Depending on these conditions we get different sets of joint probability measures different types of independence, respectively. We show that some of these cases lead to types of independence as described in [] such as strong independence, rom set independence unnown interaction. Further we investigate how the joint possibility measure fit into this scheme, if the marginal focal sets are nested. For each discussed case the upper probability of failure of the beam given in Section 3 will be computed.

2 In the last section we show in a summary how the sets of joint probability measures the upper probabilities are related to each other. Sets of marginal probability measures generated by weighted focal sets We consider two uncertain values or parameters λ λ. The possible realizations ω of an uncertain parameter λ belong to a measurable space (Ω, C ) with σ-algebra C. Here the uncertainty of a parameter λ is always modeled by a finite class A {A,..., An } C of weighted focal sets or rom sets. These focals are weighted by a map m : A [0, ] : A m (A) with A A m (A). Then the upper probability or plausibility measure of a set C C is defined by P (C ) Pl (C ) A i C m (A i ) () the lower probability or belief measure by P (C ) Bel (C ) A i C m (A i ). () If the focal sets are nested, e.g. A A An, then the above plausibility measure is a possibility measure Pos with possibility density µ (ω ) Pos ({ω }) which is also the membership function of the corresponding fuzzy number. The goal of this paper is to construct joint measures from marginals which are given by weighted focal sets. To do this we must now how the upper probability lower probability can be obtained using sets of probability measures. Therefore let K i {P i : P i(ai ) } be the set of probability measures P i on the corresponding focal set A i K P A i A m (A i )P i : P i K i be the set of probability measures for the uncertain parameter λ generated by the weighted focal sets A,..., An. Then the upper probability P (C ) is obtained by solving the following optimization problem: P (C ) max{p (C ) : P K } m (A i )P i (C ) A i A with certain P i Ki. Such an optimal P i way: P i δ ω i with ω i can be chosen in the following { A i C if A i C A i arbitrary if Ai C. δ ω is the Dirac measure at ω Ω. Then P i (C ) for A i C 0 otherwise which leads to the same result as in the defining formula (). The lower probability P (C ) is obtained by: with P (C ) min{p (C ) : P K } m (A i )δ ω i (C ) ω i A i A { A i \ C A i arbitrary if A i C otherwise. Then δ ω i (C ) for A i C 0 otherwise which leads to the same result as in formula (). 3 Numerical Example As a numerical example we consider a beam supported on both ends additionally bedded on two springs, see Fig. The values of the beam rigidity EI 0 Nm of the equally distributed load f(x) N/m are assumed to be deterministic. But the values of the two spring constants λ λ are uncertain. N/m λ λ 3 m Figure : Beam bedded on two springs. The uncertainty about the possible fluctuations of the spring constants λ λ is modeled by the same

3 three focal sets A [0, 40], A [30, 40] A3 {30},, with weights m (A ) 0., m (A ) 0.3 m (A 3 ) 0.5, see Fig. The measurable spaces are (Ω, C ) (Ω, C ) (R +, B(R + )). ) m (A j A i A j N/m A 3 A A m(a 3 )0.5 m(a )0.3 m(a )0. Figure : Uncertain spring constants λ. We want to compute measures for the ris of failure of the beam. The criterion of failure is here max M(x) M f, x [0,3] where M(x) is the bending moment at a point x [0, 3] M f the moment of failure. The bending moment M also depends on the two spring constants. We define a map M : Ω Ω R : (ω, ω ) max x [0,3] M(x, ω, ω ), which is the maximal bending moment of the beam depending on values ω ω for the two spring constants. M(x, ω, ω ) is computed by the finite element method [7, 8, 3] for fixed parameter values ω ω. 4 Sets of joint probability measures 4. Preliminary definitions Let (Ω, C) be the product measurable space with Ω Ω Ω n σ-algebra C C C n. We want to write the set K of all joint probability measures which are generated by the marginal sets K K as n K P n m(a i A j ij )P. A i A j is a joint focal set in A A P ij is a probability measure on A i A j, which means again P ij (A i A j ) for all i,..., n j,..., n. The marginals of P ij are probability measures on A i A j i,ij. We denote them by P K i P j,ij, see Fig. 3. K j P j,ij A j A i m(a i A j ) P i,ij P ij m (A i ) Figure 3: Focal set A i A j. We have several possibilities to choose the weights m(a i A j ), the probability measures P ij, their marginals P i,ij P j,ij, respectively. 4. The choice of m(a i A j ) Case : The joint focal sets A i A j are chosen in a stochastically independent way. Then we get m(a i A j ) m (A i )m (A j ) for the weights of the joint focals. Case : If there is no information on how to choose the joint focal sets we allow arbitrary weights m(a i A j ), but with the restriction that must hold. n m (A i ) m(a i A j ) j n m (A j ) m(a i A j ) i If the marginal focal sets are nested we also will use a special correlation of these weights which leads to the joint possibility measure. 4.3 The choice of P ij Case A: The measures P ij on the joint focals A i A j are chosen as product measures P ij P i,ij P j,ij with P i,ij K i P j,ij K j. 3

