International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Dimension reduction methods for reliability problems Breitung, Karl Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Dimension Reduction Methods for Reliability ProblemsKarl BreitungEngineering Risk Analysis Group, Technical University of Munich, Munich, GermanyABSTRACT: In reliability problems in high dimensional spaces it is important to identify the essentialstructure of the problem. Which variables can be neglected and which not? Here statistical dimensionreduction methods help to simplify the complexity of structures. FORM/SORM can be seen as earlyexample of dimension reduction concepts. Since these approaches do not rely on the smoothnessproperties of the limit state functions, they can be applied also for problems with noise. Three suchmethods are described. Further an approach giving the SORM factor as a ratio of small surface areas isoutlined.1. INTRODUCTIONIn structural reliability problems in the last yearsmore and more problems in high dimensionalspaces have been treated. Due to the increasingcomputer power it is nowadays possible to handlesuch questions in a reasonable time. But obtain-ing numerical results for failure probabilities is notenough, even more important is to understand thestructure of the system under consideration. Whatare its essential features and what can be neglectedfor am approximate, but enough accurate descrip-tion of it? Such problems have appeared in otherfields of science already some time ago. Here someof these methods which seem to be useful for relia-bility applications will be outlined.Another problem is that LSF’s (Limit State Func-tion) not always are smooth functions but havesome additional noise which makes it difficult orimpossible to apply the standard FORM/SORMmethods which require smooth LSF’s.Both problems can be tackled using dimensionreduction concepts where the relation between andthe LSF is seen more as a statistical relation andno more as an exact functional relation. Thesemethods are in some way complementary to re-sponse surface methods (see for example Bucherand Macke (2004)).2. DIMENSION REDUCTION ANDFORM/SORMGiven is a system described by many variables X=(X1, . . . ,Xn) as a function Y = f (X) of them, cana simpler structure having much less variables befound without much loss of information about thesystem? Here Y is a function of the rv’s X1, . . . ,Xnand n is large. Dimension reduction methods nowattempt to find a function h : Rn→ Rp with p nand a function f ∗ : Rp→ R such thatf (X1, . . . ,Xn) = f∗(h1(X), . . . ,hp(X))+ ε (1)where the term ε is small in some sense. Anoverview can be found in Burges (2009). In Hur-tado (2004) dimension reduction methods werestudied the first time for structural reliability butwith another perspective, i.e. to approximate anLSF given by data points using scalar products inhigher dimensional spaces.If h is a linear function, such a function is foundby a projection on a suitable plane M with h(X) =PMX where PM is a projection matrix onto the p-dimensional plane M.If the plane is spanned by p orthonormal vectorsv1, . . . ,vp, thenf (x)≈ f ∗(vT1 x, . . . ,vTpx)+ ε (2)FORM can be seen as a linear dimension reductionmethod. The original LSF g(u) is replaced by the112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015linearized formg∗(u) = a0 +n∑i=1aiUi = a0 +aTu, (3)which is the projection on the vector y = aTu,g∗(y) = a0 + y, g(u) ≈ g∗(y). This is a reduc-tion from n to one dimension. In the same wayin SORM the limit state function is replaced by aquadratic form of normal random variables.Now, the reduction to one dimension is not al-ways possible, so for more complex problems itseem to be useful to make reductions where p isstill small compared with n, but greater than unity.The classical counterexample for FORM/SORMis a LSF in the formg(u1, . . . ,un) = β −k∑i=1(aTi u)2, (4)where the ai are unit vectors. Here there is nounique design/beta point and the approximations ofFORM/SORM can not be derived. But using di-mension reduction methods it is easily possible toidentify the relevant variables of this LSF.3. DIMENSION REDUCTION METHODSHere some dimension reduction methods are de-scribed. In the following it is assumed that Y =g(U) is a function of a standard normal random vec-tor U. Such methods are for example SIR (SlicedInverse Regression, Li (1991)) and SAVE (SlicedAverage Variance Estimates, Cook and Weisberg(1991)). These two approaches are, respectively,based on the inverse mean of U given Y and the in-verse conditional variance of U given Y . The ideaof these methods is that if Y is a function of thevTi U vectors only as in eq. (2), then the inverse re-gression, i.e. estimating the U’s from the y’s willafflict only these variables.Let for example beY = g(U1, . . . ,Up,0, . . . ,0) (5)with 1 ≤ k < n. Then only changes in the first pvariables will have an influence of Y . In the sameway if one now considers variations of Y then theconditional mean IE(U1, . . . ,Up,Up+1, . . . ,Un|Y )will vary only in the first k variables, i.e. lie in theplane spanned by the first p unit vectors.In the same way, ifY = g(vT1 U, . . . ,vTpU,0, . . . ,0)i.e. a function of vT1 U, . . . ,vTpU only, then thecurve of the inverse regression mean IE(U|y) lies inthe subspace M spanned by the vectors v1, . . . ,vp.Since the curve IE(U|y) lies in M, the vectors con-necting the points IE(U|yi) for various values of yimust lie in the plane M. So if we can constructenough such points IE(U|yi) from their differencesIE(U|yi)− IE(U|y j) we can find the vectors whichspan M. Certainly there are exceptions where thesedifferences fail to span the whole subspace.Figure 1: The variation of IE(U|y) in the plane MWe cannot find the values of IE(U|y)) for a fixedy, but we can approximate them by taking slices.This means, we can find the values IE(U|y ∈ Si)with Si = [a,b] an interval. These IE(U|y ∈ Si)points lie in the subspace M too.The SIR method estimates directly the matrixcov(IE(U|y)) using slices. Given now a set of data(yk,uk) with k = 1, . . . ,m where y is a function ofthe u. First the real axis is partioned into h slicesS1, . . . ,Sh. Let mi the number of data in slice Si andu¯i = m−1i ∑y j∈Siu j (6)be the sample mean of the vectors u j with y j ∈ Si.Then the estimator of the covariance matrix is givenby:Ŝ1 = m−1h∑i=1mi(u¯iu¯Ti ) (7)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015This is now an approximation of the matrixcov(IE(U|y)). By taking the p largest eigenval-ues and the corresponding eigenvectors one findsan approximation for the subspace M, i.e. the spacespanned by the p largest eigenvectors v1, . . . ,vp.In the alternative method SAVE one uses esti-mates about the variation of the conditional meanto find the vectors spanning M. Since due to thelaw of total variancecov(U) = IE(cov(U|y))+ cov(IE(U|y)), (8)the matrix IE(cov(U|y)) can be estimated also fromIn− cov(IE(U|y)).Let In be the n-dimensional unity matrix and Pthe projection matrix onto the subspace M, thencov(U|y) = In−P+Pcov(U|y)PIn− cov(U|y) = P−Pcov(U|y)PIn− cov(U|y) = PInP−Pcov(U|y)PIn− cov(U|y) = P(In− cov(U|y))P (9)So the matrix In−cov(U|y) is equal to its projectiononto the subspace M. If M is spanned by the first punit vectors e1, . . . ,ep then this meansIn− cov(U|y)=(Ip− cov(U1, . . . ,Up|y) 0p, n−p0n−p, p 0n−p, n−p)(10)with 0k, l the zero matrix with k rows and l columns.So the p non-zero eigenvectors of In − cov(U|y)span M.We make an estimate Sˆ2 of In − IE(cov(U|y)).Then the eigenvalues and eigenvectors of Sˆ will ap-proximate those of In− IE(cov(U|y)). Here m is thenumber of the data, the real line is partitioned intoh slices Si, and mi is the number of data in slice Si.Then an estimate of S is given by:Ŝ2 = m−1h∑i=1mi(In− cov(U|y ∈ Si))2 (11)Here we take the square of In− IE(cov(U|y)) to getnon-negative eigenvalues. If Y is a function of onlyp variables, this matrix has n− p eigenvalues ap-proximately equal to zero and the eigenvectors forthe remaining p positive eigenvalues span the sub-space M.Both methods SIR and SAVE have some weak-nesses. So SIR detects linear relations quite well,but has difficulties to find quadratic dependencies,with SAVE it is exactly vice versa. Various im-provements and new methods have been proposed.Here only one of these trying to combine the esti-mators is described. The mixed estimator derivedin Zhu et al. (2007) is thenŜ3 = (1−a)Ŝ1 +aŜ2 (12)where 0 < a < 1. The combination of these estima-tors gives a method which is able to detect linearand quadratic functional relations quite well.4. EXAMPLEA random sample of 10000 points is taken with theLSF with noise term εg(u) = 3−u1−0.3 ·u22 + ε (13)The noise term ε has a normal distribution withmean zero and variance 0.5. Using the mixed esti-mator in eq. (12) with a= 0.5 one obtains as matrixof eigenvectors (columns)0.01 0.04 0.02 1.00 −0.08−0.02 −0.01 −0.01 −0.08 −1.000.20 −0.88 0.42 0.02 −0.000.98 0.20 −0.05 −0.01 −0.010.04 −0.42 −0.90 0.04 0.01(14)The corresponding eigenvalues are:(0.00 0.00 0.01 0.52 0.86)(15)The eigenvectors with the two largest eigenvaluesare approximately equal to the first and second unitvector, they correctly identify the relevant compo-nents.5. ESTIMATING THE SORM-FACTOR ASSURFACE AREA RATIOEstimating the SORM factor cannot be done bythese methods. Here an approach is given whichreplaces the calculation of the Hessian of the LSFby estimating surface areas.