4 Case B: Now arbitrary dependencies are allowed for the measures on A i A j. Then the only restrictions on the P ij are: P ij ( A j ) P i,ij P ij (A i ) P j,ij. 4.4 The choice of P i,ij P j,ij Case a: We use always the same marginal probability measures in K i K j respectively. We denote this by: P i : P i,i P i,i P i,in P j j,j : P P j,j P j,nj. Case b: We allow arbitrary marginal probability measures P i,ij K i P j,ij K j respectively. 4.5 Case Aa Let K Aa be the set of all probability measures generated according to case Aa. A probability measure P K Aa is written for a set C C as n n P (C) m(a i A j )P ij (C) n n m (A i )m (A j )(P i P j )(C) ( n ) m (A i )P i i (P P )(C) ( n ) m (A j )P j (C) j with P K P K. So we have K Aa {P P : P K, P K }. This is the case of strong independence or type- extension, cf. [, 6] where the outcomes of two uncertain parameters are always stochastically independent. We denote this set by K S the upper lower probability by P S P S respectively. We introduce the following computational method to obtain P S (C) P S (C). P S (C) (P S (C)) is the optimal value of the objective function of the optimization problem n n m(a i A j )(δ ω i δ ω)(c) j n n m(a i A j )I C(ω, i ω j ) max (min) subject to ω i A i, i,..., n ω j Aj, j,..., n where I C is the indicator function of the set C. Note that the objective function taes only a finite set of values. For our example we have the set C {(ω, ω ) Ω : M(ω, ω ) M f }. The upper probability P S (M M f ) of failure for strong independence is depicted in Fig. 4 as a function of M f. upper probability P S M f [Nm] Figure 4: Upper probability of failure P S (M M f ). 4.6 Case Bb Let K Bb be the set of all probability measures generated according to case Bb. A probability measure P K Bb is written as with n n P (C) m(a i A j )P ij (C) n n m (A i )m (A j )P ij (C) P ij ( A j ) P i,ij K i P ij (A i ) P j,ij K j. Here the sets A i A j are selected stochastically independent, but for the measures on A i A j dependent selections are allowed. This is the case of rom set independence [, ]. Here we use the notation K R, P R P R. The upper probability P R (C) is obtained by P R (C) Pl(C) A i Aj C m (A i )m (A j ), 4

5 which is the formula for the joint plausibility measure. Alternatively it is given as in the one-dimensional case by P R (C) max{p (C) : P K R } m (A i )m (A j )P ij (C) A i Aj where the P ij are again Dirac measures on A i A j. Then we also have P Bb (C) P Ab (C), because for Dirac measures the condition in case A holds. But K Ab is a subset of K Bb. The lower probability P R (C) is the joint belief measure Bel. Here also P Bb (C) P Ab (C) holds. Computational method to obtain P R for our example: with P R (C) m(a i A j ) A i Aj C m(a i A j ) M(A i Aj ) [M f, ) M R M R M f m(a i A j ) max M(ω, ω ). (ω,ω ) A i Aj The upper probability of failure for rom set independence is depicted in Fig. 5. upper probability P R P S M f [Nm] Figure 5: Upper probability of failure P R (M M f ). P R (C) is always greater than or equal to P S (C) ( K S K R ), because the conditions for the P ij are less restrictive. 4.7 Case Bb Let K Bb be the set of probability measures generated according to case Bb. Then we have P Bb (C) max{p (C) : P K Bb } n n m ij (A i A j )P ij (C) A i Aj C m ij (A i A j ) where P ij is an appropriate Dirac measure as for P R where m ij is the solution of the following optimization problem: m ij (A i A j ) max A i Aj C subject to n m (A i ) m(a i A j ) (3) j n m (A j ) m(a i A j ). (4) i K Bb is the set of probability measures generated by the least restrictive conditions on m P ij. We will show that K Bb K U : {P : P ( Ω ) K, P (Ω ) K } holds. K U is the set of joint probability measures whose marginal probability measures belong to K K respectively. In this case the interactions between the two marginals are completely unnown []. The following theorem will show us that every P U K U belongs also to K Bb. But first we need some definitions: Let B {B,..., BN } be a partition of i Ai such that either B r Ai or Br Ai holds. Example: Let A [0, ] A [, 3]. Then the partition B {[0, ), [, ], (, 3]} of [0, 3] has the above property. Further we define for convenience: A ij : A i A j, B rs : B r B s m ij : m(a A j ). Theorem. A probability measure P U K U {P : P ( Ω ) K, P (Ω ) K } can be written as n n P U m ij P ij 5

6 with weights m ij M ir M js P U(B rs ) B rs A ij probability measures P ij (C) m ij on the focal sets A ij. M ir M js B rs A ij P U(C B rs ) The weights M ir M js are defined by M ir m (A i )R i (B r ) P (B r ) M js m (A j )Rj (Bs ) P (B s ), where R i R j are some probability measures such that P P U ( Ω ) P P U (Ω ) can be written as P i m (A i )R i P j m (A j )Rj. In the case of P (B r ) 0 or P (B s ) 0 the above weights can be chosen arbitrary, because then only P U (C B rs ) 0 is weighted. Proof. First we observe that i M ir holds: We have to prove conditions (3) (4) as well: n j m ij n j N N r s M ir M js r,s:b rs A ij M ir M js j:a ij B rs N N M ir P U (B rs ) r s N r P U(B rs ) P U(B rs ) N r m (A i )R i (B r ) P (B r ) P (B r ) N m (A i ) R(B i ) r m (A i ). r The proof of (4) is analogous. M ir P U (B r Ω ) We have shown that K U K Bb. Since K U is the biggest possible set of joint probability measures we get K U K Bb. Using the same arguments as in case Bb leads to P Ab (C) P Bb (C) P Ab (C) P Bb (C). Computational method to obtain P U (C): The set C determines the objective function. The conditions (3) (4) are always the same have to be generated only once. The upper probability P U (M M f ) for unnown interaction is depicted in Fig. 6. n i M ir n i Now we show that for all B rs holds: m (A i )R i (B r ) P (B r ) P (B r ) P (B r ). m ij P ij (C B rs ) P U (C B rs ) i j upper probability 0.5 P U P R M f [Nm] n n m ij P ij (C B rs ) n n M ir M js P U(C B rs ) P U (C B rs ) ( n i P U (C B rs ). M ir )( n j M js ) Figure 6: Upper probability of failure P U (M M f ). 4.8 The joint possibility measure Now we want to compute the joint possibility measure given by weighted marginal focal sets. Let A A A n A A A n be given nested focal sets with weights m i m (A i ). 6

7 Further we need points ω,..., ω n Ω ω,..., ω n Ω with the following property { ω i A i \ Ai+ if i < n, A n if i n. The possibility measures of these points ω i are Pos ({ω i }) i s the joint possibility measure m s Pos({(ω i, ω j )}) min{pos ({ω i }), Pos ({ω j })}. On the other h we want to have Pos({(ω i, ω j )}) with m rs m(a r A s ). (ω i,ωj ) Ar As m rs i j r s m rs We get a system of linear equations for the weights m ij : { i j i } j m rs min m r, r s r s m s for i,..., n j,..., n. The left h side is a binary matrix, exactly a lower triangular matrix with ones in the diagonal if we use an appropriate numbering. So the weights of the joint focals for the joint possibility measure are uniquely determined. Then the joint possibility measure Pos for a set C C can be obtained by Pos(C) A i Aj C m ij with the weights m ij computed by the above procedure. The joint focal sets are not nested in general, but here the sets with weights m ij > 0 are nested, because: The nested α-cuts of the density function of the joint possibility measure are among the joint focal sets. The weights of these sets, needed for the formula for Pos, can also be obtained directly from the density function. Then for these weights the above equations must also hold. Since the solution is unique they coincide with the ones computed above. We say here that there is fuzzy set independence denote the set of joint probability measures for this choice of the weights m ij by K F the upper lower probability by P F P F respectively. Remar: If we replace min by the product on the right h side, we get the weights for the joint plausibility measure. For our example we get the linear system Am b with A , m (m, m, m 3, m, m, m 3, m 3, m 3, m 33 ) T b (0., 0., 0., 0.5, 0.5, 0.5,,, ) T. The solution is m 0., m 0.3, m otherwise. The upper probability of failure P F (M M f ) for fuzzy set independence is depicted in Fig. 7. P F is sometimes greater sometimes less than P S P R respectively, but of course it is always less or equal to P U. upper probability P F M f [Nm] Figure 7: Upper probability of failure P F (M M f ). 5 Summary Conclusion We have investigated the five cases Aa, ba, Bb, Ba Bb get for the sets of joint probability measures the results K Aa K Ab K Bb K Bb K S K R K U K F K U, where K S is the set for strong independence, K R the set for rom set independence, K U the set for unnown interaction K F the set for fuzzy set independence. 7