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The SORM factor is given as (Breitung (1984),Breitung (1994)) ∏n−1i=1 (1 − βκi)−1/2, where theκi’s are the main curvatures of the limit state sur-face at the beta point. This form uses the orientationof the limit state surface by the normal vector fieldn(u)= |∇g(u)|−1∇g(u). Ths means that curvaturesof the surface bending towards the surface normalare defined as positive, bending away as negative.Therefore the negative sign. This ensures also thata sphere around the origin where the safe domainis inside the sphere has positive curvature which isthe usual convention.The following differential geometry facts aretaken from Thorpe (1979). The Gauss-Kroneckercurvature of a surface at a point is the product of itsmain curvatures. The Gauss map for an surface Goriented by the normal vector field n(u) is definedby G→ Sn,u 7→ n(u), where Sn = {x; |x| = 1} isthe unit sphere in Rn.Consider now the Lagrangian of the problemgiven by L(u,λ ) = |u|2/2 + λg(u). The surfaceG∗ = {u;L(u,β )−β 2/2 = 0} is a hypersurface inthe n-dimensional space containing the beta pointu∗. From its definition one can deduce that all maincurvatures at the beta point are positive. At the betapoint the Gauss-Kronecker curvature K(u∗) of thishypersurface is given byK(u∗) =n−1∏i=1(1−βκi) (16)So the Gauss-Kronecker curvature of this hyper-surface is equal to the square of the inverse of theSORM-factor. This shows a way to compute it fromthe geometrical properties of the surface G∗.Let W ⊂ Rn−1 a neighborhood of the origin andΨ : W → G∗ be a local parametrization of the sur-face G∗ near the beta point with Ψ(0)) = u∗ andlet N : W → Sn, z 7→ n(Ψ(z)) the Gauss map of thesurface G∗ for all points z ∈W .From the corollary on p. 144 in Thorpe (1979)one obtains (with V (.) denoting the (n − 1)-dimensional area in Rn):K(u∗) = limε→0V (N|Bε)V (G∗|Bε)(17)with Bε = {z; |z| < ε} ⊂W . So by computing theratio of the area of the surface and of the area of itsGauss map on the n-dimensional unit sphere onea can approximate the SORM factor (for detailsabout calculating the surface area for a given localparametrization see Courant and John (1974)).6. CONCLUSIONSDimension reduction methods can detect essentialstructures in high dimensional reliability problems.Combining them with response surface approachesmight allow to find more efficient failure probabil-ity estimators.7. REFERENCESBreitung, K. (1984). “Asymptotic approximations formultinormal integrals.” Journal of the EngineeringMechanics Division ASCE, 110(3), 357–366.Breitung, K. (1994). Asymptotic Approximations forProbability Integrals. Springer, Berlin Lecture Notesin Mathematics, Nr. 1592.Bucher, C. and Macke, M. (2004). “Response Surfacesfor Reliability Assessment.” Engineering Design Re-liability Handbook, E. Nikolaidis et al., eds., CRCPress LLC, Chapter 19, 19:1–23.Burges, C. (2009). “Dimension reduction: A guidedtour.” Machine Learning, 2(4), 275–365.Cook, D. and Weisberg, S. (1991). “Comment to "SlicedInverse Regression for Dimension Reduction" by K.-Ch. Li.” Journal of the American Statistical Associa-tion, 86(414), 328–332.Courant, R. and John, F. (1974). Introduction to Calcu-lus and Analysis, Vol. 2. John Wiley, New York.Hurtado, J. (2004). Structural Reliability: StatisticalLearning Perspectives. Springer.Li, K.-C. (1991). “Sliced inverse regression for dimen-sion reduction.” Journal of the American StatisticalAssociation, 86(414), 316–327.Thorpe, J. (1979). Elementary Topics in Differential Ge-ometry. Springer, New York.Zhu, L.-X., Ohtaki, M., and Li, Y. (2007). “On hy-brid methods of inverse regression-based algorithms.”Computational Statistics & Data Analysis, 51, 2621 –2635.4


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