8 For the upper probabilities for a set C C we have P Aa (C) P Bb (C) P Bb (C) P S (C) P Ab (C) P Ab (C) P R (C) P U (C) The joint possibility measure Pos(C) P F (C) does not fit into this ordering. Here only holds. Pos(C) P F (C) P U (C) Which of the above methods is preferable depends on the type of independence. The choice of the type of independence has to be part of the modelling of the joint uncertainty of the parameters. If nothing is nown about the dependence, unnown interaction (P U (C)) is the most preferable method to be on the safe side in reliability analysis. In the case of strong independence the computational effort is in general very high, so the upper bound P R (C) P S (C) can be used as a first approximation. References [] I. Couso, S. Moral, P. Walley. Examples of independence for imprecise probabilities. In Proceedings of the first international symposium on imprecise probabilities their applications, pages 30, Ghent, 999. [] A. P. Dempster. Upper lower probabilities induced by a multivalued mapping. Ann. Math. Stat., 38:35 339, 967. [3] Th. Fetz. Finite element method with fuzzy parameters. In Troch I. Breitenecer F., editors, Proceedings IMACS Symposium on Mathematical Modelling, volume, pages 8 86, Vienna, 997. ARGESIM Report. [4] Th. Fetz, M. Hofmeister, G. Hunger, J. Jäger, H. Lessman, M. Oberguggenberger, A. Rieser, R. F. Star. Tunnelberechnung Fuzzy? Bauingenieur, 7:33 40, 997. [5] Th. Fetz, J. Jäger, D. Köll, G. Krenn, H. Lessmann, M. Oberguggenberger, R. Star. Fuzzy models in geotechnical engineering construction management. Computer-Aided Civil Infrastructure Engineering, 4:93 06, 999. [6] Th. Fetz, M. Oberguggenberger, S. Pittschmann. Applications of possibility evidence theory in civil engineering. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 8(3):95 309, [7] T. J. R. Hughes. The Finite Element Method. Prentice-Hall, New Jersey, 987. [8] Y. W. Kwon H. Bang. The finite element method using matlab. CRC Press, Boca Raton, 997. [9] B. Möller. Fuzzy-Modellierung in der Baustati. Bauingenieur, 7:75 84, 997. [0] B. Möller, M. Beer, W. Graf, A. Hoffmann. Possibility theory based safety assessment. Computer-Aided Civil Infrastructure Engineering, 4:8 9, 999. [] R. L. Muhanna R. L. Mullen. Formulation of fuzzy finite-element methods for solid mechanics problems. Computer Aided Civil Infrastructure Engineering, 4:07 7, 999. [] A. Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, 990. [3] H. R. Schwarz. Methode der finiten Elemente. Teubner, Stuttgart, 99. [4] F. Tonon A. Bernardini. A rom set approach to the optimization of uncertain structures. Comput. Struct., 68(6): , 998. [5] F. Tonon A. Bernardini. Multiobjective optimization of uncertain structures through fuzzy set rom set theory. Computer-Aided Civil Infrastructure Engineering, 4:9 40, 999. [6] P. Walley. Statistical reasoning with imprecise probabilities. Chapman Hall, London, 99. [7] Z. Wang G. J. Klir. Fuzzy Measure Theory. Plenum Press, New Yor, 99. [8] L. A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Systems, :3 8,